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PhysRevB.73.104425.pdf	Dynamics of ultrafast magnetization reversal in submicron elliptical Permalloy thin ﬁlm elements Q. F. Xiao,1J. Rudge,1B. C. Choi,1Y. K. Hong,2and G. Donohoe3 1Department of Physics and Astronomy, University of Victoria, Victoria, BC, Canada V8W 3P6 2Department of Materials Science and Engineering, University of Idaho, Moscow, Idaho 83844, USA 3Department of Electrical and Computer Engineering, University of Idaho, Moscow, Idaho 83844, USA /H20849Received 26 October 2005; revised manuscript received 22 December 2005; published 20 March 2006 /H20850 The micromagnetic dynamics of ultrafast magnetization reversal in elliptical Permalloy /H20849Ni80Fe20/H20850thin ﬁlm elements is described. It is shown that coherent rotation and magnetization ringing in submicron Py elementscan be controlled by adjusting the axis ratio of the ellipse, the thickness, and the angle of the magnetic ﬁeldpulse. For the elliptical Py element with 400-nm-long axis, 200-nm-short axis, and 4.7 nm thickness, thenonuniform distribution of magnetization results from a strong in-plane, nonuniform demagnetization ﬁeldduring magnetization precession. It is the main reason magnetization ringing appears, even though the averagevalues of M yand Mzare equal to zero at the moment the pulse is terminated. The simulation results indicate that uniformity in the distribution of the magnetization during reversal is improved by reducing the length ofthe short axis from 200 to 112 nm, and reducing the thickness of the thin ﬁlm from 4.7 to 3.2 nm. Themodiﬁcation in the geometric conﬁguration of the element is found to effectively suppress the magnetizationringing. DOI: 10.1103/PhysRevB.73.104425 PACS number /H20849s/H20850: 75.40.Gb, 75.40.Mg, 75.50.Ss, 75.60.Jk I. INTRODUCTION The factors impacting magnetization reversal in a magnet can be categorized as either intrinsic or extrinsic. Intrinsicfactors are, for example, composition and crystal structurethat determine properties such as the saturation magnetiza-tion M s, the exchange interaction constant A, and the mag- netocrystalline anisotropy constant Ka. The roughness of crystal boundaries and surfaces and interfaces, the densityand orientation of steps, strains, texture, dislocation, and de-fects are collectively referred to as a magnet’s microstruc-ture. The microstructure is another intrinsic factor that seri-ously affects the magnetic properties of a magnet. Lessobvious intrinsic factors are the size and shape of a magnet.These strongly inﬂuence the distribution of magnetization M and demagnetization ﬁeld H d, and the magnetic domains in both the static equilibrium state /H20849i.e., before and after mag- netization reversal /H20850and during the process of magnetization reversal. The practical control of these factors is the subjectof numerous experimental and theoretical problems. 1–7Of the extrinsic factors, the effects of the external surroundingson the magnet, the most notable is the external ﬁeld. Itsstrength, rise time, fall time, duration, and orientation allplay key roles in the dynamic process of magnetization re-versal. Design of an appropriate external magnetic ﬁeld pulseinvolves some technical problems and is usually done byexperimental methods. Another extrinsic factor worth notingis the temperature, which is important during magnetizationreversal. 1 The advantage of a giant magnetic resistance or a mag- netic tunneling junction based system for magnetic randomaccess memory, as well as any other magnetic storage de-vice, is that it would have a quicker write time and be moreenergy efﬁcient. To obtain the fastest magnetization reversal,the ideal mechanism is a complete, coherent, rotation of themagnetization without forming any magnetic domain wallswithin the element. Since domain wall motion requires moretime to complete the magnetization reversal, avoiding do- main wall formation, and the subsequent motion of thosewalls, the magnetization reversal is faster. On the other hand,a higher normalized remanence and a modest high coercivityare required for magnetic storage elements to guarantee theaccuracy and stability of the data. How to obtain both thefastest magnetization reversal and a stable magnetization af-ter reversal in small magnetic elements is a very worthwhiletopic in ultrafast magnetization dynamics and in ultrahighdensity magnetic recording media. Many studies have beendone on this subject by numerical simulation andexperiment. 2–7Among the more successful results was a complete magnetization reversal at about 200 ps without anyfurther magnetization precession or magnetization ringing.This was accomplished in an 8 nm thin Py element withelliptical shape /H2084916 /H9262m/H110038/H9262m/H20850using a 70 Oe, transverse, shaped magnetic ﬁeld pulse which was cut off at the appro- priate time.3With the development of /H20849and advances made in/H20850electron beam lithography, the fabrication of magnetic elements smaller than a 100 nm is relatively easy. Clearly, asmaller element would allow for higher density applications,but when the size of the magnetic element is reduced tosubmicron dimensions, it has been found impossible to ob-tain a complete, coherent rotation without magnetizationringing using the previous method. The present work dem-onstrates simulation studies that yield a solution to the prob- lem. In the simulation, run on OOMMF software,8both ul- trafast magnetization reversal and suppression of themagnetization ringing were obtained in an elliptical elementof Py thin ﬁlm with submicron dimensions. The axis ratio ofthe ellipse, the thickness of the thin ﬁlm, the orientation andthe duration time of the magnetic ﬁeld pulse were variedthroughout the studies. II. MICROMAGNETIC SIMULATION MODEL The Landau–Lifshitz–Gilbert /H20849LLG /H20850equation, dM/H20849t/H20850/dt =−/H20841/H9253/H20841/H20851M/H20849t/H20850/H11003Heff/H20849t/H20850/H20852−/H9261/H20853M/H20849t/H20850/H11003/H20851M/H20849t/H20850/H11003Heff/H20849t/H20850/H20852/H20854, is a phe-PHYSICAL REVIEW B 73, 104425 /H208492006 /H20850 1098-0121/2006/73 /H2084910/H20850/104425 /H208495/H20850/$23.00 ©2006 The American Physical Society 104425-1nomenological description of magnetization dynamics.9Here the gyroscopic constant /H9253represents the precessional fre- quency, and the phenomenological damping factor /H9261drives the system towards an energy minimum after stopping en-ergy input to a system, i.e., energy relaxation. In general, /H9261is small compared to /H9253/Ms/H20849where Msis the saturation magne- tization /H20850, which implies that the energy relaxation of the sys- tem takes much longer than a few full precessional cycles ofmagnetization M. This under-damped behavior is the mag- netization ringing previously mentioned. The H effis the in- ternal effective magnetic ﬁeld of a system. Generally, thetotal energy density /H9255 totin a system mainly includes Zeeman energy /H9255zwhen an external magnetic ﬁeld is applied on the system, exchange interaction energy /H9255ex, magnetocrystalline anisotropic energy /H9255aniand demagnetization energy /H9255d, i.e., /H9255tot=/H9255z+/H9255ex+/H9255ani+/H9255d. All these energies are the function of magnetization distribution M, i.e.,/H9255i=/H9255i/H20849M/H20850. The change of each energy with respect to magnetization Mrepresents the corresponding magnetic ﬁeld Hi,/H20849Hi=−d/H9255i/dM/H20850. Therefore, theHeffis the vector sum of the applied external ﬁeld Happ, the exchange interaction Hex, the magnetocrystalline aniso- tropy ﬁeld Hani, and the demagnetizing ﬁeld Hd, i.e., Heff =Happ+Hex+Hani+Hd. Since the coherent rotation reversal time is very short compared with the energy relaxation time,the precessional motion of the magnetization vector is pri-marily governed by the ﬁrst term of the LLG equation. Fromthe LLG equation, the dynamic magnetization equilibriumcondition, which is dM/H20849t/H20850/dt=0 /H20851orM/H20849t/H20850/H11003H eff/H20849t/H20850=0/H20852, can be obtained. It means that the condition for stopping magne- tization precession is Heff=0, or the angle between Mand Heffmust equal 0° or 180°. In our study, submicron, ellipse, Py thin ﬁlm elements were chosen. In order to avoid magnetization reversal in thedirection perpendicular to the plane of the thin ﬁlm, whichwould form a Bloch wall, the thickness of Py thin ﬁlm mustbe smaller than the exchange interaction length. Due to thezero magnetocrystalline anisotropy, the exchange interactionlength of Py depends only on the equilibrium conditionbetween its exchange interaction energy and its demagneti- zation energy. For the Py thin ﬁlm, the exchange interaction length is /H9011=/H208492A/ /H92620Ms2/H208501/2.2For Py, /H9011/H110155.29 nm is ob- tained using A=13/H1100310−12J/m and Ms=860 kA/m.9On the other hand, the fundamental magnetic properties might bechanged for the thickness of being smaller than 3 nm, 10,11 therefore, the thickness was varied from 5 to 3 nm. Thelength of long axis was ﬁxed at 400 nm. The length of shortaxis was also systematically varied from 200 to 100 nm.Considering that a too small length of short axis of ellipsemight lead to the increase of magnitude of magnetic ﬁeldpulse to reverse magnetization, in practice the magnitude ofpulse would be less than 300 Oe, 12we reduce the length of short axis only down to 100 nm. The damping coefﬁcient0.01 of Py was chosen. The initial magnetization state was asaturation remnant state in the negative xdirection /H20849i.e., ini- tially in the Sstate /H20850. The magnitude of the uniform magnetic ﬁeld pulse was ﬁxed at 250 Oe and the zero rise time and falltime of pulse were chosen. To suppress ringing, the magnetic ﬁeld pulse was applied in the plane of the thin ﬁlm /H20849x-y plane /H20850at a variable angle /H9021relative to the long axis of the ellipse, so that the remanent energy as small as possible wasobtained at the moment of cutting off the pulse and the mag-netization M y=Mz=0. In order to avoid large errors from the in-plane exchange interaction, a cell size of 2 nm /H110032n m was chosen. FIG. 1. /H20849Color online /H20850Domain conﬁgurations captured at se- lected time points for Py ellipse with size of 400 nm /H11003200 nm /H110034.7 nm. Images represent the magnetization component along the xaxis /H20849Mx/H20850. FIG. 2. /H20849Color online /H20850/H20849a/H20850Temporal evolution of the average values of the three magnetization components, the demagnetizationenergy E d, the exchange energy Eex, and the magnetic ﬁeld pulse Hp at the angle of 55° for Py ellipse with size of 400 nm /H11003200 nm /H110034.7 nm. /H20849b/H20850Three-dimensional trajectory of the average magne- tization M.XIAO et al. PHYSICAL REVIEW B 73, 104425 /H208492006 /H20850 104425-2III. RESULTS AND DISCUSSION Figure 1 and Figs. 2 /H20849a/H20850and 2 /H20849b/H20850show the simulation re- sults of the elliptical element with a 400-nm-long axis, a200-nm-short axis, and a 4.7 nm thickness. In this case, themagnetic ﬁeld pulse is applied at an angle of 55° with respectto the x-axis. Figure 1 shows the magnetization distribution at different times during the reversal. The distribution isquite uniform in the initial state, but the distribution quicklybecomes nonuniform during the rotational process. The rota-tion of the magnetization in the outer /H20849marginal /H20850areas is lagging behind the central area. This indicates that the de-magnetization ﬁeld H dyin these areas is larger than that in the central area. Then, the torque, − /H20841/H9253/H20841/H20851M/H20849t/H20850/H11003Hdy/H20849t/H20850/H20852, forces the magnetization Mto rotate in the negative zdirection. It follows that MzorHdzin these areas is smaller than in the central area. The torque, − /H20841/H9253/H20841/H20851M/H20849t/H20850/H11003Hdz/H20849t/H20850/H20852, determines the rotational speed of magnetization, therefore we can conclude that the nonuniform distribution of magnetization resultsfrom the larger demagnetization ﬁeld, H dy, in the outer areas. Figure 2 /H20849a/H20850shows the temporal evolution of the average values of the normalized magnetization components /H20849Mx/Ms, My/Ms, and Mz/Ms/H20850, the demagnetization energy Ed, the ex- change energy Eex, and the magnetic ﬁeld pulse Hp, respec- tively. Although the average values My/Ms=Mz/Ms=0 at 165.6 ps, the average value of Mx/Msis 0.86 and it has a higher Edand Eexrelative to the relaxation state. From the image of the magnetization distribution at 165.6 ps in Fig. 1it can be seen that the magnetization vectors are not perfectly FIG. 3. /H20849Color online /H20850Reducing the length of the elliptical short axis from 200 to 112 nm leads to the decrease in the Mzand the backward rotation in the central part. /H20849a/H20850Temporal evolution of the average values of the Mx/Ms,My/Ms,Mz/Ms,Ed,Eexand the pulse Hpat an angle of 64°. /H20849b/H20850Three-dimensional trajectory of average magnetization M. FIG. 4. /H20849Color online /H20850Evolution of domain conﬁgurations with increasing time for Py ellipse with size of 400 nm /H11003112 nm /H110034.7 nm. Images represent the magnetization component along the xaxis /H20849Mx/H20850. FIG. 5. /H20849Color online /H20850Simultaneously reducing both the length of the elliptical short axis and the thickness of thin ﬁlm /H20849400 nm /H11003112 nm /H110033.2 nm /H20850results in a more uniform distribution of the magnetization. /H20849a/H20850Temporal evolution of the average values of Mx/Ms,My/Ms,Mz/Ms,Ed,Eexand the pulse Hpat an angle of 78°. /H20849b/H20850Three-dimensional trajectory of the average magnetization M.DYNAMICS OF ULTRAFAST MAGNETIZATION ¼ PHYSICAL REVIEW B 73, 104425 /H208492006 /H20850 104425-3aligned along the positive xdirection. As a result, when the pulse is cut off at this time, the magnetizations in each partare not at the dynamic equilibrium state, so that the ringingaround the xaxis appears and persists for a longer time, as in Figs. 2 /H20849a/H20850and 2 /H20849b/H20850./H20851Note the enlargement of the z-axis scale in/H20849b/H20850./H20852The conclusion obtained from the result is that a uniform distribution of magnetization in the entire elementduring precession is the key to suppression of the magneti-zation ringing. Inspection of the images at 80 and 100 ps in Fig. 1 clearly shows that the rotational delay occurs at the top and bottomarcs of the ellipse. Thus we tried to further reduce the lengthof short axis of the ellipse in order to ﬂatten the top andbottom arcs which led to the nonuniform distribution. Figure3 shows the simulation results of the elliptical element withdimensions of 400 nm /H11003112 nm /H110034.7 nm. In Figs. 3 /H20849a/H20850and 3/H20849b/H20850, it can be seen that there is a cave part for the average value of M z/Ms,My/Ms, and Ed, in the middle of the pre- cessional process. In addition, the reversal speed clearlyslows down. These indicate that the magnetization in someparts rotates toward the negative zdirection because of the strong torque, − /H20841 /H9253/H20841/H20851M/H20849t/H20850/H11003Hdy/H20849t/H20850/H20852, and with − Mz, even ro- tate back towards the negative xdirection. Comparing the image in Fig. 4 at 120 ps with that of Fig. 1 at 80 ps, it canbe seen that the distribution of magnetization in the middleregion is more uniform in Fig. 4 than that in Fig. 1. Therotational speed in Fig. 4 is clearly slower in the middle thanat the left and right sides, so that the two domain walls areformed. As the reversal progresses, the domain walls movetoward the middle region and the magnetizations in themiddle region rotate backward, as shown in the image cap-tured at 204 ps. Therefore, according to the dynamic prin-ciple, it can be inferred that as a result of the stronger torque −/H20841 /H9253/H20841/H20851M/H20849t/H20850/H11003Hdy/H20849t/H20850/H20852, which is against the torque − /H20841/H9253/H20841/H20851M/H20849t/H20850 /H11003Hp/H20852, the magnetizations in the middle region have − Mz around this time, whereas the left and right sides have mag- netizations of + Mzand certainly rotate in the positve xdirec- tion. This should be attributed to the weaker torque−/H20841 /H9253/H20841/H20851M/H20849t/H20850/H11003Hdy/H20849t/H20850/H20852in both parts. In the two domain walls the magnetization has a large angle change in both the x-y plane and in the zdirection. This large angle change leads to the higher exchange interaction energy corresponding to themaximum of E exin Fig. 3 /H20849a/H20850. As the rotation proceeds, the angles between the magnetization vectors and the ydirection are increasing, so that Hdyand the average value of Edare decreasing. Thus, the magnetization in the middle region ro-tates in the positive zand ydirections again because the torque, − /H20841 /H9253/H20841/H20851M/H20849t/H20850/H11003Hp/H20852is becoming dominant again. This can be seen in Figs. 3 /H20849a/H20850and 3 /H20849b/H20850as an increase in Mz/Ms, My/Ms,Ed, and the reversal speed. Finally, the action of the two dominant torques, − /H20841/H9253/H20841/H20851M/H20849t/H20850/H11003Hp/H20852and − /H20841/H9253/H20841/H20851M/H20849t/H20850 /H11003Hdz/H20849t/H20850/H20852cause the magnetizations in different parts to rotate toward the positive xdirection, so that the average values of Mz/Ms, and My/Msgo down to zero at the time the pulse is terminated. In this case the distribution of the magnetizationwas still not uniform. By comparison with the previous el-liptical element, the increase of ringing in the zdirection is due to the increase of H dyin this element with a shorter short axis. In order to prevent a large rotation of the magnetizationin the zdirection thereby increasing the uniformity of the FIG. 6. /H20849Color online /H20850Evolution of domain conﬁgurations as a function of time for the Py ellipse with size of 400 nm /H11003112 nm /H110033.2 nm. Images represent the magnetization component along the xaxis /H20849Mx/H20850. FIG. 7. /H20849Color online /H20850/H20849a/H20850Temporal evolution of the average values of the three magnetization components, the demagnetizationenergy E d, the exchange energy Eexthe magnetic anisotropy energy Eani, and the magnetic ﬁeld pulse Hpat the angle of 49.6°, calcu- lated for Py ellipse with size of 400 nm /H11003200 nm /H110034.7 nm and Ku=3.9 kJ/m3,/H20849b/H20850Three-dimensional trajectory of the average magnetization M.XIAO et al. PHYSICAL REVIEW B 73, 104425 /H208492006 /H20850 104425-4magnetization distribution, the thickness of the thin ﬁlm was properly reduced to increase the demagnetization ﬁeld in z direction. Figure 5 and Fig. 6 show the simulation results of the elliptical element with dimensions of 400 nm /H11003112 nm /H110033.2 nm. Comparison with the results from the last element shows that the rotation of the magnetization in the zdirection is clearly suppressed by reducing the thickness of the thinﬁlm. This leads to a more uniform distribution of the mag- netization during the precessional process. At 202 ps, the M y/Ms=Mz/Ms=0, Mx/Msreaches 0.97, and both the de- magnetization energy Edand the exchange interaction energy Eexare clearly smaller than those of the former two elements. This means that most of the magnetization is aligned withthe positive xaxis and meets the dynamic equilibrium con- dition at that moment, so that the ringing is largely sup-pressed after the magnetic ﬁeld pulse is cut off. From the results of suppressing ringing shown above, the reduction in the length of short axis and the thickness of thethin ﬁlm not only increases the effective magnetic aniso-tropy, more importantly, it increases the uniformity of themagnetization distribution during the reversal process, sothat the more uniform distribution of magnetization andlower exchange and demagnetization energy can be obtained.In literatures, several methods have been reported to increasethe induced magnetic anisotropy of Py thin ﬁlm. 13–15For example, the strain induced magnetic anisotropy can giverise to the magnetic anisotropy ﬁeld of Py thin ﬁlm up to92 Oe. 15Figures 7 /H20849a/H20850and 7 /H20849b/H20850show the simulation results of the elliptical element with 400-nm-long axis, 200-nm-shortaxis, 4.7 nm thickness, and the magnetic anisotropy constantofK u=3.9 kJ/m3, which was estimated from the value of the anisotropy ﬁeld given in Ref. 15. Comparing with theresults in Figs. 2 /H20849a/H20850and 2 /H20849b/H20850, the ringing cannot be sup- pressed by only increasing magnetic anisotropy. This resultindicates that the modiﬁcation of the element shape andthickness for a given material is an effective way to suppressthe magnetization ringing. IV . SUMMARY In elliptical Py thin ﬁlm elements, the nonuniform distri- bution of magnetization results from a stronger, in-plane,nonuniform demagnetization ﬁeld during the magnetizationprecession. It is the main reason magnetic ringing appears,even when the average values of M yand Mzequal zero at the moment the pulse is cut off. For the Py elliptical elementwith 400-nm-long axis, 200-nm-short axis, and 4.7 nm thick-ness, the simulation results indicate that uniformity in thedistribution of demagnetization ﬁeld during magnetizationreversal can be improved by reducing the length of short axisfrom 200 to 112 nm, and simultaneously reducing the thick-ness of the thin ﬁlm from 4.7 to 3.2 nm. The improved uni-formity of magnetization during magnetization reversal ef-fectively suppresses magnetization ringing. ACKNOWLEDGMENTS This work was supported by the Natural Sciences and Engineering Research Council /H20849NSERC /H20850of Canada and U.S. Air Force Research Laboratory /H20849AFRL /H20850under Grant No. F29601-04-1-206. 1E. Beaurepaire, M. Maret, V. Halté, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Phys. Rev. B 58, 12134 /H208491998 /H20850. 2J. Miltat, G. Albuquerque, and A. Thiaville, in Spin Dynamics in Conﬁned Magnetic Structures I, Topics Appl. Phys. /H20849Springer, Berlin, 2002 /H20850, Vol. 83. 3Th. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. Bar, and T. Rasing, Nature /H20849London /H20850418, 509 /H208492002 /H20850. 4M. Bauer, J. Fassbender, B. Hillebrands, and R. L. Stamps, Phys. Rev. B 61, 3410 /H208492000 /H20850. 5M. Bauer, R. Lopusnik, J. Fassbender, and B. Hillebrands, Appl. Phys. Lett. 76, 2758 /H208492000 /H20850. 6S. Kakaa and S. E. Russek, Appl. Phys. Lett. 80, 2958 /H208492002 /H20850. 7H. W. Schumacher, C. Chappert, P. Crozat, R. C. Sousa, P. P. Freitas, J. Miltat, J. Fassbender, and B. Hillebrands, Phys. Rev.Lett. 90, 017201 /H208492003 /H20850. 8M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0, National Institute of Standards and Technology /H20849Sept.1999 /H20850, http://math.nist.gov/oommf/. 9L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153 /H208491935 /H20850; T. L. Gilbert, Phys. Rev. 100, 1243 /H208491955 /H20850. 10U. Gradmann, in Handbook of Magnetic Materials , edited by K. H. J. Buschow /H20849Elsevier North-Holland, Amsterdam, 1993 /H20850, Vol. 1. 11W. J. M. de Jonge, P. J. H. Bloemen, and F. J. A. den Broeder, in Ultrathin Magnetic Structures , edited by J. A. C. Bland and B. Heinrich /H20849Springer, New York, 1994 /H20850, Vol. 1, p. 65. 12B. C. Choi, M. Belov, W. K. Hiebert, G. K. Ballentine, and M. R. Freeman, Phys. Rev. Lett. 86, 728 /H208492001 /H20850. 13H. Katada, T. Shimatsu, I. Watanabe, H. Muraoka, Y. Nakamura, and Y. Sugita, IEEE Trans. Magn. 37, 2334 /H208492001 /H20850. 14C. M. Fu, P. C. Kao, M. S. Tsai, H. S. Hsu, C. C. Yu, and J. C. A. Huang, J. Magn. Magn. Mater. 239,1 7 /H208492002 /H20850. 15R. Loloee, S. Urazhdin, W. P. Pratt, H. Geng, and M. A. Crimp, Appl. Phys. Lett. 84, 2364 /H208492004 /H20850.DYNAMICS OF ULTRAFAST MAGNETIZATION ¼ PHYSICAL REVIEW B 73, 104425 /H208492006 /H20850 104425-5
PhysRevLett.104.217201.pdf	Effects of Disorder and Internal Dynamics on Vortex Wall Propagation Hongki Min,1,2Robert D. McMichael,1Michael J. Donahue,3Jacques Miltat,1,2,4and M. D. Stiles1 1Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-6202, USA 2Maryland NanoCenter, University of Maryland, College Park, Maryland 20742, USA 3Mathematical and Computational Sciences Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8910, USA 4Laboratoire de Physique des Solides, Universite ´Paris Sud, CNRS, UMR 8502, 91405 Orsay, France (Received 4 March 2010; published 26 May 2010) Experimental measurements of domain wall propagation are typically interpreted by comparison to reduced models that ignore both the effects of disorder and the internal dynamics of the domain wallstructure. Using micromagnetic simulations, we study vortex wall propagation in magnetic nanowiresinduced by ﬁelds or currents in the presence of disorder. We show that the disorder leads to increases and decreases in the domain wall velocity depending on the conditions. These results can be understood in terms of an effective damping that increases as disorder increases. As a domain wall moves throughdisorder, internal degrees of freedom get excited, increasing the energy dissipation rate. DOI: 10.1103/PhysRevLett.104.217201 PACS numbers: 75.78.Fg, 72.25.Ba, 75.78.Cd The dynamics of magnetic domain wall structures driven by ﬁelds or currents is a subject of practical impor- tance related to possible schemes for nanoscale magneticmemory [ 1–3] and logic [ 4,5] devices. In these devices, information is encoded in the magnetic domains separatedby domain walls and the stored information is manipulatedby domain wall motion driven either by ﬁelds or currents. Experimentally, domain wall dynamics have been studied by the magneto-optical Kerr effect [ 6–10], resis- tance measurements using the giant magnetoresistance effect [ 11,12] or the anisotropic magnetoresistance effect [13–15], and real-space magnetic imaging by magnetic force microscopy [ 16] or spin-polarized scanning electron microscopy [ 17,18]. Typical experiments measure a do- main wall displacement and a time interval which are usedto infer an average velocity. Interpretations of these resultstypically ignore the effects of disorder. Real samples, however, display thickness ﬂuctuations and grain structure, and contain impurities and other defects. The consequences of disorder on domain wall motion have been studied theoretically in several limits.Micromagnetic simulations show that sample edge rough-ness can enhance domain wall propagation in a Ni 80Fe20 wire [ 19,20]. The dynamics of domain walls in the pres- ence of a single pinning potential [ 21] or array of pinning potentials [ 22] show the existence of a threshold ﬁeld or current to depin domain walls trapped by the pinningpotentials. Moreover, domain wall creep [ 23] is common for distributed disorder at ﬁnite temperatures. In this Letter, we describe micromagnetic simulations of domain wall propagation induced by ﬁelds or currents inthe presence of disorder throughout the ﬁlm. Our resultsindicate that disorder, which exists inevitably in real ex- periments, affects domain wall dynamics in a way that can be interpreted as an enhancement of the effective damping.This increase is signiﬁcant enough that it should affect the interpretation of most domain wall experiments. Our work adds important considerations to the extraction from ex-periment of the intrinsic damping constant and the closelyrelated nonadiabatic spin-transfer torque parameter. Magnetization dynamics in the presence of a spin cur- rent can be described by an extended Landau-Lifshitz-Gilbert equation [ 24,25] _M¼/C13H eff/C2Mþ/C11^M/C2_M/C0ðvs/C1r ÞM þ/C12^M/C2ðvs/C1r ÞM; (1) where Heffis the effective magnetic ﬁeld including the external, exchange, demagnetization, and anisotropyﬁelds, /C13is the gyromagnetic ratio, M sis the saturation magnetization, ^M¼M=Ms, and/C11is the Gilbert damping constant. The coupling between the current and the mag- netization is characterized by two parameters. The ﬁrst isthe velocity v s¼PJg/C22B=ð2eMsÞ, where Pis the polar- ization of the current, Jis the current density, gis the Lande ´factor, /C22Bis the Bohr magneton, and eis the (negative) charge of the electron. The second parameteris the nonadiabatic spin-transfer torque parameter /C12. In magnetic nanowires, the magnetization tends to point along the wires. Domain walls form between domains of oppositely directed magnetization with demagnetizationﬁelds giving them complicated structures depending onthe wire geometry [ 26,27]. The domain wall structure of interest here is a vortex wall, in which the magnetization inthe wall rotates around a vortex core and points out of theplane of the wire at the core region. This magnetizationconﬁguration is illustrated in Fig. 1(a). The conﬁguration also contains two half antivortices on each of the edges of the wire.PRL 104, 217201 (2010) PHYSICAL REVIEW LETTERSweek ending 28 MAY 2010 0031-9007 =10=104(21) =217201(4) 217201-1 /C2112010 The American Physical SocietyWhen a magnetic ﬁeld is applied to a vortex wall, the vortex core displaces to the side of the wire. If the ﬁeld isbelow a value called the Walker breakdown ﬁeld [ 28], the core then moves steadily along the wire. If the ﬁeld isabove the breakdown ﬁeld, the vortex core collides withthe edge of the wire, reverses its magnetization, and movesto the other side. The vortex core moves along the wire as itcollides with both edges, as illustrated in the ﬁrst panel ofFig.1(b). Similar motion results when a current is applied to the wire. The motion of domain walls is frequently studied in models which adopt a reduced description of domainwall structures in terms of a limited number of collectivecoordinates [ 25,28–31]. These models, however, ignore the additional degrees of freedom that may be excited duringdomain wall motion and further ignore the degree to whichthe excitation of these additional degrees of freedomchange in the presence of disorder. These effects are cap-tured in micromagnetic simulations. We compute domain wall motion through numerical solution of Eq. ( 1) using the Object Oriented Micromag- netic Framework ( OOMMF )[32]. We set up a Ni80Fe20strip with 200 nm width, 20 nm thickness, and 5 nm cell size,and choose a long enough length to allow for subsequentdomain wall propagation (typically from 10 000 to15 000 nm). In this geometry, vortex wall structures areformed as the ground state between head-to-head magneticdomains, as shown in Fig. 1(a). For material constants, we use the saturation magnetization M s¼800 kA =m, ex- change stiffness constant A¼13 pJ=m, and damping con- stant/C11¼0:01. In order to remove ﬁnite size effects, we add two features to the simulations. First, when we truncatethe inﬁnite wire we are modeling, there are unwanted fringing ﬁelds at the ends of the ﬁnite segment. We com-pensate these ﬁelds with static magnetic ﬁelds. Second, weinclude absorbing boundary conditions [ 33] to remove spin waves reﬂected back to the computational region. We model thickness ﬂuctuations by varying the satura- tion magnetization M s[34] but keeping the geometry uniform for simplicity. We choose a spatial correlationlength of 10 nm [ 35] and characterize the disorder as the ratio of ﬂuctuation standard deviation to the saturation magnetization, D¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ hðMðrÞ/C0M sÞ2ip =Ms. We limit the size of the ﬂuctuations to ensure that the magnetizationstays positive. To test this disorder model, we made com-parisons (not shown) with recent gyration experiments[35], which show a factor of 2–3 variation in resonance frequency as a vortex is scanned over a disk-shaped sam-ple. We ﬁnd that a disorder value of 0.05 gives roughly thesame variation in simulations with similar length scales.We tested different disorder models, such as random an-isotropy directions with enhanced anisotropy constants,and found that our main results remain unaltered. Figure 1(b)shows the effect of disorder on ﬁeld-induced vortex wall propagation above H W. In the absence of dis- order (upper panel), the vortex wall moves regularly fromside to side of the wire switching the sign of core magne-tization each time as it collides into the boundary. In thepresence of disorder (lower panel), the vortex wall prop-agates both irregularly and faster. The wall moves fasterbecause disorder complicates the wall motion, increasingthe ﬂuctuations of the magnetization, and hence enhancingthe total rate of energy dissipation into the lattice. Figure 2shows the local energy dissipation rate summed over the width and thickness of the wire as a function of theposition along the wire ( xaxis) and time ( yaxis). At each time, there are peaks in the dissipation rate where themagnetization changes rapidly in time as the domainwall moves, particularly around the vortex core and aroundeach of the half antivortices. This motion is an example ofan internal degree of freedom that is left out of a descrip-tion of the domain wall in terms of collective coordinates.The straight lines running left or right and slightly upindicate the emission of spin wave packets when the vortexcore collides with the boundary. This emission is muchstronger for collisions with one wire edge than the otherbecause the collisions with the edges are not symmetric.For the ﬁeld values considered, the magnetization in vortexwalls rotates with a ﬁxed handedness around the vortexcore, which when combined with the applied ﬁeld breaksthe symmetry of the vortex relative to the two edges.Because of this asymmetry, the core has a signiﬁcantlyhigher velocity approaching one edge than it does ap-proaching the other. The inset of Fig. 2(a) shows that most of the energy dissipation occurs in a 400 nm wide region around thevortex core rather than through spin wave emission. In thepresence of disorder, both energy dissipation centered FIG. 1 (color online). (a) A typical vortex wall structure in a wire with 200 nm width and 20 nm thickness. The color indicates the in-plane angle of the magnetization, and the arrows indicate the approximate magnetization direction. (b) Schematic trajec-tories of ﬁeld-induced vortex wall propagation in Ni 80Fe20ﬁlm for/C220H¼3m T above the critical ﬁeld along the xdirection with disorder D¼0and 0.05. Here the total simulation time is 100 ns. Points that the vortex core pass through are black (dark blue) or gray (orange) depending on whether the vortex core has its magnetization into or out of the plane. Insets show the domainwall displacement as a function of time.PRL 104, 217201 (2010) PHYSICAL REVIEW LETTERSweek ending 28 MAY 2010 217201-2around the core and through spin wave emission increase, as shown in the inset of Fig. 2(b). Spin wave emission is not just associated with collisions with the boundary, but ap-parently also with motion of the core through patches ofstrong disorder. However, the dominant contribution to theincreased rate of energy dissipation occurs in the localizedregion of the domain wall itself indicating the increase inthe excitement of the internal degrees of freedom of thedomain wall. Note that the enhanced damping presentedhere is quite different than the two-magnon contribution tothe linewidth as measured in ferromagnetic resonance. Wetested this by carrying out simulations of ferromagneticresonance without vortex wall structures and found a muchsmaller enhancement of the effective damping. Figure 3shows the domain wall velocity as a function of applied ﬁeld for disorder D¼0, 0.025, and 0.05. Here the domain wall velocity is estimated by ensemble averages ofup to 40 samples with different realizations of the disorder.The disorder suppresses or enhances the domain wall ve-locity depending on the ﬁeld range. At low enough ﬁelds,the domain walls are pinned in the presence of disorder.Note that the Walker breakdown ﬁeld ( H W/C250:7m T in the absence of disorder) itself is increased by the disorder. The results in Fig. 3show that even in the absence of disorder, vortex wall motion is complicated. The curve forno disorder, D¼0, shows the expected linear rise as the ﬁeld increases up to the breakdown ﬁeld, then the subse-quent decrease and increase as the ﬁeld increases further.However, the domain wall velocity as a function of ﬁeld also shows additional peaks above the breakdown ﬁeld,H>H W. Increasing disorder suppresses the peaks in the velocity curve, which may be the reason that they are notseen in experiments. We observe that the spacing of peaksincreases with increased intrinsic damping constant, andwith the increased sample width. These results suggest that the origin of the peaks is a resonance between periodic collisions and the internal excitations of vortex wall struc-tures. It would be interesting if magnetic nanowires couldbe fabricated with sufﬁciently low disorder to observe suchfeatures. The results in Fig. 3can be understood in terms of an increase in effective damping parameter due to disorder.For ﬁeld driven motion, the velocity depends strongly onthe energy dissipation rate because a translation of thedomain wall along the wire reduces the Zeeman energy.If the internal energy of the wall is not changed, the wallcan only move as this Zeeman energy is dissipated into thelattice. In the reduced models mentioned above, we expectv DW/C24/C11forH/C29HWbecause the energy dissipation rate and drift velocity of the wall increase as /C11increases, while vDW/C241=/C11forH<H Wbecause, as /C11increases, the displacement of the core toward the sample edge de-creases, and the drift velocity and the energy dissipation rate decrease [ 25,29]. We also note that H W/C24/C11. In the presence of disorder, the domain wall velocity increases asdisorder increases for H>H W, while for H<H W, the domain wall velocity decreases, exactly as would be ex-pected for an increase in the effective damping parameter. In the case of current-induced domain wall propagation, the results can also be interpreted by an enhanced effectivedamping. While we expected that the effective value of /C12 would increase as well, we ﬁnd that changing /C11alone provides the best explanation of the results. To see thisbehavior, we compare calculations of the domain wallvelocity as a function of disorder with calculations withoutdisorder but increasing damping constant, both with ﬁxed/C12. Figure 4(a)shows the domain wall velocity as a function of disorder DforJ¼2/C210 13A=m2, which is above the critical current J/C12¼0 c/C250:8/C21013A=m2. Figure 4(b) shows the domain wall velocity as a function of the damp- FIG. 2 (color online). Energy dissipation rate along xfor /C220H¼3m T with disorder (a) D¼0and (b) D¼0:05. Insets show contributions from spin wave and vortex to the energy dissipation rate, which were obtained by separatingregions near a vortex core with a diameter of 400 nm, as shown in Fig. 1(a). FIG. 3 (color online). Domain wall velocity as a function of applied ﬁeld for disorder D¼0, 0.025, and 0.05. Error bars indicate 1 standard deviation statistical uncertainty.PRL 104, 217201 (2010) PHYSICAL REVIEW LETTERSweek ending 28 MAY 2010 217201-3ing constant /C11with the same applied current density. As the disorder increases, the variation of the domain wall velocity increases or decreases depending on /C12showing a clear resemblance to the results with the enhanced damp-ing constant in the absence of disorder. We compute the disorder dependence of effective damp- ing by ﬁtting the domain wall velocity in the linear low ﬁeld and low current regime, as shown in Fig. 5. We point out that the actual values of the disorder-enhanced damp-ing rate depend on various factors such as the type of thedomain wall structures, the type of disorder, geometry of samples, and material properties. Several experiments would test the results of our calculations. One possibleexperiment is to measure the domain wall velocity with adisorder introduced in a controlled manner. Another pos- sible experiment would be the vortex gyration in a single pinning potential in which the enhanced damping could bemeasured by comparing the spectrum between free andtrapped regimes of vortex gyration. In summary, we have demonstrated that disorder affects domain wall dynamics signiﬁcantly and that the effective damping is increased by disorder and internal excitationsof the domain wall structure. From this work, we concludethat damping constants inferred from domain wall motionmeasurements are effective rather than intrinsic values, which are enhanced by the disorder in a sample. These re-sults suggest that caution is necessary in extracting funda- mental parameters from domain wall motion measure-ments. The work has been supported in part by the NIST-CNST/ UMD-NanoCenter Cooperative Agreement. The authorsthank J. J. McClelland, K. Gilmore, and June W. Lau fortheir valuable comments. [1] S. S. P. Parkin, U.S. Patent No. 6834005 (2004). [2] S. S. P. Parkin, M. 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Porter, Interagency Report No. NISTIR 6376 (National Institute of Standards and Technology, Gaithersburg, MD, 1999), http://math.nist .gov/oommf/. [33] D. V. Berkov and N. L. Gorn, J. Appl. Phys. 99, 08Q701 (2006) ; G. Consolo et al. ,IEEE Trans. Magn. 43, 2974 (2007) . [34] Locally varying Msgives the lowest order variation in both the magnetostatic interaction and vsconsistent with thickness ﬂuctuations. We ignore changes in the exchange coupling, which tests indicate does not strongly affect ourresults. [35] R. L. Compton and P. A. Crowell, Phys. Rev. Lett. 97, 137202 (2006) . FIG. 4 (color online). Domain wall velocity (a) as a function of disorder and (b) as a function of the damping constant for J¼ 2/C21013A=m2with/C12as a parameter. FIG. 5 (color online). Effective damping /C11effas a function of disorder Dfor ﬁeld-induced and current-induced vortex wall propagation.PRL 104, 217201 (2010) PHYSICAL REVIEW LETTERSweek ending 28 MAY 2010 217201-4
PhysRevB.79.134410.pdf	Current-induced resonant motion of a magnetic vortex core: Effect of nonadiabatic spin torque Jung-Hwan Moon,1Dong-Hyun Kim,2Myung Hwa Jung,3and Kyung-Jin Lee1,* 1Department of Materials Science and Engineering, Korea University, Seoul 136-701, Korea 2Department of Physics, Chungbuk National University, Chengju 361-763, Korea 3Department of Physics, Sogang University, Seoul 121-742, Korea /H20849Received 15 October 2008; revised manuscript received 2 February 2009; published 7 April 2009 /H20850 The current-induced resonant excitation of a magnetic vortex core is investigated by means of analytical and micromagnetic calculations. We ﬁnd that the radius and phase shift of the resonant motion are not correctlydescribed by the analytical equations because of the dynamic distortion of a vortex core. In contrast, the initialtilting angle of a vortex core is free from the distortion and determined by the nonadiabaticity of the spintorque. It is insensitive to experimentally uncontrollable current-induced in-plane Oersted ﬁeld and thus allowsa direct comparison with experimental results. We propose that a time-resolved imaging of the very initialtrajectory of a core is a plausible way to experimentally estimate the nonadiabaticity. DOI: 10.1103/PhysRevB.79.134410 PACS number /H20849s/H20850: 75.70.Kw, 72.25.Ba, 85.75. /H11002d I. INTRODUCTION A spin-polarized current can exert torque to a ferromagnet by transferring spin-angular momentum, i.e., spin-transfertorque. The spin-transfer torque provides full magnetizationreversal, steady-state precession motion, and domain-wallmovement. 1,2It is composed of adiabatic and nonadiabatic spin torque terms in continuously varying magnetizationsuch as a magnetic domain wall. The adiabatic spin torquearises from the conduction electron spin whose projection onthe ﬁlm plane follows the direction of a local magnetization,whereas the nonadiabatic torque arises from a mismatch ofthe direction as a result of the momentum transfer or the spinrelaxation. 3–5 Until now, the experimental threshold current density JC to steadily move a domain wall has been reported to be about 108A/cm2, too large for an application. In addition to the resonant depinning6and the use of perpendicular anisotropy,7 an increase in /H9252which is the ratio of the nonadiabatic spin torque to the adiabatic one can reduce JC.8Despite its impor- tance, however, the exact value of /H9252is still under debate even in theories;9–13/H9252=0,/H9252=/H9251, and/H9252/HS11005/H9251where /H9251is the Gilbert damping constant. This debate is also related towhich mechanism between the Landau-Lifshitz damping andthe Gilbert one is valid to describe the energy dissipationunder the current injection. 14Experimental determination of /H9252is essential to resolve this debate, but experimentally esti- mated values are also distributed; /H9252=8/H9251,6/H9252=/H9251,15/H9252/H11022/H9251,16 /H9252=2/H9251,17and/H9252/HS11005/H9251.18Since most experiments have used the same material /H20849Permalloy /H20850, this wide distribution is caused by origins irrelevant to the material itself. The wide distribution can originate from the Joule heat- ing, the edge roughness of nanowire, and the in-plane com- ponent of the current-induced Oersted ﬁeld HOein. The Joule heating signiﬁcantly affects JCand wall velocity.13,19There- fore, it is difﬁcult to precisely estimate /H9252when the Joule heating is not negligible, i.e., JC/H11022108A/cm2. In a magnetic nanowire, the edge roughness distorts and pins the domainwall 20and thus prevents a proper interpretation of experi- mental data using theories derived for an ideal nanowire. Away to avoid the above issues is to experimentally studyresonant motions of a magnetic vortex core /H20849VC/H20850in a pat- terned disk by injecting an alternating current of the order of10 7A/cm2. The magnetic vortex is an ideal system for the resonant motion study since VC can be considered as a to-pological point charge which efﬁciently responses to externalforces. 21–23It was experimentally conﬁrmed that the VC can be resonantly excited by an ac current.6,17,24,25Even in this case, however, a very small in-plane component of the ac- induced alternating Oersted ﬁeld HOeininhibits a precise esti- mation of the spin torque parameters.25,26Note that HOeinis not a current-induced ﬁeld along the thickness direction ofthe disk, but an in-plane ﬁeld caused by any geometricalsymmetry breaking of the system. The driving force due to H Oeinof only 0.3 Oe is as large as 30% of the total resonant excitation.25Such a small HOeinis difﬁcult to remove since it is caused by an uncontrollable nonuniform current distribu-tion due to a geometrical symmetry breaking such as electriccontacts or notches. Therefore, an experimental way to esti-mate the /H9252which is safe from the Joule heating and the edge roughness and also insensitive to the HOeinis highly desired. In this work, we propose that a direct imaging of the very initial trajectory of VC induced by an ac is a plausible way toexperimentally estimate /H9252, which is free from all the three issues. We ﬁnd that /H9252does not change the resonant fre- quency but affects the phase of resonant motion. The phaseshift is /H9252dependent since /H9252determines the tilting angle of the very initial trajectory measured from the direction of the electron ﬂow. On the other hand, HOeinwith a typical magni- tude does not change the tilting angle although it affects thesteady resonant motion. We ﬁnd that the initial tilting angleis only one physical observable which can be directly com-pared to the analytical result, whereas the others such as theradius and the phase shift are not correctly described by theanalytical equations because of the dynamic distortion ofVC. This paper is organized as follows. After introducing the- oretical approach /H20849Sec. II/H20850and micromagnetic simulation used in this work /H20849Sec. III/H20850, we discuss the effect of /H9252on the vortex dynamics /H20849Sec. IV/H20850. In Sec. V, we summarize this work.PHYSICAL REVIEW B 79, 134410 /H208492009 /H20850 1098-0121/2009/79 /H2084913/H20850/134410 /H208496/H20850 ©2009 The American Physical Society 134410-1II. THEORETICAL FORMULAE OF THE VORTEX CORE DYNAMICS The current-induced motion of VC is calculated using Thiele’s equation with the spin-transfer torque terms /H20851Eq. /H208491/H20850/H20852,5,27–29 G/H20849p/H20850/H11003/H20849u−r˙/H20850=−/H9254U/H20849r/H20850 /H9254r−/H9251Dr˙+/H9252Du, /H208491/H20850 where G/H20849p/H20850=−G0pezis the polarity /H20849p/H110061/H20850dependent gy- rovector, G0is obtained from the spin texture as G0=MS /H9253/H20885dVsin/H20849/H9258/H20850/H20849/H11612/H6023/H9258/H11003/H11612/H6023/H9274/H20850·ez, /H208492/H20850 /H9258/H20849/H9274/H20850is the out-of-plane /H20849in-plane /H20850angle of the magnetiza- tion, MSis the saturation magnetization, /H9253is the gyromag- netic ratio, u=u0exp /H20849i/H9275t/H20850ey,u0/H20849=/H9262BJP /eM S/H20850is the ampli- tude of adiabatic spin torque, /H9275is the angular frequency of the ac, r/H20849t/H20850=X/H20849t/H20850+Y/H20849t/H20850is the time-dependent position vector of VC, and U/H20849r/H20850is the potential well. The damping tensor D is also obtained from the integration of spin texture as D=−MS /H9253/H20885dV/H20851/H20849/H11612/H6023/H9258/H11612/H6023/H9258+ sin2/H20849/H9258/H20850/H11612/H6023/H9274/H11612/H6023/H9274/H20850/H20852. /H208493/H20850 When VC is at the static equilibrium position, G0is 2/H9266LM S//H9253and Dis diagonal and Dxx=Dyy=G0ln/H20849R//H9254/H20850/2 and Dzz=0, where Lis the thickness of disk, Ris the vortex radius, and /H9254is the core diameter. Thiele’s equation provides an analytical solution for the time-dependent position of VCwhen the VC does not change its shape in the dynamic mo-tion. In other words, the potential proﬁle Uand all param- eters such as G 0and Dshould be assumed to be constant to obtain an analytical solution. We will discuss later whetheror not this rigid VC assumption is valid. With X/H20849t/H20850=X 0exp /H20849i/H9275t/H20850,Y/H20849t/H20850=Y0exp /H20849i/H9275t/H20850, and 1 +/H20849/H9251D/G0/H208502/H110111, the solutions are in the form of X/H20849t/H20850 =X1cos/H20849/H9275t/H20850+iX2sin/H20849/H9275t/H20850and Y/H20849t/H20850=Y1cos/H20849/H9275t/H20850+iY2sin/H20849/H9275t/H20850 where X1=Ap/H9275r/H20851−/H20849/H9275r2−/H92752/H20850+2C2/H9251/H20849/H9252−/H9251/H20850/H92752/H20852, X2=Ap/H9275C/H20851/H20849/H9252−/H9251/H20850/H20849/H9275r2−/H92752/H20850+2/H9251/H9275r2/H20852, Y1=A/H9275rC/H20851/H9252/H20849/H9275r2−/H92752/H20850+2/H9251/H208491+C2/H9251/H9252/H20850/H92752/H20852, Y2=A/H9275/H20851/H208491+C2/H9251/H9252/H20850/H20849/H9275r2−/H92752/H20850−2C2/H9251/H9252/H9275r2/H20852. /H208494/H20850 Here, A=u0//H20851/H20849/H9275r2−/H92752/H208502+/H208492C/H9251/H9275r/H9275/H208502/H20852,/H9275r=/H9260/G0is the reso- nance frequency, /H9260=/H20849dU /dr/H20850/ris the effective stiffness co- efﬁcient of the potential well, and CisD/G0=ln /H20849R//H9254/H20850/2. From Eq. /H208494/H20850, one ﬁnds the radius a/H20849t/H20850=/H20881X/H20849t/H208502+Y/H20849t/H208502, and the phase shift /H9278between the phase of the core gyration and that of the ac, /H9278= tan−1/H20873X1 Y1/H20874 = tan−1/H208771− /H20849/H9275r//H9275/H208502+2C2/H9251/H20849/H9252−/H9251/H20850 C/H9252/H20851/H20849/H9275r//H9275/H208502−1/H20852+2C/H9251/H208491+C2/H9251/H9252/H20850/H20878. /H208495/H20850III. NUMERICAL MODEL In order to check the validity of the analytical solutions, the micromagnetic simulation is also performed by means ofthe Landau-Lifshitz-Gilbert equation including the spintorque terms, /H11509M /H11509t=−/H9253M/H11003Heff+/H9251 MSM/H11003/H11509M /H11509t+u·/H11612M −/H9252 Ms/H20851u·/H20849M/H11003/H11612/H20850M/H20852, /H208496/H20850 where Heffis the effective ﬁeld including the external, the magnetostatic, the exchange, and the current-induced Oer-sted ﬁeld. The current-induced dynamics of the vortex core ismicromagnetically modeled using a computational frame-work based on the fourth-order Runge-Kutta method. Themodel system is a circular Permalloy disk with the thicknessof 10 nm and the diameter of 270 nm which is vortex-favored dimension /H20851Fig. 1/H20849a/H20850/H20852. The unit cell size is 2 /H110032 /H1100310 nm 3on a two-dimensional grid. The integration time step is 0.2 ps. The ac ﬂows along the yaxis and uniform current distribution is assumed through the disk. The maxi-mum current density is 1.25 /H1100310 7A/cm2. We did not take into account the Joule heating effect since it is negligible atthis current density. 30Standard material parameters for Per- malloy are used: MS=800 emu /cm3,/H9253=1.76 /H11003107Oe−1s−1,/H9251=0.01, P=0.7, and the exchange constant Aex=1.3/H1100310−6erg /cm. IV. EFFECT OF /H9252ON THE VORTEX CORE DYNAMICS First, we assume HOein=0 in order to investigate the effect originating exclusively from /H9252on the resonant motion. We will recall the effect of HOeinin the last part. In Figs. 2/H20849a/H20850and 2/H20849b/H20850, we show analytical and modeling results of the time averaged value /H20855Y/H20849t/H20850·I0sin/H20849/H9275t/H20850/H20856/I0at various /H9252terms. To obtain analytical results, we use C=1.3 because Ris 135 nm and/H9254is 10 nm, determined from the micromagnetic conﬁgu- ration at the initial equilibrium state. /H20855Y/H20849t/H20850·I0sin/H20849/H9275t/H20850/H20856/I0 shows a peak at the resonance frequency /H9275rof 360 MHz. The peak structure is in general asymmetric regardless of /H9252 because the phase shift is dependent on the angular fre-quency /H9275/H20851see Eq. /H208495/H20850and Fig. 3/H20849a/H20850/H20852./H9275rdoes not change with/H9252whereas the peak structure becomes more asymmetric as/H9252increases. In spite of qualitative agreement, however, analytical re- sults are quantitatively different from modeling ones. This isbecause the radius of gyroscopic motion is different betweenthe modeling result and the analytic solution /H20851inset of Fig. 2/H20849b/H20850/H20852. We attribute this difference to a dynamic distortion of VC. As shown in Fig. 2/H20849c/H20850, the VC shape in the initial equi- librium state is symmetric whereas it in a dynamic motion isasymmetric. The distortion changes the gyroscopic param-eter G 0, the damping tensor D/H20849thus, the parameter C/H20850and the effective stiffness coefﬁcient /H9260since all parameters are determined by details of the spin texture /H20851see Eqs. /H208492/H20850and /H208493/H20850 and the deﬁnition of /H9260/H20852. From micromagnetic spin textures of the vortex in the dynamic motion, we ﬁnd that both Cand/H9260MOON et al. PHYSICAL REVIEW B 79, 134410 /H208492009 /H20850 134410-2increase from the initial equilibrium values because of the dynamic distortion /H20851Fig. 2/H20849d/H20850/H20852. The increase in /H9260is much larger than that of C, and responsible for the reduced radius in the modeling results because /H9260is directly connected to the resonance frequency in the theory. This increase in /H9260occurs at the very initial time stage, indicating that the assumption of the rigid VC and potential well is invalid except for thevery initial trajectory. We have checked if Thiele’s equationcan reproduce the micromagnetic simulations with signiﬁ-cantly better accuracy by considering some of the parametersof the equation as adjustable parameters /H20849e.g., G 0,D, and/H9260/H20850. However, we found that no parameter adjustment can repro-duce. It indicates that Thiele’s equation has a limitation tofully describe the long-time vortex dynamics due to its as-sumption of the ﬁxed magnetic texture. Therefore, even when H Oeinis zero, it may be difﬁcult to deduce important spin-torque parameters by directly comparing analytical so-lutions with experimental measurements of the steady gyro-scopic motion. Nevertheless, it is worthwhile investigating how /H9252in- duces the additional asymmetry in the peak structure. Figure3/H20849a/H20850shows analytical phase shifts at various /H9252terms. As /H9252 increases, the absolute value of the phase shift /H9278decreases /H20849increases /H20850for the frequency smaller /H20849higher /H20850than/H9275r.I n other words, a vertical offset /H9278/H20849/H9252/H20850−/H9278/H20849/H9252=/H9251/H20850of the phase shift increases with increasing /H9252. From Eq. /H208494/H20850with/H9251/H112701 and/H9252/H112701, one can ﬁnd that the vertical offset is approxi- mately tan−1/H20851C/H20849/H9252−/H9251/H20850/H20852and thus dependent on /H9252. This is why the peak structure becomes more asymmetric as /H9252increases. However, quantitative disagreement between analytic solu-tion and modeling result was again observed /H20851inset of Fig. 3/H20849a/H20850/H20852. The difference becomes larger as the radius of core gyration increases; i.e., the frequency approaches /H9275r. This is also caused by the dynamic distortion of VC as explainedabove. The vertical offset is /H9252dependent since the initial tilting angle/H9258intis determined by /H9252/H20851Fig.3/H20849b/H20850/H20852. VC initially moves along the direction of the electron ﬂow. Because of the im-balance of magnetostatic ﬁeld, VC experiences the centrip- etal force and starts to undergo a gyration motion. In the absence of HOein,/H9258intof the initial trajectory can be obtained from Eq. /H208491/H20850by dropping the potential gradient term since VC is initially at the bottom of the potential well where the gradient is zero. When HOeinis nonzero, the potential gradient is no longer zero and could affect /H9258int. In order to investigate the effect of HOeinon/H9258int, we perform micromagnetic simula- tions for the initial trajectory with and without taking into account HOein. We assume that an alternating HOeinis applied along the xaxis and its magnitude is 0.3 Oe which is similar with the estimated value in the experiment in Ref. 25.A s shown in Fig. 4/H20849a/H20850, the effect of HOeinon the very initial tra- jectory is negligible whereas the difference in trajectoriesbetween the two cases becomes larger and larger as the timeevolves. This insensitivity of the initial trajectory and thus /H9258inttoHOeinis valid for a different /H9252/H20849not shown /H20850. Thus, we drop the potential gradient term in Eq. /H208491/H20850to derive /H9258int. With p= +1 and the direction of the initial current along the + y axis, one ﬁnds /H9251DX˙−G0Y˙=G0u0, G0X˙+/H9251DY˙=−/H9252Du0, /H208497/H20850 where X˙and Y˙are the velocity along the xand yaxes, respectively. By solving Eq. /H208497/H20850forX˙and Y˙, X˙=G0u0D/H9251−/H9252 /H92512D2+G02, Y˙=−u0G02+/H9251/H9252D2 /H92512D2+G02. /H208498/H20850 The initial tilting angle /H9258intis given by FIG. 1. /H20849Color online /H20850/H20849a/H20850Schematics of the model system. /H20849b/H20850Magnetization of a vortex in its initial equilibrium state. The height denotes the zcomponent, whereas the color scale corresponds to the direction of the xcomponent of magnetization.CURRENT-INDUCED RESONANT MOTION OF A MAGNETIC … PHYSICAL REVIEW B 79, 134410 /H208492009 /H20850 134410-3FIG. 3. /H20849Color online /H20850/H20849a/H20850Phase shift as a function of the frequency for various /H9252terms. /H20849b/H20850Initial vortex core trajectories for various /H9252 terms /H20849/H9275=/H9275r/H20850. Inset of /H20849a/H20850shows the vertical offset of the phase shift for various /H9252terms. In the inset, open symbols are obtained from micromagnetic modeling. Inset of /H20849b/H20850describes the deﬁnition of the initial tilting angle. The results were obtained at HOein=0.FIG. 2. /H20849Color online /H20850/H20855Y/H20849t/H20850·I0sin/H20849/H9275t/H20850/H20856/I0as a function of the frequency obtained from /H20849a/H20850analytic solution of Thiele’s equation and /H20849b/H20850 micromagnetic simulation. /H20849c/H20850Comparison of the shape of vortex core, and /H20849d/H20850variation in /H9260as a function of the gyration radius. The inset of/H20849b/H20850shows the vortex core trajectories. The inset of /H20849d/H20850shows variation in the parameter C/H20849=D/G0/H20850with the time evolution. Both insets are obtained at /H9252=0 and /H9275=/H9275r.MOON et al. PHYSICAL REVIEW B 79, 134410 /H208492009 /H20850 134410-4/H9258int= tan−1/H20873X˙ −Y˙/H20874= tan−1/H20873C/H20849/H9251−/H9252/H20850 1+/H9251/H9252C2/H20874. /H208499/H20850 Note that Eq. /H208499/H20850is equivalent to the equation for the vertical phase shift with considering the sign of the initial ac and /H9251/H9252/H112701. It conﬁrms that the vertical shift originates from the /H9252-dependent /H9258int. Figure 4/H20849b/H20850shows /H9258intas a function of /H9252//H9251 for various values of C. It should be noted that /H9258int’s obtained from modeling are in excellent agreement with analyticalones in contrast to the radius and phase shift. It is becauseVC retains its equilibrium shape at the very initial time stagewhere Cand /H9260hardly change. It also conﬁrms that dropping the potential gradient term is valid to derive the initial tiltingangle. For the Permalloy disk tested in this work /H20849C=1.3 /H20850, the difference in /H9258intbetween /H9252=0 and /H9252=10/H9251is about 7° which may be too small to experimentally measure. However, the /H9252-dependent /H9258intbecomes larger as the parameter Cin- creases. The Cincreases as the disk diameter increases. Forinstance, the difference in /H9258intbetween /H9252=0 and /H9252=10/H9251is about 15° at C=2.65 corresponding to the disk diameter of 4/H9262m and the core diameter of 10 nm. The core diameter can be determined by micromagnetic calculation or magneticforce microscopy imaging for a given disk geometry. Finally, we propose that a time-resolved magnetic imag- ing with high spatial resolution such as x-ray microscopy 31 for observation of the very initial trajectory of VC in a widedisk is a possible way to experimentally estimate /H9252. The x-ray imaging technique may not give a very accurate /H9252 because of its spatial resolution /H20849/H1101515 nm /H20850. However, it can allow us to estimate a possible range of /H9252which is still useful for a better understanding of the nonadiabaticity ofspin torque. Furthermore, it can at least differentiate whetheror not /H9252is larger than /H9251. This is very important to understand the spin transport in domain walls since /H9252larger than /H9251is possible only when other mechanisms of momentum transferexists in addition to the transfer of the spin-angular momen-tum from a spin-polarized current. 16 V. SUMMARY Using analytical and micromagnetic calculations, we in- vestigate effects of the nonadiabatic spin torque on the reso-nant motion of a vortex core induced by an ac. We ﬁnd thatthe initial trajectory of a vortex core is dependent on /H9252and insensitive to an uncontrollable in-plane component of thecurrent-induced Oersted ﬁeld. A direct imaging of the veryinitial trajectory of a vortex core can be a way to experimen-tally determine /H9252. ACKNOWLEDGMENTS This work was supported by a Korea Research Founda- tion Grant funded by the Korean Government /H20849MOEHRD /H20850 /H20849Grant No. KRF-2006-311-D00102 /H20850. *Corresponding author; kj_lee@korea.ac.kr 1L. Berger, J. Appl. Phys. 55, 1954 /H208491984 /H20850;71, 2721 /H208491992 /H20850. 2J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 3G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 /H208492004 /H20850. 4S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 /H208492004 /H20850. 5A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. 69, 990 /H208492005 /H20850. 6L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner, and S. S. P. Parkin, Nature /H20849London /H20850443, 197 /H208492006 /H20850. 7S.-W. Jung, W.-J. Kim, T.-D. Lee, K.-J. Lee, and H.-W. Lee, Appl. Phys. Lett. 92, 202508 /H208492008 /H20850. 8G. Tatara, T. Takayama, H. Kohno, J. Shibata, Y. Nakatani, and H. Fukuyama, J. Phys. Soc. Jpn. 75, 064708 /H208492006 /H20850. 9S. 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PhysRevB.102.220409.pdf	PHYSICAL REVIEW B 102, 220409(R) (2020) Rapid Communications Electrically switchable Rashba-type Dzyaloshinskii-Moriya interaction and skyrmion in two-dimensional magnetoelectric multiferroics Jinghua Liang,1Qirui Cui,1and Hongxin Yang1,2,* 1Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China 2Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China (Received 5 July 2020; revised 10 October 2020; accepted 7 December 2020; published 22 December 2020) Realizing topological magnetism and its electric control are intensive topics in spintronics due to their promising applications in information storage and logic technologies. Here, we unveil that both can be achievedin two-dimensional (2D) magnetoelectric multiferroics. Using ﬁrst-principles calculations, we show that astrong Dzyaloshinskii-Moriya interaction (DMI), which is the key ingredient for the formation of exotic chiralmagnetism, could be induced in 2D multiferroics with vertical electric polarization via the Rashba effect. Weverify that such a signiﬁcant DMI can promote the stabilization of sub-10-nm skyrmions in 2D multiferroics withperpendicular magnetic anisotropy such as a CrN monolayer. In addition, the presence of both magnetizationand electric polarization in 2D multiferroics provides us with the unique opportunity for the effective electriccontrol of both the strength and chirality of the DMI and thereby the topological magnetism. As an example,we introduce four multiferroic skyrmions with different chiralities and polarities that can be manipulated by anexternal ﬁeld. DOI: 10.1103/PhysRevB.102.220409 Introduction. Topological chiral spin textures, e.g., spin spirals [ 1,2] and skyrmions [ 3–5], are of particular inter- est from both fundamental and applied points of view. Acrucial ingredient for the generation of exotic topologicalmagnetism is the antisymmetric Dzyaloshinskii-Moriya inter-action (DMI) [ 6,7] that results from spin-orbit coupling (SOC) in magnets with inversion asymmetry, favoring noncollinearspin conﬁgurations. The induction and control of the DMI areessential steps towards spintronics applications of topologicalmagnetism. In the last decade, much research has been devoted to employ ferromagnet/heavy metal (FM/HM) heterostructures,such as Fe/Ir [ 8], PdFe/Ir [ 9], and Co/Pt [ 10], with a strong interfacial DMI to realize skyrmions. Recently, withthe breakthrough in experimental realizations [ 11–15], two- dimensional (2D) intrinsic ferromagnetic thin ﬁlms, such asCrI 3[12], VSe 2[13], and Fe 3GeTe 2[15] monolayers, have attracted a lot of interest. However, the absence of a DMIin the former 2D magnets due to a symmetry constraint hashindered their application in skyrmion-based spintronics. Ina very recent work [ 16], we ﬁnd that 2D magnets with a Janus structure can acquire a very large DMI through the Fert-Levy mechanism as for FM/HM heterostructures, in whichheavy nonmagnetic atoms act as SOC-active sites for elec-tron hopping, yet it is unclear whether a strong DMI can beinduced in 2D materials via other mechanisms. We note that2D magnetoelectric multiferroics with vertical electric polar-ization (hereafter referred to as 2D multiferroics unless statedotherwise) may be excellent candidates for the realization of *Author to whom correspondence should be addressed: hongxin.yang@nimte.ac.cna Rashba effect induced DMI [ 17–19], since their structure inversion symmetry is naturally broken and intrinsic RashbaSOC additionally exists due to the electrostatic potential dif-ference generated by the vertical electric dipole. Comparing tothe Fert-Levy mechanism, a Rashba-type DMI has the advan-tage that it does not necessarily require heavy atoms to providestrong SOC, which is a beneﬁt to achieve low Gilbert dampingfor efﬁcient magnetization switching [ 20]. In addition, with the coexistence of magnetization and electric polarization inthese multiferroics, both the strength and chirality of the DMIand thereby the topological magnetism could be effectivelycontrolled by an electric ﬁeld. In this Rapid Communication, we perform ﬁrst-principles calculations and micromagnetic simulations to demonstratethat 2D multiferroics, such as the recently predicted mono-layers of CrN [ 21], Cu MP 2X6(M=Cr, V; X=S, Se) [ 22], and vacancy-doped CrI 3(Cr 8I23, i.e., 2 ×2C r I 3supercell with one I vacancy) [ 23], naturally host Rashba-type DMI that can facilitate the emergence of topological magnetism. Fur-thermore, we show how the induced DMI and skyrmion canbe controlled by an electric ﬁeld. The subject of a skyrmionand its electric control in multiferroics has been extensivelydiscussed for bulk [ 24,25] and heterostructured [ 26] materials, but still needs to be addressed for 2D multiferroics. Proposal for electric control of the DMI and topological magnetism in 2D multiferroics. We note that the two stable states with opposite spontaneous polarizations Pin a double- well energy landscape can be switched by an out-of-planeelectric ﬁeld [see Fig. 1(a)]. Concurrently, the chirality of their associated DMI is also reversed since those two states are con-nected by a horizontal reﬂection operation. Importantly, thismechanism can be utilized to manipulate topological mag-netic structures such as skyrmions. A skyrmion has degrees of 2469-9950/2020/102(22)/220409(6) 220409-1 ©2020 American Physical SocietyJINGHUA LIANG, QIRUI CUI, AND HONGXIN YANG PHYSICAL REVIEW B 102, 220409(R) (2020) FIG. 1. Concept of the switchable DMI and multiferroic skyrmions. (a) With the electric ﬁeld switching of polarization P for the two stable states in the double-well energy landscape of 2D multiferroics, the chirality (indicated by the magenta arrows) of their associated DMI is also reversed. (b) The four interswitchablemultiferroic skyrmions with different chiralities and polarities, which can be distinguished by the notations A +,A−,C+,a n dC −as well as the corresponding icons of the rotating circular arrow. freedom of chirality that indicates the spin swirling direction and is imposed by the chirality of the DMI, which in turn isdetermined by the direction of P, and polarity that speciﬁes the direction of magnetization min the skyrmion core [ 27], which can be transformed by a magnetic ﬁeld [ 28,29]. With the different combinations of directions of Pandm, we can realize four degenerate interswitchable skyrmionic states withdifferent chiralities and polarities in a single 2D multiferroicsas schematically shown in Fig. 1(b). We have introduced the notations A +,A−,C+, and C −as well as the correspond- ing icons of a rotating circular arrow to distinguish the fourskyrmions, where the characters A and C correspond to anti-clockwise and clockwise chirality and the subscripts “ +” and “−” indicate an upward and downward polarity, respectively. To switch between the four degenerated skyrmions, one canﬁrst apply a magnetic ﬁeld to ﬁx the polarity and then theelectric ﬁeld to control the chirality. The manipulation of inter-switchable multiferroic skyrmions can inspire another conceptin the design of high-density skyrmion-based memory andlogic devices. In the following, we ﬁrst take a CrN monolayeras a representative example to demonstrate the existence ofRashba-type DMI and skyrmions as well as their electriccontrol, and then extend the discussions to other thin ﬁlms. Methodology. The micromagnetic energy functional of a spin spiral for a thin ﬁlm reads [ 30] E[q]=Aq 2+Dq−K 2, (1) where qis the spiral length, Ais the exchange stiffness, D is the DMI parameter, and K=Ku−Msμ2 0/2 is the effective anisotropy corrected by the demagnetization effect with Ku being the magnetocrystalline anisotropy, Msthe saturation magnetization, and μ0the vacuum permeability. Here, we adopt the sign convention that D>0(D<0) implies ananticlockwise (clockwise) DMI, and K>0(K<0) indicates perpendicular (in-plane) anisotropy. It is convenient to intro-duce the dimensionless parameter κ=/parenleftbigg4 π/parenrightbigg2AK D2. (2) Forκ< 1, the magnetic structure of the system exhibits spin spirals as the ground state, whereas for κ> 1, an isolated metastable skyrmion could be formed in the background ofa ferromagnetic ground state [ 31–33]. We have implemented the qSO method [ 34], which treats the SOC within the ﬁrst- order perturbation theory and in a self-consistent way forthe calculation of E[q]i nE q .( 1), in conjunction with the V ASP package [ 35] to determine the magnetic parameters (see the Supplemental Material [ 30] for details and Refs. [ 34–39] therein). Rashba-type DMI in a CrN monolayer. A CrN monolayer with a hexagonal structure [see the top and side views of thestructure in Figs. 2(a) and 2(b)] is a recently predicted 2D metallic multiferroics [ 21]. The relaxed CrN monolayer has a lattice constant aof 3.16 Å and a small buckling distance hof 0.21 Å, which results in an out-of-plane electric polarizationof 6.68 pC /m [indicated by the green arrow in Fig. 2(b)]. Consistent with previous research [ 21,40], our calculations show that the buckled ferroelectric CrN monolayer has a lowerenergy of 9.08 meV /f.u. than the planar paraelectric one, which indicates that spin-phonon coupling is responsible forits vertical ferroelectricity. The structure inversion asymmetry induced by vertical ferroelectric displacement could lead to a DMI between Cratoms. To demonstrate such an effect, we calculate E[q]i n the interval of \|q\|/lessorequalslant0.1 2π a[see Fig. 2(c)]. Clearly, we can see that when SOC is neglected [black points in the upper partof Fig. 2(c)], spin spirals with qand−qare degenerate, and the ground state is the ferromagnetic state ( q=0). Once SOC is included, as expected, E[q] [red points in the upper part of Fig. 2(c)] becomes asymmetric, where the anticlockwise rotating spin spiral [see the inset in Fig. 4(b) for an illus- tration of spin chirality] is more favorable, due to the effectof the DMI. From the zoom-in plot of E[q] around q=0 [see the inset in the upper part of Fig. 2(c)], we can ﬁnd that the lowest E[q] locates at q=− 0.006 2π a, corresponding to a spiral period length of λ=2π \|q\|=52.67 nm. With the cal- culated DMI energy /Delta1EDM[q]=(E[q]−E[−q])/2 [see the lower part of Fig. 2(c)], which shows good linear dependence onq,DandAcan be determined to be 3.74 meV Å /f.u. and 142.89 meV Å2/f.u., respectively. Here, the positive sign of Dindicates that the CrN monolayer prefers an anticlockwise DMI when its polarization Ppoints along the +zdirection. The calculated Kuis 0.22 meV /f.u. and Msis 2.40μB/f.u. Then we can deduce that κis only 1.68, which is near the critical value ( κ=1) for the transition to a spiral state so that an isolated metastable skyrmion can be stabilized in the CrNmonolayer. We will return to this point later. Next, we explore the microscopic origin of DMI in a CrN monolayer. It is helpful to use the atom-resolved /Delta1E DM[q]t o examine the energy source of the DMI [ 34,41], as shown in Fig. 3(a) forq=− 0.0062π a. One can clearly see that /Delta1EDM is dominated by Cr atoms and has a negligible contribution 220409-2ELECTRICALLY SWITCHABLE RASHBA-TYPE … PHYSICAL REVIEW B 102, 220409(R) (2020) FIG. 2. (a) Top and (b) side views of the crystal structure of a CrN monolayer with a lattice constant aand buckling distance h.( c )S p i n spiral energy E[q] (upper panel) and DMI energy /Delta1EDM[q] (lower panel) calculated as a function of spiral vector length q.E[q] is given with respect to the ferromagnetic state at q=0. Black and red points are calculated without (w/o) and with SOC, respectively. from N atoms. This feature is in great contrast to that in FM/HM multilayers [ 41,42] and our previously studied 2D Janus manganese dichalcogenides [ 16], whose DMI energy is contributed mostly by the heavy nonmagnetic atoms, which istypical for the DMI generated via the Fert-Levy mechanism.However, we can identify that it is representative for theRashba-type DMI as in graphene/Co heterostructures [ 43]. To conﬁrm that the DMI is mediated by the conduction electronswith Rashba SOC, we plot the electronic band structures of a CrN monolayer with magnetization pointing along the [1 ¯10] and [ ¯110] directions [see Fig. 3(b)], which show characteristic Rashba-type k-dependent splitting. For a Rashba-type DMI, the DMI parameter can be estimated as D R=(4m∗αR ¯h2)A[17], where ¯ his the reduced Planck constant, m∗is the effective mass of the electron, αRis the Rashba coefﬁcient, and A= 142.89 meV Å2/f.u.is the calculated spin stiffness. Here, we FIG. 3. (a) Atom-resolved DMI energy /Delta1EDM[q]a tq=− 0.0062π a, where the minimum of E[q] locates. (b) Band structures of a CrN monolayer with magnetization with the magnetization direction along [1 ¯10] (black lines) and [ ¯110] (red lines). (c) Zoom-in plot of band structures for states that are around the /Gamma1point and close to the Fermi level. For these states, we can extract that the effective mass m∗=1.94me and the Rashba coefﬁcient αR=2E0/k0=18.0 meV Å with E0being the Rashba splitting at the wave vector k0. 220409-3JINGHUA LIANG, QIRUI CUI, AND HONGXIN YANG PHYSICAL REVIEW B 102, 220409(R) (2020) FIG. 4. The calculated (a) buckling distance h, (b) DMI parameter D, (c) spin stiffness A, and (d) magnetocrystalline anisotropy Kuas the out-of-plane electric ﬁeld εsweeps from 0.50 to −0.50 V /Å (black points), and vice versa (red points). The blue right axes in (b) and (c) indicate the values of magnetic parameters with converted units [ 44]. The inset in (b) presents the preferred chirality (indicated by the magenta arrows) of the spin spiral for an anticlockwise ( D>0) and clockwise ( D<0) DMI. focus on the electronic states that are around the /Gamma1point and close to the Fermi level. From the zoom-in plot in Fig. 3(c), αR=18.0 meV Å and m∗=1.94me, with mebeing the rest mass of the electron that can be extracted. One can thendeduce that D R=2.66 meV Å /f.u., which is only slightly smaller than our previous value ( D=3.74 meV Å /f.u.) cal- culated by the qSO method. Altogether, we can infer that theDMI in a CrN monolayer originates from the Rashba effect. Electric control of magnetic parameters in a CrN mono- layer. Here, we show that both the structural and magnetic properties of a CrN monolayer can be effectively controlledby an out-of-plane electric ﬁeld. Figure 4(a) presents the vari- ation of buckling distance hwhen the electric ﬁeld εsweeps from 0.50 to −0.50 V /Å (black points), and vice versa (red points). From the hysteresis loop of h, we can see that as ε changes from 0.50 to −0.50 V /Å,hﬁrst increases slightly whenε/greaterorequalslant0.40 V/Å, then decreases monotonously from 0.23 to 0.17 Å before the electric polarization switching occurs ataboutε=− 0.40 V/Å. The variation of hreﬂects the electric ﬁeld control of ferroelectric displacement. Figures 4(b)–4(d) summarize the resulting electric-ﬁeld- dependent magnetic parameters [ 44]. Notably, from Fig. 4(b), we can see that the variation trend of Dalmost follows that ofh. Within the range of \|ε\|<0.50 V/Å, the absolute value ofDcan be tuned between 3.98 and 3.19 meV Å /f.u. with the chirality [indicated schematically in the inset of Fig. 4(b) ] determined by the direction of P. Our calculations thus di- rectly validate the previously proposed electric control of boththe strength and chirality of DMI [see Fig. 1(a)]. We also ﬁnd that A[Fig. 4(c)] and K u[Fig. 4(d)] generally show opposite electric ﬁeld dependences, i.e., as εsweeps from 0.40 to −0.40 V /Å,Aalmost decreases linearly from 172.70 to 117 .90 meV Å2/f.u., while Kuis enhanced from 0.19 to0.25 meV /f.u. With the application of electric ﬁeld, Ms changes less than 1% and remains to be about 2 .40μB/f.u. in our calculations. Realization of skyrmions in 2D multiferroics. With all the magnetic parameters calculated, we can study the realizationof skyrmions in a CrN monolayer. As we have mentioned be-fore, the deduced κis only 1.68 when ε=0, which indicates the existence of an isolated metastable skyrmion. To conﬁrmthis point, we perform a micromagnetic simulation [ 30]b y relaxing a trial skyrmion spin conﬁguration on a CrN nan-odisk. After the relaxation, we ﬁnd that a metastable skyrmion(see Fig. 5) with a higher energy of 0.30 meV than the FIG. 5. The relaxed skyrmionic magnetization conﬁguration in a CrN nanodisk with a radius of 30 nm in the absence of electric ﬁeld.The out-of-plane component m zis indicated by the color map. The radius R(deﬁned as the radius of the mz=0 contour) of a relaxed skyrmion is 4.8 nm. 220409-4ELECTRICALLY SWITCHABLE RASHBA-TYPE … PHYSICAL REVIEW B 102, 220409(R) (2020) TABLE I. The calculated lattice constant a, ﬁlm thickness t, saturation magnetization Ms, DMI parameter D, exchange stiffness A, magnetocrystalline anisotropy Ku, and dimensionless parameter κfor 2D multiferroics. The radius ¯Rof a skyrmion estimated by Eq. ( 3) is given for CuVP 2Se6and CuCrP 2Se6. The sign of Dis always determined for the structure with polarization oriented along the +zdirection. Structure a(Å) t(nm) Ms(μB/f.u.) D(meV Å /f.u.) A(meV Å2/f.u.) Ku(meV/f.u.) κ ¯R(nm) CuVP 2S6 6.02 0.33 1.74 −0.07 90.96 −0.001 −310.23 CuVP 2Se6 6.38 0.33 1.75 1.70 44.70 0.15 3.46 1.2 CuCrP 2S6 5.98 0.32 2.76 −0.47 119.64 −0.05 −66.85 CuCrP 2Se6 6.35 0.34 2.77 −2.50 111.96 0.17 4.18 1.6 Cr8I23 14.01 0.32 23.27 10.71 989.41 3.95 49.69 ferromagnetic ground state can be stabilized. The radius R(deﬁned as the radius of mz=0 contour) of a relaxed skyrmion is only 4.8 nm, which is very close to ¯R=4.5n m predicted by the theoretical formula [ 45] ¯R=πD/radicalbigg A 16AK2−π2D2K. (3) For\|ε\|<0.50 V/Å,κvaries between 1.44 and 2.67, and the relaxed radius Rchanges between 1.9 and 6.9 nm. Here, we should remind that along with the electric polarizationswitching, the chirality of the skyrmion is also switched asthat of DMI. The four proposed interswitchable multiferrroicskyrmions with a tunable size [see Fig. 1(b)] can thus be realized in CrN monolayers. To generalize the proposals for Rashba-type DMI and topo- logical magnetism, we have considered other predicted 2Dmultiferroics including Cu MP 2X6(M=Cr, V; X=S,Se) [22], and vacancy-doped CrI 3(Cr 8I23)[23]. Table Isum- marizes their calculated structural and magnetic parameters,which are also electrically switchable as for a CrN monolayer.From the calculation results, we can ﬁnd that they all have aﬁnite DMI, and the sizable Din CuVP 2Se6and CuCrP 2Se6 can even lead to a relative small κof∼4.0, which is simi- lar to those in Pd/Co/Pt ( κ=5.4) [33] and Fe/W(110) ( κ= 4.8) [46] that are anticipated to host chiral magnetic solitons [33,47]. Considering that the predicted radius ¯Rof skyrmions in CuVP 2Se6and CuCrP 2Se6is only about twice their lattice constant a, the classical description of E[q]i nE q .( 1)m a y fail, and quantum effects should be taken into account [ 48], which deserves further study in the future.Conclusion. We have performed ﬁrst-principles calcu- lations and micromagnetic simulations to demonstrate theexistence of Rashba-type DMI and skyrmions in 2D mul-tiferroics. Due to the intrinsic coupling of magnetism andelectric polarization in 2D multiferroics, both the strength andchirality of the DMI, and thereby the size and chirality ofthe skyrmion, can be effectively controlled by electric ﬁeld.A signiﬁcant Rashba-type DMI may not only promote thestabilization of topological magnetism as already describedin our work, but also enable the efﬁcient current-inducedmotion of domain walls [ 49,50] as well as skyrmions [ 51] by spin-orbit torque (SOT), which is particularly importantin practical applications. Although we have focused on 2Dmultiferroics with vertical polarization, our proposal for theelectric control of DMI and topological magnetism can beeasily extended to those with in-plane polarization [ 52–54], in which the DMI can be generated through other mecha-nisms rather than the Rashba effect. Recently, Xu et al. [55] discussed the manipulation of bimerons in VOI 2monolayers with in-plane polarization. In summary, our work can openup a different route to the generation and electric control oftopological magnetism in emerging 2D spintronics. Acknowledgments. The authors thank Dr. Erjia Guo, Dr. Youguo Shi, Prof. Bin Xiang, Prof. Chunlei Gao, Prof. Mair-bek Chshiev and Prof. Albert Fert for the fruitful discussionsand collaborations on related topics. 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PhysRevLett.113.227601.pdf	Non-Gilbert-damping Mechanism in a Ferromagnetic Heusler Compound Probed by Nonlinear Spin Dynamics P. Pirro,1,T. Sebastian,1,†T. Brächer,1,2A. A. Serga,1T. Kubota,3H. Naganuma,4 M. Oogane,4Y. Ando,4and B. Hillebrands1 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany 2Graduate School Materials Science in Mainz, Gottlieb-Daimler-Straße 47, Germany 3Institute for Materials Research, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai 980-8577, Japan 4Department of Applied Physics, Graduate School of Engineering, Tohoku University, Aoba-yama 6-6-05, Sendai 980-8579, Japan (Received 21 April 2014; published 26 November 2014) The nonlinear decay of propagating spin waves in the low-Gilbert-damping Heusler film Co2Mn 0.6Fe0.4Si is reported. Here, two initial magnons with frequency f0scatter into two secondary magnons with frequencies f1andf2. The most remarkable observation is that f1stays fixed if f0is changed. This indicates, that the f1magnon mode has the lowest instability threshold, which, however, cannot be understood if only Gilbert damping is present. We show that the observed behavior is caused by interaction of the magnon modes f1andf2with the thermal magnon bath. This evidences a significant contribution of the intrinsic magnon-magnon scattering mechanisms to the magnetic damping in high- quality Heusler compounds. DOI: 10.1103/PhysRevLett.113.227601 PACS numbers: 76.50.+g, 75.30.Ds, 78.35.+c, 85.75.-d The field of spintronics has attracted huge interest in recent years due to a multitude of physical phenomena and applications related to the spin degree of freedom [1,2].T o develop this area further, intensive research efforts aredevoted to design new materials with outstanding proper-ties, like, e.g., high spin-polarization. The class of ferro-magnetic Heusler materials [3]is of special interest since these materials potentially combine high spin polarization and Curie temperature with low magnetic Gilbert-damping [4–6]. Thus, these materials allow for a long-distance spin transport by magnons —the quanta of spin waves [7,8]. However, alongside with the decrease of the Gilbert-typerelaxation, which results, in this case, from the reducedinteraction of the magnetic excitations with the electronbath, other damping mechanisms, caused by, e.g., nonlinearmagnon-magnon scattering, can play an important role[9–13]. Thus, the development of new ferromagnetic Heusler materials calls for a comprehensive study of theoccurring nonlinear spin-wave phenomena since these will dominatetheenergyredistributioninmagnetizationdynamics. In this Letter, we investigate the spin-wave damping mechanisms in the Heusler compound Co 2Mn 0.6Fe0.4Si (CMFS) [4,5,7,8] by the experimental observation and theoretical analysis of the second order spin-wave insta-bility [12,14 –18]in a microscaled magnonic waveguide. In this four-magnon scattering process, two initial magnonsat frequency f 0scatter into two magnons with frequencies f1andf2, named the unstable modes. We observe that the mode f1is a dominant unstable mode whose frequency is independent of the frequency f0 of the initial spin waves. Such a dominant mode has notbeen reported previously for metallic ferromagnetic films. We show that it is a specific feature of the low-Gilbert- damping Heusler film caused by the influence of a non- Gilbert-damping mechanism, which significantly affectsthe total relaxation rate ηof the individual spin-wave modes. It can be identified as an intrinsic relaxation caused by the interaction with the thermal magnon bath. The sample is sketched in Fig. 1(a):a5-μm-wide CMFS waveguide [30 nm thick, 70μm long, capped with Ta (5 nm)] grown on a MgO substrate with a 40 nm Cr buffer layer was patterned by electron-beam lithography and argon-ion milling. Then, a 1-μm-wide antenna (Ti-Cu) was produced on top of the CMFS waveguide by electron- beam evaporation. A microwave current with variable frequency f 0is passed through the antenna to create dynamic Oersted fields which excite coherent propagating spin waves at the same frequency. kperkparHext MW frequency (GHz) f06 8 10 12 1410-310-210-1100 ISLB)stinu .bra((b)(a) antennahdyn BLS-laser FIG. 1 (color online). (a) Schematic view of the sample: spin waves are excited in a CMFS waveguide by means of a microstripantenna. A bias field H extis applied perpendicular to the waveguide ’s long axis. (b) Brillouin light-scattering intensity IBLSin the linear regime as a function of the microwave excitation frequency f0.PRL 113, 227601 (2014) PHYSICAL REVIEW LETTERSweek ending 28 NOVEMBER 2014 0031-9007 =14=113(22) =227601(5) 227601-1 © 2014 American Physical SocietyA static external magnetic field μ0Hext¼48mT is applied perpendicular to the waveguide ’s long axis (see Fig. 1). Hence, an excitation of spin waves propagating perpendicularly to Hextis realized. The spin waves are detected at a distance of about 2μm from the antenna in the middle of the waveguide using Brillouin light-scatteringmicroscopy (BLS) [19]. Figure 1(b) shows the BLS intensity I BLS, which is proportional to the spin-wave intensity in the probing point, as a function of the micro-wave frequency f 0for a power of 0 dBm. One can see an excitation of spin-wave modes in a frequency interval ofseveral gigahertz [20,21] . Let us now turn to the frequency composition of the individual spin-wave spectra. Figure 2(a) exemplarily shows BLS spectra for a fixed microwave frequencyf 0¼10.5GHz. For a power of P¼5dBm (black squares), only the directly excited, initial spin-wave mode with frequency f0is visible. At P¼9dBm (pink circles), the amplitude of the initial spin waves exceeds theinstability threshold and the spectrum shows two additionalpeaks at f 1andf2, corresponding to the unstable spin-wave modes. The threshold character of this process is demon-strated in Fig. 2(b): if the threshold power is exceeded, I BLSatf1andf2abruptly increases. Simultaneously, the intensity increase at the excitation frequency fMWdrops below the linear increase (red dotted line) observedfor small powers. For higher powers [green diamonds inFig.2(a)], a continuum of spin-wave frequencies between f 1andf2is found, whereas the intensities at f1andf2 saturate. This supercritical regime, where many modes are populated, has already been observed in [13]. However, the further discussion will concentrate on the unstablemodes f 1andf2with the lowest instability threshold, since they provide information about intrinsic properties of thesystem.Figure 2(c)shows the evolution of the frequencies f 1and f2as a function of the frequency f0of the initial spin waves. For each f0, the power is increased until the first satellites of the four-magnon scattering appear in the BLS spectrum; hence, f1andf2are always recorded close to the threshold power. For f0>11.25GHz, the instability threshold amplitude of the initial spin waves is not reachedanymore due to the decreased excitation efficiency [13,20,21] , which is also visible in Fig. 1(b). Remarkably, f 1is practically independent of f0. This indicates that the mode f1features a reduced relaxation rate or an enhanced coupling for the four-magnon interactioncompared to competing modes. Concerning f 2, a linear increase with f0is observed, which conserves the total energy. The energy conservation ( 2f0¼f1þf2) is dem- onstrated by comparing the energy of the initial magnons (red line) with the mean energy of the pair of the unstable magnons (triangles). From the frequencies of the unstable modes, important conclusions about the relaxation mechanisms in CMFS canbe drawn. In order to do this, the theoretical description ofspin-wave instabilities in thin films developed in Ref. [17] is used in the following to analyze the influence of the damping and the four-magnon coupling on the unstablemode frequencies. This theory calculates the critical amplitude c critof the initial wave at frequency f0as a function of the different, potentially unstable modes. For asecond order instability, this critical amplitude is [16,17] : jc critj¼min ðk1;k2Þ/C18ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ηeff jeWk0k0;k1k2js /C19 ; ð1Þ where the minimization is carried out over all spin-wave modes fulfilling energy and momentum conservation. FromEq.(1), it can be seen that the unstable modes have the Microwave power (dBm) P(b) 1010 -10 -5 0 5 10 1512 IfBLS 1() 103104 IfBLS 2()IfBLS 0())stinu.bra( ytisnetni SLBISLB Microwave frequency (GHz) f0)zHG( ycneuqerf SLBf 6 7 8 9 1 01 11 21 31 41 5101102103104 +5 dBm +9 dBm +13 dBmMW Power P BLS frequency (GHz) f)stinu.bra( ytisnetni SLBISLB(a) f2 f1f0 89 1 0 1 1789101112131415 ( )/2f+f12f2 f1(c) only Gilbert damping incl. non-Gilbert dampingcalc. f , f :12 FIG. 2 (color online). (a) BLS spectra for f0¼10.5GHz at different microwave powers. The spectrum at the threshold power (P¼9dBm) shows the two unstable mode frequencies f1andf2(shaded). The different BLS intensities at f1andf2are caused by different detection efficiencies. For supercritical powers ( P> 9dBm), the intensities at f1andf2do not increase further, instead the spectrum becomes continuously broadened. (b) BLS intensity IBLSfor the modes f1,f2, andf0as a function of the microwave power P. Above the threshold power, IBLSof the unstable modes at f1,f2rises sharply. (c) Unstable mode frequencies f1andf2(circles and squares, respectively) as a function of the microwave frequency f0.f1is independent of f0and constant at f1≈7.1GHz, while f2 increases linearly with f0. The blue and green lines indicate the calculated values for f1andf2according to Eq. (1)for pure Gilbert- damping (dashed lines) and if an additional non-Gilbert-damping term (Eq. (3)is included (solid lines).PRL 113, 227601 (2014) PHYSICAL REVIEW LETTERSweek ending 28 NOVEMBER 2014 227601-2lowest ratio of their effective relaxation frequency ηeff over their individual four-magnon coupling strength jeWk0k0;k1k2j. To evaluate Eq. (1), the resonant modes need to be determined. Energy conservation can be accounted for by choosing the frequencies of the potentially unstable modes f0 1,f0 2to fulfill 2f0¼f0 1þf0 2. Subsequently, momentum conservation can be analyzed by calculating the isofre- quency curves fk0 1gand fk0 2g, which include all wave vectors with fðk0 1Þ¼f0 1,i¼1, 2. Hereby, only the in- plane components of the wave vector should be considered, since for the used film thickness, the modes with finite out-of-plane wave vector have frequencies above 20 GHz due to the exchange interaction. Thus, they can be neglected in the investigated frequency range [ fðkÞcalculated using [22], parameters see [23]]. From the isofrequency curves, all wave vector pairs ( k 0 1,k0 2) which fulfill the wave vector conservation 2k0¼k0 1þk0 2are selected [24]. In Fig. 3, this approach is demonstrated exemplarily for the exper- imental data of Fig. 2(a). The wave vector k0of the initial spin waves is calculated assuming that the antenna excitespredominantly the first waveguide mode [20,21] . For the potentially unstable modes, the quantization of the wave vector due to the finite waveguide width can be neglectedin this geometry since the different quantized modes have only small frequency differences compared to the separa- tion between the observed mode frequencies f 1,f0, andf2. With the resonant modes being identified, let us evaluate whether the coupling ~Wprovides an explanation for the observed frequencies f1andf2. Besides the direct four- magnon coupling W, also the term Tassociated with nonresonant three-magnon processes has to be included[10,17] : ~W k0k0;k1k2¼Wk0k0;k1k2þTk0k0;k1k2: ð2Þ The term Tdescribes two subsequent three-magnon scattering processes which involve a common virtual magnon state. Since three-magnon processes are purely dipolar, Tis proportional to the saturation magnetizationMs[17] which leads to an important contribution for CMFS ( Ms≈1000 kA=m). Figure 4exemplarily shows the maximum coupling strength j~Wjmaxas a function of the lower unstable mode frequency f0 1(f0 2¼2f0−f0 1) for the experimental param- eters of Fig. 2(a). For comparison, the maximum of jWj calculated without taking into account the three-magnoncorrection Tis shown. From Fig. 4, two conclusions can be deduced. Firstly, the three-magnon correction Tenhances the coupling for frequencies f 0 1significantly lower than f0 (in the range ≈6.7–9.5GHz) compared to the bare four- magnon coupling jWj. Secondly, even with the incorpo- ration of T, the complete coupling j~Wjexhibits no maximum for the observed unstable mode frequency f1. The calculations show a similar behavior for all f0in the experimentally probed range. Nevertheless, a reasonablevariation of the effective relaxation rate η effmight explain the experimental observations. Therefore, the effective relaxation frequency ηeffof the potentially unstable spin waves should be considered in detail. A magnon can relax via a large number of differentprocesses, which involve the interaction with electrons,phonons, and other magnons [15,16,25] . Let us first analyze the part of these damping processes which can be described as “viscous ”processes and which are taken into account by the Gilbert-damping constant αin the Landau-Lifshitz-Gilbert equation [16]. For CMFS, this Gilbert-damping contribution is mainly due to the elec-tron-magnon interaction [6]and was measured to be α CMFS≈0.003 [4,5]. First, we have calculated the Gilbert-relaxation rate ηGilb¼ηGilbðf;αÞusing the approach presented in Ref. [26]. Next, the resulting thresh- old cGilb crit given by Eq. (1) with ηeffðk0 1;k0 2Þ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ηðk0 1Þηðk0 2Þp [16]was derived for each potentially unsta- ble magnon pair. Figure 5shows this normalized threshold jcGilb critjas a function of f0 1(dark blue curve). In contradiction to the experimental results, the minimal threshold, in this case, is predicted for almost degenerated modes with-10 -5 0 5 10-10-50510 f0f2 f1 kpar(rad/µm)kper) mµ/dar(f2=13.9 GHz f0=10.5 GHz =7.1 GHzf1 FIG. 3 (color online). Isofrequency curves for the experimental conditions of Fig. 2(a). Three possible four-magnon scattering processes are schematically illustrated.789 1 00.20.40.60.81.0 f0 Potentially unstable mode frequency ' (GHz) f1f1)stinu.bra( dezila mroNW FIG. 4 (color online). Normalized maximum four-magnon coupling strength ~Was a function of the frequency f0 1of the lower potentially unstable mode [parameters, see Fig. 2(a)]. The dashed line represents the maximum bare four-magnon couplingWwithout the incorporation of the three-magnon correction T.PRL 113, 227601 (2014) PHYSICAL REVIEW LETTERSweek ending 28 NOVEMBER 2014 227601-3f1≈f2≈f0. The dashed blue and green lines in Fig. 2(c) show these predicted frequencies as a function of f0. They demonstrate that the pronounced mismatch between experi-ment and this theoretical prediction is present in the whole investigated range. Since the curve for the normalized jc Gilb critjand, therefore, the predicted frequencies f1andf2, are independent of the absolute value of α, the experimental observations can neither be explained if a different Gilbert- damping strength αis assumed. Thus, additional, non- Gilbert-damping mechanisms have to play a significant role for the unstable modes in CMFS. In this context, the intrinsic interaction with the thermal magnon bath has to be considered since it is a nonviscous damping mechanism. It is known to be important in high quality, low-Gilbert-damping materials [9,27 –30].F o r example, the relaxation of a magnon mode with wave vector kcan occur due to the confluence with a thermal magnon. The relaxation rate ηmagdue to this magnon- magnon scattering processes depends on the nonlinear coupling in the magnon system, the thermal magnon population, and the dispersion relation. In a first approxi- mation, ηmagis usually described as [27–29]: ηMagðkÞ¼A fðkÞjkj: ð3Þ Here, fðkÞis the frequency of the spin wave and Ais a parameter which depends on temperature, saturation mag- netization, exchange constant, geometry, and applied field [16,25] . The damping mechanism described by Eq. (3) does not influence the ferromagnetic resonance measure- ments [4,5] widely used to extract the Gilbert-damping. A precise theoretical calculation of Afor the CMFS thin film is rather involved and beyond the scope of this Letter. However, based on the general theory for nonlinear spin waves [17] and numerical evaluation of the magnon- magnon relaxation for thin films [9], we can conclude that the latter can be of the same order of magnitude as the Gilbert-damping in CMFS.Since the theoretical value of Ais unknown, the normalized threshold jcGilbþmag crit jwith ηðkÞ¼ηGilbðkÞþ ηmagðkÞis shown in Fig. 5for several values of Aas a function of f0 1. The relaxation rate ηmagfor these A is comparable with the Gilbert-relaxation rate for the considered wave vector range. It is evident that for A≥Amin≈10−7m GHz2=rad, the threshold is minimal for the frequency f1≈7.1GHz observed in the experi- ment. Thus, the inclusion of a damping term in the formof Eq. (3)explains the position of the unstable modes in Fig.2(a). In Fig. 2(c), the same analysis is applied to all values of f 0: the solid blue and green lines show the calculated values forf1andf2as a function of f0if a constant value of A¼3×10−7m GHz2=rad is assumed. The explicit jkj dependence of ηmagfixes the calculated f1close to 7.1 GHz, in excellent agreement with the experiment.Thus, the fact that a dominant unstable mode exists in the system can be confidently attributed to the influence ofthe non-Gilbert-damping process discussed above. Additional measurements at different external fields show that the dominant unstable mode frequency f 1 depends on the dispersion relation: f1increases for higher magnetic fields. Also, in this case, the positions of f1and f2can always be predicted by including ηmagwith the mentioned value of Ainto the determination of the threshold. At the same time, if the ratio of ηmag=ηGilbis small, the Gilbert-damping dominates and modes close to f0are predicted to get unstable first (compare Fig. 5). Indeed, such a behavior was observed in NiFe [13], where the Gilbert damping is significantly larger than in CMFS. To conclude, by using four-magnon instabilities as a probing tool, we have demonstrated that non-Gilbert-damping plays an important role for the spindynamics in the Heusler compound CMFS. By includingan explicitly wave vector dependent relaxation contribu-tion, all characteristics of the observed instabilities havebeen explained using one constant parameter Afor the entire investigated range. This non-Gilbert-relaxation mechanism can be attributed to the interaction of the considered magnons with the thermal magnon bath.Since this intrinsic damping cannot be reduced withoutchanging elementary material parameters like the saturationmagnetization or the exchange constant, it constitutes alower limit for the total damping of spin waves in general.Thus, for a further development of low damping ferro-magnetic compounds, magnon-magnon interactions can beanticipated to play a decisive role. We thank G. A. Melkov for fruitful discussions and our colleagues from the Nano Structuring Center of the TU Kaiserslautern for their assistance in sample preparation.We gratefully acknowledge financial support by the DFGResearch Unit 1464 and the Strategic Japanese-German789 1 00.60.70.80.91.0 Potential unstable mode frequency ' (GHz) f1)stinu.bra( dlohserht. mronAA=min A= 5Amin AA=0.5min f1 f0Gilbert damping only\|c \|Gilb crit\|c \|Gilb+Mag crit FIG. 5 (color online). Individually normalized critical threshold amplitudes ccritas a function of f0 1for pure Gilbert-damping (blue curve) and for different values of the parameter A. For A>A min≈10−7m GHz2=rad, the threshold is minimal for the experimentally observed unstable mode frequency f1≈7.1GHz.PRL 113, 227601 (2014) PHYSICAL REVIEW LETTERSweek ending 28 NOVEMBER 2014 227601-4Joint Research from JST: ASPIMATT. Thomas Brächer received support from a fellowship of the Graduate SchoolMaterials Science in Mainz (MAINZ) through DFG- funding of the Excellence Initiative (GSC 266). ppirro@physik.uni ‑kl.de †Present address: Institut für Ionenstrahlphysik und Materi- alforschung, Helmholtz-Zentrum Dresden-Rossendorf, D-01328 Dresden, Germany [1] N. Locatelli, V . Cros, and J. Grollier, Nat. Mater. 13,1 1( 2 0 1 4 ) . [2] S. A.Wolf,D. D.Awschalom,R. A.Buhrman,J. M.Daughton, S. von Molnár, M. L. Roukes, A. Y . Chtchelkanova, andD. M. Treger, Science 294,1 4 8 8( 2 0 0 1 ) . [3] T. Graf, J. Winterlik, L. Müchler, G. H. Fecher, C. Felser, and S. S. P. Parkin, in Magnetic Heusler Compounds in Handbook of Magnetic Materials , edited by K. H. J. Buschow (North Holland, Amsterdam, 2013). [4] M. Oogane, T. Kubota, Y. Kota, S. Mizukami, H. Naganuma, A. Sakuma, and Y. Ando, Appl. Phys. Lett. 96, 252501 (2010) . [5] T. Kubota, S. Tsunegi, M. Oogane, S. Mizukami, T. Miyazaki, H. Naganuma, and Y. Ando, Appl. 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[24] Because of the finite size of the microstructure, the wave- vector conservation has to be fulfilled only within auncertainty of about 3rad=μm. [25] C. E. Patton, Magnetic Oxides (John Wiley, London, 1975). [26] D. D. Stancil and A. Prabhakar, Spin Waves: Theory and Applications (Spinger, New York, 2009). [27] M. Sparks, Ferromagnetic Relaxation Theory (Mc Gray-Hill, New York, 1964). [28] E. Schlömann, Phys. Rev. 121, 1312 (1961) . [29] T. Kasuya and R. C. LeCraw, Phys. Rev. Lett. 6, 223 (1961) . [30] C. T. Boone, J. A. Katine, J. R. Childress, V. Tiberkevich, A. Slavin, J. Zhu, X. Cheng, and I. N. Krivorotov, Phys. Rev. Lett. 103, 167601 (2009) .PRL 113, 227601 (2014) PHYSICAL REVIEW LETTERSweek ending 28 NOVEMBER 2014 227601-5
PhysRevB.76.054409.pdf	Physics of complex transverse susceptibility of magnetic particulate systems Dorin Cimpoesu * Advanced Materials Research Institute (AMRI), University of New Orleans, New Orleans, Louisiana 70148, USA Alexandru Stancu Faculty of Physics, “Al. I. Cuza” University, Iasi 700506, Romania Leonard Spinu† AMRI, University of New Orleans, New Orleans, Louisiana 70148, USA and Department of Physics, University of New Orleans, New Orleans, Louisiana 70148, USA /H20849Received 8 March 2007; published 6 August 2007 /H20850 Complex transverse susceptibility is a recent proposed method for the determination of anisotropy and volume distributions in particulate magnetic media. So far, only thermal ﬂuctuations and rate-dependentdamped dynamics of the magnetic moment have been identiﬁed as reasons for the existence of the imaginarytransverse susceptibility. In this paper, we apply a more general approach to derive the complex transversesusceptibility, and we show that the hysteresis phenomenon is the most general concept behind the existence ofcomplex transverse susceptibility. In this paper, the physical origins of the imaginary part of transverse sus-ceptibility are analyzed: rate-independent hysteresis, viscous-type rate-dependent hysteresis, and thermal re-laxation effect origin. The rate-independent origin is an intrinsic contribution to the imaginary transversesusceptibility and cannot be neglected because it is a zero-temperature effect. DOI: 10.1103/PhysRevB.76.054409 PACS number /H20849s/H20850: 75.30.Gw, 75.40.Mg, 75.50.Ss, 75.50.Tt I. INTRODUCTION One of the important topics of condensed matter physics is the magnetism of nanostructured materials as ordered ar-rays of magnetic nanoparticles and nanowires, nanolitho-graphically patterned structures, and multilayers. To revealthe properties of such systems, especially magnetic aniso- tropy, one needs appropriate experimental techniques. Transverse susceptibility /H20849TS/H20850is one of the most sensitive methods in determining the magnetic anisotropy, and it wasused with great success to characterize close-packed arraysof monodisperse Fe nanoparticles, 1Fe/H20849001 /H20850square nanostructures,2exchange-biased IrMn/FeCo multilayers,3 epitaxial CrO 2and CrO 2/Cr 2O3bilayer thin ﬁlms,4,5and per- pendicular recording media. Also, there is a great interest indescribing theoretically the transverse susceptibility in mag-netic nanosized systems. 6 However, after almost 100 years since Gans7published his paper where the concept of transverse susceptibility wasﬁrst discussed and in spite of many efforts dedicated to itsstudy, the phenomenon of transverse susceptibility in mag-netic systems is still not fully understood. The starting pointin the way for experimental use of TS was the theoreticalpaper of Aharoni et al. 8where the differential reversible sus- ceptibility tensor of a magnetic system subject to coherentrotation magnetization processes was ﬁrst calculated. The di-agonalized susceptibility tensor includes the longitudinal andthe two transverse components of the susceptibility with re-spect to the applied magnetic ﬁeld. Thirty more years passeduntil Pareti and Turilli 9experimentally veriﬁed the theory of Aharoni et al.8which predicted that the plot of TS versus applied ﬁeld presents characteristic peaks located at the an-isotropy and switching ﬁelds. Subsequently, as TS was morefrequently used as a method for magnetic anisotropy inves-tigations, the interest for TS grew exponentially and manystudies contributed to the improvement of both theory and experimental method. Thus, some of the restrictions in theﬁrst theoretical approach of Aharoni et al. 8were overcome10–15and the increased sensitivity of new experi- mental TS methods allowed the investigations of magneticsystems with very low concentration of magneticmoments. 3,4,16,17A very recent development of TS constitutes the complex TS. It was ﬁrst mentioned by Papusoi18in the context of magnetic particulate systems and it spurred a lotof attention as it was presented as a new way to determinethe particle volume and anisotropy ﬁeld distributions of aparticulate medium. 19–23The complex TS as calculated in Ref. 18is a consequence of thermal relaxation of system’s magnetic particles, whose relaxation rate can be controlledby the externally applied dc ﬁeld and not by temperature asin the case of ac susceptibility. 24 In this work, we apply a more general approach to derive the complex TS. We show that the imaginary part of TS canhave three main origins which include those of Refs. 14and 18. The starting point of our approach is the observation that the existence of an out-of-phase component of TS is essen- tially associated with a lag of the output /H20849magnetization /H20850with respect to the input /H20849applied magnetic ﬁeld /H20850. Etymologically, this lag between the effect and its cause corresponds to theword “hysteresis,” and as it is shown in the following, thehysteresis phenomenon is the more general concept behindthe existence of complex TS. In studying the hysteresis inmagnetism, Bertotti 25showed that the hysteresis can be un- derstood using different ideas as lag, dissipation, memoryand branching, or metastability. As deﬁned in Ref. 25, the hysteresis is “the whole set of intimately connected phenom-ena arising from the simultaneous existence of metastablestates, dissipation, mechanisms with characteristic timescales, and thermal relaxation.” Analyzing the correspon-dence between metastability and hysteresis, it can be easilyPHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 1098-0121/2007/76 /H208495/H20850/054409 /H2084915/H20850 ©2007 The American Physical Society 054409-1observed that the metastability summarizes all the other in- terpretations of hysteresis. Various types of hysteresis canappear in a system with a complicated free-energy landscapefeaturing many local minima corresponding to metastablestates. The main situations that can occur are rate-independent hysteresis, viscous-type rate-dependent hyster-esis, and thermal relaxation. In the case of TS, so far, onlythe last two approximations were used to explain the occur-rence of an imaginary part, i.e., viscous-type rate-dependenthysteresis in Ref. 14and thermal relaxation in Ref. 18. These two situations will be revised in this paper in Secs. IV and V ,where certain clariﬁcations and corrections will be made.Before this, in Sec. III, the rate-independent hysteresis originof the complex TS will be presented and analyzed. This is anintrinsic contribution to the complex TS being a zero-temperature approximation. II. MAGNETIC SUSCEPTIBILITY TENSOR The magnetic susceptibility is a quantity which describes the capability of a magnetic material to magnetize in re-sponse to a magnetic ﬁeld. The magnetic susceptibility /H9273is deﬁned by the ratio between the induced magnetization of amagnetic sample and the inducing magnetic ﬁeld, describingthe material’s response to an applied magnetic ﬁeld. Takinginto account that both magnetization and magnetic ﬁeld arevectors, the magnetic susceptibility of a magnetic sample isnot a scalar. Response is dependent on the state of the sampleand can occur in directions other than that of the appliedﬁeld. To accommodate this, a general deﬁnition using a sec-ond rank tensor derived from derivatives of components ofmagnetization Mwith respect to components of the applied ﬁeld H: /H9273ij=/H11509Mi /H11509Hj, called the differential susceptibility, describes ferromagnetic materials, where iandjrefer to the directions /H20849e.g., x,y, and zin Cartesian coordinates /H20850of the magnetization and applied ﬁeld, respectively. The tensor thus describes the response ofthe magnetization in the ith direction from an incremental change in the jth direction of the applied ﬁeld. Generally, the susceptibility tensor should comprise nine components, but,in fact, a maximum of three can be independent, because byappropriate choice of the orientation of the body-ﬁxed coor-dinate system, the susceptibility tensor can be reduced to adiagonal form. The general tensorial character of the suscep-tibility is, in fact, related to the anisotropy of magnetic prop-erties. The diagonalization of the susceptibility tensor takesplace when the privileged directions of magnetization be-come parallel with the coordinate system’s axis. If thechange of the applied ﬁeld provides only reversible changeof magnetization /H20849i.e., processes which involve no loss of energy /H20850, we have a reversible susceptibility. In a differential susceptibility experiment, it is required to apply two magnetic ﬁelds: a dc bias ﬁeld H dc, which can be varied in a quite large range, and a small perturbing ac ﬁeldH ac, and the magnetization’s variation is measured. The sus- ceptibility is usually measured as the amplitude ratio of theﬁrst harmonic of the induced magnetization along a given direction and that of the ac ﬁeld. If the Ozaxis is in the direction of the biasing dc ﬁeld, then the diagonal elements of the tensor are the parallel sus-ceptibility /H9273zzmeasured in the ﬁeld direction and the two transverse susceptibilities /H9273xxand/H9273yymeasured perpendicu- lar to the ﬁeld direction. III. TS IMAGINARY PART: RATE-INDEPENDENT HYSTERESIS EFFECT Rate-independent hysteresis is a zero-temperature ap- proximation where the system indeﬁnitely remains in the lo-cal free-energy minimum which is initially occupied by thesystem. In this approximation, let us consider that the dc ﬁeld H dc acts along the Ozaxis of a rectangular coordinates system and the ac ﬁeld Hac=Hac,max sin/H9275tacts along the Oxaxis /H20849see Fig. 1/H20850. Let us consider a spherical particle whose mag- netization Mreverses by coherent rotation, /H20841M/H20841=Msby as- sumption and Msis the saturation magnetization, having the volume Vand the uniaxial anisotropy of constant K/H110220, with the easy axis in the xOz plane making the angle /H9258kwith the Ozaxis. For low values of the ac ﬁeld frequency, it may be assumed that the processes leading to a certain value of M have characteristic relaxation times much shorter than thetime scale over which Mvaries signiﬁcantly and the system approaches equilibrium; that is, the magnetization Mlies in a minimum of the free energy at any moment, as in the theoryof Aharoni et al. 8Because of symmetry reasons, Mwill lie in the xOz plane. If the orientation of the magnetization is deﬁned by spherical coordinate /H9258, then the particle’s free energy is E=−2 KV/H20875hdccos/H9258+hacsin/H9258+1 2cos /H20849/H9258−/H9258k/H208502/H20876, /H208491/H20850 where hdc=Hdc/Hk,hac=Hac/Hk, and Hk=2K//H92620Msbeing the particle’s anisotropy ﬁeld. In the following, we will referonly to the behavior of the TS near the positive saturation,that is, /H9258/H33528/H20849−/H9266/2,/H9266/2/H20850. The dependence of the particle’s free energy on the magnetization’s orientation is presented in Fig. 2for two orientations of the easy axis. If the particle’s easy axis is perpendicular to the direction of the dc ﬁeld /H20849/H9258k=90° /H20850, then in the absence of the ac ﬁeld, the free energy /H208491/H20850shows one minimum for hdc/H333561, located at/H9258=0, and two minima for −1 /H11021hdc/H110211, located at /H925810FIG. 1. /H20849Color online /H20850Transverse susceptibility experiment for a uniaxial single-domain ferromagnetic particle.CIMPOESU, STANCU, AND SPINU PHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 054409-2=cos−1hdc, referred hereafter as minimum 1, and /H925820 =−cos−1hdc, referred hereafter as minimum 2 /H20849see Fig. 2/H20850.I n the classical theory in which the amplitude of the ac ﬁeldtends to zero, the magnetic moment oscillates reversiblyaround one of the two minima. If h dc=1, the ﬁrst nonvanish- ing derivative at the minimum point is of the order 4, else isof the order 2. So, for h dc=1, the minimum is the broadest, and any arbitrary ac ﬁeld, however small, will cause an os-cillation of the magnetic moment and consequently thetheory of Aharoni et al. 8predicts a singular point of the TS at hdc=1. However, in any experiment, the amplitude of the ac ﬁeld is not zero in the mathematical sense and no singularityoccurs at h dc=1, and what is important to observe is that the peak of the TS occurs at a ﬁeld hdc/H110211 because the magnetic moment oscillates around the Ozaxis between the minimum corresponding to hac=hac,max and, respectively, hac=−hac,max and not only around one of the two minima corresponding to hac=0 /H20849see Fig. 2/H20850. For a particle with /H9258k=89.9°, the energy landscape is not symmetric with respect to the Ozaxis, as in the previous case. It can be seen in the Fig. 2that for hac,max =10−3, the magnetic moment oscillates only on the side /H9258/H110220. A maxi- mum of the TS occurs for a ﬁeld hdc/H110221, but the “distance” between the two minima is considerably smaller than in thecase /H9258k=90° so that the amplitude of the TS signal is much smaller. Consequently, for an ensemble of particles with adistribution of the orientation, the most important contribu-tion to the TS comes from the particles with the easy axisoriented exactly perpendicular to the dc ﬁeld, for which theenergy plot is symmetric. In this sense, the afﬁrmation that“the ﬁeld dependence of TS presents a characteristic peaklocated at the anisotropy ﬁeld” must be understood, and notthat for any particle’s orientations the TS presents a peak ath dc=1. Increasing the amplitude of the ac ﬁeld, the magnetic moment will have sufﬁcient energy to oscillate on both sidesof the Ozaxis and the maximum of the TS occur at a ﬁeld h dc/H110211.If dE d/H9258=0 , d2E d/H92582/H110220, /H208492/H20850 then the magnetic moment lies in a minimum of the free energy. The ﬁrst condition can be written as a polynomialequation of fourth order in tan /H20849 /H9258/2/H20850which has explicit solu- tions, but it is rather difﬁcult to choose the solutions which satisfy the second condition. To simplify the discussion, weshall use that m ac=sin/H9258is the normalized induced signal along the ac ﬁeld direction, and then Eq. /H208492/H20850can be written as hac=f/H20849mac,/H9258k/H20850, /H11509f /H11509mac/H110220, /H208493/H20850 where f/H20849mac,/H9258k/H20850=hdcmac /H208811−mac2+maccos 2/H9258k−1−2 mac2 2/H208811−mac2sin 2/H9258k. /H208494/H20850 The behavior of f/H20849mac,/H9258k/H20850as a function of macfor ﬁxed /H9258kis shown in Fig. 3. Its proﬁle can be decomposed into two stable branches where /H11509f//H11509mac/H110220, one for mac/H11021mac,2and the other for mac/H11022mac,1, and a central unstable branch where /H11509f//H11509mac/H110210. If hac,1c/H11021hac/H20849t/H20850/H11021hac,2cfor all t, then the change of the magnetic moment is reversible on one of the two stable branches. Otherwise, the right branch is traversed when hacdecreases from hac,max at point A1=/H20849hac,1c,mac,1/H20850 where the right branch ends and the system jumps to point A1/H11032=/H20849hac,1c,mac,1/H11032/H20850on the left branch. A similar description ap- plies when hacincreases from − hac,max . Plotting macas a func- tion of hac, we obtain the hysteresis loop shown in Fig. 4.W e note that lim hac,max→0/H20849df/dmac/H20850−1gives the transverse suscep- tibility from Ref. 8. For/H9258k=90°, we have − hac,1c=hac,2c=/H208491−hdc2/3/H208503/2, and, con- sequently, if 1 /H11022hdc/H33356/H208491−hac,max2/3/H208503/2=notation h, then the ac ﬁeld provides sufﬁcient energy to the moment to oscillate around theOzaxis /H20849cases 1 and 2 in Fig. 5/H20850, else, if hdc/H11021h, then the magnetic moment oscillates only around one of the twominima, as in the theory of Aharoni et al. 8/H20849case 3 in Fig. 5/H20850. For/H9258k=89.9°, no sudden jumps occur for hac,max =10−3and the motion of the magnetic moment is reversible. First we have calculated TS as /H9273=/H92730mac,max −mac,min 2hac,max, /H208495/H20850 where /H92730=MsV/Hk, so that the normalized susceptibility /H9273//H92730does not depend on the volume or anisotropy. The pe- riodic induced signal machas a phase shift with respect to hac, but its maximum takes place at the same moment of timeFIG. 2. /H20849Color online /H20850The particle’s free energy /H20851Eq. /H208491/H20850/H20852for /H9258k=90° /H20849left /H20850and/H9258k=89.9° /H20849right /H20850.PHYSICS OF COMPLEX TRANSVERSE SUSCEPTIBILITY … PHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 054409-3with the maximum of the ac ﬁeld /H20849Fig. 5/H20850. Thus, relation /H208495/H20850 gives a real susceptibility. As the ac ﬁeld amplitude in-creases, the position of the anisotropy peaks moves towardsmaller ﬁeld values, and the induced signal increases /H20851Fig. 6/H20849a/H20850/H20852but the TS decreases /H20851Fig. 6/H20849b/H20850/H20852because in Eq. /H208495/H20850thedenominator increases much more than the numerator. The experimental values reported in Refs. 19–22are H ac,max =5–80 Oe and Hk=12 kOe, namely, hac,max =4.2 /H1100310−4–6.6/H1100310−3. So, we can easily see from Fig. 6that ignoring the ac ﬁeld amplitude can lead to systematic errors(a) (b) FIG. 3. /H20849Color online /H20850The proﬁle of the function f/H20851Eq. /H208494/H20850/H20852and genesis of hysteresis loop under alternating ﬁeld excitation: /H20849a/H20850/H9258k =90° and /H20849b/H20850/H9258k=89.9°. When hacdecreases from hac,max , the sys- tem jumps from point A1=/H20849hac,1c,mac,1/H20850to point A1/H11032=/H20849hac,1c,mac,1/H11032/H20850. When hacincreases from − hac,max , the system jumps from point A2=/H20849hac,2c,mac,2/H20850to point A2/H11032=/H20849hac,2c,mac,2/H11032/H20850. FIG. 4. /H20849Color online /H20850Hysteresis loop under alternating ﬁeld for/H9258k=90° and hdc=0.99. The points A1andA1/H11032are the same as in Fig. 3.FIG. 5. /H20849Color online /H20850/H20849Left /H20850TS as a function of the dc ﬁeld in the absence of the thermal ﬂuctuations for /H9258k=90°, using extreme values /H20851Eq. /H208495/H20850/H20852of the induced signal /H20849thin line /H20850and the ﬁrst har- monic from the Fourier series /H20849thick line /H20850; the ac ﬁeld amplitude hac,max =0.001. /H20849Right /H20850Time evolution of the ac ﬁeld hac,o ft h e induced signal mac/H20849thin line /H20850and its ﬁrst harmonic /H20849thick line /H20850at the points indicated in the ﬁrst graph /H20849time is normalized to the period of the ac ﬁeld /H20850. (a) (b) FIG. 6. /H20849a/H20850Induced signal and /H20849b/H20850TS as a function of the dc ﬁeld and the ac ﬁeld amplitude in the absence of the thermal ﬂuc-tuations, calculated using extreme values of the induced signal /H20851Eq. /H208495/H20850/H20852for /H9258k=90°.CIMPOESU, STANCU, AND SPINU PHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 054409-4in the evaluation of the anisotropy using TS experiments. In Ref. 26, a perturbation approach was used to calculate the modiﬁcation of magnetization due to a small ac ﬁeld. Theproblem with the perturbation technique is that near the TSpeaks, the variation /H9254/H9258from Eq. /H20849A3/H20850/H20849see the Appendix /H20850is not a small deviation from equilibrium position /H92580, and, for example, the change in magnetization along the Ozaxis can- not be approximated by sin /H92580/H9254/H9258. Due to the rectangular-like form of the signal, its maxi- mum is very difﬁcult to be determined experimentally. So,we have analyzed the ﬁtting with a sinusoidal function and,respectively, the calculus of the ﬁrst harmonic from the Fou-rier series. Both methods give practically the same results for /H9258k=90° and small differences otherwise. If the ac ﬁeld is Hac/H20849t/H20850=Hac,max Im/H20849ei/H9275t/H20850=Hac,max sin/H9275t, where Im /H20849/H20850denotes the imaginary part of the complex vari- able, then we deﬁne the magnetization M/H20849t/H20850as a function of tby M/H20849t/H20850=Hac,max V/H20858 n=1/H11009 Im/H20849/H9273nein/H9275t/H20850 =Hac,max V/H20858 n=1/H11009 /H20851/H9273n/H11032sin/H20849n/H9275t/H20850+/H9273n/H11033cos /H20849n/H9275t/H20850/H20852, where /H9273n=/H9273n/H11032+i/H9273n/H11033/H20849n=1,2,3,... /H20850, and/H9273n/H11032and/H9273n/H11033can be cal- culated by /H9273n/H11032=MsV Hk1 hac,max2 T/H20885 0T mac/H20849t/H20850sin/H20849n/H9275t/H20850dt, /H9273n/H11033=MsV Hk1 hac,max2 T/H20885 0T mac/H20849t/H20850cos /H20849n/H9275t/H20850dt. Similar results are obtained if one uses Hac/H20849t/H20850=Hac,max Re/H20849ei/H9275t/H20850=Hac,max cos/H9275t, where Re /H20849/H20850denotes the real part. In Fig. 5, are presented the time evolution of the induced signal macand its ﬁrst harmonic for /H9258k=90° and hac,max =0.001. The ﬁrst harmonic and the applied ac ﬁeld are out of phase. Thus, we obtain an imaginary part of TS similar to theone described in Ref. 18, but independent of thermal relax- ation. The area of the loop described in the Fig. 4measures the amount of energy dissipated during each excitation cycle /H20849the energy loss per ac ﬁeld cycle /H20850. Losses are solely hysteretic because we have assumed that the magnetization Mlies in a minimum of the free energy at any moment. Actually, thedissipation takes place only in the Barkhausen jumps. Theenergy loss can be evaluated in terms of the imaginary partof the fundamental harmonic of susceptibility, as /H9004E= /H9266/H92620Hac,max2/H92731/H11033. By construction, the loop area is equal to/H9004E=2 /H20851E/H20849mac,1,hac,1c/H20850−E/H20849mac,1/H11032,hac,1c/H20850/H20852, where Eis the particle’s energy /H20851Eq. /H208491/H20850/H20852. Now, we have hac,max2/H92731/H11033 /H92730=1 /H9266E/H20849mac,1,hac,1c/H20850−E/H20849mac,1/H11032,hac,1c/H20850 K. That is, the product hac,max2/H92731/H11033depends on the dc ﬁeld only and does not depend on the ac ﬁeld amplitude. When thedynamics is taken into account, due to the damping, a phaseshift occurs between h acandmac, so an imaginary component of TS also appears.14 Replacing “ /H11022” sign with “ /H11005” sign in Eqs. /H208492/H20850, we obtain hdccos/H9258k+hacsin/H9258k= − cos3/H20849/H9258k−/H9258/H20850, hdcsin/H9258k−haccos/H9258k= sin3/H20849/H9258k−/H9258/H20850, /H208496/H20850 from which one further obtains /H20849hdccos/H9258k+hacsin/H9258k/H208502/3+/H20849hdcsin/H9258k−haccos/H9258k/H208502/3=1 . /H208497/H20850 This equation gives the critical ﬁelds hac,1candhac,2cwhere the Barkhausen jumps occur when hacdecreases and increases, respectively. When hacdecreases, before the jump, mac,1= sin/H92581 = − sin /H9258k/H20849hdccos/H9258k+hacsin/H9258k/H208501/3 − cos/H9258k/H20849hdcsin/H9258k−haccos/H9258k/H208501/3, and after the jump, mac,1/H11032= sin/H92581/H11032=tan/H20849/H9258k−/H92581/H11032/H20850− tan/H9258k /H20881/H208491 + tan2/H9258k/H20850/H208511 + tan2/H20849/H9258k−/H92581/H11032/H20850/H20852, where tan/H20849/H9258k−/H92581/H20850=−/H20849hdcsin/H9258k−hac,1ccos/H9258k/H208501/3 /H20849hdccos/H9258k+hac,1csin/H9258k/H208501/3, tan/H20849/H9258k−/H92581/H11032/H20850= − tan /H20849/H9258k−/H92581/H20850/H208511 + tan2/H20849/H9258k−/H92581/H20850 −/H208811 + tan2/H20849/H9258k−/H92581/H20850+ tan4/H20849/H9258k−/H92581/H20850/H20852. By expressing the energy /H20851Eq. /H208491/H20850/H20852in terms of tan /H20849/H9258k−/H9258/H20850,w e ﬁndPHYSICS OF COMPLEX TRANSVERSE SUSCEPTIBILITY … PHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 054409-5/H92731/H11033 /H92730=/H209021 /H9266hac,max2/H208771 2/H208751 1 + tan2/H20849/H9258k−/H92581/H20850−1 1 + tan2/H20849/H9258k−/H92581/H11032/H20850/H20876+1 1 + tan2/H20849/H9258k−/H92581/H20850/H208751+/H208811 + tan2/H20849/H9258k−/H92581/H11032/H20850 1 + tan2/H20849/H9258k−/H92581/H20850/H20876/H20878 if/H20849hdccos/H9258k+hac,max sin/H9258k/H208502/3+/H20849hdcsin/H9258k−hac,max cos/H9258k/H208502/3/H110221 0 elsewhere. /H20903/H208498/H20850 Relations /H208496/H20850and /H208497/H20850can be better understood using the as- troid formalism.27,28Assuming that the magnetization lies in a minimum of the free energy at any moment, its rotation isgoverned by the Stoner-Wohlfarth astroid /H20849see Fig. 7/H20850. The standard procedure in the critical curve approach is to ﬁndthe equilibrium orientation of the magnetization as the direc-tion of the tangent to the critical curve passing through thetip of the total applied ﬁeld vector /H20849in our case, the sum between the h dc, with the origin in the astroid’s center, and hac/H20850. In Fig. 7, one sees in the critical curve approach how the amplitude of the ac ﬁeld can induce the hysteretic pro-cesses described analytically for the particular case /H9258k=90°. Forhdc/H333561, when hacdecreases from hac,max /H20849point ain Fig. 7/H20850to −hac,max /H20849point b/H20850, the tangency point shifts continu- ously from point a/H11032to point b/H11032; for hac=0, the tangency point is in the astroid’s cusp, that is, the magnetic momentswitches its direction into the negative sense along with theac ﬁeld. For 1 /H11022h dc/H33356h, the vector haccuts the astroid: when the extremity of the hacvector shifts from point atopoint d, the tangency point moves continuously from a/H11032tod; decreasing in continuance, the ac ﬁeld from point dto point b, a jump will occur and the tangency point jumps into point d/H11032, from which it continuously shifts to b/H11032. Increasing the hac from point bto point a, a jump will occur at point cwhere the tangency point will jump into point c/H11032. For any arbitrary ac ﬁeld, however small, there is an interval of the dc ﬁeld forwhich the vector h accuts the astroid and, as a result, a jump of the magnetic moment will occur. The angular interval in which the magnetic moment oscillates /H20849a/H11032b/H11032in Fig. 7/H20850is maximum when the extremities of the vector hdc+hacare on the astroid, that is, hdc=/H208491−hac,max2/3/H208503/2. For lower values of the dc ﬁeld, hdc/H11021h, the vector hacdoes not cut the astroid and the tangency point shifts continuously between point a/H11032 and point b/H11032, no sudden jump of the magnetic moment oc- curs, and the amplitude of the induced signal suddenly de-creases giving rise to a peak in the TS curve. Increasing theac ﬁeld amplitude, the position of the TS peak moves towardsmaller dc ﬁeld values. IV. TS IMAGINARY PART: RATE-DEPENDENT HYSTERESIS EFFECT In the previous section, the TS was derived without any reference to the rate at which the external ﬁeld is varied. Therole of the external ﬁeld was only to bring the system to thepoint in which it becomes unstable and a spontaneousBarkhausen jump occurs. The spontaneous jump time scaleis much shorter than the external ﬁeld variation scale and,consequently, the evolution of the system was independent ofthe external ﬁeld rate. However, rate-independent hysteresisis an approximation and real magnetization processes are ratedependent due to two main reasons. First of all, Barkhausen jumps have a certain characteris- tic time scale determined by the processes on which the sys-tems dissipates energy 25as eddy currents or magnetization damping. Thus, for high ﬁeld rate values, the system ceasesto be described by spontaneous Barkhausen events and ap-proaches a forced dynamic regime driven by the externalmagnetic ﬁeld. In the present paper, we do not take intoaccount the effect of eddy currents, and in Sec. IV A, we willanalyze the occurrence of imaginary TS due the magnetiza-tion damping as a viscous-type rate-dependent hysteresis ef-fect. Second, as the temperature increases, the assumption that the system stays indeﬁnitely in a local energy minima is notanymore valid and thermal triggered Barkhausen jumps canoccur. These thermal effects have a certain time scale thatFIG. 7. /H20849Color online /H20850The astroid curve and the applied ﬁelds with/H9258k=90°, hac,max =10−3, and different values of the dc ﬁeld: /H208491/H20850 hdc=1, i.e., the total applied ﬁeld is in the exterior of the astroid and no sudden jump of the magnetic moment occurs; /H208492/H208501/H11022hdc/H11022/H208491 −hac,max2/3/H208503/2, i.e., the vector haccut the astroid; /H208493/H20850hdc=/H208491 −hac,max2/3/H208503/2, i.e., the extremities of the applied ﬁeld are on the as- troid, and the angular interval a/H11032b/H11032in which the magnetic moment oscillates is maximum; and /H208494/H20850hdc/H11021/H208491−hac,max2/3/H208503/2, i.e., the vector hacdoes not cut the astroid and no sudden jump of the magnetic moment occurs.CIMPOESU, STANCU, AND SPINU PHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 054409-6when correlated to the external ﬁeld rate can determine an imaginary TS. This made the object of Ref. 18and will be revised in Sec. IV B. A. Viscous-type rate-dependent hysteresis In the model presented previously, the TS is evaluated by employing a free-energy minimization method, namely, themodel is frequency independent because it is assumed that ineach moment the magnetization reaches an equilibrium stateindependent of the ac ﬁeld rate. In real magnetic systems, thedynamics is associated with energy dissipation, determininga lag between the magnetization’s response and the externalexcitation, which is strongly dependent on the excitation fre-quency. Consequently, knowing the frequency dependence ofthe TS becomes critical. The frequency used in the latest TSexperimental setups has increased from kilohertz tomegahertz 4or even to gigahertz range,29so, it is important to have a model in which the frequency dependence of TS istaken into account. Moreover, owing to the damping at highfrequency, a phase shift occurs between the applied ac ﬁeldand the alternating magnetization; so, an imaginary compo-nent of TS appears. A study of TS by focusing on magnetization dynamics is presented in Refs. 13and14. This model allows one to ﬁnd a general formula for TS for any magnetic system if an ex-pression for the magnetic free-energy density is known. Thestarting point of the model, as in the case of ferromagnetic resonance, 30is the magnetization’s equation of motion Landau-Lifshitz-Gilbert /H20849LLG /H20850:31 dM dt=−/H20841/H9253/H20841 1+/H92512M/H11003Heff−/H20841/H9253/H20841 1+/H92512/H9251 MsM/H11003/H20849M/H11003Heff/H20850, /H208499/H20850 where /H20841/H9253/H20841=2.211 /H11003105/H20849rad/s /H20850//H20849A/m /H20850is the gyromagnetic ratio,/H9251is the Gilbert damping constant, and Heffis the /H20849de- terministic /H20850effective ﬁeld and it incorporates the applied ﬁelds and the effects of different contributions in the freeenergy; the ﬁrst term on the right-hand side of the LLG equa-tion describes the gyromagnetic precession, and the secondterm describes the damping. The dynamics of the magneti-zation is governed by the relation that exists between thefrequency of the external excitation and the frequency ofresonance of the magnetic moment, which depends on thephenomenological damping term in the LLG equation. Inorder to preserve the magnetization’s magnitude, the LLGequation can be expressed better in spherical coordinates/H20849 /H9258,/H9272/H20850, and the singularity of the spherical coordinates along the polar axis can be avoided by an adequate choice of the Ozaxis. The dynamic susceptibility tensor in spherical coor- dinates is14/H9273/H9258/H9272=1 /H9275r2−/H92752+i/H9275/H9004/H9275/H20898/H92532/H208491+/H92512/H20850 /H92620/H20900F/H9272/H9272 sin2/H9258−F/H9258/H9272 sin/H9258 −F/H9258/H9272 sin/H9258F/H9258/H9258/H20901 +i/H9275/H9253Ms/H20875/H92511 −1/H9251/H20876/H20899, /H2084910/H20850 where /H9275r=/H9253/H208811+/H92512 /H92620Mssin/H9258/H20881F/H9258/H9258F/H9272/H9272−F/H9258/H92722 and /H9004/H9275=/H9251/H9253 /H92620Ms/H20873F/H9258/H9258+F/H9272/H9272 sin2/H9258/H20874, with F/H9258/H9258,F/H9258/H9272, and F/H9272/H9272being the second derivatives of the free-energy density Fat the equilibrium position /H20849/H9258,/H9272/H20850in the absence of the ac ﬁeld. The susceptibility tensor in the labo- ratory system is given by /H9273xyz=Tt·/H9273/H9258/H9272·T, where T=/H20875cos/H9258cos/H9272cos/H9258sin/H9272− sin/H9258 − sin/H9272 cos/H9272 0/H20876. From Eq. /H2084910/H20850, we can see that the susceptibility tensor is related to the second derivative of the free-energy density F and is a measure of the “curvature” of F. We can also ob- serve that the components of the susceptibility tensor arecomplex, the apparition of the imaginary part of TS being adirect consequence of the damping, in the absence of thedamping the imaginary part being zero. So, an imaginarypart of TS can occur even in the absence of thermal relax-ation due only to the damping. Actually, thermal ﬂuctuationsand dissipation are related manifestations of one and thesame interaction of the magnetic moment with its environ-ment. We note that, however, this model does not take intoaccount the amplitude of the ac ﬁeld. B. Thermal relaxation effect Thermal relaxation is taken into account using a two-level model in Refs. 10and11where only a real susceptibility is predicted, and it is shown that the temperature can changethe position of the coercivity peak while the anisotropy peaksmaintain their position, though their shape is altered. MonteCarlo simulations of the TS for a hexagonal array of dipolarinteracting nanoparticles with random anisotropy are pre-sented in Ref. 6, where only a real susceptibility is analyzed. However, the effect of the ac ﬁeld amplitude is not discussed,even if in simulations h ac,max =10−2is taken and the obtained shift of the anisotropy peak toward lower ﬁeld values is as-signed only to thermal ﬂuctuations. On the other hand,Papusoi 18used also the two-level model, but an imaginaryPHYSICS OF COMPLEX TRANSVERSE SUSCEPTIBILITY … PHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 054409-7TS is predicted because during a TS experiment, the increas- ing dc ﬁeld shifts the particles’ relaxation time toward lowervalues, making the occurrence of the thermal relaxation onthe time scale of the ac ﬁeld period possible. This mecha-nism is similar to that giving rise to a complex acsusceptibility, 24where the role of the transverse ﬁeld is played by the temperature. For a system of identical particleshaving the easy axis perpendicular to the dc ﬁeld, the thermalrelaxation gives rise to a secondary peak in the real part ofthe TS and to a peak in its imaginary part. The complexsusceptibility is now used to separate volume and anisotropyﬁeld dispersions 19–22by ﬁtting the experimental data with the expressions obtained in Ref. 18. In Ref. 23, the stochastic Landau-Lifshitz-Gilbert /H20849SLLG /H20850 equation32is used in order to describe an array of 400 “V oronoi” cells with intergranular exchange and magneto-static interactions, with different average diameters and dis-persions, and a Gaussian anisotropy ﬁeld distribution. It isshown that the volume dispersion only weakly affects thepeak shape, according the model from Ref. 18, while the dependence on both anisotropy ﬁeld dispersion and magne-tization is substantial. The magnetostatic interactions distortthe theory from Ref. 18, but the most substantial change is due to intergranular exchange. Recently, our micromagneticanalysis based on the SLLG equation showed a complexbehavior of the real and imaginary parts of TS for a systemof interacting particles. 33–35 In the model presented in Ref. 18, the dc ﬁeld acts along theOzaxis of a rectangular coordinates system and the ac ﬁeld acts in the plane xOy, with the azimuth /H9272ac/H20849Fig. 8/H20850. Initially, a system of identical particle is taken into account.Each particle has the volume Vand a uniaxial anisotropy of constant K, the easy axis being conﬁned in the xOz plane, with the polar coordinate /H9258k. The particle magnetization’s spherical coordinates are denoted by /H20849/H9258,/H9272/H20850. In order to better clarify the method used in Ref. 18and because in Ref. 18 some typographical errors seem to be present, a detailed deri-vation of TS formula is exposed in the Appendix. Unfortu-nately, these errors propagated in the subsequent papers 19–23 dealing with the determination of volume and anisotropy ﬁeld distributions in recording media using the model fromRef. 18. The complex TS for a system of particles with the easy axis oriented perpendicular to the dc ﬁeld direction /H20849 /H9258k =90° /H20850is given by /H20849see the Appendix /H20850/H9273r =/H20902/H208738/H9252/H926021−hdc2 /H92752+4/H92602+hdc2 1−hdc2/H20874cos2/H9272ac+ sin2/H9272ac,hdc/H110211 1 hdc−1cos2/H9272ac, hdc/H333561,/H20903 /H2084911/H20850 /H9273i=/H20902−4/H9275/H9252/H92601−hdc2 /H92752+4/H92602cos2/H9272ac,hdc/H110211 0, hdc/H333561,/H20903/H2084912/H20850 where /H9260=f0exp /H20851−/H9252/H208491−hdc/H208502/H20852,f0is the attempt frequency and it is of the order of 109–1013Hz, and /H9252=KV/kBTis the thermal factor. In Ref. 18, Eq. /H2084911/H20850is approximated by /H9273r=/H209028/H9252/H926021−hdc2 /H92752+4/H92602cos2/H9272ac+ sin2/H9272ac,hdc/H110211 /H9254/H20849hdc−1 /H20850cos2/H9272ac, hdc/H333561,/H20903 where /H9254is the Dirac delta “function.” However, this approxi- mation does not give the correct variation of /H9273r/H20849hdc/H20850in the proximate neighborhood of the point hdc=1. For a particle system with volume and anisotropy ﬁeld distributions, /H9273r0=/H20885 0hdc01 hkcos2/H9272ac hdc0 hk−1G/H20849hk/H20850dhk +/H20885 hdc0/H11009/H209028/H92520/H208751−/H20873hdc0 hk/H208742/H208761 v¯ /H11003/H20875/H20885 0/H11009/H92602 /H92752+4/H92602v2F/H20849v/H20850dv/H20876cos2/H9272ac +1 hk/H20873hdc0 hk/H208742 1−/H20873hdc0 hk/H208742cos2/H9272ac+1 hksin2/H9272ac/H20903G/H20849hk/H20850dhk, /H2084913/H20850 /H9273i0=−4/H92520/H92751 v¯cos2/H9272ac/H20885 hdc0/H11009/H208751−/H20873hdc0 hk/H208742/H20876G/H20849hk/H20850 /H11003/H20875/H20885 0/H11009/H9260 /H92752+4/H92602v2F/H20849v/H20850dv/H20876dhk, /H2084914/H20850 where hac0=hac/Hk0,hdc0=hdc/Hk0,hk=Hk/Hk0,/H92730 =/H11509mac//H11509hac0,/H92520=K0V0/kBT, and Hk0=2K0//H92620Msbeing the most probable value of the anisotropy ﬁeld distribution,G/H20849h k/H20850is the normalized anisotropy ﬁeld distribution, v =V/V0,v¯is the mean normalized volume, V0is the most probable value of the particle volume distribution, F/H20849v/H20850is the normalized particle volume distribution.FIG. 8. /H20849Color online /H20850Sketch of the geometry used to calculate the complex TS.CIMPOESU, STANCU, AND SPINU PHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 054409-8We note that in Ref. 18, the lower and upper limits of integrations in Eq. /H2084913/H20850are 0 and /H11009, respectively, for both integral over anisotropy ﬁeld distribution. Also, in Ref. 18, both lower limits of integration are equal to zero in Eq. /H2084914/H20850 and/H9252is used instead of /H92520. In the next section, these results are compared with the results obtained by numerical integration of stochasticLandau-Lifshitz-Gilbert equation, and an analysis of the in-ﬂuence of the ac ﬁeld amplitude and of the interactions be-tween particles on the complex TS method for the determi-nation of volume and anisotropy ﬁeld distributions inperpendicular recording media 19–22is also presented. V. LANGEVIN SIMULATION OF TS In the presence of thermal agitation, it is supposed that the dissipative term from the LLG equation describes only thestatistical /H20849ensemble /H20850average of rapidly ﬂuctuating random forces and that for an individual particle the effective ﬁeldmust be augmented by a stochastic thermal ﬁeld H thwhich is assumed to be a normal /H20849Gaussian /H20850random process with the following statistical properties:32 /H20855Hth,i/H20849t/H20850/H20856=0 , /H20855Hth,i/H20849t/H20850Hth,j/H20849t/H11032/H20850/H20856=2D/H9254ij/H9254/H20849t−t/H11032/H20850, D=/H9251kBT /H9253/H92620MsV, where /H20855/H20856means the statistical average over different realiza- tions of the ﬂuctuating ﬁeld, and iand jare the Cartesian indices. The Kronecker /H9254ijexpresses the assumption that the different components of thermal ﬁeld are uncorrelated andthe Dirac /H9254expresses that Hth,i/H20849t/H20850andHth,j/H20849t/H11032/H20850are correlated only for time intervals t−t/H11032much shorter than the time re- quired for an appreciable change of Maccording to the equa- tion /H20849LLG /H20850/H20849i.e., the random thermal forces are reduced to a “purely random” process, with a “white” spectrum /H20850. The constant Dmeasures the strength of the thermal ﬂuctuations and it is determined on the grounds of statistical-mechanicalconsiderations. For a particle system, it is also assumed thatthe ﬂuctuating ﬁelds acting on different magnetic momentsare independent. The stochastic ﬁeld changes the determin-istic motion of the magnetization into a random walk. As aresult, the LLG equation is converted into a stochastic dif-ferential equation of Langevin type with multiplicativenoise: 32 dM dt=−/H20841/H9253/H20841 1+/H92512M/H11003/H20849Heff+Hth/H20850 −/H20841/H9253/H20841 1+/H92512/H9251 MsM/H11003/H20851M/H11003/H20849Heff+Hth/H20850/H20852. /H2084915/H20850 In this paper, we integrate the stochastic Landau-Lifshitz- Gilbert /H20849SLLG /H20850equation numerically by means of the Heun numerical integration scheme.36No temperature depen- dences of the anisotropy constant and saturation magnetiza-tion are taken. The magnetic properties follow from averagesover many numerical realizations of the dynamic process. In order to test our program and to determine the time step sizein numerical integration, we have considered a system ofidentical, noninteracting Stoner-Wohlfarth particles and wehave compared our numerical results with the explicit solu-tions available in this case both for T=0 K and T/HS110050 K. The results obtained in this way are, in fact, identical to thosecalculated for one particle, because time average and averageover the statistical ensemble are the same /H20849ergodic hypoth- esis /H20850. In our simulation, we have considered a system of 4096 noninteracting spherical particles, and the desired quantitiesare averaged over 10–30 cycles of the ac ﬁeld, after theattenuation of the transitory effects which initially appeared.The results obtained by duplication of the number of par-ticles practically coincide with the former, but the computa-tion time increases signiﬁcantly. A. Identical particles In Fig. 9, we compare our numerical results for a system of identical particles, obtained for hac,max =10−3and/H9251=0.1, with relations /H2084911/H20850and /H2084912/H20850obtained in Ref. 18 /H20849referred hereafter as the CP model /H20850. The parameters’ values used in the simulations are Ms=300 kA/m, K=200 kJ/m3,T =300 K, f=1 MHz, and /H9272ac=0. The results presented in Fig. 9conﬁrm the results of the CP model with certain differ- ences, which can be explained in experimental conditions.However, as the SLLG method generates the stochastic tra-jectories for each individual magnetic moment, it providesmuch more insight into the dynamics of the system. Theprincipal difference is that we do not obtain two peaks forthe real part of TS due to no-vanishing amplitude of the acﬁeld, while Eq. /H2084911/H20850always predicts two peaks, one of them being the anisotropy peak from the theory of Aharoni et al. 8, athdc=1. In our results, both the position and the amplitude of the TS peaks depend on hac,max , so that the comparison of these quantities between the two models is, in some ways,superﬂuous. However, we can see that for /H9258k=90° and hac,max =10−3, as the particles’ diameter dincreases, the real TS peak is always at higher values than that given by Eq.FIG. 9. /H20849Color online /H20850TS curves at T=300 K for two values of the particles’ diameter, d=50 nm /H20849left /H20850andd=100 nm /H20849right /H20850, ob- tained with the CP model /H20849symbols /H20850and SLLG with hac,max =10−3 /H20849line /H20850.PHYSICS OF COMPLEX TRANSVERSE SUSCEPTIBILITY … PHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 054409-9/H2084911/H20850, while the imaginary TS peak is moving from higher values to smaller values than that given by Eq. /H2084912/H20850. In both models, an increase of the diameter dsharpens the TS peaks and shifts them to higher values of hdc, but in our model, the amplitude of real TS as a function of particles’ diameter hasa maximum /H20849see Figs. 10and 11/H20850, while in Ref. 18,i ti s increasing toward an inﬁnite value given by the theory ofAharoni et al. 8 When /H9258kdecreases from 90°, both real and imaginary parts of the TS undergo a decrease /H20849see Fig. 10where the scale of the vertical axes for /H9258k=90° is twice larger than that for/H9258k=89.9° /H20850. This decrease is very abrupt for the biggest particles but is smaller for the smallest particles because forthese particles the effect of thermal relaxation is more impor-tant. Therefore, the TS is dominated by the particles havingthe easy axis perpendicular to the dc ﬁeld /H20849 /H9258k=90° /H20850only if the particles are sufﬁciently large. In order to estimate the role of the temperature in the dynamics of the magnetic moment, we also present in Fig.12some typical time evolutions of the projection m acof the magnetic moment on the direction of the ac ﬁeld. Due to therandom character of the thermal ﬂuctuations, the magneticmoment does not exactly follow a certain path on the energysurface, from one minimum to another one passing throughthe saddle point with the smallest energy barrier, like in thetwo-level model. In fact, there is always a probability for the magnetic moment to follow any other path, the path with thelowest energy barrier being the most probable. As weshowed in Sec. III, if h /H33355hdc/H333551 and T=0 K, then the ac ﬁeld provides sufﬁcient energy to the magnetic moment tooscillate around the Ozaxis. For T/H110220 K, the magnetic mo- ment effectuates random oscillations by thermal activationclose to the bottom of the energy minima, but also a numberof energy barrier crossings followed by a return to the origi-nal energy minima, from one part of the Ozaxis to other, during a cycle of the ac ﬁeld can be observed even for h ac,1c/H11021hac/H11021hac,2cif the dimension of the particle is sufﬁ- ciently small. Consequently, the induced signal macdoes not have a rectangular-like form as in the case T=0 K and it is approximately in phase with the ac ﬁeld, and its mean valueover many cycles of the ac ﬁeld is less than that obtained forT=0 K /H20851see Fig. 12/H20849a/H20850/H20852. Decreasing the dimension of the particle, the magnetic moment effectuates a considerablenumber of overbarrier rotations during a cycle of the ac ﬁeld.Similar features are encountered for /H9258k=89.9°, but in this case, because the energy landscape is not symmetric, thehigher energy minimum is less frequented by the magneticmoment and the induced signal is more distorted from itsideal sinusoidal form. Forh dc/H33355handT=0 K, the magnetic moment oscillates only around minimum 1 or minimum 2. However, for T /H110220 K, in continuance, the magnetic moment can pass from one part of the Ozaxis to the other by means of the energy gained from the thermal ﬁeld, the induced signal increaseswith the decrease of the dc ﬁeld, and, consequently, the po-sition of the anisotropy peaks moves toward smaller ﬁeldvalues; that is, the effect of no-vanishing amplitude of the acFIG. 10. /H20849Color online /H20850TS curves at T=300 K for different values of the particles’ diameter dfor/H9258k=90° /H20849left /H20850and/H9258k=89.9° /H20849right /H20850. The curves for T=0 K are also presented for comparison. FIG. 11. The dependence of the maximum of the real part /H20849up/H20850 and the minimum of the imaginary part /H20849down /H20850of TS on particles’ diameter d.(a) (b) (c)( d) FIG. 12. Time evolution of the induced signal mac/H20849time is nor- malized to the period of the ac ﬁeld /H20850for one particle with d =50 nm at T=300 K for hac=0.001, /H9251=0.1, and for different values of dc ﬁeld: /H20849a/H20850hdc=0.97, /H20849b/H20850hdc=0.954, i.e., Re /H20849/H9273/H20850is maximum, /H20849c/H20850 hdc=0.948, i.e., Im /H20849/H9273/H20850is minimum, and /H20849d/H20850hdc=0.942, i.e., the phase shift between macandhacis maximum.CIMPOESU, STANCU, AND SPINU PHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 054409-10ﬁeld is ampliﬁed by the thermal agitation. When the dc ﬁeld decreases, the energy minima become narrower and the am-plitude of the random oscillations around the two minimadecreases, the energy barriers increase, and the probability topass from one minimum to other decreases, so that inducedsignal shifts into a rectangular-like form, and its amplitudeincreases exhibiting a maximum in the amplitude and, sub-sequently, in the phase shift behind the ac ﬁeld /H20851see Figs. 12/H20849b/H20850–12/H20849d/H20850/H20852. At low values of h dc, the magnetic moment oscillates close to the bottom of the energy minima, with the overbar-rier relaxation mechanism being blocked and the inducedsignal is very small and in phase with the ac ﬁeld. The minimum of the imaginary TS /H20851Eq. /H2084912/H20850/H20852for the par- ticles with the easy axis perpendicular to the dc ﬁeld takesplace when 2 /H9260/H11229/H9275, that is, when the energy barrier in the absence of the ac ﬁeld is /H9004E0/kBT/H11229ln/H208492f0//H9275/H20850. From this relation, it can be seen that increasing the frequency of the excitatory ac ﬁeld, the position of the imaginary TS peakmoves toward higher dc ﬁeld values, because increasing thefrequency decreases the probability of surmounting the en-ergy barrier. This increase is conﬁrmed by our simulations. If /H9270=2/H9260is the relaxation time, then the above relation can be written as /H9270/H11229Tac/2/H9266, where Tacis the period of hac; that is, the minimum of the imaginary part /H20851Eq. /H2084912/H20850/H20852takes place when the relaxation time is 2 /H9266times smaller than the period Tacof the ac ﬁeld. In order to test this relation, we have counted in our numerical simulations the number of cross-ings of the magnetic moment of one particle from one part oftheOzaxis to the other, and we have normalized it to the number of the cycles of the ac ﬁeld, so that it does notdepend on the number of cycles over which m acis averaged. ForT=0 K, when hdcdecreases, the number of crossings nis equal to 1 until the norm of TS reaches its maximum valueand is equal to 0 after that. Taking into account the tempera-ture, the magnetization reversal can take place by thermalactivation over the energy barriers, and when h dcdecreases, the number of crossing ndecreases continuously from an inﬁnite value /H20849forhdc=1 when there is no energy barrier /H20850to 0/H20849when the energy barrier is very high and the probability of crossing is practically zero /H20850. In Fig. 13, presented are the number of crossings nrwhen the real part of TS reaches its maximum value and the number of crossings niwhen the imaginary part reaches its minimum value as a function ofparticle’s diameter. We can see that n iincreases from 1 to a limit value of approximately 2.2 as the particle’s dimensiondecreases, while in the case of n r, the increase is more abrupt. So, the micromagnetic analysis showed the complexbehavior of the dynamics of the magnetic moment when theenergy barrier is smaller than k BT, the domain in which the two-level model is not valid.37 We note that the minimum values of the imaginary part and the phase shift do not take place for the same value ofh dc. The lag behind hacreaches the extreme value when the magnetic moment oscillates mainly around one minimum,but there is a small probability to pass from one minimum toanother during a number of cycles of the ac ﬁeld, so that n /H110211/H20851Fig. 12/H20849d/H20850/H20852. So, the temperature has two antagonistic effects on the phase shift: for h /H33355hdc/H333551, thermal ﬂuctua- tions decrease the phase shift, while for hdc/H33355h, thermal ﬂuctuations increase the phase shift.B. Particle distributions From this point on, we are discussing the properties of a system of aligned particles lying in a regular network formedwithin one layer, the system being generated using lognormaldistributions for volume and for the anisotropy constant. Theresults obtained for a noninteracting system are qualitativelysimilar of those obtained using the CP model. However, ourresults show the importance of the ac ﬁeld amplitude. Anincrease of the most probable volume V 0shifts TS’s peaks to higher ﬁelds and sharpens the peaks. An increase of the mostprobable anisotropy constant K 0also shifts the peaks to higher ﬁelds but their width increases. However, the width’srelative variation is not so important in both cases. This factis an important source of errors for the ﬁtting procedure de-scribed in Refs. 19–22. The ﬂattening of the volume distri- bution determines a decrease of imaginary TS and shifts itspeak toward higher ﬁelds while its width increases /H20849see Fig. 14/H20850. The most important factor in Eqs. /H2084913/H20850and /H2084914/H20850is /H9268k: both the amplitude and the width of the TS peak stronglydepend on it /H20849see Fig. 14/H20850. As a consequence, /H9268kis deter- mined with the smallest errors by ﬁtting procedure. By sys-FIG. 13. /H20849Color online /H20850The number of crossings of the mag- netic moment of one particle from one part of the Ozaxis to the other, normalized to the number of the cycles of the ac ﬁeld, as afunction of particle’s diameter: when the real part of TS reaches itsmaximum value /H20849n r/H20850and when the imaginary part reaches its mini- mum value /H20849ni/H20850. FIG. 14. /H20849Color online /H20850TS curves for different volume disper- sions/H9268V/H20849left /H20850and anisotropy dispersions /H9268k/H20849right /H20850forT=300 K andhac0max=0.001.PHYSICS OF COMPLEX TRANSVERSE SUSCEPTIBILITY … PHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 054409-11tematic simulations, it was observed that a small angular dispersion of the easy axis affects the signal’s amplitudeonly. Figure 15shows the ac ﬁeld amplitude dependence of TS for two temperatures. The curves given by the CP model arealso presented. We observe that only at low temperatures, asthe ac ﬁeld amplitude decreases, the CP model gives a betterdescription of the results obtained with the SLLG equation.The effect of the increase of ac ﬁeld amplitude is similar to adecrease of /H9268VandV0. In order to see how the magnetostatic interactions be- tween particles inﬂuence the TS curves, we have generatedsamples with increasing packing fraction. The magnetostaticinteractions between particles are controlled by the networkcharacteristic length Dand periodic boundary conditions are used. The fast Fourier transform technique is applied in orderto compute the ﬁeld due to all particles. As the intensity ofthe interactions increases, the position of the imaginary TSpeak moves toward smaller ﬁelds values /H20849Fig. 16/H20850and its amplitude decreases; that is, the effect of the interactions issomehow similar to the ac ﬁeld amplitude. In Ref. 34, a systematic quantitative analysis of the inﬂu- ence of the ac ﬁeld amplitude and of the interactions betweenparticles on the complex TS method for the determination ofvolume and anisotropy ﬁeld distributions in perpendicularrecording media is presented. Following a similar methodol-ogy as described in Refs. 19–22, we have ﬁtted the “experi- mental” data produced by our micromagnetic model with therelation from the CP model. Because experimentally a signalproportional to the magnetization and not its real value ismeasured and because, as we mentioned in previous sec-tions, the measured susceptibility is not equal to the differ-ential susceptibility given by CP model, we have ﬁtted thenormalized imaginary TS curves even if in this way impor-tant information contained in the TS amplitude is lost. Themost affected by the ac ﬁeld amplitude and the interactionsbetween particles are the volume distribution’s parameters.In this case, the error generated by not taking into accountthe ﬁnite value of the ac ﬁeld amplitude and by ignoring theinterparticle interactions can be up to 55%. Similar resultshave been obtained for wider distributions, but the errors with which /H9268Vis obtained decrease if /H9268Vincreases. However, for too large volume and anisotropy distributions, the peaksin the TS signal disappear. Assuming that the values of V 0 andK0are available through another measurement, we also tried to ﬁt only /H9268Vand/H9268k. In this case, the error with which /H9268Vis obtained increases because its effect is covered by the effect of the ac ﬁeld amplitude. VI. CONCLUSIONS We have analyzed the origins of the imaginary part of TS: rate-independent hysteresis, viscous-type rate-dependenthysteresis, and thermal relaxation effect origin. The rate-independent origin is an intrinsic contribution to the imagi-nary TS and cannot be neglected because it is a zero-temperature effect. We have shown that because in theexperimental procedure the amplitude of the ac ﬁeld is notzero in the mathematical sense, a sinusoidal excitation givesrise to a distorted output, an irreversible susceptibility ismeasured, and the lag of the magnetization with respect tothe applied ﬁeld can be associated with a complex suscepti-bility. Starting from the energy loss per ac ﬁeld cycle, anexpression for the imaginary TS is obtained at T=0 K, as- suming that in each moment the system lies in a minimum ofthe free energy. Also, using the critical curve formalism, wehave shown how the ac ﬁeld can induce a hysteresis and,consequently, imaginary TS. The magnetization dynamics inthe framework of the LLG equation was used to analyze theoccurrence of imaginary TS as a consequence of magnetiza-tion damping, which determines a lag between the magneti-zation’s response and the applied excitation. This lag isstrongly dependent on the excitation frequency. The modeldeveloped in Ref. 18, using a two-level approximation to take into account the temperature effects, was revised andcertain clariﬁcations and corrections have been made. Ther-FIG. 15. /H20849Color online /H20850TS curves for /H9268V=0.1, /H9268k=0.01, and different values of the ac ﬁeld amplitude for T=50 K /H20849left /H20850andT =300 K /H20849right /H20850; with symbols, the curves predicted by the CP model are also presented.FIG. 16. /H20849Color online /H20850Imaginary TS curves for different values of the packing fraction and different values of the ac ﬁeld amplitude/H20849temperature T=300 K /H20850; with symbols, the curve predicted by the CP model is also presented.CIMPOESU, STANCU, AND SPINU PHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 054409-12mal ﬂuctuation effects on TS have also been considered us- ing the stochastic LLG equation, and our results have beencompared with those from Ref. 18, the differences between them being analyzed. The SLLG analysis showed complexbehavior of the dynamics of the magnetic moment when theenergy barrier is smaller than k BT, the domain in which the two-level model is not valid.37We have shown that the ther- mal energy assists the switch discussed in the case T=0 K and causes a shift of the anisotropy peak toward lower ﬁeldvalues and that the imaginary part of the TS is a cumulativeeffect of no-vanishing amplitude of the ac ﬁeld, which isampliﬁed by the thermal agitation. Finally, an analysis of theinﬂuence of the ac ﬁeld amplitude and of the interactionsbetween particles on the complex TS method for the deter-mination of volume and anisotropy ﬁeld distributions in per-pendicular recording media, developed in Refs. 19–22,i s presented. ACKNOWLEDGMENTS Work at AMRI was supported by DARPA under Grant No. HR0011-05-1-0031. This work was partially supportedby Romanian CNCSIS under the Grant A /H20849RELSWITCH /H20850. APPENDIX Let us consider a system of identical particles of volume Vand uniaxial anisotropy of constant K/H110220, the easy axes being conﬁned in the xOz plane, having the polar coordinate /H9258k. The particle magnetization’s spherical coordinates are de- noted by /H20849/H9258,/H9272/H20850/H20849see Fig. 8/H20850. The particle’s free energy is given by E=−2 KV/H20875hdccos/H9258+hacsin/H9258cos /H20849/H9272−/H9272ac/H20850 +1 2/H20849sin/H9258cos/H9272sin/H9258k+ cos/H9258cos/H9258k/H208502/H20876, and the necessary equilibrium condition, /H11509E /H11509/H9258=0 ,/H11509E /H11509/H9272=0 , /H20849A1 /H20850 has to be true for all hac. For hac=0, the extreme points satisfy sin/H9272=0 , 2hdcsin/H9258+ sin 2 /H20849/H9258/H11007/H9258k/H20850=0 , /H20849A2 /H20850 where “ /H11002” sign correspond to /H9272=0 and “ /H11001”t o/H9272=/H9266.I tc a n be shown that there are two minima with /H927210=0 and /H927220=/H9266 separated by a saddle point with /H927230=/H9266. The polar angles corresponding to these three points are noted with /H925810,/H925820, and /H925830. The extreme points in the presence of a small ac ﬁeld could be approximated by the ﬁrst terms in the Taylor seriesexpansions: /H9258i/H20849hac/H20850/H11015/H9258i0+hac/H9254/H9258i,/H9272i/H20849hac/H20850/H11015/H9272i0+hac/H9254/H9272i, where /H9254/H9258i= lim hac→0d/H9258i dhac=±cos/H9258i0cos/H9272ac cos 2 /H20849/H9258i0/H11007/H9258k/H20850+hdccos/H9258i0,/H20849A3 /H20850 /H9254/H9272i= lim hac→0d/H9272i dhac=sin/H9272ac cos /H20849/H9258i0/H11007/H9258k/H20850sin/H9258k, /H20849A4 /H20850 where the ﬁrst sign corresponds to i=1 and the second to i =2,3. The variations /H9254/H9258iand/H9254/H9272iare obtained by differenti- ating relations /H20849A1/H20850with respect to hac. When the energy barriers /H9004Eare very small in comparison with kBT/H20849kBis the Boltzmann constant and the Tis the temperature /H20850, the ther- mal agitation causes continual changes in the orientation ofthe magnetic moment. When /H9004E/k BTis large enough, a discrete-orientation model can be used: if the occupationprobabilities of the two energy levels are denoted by n 1and n2=1− n1and if a particle in orientation ihas probability /H9260ij per unit time of jumping to orientation j, then the approach to statistical equilibrium is described by the following masterequation: dn 1 dt=−/H926012n1+/H926021n2. /H20849A5 /H20850 The rates of thermally activated transitions of noninteracting particles are generally assumed to be given by the Néel-Brown /H20849or Arrhenius /H20850expression /H9260ij=f0exp/H20873−E3−Ei kBT/H20874, /H20849A6 /H20850 where E1andE2are the energies of the two local minima, andE3is the energy of the saddle point separating them. This simple model breaks down as /H9004E/kBTdecreases, because the distribution of the particle orientation will no longer be suf-ﬁciently concentrated near the two minima. Aharoni 37has shown, in the case of uniaxial anisotropy and zero magneticﬁeld, that the high-energy-barrier approximation is still agood approximation when the barriers are of the order ofk BT. However, when the dc ﬁeld is in the vicinity of Hk, the energy barrier becomes very small in comparison with kBT. In the presence of the ac ﬁeld on besides the explicit time dependence of the occupation probabilities, there is an im-plicit time dependence via ac ﬁeld, n 1(hac/H20849t/H20850,t), and in ﬁrst approximation, we can write n1„hac/H20849t/H20850,t…/H11015n1/H208490,t/H20850+hac/H20849t/H20850/H9254n1/H20849t/H20850, /H20849A7 /H20850 where /H9254n1/H20849t/H20850=/H11509n1/H208490,t/H20850//H11509hac. The normalized magnetic mo- ment along the ac ﬁeld is mac„hac/H20849t/H20850,t…=n1„hac/H20849t/H20850,t…sin/H92581„hac/H20849t/H20850…/H20851cos/H92721„hac/H20849t/H20850…cos/H9272ac + sin/H92721„hac/H20849t/H20850…sin/H9272ac/H20852 +n2„hac/H20849t/H20850,t…sin/H92582„hac/H20849t/H20850… /H11003/H20851cos/H92722„hac/H20849t/H20850…cos/H9272ac + sin/H92722„hac/H20849t/H20850…sin/H9272ac/H20852, which gives for the TS the expressionPHYSICS OF COMPLEX TRANSVERSE SUSCEPTIBILITY … PHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 054409-13/H9273= lim hac→0/H11509mac„hac/H20849t/H20850,t… /H11509hac =/H20849sin/H925810+ sin/H925820/H20850cos/H9272ac/H9254n1/H20849t/H20850+/H20851/H20849cos/H925810/H9254/H92581 + cos/H925820/H9254/H92582/H20850cos/H9272ac+/H20849sin/H925810/H9254/H92721 + sin/H925820/H9254/H92722/H20850sin/H9272ac/H20852n1/H208490,t/H20850− cos/H925820cos/H9272ac/H9254/H92582 − sin/H925820sin/H9272ac/H9254/H92722. /H20849A8 /H20850 The master equation /H20851Eq. /H20849A5/H20850/H20852describing the time evolution of the occupation probability of the ﬁrst energy minimum is dn1„hac/H20849t/H20850,t… dt=− /H20851/H926012„hac/H20849t/H20850…+/H926021„hac/H20849t/H20850…/H20852n1„hac/H20849t/H20850,t… +/H926021„hac/H20849t/H20850…, /H20849A9 /H20850 and in ﬁrst approximation, one can write /H9260ij„hac/H20849t/H20850…/H11015/H9260ij/H208490/H20850+hac/H20849t/H20850/H9254/H9260ij, Ei„hac/H20849t/H20850…/H11015Ei/H208490/H20850+hac/H20849t/H20850/H9254Ei, where /H9254/H9260ij=d/H9260ij/H208490/H20850 dhac=−/H9260ij/H208490/H20850/H9254E3−/H9254Ei kBT, /H20849A10 /H20850 /H9254Ei=dEi/H208490/H20850 dhac=/H110072 sin/H9258i0cos/H9272acKV. /H20849A11 /H20850 In the above relations, it is assumed that the magnetic mo- ment is only in an extreme of the free energy and there is nodistribution of the particles’ orientation around minima dueto thermal ﬂuctuations, and the energy is taken as a functionofh aconly. For brevity, in the what follows, we shall note with superscript “0” the values of different quantities at hac=0. Thus, in ﬁrst approximation, the master equation /H20851Eq. /H20849A9/H20850/H20852can be written as dn1„hac/H20849t/H20850,t… dt=− /H20849/H9260120+/H9260210/H20850n10/H20849t/H20850+/H9260210 +hac/H20849t/H20850/H20851−/H20849/H9260120+/H9260210/H20850/H9254n1/H20849t/H20850 +/H9254/H926021−/H20849/H9254/H926012+/H9254/H926021/H20850n10/H20849t/H20850/H20852. /H20849A12 /H20850 Using the approximation dn1„hac/H20849t/H20850,t… dt/H11015dn10/H20849t/H20850 dt+hac/H20849t/H20850d„/H9254n1/H20849t/H20850… dt+dhac/H20849t/H20850 dt/H9254n1/H20849t/H20850 and with hac=hac,max exp /H20849i/H9275t/H20850, Eq. /H20849A12 /H20850turns into d„/H9254n1/H20849t/H20850… dt=− /H20849/H9260120+/H9260210+i/H9275/H20850/H9254n1/H20849t/H20850+/H9254/H926021 −/H20849/H9254/H926012+/H9254/H926021/H20850n10/H20849t/H20850. Neglecting the time dependence of n10and with initial condi- tion/H9254n1/H208490/H20850=0, one obtains /H9254n1/H20849t/H20850=/H20851e−/H20849/H9260120+/H9260210/H20850te−i/H9275t−1 /H20852/H20849/H9254/H926012+/H9254/H926021/H20850n10−/H9254/H926021 /H9260120+/H9260210+i/H9275. In view of the exponential variation of transition rates /H20851Eq. /H20849A6/H20850/H20852, the ﬁrst term from the second member can be ne- glected when the energy barrier is comparable with kBT. Us- ing Eqs. /H20849A10 /H20850and /H20849A11 /H20850, one obtains /H9254n1/H20849t/H20850=/H9252/H20849/H9260120g1+/H9260210g2/H20850n10−/H9260210g2 /H9260120+/H9260210+i/H9275, where gi=/H20849/H9254E3−/H9254Ei/H20850/KV. Because the time dependence of n10is neglected, we have − /H20849/H9260120+/H9260210/H20850n10+/H9260210=0, and thus the real and imaginary components of Eq. /H20849A8/H20850, using Eq. /H20849A2/H20850, can be written as /H9273r=/H20902/H9252/H20849sin/H925810+ sin/H925820/H20850cos/H9272ac/H9260120/H9260210/H20849g1−g2/H20850 /H92752+/H20849/H9260120+/H9260210/H208502 +/H20851/H20849cos/H925810/H9254/H92581+ cos/H925820/H9254/H92582/H20850cos/H9272ac+/H20849sin/H925810/H9254/H92721+ sin/H925820/H9254/H92722/H20850sin/H9272ac/H20852n10 − cos/H925820cos/H9272ac/H9254/H92582− sin/H925820sin/H9272ac/H9254/H92722, ifhdc/H11021hcr/H20849/H9258k/H20850 cos/H925810cos/H9272ac/H9254/H92581+ sin/H925810sin/H9272ac/H9254/H92721, ifhdc/H33356hcr/H20849/H9258k/H20850,/H20903/H20849A13 /H20850 /H9273i=/H20902/H9252/H20849sin/H925810+ sin/H925820/H20850cos/H9272ac/H9275/H9260120g2−/H20849/H9260120g1+/H9260210g2/H20850n10 /H92752+/H20849/H9260120+/H9260210/H208502,ifhdc/H11021hcr/H20849/H9258k/H20850 0, ifhdc/H33356hcr/H20849/H9258k/H20850,/H20903/H20849A14 /H20850CIMPOESU, STANCU, AND SPINU PHYSICAL REVIEW B 76, 054409 /H208492007 /H20850 054409-14where hcr/H20849/H9258k/H20850=/H20849sin2/3/H9258k+ cos2/3/H9258k/H20850−3/2 is the critical ﬁeld /H20851for an orientation /H9258kof the easy axis, there are two minima of the free energy when hdc/H11021hcr/H20849/H9258k/H20850/H20852.These two equations are equivalent to Eqs. /H2084911/H20850and /H2084912/H20850 from Ref. 18. For/H9258k=90° and −1 /H11021hdc/H110211, we have E10=E20 =− /H208491+hdc2/H20850KV, E30=−2hdcKV, and /H9260120=/H9260210=/H9260 =f0exp /H20851−/H9252/H208491−hdc/H208502/H20852, so that the complex susceptibility is given by Eqs. /H2084911/H20850and /H2084912/H20850. *Permanent address: Faculty of Physics, “Al. I. Cuza” University, Iasi 700506, Romania. †LSpinu@uno.edu 1P. Poddar, J. L. Wilson, H. Srikanth, D. F. Farrell, and S. A. Majetich, Phys. Rev. B 68, 214409 /H208492003 /H20850. 2P. Vavassori, D. Bisero, F. Carace, A. di Bona, G. C. Gazzadi, M. Liberati, and S. Valeri, Phys. Rev. B 72, 054405 /H208492005 /H20850. 3L. Spinu, A. Stancu, Y . Kubota, G. Ju, and D. Weller, Phys. Rev. B68, 220401 /H20849R/H20850/H208492003 /H20850. 4L. Spinu, H. Srikanth, A. Gupta, X. W. Li, and G. Xiao, Phys. Rev. B 62, 8931 /H208492000 /H20850. 5N. A. Frey, S. Srinath, H. Srikanth, M. Varela, S. Pennycook, G. X. Miao, and A. Gupta, Phys. Rev. B 74, 024420 /H208492006 /H20850. 6D. Kechrakos and K. 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PhysRevB.94.054434.pdf	PHYSICAL REVIEW B 94, 054434 (2016) Longitudinal spin dynamics in ferrimagnets: Multiple spin wave nature of longitudinal spin excitations V . N. Krivoruchko Donetsk Institute for Physics and Engineering, NAS of Ukraine, 46, Nauki Avenue, 03680, Kiev, Ukraine (Received 19 May 2016; revised manuscript received 30 July 2016; published 30 August 2016) Motivated by the existing controversy about the physical mechanisms that govern longitudinal magnetization dynamics under the effect of ultrafast laser pulses, in this paper we study the microscopic model oflongitudinal spin excitations in a two-sublattice ferrimagnet using the diagrammatic technique for spin operators.The diagrammatic approach provides us with an efﬁcient procedure to derive graphical representations forperturbation expansion series for different spin Green’s functions and thus to overcome limitations typical forphenomenological approaches. The inﬁnite series involving all distinct loops built from spin wave propagatorsare summed up. These result in an expression for the longitudinal spin susceptibility χ zz(q,ω) applicable in all regions of frequency ωand wave vector qspace beyond the hydrodynamical and critical regimes. A strong renormalization of the longitudinal spin oscillations due to processes of virtual creation and annihilation oftransverse spin waves has been found. We have shown that the spectrum of longitudinal excitations consists of aquasirelaxation mode forming a central peak in χ zz(q,ω) and two (acoustic and exchange) precessionlike modes. As the main result, it is predicted that both acoustic and exchange longitudinal excitations are energetically abovesimilar modes of transverse spin waves at the same temperature and wave vector. The existence of the exchangelongitudinal mode at such frequencies can result in a new form of excitation behavior in ferrimagnetic system,which could be important for understanding the physics of nonequilibrium magnetic dynamics under the effectof ultrafast laser pulses in multisublattice magnetic materials. DOI: 10.1103/PhysRevB.94.054434 I. INTRODUCTION Ultrafast laser-induced magnetic switching in ferrimagnets has become a hot topic of modern magnetism. Uncovering thephysical mechanisms that govern ultrafast spin dynamics iscritical for understanding fundamental limits of ultrafast spin-based electronics. It is of fundamental importance to realizethe time evolution of magnetic moments at high temperaturesand time scales approaching femtoseconds. A recent review on the state of the art of ultrafast spin dynamics and its prospects is given by Kirilyuk et al. [1]. Ultrafast magnetization dynamics induced by femtosecond laser pulses have been investigated in different ferrimagneticalloys [ 2–9]. Several hypotheses have been put forward to explain the observed magnetization switching: the crossingof the angular momentum compensation point [ 3] and the inverse Faraday effect [ 2] and its combination with ultrafast heating [ 4], among others [ 5,10,11]. The ﬁrst attempts to de- scribe longitudinal magnetization dynamics in two-sublatticesystems have been proposed recently in Refs. [ 12–17]. It has been shown that spin dynamics simulations within thephenomenological Landau-Lifshitz-Bloch [ 12–14], Landau- Lifshitz-Gilbert [ 15], or Landau-Lifshitz-Baryakhtar formula- tions [ 6,16,17] can be used to describe ultrafast laser-induced demagnetization in multisublattice magnets when a sufﬁ-ciently large and fast dissipation of spin angular momentum isassumed. However, in spite of the progress in phenomenological description, to achieve an exhaustive theoretical explanation ofultrafast magnetization dynamics, ﬁrst-principles calculations based on quantum-mechanical theory are indispensable. In particular, at present it is commonly accepted that the shorttime scale of the laser pulse and high temperatures followingthe excitation lead to processes where longitudinal magnetiza-tion dynamics becomes pronounced [ 18]. Naturally, any model of an ultrafast magnetization switching should correctly repro- duce equilibrium longitudinal dynamics of the system in the limit of a long-time evolution. Surprisingly enough, whereasdynamics of transverse spin components in multisublatticemagnets is well understood theoretically, distinctive featuresof longitudinal dynamics are still a puzzling point even inthe system equilibrium state. Dynamics of longitudinal spincomponents implies a different physical picture than transverse spin oscillations and poses speciﬁc theoretical problems. In fact, this is a very important fundamental question that hasnot been addressed theoretically to date. Obviously, in-depthknowledge of the physics of longitudinal spin dynamics canshed light on the role of the exchange interaction in the ultrafastdynamics of spin subsystems in complex materials [ 1]. Note that according to the ﬁrst publications on this issue [ 19,20], the longitudinal spin mode arises because of a virtual process of coherent creation and annihilation ofordinary spin waves. In more recent studies based on a dia-grammatic approach for spin operators [ 21], for a ferromagnet the longitudinal excitations have been found at frequenciesω(q)=[ε(k)−ε(k±q)] and have been interpreted in terms of processes of creation and annihilation of two transversespin waves with energies ε(k) and ε(k±q) (here the wave vector kis a variable). The excitation processes are controlled, however, by the occupation factor, determined through theBose distribution function n(ε(k)), which makes the spin waves with k∼0 the dominant ones. Whereas the underlying magnetization dynamics of mul- tisublattice magnets has become a hot topic at present,complete understanding of this phenomenon in ferrimagnetsis still missing. In the present paper, a basic understanding ofthe longitudinal spin dynamics in ferrimagnetic materials is addressed. To this end, we used a general theoretical treatment 2469-9950/2016/94(5)/054434(13) 054434-1 ©2016 American Physical SocietyV . N. KRIVORUCHKO PHYSICAL REVIEW B 94, 054434 (2016) aiming to overcome limitations of the models [ 6,12–17] and to reveal a unique physical picture common to different multisublattice magnets. Namely, we consider the micro-scopic Heisenberg model of a two-sublattice ferrimagnet. Theconvenient version of the diagrammatic technique for spinoperators has been used to provide quantitative calculation ofmagnetization excitation frequencies and relaxation times. Adetailed account of this diagrammatic technique is given in the textbooks [ 22,23]. We start in the next section with the formulation of a simple model for a two-sublattice ferrimagnet. In Sec. III,a diagrammatic representation of the longitudinal spin Green’sfunctions (GFs) is formulated. Since, as expected, the quantumdynamics of longitudinal spin components is generated byvirtual processes of creation and annihilation of spin waves,the mathematical problem reduces to summing up all the loopdiagrams describing these processes. The complexity of theproblem arises from the fact that a commutator of two spinoperators is not a cnumber. Therefore, the series of the loop diagrams turns out to be rather complicated and containsfour different types of loops. To sum up these series, weuse a method called generalized random-phase approximation(RPA) elaborated on earlier (see [ 24,25]). By summing up all one- and two-loop diagrams in Sec. IV, we arrive at an expression for the longitudinal GF, the denominator of whichincludes all terms of 1 /zorder ( zis the ﬁrst coordination number of a relevant magnetic lattice). Analytic continuationof the Matsubara-type spin GFs onto the real-frequency axisdetermines the longitudinal susceptibility χ zz(q,ω)o ft h e ferrimagnet. The susceptibility is examined as a function offrequency ωand wave vector q. This analysis shows that the dynamics of longitudinal spin components corresponds to afew virtual processes of creation and annihilation of transversespin waves. The dynamics was studied and discussed indetail within an approximation of a quadratic spin wavedispersion law. The last section summarizes our main results.Some cumbersome mathematical details are expounded inAppendixes A–C. II. THE MODEL To capture the main physics, we consider the simple isotropic model for a two-sublattice ferrimagnet. In theabsence of any external inﬂuences, the atomistic spin modelis described purely by exchange interactions, given by theHeisenberg Hamiltonian: H=/summationdisplay f,gJfg/bracketleftbigg1 2/parenleftbig S+ 1fS− 2g+S− 1fS+ 2g/parenrightbig +Sz 1fSz 2g/bracketrightbigg .( 1 ) Here S1fandS2gare the spin operators on an fth and gth sites of sublattices 1 and 2, respectively, and the circular spinoperators are deﬁnite as usual, S ±=(Sx±Sy);Jfgstands for the exchange integral between spins. We will suggest thatS 1>S 2andJfg>0, i.e., the sublattices, are in antiparallel orientation. We rewrite the Hamiltonian (1) in the form H=E0+ H0+Hint.H e r e E0=−J0S1S2Nis the ground-state en- ergy ( Nis the number of magnetic unit cells); H0standsfor the Hamiltonian of the molecular ﬁeld of a standard structure: H0=−y/summationdisplay fSz 1f−x/summationdisplay gSz 2g, (2a) where y=/angbracketleftSz 2/angbracketrightJ0,x=/angbracketleftSz 1/angbracketrightJ0andJ0=Jq=0is the Fourier transform of the exchange interaction. The interaction Hamil-tonian H intis of the form Hint=/summationdisplay f,gJfg/bracketleftbigg1 2/parenleftbig S+ 1fS− 2g+S− 1fS+ 2g/parenrightbig +/parenleftbig Sz 1f−/angbracketleftbig Sz 1/angbracketrightbig/parenrightbig/parenleftbig Sz 2g−/angbracketleftbig Sz 2/angbracketrightbig/parenrightbig/bracketrightbigg . (2b) In the zeroth-order approximation of a self-consistent ﬁeld we have /angbracketleftSz 1/angbracketright(0)=b1(βy0S1),/angbracketleftSz 2/angbracketright(0)=b2(βx0S2),b(z)= SBS(z), where BS(z) stands for the Brillouin function, y0= b2J0,x0=b1J0, andβ−1=Tis temperature. In an antiferro- magnetic ground state, the mean value of a magnetic unit-cellmagnetization is equal to M=μ B(g1b1−g2b2), where μB is the Bohr magneton and g1(2)stands for the gfactor of the sublattices. As is known [ 22,23,25], within the microscopic (Green’s function method) approach, the study of a system’s longi-tudinal magnetization dynamics is based on the dynamicsusceptibility χ zz(q,ω) calculation as a function of frequency ωand momentum q, which in turn is reduced to the retarded spin GFs calculation: χzz(q,ω)→G(R)(q,ω). In our case, the calculation of the system’s longitudinal susceptibility χzz(q,ω) corresponds to the calculation of the retarded longitudinal spin GFGzz(R) tot(q,ω): χzz(q,ω)=/angbracketleftbig/angbracketleftbig/hatwideTMz tot(t)/vextendsingle/vextendsingleMz tot(0)/angbracketrightbig/angbracketrightbig/vextendsingle/vextendsingle q,ω =−μ2 B v0Gzz(R) tot(q,ω). (3) Here Mz totis azcomponent of the total magnetization Mtot=μB(g1S1+g2S2), and vostands for the volume of a primitive magnetic cell. The symbol /angbracketleft/angbracketleft· · · /angbracketright/angbracketright q,ωdenotes the Fourier transform of the trace of ρ0(···), with ρ0= exp(−βH 0)/Sp[exp(−βH 0)];/hatwideTstands for the time-ordering operator. There are theorems proving that the poles ofthe retarded GFs correspond to the natural frequencies ofmagnetization excitations that are transverse magnetizationoscillations of the spins or ordinary spin waves and longitudi-nal spin oscillations. In turn, the retarded GFs of the system canbe obtained from the temperature GFs by analytic continuationfrom the Matsubara frequencies iω nonto the real axis iωn→ ω+iδ,(δ→0) (for more details, see, e.g., [ 22,23,25]). In our case of a two-sublattice system, the total GF Gzz(R) tot(q,ω) can be reduced to four sublattice longitudinal GFs Gzz ij(q,iωn)(i,j=1,2) as follows: Gzz tot(q,iωn)=Gzz tot(q) =/angbracketleftbig/angbracketleftbig/hatwideT/parenleftbig g1δSz 1−g2δSz 2/parenrightbig/vextendsingle/vextendsingle/parenleftbig g1δSz 1−g2δSz 2/parenrightbig/angbracketrightbig =g2 1Gzz 11(q)−g1g2/bracketleftbig Gzz 12(q)+Gzz 21(q)/bracketrightbig +g2 2Gzz 2(q), (4) 054434-2LONGITUDINAL SPIN DYNAMICS IN FERRIMAGNETS: . . . PHYSICAL REVIEW B 94, 054434 (2016) where δSz i=Sz i−/angbracketleftSz i/angbracketrightandGzz ij=/angbracketleft /angbracketleft/hatwideTδSz i\|δSz j/angbracketright/angbracketright. Here and below we use the notation q={q,iωn}, where qstands for the momentum and the Matsubara frequency iωn=i2πnT (n=0,±1,±2,...). Thus, the calculation of the dynamic susceptibility χzz(q,ω) reduces to the calculation of the sublattice longitudinal GFs Gzz ij(q,iωn). III. GREEN’S FUNCTIONS OF LONGITUDINAL SPIN COMPONENTS To calculate the sublattice GFs we use the Larkin equation derived earlier in the framework of the diagrammatic techniquefor spin operators. Without dwelling on the details of summingup the diagram series (Refs. [ 22,23,25] contain technical details concerning the construction of the spin diagramtechnique for the Heisenberg magnet), we present here theﬁnal analytic results. One can show that the graph series fortheG zz ij(q) functions of a two-sublattice ferrimagnet can be presented analytically in the form Gzz 11(q)=/Sigma1z 11(q)/braceleftbig/bracketleftbig 1−Jq/Sigma1z 12(q)/bracketrightbig/bracketleftbig 1−Jq/Sigma1z 21(q)/bracketrightbig −J2 q/Sigma1z 11(q)/Sigma1z 22(q)/bracerightbig−1(5) forGzz 11(q) and Gzz 21(q)=/braceleftbig/bracketleftbig 1−Jq/Sigma1z 12(q)/bracketrightbig /Sigma1z 21(q)+Jq/Sigma1z 11(q)/Sigma1z 22(q)/bracerightbig ×/braceleftbig/bracketleftbig 1−Jq/Sigma1z 12(q)/bracketrightbig/bracketleftbig 1−Jq/Sigma1z 21(q)/bracketrightbig −J2 q/Sigma1z 11(q)/Sigma1z 22(q)/bracerightbig−1(6) forGzz 21(q). One can obtain expressions for the functions Gzz 22(q) and Gzz 12(q) from Eqs. ( 5) and ( 6), respectively, by substituting 1 →2→1. In terms of the diagrammatic technique, the quantity /Sigma1z ij(q) is called the irreducible (by Larkin’s method of isolating the irreducible diagrams) parts.Note that the irreducibility is understood here in the sense that/Sigma1 z ij(q) is represented by the collection of all diagrams from the series for Gzz ij(q) that cannot be cut across a line of interaction Jq. Summarizing up the results, for the GF Gzz tot(q) we obtain the following general expression: Gzz tot(q)=N(q) D(q). (7) Here we specify the numerator as N(q)=g2 1/Sigma1z 11(q)−g1g2/bracketleftbig /Sigma1z 12(q)+/Sigma1z 21(q) −2Jq/Sigma1z 12(q)/Sigma1z 21(q)+2Jq/Sigma1z 11(q)/Sigma1z 22(q)/bracketrightbig +g2 2/Sigma1z 22(q) (8) and the denominator as D(q)=/bracketleftbig 1−Jq/Sigma1z 12(q)/bracketrightbig/bracketleftbig 1−Jq/Sigma1z 21(q)/bracketrightbig −J2 q/Sigma1z 11(q)/Sigma1z 22(q). (9) By calculating the longitudinal GF irreducible parts /Sigma1z ij(q) we will use the approximation when all ladder diagrams withantiparallel lines have been summed. Typically, such summingcorresponds to the so-called RPA and rather well describesthe ground state and dynamics of magnetic systems (see,for example, Refs. [ 24,25]). This approximation was used,in particular, by Izyumov at al. [21] to study the longitudinal spin dynamics in the Heisenberg ferromagnet. It can be shown that in the case of the zeroth order of a large interaction radius (or of the zeroth order of the parameter1/z) one obtains /Sigma1 z 11(q,iωn)=δn,0b/prime 1,/Sigma1z 22(q,iωn)=δn,0b/prime 2, /Sigma1z 12(q,iωn)=/Sigma1z 21(q,iωn)=0 (here b/primestands for the ﬁrst derivative of the Brillouin function and δn,0=δωn,0is the Kro- necker symbol for the corresponding frequency difference).Within this approximation we get [ 20] Gzz(0) 11(q,iωn)=b/prime 1 1−(βJq)2b/prime 1b/prime 2δn,0, Gzz(0) 22(q,iωn)=b/prime 2 1−(βJq)2b/prime 1b/prime 2δn,0, Gzz(0) 12(q,iωn)=Gzz(0) 21(q,iωn)=b/prime 1b/prime 2 1−(βJq)2b/prime 1b/prime 2δn,0. Note that within this approximation we deal with static ﬂuctuations of the longitudinal spin components, which arecharacterized by the Brillouin function derivatives and whichare responsible for the distinction between the isolated andisothermal susceptibilities of the system (see below). Weemphasize in this relation that the traditional representationof spin operators by Bose operators (e.g., the Holstein-Primakov or Dyson-Maleev representations) only accountsfor the dynamic ﬂuctuations in the magnetization, i.e., areduction in the magnetization of the sublattice owing tothermal excitation of spin waves. Static ﬂuctuations of thelongitudinal components of the spin are entirely neglected interms of these representations. This circumstance, in particular,dictates the need to use spin operator diagram techniquesin order to correctly describe properties of the systems atﬁnite temperatures. The difference in the nature of Bose andspin operators takes on special signiﬁcance if we considermagnetic system behavior at high enough temperatures. Inthis temperature region the basic fact of the difference in thenumber of states [ﬁnite (2 S+1) for spin operator states and an inﬁnite number of states for Bose operators] plays a crucialrole already in dynamic. Temperature renormalization of the spectrum and damping of spin wave excitations owing to the scattering of spinwaves on (longitudinal) ﬂuctuations in the magnetization bothappear in the ﬁrst approximation with respect to the reciprocalof the interaction radius. Note that the “standard” dampingof spin wave oscillations owing to their scattering on oneanother occurs only in the next (second) approximation ofthe perturbation theory. (Spin wave relaxation in rare-earthferrite garnets has been studied previously [ 26].) To restore the dynamical characteristics, we need to calculate the irreducibleparts/Sigma1 z ij(q) within a high-order approximation. The graphs for the “uncuttable” parts /Sigma1z ij(q)o ft h e next order (one-loop order) are shown in Fig. 1.T h e thick lines represent the “dressed” transverse GFs Gij(q)= −1/2/angbracketleft/angbracketleft/hatwideTS+ ig\|S− jf/angbracketright/angbracketright\|q,ω, and the open points indicate vertices corresponding to the operators Sz i(i=1,2). The dressed transverse GFs (thick lines) of spin wave lines are a resultof the summation graphical series for the transverse GFs in theHartree-Fock approximation shown in Fig. 2(here the wavy 054434-3V . N. KRIVORUCHKO PHYSICAL REVIEW B 94, 054434 (2016) FIG. 1. One-loop order diagrams for irreducible parts of the longitudinal spin Green’s functions. Here and in Figs. 2,3,a n d 4 external hollow vertices correspond to spin operators Sz i(i=1,2); the thick solid line represents the dressed Green’s function G11(q); the thick dashed line describes the dressed Green’s function G22(q), and the double solid line illustrates the dressed G12(q)o rG21(q) Green’s function. Only the diagrams describing the Kubo susceptibility areshown explicitly. line is the graphical representation of the interaction Jq). The analytical solution of the graphical equations for the transverseGFs is given in Appendix A. In accordance with the rules for the diagram techniques, a block encompassing Noperators S z is to be compared to the ( N−1)th derivative of the function b(z)o ft h ef o r m ∼δn,0b(N−1)(z). The analytical expression of these diagrams contains a singular discrete frequency part∼δ n,0and thus does not depend on the thermodynamic time (see Refs. [ 22,23] for more details). Let us recall here that the singular contribution to the temperature GF arises due to thedistinction between the isolated and isothermal susceptibilitiesof the system [ 27,28]. As was shown in these studies, the intensity of the singular contribution coincides with the dis-tinction between the isothermal and isolated susceptibilities atthe zeroth frequency. In accordance with the general analysis ofdifferent susceptibilities [ 27,28] the distinction between them points to the nonergodicity of the system. We are interestedin the Kubo (or isolated) susceptibility of the system derivedfrom the quantity G zz tot(q,iωn) by analytic continuation from FIG. 2. Graphical representation of the system of equations for dressed transverse (a) G11(q)a n dG21(q)a n d( b ) G22(q)a n dG12(q) Green’s functions. The thin solid line represents the undressed (initial)Green’s function K 11(q); the thin dashed line describes the initial (undressed) Green’s function K22(q). Here and in Fig. 4a wavy line corresponds to the interaction Jq. the Matsubara frequencies onto the real axis iωn→ω+iδ (δ→0). For this reason the diagrams which do not depend on the thermodynamic time (i.e., ∼δn,0)i nF i g . 1(and in Fig. 3 below) are not shown. Returning to the graphs in Fig. 1and following the rules of the diagram techniques, one can obtain their analyticexpressions. For the diagrams in Fig. 1we write the result in the form /Sigma1 z 11(q)=/Pi1(q)+(∼δn,0),/Sigma1z 12(q)=/Sigma1z 21(q)= /Phi1(q)+(∼δn,0), and/Sigma1z 22(q)=B(q)+(∼δn,0). Here ( ∼δn,0) stands for the analytic expressions of the diagrams which donot contribute to the isolated system susceptibilities (as alreadymentioned, these diagrams are not shown in Fig. 1explicitly). Then the one-loop graph’s contribution to the longitudinalGFs is /Pi1(q)=N −1β−1/summationtext pG11(p)G11(p−q), (10) /Phi1(q)=N−1β−1/summationtext pG12(p)G21(p−q), (11) B(q)=N−1β−1/summationtext pG22(p)G22(p−q). (12) Explicit expressions for the transversal GFs Gij(q)a r eg i v e n by Eqs. ( A3)–(A5). All possible two-loop diagrams related to the Kubo (isolated) susceptibility are depicted in Fig. 3. There two external vertices of the GFs (1 and 2) are represented byopen points with incoming and outgoing Green’s lines. Otherarrangements of external vertices do not exist. The hatchedsquares here represent graphically the effective four-point 054434-4LONGITUDINAL SPIN DYNAMICS IN FERRIMAGNETS: . . . PHYSICAL REVIEW B 94, 054434 (2016) FIG. 3. Two-loop order diagrams for irreducible parts of the longitudinal spin Green’s functions. The hatched squares representgraphically the effective four-point vertexes /Gamma1 ii,ij(k1,k2\|k1−q,k 2+ q). Only the diagrams describing the Kubo susceptibility are shown. vertexes /Gamma1ii,ij(k1,k2\|k1−q,k 2+q). The calculation of these vertexes is a key point now to proceed further. The equations for the vertexes /Gamma1ii,ij(k1,k2\|k1−q,k 2+q) are presented graphically in Fig. 4. Here, as in Fig. 2, the wavy line is the graphical representation of the interaction Jq.T h e graphical equation for the vortex /Gamma111,12(k1,k2\|k1−q,k 2+q) shown in Fig. 4(a) corresponds to the analytical expression of the following form: /Gamma111,12(k1,k2\|k1−q,k 2+q) =Jk2+q+β−1N−1/summationtext k3Jk3+qG11(k3)G21(k3+q) ×/Gamma111,12(k3+q,k 2\|k3,k2+q). This integral equation can be transformed into an algebraic one. To this end, let us ﬁrst multiply both sides by Jk1G11(k1− q)G21(k1) and then sum up over k1. This results in a linear FIG. 4. Graphical representation of the equation for the ef- fective four-point vortex: (a) /Gamma111,12(k1,k2\|k1−q,k 2+q)a n d( b ) /Gamma122,21(k1,k2\|k1−q,k 2+q). equation with the solution /Gamma111,12(k1,k2\|k1−q,k 2+q)=Jk2+q 1−Q(q), where Q(q)=N−1β−1/summationtext pG11(p)G21(p−q). (13) The graphical equation for another vortex /Gamma122,21(k1,k2\|k1− q,k 2+q) shown in Fig. 4(b) leads to the analytical expression /Gamma122,21(k1,k2\|k1−q,k 2+q) =Jk2+q+β−1N−1/summationtext k3Jk3+qG22(k3)G12(k3+q) ×/Gamma122,21(k3+q,k 2\|k3,k2+q). This integral equation can also be rewritten in an algebraic form. Like in the previous case, we multiplied both sides ofthe equation by J k1G22(k1−q)G12(k1) and then summed up overk1. The solution can be presented as /Gamma122,21(k1,k2\|k1−q,k 2+q)=Jk2+q 1−/Lambda1(q), where /Lambda1(q)=N−1β−1/summationtext pG22(p)G12(p−q). (14) Using the obtained expressions for the vertexes and summing up all the contributions, we found the following analyticexpressions for the two-loop diagrams shown in Fig. 3: /Sigma1z(2) 11(q)=2Q(q)/bracketleftbigg/Pi1(q) 1−Q(q)+/Phi1(q) 1−/Lambda1(q)/bracketrightbigg , /Sigma1z(2) 12(q)=/Sigma1z(2) 21(q) =Q(q)/Phi1(q)+/Pi1(q)/Lambda1(q) 1−Q(q)+B(q)Q(q)+/Phi1(q)/Lambda1(q) 1−/Lambda1(q), /Sigma1z(2) 22(q)=2/Lambda1(q)/bracketleftbigg/Phi1(q) 1−Q(q)+B(q) 1−/Lambda1(q)/bracketrightbigg . 054434-5V . N. KRIVORUCHKO PHYSICAL REVIEW B 94, 054434 (2016) The total contribution of the one- and two-loop graphs to the irreducible elements /Sigma1z ij(q) can be written as follows: /Sigma1z 11(q)=/Pi1(q)1+Q(q) 1−Q(q)+2Q(q)/Phi1(q) 1−/Lambda1(q), (15) /Sigma1z 12(q)=/Sigma1z 21(q)=/Phi1(q)+Q(q)/Phi1(q)+/Pi1(q)/Lambda1(q) 1−Q(q) +B(q)Q(q)+/Phi1(q)/Lambda1(q) 1−/Lambda1(q), (16) /Sigma1z 22(q)=B(q)1+/Lambda1(q) 1−/Lambda1(q)+2/Lambda1(q)/Phi1(q) 1−Q(q). (17) Thus, to ﬁnd the isolated (Kubo) susceptibility χzz(q,ω) it is necessary to calculate ﬁve functions, /Pi1(q),/Phi1(q), B(q),Q(q), and /Lambda1(q), given by formulas ( 10)–(14), re- spectively. After simple but cumbersome calculations, onecan obtain the analytical expressions for these functions.They are presented in the explicit form in Appendix B.IV . LONGITUDINAL SPIN DYNAMICS IN FERRIMAGNET The excitation spectrum of the system is determined by the poles of the analytic continuation iωn→ω+iδof the temperature GF Gzz tot(q,iωn). The real part of the pole is the energy of a quasiparticle excitation, while the imaginary partcharacterizes broadening of the energy level, i.e., a quasi-particle damping. Thus, we should investigate whether theG zz tot(q,iωn) denominator has solutions that would determine the longitudinal wave excitations. To this end, let us examineEq. ( 9) more closely. First of all, before proceeding to an estimation of this expression in some limiting cases, letus qualitatively analyze the origin of the longitudinal spinexcitations in the system under consideration. It is informativeto start with the one-loop order approximation. In this approachthe denominator ( 9) can be rewritten in the form D(q)=[1−J q/Phi1(q)]2−J2 q/Pi1(q)B(q). (18) After summing up over the discrete Matsubara frequency for the function /Phi1(q)=/Phi1(q,iωn), Eq. ( 11), we obtained /Phi1(q)=b2 1b2 2 N/summationtext pJp ε1p+ε2pJp−q ε1p−q+ε2p−q ×/braceleftbiggn1(ε1p)−n1(ε1p−q) iωq−ε1p+ε1p−q−n2(ε2p)−n2(ε2p−q) iωq+ε2p−ε2p−q+1+n1(ε1p−q)+n2(ε1p) iωq+ε2p+ε1p−q−1+n1(ε1p)+n2(ε2p−q) iωq−ε1p−ε2p−q/bracerightbigg .(19) [Here and below we introduce the designation εi(q)=εiq.] The expressions for /Pi1(q) andB(q), Eqs. ( 10) and ( 12), are represented by Eqs. ( B1) and ( B2) in Appendix B, and as one can see, they have similar structures. The analysis of Eq. ( 19), as well as Eqs. ( B1) and ( B2), shows that the dynamics of longitudinal spin components isdue to a few processes of virtual creation and annihilation of transverse spin wave modes. Namely, the ﬁrst channel, the ﬁrst item in the braces on the right-hand side (rhs) of Eq. ( 19), represents the processes of creation and annihilation of twospin waves with energies ε 1pandε1p−q, which correspond to in-phase sublattice magnetization precession. This channel iscontrolled by the occupation factor n 1(ε1p), which makes the spin waves with p∼0 the dominant ones. Simple comparison reveals that this channel is a direct analog of the magnetizationlongitudinal dynamics in a ferromagnet found by Izyumovet al. [21]. There is also the second channel, the second term on the r h so fE q .( 19), with characteristic energy at ω(q)=ε 2p− ε2p−q. It corresponds to virtual creation and annihilation of exchange spin waves with antiphase precession of sublatticemagnetization. This channel is also controlled by the relatedoccupation factor determined through the Bose distributionfunctions n 2(ε2p), which makes the spin waves with p∼0t h e dominant ones. It is not difﬁcult to see that the last two items in the braces in Eq. ( 19) present the third channel of longitudinal excita- tions. Namely, there is a two-spin-wave creation/annihilationprocess at frequency ω(q)=ε 1p+ε2p−qwhich corresponds to creation or annihilation of one acoustic and one exchangetransverse mode. This channel remains in force even in theabsence of thermal excitations, i.e., when n 1(ε1p)∼0 and/or n2(ε2p)∼0. All listed mechanisms of the longitudinal spin excitations remain valid in the high-loop approximation, too. Indeed,calculation of the functions Q(q) and /Lambda1(q) leads us to the ﬁnal expressions given by Eqs. ( B3) and ( B4), respec- tively, in Appendix B. Analyzing these functions, we see that all conclusions made above remain in force. Thus,the structure of the denominator D(q,ω) and, in particular, the equation Re D(q,ω)=0 that determines the dispersion law point to a strong renormalization of longitudinal spin excitation frequency due to a few virtual transverse spin wavecreation/annihilation processes. Accordingly, as will be shownbelow, the energy of longitudinal spin vibration strongly differsfrom a simple algebraic sum of acoustic and/or exchange modeenergy. Let us now examine Eq. ( 18) more closely. The main physics can be captured in a long-wave limit ( ak)/lessmuch1. Within this approximation the energy of transverse spin waveexcitations reads [see Eqs. ( A6) and ( A7)] ε 1k=D(ak)2,D =2b1b2 b1−b2J0, (20) and ε2k=(b1−b2)J0+D(ak)2(21) for in-phase and antiphase precession of sublattice magnetiza- tion, respectively. Here we used the quadratic expansion whenevaluating the quantity J 0−Jk≈zJ(ak)2=J0(ak)2(herez is the number of nearest neighbors, and astands for the lattice spacing). 054434-6LONGITUDINAL SPIN DYNAMICS IN FERRIMAGNETS: . . . PHYSICAL REVIEW B 94, 054434 (2016) As already mentioned, the poles of χzz(q,ω) deﬁne the energy of longitudinal excitations in the system. Due torather complex dependence of the function on frequency, weshall consider real and imaginary parts of D(q,ω) only in some limiting cases. Namely, we investigate this functionnear the singularities of /Phi1(q),/Pi1(q), and B(q). That is when one of the equations ω(q)=ε 1p−ε1p−q,o rω(q)= ε2p−ε2p−q,o rω(q)=ε1p+ε2p−qis fulﬁlled, i.e., when the related virtual two-spin-wave creation/annihilation processesare most effective. For reasons which will be explainedbelow, we will call the branch of longitudinal excitations dueto creation and annihilation of the same (both acoustic orboth exchange) spin waves the “acoustic” longitudinal spinexcitations; we will call the branch of longitudinal excitationsdue to creation and annihilation of different (one acousticand one exchange) transverse spin waves the “exchange”longitudinal spin excitations. A. Acoustic longitudinal spin excitations Let us start with the singularity in χzz(q,ω) due to virtual processes of creation and annihilation of two spinwaves with in-phase or antiphase sublattice magnetizationprecession (the ﬁrst and second channels of longitudinalexcitations). Quantitative analysis given in Appendix Cshows that these processes will provide the main contribution in thedenominator D(q,ω) if the parameter a ±=q 2±ω 2Dq(22) is small enough, i.e., \|a±\|/lessmuch 1. (Hereinafter we suppose that the conventional analytic continuation is made, i.e., iωn→ ω+iδ.) In this approximation, i.e., ignoring the third channel, after simple algebra, the real and imaginary parts of expression(18) can be written as ReD(q,ω)≈1−J qb2 1b2 2 (b1−b2)2{2R e (λ+ 1+λ− 1−λ+ 2−λ− 2) −Jq(b1+b2)2[Re(λ+ 1+λ− 1)R e (λ+ 2+λ− 2) −Im(λ+ 1+λ− 1)I m (λ+ 2+λ− 2)]}, (23) ImD(q,ω)≈−Jqb2 1b2 2 (b1−b2)2{2I m (λ+ 1+λ− 1−λ+ 2−λ− 2) −Jq(b1+b2)2[Re(λ+ 1+λ− 1)I m (λ+ 2+λ− 2) +Im(λ+ 1+λ− 1)R e (λ+ 2+λ− 2)]}. (24) The explicit form of the real and imaginary parts of additional functions λ± i=λ± i(q,ω)[ (C1)–(C3)] introduced here are presented in Appendix C. After simple but cumbersome calculations (some details can be found in Appendix C), we obtain for the leading part of the longitudinal spinsusceptibility χ zz(q,ω)∼i(b1−b2)4 b2 1b2 2/parenleftbiggJ0 Jq/parenrightbigg2N(q,ω)/parenleftbig ω2−/Omega12res/parenrightbig/parenleftbig ω2−/Omega12 dif/parenrightbig, (25)where the characteristic frequencies are /Omega12 res(dif)/(Dq)2=6+iA(T) ±/braceleftbigg [6+iA(T)]2−3π(b1−b2)2 b1b2[π −i6(b1+b2)]arctg/parenleftbigg2b1b2 (b1−b2)2/parenrightbigg/bracerightbigg1/2 , (26) where A(T)=π(b1+b2)/bracketleftbigg 1+3(b1−b2)2 4b1b2arctg/parenleftbigg2b1b2 (b1−b2)2/parenrightbigg/bracketrightbigg . In particular, in a temperature region near the Curie one, when b1≈b2and is small, the expressions for longitudinal vibrations acquire a simple form, /Omega1res≈3.46Dq+iγ(q,T), (27) γ(q,T)∼3 2π2(b1+b2)2 b1b2Dq. For the diffusive mode, one obtains /Omega1dif≈0.8π(b1−b2)√b1b2Dq{a(T)−ic(T)}1/2, (28) where a(T)=6π1−(b1+b2)2 36+π2(b1+b2)2, c(T)=(b1+b2)36+π2 36+π2(b1+b2)2. Thus, in the ( q,ω) region where the condition \|a±\|/lessmuch 1 is fulﬁlled, the function χzz(q,ω) possesses two types of resonances. There is a peak which characterizes a preces-sionlike motion with the frequency ±/Omega1 res∼Dqand damping γ(q)∼Dq; both these functions linearly depend on the wave vector (the acoustic branch longitudinal spin excitations). Thisallows us to afﬁrm that in the system the wavelike vibrationsof spin longitudinal components exist, although with strongattenuation. At the same time, there is also a quasidiffusive poleat±i/Omega1 dif, i.e., the quasirelaxation mode connected to diffusion of longitudinal ﬂuctuations, which forms the central peak inthe spectral function 1 ωImχzz(q,ω). Thus, in the frequency and wave vector region under consideration, Eq. ( 22), the ferrimagnet spectral function behavior is similar to those inthe case of a ferromagnet [ 21]. The susceptibility and the longitudinal spin ﬂuctuations at temperature close to T Ccan be directly determined by inelastic scattering of polarized neutrons. In these experiments,a spin polarization analysis of the data gives the possibility todistinguish contributions from longitudinal and transverse spinmodes (for more details see, e.g., [ 29] and references therein). Note here, to avoid confusion, that applicability the results obtained in the ( q,ω) area for the frequency is determined by the conditions ω(q)≈ε 1p−ε1p−qandω(q)≈ε2p−ε2p−q, while a restriction on the wave vector is conditioned byproximity to the hydrodynamic regime, where the physics ofthe magnetization dynamics is determined by the magnetiza-tion conservation laws [ 30]. As is already known, the RPA 054434-7V . N. KRIVORUCHKO PHYSICAL REVIEW B 94, 054434 (2016) results remain valid beyond the hydrodynamic regime, i.e., when the wave vector is larger than the inverse correlationlength ξ:q> 1/ξ∼(1−T/T C)1/2, and, of course, beyond the critical region (for more details see, e.g., Refs. [ 21,25]). (A second-order phase transition takes place in the system ata temperature T C=1/3[S1S2(S1+1)(S2+1)]1/2J0[20].) B. Exchange longitudinal spin excitations We are now in a position to consider a central result of the paper, that is, longitudinal excitations due to virtual creationand annihilation transverse spin waves of different branches(the third channel of longitudinal excitations). Quantitativeanalysis (see Appendix Cfor details) indicates that these virtual processes will cause a singularity in χ zz(q,ω)i ft h e parameter b±=q 2+/bracketleftbiggq2 4+(b1−b2)J0±ω 2D/bracketrightbigg1/2 (29) is small enough, i.e., \|b±\|/lessmuch 1. In this approximation, after simple algebra, the leading part of the denominator ( 18) that determines the dispersion law and damping of longitudinalspin excitations can be written as D(q,ω q)≈1−2Jq/Phi1(q,ωq). (30)Calculations show that now the leading part of the longitudinal spin susceptibility reads χzz(q,ω)∼(b1−b2)4 2b2 1b2 2J0 JqN(q,ω) ±ω+/Omega1exc+iγexc. (31) The quantity /Omega1exc=/Omega1exc(q,T) should be considered the fre- quencies of collective vibrations of magnetization longitudinalcomponents, which in this ( q,ω) region are described by the following expression (see Appendix C): /Omega1 exc(q,T)≈(b1−b2)J0/bracketleftbigg 1+2l nT (b1−b2)J0+π2 2+f(T)q/bracketrightbigg , (32) where f(T)≈4J2 0 TJq+ln/bracketleftbigg(b1+b2)J0+D TDq/bracketrightbigg −2/parenleftbigg 1+3D 4T/parenrightbigg . (33) By analogy to the terminology for the transverse spin wave oscillations, we will call this type of longitudinal excitationthe exchange longitudinal spin excitations. Note that the ex-change longitudinal mode is energetically above the exchangemode of the transverse spin wave ( 21) at the same temperature and wave vector, and liner depends on q. For the damping of the exchange longitudinal mode, we obtained γ(q,/Omega1 exc)∼2πb1b2J2 0 T/braceleftbigg 1−κ2 0+T Dln/bracketleftbigg(b1−b2)J0+D (b1−b2)J0+Dκ2 0/bracketrightbigg +T Dln/parenleftbigg κ2 0/bracketleftbigg4b1b2 (b1−b2)2/bracketrightbigg2/parenrightbigg/bracerightbigg , (34) where κ2 0=(b1−b2)2 4b1b2/bracketleftbigg 2l nT (b1−b2)J0+π2 2+f(T)q/bracketrightbigg .(35) Thus, the ratio γ(q,/Omega1 exc)//Omega1 exc(q,T) is approximately γ(q,/Omega1 exc)//Omega1 exc(q,T)∼b1b2 b1−b2J0 T, and at low temperature T< J0it is large enough. But at T∼TCb1/lessmuchS1,b2/lessmuchS2, andγ(q,/Omega1 exc)//Omega1 exc(q,T)∼(b1/S1)(b2/S2) b1−b2/lessmuch1i ss m a l l e rt h a n unity. This allows us to afﬁrm that in a ferrimagnet wavelikeexcitations of longitudinal components of magnetization exist,although with strong attenuation at low temperature. As follows from direct examination of the functions Q(q) [Eq. ( B3)] and /Lambda1(q)[ E q .( B4)], the physical mechanism of the spin longitudinal excitations remains valid in the highapproximation, too. We investigated these functions in thefrequency ωand momentum qdomains where parameters (22) and ( 29) are small enough and found that the account of the two-loop diagrams has not caused principal correctionsto the energy and damping of longitudinal spin excitations.(The related analysis will be published elsewhere.) V . DISCUSSION It is now commonly accepted that the ultrafast magneti- zation process proceeds with several important characteristictime scales [ 1,31–33]: (i) a femtosecond demagnetization, (ii) a picosecond recovery, and (iii) a nanosecond magneti-zation precession and relaxation, traditionally characterized by the ferromagnetic resonance frequency and the Landau-Lifshitz-Gilbert (LLG) damping parameter. It is also generallyrecognized that the physics of magnetization changes onfemtosecond time scales requires understanding the role of dif-ferent subsystems (photons, phonons, electrons) in the angularmomentum transfer [ 1,31,32], too. Concerning multisublattice magnetic materials, an additional question arises about theeffect which the magnetization or angular momentum com-pensation point plays in ultrafast longitudinal magnetizationdynamics (see, e.g., [ 7,8] and references therein). The ﬁrst attempts to describe longitudinal magnetization dynamics in two-sublattice systems have been proposed recently [ 13,14,16,17]. Based on the existing experimental results and from a general point of view, it was suggestedthat the longitudinal relaxation and the transverse relaxation(the LLG damping) are independent quantities. That is, asin the case of nanoscale magnetism (see reviews [ 34,35] and references therein), there is a longitudinal relaxationtime which ﬁts a direct path to the thermal bath and a so-called transverse time which represents scattering into the transverse magnetization components. The main featureof the phenomenological dynamic equations proposed thatmade them suitable for the ultrafast magnetization dynamicsis the presence of a longitudinal relaxation term comingfrom the strong exchange interaction between spins. Sincethe exchange ﬁelds are large (10–100 T), the corresponding 054434-8LONGITUDINAL SPIN DYNAMICS IN FERRIMAGNETS: . . . PHYSICAL REVIEW B 94, 054434 (2016) characteristic longitudinal relaxation time scale is of the order of 10–100 fs and thus manifests itself in the ultrafast processes. On the other hand, it is obvious that for exhaustive understanding of the underlying physics of ultrafast magne-tization dynamics a microscopic description of longitudinalmagnetization dynamics in the equilibrium state of the systemis strongly required. Distinctive features of the longitudinaldynamics in the multisublattice magnet equilibrium state haveto be a datum point in attempts to understand the dynamics of aferrimagnet after femtosecond laser pulse heating far beyondan equilibrium state. Keeping in mind that the Heisenbergmodel deals exclusively with spin degrees of freedom, let usmake some estimation concerning the suggestions made in thephenomenological models with the same approximation. In the models in Refs. [ 16,17] the authors suggest that the dynamics of the length of the sublattice magnetizationsis pertinent to the time scale of the exchange interaction andthat on this time scale the conventional transverse dynamics ofthe angular momentum is negligible. Particularly, in this ap-proximation longitudinal exchange relaxation in magnets withonly one sublattice is not possible. However, following ourconsideration, in two-sublattice magnets there are additionalequal-in-value channels of the longitudinal mode relaxationby which energy is scattered into the transverse magnetizationcomponent. Note also that according to Izyumov et al. [21], in one-sublattice magnets a longitudinal spin mode relaxes dueto a virtual process of creation and annihilation of ordinaryspin waves, too. In Ref. [ 11] the authors, using conventional suggestions about the spin wave spectrum in a ferrimagnetic system, discuss the role which the magnetization or angular momentum compensation point plays in thermally induced magnetizationswitching. The authors consider the classical spin Heisenbergmodel, and thus, the thermal equilibrium distribution of thespin ﬂuctuations rests on the classical limit (see the Supple-mentary Information in [ 11]). Based on these suggestions, the authors conclude that the switching is caused by the excitationof two-magnon bound states. However, the microscopic calculation using the diagram technique for spin operators [ 36] shows that near the angular momentum compensation point T Lthe system behavior is not the classical one. In particular, the energy of the in-phase precession of the magnetizations of the sublatticesbecomes higher than that of the out-of-phase precession; nearT Lﬂuctuations in the weak sublattice magnetization have a signiﬁcant inﬂuence on the resonance properties of thesystem, with different temperature behaviors of the in-phaseand out-of-phase precessions of the sublattice magnetization.Also, the occupation numbers for the acoustic magnonsbecome large, and a ﬁnite number of spin operator states(2S+1) takes on special signiﬁcance. In other words, the difference in quantum nature of Bose and spin operators isfundamental for understanding the magnetization dynamics inthe neighborhood of the compensation points, too. Based on our results, thermally induced magnetization switching most likely comes from emission/absorption of twotransverse spin waves which are not in bound states. However,we stress that in this paper we consider a ferrimagnet without acompensation point. A ferrimagnetic system with compensa-tion points requires special consideration, in particular becauseat high enough temperature a weak sublattice dynamics is, in fact, a paramagnetic precession of magnetization in theexchange ﬁeld of a strong sublattice (for more details see [ 36] and reference therein). Note in this connection that the case ofa compensated antiferromagnet ( S 1=S2,g1=g2) also calls for a special analysis in view of the strong dependence of itsproperties on the anisotropic interactions. In conclusion, in this paper, in the framework of a quantum- mechanical approach, the general expression for longitudinalspin susceptibility χ zz(q,ω) of a two-sublattice Heisenberg ferrimagnet was obtained with the aim to overcome limitationstypical of the phenomenological approaches. Our microscopicanalysis utilizes the diagram techniques for the spin operators,which formally yield analytic expressions that are valid overthe entire temperature range of the magnetically ordered stateof the system. Namely, the results are applicable in the ( q,ω) space beyond both hydrodynamical and critical regimes. Strong renormalization of the magnetization longitudinal vibration due to a few channels of virtual creation and anni-hilation of transverse spin waves has been found. Videlicet,there is a process of creation and annihilation of two spin(acoustic) waves at frequencies ω(q)=(ε 1k−ε1k±q) which corresponds to in-phase sublattice magnetization precessionand is in close analogy to a ferromagnet case. There is alsoa channel of two spin wave excitations with the energiesε 2kandε2k±q, which correspond to out-of-phase sublattice magnetization precession (exchange spin waves). The thirdchannel is a two-spin-wave creation/annihilation process atfrequency ω(q)=(ε 1k+ε2k±q). (In all these processes the wave vector kis a variable.) The ﬁrst two channels are controlled by the occupation factor determined through thespin wave Bose distribution function. However, the processesof creation or annihilation of one acoustic and one exchangemode (the third channel) remain effective even in the absenceof transverse excitations, i.e., when n(ε k)→0, and, in our opinion, most likely provide the main contribution to thethermally induced magnetization reversal. We have shown that the spectrum of longitudinal excitations consists of a quasirelaxation mode forming a central peakinχ zz(q,ω) and two (acoustic and exchange) precessionlike modes. As the main result, it is predicted that both acousticand exchange longitudinal excitations are energetically abovesimilar modes of transverse spin waves at the same temperatureand wave vector. The existence of such a high-energy exchangelongitudinal mode reveals the possibility for a new formof excitation behavior in ferrimagnetic materials. Also, ouranalysis indicates that in a temperature region near the Curieone the main contribution to the longitudinal magnetizationrelaxation comes from the high-frequency spin waves. Thisprocess occurs due to a strong exchange ﬁeld. As a result, thelongitudinal relaxation time (the inverse longitudinal relax-ation rate) is much faster than transverse spin wave damping. The existing experimental results indicate that in a ferri- magnetic system the ultrafast magnetization reversal occursdue to intrinsic material properties, but so far the microscopicmechanism responsible for the reversal has not been identiﬁed.We hope that the results obtained in this paper within a consis-tent microscopic theory will be important for understanding thephysics of nonequilibrium magnetic dynamics under the effectof ultrafast laser pulses in multisublattice magnetic materials. 054434-9V . N. KRIVORUCHKO PHYSICAL REVIEW B 94, 054434 (2016) ACKNOWLEDGMENTS The author is grateful for valuable discussions with V . G. Bar’yakhtar and B. A. Ivanov. This work is partly supported bythe European Union Horizon 2020 Research and InnovationProgramme under Marie Sklodowska-Curie Grant AgreementNo. 644348 (MagIC). APPENDIX A: TRANSVERSE GFS IN THE HARTREE-FOCK APPROXIMATION To calculate the transverse GFs, Gij(q)= −1/2/angbracketleft/angbracketleft/hatwideTS+ ig\|S− jf/angbracketright/angbracketright\|q,ω(i,j=1,2), we use the Larkin equation derived earlier in the framework of the diagrammatictechnique for spin operators (see, for instance, [ 22,23]). In that case, each connected diagram for the transverse GFscan be represented in the form of single-cell blocks /Sigma1 ij(q) joined by the interaction lines Jq(the Fourier transform of the exchange interaction). It can be shown that the total(inﬁnite) graph series for transverse GFs of a two-sublatticeferrimagnet obeys the system of equations G ii(q)=/Sigma1ii(q)+/Sigma1ii(q)JqGji(q), Gji(q)=/Sigma1ji(q)+/Sigma1jj(q)JqGii(q). In terms of the diagrammatic technique, the quantity /Sigma1ij(q) should be called the part uncuttable across a line of interactionJ q. All “cuttable” parts are compressed into the second term on the right-hand sides of these equations. Dealing with transversespin wave excitations, we are interested in summation ofgraphical series for the GFs in the Hartree-Fock approxima-tion. In this approximation, a graphical representation of thesystems of equations for the transverse G 11(q) and G12(q) and also G22(q) and G21(q) GFs are shown in Figs. 2(a) and2(b), respectively. Here thick (thin) lines represent dressed [undressed, Kii(q)] GFs; the wavy line corresponds to the interaction Jq. The open points indicate vertices corresponding to the operators Sz i(i=1,2). These equations in the Fourier transformed form read G11(q)=b1K11(iωn)+b1K11(iωn)JqG21(q), (A1) G21(q)=−b2K22(iωn)JqG11(q). (A2) The system of equations for the next pair of coupled functions, G22(q) andG12(q), possesses a very similar structure. HereK11(iωn)=1/(iωn+y0) and K22(iωn)=1/(iωn−x0)a r e (undressed) GFs in the mean-ﬁeld approximation. We usedthe notations b 1≡b1(βyS 1)=/angbracketleftSz 1/angbracketright(0)andb2≡b2(βxS 2)= /angbracketleftSz 2/angbracketright(0), which are the magnetization of the ﬁrst and second sublattices, respectively. Using these equations, for the GFs describing transverse spin precessions we obtained G11(q)=b1K−1 22(iωn) (iωn+ε1q)(iωn−ε2q), (A3) G12(q)=G21(q)=−b1b2Jq (iωn+ε1q)(iωn−ε2q),(A4) G22(q)=−b2K−1 11(iωn) (iωn+ε1q)(iωn−ε2q). (A5) Here and below, for shorthand notation, we introduce the designation εi(q)=εiq. These GFs describe the propagation of spin waves with the momentum qand energy ε1q=−1 2(b1−b2)J0+1 2/bracketleftbig (b1−b2)2J2 0 +4b1b2/parenleftbig J2 0−J2 q/parenrightbig/bracketrightbig1/2(A6) for an acoustic spin wave with an in-phase precession of the sublattice magnetization and energy ε2q=1 2(b1−b2)J0+1 2/bracketleftbig (b1−b2)2J2 0 +4b1b2/parenleftbig J2 0−J2 q/parenrightbig/bracketrightbig1/2(A7) for an exchange spin wave with an antiphase precession of the sublattice magnetization. In a long-wave limit ( aq)/lessmuch1, the asymptotic expansions provide expressions ( 20) and ( 21), respectively. APPENDIX B: ANALYTICAL EXPRESSIONS FOR THE LOOP DIAGRAMS To calculate the loop diagrams /Phi1(q),/Pi1(q),B(q),Q(q), and/Lambda1(q), where q={q,iωn}, we substitute the spin wave GFs (A3)–(A5) into Eqs. ( 10)–(14) and sum up over the discrete Matsubara frequencies. Simple but cumbersome calculationsgive, for the function /Phi1(q), analytical expression ( 19). For functions /Pi1(q),B(q),Q(q), and /Lambda1(q) similar calculations yield /Pi1(q)=b2 1 N/summationtext p/braceleftbiggε1p−x ε1p+ε2pε1p−q−x ε1p−q+ε2p−qn1(ε1p)−n1(ε1p−q) iωq−ε1p+ε1p−q−ε2p+x ε1p+ε2pε1p−q−x ε1p−q+ε2p−q1+n1(ε1p−q)+n2(ε2p) iωq+ε2p+ε1p−q +ε1p−x ε1p+ε2pε1p−q+x ε1p−q+ε2p−q1+n1(ε1p)+n2(ε2p−q) iωq−ε1p−ε2p−q+ε2p+x ε1p+ε2pε2p−q+x ε1p−q+ε2p−qn2(ε2p−q)−n2(ε2p) iωq+ε2p−ε2p−q/bracerightbigg , (B1) B(q)=b2 2 N/summationtext p/braceleftbiggε2p−y ε1p+ε2pε2p−q−y ε1p−q+ε2p−qn2(ε2p−q)−n2(ε2p) iωq−ε2p+ε2p−q−ε1p+y ε1p+ε2pε2p−q−y ε1p−q+ε2p−q1+n1(ε1p)+n2(ε2p−q) iωq−ε1p−ε2p−q −ε2p−y ε1p+ε2pε2p−y ε1p−q+ε2p−q1+n1(ε1p−q)+n2(ε2p) iωq+ε2p+ε1p−q+ε1p+y ε1p+ε2pε1p−q+y ε1p−q+ε2p−qn1(ε1p)−n1(ε1p−q) iωq+ε1p−ε1p−q/bracerightbigg , (B2) 054434-10LONGITUDINAL SPIN DYNAMICS IN FERRIMAGNETS: . . . PHYSICAL REVIEW B 94, 054434 (2016) Q(q)=b2 1b2 N/summationtext pJ2 p ε1p+ε2p/braceleftbiggε1p−q−x ε1p−q+ε2p−q/parenleftbiggn1(ε1p)−n1(ε1p−q) iωq−ε1p+ε1p−q+1+n1(ε1p−q)+n2(ε2p) iωq+ε2p+ε1p−q/parenrightbigg +ε1p−q+x ε1p−q+ε2p−q/parenleftbigg1+n1(ε1p)+n2(ε2p−q) iωq−ε1p−ε2p−q+n2(ε2p)−n2(ε2p−q) iωq+ε2p−ε2p−q/parenrightbigg/bracerightbigg , (B3) /Lambda1(q)=b1b2 2 N/summationtext pJ2 p ε1p+ε2p/braceleftbiggε2p−q−y ε1p−q+ε2p−q/parenleftbigg1+n1(ε1p)+n2(ε2p−q) iωq−ε1p−ε2p−q−n2(ε2p−q)−n2(ε2p) iωq−ε2p−q+ε2p/parenrightbigg +ε1p−q+y ε1p−q+ε2p−q/parenleftbigg1+n1(ε1p−q)+n2(ε2p) iωq+ε1p−q+ε2p+n1(ε1p)−n1(ε1p−q) iωq−ε1p+ε1p−q/parenrightbigg/bracerightbigg . (B4) Heren1(ε1p) andn2(ε2p) are the Bose distribution functions for excitations with in-phase and out-of-phase sublattice oscillations, respectively (i.e., acoustic and exchange spin waves). APPENDIX C: CALCULATION OF THE LOOP DIAGRAMS To ﬁnd the asymptotic of loop diagrams /Phi1(q),/Pi1(q),B(q),Q(q), and /Lambda1(q) we will use the approximation ε1p+ε2p≈ (b1−b2)J0,ε1p−x≈−b2J0,ε2p+x≈b1J0,ε2p−y≈−b2J0, andε1p+y≈b2J0in Eqs. ( 19) and ( B1)–(B4) for the factors at the singularities. Then the quantities on the left-hand sides of Eqs. ( 19) and ( B1)–(B4) can be expressed in terms of the universal functions deﬁned as λi(q,ω)=1 N/summationtext pni(εip) ω−εip+εip−q+iδ, (C1) λ0(q,ω)=1 N/summationtext p1 ω−ε1p−ε2p−q+iδ, (C2) λ0i(q,ω)=1 N/summationtext pni(εip) ω−εip+εjp−q+iδ,i/negationslash=j, (C3) where i,j=1,2. Namely, after simple algebra we can write /Phi1(q),Q(q), and B(q) in the form /Phi1(q)=b2 1b2 2 (b1−b2)2(λ+ 1+λ− 1−λ+ 2−λ− 2−λ+ 0−λ− 0−λ+ 01−λ− 01−λ+ 02−λ− 02), /Pi1(q)=b2 1 (b1−b2)2/braceleftbig b2 2(λ+ 1+λ− 1)−b2 1(λ+ 2+λ− 2)−b1b2(λ+ 0+λ− 0+λ+ 01+λ− 01+λ+ 02+λ− 02)/bracerightbig , B(q)=b2 2 (b1−b2)2/braceleftbig b2 1(λ+ 1+λ− 1)−b2 2(λ+ 2+λ− 2)−b1b2(λ+ 0+λ− 0+λ+ 01+λ− 01+λ+ 02+λ− 02)/bracerightbig , with similar expressions for Q(q) and/Lambda1(q) (not shown here). We also introduce here the shorthand notation λ± i=λi(q,±ω). The real part Re λi(q,ω) is deﬁned by Eqs. ( C1)–(C3), implying that its principal value has been taken for this case. The imaginary part of λi(q,ω) reads Imλi(q,ω)=−π N/summationtext pni(εip)δ(ω−εip+εip−q), (C4) Imλ0(q,ω)=−π N/summationtext pδ(ω−ε1p−ε2p−q), (C5) Imλ0i(q,ω)=−π N/summationtext pni(εip)δ(ω−εip+εjp−q),i/negationslash=j. (C6) The main physics can be captured in a long-wave limit, ( ak)/lessmuch1, when the spin wave energies possess a quadratic dispersion law, Eqs. ( 20) and ( 21). Within this approximation, integration over the angle between the momentums pandqleads us to an intermediate result for the real parts of ( C1)–(C3): Reλi(a±)=1 2Dq/integraldisplay1 0ni(εip)pdp ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglep+a± p−a±/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (C7) Reλ 0(b±)=1 2Dq/integraldisplay1 0pdp ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglep2−pq+b± p2+pq+b±/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (C8) Reλ 0i(b±)=1 2Dq/integraldisplay1 0ni(εip)pdp ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglep 2+pq+b± p2−pq+b±/vextendsingle/vextendsingle/vextendsingle/vextendsingle,i/negationslash=j. (C9) 054434-11V . N. KRIVORUCHKO PHYSICAL REVIEW B 94, 054434 (2016) Here we introduce the parameters a±=q 2±ω 2Dqandb±=q 2+[q2 4+(b1−b2)J0±ω 2D]1/2. Like for the imaginary parts, we obtain Imλi(a±)=π 2Dq/integraldisplay1 \|a±\|ni(εip)pdp, (C10) Imλ0(b±)=π 2Dq/integraldisplay1 \|b±\|pdp=π 4Dq[1−(b±)2], (C11) Imλ0i(b±)=π 2Dq/integraldisplay1 \|b±\|ni(εip)pdp, i /negationslash=j. (C12) Below we estimate these integrals for the regimes when the parameter \|a±\|or\|b±\|is small and one can ﬁnd dominant contributions in expressions ( C7)–(C12). (i)\|a±\|/lessmuch 1. If\|a±\|is small, which follows from expressions ( C7) and ( C10), the functions λ± 1andλ± 2give the main contribution. The required asymptotic expansions are found to be Re(λ+ 1+λ− 1)≈T D2q2b1b2 (b1−b2)2/bracketleftbigg qarctg2b1b2 (b1−b2)2−2b1b2 3(b1−b2)2/parenleftbigg q2+ω2 D2q2/parenrightbigg/bracketrightbigg , Re(λ+ 2+λ− 2)≈T D2q/parenleftbiggπ2 4−q/parenrightbigg , Im(λ+ 1+λ− 1)≈πT 2D2q2b1b2 (b1−b2)2/bracketleftbigg arctg2b1b2 (b1−b2)2−q2b1b2 (b1−b2)2/bracketrightbigg , Im(λ+ 2+λ− 2)≈πT 2D2q/bracketleftbigg 3−2q+1 4/parenleftbigg q2+ω2 D2q2/parenrightbigg/bracketrightbigg . Using these asymptotes, the main contribution to the longitudinal susceptibility takes the form of expression ( 25), where the characteristic frequencies are determined by Eq. ( 26). (ii)\|b±\|/lessmuch 1. Now the functions λ± 0andλ± 0igive a substantial contribution. In this ( ω,q) region the denominator, Eq. ( 18), reads D(q,ωq)≈1−2Jq/Phi1(q,ωq)=1+2Jqb2 1b2 2 (b1−b2)2(λ+ 0+λ− 0+λ+ 01+λ− 01+λ+ 02+λ− 02). (C13) The real parts of the functions λ± 0andλ± 0iare equal to Re(λ+ 0+λ− 0)≈− 3/4D, Re(λ+ 01+λ− 01)≈T D2q/bracketleftbigg qln/parenleftbiggD+(b1−b2)J0 T/parenrightbigg −2l n(b1−b2)J0 T+(b1−b2)J0±ω (b1−b2)J0/bracketrightbigg , Re(λ+ 02+λ− 02)≈T D2q/parenleftbiggπ2 4−q/parenrightbigg . For the excitation damping (the pole imaginary part on the mass surface) we have γ(q,/Omega1 exc)≈(b1−b2)Jq2D2q TIm(λ+ 0+λ− 0+λ+ 01+λ− 01+λ+ 02+λ− 02). For the imaginary parts ( C11) and ( C12) with ω=/Omega1exc, calculations yield Imλ0(q,/Omega1 exc)∼π 2Dq/bracketleftbig 1−κ2 0/bracketrightbig , Imλ01(q,/Omega1 exc)∼π 2DqT Dln/bracketleftbigg(b1−b2)J0+D (b1−b2)J0+Dκ2 0/bracketrightbigg , Imλ02(q,/Omega1 exc)∼π 2DqT Dln/braceleftbigg4b1b2 (b1−b2)2/bracketleftbigg 2l nT (b1−b2)J0+π2 2+f(T)q/bracketrightbigg/bracerightbigg , where the functions κ2 0andf(T) are determined by the analytical forms ( 35) and ( 33), respectively. Summarizing the results, we obtain the characteristic frequencies deﬁned by Eqs. 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PhysRevB.82.134431.pdf	Thermal ﬂuctuation ﬁeld for current-induced domain wall motion Kyoung-Whan Kim and Hyun-Woo Lee PCTP and Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea /H20849Received 18 May 2010; revised manuscript received 23 August 2010; published 20 October 2010 /H20850 Current-induced domain wall motion in magnetic nanowires is affected by thermal ﬂuctuation. In order to account for this effect, the Landau-Lifshitz-Gilbert equation includes a thermal ﬂuctuation ﬁeld and literatureoften utilizes the ﬂuctuation-dissipation theorem to characterize statistical properties of the thermal ﬂuctuationﬁeld. However, the theorem is not applicable to the system under ﬁnite current since it is not in equilibrium. Toexamine the effect of ﬁnite current on the thermal ﬂuctuation, we adopt the inﬂuence functional formalismdeveloped by Feynman and Vernon, which is known to be a useful tool to analyze effects of dissipation andthermal ﬂuctuation. For this purpose, we construct a quantum-mechanical effective Hamiltonian describingcurrent-induced domain wall motion by generalizing the Caldeira-Leggett description of quantum dissipation.We ﬁnd that even for the current-induced domain wall motion, the statistical properties of the thermal noise isstill described by the ﬂuctuation-dissipation theorem if the current density is sufﬁciently lower than theintrinsic critical current density and thus the domain wall tilting angle is sufﬁciently lower than /H9266/4. The relation between our result and a recent result /H20851R. A. Duine, A. S. Núñez, J. Sinova, and A. H. MacDonald, Phys. Rev. B 75, 214420 /H208492007/H20850/H20852, which also addresses the thermal ﬂuctuation, is discussed. We also ﬁnd interesting physical meanings of the Gilbert damping /H9251and the nonadiabaticy parameter /H9252; while /H9251charac- terizes the coupling strength between the magnetization dynamics /H20849the domain wall motion in this paper /H20850and the thermal reservoir /H20849or environment /H20850,/H9252characterizes the coupling strength between the spin current and the thermal reservoir. DOI: 10.1103/PhysRevB.82.134431 PACS number /H20849s/H20850: 75.78.Fg, 75.60.Ch, 05.40.Ca I. INTRODUCTION Current-induced domain wall /H20849DW/H20850motion in a ferro- magnetic nanowire is one of representative examples tostudy the effect of spin-transfer torque /H20849STT/H20850. The motion of DW is generated by the angular momentum transfer betweenspace-time-dependent magnetization m /H6023/H20849x,t/H20850and conduction electrons, of which spins interact with m/H6023by the exchange coupling. This system is usually described by the Landau-Lifshitz-Gilbert /H20849LLG/H20850equation, 1–3 /H11509m/H6023 /H11509t=/H92530H/H6023eff/H11003m/H6023+/H9251 msm/H6023/H11003/H11509m/H6023 /H11509t+jp/H9262B ems/H20875/H11509m/H6023 /H11509x−/H9252 msm/H6023/H11003/H11509m/H6023 /H11509x/H20876, /H208491/H20850 where /H92530is the gyromagnetic ratio, jpis the spin-current density, ms=/H20841m/H6023/H20841is the saturation magnetization, and /H9262Bis the Bohr magneton. /H9251is the Gilbert damping coefﬁcient, and /H9252 is the nonadiabatic coefﬁcient representing the magnitude ofthe nonadiabatic STT. 4In Eq. /H208491/H20850, the effective magnetic ﬁeld Heffis given by H/H6023eff=A/H116122m/H6023+H/H6023ani+H/H6023th, /H208492/H20850 where Ais stiffness constant, H/H6023anidescribes the effect of the magnetic anisotropy, and H/H6023this the thermal ﬂuctuation ﬁeld describing the thermal noise. In equilibrium situations, the magnitude and spatiotemporal correlation of H/H6023thare gov- erned by the ﬂuctuation-dissipation theorem,5–7 /H20855Hth,i/H20849x/H6023,t/H20850Hth,j/H20849x/H6023/H11032,t/H11032/H20850/H20856=4/H9251kBT /H6036/H9267/H9254/H20849x/H6023−x/H6023/H11032/H20850/H9254/H20849t−t/H11032/H20850/H9254ij,/H208493/H20850 where /H20855¯/H20856represents the statistical average, i,jdenote x,y, orzcomponent, kBis the Boltzmann constant, Tis the tem-perature, and /H9267=ms//H9262Bis the spin density. Equation /H208493/H20850 plays an important role for the study of the magnetizationdynamics at ﬁnite temperature, 8 Equation /H208493/H20850has been also used in literature9–13to exam- ine effects of thermal ﬂuctuations on the current-inducedDW motion. In nonequilibrium situations, however, the ﬂuctuation-dissipation theorem does not hold generally.Since the system is not in equilibrium any more when thecurrent is applied, it is not clear whether Eq. /H208493/H20850may be still used. Recalling that H /H6023this estimated to affect the magnetiza- tion dynamics considerably in many experimentalsituations 14–17of the current-driven DW motion, it is highly desired to properly characterize H/H6023thin situations with non- zero jp. Recently, Duine18attempted this characterization and showed that Eq. /H208493/H20850is not altered by the spin current up to ﬁrst order in the spin-current magnitude. This analysis how-ever is limited to situations where the spin-ﬂip scattering isthe main mechanism responsible for /H9252. In this paper, we generalize this analysis by using a completely different ap-proach which does not assume any speciﬁc physical origin of /H9252. Htharises from extra degrees of freedom /H20849other than mag- netization /H20850, which are not included in the LLG equation. The extra degrees of freedom /H20849phonons for instance /H20850usually have much larger number of degrees of freedom than magnetiza-tion and thus form a heat reservoir. Thus properties of H thare determined by the heat reservoir. The heat reservoir playsanother role. In the absence of the extra degrees of freedom,the Gilbert damping coefﬁcient /H9251should be zero since the total energy should be conserved when all degrees of free-dom are taken into account. Thus the heat reservoir is re-sponsible also for ﬁnite /H9251. These dual roles of the heat res- ervoir are the main idea behind the Einstein’s theory of thePHYSICAL REVIEW B 82, 134431 /H208492010/H20850 1098-0121/2010/82 /H2084913/H20850/134431 /H2084916/H20850 ©2010 The American Physical Society 134431-1Brownian motion.19There are also claims that /H9251is correlated with/H9252/H20849Refs. 18and20–22/H20850in the sense that mechanisms, which generate /H9252, also contribute to /H9251. Thus the issue of H/H6023th and the issue of /H9251and/H9252are mutually connected. Recalling that the main mechanism responsible for /H9251varies from ma- terial to material, it is reasonable to expect that the main mechanism for H/H6023thand/H9252may also vary from material to material. Recently, various mechanisms of /H9252were examined such as momentum transfer,23–25spin mistracking,26,27spin- ﬂip scattering,18,21,22,25,28and the inﬂuence of a transport current.29This diversity of mechanisms will probably apply toH/H6023thas well. Instead of examining each mechanism of H/H6023thone by one, we take an alternative approach to address this issue. In1963, Feynman and Vernon 30proposed the so-called inﬂu- ence functional formalism, which allows one to take accountof damping effects without detailed accounts of dampingmechanisms. This formalism was later generalized by Smith and Caldeira. 31This formalism has been demonstrated to be a useful tool to address dissipation effects /H20849without speciﬁc accounts of detailed damping mechanisms /H20850on, for instance, quantum tunneling,32nonequilibrium dynamic Coulomb blockade,33and quantum noise.34To take account of damp- ing effects which are energy nonconserving processes in gen-eral, the basic idea of the inﬂuence functional formalism is tointroduce inﬁnite number of degrees of freedom /H20849called en- vironment /H20850behaves like harmonic oscillators which couple with the damped system. /H20851See Eq. /H2084913/H20850./H20852Caldeira and Leggett 32suggested the structure of the spectrum of environ- ment Eq. /H2084913/H20850and integrated out the degrees of freedom of environment to ﬁnd the effective Hamiltonian describing theclassical damping Eq. /H2084912/H20850. For readers who are not familiar with the Caldeira-Leggett’s theory of quantum dissipation,we present the summary of details of the theory in Sec. II B. In order to address the issue of H /H6023th, we follow the idea of the inﬂuence functional formalism and construct an effectiveHamiltonian describing the magnetization dynamics. The ef-fective Hamiltonian describes not only energy-conservingprocesses but also energy-nonconserving processes such asdamping and STT. From this approach, we ﬁnd that Eq. /H208493/H20850 holds even in nonequilibrium situations with ﬁnite j p, pro- vided that jpis sufﬁciently smaller than the so-called intrin- sic critical current density23so that the DW tilting angle /H9278 /H20849to be deﬁned below /H20850is sufﬁciently smaller than /H9266/4. We remark that in the special case where the spin-ﬂip scattering mechanism of /H9252is the main mechanism of H/H6023th, our ﬁnding is consistent with Ref. 18, which reports that the spin ﬂip scat- tering mechanism does not alter Eq. /H208493/H20850at least up to the ﬁrst order in jp. But our calculation indicates that Eq. /H208493/H20850holds not only in situations where the spin ﬂip scattering is the dominant mechanism of H/H6023thand/H9252but also in more diverse situations as long as the heat reservoir can be described bybosonic excitations /H20849such as electron-hole pair excitations or phonon /H20850, i.e., the excitations effectively behave like har- monic oscillators to be described by Caldeira-Leggett’stheory. We also remark that in addition to the derivation ofEq./H208493/H20850in nonequilibrium situations, our calculation also re- veals an interesting physical meaning of /H9252, which will be detailed in Sec. III.This paper is organized as follows. In Sec. II, we ﬁrst introduce the Caldeira-Leggett’s version of the inﬂuencefunctional formalism and later generalize this formalism sothat it is applicable to our problem. This way, we construct aHamiltonian describing the DW motion. In Sec. III, some implications of this model is discussed. First, a distinct in-sight on /H9252is emphasized. Second, as an application, statisti- cal properties of the thermal ﬂuctuation ﬁeld are calculatedin the presence of nonzero j p, which veriﬁes the validity of Eq./H208493/H20850when jpis sufﬁciently smaller than the intrinsic criti- cal density. It is believed that many experiments16,17are in- deed in this regime. Finally, in Sec. IV, we present some concluding remarks. Technical details about the quantumtheory of the DW motion and methods to obtain solutions areincluded in Appendices. II. GENERALIZED CALDEIRA-LEGGETT DESCRIPTION A. Background Instead of full magnetization proﬁle m/H6023/H20849x,t/H20850, the DW dy- namics is often described2,23,35–37by two collective coordi- nates, DW position x/H20849t/H20850and DW tilting angle /H9278/H20849t/H20850. When expressed in terms of these collective coordinates, the LLGEq./H208491/H20850reduces to the so-called Thiele equations, dx dt=jp/H9262B ems+/H9251/H9261d/H9278 dt+/H92530K/H9261 mssin 2/H9278+/H9257x/H20849t/H20850,/H208494a/H20850 /H9261d/H9278 dt=−/H9251dx dt+/H9252jp/H9262B ems+/H9257p/H20849t/H20850. /H208494b/H20850 Here Kis the hard-axis anisotropy, /H9261is the DW thickness. /H9257x/H20849t/H20850and/H9257p/H20849t/H20850are functions describing thermal noise ﬁeld Hth,i/H20849x,t/H20850. By deﬁnition, the statistical average of the thermal noise ﬁeld Hth,i/H20849x,t/H20850is zero and similarly the statistical aver- ages of /H9257x/H20849t/H20850and/H9257p/H20849t/H20850should also vanish regardless of whether the system is in equilibrium. The question of theircorrelation function is not trivial however. If the thermalnoise ﬁeld H th,i/H20849x,t/H20850satisﬁes the correlation in Eq. /H208493/H20850,i tc a n be derived from Eq. /H208493/H20850that/H9257x/H20849t/H20850and/H9257p/H20849t/H20850satisfy the cor- relation relation12 /H20855/H9257i/H20849t/H20850/H9257j/H20849t/H11032/H20850/H20856/H11008/H9251kBT/H9254ij/H9254/H20849t−t/H11032/H20850, /H208495/H20850 for/H20853i,j/H20854=/H20853x,p/H20854. But as mentioned in Sec. I, Eq./H208493/H20850is not guaranteed generally in the presence of the nonzero current.Then Eq. /H208495/H20850is not guaranteed either. The question of what should be the correlation function /H20855 /H9257i/H20849t/H20850/H9257j/H20849t/H11032/H20850/H20856in such a situ- ation will be discussed in Sec. III. When the spin-current density jpis sufﬁciently smaller than the so-called intrinsic critical density /H20841e/H92530K/H9261//H9262B/H20841,23/H9278 stays sufﬁciently smaller than /H9266/4. In many experimental situations,38–40this is indeed the case,41so we will conﬁne ourselves to the small /H9278regime in this paper. Then, one can approximate sin 2 /H9278/H110152/H9278to convert the equations into the following form:42 dx dt=vs+/H9251S 2KMdp dt+p M+/H92571/H20849t/H20850, /H208496a/H20850KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-2dp dt=−2/H9251KM Sdx dt+2/H9252KM Svs+/H92572/H20849t/H20850, /H208496b/H20850 where p=2KM/H9261/H9278/S,Sis the spin angular momentum at each individual magnetic site, and vs=jp/H9262B/emsis the adia- batic velocity,43which is a constant of velocity dimension and proportional to jp. The yet undetermined constant Mis the effective DW mass42–44which will be ﬁxed so that the new variable pbecomes the canonical conjugate to x./H92571/H20849t/H20850 and/H92572/H20849t/H20850are the same as /H9257x/H20849t/H20850and/H9257p/H20849t/H20850except for propor- tionality constants. When the thermal noises /H92571/H20849t/H20850and/H92572/H20849t/H20850are ignored, one obtains from Eq. /H208496/H20850the time dependence of the DW posi- tion, x/H20849t/H20850=x/H208490/H20850+/H9252 /H9251vst+S 2KM/H92512/H208491−e−2K/H9251t/S/H208491+/H92512/H20850/H20850 /H11003/H20851/H9251p/H208490/H20850−Mvs/H20849/H9251−/H9252/H20850/H20852. /H208497/H20850 Note that after a short transient time, the DW speed ap- proaches the terminal velocity /H9252vs//H9251. Thus the ratio /H9252//H9251is an important parameter for the DW motion. When the ther-mal noises are considered, they generate a correction to Eq./H208497/H20850. However, from Eq. /H208496/H20850, it is evident that the statistical average of x/H20849t/H20850should still follow Eq. /H208497/H20850. Thus as far as the temporal evolution of the statistical average is concerned, wemay ignore the thermal noises. In the rest of Sec. II,w ea i m to derive a quantum mechanical Hamiltonian, which repro-duces the same temporal evolution as Eq. /H208497/H20850in the statistical average level. In Sec. III, we use the Hamiltonian to derive the correlation function /H20855 /H9257i/H20849t/H20850/H9257j/H20849t/H11032/H20850/H20856in the presence of the nonzero current. Now, we begin our attempt to construct an effective Hamiltonian that reproduces the DW dynamics Eq. /H208496/H20850/H20851or equivalently Eq. /H208497/H20850/H20852. We ﬁrst begin with the microscopic quantum-mechanical Hamiltonian Hs-d, Hs-d=−J/H20858 iS/H6023i·S/H6023i+1−A/H20858 i/H20849S/H6023i·zˆ/H208502+K/H20858 i/H20849S/H6023i·yˆ/H208502+HcS, /H208498/H20850 which has been used in previous studies20of the DW dynam- ics. Here Jrepresents the ferromagnetic exchange interac- tion, AandKrepresent longitudinal /H20849easy-axis /H20850and trans- verse/H20849hard-axis /H20850anisotropy, respectively. The last term HcS represents the coupling of the spin system with the spin- polarized current, HcS=−/H20858 i,/H9251=↑,↓/H20851t/H20849ci/H9251†ci+1/H9251+ci+1/H9251†ci/H9251/H20850−/H9262ci/H9251†ci/H9251/H20852−JH/H20858 iS/H6023ci·S/H6023i, /H208499/H20850 where JHis the exchange interaction between conduction electron and the localized spins, ci/H9251is the annihilation opera- tor of the conduction electron at the site i,S/H6023ciis the electron- spin operator, tis the hopping integral, and /H9262is the chemical potential of the system.Recently Kim et al.43analyzed Hs-din detail in the small tilting angle regime and found that Hs-dcontains gapless low-lying excitations and also high-energy excitations with aﬁnite energy gap. The gapless excitations of H s-dare de- scribed by a simple Hamiltonian H0, H0=vsP+P2 2M/H2084910/H20850 while the high-energy excitations have a ﬁnite energy gap 2S/H20881A/H20849A+K/H20850.I nE q . /H2084910/H20850,Pis the canonical momentum of the DW position operator Q, and M=/H60362 K/H208812A Ja4is the effective DW mass called Döring mass.44Here, ais the lattice spacing between two neighboring spins. /H20849See, for details, Appendix A./H20850Below we will neglect the high energy excitations and focus on the low-lying excitations described by Eq. /H2084910/H20850. For the analysis of the high-energy excitation effects on the DW,See Ref. 42. From Eq. /H2084910/H20850, one obtains the following Heisenberg’s equation of motion: dQ dt=vs+P M, /H2084911a/H20850 dP dt=0 . /H2084911b/H20850 Note that the current /H20849proportional to vs/H20850appears in the equa- tion fordQ dt. Thus the current affects the DW dynamics by introducing a difference between the canonical momentum P and the kinematic momentum P+Mvs. In this sense, the ef- fect of the current is similar to a vector potential /H20851canonical momentum P/H6023and kinematic momentum P/H6023+/H20849e/c/H20850A/H6023/H20852. The vector potential /H20849difference between the canonical momen- tum and the kinetic momentum /H20850allows the system in the initially zero momentum state to move without breaking thetranslational symmetry of the system. In other words, thecurrent-induced DW motion is generated without any forceterm in Eq. /H2084911b/H20850violating the translational symmetry of the system. This should be contrasted with the effect of the mag-netic ﬁeld or magnetic defects, which generates a force termin Eq. /H2084911b/H20850. The solution of Eq. /H2084911/H20850is trivial, /H20855Q/H20849t/H20850/H20856=/H20855Q/H208490/H20850/H20856 +/H20849/H20855P/H208490/H20850/H20856/M+ vs/H20850t. Here, the statistical average /H20855¯/H20856is de- ﬁned as /H20855¯/H20856=Tr/H20849/H9267¯/H20850/Tr/H20849/H9267/H20850, where /H9267denotes the density matrix at t=0. Associating /H20855Q/H20849t/H20850/H20856=x/H20849t/H20850,/H20855P/H20849t/H20850/H20856=p/H20849t/H20850, one ﬁnds that Eq. /H2084911/H20850is identical to Eq. /H208496/H20850if/H9251=/H9252=0. This implies that the effective Hamiltonian H0/H20851Eq./H2084910/H20850/H20852fails to capture effects of nonzero /H9251and/H9252. In the next three sections, we attempt to resolve this problem. B. Caldeira-Leggett description of damping To solve the problem, one should ﬁrst ﬁnd a way to de- scribe damping. A convenient way to describe ﬁnite dampingwithin the effective Hamiltonian approach is to adopt theCaldeira-Leggett description 32of the damping. Its main idea is to introduce a collection of additional degrees of freedom/H20849called environment /H20850and couple them to the original dy- namic variables so that energy of the dynamic variables canTHERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-3be transferred to the environment. For instance, for a one- dimensional /H208491D/H20850particle subject to damped dynamics, dx dt=p M, /H2084912a/H20850 dp dt=−dV/H20849x/H20850 dx−/H9253dx dt. /H2084912b/H20850 Caldeira and Leggett32demonstrated that its quantum- mechanical Hamiltonian can be constructed by adding damp-ing Hamiltonian H 1to the undamped Hamiltonian H0 =P2/2M+V/H20849Q/H20850. The damping Hamiltonian H1contains a collection of environmental degrees of freedom /H20853xi,pi/H20854be- having like harmonic oscillators /H20851see Eq. /H2084914/H20850/H20852, which couple to the particle through the linear coupling term /H20858iCixiQbe- tween Qand the environmental variables xi. Here, Ciis the coupling constant between xiandQ. The implication of the coupling is twofold: /H20849i/H20850the coupling to the environment gen- erates damping, whose precise form depends on Ci,mi, and /H9275i. It is demonstrated in Ref. 32that the coupling generates the simple damping of the form in Eq. /H2084912b/H20850ifCi,mi, and/H9275i satisfy the following relation of the spectral function J/H20849/H9275/H20850: J/H20849/H9275/H20850/H11013/H9266 2/H20858 iCi2 mi/H9275i/H9254/H20849/H9275−/H9275i/H20850=/H9253/H9275. /H2084913/H20850 /H20849ii/H20850The coupling also modiﬁed the potential Vby generating an additional contribution − /H20858iCi2Q2/2mi/H9275i2. This implies that V/H20849x/H20850in Eq. /H2084912b/H20850should not be identiﬁed with V/H20849Q/H20850inH0 /H20849even though the same symbol Vis used /H20850but should be iden- tiﬁed instead with the total potential that includes the contri-bution from the environmental coupling. If we express thetotal Hamiltonian Hin terms of the effective V/H20849x/H20850that ap- pears in Eq. /H2084912b/H20850, it reads H=H 0+H1, /H2084914a/H20850 H0=P2 2M+V/H20849Q/H20850, /H2084914b/H20850 H1=/H20858 i/H20875pi2 2mi+1 2mi/H9275i2/H20873xi+Ci mi/H9275i2Q/H208742/H20876./H2084914c/H20850 By identifying x/H20849t/H20850=/H20855Q/H20849t/H20850/H20856,p/H20849t/H20850=/H20855P/H20849t/H20850/H20856, the equations of motion obtained from Eqs. /H2084913/H20850and/H2084914/H20850reproduce Eq. /H2084912/H20850. C. Generalization to the DW motion: /H9251term Here we aim to apply the Caldeira-Leggett approach to construct an effective Hamiltonian of the DW dynamics sub-ject to ﬁnite damping /H20849 /H9251/HS110050/H20850. To simplify the problem, we ﬁrst focus on a situation, where only /H9251is relevant and /H9252is irrelevant. This situation occurs if there is no current /H20849vs =0/H20850. Then Eq. /H208496/H20850reduces to dx dt=/H9251S 2KMdp dt+p M, /H2084915a/H20850dp dt=−2/H9251KM Sdx dt. /H2084915b/H20850 Note that /H9252does not appear. Note also that these equations are slightly different from Eq. /H2084912/H20850, where a damping term is contained only in the equation ofdp dt. However, in the equa- tions of the DW /H20851Eq./H2084915/H20850/H20852, damping terms appear not only in the equation ofdp dt/H20851Eq./H2084915b/H20850/H20852but also in the equation ofdx dt /H20851Eq./H2084915a/H20850/H20852. Thus the Caldeira-Leggett description in the preceding section is not directly applicable and should be generalized.To get a hint, it is useful to recall the conjugate relation between QandP. The equations ofdQ dtanddP dtare obtained by differentiating Hwith respect to Pand − Q, respectively. Of course, it holds for /H20849xi,pi/H20850, also. Thus, one can obtain another set of Heisenberg’s equation of motion by exchang-ing/H20849Q,x i/H20850↔/H20849−P,−pi/H20850. By this canonical transformation, the position coupling /H20858iCixiQchanges to a momentum coupling term, and the damping term in the equation ofdP dtis now in that ofdQ dt. This mathematical relation that the momentum coupling generates a damping term in the equation ofdQ dt makes it reasonable to expect that the momentum coupling /H20858iDipiPis needed45to generate the damping in the equation fordQ dt. Here Diis the coupling constant between Pandpi. The reason why, in the standard Caldeira-Leggett approach,the damping term appears only in Eq. /H2084912b/H20850is that Eq. /H2084914/H20850 contains only position coupling terms /H20858 iCixiQ. It can be eas- ily veriﬁed that the implications of the momentum couplingare again twofold: /H20849i/H20850the coupling indeed introduces the damping term in the equation ofdQ dt./H20849ii/H20850it modiﬁes the DW mass. The mass renormalization arises from the fact that inthe presence of the momentum coupling /H20858 iDipiP, the kine- matic momentum midxi dtof an environmental degree of free- dom xiis now given by /H20849pi+DimiP/H20850instead of pi. Then the term/H20858i/H20851pi2 2mi+DipiP/H20852can be decomposed into two pieces, /H20858i/H20849pi+DimiP/H208502 2mi, which is the kinetic energy associated with xi, and/H20851−/H20858iDi2mi 2/H20852P2. Note that the second piece has the same form as the DW kinetic termP2 2M. Thus this second piece generates the renormalization of the DW mass. Due to thismass renormalization effect, Min Eq. /H2084915/H20850should be inter- preted as the renormalized mass that contains the contribu-tion from the environmental coupling. If MinH 0in Eq. /H2084910/H20850 is interpreted as the renormalized mass, the environmentHamiltonian H 2for the DW dynamics becomes H2=/H20858 i/H208751 2mi/H20849pi+DimiP/H208502+1 2mi/H9275i2/H20873xi+Ci mi/H9275i2Q/H208742/H20876. /H2084916/H20850 Here,/H20858i/H20849pi+DimiP/H208502/2micoupling is equivalent to the origi- nal form /H20858i/H20849pi2/2mi+DipiP/H20850under the mass renormalization 1/M→1/M−/H20858iDi2mi/2. Note that in H2, the collective co- ordinates QandPof the DW couple to the environmental degrees of freedom /H20853xi,pi/H20854through two types of coupling, /H20858iCixiQand/H20858iDipiP. Finally, one obtains the total Hamiltonian describing the DW motion in the absence of the current,KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-4H=H0/H20841vs=0+H2=P2 2M+/H20858 i/H208751 2mi/H20849pi+DimiP/H208502 +1 2mi/H9275i2/H20873xi+Ci mi/H9275i2Q/H208742/H20876. /H2084917/H20850 Now, the renormalized mass Min the above equation is iden- tical to the mass in Eq. /H2084915/H20850. To make the physical meaning ofxiclearer, we perform the canonical transformation, xi→−Ci mi/H9275i2xi,pi→−mi/H9275i2 Cipi. /H2084918/H20850 Deﬁning /H9253i=CiDi /H9275i2, and redeﬁning a new miasmi/H20849new/H20850=Ci2 mi/H9275i4, the Hamiltonian becomes simpler as H=P2 2M+/H20858 i/H208751 2mi/H20849pi−/H9253iP/H208502+1 2mi/H9275i2/H20849xi−Q/H208502/H20876. /H2084919/H20850 Now, the translational symmetry of the system and the physi- cal meaning of xibecome obvious. The next step is to impose proper constraints on /H9253iandmi, so that the damping terms arising from Eq. /H2084919/H20850agree exactly with those in Eq. /H2084915/H20850. For this purpose, it is convenient to introduce Laplace transformed variables Q˜/H20849/H9261/H20850,P˜/H20849/H9261/H20850,x˜i/H20849/H9261/H20850, p˜i/H20849/H9261/H20850, where Q˜/H20849/H9261/H20850=/H208480/H11009e−/H9261t/H20855Q/H20849t/H20850/H20856dt, and other transformed variables are deﬁned in a similar way. Then the variables x˜i andp˜ican be integrated out easily /H20849see Appendix B /H20850. After some tedious but straightforward algebra, it is veriﬁed thatwhen the following three constraints on /H9253i,/H9275i,miare satis- ﬁed for any positive /H9261, /H20858 i/H9253i/H9275i2 /H92612+/H9275i2=0 , /H2084920a/H20850 /H20858 i/H9253i2/H9261 mi/H20849/H92612+/H9275i2/H20850=/H9251S 2KM, /H2084920b/H20850 /H20858 imi/H9275i2/H9261 /H92612+/H9275i2=2/H9251KM S, /H2084920c/H20850 the DW dynamics satisﬁes the following equation: /H20898/H9261 −1 M−/H9251S/H9261 2KM 2/H9251KM S/H9261 /H9261/H20899/H20873Q˜ P˜/H20874 =/H20873/H20855Q/H208490/H20850/H20856 /H20855P/H208490/H20850/H20856/H20874+/H20898−/H9251S 2KM/H20855P/H208490/H20850/H20856 2/H9251KM S/H20855Q/H208490/H20850/H20856/H20899, /H2084921/H20850 which is nothing but the Laplace transformation of the DW equation /H20851Eq./H2084915/H20850/H20852if/H20855Q/H20856and/H20855P/H20856are identiﬁed with xandp. Thus we verify that the Hamiltonian Hin Eq. /H2084919/H20850indeed provides a generalized Caldeira-Leggett-type quantumHamiltonian for the DW motion. As a passing remark, we mention that in the derivation of Eq. /H2084921/H20850, the environmental degrees of freedom at the initial moment /H20849t=0/H20850are assumed to be in their thermal equilibrium so that /H20855xi/H208490/H20850/H20856=/H20855Q/H208490/H20850/H20856, /H2084922a/H20850 /H20855pi/H208490/H20850/H20856=/H9253i/H20855P/H208490/H20850/H20856. /H2084922b/H20850 Equation /H2084922/H20850can be understood as follows. First, one ob- tains Eq. /H2084922/H20850by following Appendix D which describes the statistical properties of Eq. /H2084919/H20850at high temperature. In Ap- pendix D, /H20855xi/H208490/H20850−Q/H208490/H20850/H20856=/H20855pi/H208490/H20850−/H9253iP/H208490/H20850/H20856is reduced to an in- tegration of an odd function so it is shown to vanish. Thesecond way is probably easier to understand and does notrequire the classical limit or high-temperature limit. TheHamiltonian /H20851Eq./H2084919/H20850/H20852is symmetric under the canonical transformation Q/H208490/H20850→−Q/H208490/H20850,P/H208490/H20850→−P/H208490/H20850,x i/H208490/H20850→−xi/H208490/H20850, and pi/H208490/H20850→−pi/H208490/H20850. Due to this symmetry, one obtains /H20855xi/H208490/H20850−Q/H208490/H20850/H20856=/H20855Q/H208490/H20850−xi/H208490/H20850/H20856and/H20855pi/H208490/H20850−/H9253iP/H208490/H20850/H20856=/H20855/H9253iP/H208490/H20850 −pi/H208490/H20850/H20856, which lead to /H20855xi/H208490/H20850/H20856=/H20855Q/H208490/H20850/H20856and/H20855pi/H208490/H20850/H20856=/H9253i/H20855P/H208490/H20850/H20856, respectively. Here physical origin of the momentum coupling /H20849/H9253/H20850be- tween the DW and environment deserves some discussion.Equation /H2084919/H20850is reduced to the original Caldeira-Leggett Hamiltonian if /H9253i=0. However, Eq. /H2084920b/H20850implies that the momentum coupling as well as the position coupling is in-dispensable to describe the Gilbert damping. To understandthe origin of the momentum coupling /H9253i, it is useful to recall that since P/H11008/H9278/H11008/H20849tilting/H20850, one can interpret Pand Qas transverse and longitudinal spin ﬂuctuation of the DW state,respectively. /H20849See, for explicit mathematical relation, Appen- dix A. /H20850Thus, if there is rotational symmetry on spin interac- tion with the heat bath /H20849or environment /H20850, the existence of the position coupling requires the existence of the momentumcoupling. Thus the appearance of the damping terms both inEqs./H2084915a/H20850and/H2084915b/H20850is natural in view of the rotational sym- metry of the spin exchange interaction and also in view ofthe physical meaning of PandQas transverse and longitu- dinal spin ﬂuctuations. D. Coupling with the spin current: /H9252term In this section, we aim to construct a Caldeira-Leggett- type effective quantum Hamiltonian that takes account of notonly /H9251but also /H9252. Since /H9252becomes relevant only when there exists ﬁnite spin current, we have to deal with situations withﬁnite current /H20849 vs/HS110050/H20850. Then the system is notin thermal equi- librium. As demonstrated in Eq. /H2084910/H20850, the spin current couples with the DW linear momentum, i.e., vsP. Here, adiabatic velocity vsacts as the coupling constant proportional to spin current. The spin current may also couple directly to the environmen-tal degrees of freedom. Calling this coupling constant v, one introduces the corresponding coupling term /H20858ivpi. Later we ﬁnd that this coupling is crucial to account for nonzero /H9252.A t this point we will not specify the value of v. Now, the total effective Hamiltonian in the presence of the spin current ob-tained by adding the coupling term /H20858 ivpito Eq. /H2084919/H20850. Then,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-5Htot=H+Hcurrent =P2 2M+vsP+/H20858 ivpi +/H20858 i/H208751 2mi/H20849pi−/H9253iP/H208502+1 2mi/H9275i2/H20849xi−Q/H208502/H20876./H2084923/H20850 In order to illustrate the relation between Eqs. /H208496/H20850and /H2084923/H20850, we consider a situation, where the current is zero until t=0 and turned on at t=0 to a ﬁnite value. This situation is described by the following time-dependent Hamiltonian: Htot=P2 2M+vs/H20849t/H20850P+/H20858 iv/H20849t/H20850pi +/H20858 i/H208751 2mi/H20849pi−/H9253iP/H208502+1 2mi/H9275i2/H20849xi−Q/H208502/H20876,/H2084924/H20850 where vs/H20849t/H20850=vs/H9008/H20849t/H20850andv/H20849t/H20850=v/H9008/H20849t/H20850. And /H9008/H20849t/H20850is /H9008/H20849t/H20850=/H208771fort/H110220, 0fort/H110210./H20878 /H2084925/H20850 To make a quantitative comparison between Eqs. /H208496/H20850and /H2084924/H20850, one needs to integrate out environmental degrees of freedom /H20853xi,pi/H20854, which requires one to specify their initial conditions. Since the system is in thermal equilibrium untilt=0, we may still impose the constraint in Eq. /H2084922/H20850to exam- ine the DW dynamics for t/H110220. By following a similar pro- cedure as in Sec. II C and by using the constraints in Eq. /H2084920/H20850, 46one ﬁnds that the effective Hamiltonian H/H20851Eq./H2084924/H20850/H20852 predicts/H20855Q/H20849t/H20850/H20856=/H20855Q/H208490/H20850/H20856+vt+S 2KM/H92512/H208491−e−2K/H9251t/S/H208491+/H92512/H20850/H20850 /H11003/H20851/H9251/H20855P/H208490/H20850/H20856−M/H9251/H20849vs−v/H20850/H20852. /H2084926/H20850 This is exactly the same as Eq. /H208497/H20850if /H9252 /H9251=v vs. /H2084927/H20850 So by identifying vwith vs/H9252//H9251, we obtain a Caldeira- Leggett-type effective quantum Hamiltonian of the DW dy-namics. One needs to consider an external force on Eq. /H208496b/H20850/H20851or Eq./H208494b/H20850/H20852when the translational symmetry of the system is broken by some factors such as external magnetic ﬁeld andmagnetic defects. To describe this force, one can add a posi-tion dependent potential V/H20849Q/H20850/H20849Ref. 47/H20850to Eq. /H2084924/H20850. Consid- ering the Heisenberg’s equation, the potential V/H20849Q/H20850generates the term − V /H11032/H20849Q/H20850in Eq. /H208496b/H20850. III. IMPLICATIONS A. Insights on the physical meaning of /H9252 Equation /H2084927/H20850provides insights on the physical meaning of/H9252./H9252depends largely on the coupling between the envi- ronment and current, not on the damping form. Recallingthat vsdescribes the coupling between the current and the DW, we ﬁnd that /H9252//H9251, which describes the asymptotic be- havior of the DW motion, is the ratio between the current-magnetization /H20849DW in the present case /H20850coupling and current-environment coupling. That is, /H9252 /H9251=/H20849Coupling between the current and the environment /H20850 /H20849Coupling between the current and the DW /H20850. /H2084928/H20850 To make the physical meaning of Eq. /H2084928/H20850more transpar- ent, it is useful to examine consequences of the nonzero cou-pling vbetween the current and the environment. One of the immediate consequences of the nonzero vappears in the ve- locities of the environmental degrees of freedom. It can beveriﬁed easily that the initial velocities of environmental co- ordinates are given by exactly v,/H20855x˙i/H208490/H20850/H20856=v. Recalling that the terminal velocity of the DW, /H20855Q˙/H20849t/H20850/H20856approaches vs/H9252//H9251, one ﬁnds from Eq. /H2084927/H20850that the terminal velocity of the DW is nothing but the environment velocity. This result is verynatural since the total Hamiltonian H tot/H20851Eq./H2084923/H20850/H20852is Galilean invariant and the total mass of the environment /H20849or reservoir /H20850 is much larger than the DW mass.48A very similar conclu- sion is obtained by Garate et al.29By analyzing the Kamber- sky mechanism,49which is reported50to be the dominant damping mechanism in transition metals such as Fe, Co, Ni,they found that the ratio /H9252//H9251is approximately given by the ratio between the drift velocity of the Kohn-Sham quasipar- ticles and vs. Since the collection of Kohn-Sham quasiparti-cles play the role of the environment in case of the Kamber- sky mechanism, the result in Ref. 29is consistent with ours. It is interesting to note that our calculation, which is largelyindependent of details of damping mechanism, reproducesthe result for the speciﬁc case. 29This implies that the result in Ref. 29can be generalized if the drift velocity of the Kohn-Sham quasiparticles is replaced by the general cou-pling constant vbetween the current and the environment. Our claim that the origin of /H9252is the direct coupling be- tween the current and environment has an interesting con-ceptual consistency with the work by Zhang and Li. 4Zhang and Li derived the nonadiabatic term by introducing a spin-relaxation term in the equation of motion of the conduction electrons. A clear consistency arises from generalizing thespin relaxation in Ref. 4to the coupling with environment in our work. In Ref. 4, Gilbert damping /H20849 /H9251/H20850and the nonadia- batic STT /H20849/H9252/H20850are identiﬁed as the spin relaxation of magne- tization and conduction electrons, respectively. Generalizingthe spin relaxation to environmental coupling, /H9251and/H9252areKYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-6now identiﬁed as the coupling of the environment with the magnetization /H20849i.e., the DW in our model /H20850and the coupling of the environment with current, respectively. It is exactlyhow we identiﬁed /H9251and/H9252, and this gives the conceptual consistency between our work and Ref. 4. As an additional comment, while some magnitudes and origins of /H9252claimed in different references, such as Refs. 4and29, seem to be based on completely independent phenomena, our work andinterpretation on /H9252provide a connection between them through the environmental degrees of freedom. B. Effect of environment on stochastic forces Until now, our considerations has been limited to the evo- lution of the expectation values /H20855Q/H20849t/H20850/H20856and/H20855P/H20849t/H20850/H20856and thus thermal ﬂuctuation effects have been ignored. In this section,we address the issue of thermal ﬂuctuations. For this pur-pose, we need to go beyond the expectation values and so wederive the following operator equations from the Hamil- tonian Eq. /H2084923/H20850: Q˙= vs+/H9251S 2KMP˙+P M+/H92571/H20849t/H20850, /H2084929a/H20850 P˙=−2/H9251KM SQ˙+2/H9251KM Sv+/H92572/H20849t/H20850, /H2084929b/H20850 where /H92571/H20849t/H20850=/H20858 i/H9253i/H9275i/H20873/H9004xisin/H9275it−/H9004pi mi/H9275icos/H9275it/H20874,/H2084930a/H20850 /H92572/H20849t/H20850=/H20858 i/H20849mi/H9275i2/H9004xicos/H9275it+/H9275i/H9004pisin/H9275it/H20850./H2084930b/H20850 Here/H9004xi/H11013xi/H208490/H20850−Q/H208490/H20850and/H9004pi/H11013pi/H208490/H20850−/H9253iP/H208490/H20850. The deriva- tion of Eqs. /H2084929/H20850and/H2084930/H20850utilizes constraints Eqs. /H2084920/H20850and /H2084927/H20850. We remark that the result in Sec. II D can be recovered from Eqs. /H2084929/H20850and/H2084930/H20850by taking the expectation values of the operators. When Eq. /H2084929/H20850is compared to Eq. /H208496/H20850,i ti s evident that /H92571/H20849t/H20850and/H92572/H20849t/H20850deﬁned in Eq. /H2084930/H20850carry the information about the thermal noise. It is easy to verify thatthe expectation values of /H92571/H20849t/H20850and/H92572/H20849t/H20850vanish, thus repro- ducing the results in the earlier section. Here it should benoticed that Eq. /H2084930/H20850relates /H92571/H20849t/H20850and/H92572/H20849t/H20850in the nonequi- librium situations /H20849after the current is turned on or t/H110220/H20850to the operators /H9004xiand/H9004pi, which are deﬁned in the equilib- rium situation /H20849right before the current is turned on or t=0/H20850. Thus by combining Eq. /H2084930/H20850with the equilibrium noise char- acteristics of /H9004xiand/H9004pi, we can determine the thermal noise characteristic in the nonequilibrium situation /H20849t/H110220/H20850. To extract information about the noise, one needs to evaluate the correlation functions /H20855/H20853/H9257i/H20849t/H20850,/H9257j/H20849t/H20850/H20854/H20856 /H20849i,j=1,2/H20850, where /H20853,/H20854denotes the anticommutator. Due to the relations in Eq./H2084930/H20850, the evaluation of the correlation function reduces to the expectation value evaluation of the operator products/H20853x i/H208490/H20850,pj/H208490/H20850/H20854,xi/H208490/H20850xj/H208490/H20850, and pi/H208490/H20850pj/H208490/H20850in the equilibrium situation governed by the equilibrium Hamiltonian /H20851Eq. /H2084919/H20850/H20852. In the classical limit /H20849/H6036→0, see the next paragraph to ﬁnd out when the classical limit is applicable /H20850, Eq./H2084919/H20850is just acollection of independent harmonic oscillators of /H20853/H9004xi,/H9004pi/H20854. Hence, the equipartition theorem determines their correla-tions, /H20855/H9004x i/H20856=/H20855/H9004pi/H20856=/H20855/H9004xi/H9004pi/H20856=0 , /H2084931a/H20850 /H20855/H9004xi/H9004xj/H20856=kBT mi/H9275i2/H9254ij, /H2084931b/H20850 /H20855/H9004pi/H9004pj/H20856=mikBT/H9254ij. /H2084931c/H20850 Equation /H2084920/H20850and/H2084931/H20850give the correlations of /H92571/H20849t/H20850and /H92572/H20849t/H20850. After some algebra, one straightforwardly gets /H20855/H9257i/H20849t/H20850/H20856=0 , /H2084932a/H20850 /H20855/H92571/H20849t/H20850/H92572/H20849t/H11032/H20850/H20856=0 , /H2084932b/H20850 /H20855/H92571/H20849t/H20850/H92571/H20849t/H11032/H20850/H20856=/H9251S 2KMkBT/H9254/H20849t−t/H11032/H20850, /H2084932c/H20850 /H20855/H92572/H20849t/H20850/H92572/H20849t/H11032/H20850/H20856=2/H9251KM SkBT/H9254/H20849t−t/H11032/H20850. /H2084932d/H20850 These relations are consistent with Eq. /H208495/H20850when/H92571/H20849t/H20850and /H92572/H20849t/H20850in Eq. /H2084930/H20850are identiﬁed with those in Eq. /H208496/H20850. Thus they conﬁrm that the relations /H20851Eq./H2084932/H20850/H20852assumed in many papers9–13indeed hold rather generally in the regime where the tilting angle remains sufﬁciently smaller than /H9266/4. Next we consider the regime where the condition of the classical limit is valid. Since statistical properties of the sys- tem at ﬁnite temperature is determined bykBT /H6036, the classical limit/H20849/H6036→0/H20850is equivalent to the high-temperature limit /H20849T →/H11009/H20850. Thus, in actual experimental situations, the above cor- relation relations, Eq. /H2084932/H20850, will be satisﬁed at high tempera- ture. In this respect, we ﬁnd that most experimental situa-tions belong to the high-temperature regime. See AppendixD for the estimation of the “threshold” temperature, abovewhich Eq. /H2084932/H20850is applicable. In Appendix D, the correlations in the high temperatures are derived more rigorously. Finally we comment brieﬂy on the low-temperature quan- tum regime. In this regime, one cannot use the equipartitiontheorem since the system is not composed of independentharmonic oscillators, that is, /H20851/H9004x i,/H9004pj/H20852=i/H6036/H20849/H9254ij+/H9253j/H20850. Note that the commutator contains an additional term i/H6036/H9253j. Here, the additional term i/H6036/H9253jcomes from the commutator /H20851−Q, −/H9253jP/H20852. Then, Eq. /H2084932/H20850, which is assumed in other papers,9–13 is not guaranteed any more. IV . CONCLUSION In this paper, we examine the effect of ﬁnite current on thermal ﬂuctuation of current-induced DW motion by con-structing generalized Caldeira-Leggett-type Hamiltonian ofthe DW dynamics, which describes not only energy-conserving dynamics processes but also the Gilbert dampingand STT. Unlike the classical damping worked out by Cal-deira and Leggett, 32the momentum coupling is indispensable to describe the Gilbert damping. This is also related to theTHERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-7rotational symmetry of spin-interaction nature. It is demon- strated that the derived Caldeira-Leggett-type quantum-mechanical Hamiltonian reproduces the well-known DWequations of motion. Our Hamiltonian also illustrates that the nonadiabatic STT is closely related with the coupling of the spin current tothe environment. Thus, the environmental degrees of free-dom are responsible for both the Gilbert damping /H20849 /H9251/H20850and the nonadiabatic STT /H20849/H9252/H20850. By this process, the ratio of /H9252and/H9251 was derived to be the ratio of current-DW coupling and current-environment coupling. The nonadiabatic term isnothing but the result of the direct coupling between thecurrent and environment in our theory. By using the Calderia-Leggett-type Hamiltonian, which describes the time evolution of the system, we obtained theexpression of stochastic forces caused by thermal noise inthe presence of the ﬁnite current. By calculating the equilib-rium thermal ﬂuctuation at high temperature, we verify thatwhen j pis sufﬁciently smaller than the intrinsic critical den- sity, jpdoes not modify the correlation relations of thermal noise unless the temperature is extremely low. The upperbound of the critical temperature, below which the aboveconclusion does not apply, is obtained by reexamining thesystem with Feynman path integral. The bound is muchlower than the temperature in most experimental situations. Lastly we remark that the Joule heating 51is an important factor that affects the thermal ﬂuctuation ﬁeld since it raisesthe temperature of the nanowire. The degree of the tempera-ture rise depends on the thermal conductivities and heat ca-pacities of not only the nanowire but also its surroundingmaterials such as substrate layer materials of the nanowire.Such factors are not taken into account in this paper. Simul-taneous account of the Joule heating dynamics and the ther-mal ﬂuctuation ﬁeld /H20849in the presence of current /H20850goes beyond the scope of the paper and may be a subject of future re-search. ACKNOWLEDGMENTS We acknowledge critical comment by M. Stiles, who pointed out the importance of the momentum coupling andinformed us of Ref. 29. This work was ﬁnancially supported by the NRF /H20849Grants No. 2007-0055184, No. 2009-0084542, and No. 2010-0014109 /H20850and BK21. K.W.K. acknowledges the ﬁnancial support by the TJ Park. APPENDIX A: EFFECTIVE HAMILTONIAN OF THE DW MOTION FROM 1D s-dMODEL (Ref. 52) The starting point is 1D s-dmodel, Hs-d=−J/H20858 iS/H6023i·S/H6023i+1−A/H20858 i/H20849S/H6023i·zˆ/H208502+K/H20858 i/H20849S/H6023i·yˆ/H208502+HcS, /H20849A1/H20850 as mentioned in Sec. II A. In order to consider the DW dynamics, one ﬁrst introduce the classical DW proﬁle initially given by /H20855S/H6023i·xˆ/H20856=Ssin/H9258/H20849zi/H20850, /H20849A2a/H20850/H20855S/H6023i·yˆ/H20856=0 , /H20849A2b/H20850 /H20855S/H6023i·zˆ/H20856=Scos/H9258/H20849zi/H20850, /H20849A2c/H20850 where ziis the position of the ith localized spin, and /H9258/H20849z/H20850 =2 cot−1e−/H208812A/Ja2/H20849z−q/H20850. Here qis the classical position of the DW. Small quantum ﬂuctuations of spins on top of the clas-sical DW proﬁle can be described by the Holstein-Primakoffboson operator b i, to describe magnon excitations. Kim et al.43found eigenmodes of these quantum ﬂuctuations in the presence of the classical DW background, which amount toquantum mechanical version of the classical vibration eigen-modes in the presence of the DW background reported longtime ago by Winter. 53The corresponding eigenstates of this Hamiltonian are composed of spin-wave states with the ﬁnite eigenenergy Ek=/H20881/H20849JSa2k2+2AS/H20850/H20849JSa2k2+2AS+2KS/H20850 /H20849/H113502S/H20881A/H20849A+K/H20850/H20850and so-called bound magnon states with zero energy Ew=0. Here, kis the momentum of spin wave states and ais the lattice spacing between two neighboring spins. Let akandbwdenote proper linear combinations of bi andbi†, which represent the boson annihilation operators of ﬁnite-energy spin-wave states and zero-energy bound mag-non states, respectively. In terms of these operators, Eq. /H208498/H20850 reduces to H s-d=P2 2M+/H20858 kEkak†ak+HcS, /H20849A3/H20850 where higher-order processes describing magnon-magnon in- teractions are ignored. Here Mis the so-called Döring mass,44deﬁned as M=/H60362 K/H208812A Ja4, and Pis deﬁned as −i/H6036/H208492AS2 Ja4/H208501/4/H20849bw†−bw/H20850. According to Ref. 43,Pis a translation generator of the DW position, that is, exp /H20849iPq 0//H6036/H20850shifts the DW position by q0. Thus Pcan be interpreted as a canonical momentum of the DW translational motion. The ﬁrst term inEq./H20849A3/H20850, which amounts to the kinetic energy of the DW translational motion, implies that Mis the DW mass. We identify this Mwith the undetermined constant Min Eq. /H208496/H20850. According to Ref. 43,Pis also proportional to the degree of the DW tilting, that is, /H20849b w†−bw/H20850/H11008Siy. In the adiabatic limit, that is, when the DW width /H9261is sufﬁciently large in view of the electron dynamics, the re-maining term H cScan be represented in a simple way in terms of the bound magnon operators and the adiabatic ve-locity of the DW, 20,43 HcS=vsP. /H20849A4/H20850 Then the effective s-dHamiltonian of the DW motion be- comes Hs-d=P2 2M+vsP+/H20858 kEkak†ak. /H20849A5/H20850 Note that the bound magnon part and the spin-wave part are completely decoupled in Eq. /H20849A5/H20850since Pcontains only the bound magnon operators, which commute with the spin-wave operators. The DW position operator should satisfy the following two properties: geometrical relation /H20855Q/H20856−q=a 2S/H20858i/H20855S/H6023i·zˆ/H20856andKYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-8canonical relation /H20851Q,P/H20852=i/H6036. Then, one can show that Q =q−/H20849Ja4 32AS2/H208501/4/H20849bw†+bw/H20850satisﬁes these two properties. Note thatQis expressed in terms of the bound magnon operators. Then as far as the Heisenberg equations of motion for Qand Pare concerned, the last term in Eq. /H20849A5/H20850does not play any role. This term will be ignored from now on. Thus, the ef-fective Hamiltonian for the DW motion is reduced to H 0=P2 2M+vsP /H20849A6/H20850 so we got Eq. /H2084910/H20850, the effective Hamiltonian of the DW motion. APPENDIX B: SOLUTION FOR A GENERAL QUADRATIC DAMPING This section provides the solution of the equation of mo- tion for a general quadratic damping. This is applicable notonly for the generalized Caldeira-Leggett description in thispaper but also for any damping type which quadraticallyinteracts with the DW. In general, let us consider a general quadratic damping Hamiltonian, H=P 2 2M+vsP+/H20858 i/H9273iTAi/H9273i, /H20849B1/H20850 where /H9273i=/H20849QPx ipi/H20850T, and Aii sa4/H110034 Hermitian matrix. Now, one straightforwardly gets the corresponding coupledequations, dQ dt=P M+vs+/H20858 i/H20849B21iQ+B22iP+B23ixi+B24ipi/H20850, /H20849B2a/H20850 dP dt=−/H20858 i/H20849B11iQ+B12iP+B13ixi+B14ipi/H20850,/H20849B2b/H20850 dxi dt=B41iQ+B42iP+B43ixi+B44ipi, /H20849B2c/H20850 dpi dt=−/H20849B31iQ+B32iP+B33ixi+B34ipi/H20850./H20849B2d/H20850 Here, Bii sa4 /H110034 real symmetric matrix deﬁned as Bi =2 Re /H20851Ai/H20852, and Bjkiis the element of Biinjth row and kth column. With the Laplace transform of the expectation values of each operator, for example, Q˜/H20849/H9261/H20850/H11013L/H20851Q/H20849t/H20850/H20852/H20849/H9261/H20850=/H20885 0/H11009 /H20855Q/H20849t/H20850/H20856e−/H9261tdt, /H20849B3/H20850 the set of coupled equations transforms as /H9261Q˜−/H20855Q/H208490/H20850/H20856=P˜ M+vs /H9261+/H20858 i/H20849B21iQ˜+B22iP˜+B23ix˜i+B24ip˜i/H20850, /H20849B4a/H20850/H9261P˜−/H20855P/H208490/H20850/H20856=−/H20858 i/H20849B11iQ˜+B12iP˜+B13ix˜i+B14ip˜i/H20850, /H20849B4b/H20850 /H9261x˜i−/H20855xi/H208490/H20850/H20856=B41iQ˜+B42iP˜+B43ix˜i+B44ip˜i,/H20849B4c/H20850 /H9261p˜i−/H20855pi/H208490/H20850/H20856=−/H20849B31iQ˜+B32iP˜+B33ix˜i+B34ip˜i/H20850. /H20849B4d/H20850 Rewriting these in matrix forms, the equations become sim- pler as /H9261/H20873Q˜ P˜/H20874−/H20898/H20855Q/H208490/H20850/H20856+vs /H9261 /H20855P/H208490/H20850/H20856/H20899=/H20902/H2089801 M 00/H20899+/H20858 i/H20873B21iB22i −B11i−B12i/H20874/H20903 /H11003/H20873Q˜ P˜/H20874+/H20858 i/H20873B23iB24i −B13i−B14i/H20874 /H11003/H20873x˜i p˜i/H20874, /H20849B5a/H20850 /H9261/H20873x˜i p˜i/H20874−/H20873/H20855xi/H208490/H20850/H20856 /H20855pi/H208490/H20850/H20856/H20874=/H20873B41iB42i −B31i−B32i/H20874/H20873Q˜ P˜/H20874 +/H20873B43iB44i −B33i−B34i/H20874/H20873x˜i p˜i/H20874. /H20849B5b/H20850 From Eq. /H20849B5b/H20850, one can calculate /H20849x˜ip˜i/H20850Tin terms of Q˜and P˜, /H20873x˜i p˜i/H20874=/H20873/H9261−B43i−B44i B33i/H9261+B34i/H20874−1/H20873B41iB42i −B31i−B32i/H20874/H20873Q˜ P˜/H20874 +/H20873/H9261−B43i−B44i B33i/H9261+B34i/H20874−1/H20873/H20855xi/H208490/H20850/H20856 /H20855pi/H208490/H20850/H20856/H20874. /H20849B6/H20850 From Eqs. /H20849B5a/H20850and/H20849B6/H20850, one ﬁnally gets the equation of /H20849Q˜P˜/H20850T,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-9/H20900/H20898/H9261−1 M 0/H9261/H20899−/H20858 i/H20877/H20873B21iB22i −B11i−B12i/H20874+/H20873B23iB24i −B13i−B14i/H20874/H20873/H9261−B43i−B44i B33i/H9261+B34i/H20874−1/H20873B41iB42i −B31i−B32i/H20874/H20878/H20901/H20873Q˜ P˜/H20874 =/H20898/H20855Q/H208490/H20850/H20856+vs /H9261 /H20855P/H208490/H20850/H20856/H20899+/H20858 i/H20873B23iB24i −B13i−B14i/H20874/H20873/H9261−B43i−B44i B33i/H9261+B34i/H20874−1/H20873/H20855xi/H208490/H20850/H20856 /H20855pi/H208490/H20850/H20856/H20874. /H20849B7/H20850 Inverting the matrix in front of /H20849Q˜P˜/H20850T, one can get the solu- tion of /H20849Q˜P˜/H20850T. Then, ﬁnally, the solution /H20849/H20855Q/H20856/H20855P/H20856/H20850Tis ob- tained by the inverse Laplace transform of /H20849Q˜P˜/H20850T, /H20873/H20855Q/H20849t/H20850/H20856 /H20855P/H20849t/H20850/H20856/H20874=L−1/H20875/H20873Q˜/H20849/H9261/H20850 P˜/H20849/H9261/H20850/H20874/H20876. /H20849B8/H20850 APPENDIX C: SOLUTION OF EQ. ( 24) In the special case that current is applied at t=0,/H9008/H20849t/H20850in Eq./H2084925/H20850becomes Heaviside step function. This is the case we are interested in. In a real DW system, the DW velocityjumps from 0 to a ﬁnite value at the moment that the spincurrent starts to be applied. This jumping comes from thediscontinuity in Eq. /H2084925/H20850which makes the Hamiltonian dis- continuous. Right before the current is applied, the DW re-mains on the stable /H20849or equilibrium /H20850state described by Eq. /H2084922/H20850. Suppose that Eq. /H2084920a/H20850also holds for /H9261=0. Then, Eq. /H2084924/H20850 transforms as /H20849up to constant /H20850 H tot=P2 2M+vs/H20849t/H20850P+/H20858 i/H208751 2mi/H20849pi−/H9253iP+miv/H20849t/H20850/H208502/H20876 +/H20858 i1 2mi/H9275i2/H20849xi−Q/H208502. /H20849C1/H20850 Performing the canonical transform pi→pi−miv/H20849t/H20850, one can transform this Hamiltonian in the form of Eq. /H20849B1/H20850, Htot=P2 2M+vs/H20849t/H20850P+/H20858 i/H208751 2mi/H20849pi−/H9253iP/H208502+1 2mi/H9275i2/H20849xi−Q/H208502/H20876. Here, one of the constraints Eq. /H2084920a/H20850is generalized to hold even for /H9261=0, so that /H20858i/H9253i=0. Note that the discontinuity due tov/H20849t/H20850is absorbed in the new pi. Thus, Eq. /H2084922b/H20850should be written as /H20855pi/H208490+/H20850/H20856=/H20855pi/H208490−/H20850/H20856+miv=/H9253i/H20855P/H208490/H20850/H20856+miv./H20849C2/H20850 The initial condition of xiis the same as Eq. /H2084922a/H20850. Now, using these initial conditions and Eqs. /H20849B7/H20850and/H20849B8/H20850under the constraints in Eq. /H2084920/H20850, one gets the solution of this sys- tem as Eq. /H2084926/H20850.APPENDIX D: CORRELATIONS OF STOCHASTIC FORCES AT HIGH TEMPERATURE This section provides the quantum derivation of correla- tion relations of stochastic forces at high temperature. Theclassical correlation relations in Eq. /H2084932/H20850are valid quantum mechanically at high temperature. Since Eq. /H2084931/H20850implies Eq. /H2084932/H20850, it sufﬁces to show Eq. /H2084931/H20850in this section. The basic strategy is studying statistical properties of the HamiltonianEq./H2084919/H20850/H20849under quadratic potential bQ 2/H2085054by the Feynman path integral along the imaginary-time axis. The Feynmanpath integral of a system described by a quadratic Lagrang-ian is proportional to the exponential of the action valueevaluated at the classical solution. Hence, the key point ofthe procedure is to get the classical solution with imaginarytime. 1. General relations a. Classical action under high-temperature limit Deﬁne a column vector /H9273=/H20849Qx1x2¯/H20850T. Let the Euclidean Lagrangian of the system be LE=1 2/H9273˙TA/H9273˙+1 2/H9273B/H9273, where A andBare symmetric matrices. /H20849The symbols “ A” and “ B” are not the same as those in Appendix B. /H20850Explicitly, L =1 2/H20858nmx˙nAnmx˙m+1 2/H20858nmxnBnmxm. Here x0/H11013Q./H11509LE /H11509x˙n=/H20858mAnmx˙m =A/H9273˙and/H11509LE /H11509xn=/H20858mBnmxm=B/H9273lead to the classical equation of motion, A/H9273¨=B/H9273. /H20849D1/H20850 The classical action value Sc/H20849evaluated at the classical path /H20850 is then, Sc=/H208480/H9270LEdt=1 2/H208480/H9270/H20849/H9273˙TA/H9273˙+/H9273B/H9273/H20850dt=1 2/H9273TA/H9273˙/H208410/H9270 +/H208480/H9270/H20849−/H9273TA/H9273¨+/H9273B/H9273/H20850dt=1 2/H9273TA/H9273˙/H208410/H9270. Here, /H9270=/H6036/kBT. Now, the only thing one needs is to ﬁnd /H9273˙at boundary points. In the case of Eq. /H2084919/H20850,Ais invertible. Hence, the equa- tion becomes /H9273¨=A−1B/H9273. Suppose that A−1Bis diagonaliz- able, that is A−1B=C−1DC. Here Dnm=/H9261n/H9254nmis diagonal ma- trix and /H9261nisnth eigenvalue of A−1B. Deﬁne a new vector /H9264=C/H9273. Finally, we get the equation, /H9264¨=/H20898/H926100¯ 0/H92611¯ ]]/GS/H20899/H9264. /H20849D2/H20850 Imposing the boundary condition /H9273/H208490/H20850=/H9273i,/H9273/H20849/H9270/H20850=/H9273fand de- ﬁning the corresponding /H9264i=C/H9273i,/H9264f=C/H9264f, then one gets the solution of /H9264and its derivative straightforwardly,KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-10/H9264n=/H9264fn+/H9264in 2cosh/H20881/H9261n/H20873t−/H9270 2/H20874 cosh/H20881/H9261n/H9270 2+/H9264fn−/H9264in 2sinh/H20881/H9261n/H20873t−/H9270 2/H20874 sinh/H20881/H9261n/H9270 2, /H20849D3/H20850 /H9264˙n=/H20881/H9261n/H20900/H9264fn+/H9264in 2sinh/H20881/H9261n/H20873t−/H9270 2/H20874 cosh/H20881/H9261n/H9270 2 +/H9264fn−/H9264in 2cosh/H20881/H9261n/H20873t−/H9270 2/H20874 sinh/H20881/H9261n/H9270 2/H20901. /H20849D4/H20850 Now, /H9264˙at boundary points are obtained as /H9264˙n/H208490/H20850=/H20881/H9261n/H20873−/H9264fn+/H9264in 2tanh/H20881/H9261n/H9270 2+/H9264fn−/H9264in 2coth/H20881/H9261n/H9270 2/H20874, /H20849D5/H20850 /H9264˙n/H20849/H9270/H20850=/H20881/H9261n/H20873/H9264fn+/H9264in 2tanh/H20881/H9261n/H9270 2+/H9264fn−/H9264in 2coth/H20881/H9261n/H9270 2/H20874. /H20849D6/H20850 If/H20881/H20841/H9261n/H20841/H9270 2=/H20881/H20841/H9261n/H20841/H6036 2kBT/H112701, tanh/H20881/H9261n/H9270 2/H11015/H20881/H9261n/H9270 2. Then, /H9264˙n/H208490/H20850/H11015−/H9264fn+/H9264in 2/H9261n/H9270 2+/H9264fn−/H9264in /H9270, /H20849D7/H20850 /H9264˙n/H20849/H9270/H20850/H11015/H9264fn+/H9264in 2/H9261n/H9270 2+/H9264fn−/H9264in /H9270. /H20849D8/H20850 In matrix form, /H9264˙/H208490/H20850/H11015−D/H9264f+/H9264i 2/H9270 2+/H9264f−/H9264i /H9270=−DC/H9273f+/H9273i 2/H9270 2+C/H9273f−/H9273i /H9270, /H20849D9/H20850 /H9264˙/H20849/H9270/H20850/H11015D/H9264f+/H9264i 2/H9270 2+/H9264f−/H9264i /H9270=DC/H9273f+/H9273i 2/H9270 2+C/H9273f−/H9273i /H9270. /H20849D10/H20850 Using A−1B=C−1DC, it leads to /H9273˙/H208490/H20850/H11015−A−1B/H9273f+/H9273i 2/H9270 2+/H9273f−/H9273i /H9270, /H20849D11/H20850 /H9273˙/H20849/H9270/H20850/H11015A−1B/H9273f+/H9273i 2/H9270 2+/H9273f−/H9273i /H9270. /H20849D12/H20850 Finally one can obtain the classical action,Sc=1 2/H9273TA/H9273˙/H208410/H9270=/H20873/H9273f+/H9273i 2/H20874T B/H20873/H9273f+/H9273i 2/H20874/H9270 2 +/H20873/H9273f−/H9273i 2/H20874T A/H20873/H9273f−/H9273i 2/H208742 /H9270. /H20849D13/H20850 This is valid even if some eigenvalues are zero. /H20849By taking limit of /H9261i→0, cosh and sinh becomes constant and linear, respectively. /H20850 b. Propagator and its derivatives The propagator is given by the Feynman path integral, K/H20849/H9273f,/H9273i;/H9270/H20850=/H20855/H9273f/H20841e−H/kBT/H20841/H9273i/H20856=/H20848D/H9273e−/H20848LEdt//H6036, where D/H9273=/H20863iDxi. For quadratic Lagrangian, it is well known that/H20848D /H9273e−/H20848LEdt/H6036=F/H20849/H9270/H20850e−Sc//H6036. Here F/H20849/H9270/H20850is a smooth function de- pendent on /H9270only. Now we aim to calculate K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850. It is easy to obtain the corresponding classical action by replacing /H9273f =/H9273i+/H9254/H9273in Eq. /H20849D13/H20850, Sc/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850=/H9270 2/H9273iTB/H9273i+/H9270 2/H9273iTB/H9254/H9273+/H9270 8/H9254/H9273TB/H9254/H9273 +1 2/H9270/H9254/H9273TA/H9254/H9273. /H20849D14/H20850 Then, K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850is/H20849up to second order of /H9254/H9273/H20850, K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850=F/H20849/H9270/H20850e−Sc//H6036=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i /H11003/H208751−1 /H6036/H20873/H9270 2/H9273iTB/H9254/H9273+/H9270 8/H9254/H9273TB/H9254/H9273 +1 2/H9270/H9254/H9273TA/H9254/H9273/H20874+1 2/H60362/H20873/H9270 2/H9273iTB/H9254/H9273/H208742/H20876. Zeroth order: F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i. First order:−/H9270 2/H6036F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H9273iTB/H9254/H9273 =−/H9270 2/H6036F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20858nmxinBnm/H9254xm. Second order: F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1 /H6036/H20873/H9270 8/H9254/H9273TB/H9254/H9273 +1 2/H9270/H9254/H9273TA/H9254/H9273/H20874+1 2/H60362/H20873/H9270 2/H9273iTB/H9254/H9273/H208742/H20878 =F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1 2/H6036/H20858 nm/H9254xn/H20873/H9270 4Bnm+1 /H9270Anm/H20874/H9254xm +/H92702 8/H60362/H20873/H20858 klmnxikBkn/H9254xnxilBlm/H9254xm/H20874/H20878. /H20849D15/H20850 By the relation, K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850=K/H20849/H9273i,/H9273i;/H9270/H20850+/H20858m/H11509K /H11509xfm/H9254xm +/H20858nm1 2/H115092K /H11509xfn/H11509xfm/H9254xn/H9254xm+O/H20849/H9254/H92733/H20850, K/H20849/H9273i,/H9273i;/H9270/H20850=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i, /H20849D16/H20850THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-11/H20879/H11509K /H11509xfm/H20879 /H9273i=/H9273f=−/H9270 2/H6036F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20858 nBnmxin,/H20849D17/H20850 /H20879/H115092K /H11509xfn/H11509xfm/H20879 /H9273i=/H9273f=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1 /H6036/H20873/H9270 4Bnm+1 /H9270Anm/H20874 +/H92702 4/H60362/H20873/H20858 klBknxikBlmxil/H20874/H20878 =F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1 /H6036/H20873/H9270 4Bnm+1 /H9270Anm/H20874 +/H92702 4/H60362/H20873/H20858 kBknxik/H20874/H20873/H20858 kBkmxik/H20874/H20878./H20849D18/H20850 c. Correlations Statistical average of an operator Ais given byTr/H20849Ae−H/kBT/H20850 Tr/H20849e−H/kBT/H20850. What we want to ﬁnd are the averages of /H9004xn/H9004xm,/H9004pn/H9004pm, and/H20853/H9004xn,/H9004pm/H20854for/H9004xn/H11013xn−Qand/H9004pn/H11013pn−/H9253nP, Tr/H20849/H9004xn/H9004xme−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841/H9004xn/H9004xme−H/kBT/H20841/H9273i/H20856 =/H20885d/H9273i/H20849xin−Qi/H20850/H20849xim−Qi/H20850/H20855/H9273i/H20841e−H/kBT/H20841/H9273i/H20856 =/H20885d/H9273i/H20849xin−Qi/H20850/H20849xim−Qi/H20850K/H20849/H9273i,/H9273i;/H9270/H20850, /H20849D19/H20850 Tr/H20849/H9004pn/H9004pme−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841/H9004pn/H9004pme−H/kBT/H20841/H9273i/H20856 =−/H60362/H20885/H20879d/H9273i/H20873/H11509 /H11509xfn−/H9253n/H11509 /H11509Qf/H20874 /H11003/H20873/H11509 /H11509xfm−/H9253m/H11509 /H11509Qf/H20874K/H20849/H9273f,/H9273i;/H9270/H20850/H20879 /H9273i=/H9273f, /H20849D20/H20850 Tr/H20849/H9004xn/H9004pme−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841/H9004xn/H9004pme−H/kBT/H20841/H9273i/H20856 =−i/H6036/H20885/H20879d/H9273i/H20849xin−Qi/H20850/H20873/H11509 /H11509xfm −/H9253m/H11509 /H11509Qf/H20874K/H20849/H9273f,/H9273i;/H9270/H20850/H20879 /H9273i=/H9273f, /H20849D21/H20850 Tr/H20849e−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841e−H/kBT/H20841/H9273i/H20856=/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850, /H20849D22/H20850 where d/H9273i=/H20863ndxin.2. Correlations under quadratic potential Under potential bQ2, the matrices AandBcorresponding the Hamiltonian Eq. /H2084919/H20850are A=/H20898MM /H92531 M/H92532 ¯ M/H92531M/H925312+m1M/H92531/H92532¯ M/H92532M/H92532/H92531M/H925322+m2¯ ]] ] /GS /H20899,/H20849D23/H20850 B=/H20898b+/H20858 nmn/H9275n2 −m1/H927512−m2/H927522¯ −m1/H927512m1/H927512 0 ¯ −m2/H9275220 m2/H927522¯ ]] ] /GS/H20899./H20849D24/H20850 Then, e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273iis written as e−/H20849/H9270/2/H6036/H20850/H20851/H20858nmn/H9275n2/H20849Qi−xin/H208502+bQi2/H20852. a. x-x correlations Since K/H20849/H9273,/H9273;/H9270/H20850is an even function of /H20849xn−Qi/H20850,i ti s trivial that Tr /H20849/H9004xn/H9004xme−H/kBT/H20850=0 unless n=m. For n=m,T r /H20849/H9004xn2e−H/kBT/H20850=/H20848d/H9273i/H20849xin−Qi/H208502K/H20849/H9273i,/H9273i;/H9270/H20850. Thus, Tr/H20849/H9004xn2e−H/kBT/H20850 Tr/H20849e−H/kBT/H20850=/H20885dxin/H20849xin−Qi/H208502e−/H20849/H9270/2/H6036/H20850mnwn2/H20849Qi−xin/H208502 /H20885dxine−/H20849/H9270/2/H6036/H20850mnwn2/H20849Qi−xin/H208502 =/H6036 /H9270mnwn2=kBT mnwn2. /H20849D25/H20850 So, ﬁnally one gets /H20855/H9004xn/H9004xm/H20856=kBT mnwn2/H9254nm. b. x-p correlations Explicitly rewriting the derivative of K, /H20879/H11509K /H11509xfm/H20879 /H9273i=/H9273f=−/H9270 2/H6036K/H20849/H9273i,/H9273i;/H9270/H20850mm/H9275m2/H20849xim−Qi/H20850for/H20849m /HS110050/H20850, /H20849D26/H20850 /H20879/H11509K /H11509Qf/H20879 /H9273i=/H9273f=−/H9270 2/H6036K/H20849/H9273i,/H9273i;/H9270/H20850/H20877bQi2+/H20858 nmn/H9275n2/H20849Qi−xin/H20850/H20878. /H20849D27/H20850 Using the above relations,KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-12Tr/H20849/H9004xn/H9004pme−H/kBT/H20850=−i/H6036/H20885/H20879d/H9273i/H20849xin−Qi/H20850/H20873/H11509 /H11509xfm−/H9253m/H11509 /H11509Qf/H20874K/H20849/H9273f,/H9273i;/H9270/H20850/H20879 /H9273i=/H9273f =−i/H9270 2/H20885d/H9273i/H20849xin−Qi/H20850/H20877/H9253mbQi2+/H9253m/H20858 lml/H9275l2/H20849Qi−xil/H20850+mm/H9275m2/H20849Qi−xim/H20850/H20878K/H20849/H9273i,/H9273i;/H9270/H20850 =−i/H9270 2/H20885d/H9273i/H20849xin−Qi/H20850/H20877/H9253m/H20858 lml/H9275l2/H20849Qi−xil/H20850+mm/H9275m2/H20849Qi−xim/H20850/H20878K/H20849/H9273i,/H9273i;/H9270/H20850 =i/H9270 2/H9253m/H20858 lml/H9275l2Tr/H20849/H20849/H9004xin/H9004xile−H/kBT/H20850/H20850+mm/H9275m2Tr/H20849/H9004xin/H9004xime−H/kBT/H20850. /H20849D28/H20850 In the third line, it is used that /H20848dxin/H20849xin−Qi/H20850 /H11003/H20851even function of /H20849xin−Qi/H20850/H20852=0. One can now write the x-pcorrelations in terms of x-x correlations. /H20855/H9004xn/H9004pm/H20856=i/H9270 2/H20873/H9253m/H20858 lml/H9275l2/H20855/H9004xin/H9004xil/H20856+mm/H9275m2/H20855/H9004xin/H9004xim/H20856/H20874 =i/H9270kBT 2/H20873/H9253m/H20858 l/H9254nl+/H9254nm/H20874=i/H6036 2/H20849/H9253m+/H9254nm/H20850,/H20849D29/H20850 which is purely imaginary. Thus, /H20855/H20853/H9004xn,/H9004pm/H20854/H20856=/H20855/H9004xn/H9004pm/H20856 +/H20855/H9004xn/H9004pm/H20856/H11569=0. c. p-p correlations It is convenient to calculate /H20848d/H9273i/H115092K /H11509xfn/H11509xfm/H20841/H9273i=/H9273f. The trickiest part is /H20848d/H9273i/H20858kBknxik/H20858kBkmxikK/H20849/H9273i,/H9273i;/H9270/H20850, n/HS110050,m/HS110050:/H20858 kBknxik/H20858 kBkmxik=mn/H9275n2/H20849xin−Qi/H20850mm/H9275m2/H20849xim −Qi/H20850, n=0 , m/HS110050:/H20858 kBknxik/H20858 kBkmxik=/H20873/H20858 kmk/H9275k2/H20849Qi−xik/H20850 +bQi/H20874mm/H9275m2/H20849xim−Qi/H20850, n=0 , m=0 :/H20858 kBknxik/H20858 kBkmxik=/H20873/H20858 kmk/H9275k2/H20849Qi−xik/H20850+bQi/H20874 /H11003/H20873/H20858 kmk/H9275k2/H20849Qi−xik/H20850+bQi/H20874. After integrating over xik, odd terms with respect to /H20849xik −Q/H20850vanish. Taking only even terms, one obtains n/HS110050,m/HS110050→mn2/H9275n4/H20849xin−Qi/H208502/H9254nm=mm/H9275m2/H20849xin−Qi/H208502Bnm, n=0 , m/HS110050→−mm2/H9275m4/H20849Qi−xim/H208502=mm/H9275m2/H20849xim−Qi/H208502Bnm,n=0 , m=0→/H20858 kmk2/H9275k4/H20849Qi−xik/H208502+b2Qi2. Integrating out and using the identity /H20848duu2e−u2/2/H9251 =/H9251/H20848due−u2/2/H9251for/H9251/H110220, one ﬁnds n/HS110050,m/HS110050:/H20885d/H9273imm/H9275m2/H20849xin−Qi/H208502BnmK/H20849/H9273i,/H9273i;/H9270/H20850 =/H6036 /H9270Bnm/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850, n=0 , m/HS110050:/H20885d/H9273imm/H9275m2/H20849xim−Qi/H208502BnmK/H20849/H9273i,/H9273i;/H9270/H20850 =/H6036 /H9270Bnm/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850, n=0 , m=0 :/H20885d/H9273i/H20875/H20858 kmk2/H9275k4/H20849Qi−xik/H208502+b2Qi2/H20876K/H20849/H9273i,/H9273i;/H9270/H20850 =/H6036 /H9270/H20873/H20858 kmk/H9275k2+b/H20874/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850 =/H6036 /H9270Bnm/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850. The result is/H6036 /H9270Bnm/H20848d/H9273iKindependent of the cases. Finally, one can obtain /H20885/H20879d/H9273i/H115092K /H11509xfn/H11509xfm/H20879 /H9273i=/H9273f=/H20877−1 /H6036/H20873/H9270 4Bnm+1 /H9270Anm/H20874 +/H9270 4/H6036Bnm/H20878/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850 =−Anm /H9270/H6036/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850,/H20849D30/H20850 or equivalently,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-13/H20883/H115092 /H11509xn/H11509xm/H20884=−kBT /H60362Anm=−kBT /H60362/H20849M/H9253n/H9253m+mn/H9254nm/H20850, /H20849D31/H20850 where /H92530=1,m0=0. Finally, p-pcorrelation is obtained /H20855/H9004pn/H9004pm/H20856=−/H60362/H20883/H20873/H11509 /H11509xn−/H9253n/H11509 /H11509Q/H20874/H20873/H11509 /H11509xm−/H9253m/H11509 /H11509Q/H20874/H20884 =kBT/H20849Anm−/H9253mAn0−/H9253nAm0+/H9253n/H9253mA00/H20850 =kBT/H20849M/H9253n/H9253m+mn/H9254mn−M/H9253m/H9253n−M/H9253n/H9253m +M/H9253n/H9253m/H20850=mnkBT/H9254mn. /H20849D32/H20850 The above three results of x-x,x-p, and p-pcorrelations are the same as Eq. /H2084931/H20850.3. Sufﬁcient condition for “high” temperature We assumed the high-temperature approximationkBT /H6036 /H11271/H20881/H20841/H9261n/H20841 2. Indeed, the temperature should satisfykBT /H6036/H11271/H20881/H9261M 2, where /H9261Mis the absolute value of maximum eigenvalue of A−1B. It is known that, for eigenvalue /H9261of a matrix A,/H20841/H9261/H20841is not greater than maximum column /H20849or row /H20850sum,55 /H20841/H9261/H20841/H11349max j/H20858 i/H20841aij/H20841/H11013/H20648A/H20648. /H20849D33/H20850 According to the above deﬁnition of /H20648·/H20648, It is not hard to see that/H20648AB/H20648/H11349/H20648A/H20648/H20648B/H20648. The above argument says /H9261M/H11349/H20648A−1B/H20648/H11349/H20648A−1/H20648/H20648B/H20648. /H20849D34/H20850 It is not hard to obtain A−1with the following LDU factorization. /H20898MM /H92531 M/H92532 ¯ M/H92531M/H925312+m1M/H92531/H92532¯ M/H92532M/H92532/H92531M/H925322+m2¯ ]] ] /GS /H20899=/H2089810 0 ¯ /H9253110 ¯ /H9253201 ¯ ]] ]/GS/H20899/H20898M 0 0¯ 0m10¯ 00 m2¯ ]]]/GS/H20899/H208981/H92531/H92532¯ 01 0 ¯ 00 1 ¯ ]] ]/GS/H20899. /H20849D35/H20850 Inverting the factorized matrices, A−1=/H208981/H92531/H92532¯ 01 0 ¯ 00 1 ¯ ]] ]/GS/H20899−1 /H20898M 0 0¯ 0m10¯ 00 m2¯ ]]]/GS/H20899−1 /H2089810 0 ¯ /H9253110 ¯ /H9253201 ¯ ]] ]/GS/H20899−1 =/H208981−/H92531−/H92532¯ 01 0 ¯ 00 1 ¯ ]] ]/GS/H20899/H208981 M0 0¯ 01 m10¯ 001 m2¯ ]]] /GS/H20899 /H11003/H2089810 0 ¯ −/H9253110 ¯ −/H9253201 ¯ ]] ] /GS/H20899=/H208981 M+/H20858 n/H9253n2 mn−/H92531 m1−/H92532 m2¯ −/H92531 m11 m10¯ −/H92532 m201 m2¯ ]] ] /GS/H20899. /H20849D36/H20850KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-14Thus, the maximum column sum of A−1is /H20648A−1/H20648= max n/H208731 M+/H20858 i/H9253i2 mi+/H20858 i/H20841/H9253i/H20841 mi,1+/H20841/H9253n/H20841 mn/H20874./H20849D37/H20850 If/H9253iare on the order of 1 or larger,1 M+/H20858i/H9253i2 mi+/H20858i/H20841/H9253i/H20841 miis the maximum value. And, in this limit, it is smaller than1 M +2/H20858i/H9253i2 mi. So one can get /H20648A/H20648/H113491 M+2/H20858 i/H9253i2 mi. /H20849D38/H20850 Since Bis given by B=/H20898b+/H20858 nmn/H9275n2 −m1/H927512−m2/H927522¯ −m1/H927512m1/H927512 0 ¯ −m2/H9275220 m2/H927522¯ ]] ] /GS/H20899,/H20849D39/H20850 the maximum column sum of Bis /H20648B/H206481= max n/H20873b+2/H20858 imiwi2,2mnwn2/H20874/H11349/H20841b/H20841+2/H20858 imiwi2. /H20849D40/H20850 Finally, one obtains the upper bound of /H9261M, /H9261M/H11349/H20648A−1/H20648/H20648B/H20648/H11349/H208731 M+2/H20858 i/H9253i2 mi/H20874/H20873/H20841b/H20841+2/H20858 imi/H9275i2/H20874. /H20849D41/H20850 In order to evaluate the expression on the right-hand side of the inequality Eq. /H20849D41/H20850, we use the constraints Eq. /H2084920/H20850.T o convert the summations to known quantities, we generalizethe constraint to the Caldeira-Legget-type continuous formwith the following deﬁnitions of spectral functions, J p/H20849/H9275/H20850/H11013/H9266 2/H20858 i/H9253i2/H9275i mi/H9254/H20849/H9275i−/H9275/H20850=/H9251S 2KM/H9275,/H20849D42/H20850 Jx/H20849/H9275/H20850/H11013/H9266 2/H20858 imi/H9275i3/H9254/H20849/H9275i−/H9275/H20850=2/H9251KM S/H9275./H20849D43/H20850 Checking the constraints,/H20858 i/H9253i2/H9261 mi/H20849/H92612+/H9275i2/H20850=2/H9261 /H9266/H20885d/H9275Jp/H20849/H9275/H20850 /H9275/H20849/H92612+/H92752/H20850 =2/H9261 /H9266/H9251S 2KM/H20885d/H92751 /H92612+/H92752=/H9251S 2KM, /H20849D44/H20850 /H20858 imi/H9275i2/H9261 /H92612+/H9275i2=2/H9261 /H9266/H20885d/H9275Jx/H20849/H9275/H20850 /H9275/H20849/H92612+/H92752/H20850 =2/H9261 /H92662/H9251KM S/H20885d/H92751 /H92612+/H92752=2/H9251KM S. /H20849D45/H20850 Finally, /H20858 i/H9253i2 mi=2 /H9266/H20885d/H9275/H9251S 2KM=/H9251S /H9266KM/H9275c, /H20849D46/H20850 /H20858 imi/H9275i2=2 /H9266/H20885d/H92752/H9251KM S=4/H9251KM S/H9266/H9275c,/H20849D47/H20850 where /H9275cis the critical frequency of the environmental exci- tations. Therefore, /H9261M/H11349/H208491 M+2/H9251S /H9266KM/H9275c/H20850/H20849/H20841b/H20841+8/H9251KM S/H9266/H9275c/H20850. Hence, one ﬁ- nally ﬁnds that the sufﬁcient condition of the high tempera- ture is T/H11271Tc, where the critical temperature Tcis deﬁned as Tc/H11013/H6036 2kB/H20881/H208731 M+2/H9251S /H9266KM/H9275c/H20874/H20873/H20841b/H20841+8/H9251KM S/H9266/H9275c/H20874. /H20849D48/H20850 Now, we check if the above condition is satisﬁed in ex- perimental situations. Ignoring /H20841b/H20841, the critical temperature becomes /H20881/H208491+2/H9251S /H9266K/H9275c/H208502/H9251K S/H9266/H9275c. Since the environmental excita- tion is caused by magnetization dynamics, one can note thatthere is no need to consider the environmental excitationwith frequencies far exceeding the frequency scale of mag-netization dynamics. 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PhysRevLett.125.247201.pdf	Stacking Domain Wall Magnons in Twisted van der Waals Magnets Chong Wang ,1Yuan Gao,1,2Hongyan Lv,1,3Xiaodong Xu,4,5and Di Xiao1 1Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA 2International Center for Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China 3Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031, China 4Department of Physics, University of Washington, Seattle, Washington, DC 98195, USA 5Department of Materials Science and Engineering, University of Washington, Seattle, Washington, DC 98195, USA (Received 20 July 2020; accepted 6 November 2020; published 8 December 2020) Using bilayer CrI 3as an example, we demonstrate that stacking domain walls in van der Waals magnets can host one-dimensional (1D) magnon channels, which have lower energies than bulk magnons.Interestingly, some magnon channels are hidden in magnetically homogeneous background and can only be inferred with the knowledge of stacking domain walls. Compared to 1D magnons confined in magnetic domain walls, 1D magnons in stacking domain walls are more stable against external perturbations. We show that the relaxed moir´ e superlattices of small-angle twisted bilayer CrI 3is a natural realization of stacking domain walls and host interconnected moir´ e magnon network. Our Letter reveals the importance of stacking domain walls in understanding magnetic properties of van der Waals magnets and extends thescope of stacking engineering to magnetic dynamics. DOI: 10.1103/PhysRevLett.125.247201 The physical properties of two-dimensional van der Waals (vdW) materials depend sensitively on the stackingarrangement between adjacent layers. Consequently, the modulation of stacking can strongly modify the local electronic properties. For example, in bilayer graphene,the stacking domain walls between the AB and BA stackings [1,2] induce a variation in the electronic Hamiltonian, which can give rise to one-dimensional (1D) topologically protected electronic states [3–6]. Similarly, in transition metal dichalcogenides, electrons can be confined in stacking domain walls, which can be controlled experimentally via strain engineering [7]. Recently, it has been realized that the stacking depend- ence also extends to magnetic properties. For example, inbilayer CrI 3, the interlayer exchange coupling changes sign as the stacking is varied [8–13]. Therefore, a stacking domain wall also induces a modulation in the spinHamiltonian. It is thus natural to expect stacking domain walls to host 1D spin wave (magnon) channels. Previously, confined 1D magnons have been proposed to exist inmagnetic domain walls [14–18]. However, magnetic domain walls are generally fragile with respect to external perturbations and may even have their own dynamics. On the other hand, stacking domain walls, whose energy scaleis at least one order of magnitude larger than magnetic domain walls, provide a more stable platform to host 1D magnons. In this Letter, using bilayer CrI 3as an example, we study 1D magnons in stacking domain walls in vdW magnets.We show that, quite generally, all stacking domain walls of bilayer CrI 3can host 1D magnons, which have lower energies than bulk magnons. The existence of these 1D magnons can be adiabatically traced back to the Goldstone modes of the spin Hamiltonian. Interestingly, we find thatsome magnon channels are hidden in magnetically homo-geneous background and can only be inferred with the knowledge of stacking domain walls. These domain walls are naturally realized in moir´ e superlattices in twisted bilayer magnets with small twist angles. Moir´ e magnons have been recently studied in Refs. [19–21]. However, these works have ignored lattice relaxation, and theinformation of stacking domain walls is not utilized in the construction of the spin Hamiltonian. With a full account of lattice relaxation, we calculate the stackingand magnetic moir´ e pattern in small-angle twisted bilayer CrI 3(Fig. 3). In this system, the stacking domain walls and corresponding 1D magnons are interconnected, forming amagnon network, which will dominate low-energy spin andthermal transport. Our Letter reveals the importance of stacking domain walls in understanding magnetic proper- ties of vdW magnets and extends the scope of stackingengineering to magnetic dynamics. Stacking domain wall. —CrI 3is a layered magnetic material whose magnetic order can survive down to the monolayer limit [22]. The Cr atoms in a monolayer CrI 3 forms a hexagonal lattice with lattice constant 6.9 Å. Monolayer CrI 3is a ferromagnet with an out-of-plane easy axis. Bilayer CrI 3has two stable stackings, i.e.,PHYSICAL REVIEW LETTERS 125, 247201 (2020) 0031-9007 =20=125(24) =247201(6) 247201-1 © 2020 American Physical Societyrhombohedral and monoclinic stackings. Both stackings have roughly the same energy, but rhombohedral stacking strongly favors interlayer ferromagnetic (FM) configura- tion, while monoclinic stacking weakly favors interlayer antiferromagnetic (AFM) configuration [8–13]. The stacking configuration is described by the relative in-plane displacement bbetween the top and bottom layer [Fig. 1(a)].bis only defined modulo lattice translations. All possible values of bconstitute the stacking space, which coincides with the unit cell of monolayer CrI 3. We first consider a 1D domain wall separating two semi-infiniteplanes with different stackings, which are described by a continuous variation of the stacking vector b. In our study of each domain wall, we choose our coordinates such thatthe domain wall always lies in the zdirection at x∼0. The yaxis is the easy axis, pointing to the out-of-plane direction. x≪0is the left stacking domain, described bybð−∞Þ¼b left. Similarly, bðþ∞Þ¼brightis the stacking vector of the right stacking domain. Around x¼0,bðxÞ changes rapidly from blefttobright, passing through a series of unstable stackings. In CrI 3, domain walls can appear between two rhombohedral stackings (denoted as RRdomain wall), between rhombohedral and monoclinic stackings (RM domain wall), and between two monoclinic stackings (MM domain wall).To quantitatively characterize bðxÞfor each domain wall, we employ continuous elastic theory. Since the energyscale of different stackings is larger than the energy scale ofmagnetism by at least one order of magnitude, we ignore the influence of magnetism at this stage. The stacking energy of the domain wall is a summation of the interlayerpotential energy and the elastic energy E str¼1 jPjZ∞ −∞/C20 VðbÞþBþG 4ð∂xbxÞ2þG 4ð∂xbzÞ2/C21 dx; ð1Þ where VðbÞis interlayer potential energy per unit cell [Fig. 1(a)],jPjis the area of the unit cell, B¼54307 meV per unit cell is the bulk modulus of monolayer CrI 3, and G¼39248 meV per unit cell is the shear modulus [23].I n Eq.(1), we have assumed that the absolute displacements of the top and the bottom layer are /C6bðxÞ=2. The elastic energy Estrcan be minimized by solving the Euler-Lagrangian equation δEstr=δb¼0. For the RM domain wall, Vis roughly reflection symmetric with respect to the line connecting rhombohedral and mono-clinic stackings [yellow line in Fig. 1(a)], and we can safely assume bðxÞsimply takes this straight path [ b xðxÞ¼0]. Furthermore, Vcan be approximated as a cosine function on the path of the stacking vector: the two minima of thecosine function at /C6πcorrespond to rhombohedral and monoclinic stackings; the maximum of the cosine function is in the middle between the two stable stackings. With thisassumption, the Euler-Lagrangian equation reduces to thesine-Gordon equation and admits a soliton solution b z¼2ðbz right−bz leftÞarctan ½expðx=wÞ/C138=πþbz left:ð2Þ The characteristic width w¼jbleft−brightjﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ G=V 0p =πis roughly 8.8 Å. Here, V0is the barrier of Valong the path of bðxÞ. Notice that the range of xfor which arctan ½expðx=wÞ/C138varies significantly is roughly 6w. In contrast, for the RR and MM domain walls, bðxÞhas to bypass a high-energy barrier [brown dot in Fig. 1(a)], and analytic solutions cannot be obtained. We instead numeri- cally solve the Euler-Lagrangian equation and present bðxÞ as the red line and cyan line in Fig. 1(a)for the two types of domain walls. Nevertheless, it is possible to fit bzto Eq. (2) and the characteristic widths for the RR and MM domain walls are 9.5 and 7.5 Å, respectively. All three types ofstacking domain walls are plotted in Fig. 1(c). Note that we are studying shear domain walls and therefore b left−bright is always along the zdirection. 1D magnon channel. —The variation of the stacking vector binduces a variation in interlayer exchange coupling. In Fig. 1(b), we plot interlayer exchange cou- plings for different stackings together with paths of bðxÞfor the three types of stacking domain walls. To study themagnetic properties of the domain walls, we focus on the FIG. 1. (a) Interlayer potential energy per unit cell as a function of stackings bfor bilayer CrI 3calculated with density functional theory [23]. Important stackings are rhombohedral (red dot), monoclinic (blue dot), AA(green dot) and AC(brown dot) stackings. Red, cyan, and yellow lines represent the paths of thestacking vector bfor RR, MM, and RM domain walls, respec- tively. Inset: definition of the stacking vector b; red (blue) lattice denotes the top (bottom) layer. (b) The energy difference per unitcell between interlayer FM and interlayer AFM CrI 3as a function of stackings. (c) The top view of the three types of stackingdomain walls, with the stacking vectors denoted by purplearrows. The stacking of the left (right) domain is sketched onthe left (right) side. (d) JðxÞacross different types of stacking domain walls.PHYSICAL REVIEW LETTERS 125, 247201 (2020) 247201-2long-wavelength continuum limit and adopt the following micromagnetics energy: Emag¼Z/C18X α;β;lA 2ð∂βmα lÞ2−X lK 2ðmy lÞ2−X αJmα 1mα2/C19 d2r; ð3Þ where αandβare Cartesian indices, lis the layer index, and mis the unit vector pointing in the direction of magneti- zation. The first and second term in Emagis intralayer FM coupling and magnetic anisotropy, where A≈5.3meV and K≈0.032meV=Å2for monolayer CrI 3. The last term describes the interlayer exchange coupling, where J depends on x. The value of JðxÞ≡J½bðxÞ/C138is extracted from Fig. 1(b) and is plotted in Fig. 1(d) for different domain walls. We start with the magnetic properties of the RR domain wall. Rhombohedral stacking strongly favors interlayer FMconfiguration. In the domain wall, the stacking vectorpasses through an region favoring interlayer AFM con-figuration [Figs. 1(b) and1(d)]. However, this interlayer AFM tendency is punished by both intralayer FM coupling(∝A) and the magnetic anisotropy ( ∝K). To quantitatively characterize the competition, we carry out simulations ofthe Landau-Lifshitz-Gilbert equation with the dampingterm. Our simulation shows that the weak interlayerAFM coupling within the domain wall does not changethe direction of the magnetization and the ground state is simply described by a uniform, out-of-plane m. The variation of interlayer exchange coupling, although not manifested in the magnetic ground state, willenter the equation of motion of magnetization dynamics,∂ tml¼−γml×Hl, where the effective magnetic field is Hl¼A∇2mlþKmyˆyþJm¯l(¯1¼2,¯2¼1, and γis the gyromagnetic ratio). Since mis a unit vector, the first order variation of magnetization is written as δml¼ δmx lˆxþδmz lˆz. The ground state respects mirror symmetry in the ydirection ( My), so it is natural to decouple the mirror eigenspaces by defining δm/C6¼ðδm1/C6δm2Þ=2. With the notation δmþ /C6¼δmx /C6þiδmz /C6, the dynamical equation can be transformed into two decoupledSchrödinger type equations, iγ −1∂tδmþ þ¼½−A∂2xþKþAk2z/C138δmþ þ; iγ−1∂tδmþ−¼½−A∂2xþKþ2JðxÞþAk2z/C138δmþ−;ð4Þ where we have assumed mlbehaves like a plane wave in the zdirection with wave vector kz. We see that δmþis blind to the interlayer coupling. For δm−, since the variation of JðxÞ serves as a trapping potential [Fig. 1(d)], despite the magnetization of the ground state is uniform across theRR domain wall, 1D magnon solutions generally exist andare confined in the domain wall. In Fig. 2(a), we present the profile of the 1D magnon with the lowest energy. It is acircular motion isotropic in the x–zplane and the magnon profile is independent of k z. The frequency of this 1D magnon mode at kz¼0is calculated to be 0.3γK. Here γK is the bulk magnon frequency gap and is experimentallymeasured [34] as∼0.3meV, from which we deduce the energy of the 1D magnon is ∼0.09meV. Therefore, the low-energy magnon transport should be dominated by this1D magnon channel. To gain more insight into the existence of this 1D magnon, we assume the width and depth of the trapping potential JðxÞcan be artificially tuned. For a weak trapping potential, the magnetic ground state is a uniform interlayerFM configuration across the RR domain wall [Fig. 2(c)], which is the case for CrI 3. The degree of freedom δm− describes the deviation from interlayer FM configuration and the frequency of the corresponding magnon is deter-mined by the energy cost of such deviation. Since in the domain wall the interlayer exchange coupling JðxÞ FIG. 2. (a) Bounded magnons in the RR (top) and MM (bottom) domain walls. Bottom: solid (dashed) lines correspond tokz¼0.0(0.05) 1/Å. Insets: motions of the magnons. (b) Mag- netic ground state of MM domain wall. Top: magnetization;bottom: azimuth angles of the top layer magnetization. Bottominset: spatial varying coordinates. ˆe ris along the direction of m; (ˆeθ,ˆeϕ) are defined as the unit vectors in the direction of increasing θandϕ. (c) Schematic plots of confined magnons (top) and magnetic ground states (bottom) for weak (red) andstrong (blue) trapping potentials JðxÞ. Top: a Bloch sphere representing the direction of magnetization. For weak trappingpotential, the magnon is a circular motion around the north pole(y¼1); for strong trapping potential, the confined magnon is a Goldstone mode oscillating on the latitude line (constant y). (d) Magnetic ground state of the RM domain wall. Azimuthangles of the magnetization of the top layer ( ϕ 1) and bottom layer (ϕ2) are plotted in the bottom panel.PHYSICAL REVIEW LETTERS 125, 247201 (2020) 247201-3becomes smaller, such energy cost is lower in the domain wall. Therefore, a confined mode should exist in the domain wall, which is manifested as a bound state duetoJðxÞin Eq. (4). For increasingly stronger trapping potential, there will be a critical point where the frequency of this 1D magnon at k z¼0becomes zero. Beyond this critical point, the magnetic ground state deviates from theinterlayer FM configuration in the domain wall [Fig. 2(c)]. However, the 1D magnon mode does not disappear. The magnetic energy described by Eq. (3)is actually invariant under a global (independent of x) rotation about the yaxis. This continuous symmetry immediately gives rise to a Goldstone magnon mode, which costs no energy at k z¼0 and is still localized in the domain wall [Fig. 2(c)]. It is worth noting that arbitrarily weak trapping potentials can confine 1D magnons. However, if JðxÞis always positive, the frequency of the 1D magnon will be larger than the bulk magnon gap γK. Similar to the RR domain wall, the MM stacking domain wall appears between two magnetically identical (AFM) domains. However, since the AFM interlayer coupling inCrI 3is much weaker than the FM interlayer coupling, the magnetization now tilts away from the yaxis in the MM stacking domain wall. We parametrize the magnetization of the top layer m1ðxÞin spherical coordinates θ(polar angle) andϕ(azimuth angle) and present the magnetic ground state in Fig. 2(b). The Goldstone mode argument discussed above immediately gives rise to a zero-energy 1D magnonmode trapped in the MM domain wall. Nevertheless, it is instrumental to write down the magnetic dynamical equa- tions. We expand δm lasδml¼δmθ lˆeθ lþδmϕ lˆeϕ l, where (ˆeθ l,ˆeϕ l) is the local unit vectors associated with ml [Fig. 2(b)]. Instead of My, the ground state now respects twofold rotation symmetry in the xdirection ( C2x). We decouple the eigenspaces of C2xby defining δ˜mθðϕÞ /C6¼ ðδmθðϕÞ 1/C6δmθðϕÞ 2Þ=2. The dynamical equation for δ˜mþis γ−1∂t˜mθ þ¼ðA∂2x−Uϕ−Ak2zÞδ˜mϕ þ; γ−1∂t˜mϕ þ¼ð−A∂2xþUθþAk2zÞδ˜mθ þ; ð5Þ where Uϕ¼−Kcosð2ϕÞ and Uθ¼−Að∂xϕÞ2þ Ksin2ðϕÞ−J½1−cosð2ϕÞ/C138. Every term in UϕandUθis a trapping potential, which is a combined effect of varyingmagnetization and varying JðxÞ. These trapping potentials serve as an alternative explanation for the 1D zero-energy Goldstone magnon mode in the MM domain wall. Theprofile of the lowest-energy 1D magnon of δ˜m þis shown in Fig. 2(a).F o r kz¼0, the magnon is perfectly polarized: δ˜mϕ þ¼0, consistent with the Goldstone mode oscillating around the yaxis. This perfect polarization is reduced by finite kz[Fig. 2(a)]. Equation (5)shows that, even when ϕ¼π=2across the domain wall [e.g., for weaker variation ofJðxÞ] and the Goldstone mode argument fails, MMdomain walls can still support 1D magnons due to the variation of JðxÞalone. Forδ˜m−, whether the variation of Jandϕserves as a trapping potential depends on the specific parameters. Nevertheless, for bilayer CrI 3,δ˜m−also has a 1D magnon solution with energy higher than the 1D magnon mode ofδ˜m þ, but lower than the bulk magnons [23]. Finally, we investigate the properties of the RM domain wall, which is both a stacking and a magnetic domain wall. The magnetic ground state is presented in Fig. 2(d): the magnetization of the top layer slightly tilts away from þy direction in the domain wall, while the magnetization of the bottom layer still roughly maintains the Walker ’s profile [35]. The Goldstone mode argument is also applicable to the RM domain wall and we have verified numerically such a 1D magnon exists. Therefore, we conclude that all three types of stacking domain walls in bilayer CrI 3support 1D magnon channels. Equation (3)is oversimplified since it does not include all symmetry-allowed magnetic interactions. Especially, weak Dzyaloshinskii-Moriya interaction [36,37] (DMI) is possible in this system. With DMI, the magnons in Fig. 2(a) will be generally reflection ( x→−x) asymmetric, which is another degree of freedom to be utilized in device designing [18,38] . Some of the 1D magnons (e.g., the 1D δm− magnon in RR domain wall) do not carry net magnetic moments. For spintronics applications, a source that differ- entiates the top layer and the bottom layer (e.g., by proximity effects) is needed to excite these magnons [23]. Moir´ e magnon network. —A natural realization of stack- ing domain walls is by twisting the magnetic bilayer.Twisted bilayer materials create a moir´ e pattern, which is a periodic modulation of stackings. After being twisted, the structure will generally relax to lower its energy. Forlarge twist angle, lattice relaxation can be ignored, and novel spin textures may appear [19]. For small twist angles, the stable stackings will grow and form domains, whileunstable stackings will shrink and eventually only appeararound the domain walls. To understand at which twist angles large domains of rhombohedral and monoclinic stackings appear and to obtain a real space pattern of these domains, we calculate the lattice relaxations of the twisted bilayer CrI 3using the method introduced in Ref. [24]. We present the results of lattice relaxation for several twisting angles in the Supplemental Material [23] and estimate large stacking domains emerge for twist angle smaller than 1.3°. Figure 3 shows the stacking and magnetic domain patterns in the small-angle limit. Since the AFM monoclinic stacking isactually slightly energetically higher than the FM rhombohedral stacking (by about 15 meV), the magnetic domain pattern consists of isolated AFM domains andinterconnected FM domains. These domains are useful bythemselves. For example, using twisted bilayer CrI 3as the substrate, electrons experience periodic exchangePHYSICAL REVIEW LETTERS 125, 247201 (2020) 247201-4couplings in the real space and accumulate Berry phase as they move along, which may be useful to realize thetopological Hall effect [39]. Figure 3shows that, in small-angle twisted bilayer CrI 3, all three types of stacking domain walls appear. Theinterconnected stacking domain walls give rise to amagnon network, which might be observed via inelastictunneling spectroscopy [40]. A similar coherent conducting network, but for electrons, can be found in twisted bilayer graphene with an out-of-plane electric field [41–47]. Similarly, the 1D magnons in twisted bilayer CrI 3will provide a platform for low-energy coherent spin and thermal transport. Finally, we note that beyond thecontinuum limit, magnons in CrI 3monolayers are pre- dicted to be topological, such that the boundary between FM and AFM regions will host chiral magnon states in theDirac gap [48,49] . In summary, we propose stacking domain walls in vdW magnets can support 1D magnon channels. These chan- nels can live on uniform magnetic ground states and arerobust against external perturbations. They can be realizedin naturally occurring stacking faults or through carefulstrain engineering [7]. We show that a realistic and highly tunable playground of such 1D magnons is twisted bilayer magnets with small twist angles, where their implications in spin and thermal transport phenomena are yet to beuncovered. We acknowledge useful discussions with Pablo Jarillo- Herrero, H´ ector Ochoa, and Wenguang Zhu. This work is supported by AFOSR MURI 2D MAGIC (FA9550-19-1-0390). The understanding of moir´ e magnon network is partially supported by DOE Award No. DE-SC0012509. Y. 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PhysRevB.73.172408.pdf	Role of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets in the s-dmodel M. Fähnle, R. Singer, and D. Steiauf Max-Planck-Institut für Metallforschung, Heisenbergstrasse 3, 70569 Stuttgart, Germany V . P. Antropov Department of Physics and Astronomy, Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA /H20849Received 9 August 2005; revised manuscript received 14 February 2006; published 12 May 2006 /H20850 In the literature on the s-dmodel for the magnetization dynamics in systems without and with spin-polarized transport currents, it is generally assumed that the adiabatic part of the s-electron magnetization is everywhere parallel to the d-electron magnetization. We present model calculations within the ab initio spin-density functional theory which suggest that this assumption is not valid for systems with strongly noncollinearmagnetization conﬁgurations. DOI: 10.1103/PhysRevB.73.172408 PACS number /H20849s/H20850: 75.45. /H11001j, 72.25.Ba In recent years there has been an extensive research ac- tivity to achieve a basic understanding of ultrafast magneti-zation processes in magnetically ordered materials. Specialemphasis thereby has been given to the damping of the mag-netization dynamics and to the inﬂuence of spin-polarizedtransport currents which may induce precession and reversalof the magnetization /H20849for a review, see Ref. 1 /H20850. One of the frequently used models to describe the dynamics of the mag-netization ﬁeld M/H20849r,t/H20850in ferromagnets is the s-dmodel where “ s” and “ d” denote the itinerant electronic states /H20851s and p“conduction electrons” which make a contribution m/H20849r,t/H20850toM/H20852and the localized electronic states /H20851forming the density M d/H20849r,t/H20850of the “local” moments /H20852, M/H20849r,t/H20850=Md/H20849r,t/H20850+m/H20849r,t/H20850. /H208491/H20850 In this Brief Report we present test calculations within the ab initio spin-density functional electron theory which suggest that one of the basic assumptions which is generally used inthes-dmodel is not valid for systems with strongly noncol- linear magnetization conﬁgurations. Such systems /H20849e.g., vortices 2or very narrow domain walls in wires3/H20850probably become more and more important for future technologicalapplications, and we therefore think that it is essential topoint out the limitation of this assumption for future theoryon the spin dynamics in strongly noncollinear systems. Inour presentation we refer to the variant of the s-dmodel introduced by Zhang and Li. 4 In general, m/H20849r,t/H20850will not be parallel to Md. One reason for this is the spin-ﬂip scattering which the conduction elec- trons experience; another one will be discussed below. Thisnonparallel alignment generates a torque density T/H20849r,t/H20850ex- erted on M d/H20849r,t/H20850via the s-dexchange interaction, T=−SJex /H6036MdMd/H11003m, /H208492/H20850 where Sis the effective spin quantum number related to Md andJexdenotes the s-dexchange coupling constant. For the equation of motion of Md/H20849r,t/H20850=Mded/H20849r,t/H20850a Gilbert-type equation has been suggested:4/H11509Md/H20849r,t/H20850 /H11509t=−/H9253/H20849Md/H11003Heff,d/H20850+1 MdMd/H11003/H9251/H11509Md /H11509t+T/H20849r,t/H20850. /H208493/H20850 In Eq. /H208493/H20850Heff,dand/H9251are the effective ﬁeld experienced by Mdand the damping constant that would describe the damp- ing of Mdwhen switching off m, i.e., the damping due to direct relaxation processes of the delectrons. Finally, m/H20849r,t/H20850 can be determined in a phenomenological approach from a local generalized continuity equation,4 /H11509m /H11509t=Tm, /H208494/H20850 where Tm=−/H11633·J−T−/H9003re /H208495/H20850 is the total torque density acting on m. Thereby Jis the expectation value of the spin-current-density tensor operator Jˆ, Jˆ=/H20858 s,s/H11032/H9274ˆ s+/H20849r/H20850/H9268ˆss/H11032/H20002vˆ/H9274ˆs/H11032/H20849r/H20850. /H208496/H20850 In Eq. /H208496/H20850/H9274ˆsis the scomponent of the spin ﬁeld operator, /H9268 is the vector of Pauli matrices, and vˆis the velocity operator. The quantity /H9003rerepresents the spin relaxation due to spin- ﬂip scattering of the conduction electrons. To solve Eqs. /H208494/H20850and /H208495/H20850, phenomenological Ansätz are made for m,J, and /H9003re. First, an adiabatic magnetization m0/H20849r,t/H20850is deﬁned as the conduction electron magnetization which would arise if the conduction electron spin relaxation was so fast that m/H20849r,t/H20850could follow Md/H20849r,t/H20850instanta- neously. The actual m/H20849r,t/H20850then is given by m/H20849r,t/H20850=m0/H20849r,t/H20850+/H9254m/H20849r,t/H20850, /H208497/H20850 which is an exact relation because it is nothing else but a deﬁnition of the nonadiabatic contribution /H9254m/H20849r,t/H20850.I na n analogous manner J/H20849r,t/H20850may be represented asPHYSICAL REVIEW B 73, 172408 /H208492006 /H20850 1098-0121/2006/73 /H2084917/H20850/172408 /H208493/H20850 ©2006 The American Physical Society 172408-1J/H20849r,t/H20850=J0/H20849r,t/H20850+/H9254J/H20849r,t/H20850. /H208498/H20850 With this deﬁnition of /H9254m, the spin relaxation term is ap- proximated by a simple relaxation time Ansatz /H9003rel=/H9254m//H9270sf with the relaxation time /H9270sffor the spin-ﬂip scattering of conduction electrons. In the papers on the s-dmodel only the contributions to J of those itinerant states that are related to a transport currentare taken into account. In reality, however, all itinerant statescontribute to J. For situations without transport current, /H11633·J then describes the dependence of the kinetic energy term onthe relative orientations of the magnetic moments, and it hasbeen emphasized repeatedly /H20849see, e.g., Refs. 1, 5, and 6 /H20850that this kinetic exchange term is very essential. If the kineticexchange term is neglected, then m 0/H20849r,t/H20850is parallel to Md/H20849r,t/H20850, m0/H20849r,t/H20850=/H20841m0/H20841ed/H20849r,t/H20850, /H208499/H20850 and then J0/H20849r,t/H20850is approximated by J0/H20849r,t/H20850=−/H9262BP/e0je/H20002ed/H20849r,t/H20850, /H2084910/H20850 where /H9262B,e0,P, and jeare Bohr’s magneton, the elementary charge, the modulus of the spin-current polarization, and thecharge-current density. The nonadiabatic spin-current densityis often neglected. 4 In all papers on the s-dmodel the kinetic energy term for the conduction electrons has been neglected and—as aconsequence—the approximations /H208499/H20850,/H2084910/H20850for the adiabatic part of the magnetization and the spin-current density havebeen adopted. We will now perform model calculationswhich suggest that these assumptions are not justiﬁed for thecase of strongly noncollinear magnetization conﬁgurations.In this Brief Report we conﬁne ourselves to pointing at theroot of the problems which will appear when applying the s- dmodel to strongly noncollinear conﬁgurations. To demonstrate the problems we consider a supercell of hcp Co, fcc Ni, or bcc Fe /H20849at the experimental lattice con- stant /H20850containing 16 atoms /H2084932 atoms in the case of Ni /H20850with the magnetic moments ferromagnetically aligned along the z direction in the initial conﬁguration. Then we rotate the cen-tral atomic moment out of the zdirection by an angle /H9277 while ﬁxing all the other moments in their initial direction. Increasing /H9277we thus arrive at a system with strong atomic- scale noncollinearity. Our calculations involve some approximations./H208491/H20850We apply the atomic-sphere approximation /H20849ASA /H20850for the potential because we use the tight-binding linear-mufﬁn-tin orbital method /H20849TB LMTO /H20850in the ASA. 7For the inves- tigation of magnetic properties this does not represent a ma-jor approximation. /H208492/H20850We apply in addition the atomic-sphere approximation for the spin direction. 6,8For a detailed comparison of the results for noncollinear spin systems as obtained by calcula-tions with and without spin ASA see Ref. 9. The use of thespin ASA has some effect on the quantitative results but thequalitative results are unaffected. /H208493/H20850We use the ab initio density functional electron theory in the local spin-density approximation 10/H20849LSDA /H20850where the effect of the kinetic exchange term is fully taken into ac-count. In the LSDA the exchange-correlation energy is the same for s,p, and dstates, and it is a strictly local quantity which wants to align the s,p, and dcontributions, whereas the kinetic exchange term may lead to a noncollinearityamong them. Taking into account an orbital dependence ofthe exchange-correlation energy /H20849e.g., along the lines of Ref. 5/H20850would have some effect on the quantitative results but we are convinced that the basic statements would be unaffected. Altogether, we think that the basic ﬁnding of our paper would not be affected if one were able to abandon all theseapproximations. The total magnetic moment density M/H20849r,t/H20850contains con- tributions M /H9263/H20849r,t/H20850with/H9263=s,p,doriginating from the elec- tronic s,p, and dstates, whereby Md/H20849r,t/H20850corresponds to the density of the “local moments” and Ms/H20849r,t/H20850+Mp/H20849r,t/H20850corre- sponds to “conduction electron magnetization” m/H20849r,t/H20850of the s-dmodel. Integrating M/H9263/H20849r,t/H20850over the atomic spheres cen- tered at the atomic sites iof the supercell we obtain the /H9263 contributions M/H9263,ito the total atomic magnetic moments Mi=/H20858/H9263M/H9263,i. By means of constraining ﬁelds9,11we ﬁx the orientation of the total moments Mi/H20849i.e., the central moment with angle /H9277against the zaxis/H20850, and then we calculate the resulting angles /H9277/H9263that the M/H9263form with the zaxis. Figure 1 /H20849top/H20850shows for Co the angles /H9277/H9263for the central site of the supercell as function of the angle /H9277of the total moment at that site. Whereas Mdfollows the direction /H9277 nearly perfectly, MsandMpare nearly independent of /H9277. For /H9277=0Msand Mpare antiparallel to Md. For nonzero /H9277a noncollinearity develops, in contrast to the approximationgiven by Eq. /H208498/H20850which is adopted in all papers on the s-d FIG. 1. Top: The angles /H9277/H9263with/H9263=s,p, and dwhich the M/H9263of the central Co atom form with the zdirection, as function of the prescribed angle /H9277for the total magnetic moment. Please note that we present /H9266−/H9277s,pand/H9277d. Thus the alignment of MsandMpis antiparallel with Mdfor/H9277=0. Bottom: The magnitudes M/H9263=/H20841M/H9263/H20841 of the central Co atom as function of the same prescribed angle /H9277.BRIEF REPORTS PHYSICAL REVIEW B 73, 172408 /H208492006 /H20850 172408-2model. Our results for Fe and Ni look very similar in this respect. Figure 1 /H20849bottom /H20850shows in addition the absolute values M/H9263to demonstrate that the magnitude Mddepends sensitively9on the angle /H9277and that Msand Mpare much smaller than Md. Our test calculations for a system with strong atomic- scale magnetic noncollinearity suggest that for a realistics-dmodel of strongly noncollinear magnetization conﬁgura- tions like vortex cores or very narrow domain walls the ef- fect of the kinetic exchange term has to be taken into ac-count. To estimate the relevance of the effect, we assume forthe moment that it is primarily the noncollinearity /H20849rather than the nonparallelism /H20850which matters and that the noncol- linearity becomes relevant if it corresponds to an angle /H9004 /H9277 =/H9277s−/H9277d/H20849modulo /H9266/H20850which is larger than a value /H9004/H9277c. With the help of Fig. 1 /H9004/H9277cmay be related to a critical value /H9277cof the relative angle between the dmoments at two adjacent sites. For a 180/H11568domain wall, e.g., this means that the mis- alignment becomes relevant for a critical domain wall widthof about /H9266a//H9277cwhere ais the lattice spacing. If we assume /H9004/H9277c=/H9266/20 then according to Fig. 1 this corresponds to /H9277c /H110150.09 and leads to a critical domain wall width of about 35a. Taking into account that domain walls in quasi-one- dimensional Fe nanostripes have a width of about aonly3it becomes obvious that for inhomogeneous magnetizationconﬁgurations in nanostructured materials the importance ofthe noncollinearity probably is the rule rather than the excep-tion. To work out the improved theory in detail is a very big challenge which goes far beyond the scope of the presentpaper. Our objective was to point at the root of the problemsthat will appear when applying the s-dmodel to strongly noncollinear conﬁgurations. We end by sketching some stepsfor an improved theory. As discussed by Eqs. /H208491/H20850–/H208498/H20850, theconventional s-dmodel makes an Ansatz , respectively, for the adiabatic parts m 0andJ0, relating these quantities to the given ﬁeld Md/H20849r,t/H20850via Eqs. /H208499/H20850,/H2084910/H20850, and then /H9254m/H20849r,t/H20850is calculated by means of Eq. /H208494/H20850and/H9254J/H20849r,t/H20850is often ne- glected. In the same spirit the improved theory could make anAnsatz form0andJ0as a basis for the succeeding calcu- lation of /H9254m, which of course will be also affected by the improvement of the theory. To do this, the simple Ansatz /H208499/H20850 for m0/H20849Md(r,t/H20850)has to be replaced by a functional m0/H20851Md/H20849r,t/H20850/H20852which may be obtained with the help of the ab initio density functional electron theory and which hopefully may be cast into an approximate analytical form. Accord-ingly, the direction ﬁeld e d/H20849r,t/H20850in Eq. /H2084910/H20850has to be replaced bym0/H20851Md/H20849r,t/H20850/H20852//H20841m0/H20841. Finally, the kinetic exchange of the nonadiabatic magnetization /H9254mof the itinerant states has to be taken into account in Eq. /H208495/H20850via an Ansatz for/H11633·/H9254J, e.g., /H11633·/H9254J=C/H20849m0/H11003/H9004/H9254m+/H9254m/H11003/H9004m0/H20850. It is clear that all these manipulations will yield additional terms in the torque T/H20849r,t/H20850 appearing in the equation of motion /H208493/H20850forMd/H20849r,t/H20850with mathematical structures different from the ones of the con- ventional theory. Experience tells /H20849see, e.g, Refs. 4 and 12 /H20850 that terms with a new mathematical structure in an equationof motion do not change results just quantitatively but addalso new physics. To conclude, our test calculations suggest that in systems with strong magnetic noncollinearity it is essential to takeinto account the effect of the kinetic exchange term whichhas been neglected in former variants of the s-dmodel and which causes a noncollinearity between the conduction elec-tron magnetization and the dmagnetization. It may well be that this will not just change the quantitative results but willalso add new physics to the dynamics of, e.g., vortices orvery narrow domain walls. Electronic address: faehnle@mf.mpg.de 1M. D. Stiles and J. Miltat, in Spin Dynamics in Conﬁned Mag- netic Structures III , edited by B. Hillebrands and A. Thiaville /H20849Springer, Berlin, 2005 /H20850. 2J. Miltat and A. Thiaville, Science 298, 555 /H208492002 /H20850. 3M. Pratzer, H. J. Elmers, M. Bode, O. Pietzsch, A. Kubetzka, and R. Wiesendanger, Phys. Rev. Lett. 87, 127201 /H208492001 /H20850. 4S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 /H208492004 /H20850. 5V . P. Antropov, B. N. Harmon, and A. N. Smirnov, J. Magn. Magn. Mater. 200, 148 /H208491999 /H20850. 6O. Grotheer, C. Ederer, and M. Fähnle, Phys. Rev. B 62, 5601/H208492000 /H20850. 7O. K. Andersen and O. Jepsen, Phys. Rev. Lett. 53, 2571 /H208491984 /H20850. 8H. Köhler, J. Sticht, and J. Kübler, Physica B 172,7 9 /H208491991 /H20850. 9R. Singer, M. Fähnle, and G. Bihlmayer, Phys. Rev. B 71, 214435 /H208492005 /H20850. 10J. P. Perdew and Y . Wang, Phys. Rev. B 45, 13244 /H208491992 /H20850/H20849cor- relation energy /H20850;33, 8800 /H208491986 /H20850/H20849exchange part /H20850. 11P. H. Dederichs, S. Blügel, R. Zeller, and H. Akai, Phys. Rev. Lett. 53, 2512 /H208491984 /H20850. 12D. Steiauf and M. Fähnle, Phys. Rev. B 72, 064450 /H208492005 /H20850.BRIEF REPORTS PHYSICAL REVIEW B 73, 172408 /H208492006 /H20850 172408-3
PhysRevB.81.134408.pdf	Effect of crystalline defects on domain wall motion under ﬁeld and current in nanowires with perpendicular magnetization F. Garcia-Sanchez,1H. Szambolics,1,2A. P. Mihai,3,4L. Vila,3,4A. Marty,3,4J.-P. Attané,3,4J.-Ch. Toussaint,2,5and L. D. Buda-Prejbeanu1 1SPINTEC, UMR-8191, CEA-INAC/CNRS/UJF-Grenoble 1/Grenoble-INP, 17 Rue des Martyrs, 38054 Grenoble Cedex 9, France 2Institut Néel, CNRS, 25 Rue des Martyrs, 38042 Grenoble, France 3CEA, Inac, SP2M, Nanostructures et Magnétisme, 38054 Grenoble, France 4Université Joseph Fourier, BP 53-38041 Grenoble, France 5Institute Polytechnique de Grenoble, 46, Avenue Félix Viallet, 38031 Grenoble, France /H20849Received 16 November 2009; revised manuscript received 4 February 2010; published 7 April 2010 /H20850 In this work, we present a micromagnetic study of a magnetic domain wall dynamics inﬂuenced by magnetic ﬁelds and/or spin-polarized currents in systems characterized by perpendicular magnetization including crys-talline defects. The dynamics in two different systems is studied: one showing the effect of defects in the ﬂowregime, and the second one presenting the thermally activated depinning from a single defect. In the latter case,the thermal depinning can be analyzed within the framework of a single energy barrier process. In suchsituation, the current density can be assimilated to an applied ﬁeld. We found that the energy barrier dependslinearly on the applied ﬁeld and on the injected current. DOI: 10.1103/PhysRevB.81.134408 PACS number /H20849s/H20850: 75.40.Gb, 75.40.Mg, 75.75. /H11002c I. INTRODUCTION The perspective of new applications based on domain walls1has renewed the interest in the dynamics of these magnetic objects. Their control through spin polarized cur-rent, theoretically predicted by Berger, 2seems to be the most appealing choice because it allows device miniaturization.The recent progress in understanding and mastering of thecurrent-induced domain wall /H20849DW /H20850motion expose the im- portance of DW pinning due to defects and the thermal acti-vation phenomenon. First, the control of DW pinning and depinning on spe- ciﬁcally designed artiﬁcial defects /H20849i.e., geometrical constric- tions /H20850is a prerequisite to the use of DW motion in any memory devices. In that case, the total extrinsic pinningstrength is the result of the addition of the pinning due toartiﬁcial defects and that due to natural defects. Second, perpendicularly magnetized systems appear to be more interesting than their in-plane magnetized counterpart,since they are developing thinner DW. More importantly,they are exhibiting higher efﬁciencies of spin transfer andDW velocities. 3,4Since in these materials, the magnetization reversal by an applied ﬁeld is mainly controlled by DW pin-ning, one can expect that properties like critical currents maybe linked to the strength of DW pinning on natural defects, 5,6 in a similar fashion to coercive ﬁelds. The types of DWs appearing in these materials, Bloch and Néel walls, makethem especially appealing for comparison with the one-dimensional /H208491D/H20850analytical models of current driven DW motion. 7,8In these models, the effect of defects6and temperature9–11has been also incorporated. Third, the DW behavior at ﬁnite temperature is affected by thermal activation, hence motion is observed even forexcitations below the zero temperature critical current andﬁeld values. 5The thermally activated depinning is, therefore, the dominating effect on the velocity. As a result of its sto-chastic nature, the DW behavior is random. 12Such effectshave to be studied carefully for current-induced DW motion, especially since stochasticity was evidenced in recent DWdepinning experiments. 1,13Thus, the role of thermal activa- tion in DW depinning cannot be neglected. The goal of the present work is to shed a light on the randomness of the wall behavior. This study focuses on sys-tems characterized by strong magnetocrystalline perpendicu-lar anisotropy: CoPt multilayers and FePt L1 0layers. In both systems, the DW dynamics is dominated by the natural crys-talline defects occurring randomly along the sample. By using micromagnetic modeling, we studied the viscous motion and the thermally activated depinning of a DW, usingan external excitation in the form of either a magnetic ﬁeldor a spin-polarized current. Both the spin transfer relatedeffects and the thermal ﬂuctuations were implemented in themicromagnetic solver WALL_ST. 14In general, the combined effect of the charge current and the spin current on a DW canbe split in two terms, added to the classical Landau-Lifshitz-Gilbert equation, and the ﬁnal equation yielding: /H11509M /H11509t=−/H92530M/H11003Heff+/H9251 Ms/H20873M/H11003/H11509M /H11509t/H20874 −/H20849u·/H11612/H20850M+/H9252 MSM/H11003/H20851/H20849u·/H11612/H20850M/H20852, /H208491/H20850 where uis a vector pointing in the current direction with an absolute value given by u=JappPg/H9262B/2/H20841e/H20841Ms/H20849with the dimen- sion of a velocity /H20850. The ﬁrst added term is known as the adiabatic term and the second as the nonadiabatic spin torqueterm. The origin of the nonadiabatic spin torque term, to-gether with the value of the nonadiabatic spin transfer pa-rameter, /H9252, is still under discussion. Different physical ef- fects have been proposed, such as spin transfer,7,15–18and momentum transfer.6,19The/H9252value in narrow DWs is ex- pected to be large, due to their large gradients6and to the small dimensions that become comparable to, or smaller thanPHYSICAL REVIEW B 81, 134408 /H208492010 /H20850 1098-0121/2010/81 /H2084913/H20850/134408 /H208497/H20850 ©2010 The American Physical Society 134408-1the Fermi length19or the Larmor precession length.18The value of /H9252is difﬁcult to measure, and it depends on the model used to interpret the experimental data. Nevertheless,two scenarios seem to be the most plausible in both perpen-dicularly and longitudinally magnetized systems: /H9252/H11015/H9251 /H20849Ref. 13/H20850or/H9252much larger than /H9251.20Across this paper, the value of /H9252is varied depending on the material considered. The following section, concerning CoPt nanowires, aims toexplore a rather general case. /H9252was set to 0.02, a value slightly larger than /H9251. In the third section, a FePt system is studied with a /H9252value varying from 0 to 1. For all simula- tions reported here the polarization Pis chosen to be equal to 1. The paper is organized as follows. In Sec. II, we study the inﬂuence of a random distribution of magnetocrystalline an-isotropy values on the displacement of a DW. In Sec. III,w e present the simulations of the thermally activated depinningfrom a single crystalline defect, calculating the dependenceon the ﬁeld and the current of the energy barrier. Finally, theconclusions are given in the Sec. IV. II. DOMAIN WALL MOTION AND RANDOM ANISOTOPY Most of the present theories and numerical results con- cerning DW motion deal with ideal systems, which might befar from real samples. For instance, it was shown that inPermalloy nanowires the characteristic defects—the surfaceroughness-prevented formation of antivortices. 21This re- sulted in a faster motion of the walls, compared to idealnanowires. We are interested in checking if the same behav-ior is observed when introducing a random anisotropy distri-bution in a CoPt nanowire, with out-of-plane magnetizationorientation. Apparently, in such materials, crystalline defectsact as very important pinning sites due to the narrowness ofthe wall. 22In order to address the effect of crystalline defects on the motion of a Bloch wall, we studied their direct inﬂu-ence on the wall structure, and on the average wall velocitytaking into account the possibility of pinning. Following this idea, we considered a wire of size 512/H11003120/H1100311 nm 3, divided in regular cells of 4/H110034/H1100311 nm3. We used the following material parameters: the exchange stiffness Aex=1·10−11J/m, the saturation magnetization MS=254·103A/m, the damping constant /H9251=0.01 /H20849Ref. 23/H20850and the nonadiabatic parameter /H9252=0.02. For the ideal wire, the magnetocrystalline anisotropy /H20849MC /H20850 constant is Ku,ideal=1.27·105J/m3. The direction of the MC anisotropy ﬁeld was kept constant along the zdirection, perpendicular to the wire plane /H20851Fig. 1/H20849a/H20850/H20852. At the same time, the value of the magnetocrystalline anisotropyconstant K uwas varied in each discretization cell. Two types of distributions of Ku, were generated. In the ﬁrst /H20849second /H20850 sample D1 /H20849D2/H20850,Kuwas varied in the range /H208510.5, 1 /H20852·Ku,ideal /H20849/H208510.5, 1.5 /H20852·Ku,ideal /H20850/H20850, respectively. With an average aniso- tropy of /H1101175% of the Ku,ideal, sample D1 corresponds to a softer material than the initial one. In the second conﬁgura-tion /H20849sample D2 /H20850, the average anisotropy value is very close to the K u,ideal. It has been determined experimentally that the length scale of the modulation of Kuis determined by the grain size, measured to be around 5–10 nm in polycrystallineﬁlms.23Therefore, associating anisotropy cells with the dis- cretization cells seems to be an appropriate choice. The Ku distribution for the sample D1 is depicted in Fig. 1/H20849a/H20850. We compare results obtained for an ideal /H20849uniform Ku/H20850 CoPt-like nanowire to those obtained for wires with a ran-dom variation of K u. In Figs. 1/H20849b/H20850and1/H20849c/H20850, the dependence of the domain wall position as a function of time is shown,for J app=5·1010A/m2and Japp=50·1010A/m2, respec- tively. Two kinds of behavior are identiﬁed: /H20849i/H20850For low-current density /H20851Fig. 1/H20849b/H20850/H20852, the domain wall displacement seems to be sensitive to the presence of therandom anisotropy distribution, exhibiting several plateaus/H20849red and green curves online /H20850. Furthermore, if the pinning potential is strong enough, the current values are too weak torelease the DW, which stays pinned in a certain location forseveral tens of nanoseconds /H20849green curve online /H20850. At this point, it is worth to note that the same kind of stochasticbehavior for small values of the ﬁeld is suggested by recentexperimental results 12,22of ﬁeld-induced depinning. /H20849ii/H20850In the high-current regime /H20851Fig. 1/H20849c/H20850/H20852, the curves look similar to the ideal one, which correspond to the ﬂow re-gime. As the current is increased, the curves are gettingsmoother suggesting that the domain wall motion is less af-fected by the pinning. This behavior is reminiscent to thatobtained for ﬁeld-induced dynamics, where current-inducedDW motion undergoes a transition between a thermally acti-vated depinning regime and a ﬂow regime, where the dy-namics is mainly governed by precession. In Fig. 2, we show the time evolution of the wall position for current range between 5·10 10A/m2and 50·1010A/m2 for the sample D2. A critical current above which DW ther- mally assisted depinning occurs is identiﬁed around5·10 10A/m2. To get rid of the natural pinning, a critical current above 20·1010A/m2must be injected. This value can be viewed as a characteristic for the transition betweenthe low and high-current regimes. In Fig. 3, we plot the magnetization conﬁgurations for the ideal wire and the sample D1. These conﬁgurations show0 2 55 07 5 1 0 00.51.01.52.0wall position (µm) time(ns)sample D2sample D1ideal wire 01 0 2 0 3 0 4 0 5 00246wall position (µm) time (ns)sample D2sample D1ideal wire(a) (b) (c) 1.27·105 0.65·105KuJapp xy z FIG. 1. /H20849Color online /H20850/H20849a/H20850Magnetocrystalline anisotropy pattern of sample D1. Time evolution of the DW position for /H20849b/H20850 Japp=5·1010A/m2and /H20849c/H20850Japp=50·1010A/m2.GARCIA-SANCHEZ et al. PHYSICAL REVIEW B 81, 134408 /H208492010 /H20850 134408-2that the DW structure in the sample D1 is far from being uniform, since very signiﬁcant changes in the orientation ofthe magnetization occur inside the wall. Here, the DW isneither pure Bloch nor pure Néel wall type, nor the transitionbetween these two, as in the case of the ideal wire. It is well known that the wall velocity is intimately related to the domain wall structure. If the domain wall is of pureBloch or pure Néel type, the wall has zero velocity. On theother hand, a transition from one conﬁguration to the othercauses the velocity to oscillate between minimal and maxi-mal values. These observations were valid for the ideal sys-tems, but, as we reported earlier, the magnetization orienta-tion is not uniform inside the domain wall, in a wire with anisotropy distribution. From the displacement versus timecurves in case of high-current densities, where pinning isnegligible, the effective velocity has a similar value to theideal wire. These values seem to be slightly affected by thedisorder. For example, if J app=50·1010A/m2, the velocity in the ideal wire is 113 m/s, whereas for D1 /H20849D2/H20850sample a value of 126 m/s /H20849103 m/s /H20850is obtained. The same holds for the smaller current of Japp=20·1010A/m2, where the values are: for the ideal wire 51m/s, with D1 37m/s and with D245m/s. During its displacement, the domain wall will try to avoid the regions with high anisotropy and in counterbalance tofavor the regions with low anisotropy. As a consequence, thedomain wall movement might be stopped, and the domainwall pinned if the energy barrier to overcome is too high. III. THERMAL DEPINNING FROM SINGLE DEFECT In this section, we study the effect of a single crystalline defect, to gain a deeper insight on the pinning and depinningprocesses. For this purpose, we simulated wires tailored inFePt/MgO thin ﬁlms, with very large crystal anisotropy. 24 The simulated sample has a size of 80 /H1100350/H110035n m3and includes a defect in the center with a size of5/H1100310/H110035n m 3/H20851Fig. 4/H20849a/H20850/H20852. The material parameters are: Ku =5·106J/m3,Aex=6.9·10−12J/m and MS=1.03·106A/m. For the damping parameter, a value of /H9251=0.1 was used.25 The easy axis of the material is perpendicular to sampleplane /H20849zdirection /H20850. Such materials are known to form narrow domain walls /H20849Bloch parameter /H9004=/H20881Aex/Ku=1.17 nm /H20850,s o in order to ensure a good accuracy, the system has beendiscretized in cells of size /H9004 x/H11003/H9004 y/H11003/H9004 z=0.5/H110031/H110035n m3. The defect is modeled as a region with a lower anisotropyconstant K def=0.5 Ku, keeping the rest of parameters un- changed. The calculated depinning ﬁeld at 0 K, for a domainwall pinned at the defect in this sample is found to be /H92620Hdep/H20849T=0 K /H20850=0.225 T, which is of the same order of magnitude as the coercitive ﬁeld measured in these materials.02 5 5 0 7 5 1 0 00.51.01.52.0 Japp(A/m2) 51 010 10 1010 15 1010 20 1010 50 1010wall position (/CID80m) time(ns) FIG. 2. /H20849Color online /H20850Time evolution of the domain wall posi- tion for several current values, for the sample D2. (a) (b) 0n s 5n s 15 ns 2n s 3n s 3.5 nsMx/MS FIG. 3. /H20849Color online /H20850Magnetization distribution at several times, for the ideal wire /H20849a/H20850, and for the sample D1 /H20849b/H20850. The injected current is Japp=50·1010A/m2. The area represented is 240/H11003120 nm2. xz yKuKu/2(a) (b) (c)Mz/MSJappHapp FIG. 4. /H20849Color online /H20850/H20849a/H20850Schematic view of the system and coordinates. /H20849b/H20850Equilibrium state under applied ﬁeld /H92620Happ=0.18 T at T=0 K. /H20849c/H20850Conﬁguration at T=400 K corre- sponding to one depinning event.EFFECT OF CRYSTALLINE DEFECTS ON DOMAIN WALL … PHYSICAL REVIEW B 81, 134408 /H208492010 /H20850 134408-3A magnetic ﬁeld lower than the depinning ﬁeld is applied in thezdirection and the sample temperature Twas kept 400 K. The temperature was included in the form of a Gaussiandistributed thermal ﬁeld H th, which is added to the total ef- fective ﬁeld. The thermal ﬂuctuations have the followingproperties: 26 /H20855Hth,i/H20849t/H20850/H20856=0 , /H208492/H20850 /H20855Hth,i/H20849t/H20850Hth,i/H20849t/H11032/H20850/H20856=2/H9251kBT /H92530/H92620MS/H9004x/H9004y/H9004z/H9254ij/H9254/H20849t−t/H11032/H20850, /H208493/H20850 where kBis the Boltzmann constant. The initial conﬁgura- tion, corresponding to the zero temperature equilibrium, con-tains a DW pinned on the defect. The DW character is notpurely Bloch type, but, due to the presence of the defect, is atwo-dimensional one /H20851Fig. 4/H20849b/H20850/H20852. The magnetization in the defect region as a function of time is plotted in Fig. 5, for several depinning events. During the initial stages, the mag-netization evolves under the effect of thermal ﬂuctuations,and, after several attempts, DW successfully overcomes thebarrier, and gets out of the box deﬁned by the defect /H20851Fig. 4/H20849c/H20850/H20852. We deﬁne the depinning time as the one at which the value of /H20855M z/MS/H20856becomes larger than 0.95 /H20849Fig. 5/H20850. After the depinning, the DW will move under the effect of theapplied excitation, but this motion is not relevant for theexperiments of thermal depinning. The dynamics prior todepinning is in the range of nanoseconds, while the experi-ments of thermal depinning can extend to the scale of sec-onds. However, due to the limitations in the computing time,we restricted our computations to 100 ns. We repeated thesimulations several hundred times /H20849at least 200 /H20850, to ensure enough statistics, for plotting the distribution of number ofevents with determined depinning time. This allows theevaluation of the cumulative distribution function, whichgives the probability of being depinned after time t. The analysis of the thermally activated processes is in general a complicated task. However, if the processcorresponds to the crossing of a single energy barrierand that barrier is much larger than the thermal energy,/H20849i.e., E B/H11271kBT, where EBis the energy barrier that the DW has to overcome /H20850, the process can be described by an expo- nential probability law.26,27In that case, the cumulative dis- tribution function is F/H20849t/H20850=1−e x p/H20873−t /H9270/H20874, /H208494/H20850 where /H9270is the Arrhenius-Néel relaxation time, given by /H9270=/H92700exp/H20849EB/kBT/H20850, where /H92700is the attempt frequency. Figure 6/H20849a/H20850shows the cumulative probability for different applied ﬁelds. For the smallest ﬁeld value at T=400 K, the simulations conﬁrm the exponential law, in accordance withexperimental results. 12In this case, it is clear that thermal activation is observed in our simulation with an apparentsingle energy barrier process. This also holds well for thecurve calculated at T=300 K and /H92620Happ=0.18 T /H20851Fig.6/H20849a/H20850, open circles /H20852. The exponential law for the cumulative prob- ability does not hold for larger ﬁeld values, at T=400 K. This fact can be explained assuming two different timescales in the depinning, one corresponding to the time scaleof thermal activation, and the other corresponding to the timein which the successful event takes place /H20849see Fig. 5/H20850. When both time scales are comparable, the exponential law is notobeyed. In Fig. 6/H20849b/H20850, we plot the time parameter /H9270as a function of the applied ﬁeld, for T=400 K. If the cumulative distribu- tion follows Eq. /H208492/H20850, the average depinning time becomes equal to /H9270. Increasing the applied ﬁeld reduces the average depinning time. The ﬁtting of the logarithm of /H9270versus ﬁeld FIG. 5. /H20849Color online /H20850Time evolution of mean value of Mz/MS calculated inside the defect for three different events. The applied ﬁeld is /H92620Happ=0.18 T at T=400 K. The black line is the value /H20855Mz/MS/H20856=0.95, which corresponds to the depinning criterion. ( ( FIG. 6. /H20849a/H20850Cumulative probability of depinning as a function of time for different applied ﬁeld values and no current. /H20849b/H20850Depinnig time constant variation with the applied ﬁeld at T=400 K.GARCIA-SANCHEZ et al. PHYSICAL REVIEW B 81, 134408 /H208492010 /H20850 134408-4yields the dependence of EBon the applied ﬁeld, considering that the attempt frequency /H92700is independent or slightly de- pendent on the ﬁeld values. For quite a large range of appliedﬁelds, we found a linear dependence of the energy barrier onthe ﬁeld value. Thus, we can express the energy barrier as: E B=2MsV/H20849Hdep−Happ/H20850=aHHapp+bH, /H208495/H20850 where Vis the activation volume. This expression represents the weak pinning limit,28and was conﬁrmed experimentally.12However, there are differences in the time scales and the actual value of depinning ﬁelds. The ﬁrst onecan be assigned to the difﬁculties in extending our calcula-tion beyond the nanosecond time scale, the latter to the lackof an accurate model for the defect. Nevertheless, the physicsof the system is preserved, and the same behavior is ob-tained. From our simulations we deduced an activation vol-ume of V=327 nm 3. We can deﬁne the length of the energy barrier dividing the activation volume by the cross section ofthe defect. The length is equal to 6.55 nm, which is largerthan the length of the defect /H208495n m /H20850. This indicates that en- ergy barrier is comparable to the difference in the Zeemanenergy corresponding to the defect magnetization reversal,but the thermal mechanism is still an inhomogeneous pro-cess. In the following, we ﬁxed the magnetic ﬁeld and applied current densities ﬂowing parallel to the long axis of the wire/H20849xdirection /H20850. Figure 7/H20849a/H20850shows the cumulative probabilityfor different current densities, for /H92620Happ=0.155 T. Since the origin of the nonadiabatic term can be assigned to themistracking of the electron spins with respect to the localmagnetization direction, the nonadiabatic term is expected tohave larger values in perpendicular anisotropy materials,where the domain walls are narrow and comparable to thecharacteristic lengths of the electrons. For this reason wechoose a value of /H9252=1 throughout this section. An appreciable effect on the average depinning time is observed even at low-current densities. Moreover, an asym-metric behavior of the probability with respect to the currentpolarity is present. The effect of negative current is to slowdown the depinning, while positive current reduces the de-pinning time. This result is also consistent with the experi-mental observations. 13In Fig. 7/H20849b/H20850, we plot the time constant versus the current density for two different applied ﬁeld val-ues. Again, we observed a linear dependence of the energybarrier on the current density E B/H20849Japp/H20850=aJJapp+bJ, /H208496/H20850 which was also obtained experimentally.13The slope of the curves in Fig. 7/H20849b/H20850is slightly different for the two ﬁeld val- ues. In general aJis ﬁeld dependent because the ﬁeld value will alter the gradient of the magnetization according to theEq. /H208491/H20850. The linear dependence and the parameter a Jhave been derived from the Fokker-Planck equation, in a 1Dmodel of thermal activation. 10,11 In the one dimensional model of DW motion without ther- mal activation, the nonadiabatic term is equivalent to an ap-plied ﬁeld. 7This equivalence suggests an experimental pro- cedure to measure the value of /H9252. In the case of thermal activation, we found that the effects of currents and ﬁelds aresimilar, as ln /H20849t/H20850is proportional to both H appandJapp. We can thus establish equivalence between two sets of current andﬁeld values, which have the same depinning probability dis-tributions and, therefore, equal average depinning times. Thisis shown in Fig. 8for two different sets of parameters. In this case, the equivalent ﬁeld for the given current is the differ-ence between the corresponding applied ﬁeld values of thesets. We deﬁne the efﬁciency /H9264as the ratio between the equivalent ﬁeld and the current density as7 (a) (b) FIG. 7. /H20849a/H20850Cumulative probability of depinning as a function of time at T=400 K for /H92620Happ=0.155 T with /H9252=1 and different in- jected current values. /H20849b/H20850Depinnig time constant variation with the injected current at T=400 K. FIG. 8. Equivalence between injected the effects of current and applied ﬁeld.EFFECT OF CRYSTALLINE DEFECTS ON DOMAIN WALL … PHYSICAL REVIEW B 81, 134408 /H208492010 /H20850 134408-5/H9264=/H9252P/H6036 2/H20841e/H20841MS/H9004. /H208497/H20850 However, this 1D formula for the efﬁciency is not valid for our nanowire, since we are studying thermal activation inthe framework of a three-dimensional /H208493D/H20850micromagnetic model. In our simulations, we found that there is a linear relation between the energy barrier and the applied ﬁeldand/or injected current. Therefore, it is easy to establish theequivalence for two sets of currents and applied ﬁelds withequal energy barrier. We calculated the efﬁciency as the ratiobetween the parameters of the energy barrier, thusobtaining a value of /H9264=aJ/aH=2.4·10−13Tm2/A for /H92620Happ=0.155 T. Finally, we also calculated the /H9270dependence on the cur- rent density in the adiabatic case, /H9252=0, and in the case /H9252=/H9251, and the results are shown in Fig. 9. For the case /H9252=/H9251and intermediate current densities, a linear dependence of the logarithm of average depinning time on current isagain obtained. Equation /H208497/H20850and Ref. 11indicate a direct proportionality of the efﬁciency with /H9252. In agreement with that fact, the slope in Fig. 9for/H9252=1 is 10 times the slope for the case /H9252=/H9251. The adiabatic /H20849/H9252=0/H20850case can be separated in two regions: one with zero slope, in agreement with the1D formula, 11and one with appreciable reduction for large densities, comparable to the total critical current/H20851J c/H20849T=0 K /H20850=7.75·1011A/m2/H20852. In the adiabatic case, there is an additional contribution to the pinning known as intrin-sic pinning 6originated, in perpendicular anisotropy materi- als, by the in-plane anisotropy. The reduction of the depin-ning time is due to the modiﬁcation of the two-dimensional /H208492D/H20850equilibrium conﬁguration produced by the applied current. IV . CONCLUSIONS Determination of /H9252by dynamical measurements requires the knowledge of whether the DW motion is beyond thevelocity breakdown, 7similar to the ﬁeld case Walker limit.29 For current densities below that breakdown, the 1D modelpredicts a velocity proportional to /H9252. Beyond that limit, the value of /H9252can be obtained from the oscillation frequency of the observed structural changes, similar to those observed inFig. 3/H20849a/H20850. From the evolution shown in Fig. 3/H20849b/H20850, it is clear that those oscillations will be perturbed by the presence ofdefects and the value of /H9252will be not accessible. So the role played by the defect is twofold: ﬁrst, it determines the criti-cal current for the motion through extrinsic pinning, and sec-ond, it suppresses the appearance of a single frequency DWprecession. In the second case studied the model reproduces qualita- tively the experiments and the analytical theory of thermaldepinning due to defects. Namely, there is a linear depen-dence of the energy barrier on the applied ﬁeld and current,and an asymmetric behavior with respect to the current. Theexponential law indicates that the thermal depinnig is welldescribed by a single barrier process. In addition, we foundan equivalence between current and ﬁeld, which can be ex-pressed in the form H eq=/H9264Japp. However, in order to address the question about the real value of /H9252, the proportionality factor in the linear dependence of the energy barrier has to beextracted from the theory and compared to experiment. Thisfactor has been studied in 1D system 10,11from a quadratic pinning potential, but an equivalent derivation is beyond thescope of this article. Finally, the results suggest that thenonadiabatic term is necessary to obtain non-negligible ef-fect of the current for low densities. This represents an addi-tional proof of the necessity to include a nonadiabatic term inspin transfer theories. ACKNOWLEDGMENTS This work has been supported by the French National Research Agency /H20849ANR /H20850through the ISTRADE /H20849Project No. ANR-07-NANO-037-04 /H20850and part of numerical simulations has been carried out with the resources of IDRIS /H20849Project No. 096048 /H20850. 1M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin, Science 320, 209 /H208492008 /H20850. 2L. Berger, J. Appl. Phys. 55, 1954 /H208491984 /H20850. 3S. Fukami, T. Suzuki, N. Ohshima, K. Nagahara and N. Ishiwata, J. Appl. Phys. 103, 07E718 /H208492008 /H20850. 4T. A. Moore, I. M. Miron, G. Gaudin, G. Serret, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. V ogel, and M. Bonﬁm, Appl.Phys. Lett. 93, 262504 /H208492008 /H20850. 5D. Ravelosona, D. Lacour, J. A. Katine, B. D. Terris and C. Chappert, Phys. Rev. Lett. 95, 117203 /H208492005 /H20850. 6G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 /H208492004 /H20850. 7A. Thiaville, Y . Nakatani, J. Miltat, and Y . Suzuki, Europhys. 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PhysRevB.95.174416.pdf	PHYSICAL REVIEW B 95, 174416 (2017) Hybrid magnetic skyrmion H. Z. Wu,1B. F. Miao,1,2L. Sun,1,2D. Wu,1,2and H. F. Ding1,2,* 1National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, 22 Hankou Road, Nanjing 210093, P .R. China 2Collaborative Innovation Center of Advanced Microstructures, Nanjing University, 22 Hankou Road, Nanjing 210093, P .R. China (Received 8 November 2016; revised manuscript received 20 April 2017; published 11 May 2017) We introduce the concept of the hybrid magnetic skyrmion (HMS) by patterning arrays of magnetic nanodisks on top of thin ﬁlms that possess the Dzyaloshinskii-Moriya interaction (DMI). The practical feasibility of themethod is validated by micromagnetic simulations and computed Skyrmion number per unit cell. The HMS hasenhanced stability and mobility with respect to existing skyrmion conﬁgurations. The skyrmion Hall effect issubstantially reduced, in comparison with skyrmion stabilized by the DMI only. We created isolated skyrmionsvia current pulses and drove them by a continuous current at enhanced speed of 750 m /s. DOI: 10.1103/PhysRevB.95.174416 Magnetic skyrmions possess a nontrivial spin texture and carry a topological charge [ 1]. They are relatively stable and inert under small perturbations due to the topologicalprotection. Since the ﬁrst observation of magnetic skyrmionsin bulk magnets with broken inversion symmetry [ 2], there has been considerable attention due to their fundamental interestand potential for applications [ 3–13]. For instance, skyrmions are promising candidates for future information technologydue to their small size and to the small current densities neededto displace them [ 14–16]. Also, in contrast to domain walls, the ﬂexibility of skyrmions allows them to be less hindered bydefects [ 14,17,18]. Skyrmion states are generally explained by the existence of the Dzyaloshinskii-Moriya interaction(DMI) in systems lacking inversion symmetry. Skyrmionswere typically found to be stable at low temperature inmagnetic ﬁelds, which impedes their physical explorationand application [ 2–4]. Although major effort has been made to extend the phase diagram to higher temperatures, it wasonly recently that DMI-induced, room temperature skyrmionswere reported [ 19–24]. As the DMI is commonly weak, there is only a small group of materials where skyrmions arisenaturally from the inherent DMI [natural skyrmion materials,Fig.1(a)]. Alternatively, there is a growing group of artiﬁcially designed magnetic superstructures that enable the existence ofskyrmions without the need of a microscopically built-in DMImechanism [artiﬁcial skyrmion materials, Fig. 1(b)][25,26]. Such artiﬁcial skyrmion crystals have been realized experi-mentally at room temperature [ 27–29]. Numerical simulations also suggest artiﬁcial skyrmion crystals have similar dynamicbehavior to that of the natural skyrmion crystal derived fromthe DMI [ 30,31]. Artiﬁcial skyrmions, without the limitation of a DMI, signiﬁcantly expand the pool of prospective materialsfor skyrmion-based systems. The mobility of the artiﬁcialskyrmion, however, is restricted by the patterned disks. Another interesting phenomenon associated with skyrmion is the skyrmion Hall effect [ 14,32]. Owning to its speciﬁc magnetic conﬁguration, skyrmion does not only move forwardbut also rotates when it is driven by a spin-polarized current.This leads to a transverse movement besides the motion along *hfding@nju.edu.cnthe current direction, similar as the Hall effect [ 33]. The effect, however, signiﬁcantly limits the speed of the skyrmion motionas the skyrmion can be driven out of the track when a highcurrent is used, resulting in a loss of information. To achieve ahigh-speed skyrmion-based spintronics device, the skyrmionHall effect needs to be suppressed effectively [ 32]. Herein, we present the concept of the hybrid magnetic skyrmion (HMS), which is based on the advantages of bothnatural and artiﬁcial skyrmion materials. A natural approachto seek an improvement beyond what each of the componentscan give separately would be to stack the natural and theartiﬁcial skyrmion crystals on each other in close proximity.This leads to an increase in stability, but the skyrmion mobilityis compromised. By inserting a spacer with suitable thickness,we decouple the two original materials from direct coupling toRKKY-type. Remarkably, we ﬁnd that not only the skyrmionmobility is signiﬁcantly increased but also the skyrmion Halleffect is effectively suppressed due to the attraction of theskyrmion by the disks. The proposed HMS can be drivenby an electrical current to a speed of 750 m /s, which is two times larger than the skyrmion with DMI only. Wefurther demonstrate the viability of information encoding andpropagation with the HMS in a nanotrack. With a verticallyinjected current pulse, isolated skyrmions can be createddue to the spin transfer torque effect [ 34,35]. With a steady current, the skyrmions can then be driven with relatively highspeed. As individual skyrmion can be detected by a simpleHall geometry via the topological Hall effect [ 9,36,37], the proposed hybrid structure enables the construction of the HMSracetrack memory that can work at high temperature and withhigh speed. There are two types of DMI, the bulk-type and the interfacial-type. Typically, the bulk-type DMI favors theformation of Bloch-type (vortex-type) skyrmions, whilethe interfacial DMI results in Néel-type skyrmions. Here, wefocus on the simulations of bulk-type DMI-induced skyrmions.The Hamiltonian of the DMI term can be written as H DM= −/arrowrighttophalf D12·(/arrowrighttophalf S1×/arrowrighttophalf S2), where the Dzyaloshinskii-Moriya vector /arrowrighttophalf D12is parallel with the unit vector joining spin/arrowrighttophalf S1and/arrowrighttophalf S2 [See the inserted sketch in Fig. 1(a)]. The spin rotates along the plane perpendicular to the radial direction and forms aBloch-type skyrmion. Figure 1(a) presents a schematic of 2469-9950/2017/95(17)/174416(6) 174416-1 ©2017 American Physical SocietyH. Z. WU, B. F. MIAO, L. SUN, D. WU, AND H. F. DING PHYSICAL REVIEW B 95, 174416 (2017) FIG. 1. (a) Schematic of the spin conﬁguration in a Bloch-type, DMI-induced skyrmion crystal with arrows representing direction of local moments. The DMI vector/arrowrighttophalf D12is parallel to the direction joining neighboring sites. (b) Schematic of an artiﬁcial skyrmion crystal, with ordered arrays of vortices on top of a ﬁlm with perpendicularanisotropy. (c) The proposed HMS crystal, in which ordered arrays of vortices are placed on top of a ﬁlm with DMI. The exchange coupling of the capping vortices stabilizes the skyrmion crystalbeneath. a Bloch-type skyrmion crystal with hexagonal symmetry, where the arrows represent the direction of the local moment.Typically, this type of skyrmion crystal only exists at lowtemperature and under a magnetic ﬁeld. On the other hand,artiﬁcial skyrmion crystal can be realized at higher temperaturevia patterning an array of magnetic vortices onto the surfaceof a perpendicularly magnetized ﬁlm [Fig. 1(b)]. By patterning arrays of magnetic disks onto a substrate with DMI, we show the concept of the HMS [Fig. 1(c)]. Remarkably, the hybrid has a signiﬁcantly enhanced stabilityin comparison with skyrmions stabilized with only a DMIof the same magnitude. With proper tuning of the exchangecoupling between the ﬁlm and the disks, the skyrmionsbeneath can be moved by a spin-polarized current. Thus, HMSbeneﬁt from both the advantages of the natural DMI-inducedskyrmion and the artiﬁcial skyrmion. The micromagnetic simulations are performed via the OOMMF code, including a bulk-type DMI [ 38] and the ther- mal ﬂuctuations [ 39]. For the calculation of all temperature- and ﬁeld-dependences of topological charge density, thedimensions of the magnetic substrate with DMI are 210 × 240×1n m 3with cell size 2 ×2×1n m3. Material parameters are used for FeGe in the calculations as follows: experimentallyreported temperature dependent saturation magnetization [ 40] andMsub=3.3×105A/m at 0 K. The exchange constant Asub=7.54×10−12J/m, is calculated from Curie temper- ature. Similar approach has been used in Ref. [ 41]. The DMI constant D=1.35 mJ/m2is determined from the helical period 70 nm [ 42]. We also include perpendicular magneto- crystalline anisotropy Ku=1.5×105J/m3. As FeGe is 1-nm thick, the magnetic anisotropy may originate from the surfaceanisotropy and/or the magnetoelastic anisotropy caused by thelattice mismatch etc. In the calculations for the HMS, an array of vortex disks with 60-nm diameter, 70-nm disk spacing,and 4-nm thickness are patterned onto the DMI material. Thesaturation magnetization and exchange constant are M disk= 1.4×106A/m and Adisk=2.5×10−11J/m (corresponding to the values of Co), respectively. The coupling between thedisks and DMI substrate changes depending on the geometry(see below). For simulations of the spin transfer torque with current- in-plane geometry, we consider both adiabatic and nonadia-batic terms in the Landau-Lifshitz-Gilbert equation: τ adiab.= um×(m×∂m ∂x) and τnonadiab .=βu(m×∂m ∂x), where u= γ(¯hjP/ 2eM S),xis the direction of the electron velocity, γis the gyromagnetic ratio, MSis the saturation magnetization, jis the current density, Pis the spin polarization, and βis the nonadiabatic damping coefﬁcient [ 43,44]. In the simulation, the Gilbert damping coefﬁcient αis set to 0.05, and the nonadiabatic damping βis set to 0.08. Current ﬂows only within the ﬁlm with DMI. For the simulationof the nucleation of skyrmions with vertically injected spinpolarized current, we consider the in-plane torque written asτ IP=u tm×(mP×m), where mpis the current polarization vector, and tis the FeGe thickness. In the calculation, the out-of-plane ﬁeld-like torque is set to zero. Figure 2shows the comparison of the stability of the skyrmion before and after capping with the magnetic disks.In Fig. 2(a), we present temperature dependent snapshots of the magnetic conﬁguration for a 1-nm FeGe ﬁlm withDMI. The system was originally relaxed from the randomstate. At 0 K, a hexagonal skyrmion lattice is formed whena 250 mT ﬁeld is applied normal to the ﬁlm plane. Theskyrmion size (deﬁned as the diameter of the line wherem z=0) is∼40 nm with a ∼70−nm lattice spacing in good agreement with the experimentally reported value [ 6]. With increasing temperature, the hexagonal order and the skyrmionitself distort due to thermal ﬂuctuations. Also, the backgroundbecomes blurred and no skyrmion can be found above 280 K.In contrast, the system after capping with the disk array(60 nm in diameter and 70 nm in spacing) shows stronglyenhanced stability, where the skyrmion conﬁguration shows noapparent change up to 275 K. We note that, strictly speaking,the micromagnetic approach provides a correct description ofmagnetic systems only for low temperatures [ 45]. Thereby, the topological charge dependencies shown in Figs. 2(c),2(d), and Fig.3(d) serve only as a rough estimation of the predicted effect of the enhancement of the skyrmion phase. A more preciseestimation of the temperature effects could be achieved withatomistic models based on Monte Carlo simulations [ 46,47] or with an advanced micromagnetic approach adapted forhigh temperatures and based on the Landau-Lifshitz-Blochequation [ 48,49]. 174416-2HYBRID MAGNETIC SKYRMION PHYSICAL REVIEW B 95, 174416 (2017) FIG. 2. Temperature-dependent snapshots of the magnetic conﬁguration of the ﬁlm with DMI only (a) and the ﬁlm with DMI and capping disk (b). Red/blue represents moment pointing out-of-plane/into-the-plane, respectively. Image sizes are 210 ×240 nm2. Temperature- and ﬁeld-dependence of Sin the thin ﬁlm (c) without capping vortices and (d) with capping vortices. The stability is greatly enhanced in the latter case. Solid/dash lines in (c) and (d) represent isolines with topological number of 80% and 50% of the maximum value, respectively. In order to compare the stability quantitatively, we compute the integrated skyrmion number S=1 4π/integraltext m·(∂m ∂x×∂m ∂y)dxdy [2,14,50]. The temperature- and ﬁeld-dependence of Sof a 1-nm FeGe ﬁlm with DMI (without capping) is summarizedin Fig. 2(c), which is similar to that reported by Huang et al [8]. As a skyrmion unit possesses a skyrmion number Sof 1 or−1 (in our case, S=1), an ordered skyrmion crystal would haveS=12 for such a given area [red region in Fig. 2(c)]. To better illustrate the temperature- and ﬁeld-dependence ofS, we deﬁne two boundaries where the skyrmion number of the given area has 80% (solid line) and 50% (dash line) ofthe maximum value. The skyrmion crystal without cappingdisks is stable only between 180 and 350 mT and <180 K with the 80%-boundary deﬁnition. In comparison, the areawith large value of Sfor the system after capping with disk array is greatly extended [Fig. 2(d)]. Given the same deﬁnition of an 80% boundary, the skyrmion phase is stable for0−420 mT and <260 K, which is signiﬁcantly larger than that without a disk capping. We, thus, demonstrate that the vortexpatterning onto the magnetic material with DMI can indeedenhance signiﬁcantly the stability of the skyrmion crystalbeneath. As has been reported, DMI-induced skyrmions can be moved at low current density via the spin transfer torquemechanism. Thus, they have been proposed as candidates forinformation carriers [ 14–16]. As a reference, we ﬁrst compute the current-driven motion of skyrmions in a nanotrack withoutcapping disks. The dimension of the nanotrack is 1400 × 100×1n m 3. An in-plane current of spin polarization P=0.4 is injected with electron ﬂow along the nanotrack. We ﬁnd thatthe current threshold is small, well below 1 .0×10 10A/m2, and the velocity is linearly proportional to the injected currentdensity [blue triangles in Fig. 3(a)], similar to previous ﬁndings [14]. Under a current density of 4 .0×10 12A/m2, skyrmions can reach a velocity of ∼320 m/s. Upon further increasing the current density, we ﬁnd that the skyrmion will deviate from thecenter of the track and slip away at the edge of the track dueto the skyrmion Hall effect [see blue triangles in Fig. 3(b)], in agreement with previous ﬁndings [ 14,32]. Hence, the velocity is limited. As a comparison, we also computed the current-driven dynamics for the skyrmion with disk capping. As mentionedabove, the stability of a skyrmion crystal with disk cappingis extended, but its mobility is compromised, as is the casefor the artiﬁcial skyrmion [ 25]. To remove this obstacle, we reduced the magnetic coupling between the ﬁlm withDMI and the vortices by inserting a nonmagnetic spacerin between. In such a conﬁguration, the magnetic couplingchanged from direct coupling to RKKY-type whose strengthcan be tuned by varying the spacer thickness [ 51,52]. The typical maximum value for the ferromagnetic (FM) couplingis of the order of 10 −4J/m2[51,53,54]. In our simulations, we inserted 6-nm Cu and choose the RKKY coupling strengthto be A RKKY=4.0×10−5J/m2. Remarkably, we ﬁnd that 174416-3H. Z. WU, B. F. MIAO, L. SUN, D. WU, AND H. F. DING PHYSICAL REVIEW B 95, 174416 (2017) FIG. 3. (a) Skyrmion velocity νas a function of in-plane current density Jfor skyrmions with DMI only (blue triangle) and the HMS with magnetic disks lifted above (red circle). The velocity of skyrmions with DMI only is limited by the skyrmion Hall effect. (b) The trajectories for skyrmion with DMI only and HMS driven by current with 1 .0×1013A/m, respectively. Time interval for neighboring position is set as 0.1 ns. (c) The threshold current that enables skyrmion motion versus the RKKY coupling strength between magnetic disk and DMI material. (d) Temperature- and ﬁeld-dependence of Sof the HMS with lifted magnetic disks. Solid/dash line represents isolines with topological number of 80% and 50% of the maximum value, respectively. skyrmions stabilized with this conﬁguration can be moved when the current is >5.0×1012A/m2. Above that, the velocity of the skyrmions show a linear dependence with slopesimilar to that without vortex capping [red circles in Fig. 3(a)]. Equally interesting, we ﬁnd that the vortex can also serve asan attracting center, signiﬁcantly reducing the skyrmion Halleffect [ 14,32,55]. Thus, the skyrmion can be moved with speed up to 750 m /s, which is two times the speed of the skyrmion without vortex capping. Figure 3(b) presents the trajectories for DMI-induced skyrmion and HMS under current withdensity of 1 .0×10 13A/m2, respectively. The time interval for neighboring position is 0.1 ns. Obviously, the skyrmionHall effect of DMI-induced skyrmion is strongly suppressedin the HMS system. With this current density, skyrmion movesat the speed of 600 m /s which results in a reading speed of 8 Gb/s for a single track, given that the separation betweeneach skyrmion is ∼70 nm. One can expect that the reading speed can be readily increased when multiple tracks are usedsimultaneously. We also calculate the relationship betweenthe lowest current needed to drive the HMS and the RKKYcoupling strength. The result presented in Fig. 3(c) clearly indicates that higher threshold current is needed when RKKYcoupling strength increases. In addition, the skyrmion stateis stable within (190 mT, 310 mT) and below 220 K in thisgeometry [Fig. 3(d)], which is extended by 25% compared with the skyrmion phase without vortex capping [Fig. 2(c)]. We note here that the simulations are for Co disks on an FeGe ﬁlm butthe generality of the method is not limited to these materials. Since the swirling spin structure of the disk and the interlayerexchange coupling favor the stability of the skyrmion, similareffect would be expected for other materials. We applied ourmethod for Co disks on an Fe 0.5Co0.5Si and found similar results. Our method would be especially useful for the DMIinduced skyrmion material with the phase stability close toroom temperature. For a skyrmion-based information technology, controllable creation, manipulation, and detection of the skyrmion arenecessary. We demonstrate that skyrmions can be nucleatedand driven steadily in a prototype skyrmion-based racetrackmemory. The detection of skyrmions can be achieved bythe measurement of the topological Hall effect, as has beenreported previously [see Fig. 4(a)][9,36]. The ﬁlm with DMI is relaxed from saturation state at −250 mT along −zdirection, and the capping disks are in vortex states with +zpolarity and counterclockwise circulation. A vertical spin polarizedcurrent pulse is injected through a Co nanopillar with 20-nmdiameter at position ( x=50 nm,y=50 nm) [Fig. 4(b),r e d pulses], and an in-plane spin polarized current is applied alonga nanotrack to drive the movement of the skyrmions [Fig. 4(b), blue lines]. After applying a 0.4-ns current pulse with densityJ=5.0×10 12A/m2, the magnetization beneath the Co nanopillar switches and results in a skyrmion state [Fig. 4(c), 1], which moves steadily at ∼600 m/s under a continuous current of density J=1.0×1013A/m2. Thereafter, the chain 174416-4HYBRID MAGNETIC SKYRMION PHYSICAL REVIEW B 95, 174416 (2017) FIG. 4. (a) Schematic of skyrmion race-track memory. (b) Time-dependent current density used for the information encoding. Red current pulse is injected vertically through Co nanopillar to nucleate an isolated skyrmion, and blue steady current is applied along the long axis of the nanotrack to displace the skyrmions. (c) Snapshots of magnetic conﬁguration (top view) in HMS nanotrack at different time sequences, as alsonoted in (b). of skyrmions, created by several pulses, moves along the same trajectory as isolated skyrmions with the same velocity[Fig. 4(c), 2]. The nucleation of skyrmion (i.e., writing of information) can be repeated reliably and with given distances[Fig. 4(c), 3]. After six current pulses, we observe a chain of six skyrmions (encoded as “11011101”) moving steadily withseparate distances along the nanotrack [Fig. 4(c),4 ] .D u et o the coupling between skyrmion and capping disks, the shapeof the moving skyrmion has a slight distortion depending onthe relative position with the disks, compared with the staticHMS presented in Fig. 2. In summary, we demonstrate a new concept of hybrid magnetic skyrmion using micromagnetic simulations. Bypatterning an array of nanodisks onto a magnetic ﬁlm withDMI, the stability of the skyrmion state can be signiﬁ-cantly enhanced due to the coupling from the chiral spinsof the vortices above. Also, the skyrmions beneath can move under the action of a spin-polarized current when thevortices are lifted at a certain distance by a nonmagneticspacer, where the skyrmion Hall effect is effectively sup-pressed. 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PhysRevB.86.014438.pdf	PHYSICAL REVIEW B 86, 014438 (2012) Non-Adlerian phase slip and nonstationary synchronization of spin-torque oscillators to a microwave source G. Finocchio,1,M. Carpentieri,2A. Giordano,1and B. Azzerboni1 1Department of Fisica della Materia e Ingegneria Elettronica, University of Messina, C.da di Dio, I-98100, Messina, Italy 2Department of Elettronica, Informatica e Sistemistica, University of Calabria, Via P . Bucci 42C, I-87036, Rende (CS), Italy (Received 5 January 2012; published 30 July 2012) The nonautonomous dynamics of spin-torque oscillators in the presence of both microwave current and ﬁeld at the same frequency can exhibit complex nonisochronous effects. A nonstationary mode hopping betweenquasiperiodic mode (frequency pulling) and periodic mode (phase locking) and a deterministic phase slipcharacterized by an oscillatory synchronization transient (non-Adlerian phase slip) after the phase jump of ±2π were predicted. In the latter effect, a wavelet-based analysis reveals that in the positive and negative phase jumpthe synchronization transient occurs at the frequency of the higher and lower sideband frequency, respectively.The non-Adlerian phase slip effect, even if discovered in STOs, is a general property of nonautonomous behaviorvalid to any nonisochronous auto-oscillator in regime of moderate and large force locking. DOI: 10.1103/PhysRevB.86.014438 PACS number(s): 75 .75.−c, 85.75.−d I. INTRODUCTION In the last decade, a class of nonlinear auto-oscillators, spin-torque-oscillators (STOs),1has been extensively studied experimentally2–5and theoretically.6–8The STO is promising from a technological point of view, being one of the smallestauto-oscillators observed in nature. It also exhibits complexnonlinear dynamics arising from the intrinsic coupling be-tween the oscillator phase and power (the effective magneticﬁeld depends on the spatial distribution of the magnetization). 7 The main properties of STOs are frequency tunability on bias current and ﬁeld, narrow linewidth, and high output power.In addition, the nonautonomous dynamical behavior of STOs(in the presence of microwave current or ﬁeld) can exhibitnonlinear frequency amplitude modulation, frequency pulling,frequency locking, hysteretic and fractional synchronization,and stochastic resonance. 9–14Analytical, semianalytical, and micromagnetic simulations have been used for the predictionor the explanation of those results, which are mainly in theregime of a “weak” microwave signal, where the oscillatorbehavior is characterized by isochronous dynamical response,and it is possible to neglect the difference between theinstantaneous and the stationary (no microwave signal) oscilla-tion power. 7,15–17In the nonisochronous regime (“moderate” or “large” microwave signal), analytical theories fail and acomplete numerical approach is necessary; synchronizationregions are not symmetric and can be overlapped, and strongnonstationary time domain behaviors (transitions to chaosthrough period doublings of the orbit, unstable intermittenttransition from synchronization to chaos, phase slip) areachieved. 18 We studied the nonisochronous dynamical behavior of STOs in presence of a microwave signal composed by thesimultaneous application of microwave current density J AC and ﬁeld hAC, both at the same frequency. The key result of this paper is the identiﬁcation of two dynamical effects in STOs: (i) nonstationary hopping betweenquasistationary Q(frequency pulling) and periodic P(phase locking) mode 19(the power spectrum is characterized by two modes with power of the same order, one at the frequency of the microwave source and one near the frequency ofthe self-oscillation mode); and (ii) phase slip characterized by an oscillatory resynchronization transient after a ±2π phase jump of the oscillator phase (the power spectrum is characterized by one mode at the frequency of the microwave source and two sidebands). In analogy to the results and theformalism presented in Ref. 20, we called this latter effect non-Adlerian phase slip. A wavelet-based time-frequencystudy shows that when the phase slip occurs, the sidebandmodes are nonstationary, being the higher and the lowersideband frequencies related to the oscillator phase jump of+2πand−2π, respectively. The paper is organized as follows: Sec. IIintroduces the details of the device studied and the numerical implementationof the model, and Secs. IIIandIVdescribe the results and the conclusions of our study. II. NUMERICAL DETAILS We studied the dynamical behavior of exchange biased spin valves composed of IrMn (8 nm) /Py (10 nm) (polarizer) /Cu (10 nm) /Py (4 nm) (free layer) with an elliptical cross sectional area (120 ×60 nm) [see inset of Fig. 1(a)]. A Cartesian coordinate system was introduced, where the xand the y axes are related to the easy and the hard in-plane axes ofthe ellipse, respectively. Our numerical experiment was basedon the numerical solution of the Landau–Lifshitz–Gilbert–Slonczweski (LLGS) equation. 1In addition to the standard effective ﬁeld (external, exchange, self-magnetostatic), theOersted ﬁeld and the magnetostatic coupling with the polarizerwere taken into account. The time step used was 32 fs. Fora complete description of the numerical techniques see alsoRef. 21. Typical parameters for the Py were used: saturation magnetization M S=650×103A/m, exchange constant A=1.3×10−11J/m, damping parameter α=0.02, and polarization factor η=0.3. The free layer was discretized in computational cells of 5 ×5×4n m3(the exchange length islex=/radicalBig 2A μ0M2 S≈7 nm). The bias ﬁeld was applied out of plane ( zdirection) with a tilted angle of 10◦along the x axis to control the in-plane component of the magnetization[see Fig. 1(a)]. The polarizer was considered ﬁxed along the 014438-1 1098-0121/2012/86(1)/014438(5) ©2012 American Physical SocietyFINOCCHIO, CARPENTIERI, GIORDANO, AND AZZERBONI PHYSICAL REVIEW B 86, 014438 (2012) FIG. 1. (Color online) (a) Auto-oscillation frequency f0as function of the current density ( H=200 mT) (inset: sketch of the studied device); (b) integrated output power as a function of the current density; (c) locking regions computed for J=−5a n d−8×107A/cm2(only a microwave current is applied); (d) intrinsic phase shift /Psi1Ibetween the phase of self-oscillation and the microwave current in the locking region of Fig. 1(c)(bothJ=−5a n d−8×107A/cm2)f o rJM=1a n d2 ×107A/cm2. xdirection. To study the locking, we considered a microwave current JAC=JMsin(2πfACt+π/2) (JM/lessorequalslant2×107A/cm2) and a microwave ﬁeld linearly polarized at π/4i nt h e xyplanehAC=hMsin(2πfACt+π/4)ˆx+hMsin(2πfACt+ π/4)ˆy(hM/lessorequalslant3 mT). This microwave ﬁeld can be generated by using the experimental technique developed in Ref. 12.T h e computational data presented in the rest of the article wereobtained with with no thermal effects. III. RESULTS AND DISCUSSIONS A. Free running data First we characterized the STO in the free running regime (no microwave signal). Persistent magnetization oscillationwas observed in a wide range of current density and forout-of-plane bias ﬁeld larger than 180 mT. Here we discussin detail data for a bias ﬁeld of 200 mT, but qualitativesimilar results have been also achieved for 180 and 220 mT.Figures 1(a)and1(b) display current density Jdependence of the oscillation frequency ( f 0) and the integrated output power for the giant magnetoresistive (GMR) signal ( H=200 mT). Thef0vsJcurve is characterized by red shift and an in-plane oscillation axis up to J1=− 3.4×107A/cm2, while for \|J\|>\|J1\| the magnetization precesses around an out-of-planeaxis (blue shift). The discontinuities observed in the data are related to oscillation axis jumps.22,24 B. Isochronous synchronization We systematically studied the locking region to the ﬁrst harmonic (the same of the self-oscillation) as function of JM andhMin the blue shift region. Figure 1(c)shows the locking region computed for J=−5 and−8×107A/cm2and related to a microwave current only ( hM=0 mT). The response can be considered in the regime of a “weak” microwave signal; infact, the locking regions are symmetric and the locking bandis linearly dependent on J M. As already predicted,23it is also found an intrinsic phase shift /Psi1Ibetween the phase of the self-oscillation /Psi1and the phase of the microwave current /Psi1E in the whole locking region. Figure 1(d) summarizes /Psi1Ias function of fAC; as can be observed a linear relationship is achieved and, depending on fACandJM,/Psi1I, can also assume the value 0 or π/2. C. Nonisochronous synchronization As will be discussed, when both microwave current and ﬁeld are applied simultaneously at the same frequency, thenonautonomous response becomes more complicated and 014438-2NON-ADLERIAN PHASE SLIP AND NONSTATIONARY ... PHYSICAL REVIEW B 86, 014438 (2012) FIG. 2. (Color online) (a) Frequency of the main excited mode (the mode with the largest power) as a function of the microwave frequency computed for J=− 5×107A/cm2,JM=1×107A/cm2,a n d hM=1 mT; (b) locking region computed for J=− 5×107A/cm2; (c) power spectra for Pmode ( fAC=5.75 GHz, point 1), P/Q nonstationary mode ( fAC=5.95 GHz, point 3), and non-Alderian phase slip (fAC=5.2 GHz, point 2), JM=1×107A/cm2,a n dhM=1 mT as indicated in (b); (d) wavelet transform of the time trace of the GMR signal related to the P/Q mode of (c). nonisochronous effects are observed in the locking region. Figure 2(a) shows the dependence of the frequency of the excited mode with the larger power as a function of themicrowave frequency for J=− 5×10 7A/cm2,JM= 1×107A/cm2, and hM=1 Mt. The presence of the microwave ﬁeld gives rise to an increasing of the locking region(from 150 MHz at h M=0m Tu pt o1 . 3G H zf o r hM=1m T ) and to a behavior different from the one described in Figs. 1(c) and1(d). We performed a systematic study to better understand the origin of this dynamical behavior, and Fig. 2(b)summarizes the locking region ( J=− 5×107A/cm2) computed up to JM=1×107A/cm2(hM=0 mT) and then increasing hMup to 3 mT, maintaining ﬁxes JM=1×107A/cm2. The border lines have been computed considering the lastmicrowave frequency where a mode at the frequency of themicrowave source with large power is excited. This lockingregion is asymmetric, and at some microwave frequencies thepower spectrum cannot be identiﬁed as a regular Pmode [see Fig. 2(c)]. The results obtained for J=−5×10 7A/cm2 (JM=1×107A/cm2andhM=1 mT) are discussed in detail. Figure 2(c) shows the power spectra related to points 1–3 depicted in Fig. 2(b). The spectrum for fAC=5.75 GHz (point 1) is a regular Pmode, and the spectra at 5.95 GHz (point 3) and 5.2 GHz (point 2) are evidence ofthe strong nonisochronisms; in fact together with the modewith the large power at the microwave frequency (locking regime), an additional large power mode or two sidebands areexcited, respectively. Qualitative similar results have been alsoobtained up to J=−9.5×10 7A/cm2and for hM=2–3 mT andJM=1.5 and 2 ×107A/cm2. The origin of this nonisochronism can be understood by means of a time-frequency study of the GMR-signal basedon the wavelet transform. We used the complex Morlet as themother wavelet: ψ u,s=1√sπfBej2πfc(t−u s)e−(t−u s)2/fB, (1) where fCandfBare two parameters that characterize the mother wavelet function, and sanduare the scale and the shift parameters, respectively. The wavelet transform Wr(u,s) is computed as Wr(u,s)=1√s/integraldisplay+∞ −∞r(t)ψ∗/parenleftbiggt−u s/parenrightbigg dt, (2) where r(t) is the time domain GMR signal, and ψ∗is the complex conjugate of the mother wavelet (see Ref. 24for all the computational details). Figure 2(d) displays the wavelet scalogram [the modulus of Wr(u,s)] for the time trace related to point 3 in Fig. 2(b), withfC=1 andfB=300 (the amplitude increases from black to white). The wavelet analysis shows a 014438-3FINOCCHIO, CARPENTIERI, GIORDANO, AND AZZERBONI PHYSICAL REVIEW B 86, 014438 (2012) mode hopping between the two modes; that is in some time ranges (for example, 5.5–9 ns) the Pmode (phase locking) is excited while in other time ranges (for example, 9.5–11.5 ns)theQmode (frequency pulling) is excited (the main mode has the frequency near the self-oscillation mode and a low powerpeak at the microwave frequency). We refer to this behavioras nonstationary P/Q mode. The relative power of the two modes depends on how long they are excited, for example, atf AC=5.95 GHz the power of the Pmode is larger than that of the Qmode. In several nonlinear systems a diffuse nonisochronous effect called phase slip is observed.25,26It is characterized by brief periods of resynchronization after a phase jumpof±2πof the oscillator phase. Commonly, after the phase jump the resynchronization transient occurs asymptotically(Adlerian phase slip). 23In contrast, here we observe in the locking region the Pmode coupled with two sidebands. The wavelet transform of those time domain GMR signalsdisplayed in Fig. 3(a)forf AC=5.2 GHz [point 2 of Fig. 2(b)] indicates the nonstationary excitation of the sideband modes,the result of which is completely different from the modulationprocesses where the sideband modes are stationary. 9,15In particular, the time evolution of the oscillation phase /Psi1is FIG. 3. (a) Wavelet scalogram (white/black color corresponds to the largest/smallest wavelet coefﬁcient amplitude) for J= −5×107A/cm2andJM=1×107A/cm2andhM=1m Ta tt h e frequency of f=5.2 GHz. (b) Time domain evolution of the phase of the oscillator. Inset: example of energy landscape where it is possible to observe positive or negative phase jump.characterized by the occurrence of a phase slip with an oscillatory synchronization transient we called non-Adlerianphase slip [Fig. 3(b)summarizes those computations for 2 ns]. The origin of the nonstationary sidebands is related to thefact that the higher and lower sideband modes are connectedto the resynchronization frequency of a +2πor−2πphase slip. From a theoretical point of view, the linewidth of a regular Pmode coincides with the one of the microwave source. The phase slip introduces an additional intrinsic dissipationmechanism that can be seen as an enhancement of the oscillatorlinewidth and should be taken into account in the design of ar-rays of STOs, which should work synchronized. The presenceof the phase slip can also explain, for example, the experi-mental evidence of the linewidth enhancement in the powerspectrum of STOs near the boundary of the locking region[see Fig. 3(d) of Ref. 13] and the thermally induced sideband observed in Ref. 27. The phase slip achieved here is periodic (deterministic phase slip) at the same frequency of the microwave source anddifferent from the one obtained in a nonlinear system wherethe thermal ﬂuctuations are responsible for the phase jumpsamong different energetic minima (stochastic phase slip). Tounderstand the origin of the observed deterministic phase slip,we introduce the simple scenario displayed in the inset ofFig. 3(b); when the oscillation induced by the microwave source is large enough to overcome the energy barrier thatseparates two different minima (this is achieved for some valueoff AC), the phase slip occurs. In addition, our computations suggest that the disconti- nuities in the free running data reduce the amplitude of themicrowave source to apply in order to excite the phase slip andthe nonstationary P/Q mode. In fact, the synchronization data (by using the same microwave signals) achieved for an out-of-plane bias ﬁeld larger than 400 mT, where discontinuitiesin the free running data disappear, are similar to the resultsalready published and described analytically in the regime ofa “weak” microwave signal (see, for example, Ref. 7for a review). FIG. 4. (Color online) Time evolution of the oscillator phase for different excited modes: Qmode, nonstationary P/Q mode, non- Adlerian and Adlerian phase slip, and Pmode. 014438-4NON-ADLERIAN PHASE SLIP AND NONSTATIONARY ... PHYSICAL REVIEW B 86, 014438 (2012) IV . SUMMARY AND CONCLUSIONS Figure 4summarizes qualitatively the time dependence of the oscillator phase for the Qmode (frequency pulling), where /Psi1Iincreases linearly; the Pmode (phase locking), where /Psi1Iis constant; the nonstationary P/Q region, where a time domain mode hopping exists between QandPmodes (the phase can be constant or increase linearly); and the Adlerian andnon-Adlerian phase slip (abrupt jumps of 2 πin the oscillator phase are observed together to asymptotic and oscillatoryresynchronization). In summary, in the nonautonomous behavior of STOs by means of micromagnetic simulations and time-frequencydomain analysis based on wavelet transform, the presence ofthe nonstationary P/Q modes and non-Adlerian phase slip was found. The key ﬁnding was the possibility of drivingdeterministic phase slip by means of the application of a moderate microwave source that could be easily generatedexperimentally. We think our results can stimulate futureexperimental studies of the nonisochronisms in self-oscillatorssuch as nonstationary behavior and phase slip not thermallyactivated. Finally, we identiﬁed a phase slip with an oscillatoryresynchronization transient, non-Adlerian phase slip, that evenif discovered in STOs, is a general property of nonautonomousbehavior valid for any auto-oscillator in regime of moderateand large amplitude microwave signal. ACKNOWLEDGMENTS This work was supported by Spanish Project under Contract No. MAT2011-28532-C03-01. 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Siracusano, V . Tiberkevich, I. N. Krivorotov,L. Torres, and B. Azzerboni, Phys. Rev. B 81, 184411 (2010). 9M. R. Pufall, W. H. Rippard, S. Kaka, T. J. Silva, and S. E. Russek, Appl. Phys. Lett. 86, 082506 (2005). 10W. H. Rippard, M. R. Pufall, S. Kaka, T. J. Silva, and S. E. Russek, P h y s .R e v .L e t t . 92, 027201 (2004). 11B. Georges, J. Grollier, M. Darques, V . Cros, C. Deranlot, B. Marcilhac, G. Faini, and A. Fert, P h y s .R e v .L e t t . 101, 017201 (2008). 12P. Tabor, V . Tiberkevich, A. Slavin, and S. Urazhdin, Phys. Rev. B 82, 020407(R) (2010). 13S. Urazhdin, P. Tabor, V . Tiberkevich, and A. Slavin, Phys. Rev. Lett.105, 104101 (2010). 14G. Finocchio, I. N. Krivorotov, X. Cheng, L. Torres, and B. Azzerboni, P h y s .R e v .B 83, 134402 (2011). 15G. Consolo, V . Puliaﬁto, G. Finocchio, L. Lopez-Diaz, R. Zivieri, L. Giovannini, F. Nizzoli, G. Valenti, and B. Azzerboni, IEEE Trans. Magn. 46, 3629 (2010). 16R. Bonin, G. Bertotti, C. Serpico, I. D. Mayergoyz, and M. d’Aquino, E u r .P h y s .J .B 68, 221 (2009).17M. Carpentieri, G. Finocchio, B. Azzerboni, and L. Torres, Phys. Rev. B 82, 094434 (2010). 18See P. Maffezzoni, IEEE Trans. Comput.-Aided-Design Integrated Circuits Syst. 29, 1849 (2010), and references therein. 19In the regime of isochronisms, an STO with f0as a stationary oscillation frequency can exhibit different behavior depending onthe frequency f ACof the microwave signal such as frequency pulling and phase locking. The frequency pulling occurs when the fAC approaches f0and a quasistationary mode or Qmode is excited (the power spectrum is characterized by two modes, one near f0 and one with lower power at fAC). The phase locking occurs when the oscillation frequency f0is locked at the frequency of the external microwave source fACand a periodic mode or Pmode is excited (the power spectrum is characterized by a single mode at fAC). 20Y . Zhou, V . Tiberkevich, G. Consolo, E. Iacocca, B. Azzerboni, A. Slavin, and J. ˚Akerman, Phys. Rev. B 82, 012408 (2010). 21A. Romeo, G. Finocchio, M. Carpentieri, L. Torres, G. Consolo, and B. Azzerboni, Physica B 403, 464 (2008); E. Martinez, L. Torres, L. Lopez-Diaz, M. Carpentieri, and G. Finocchio, J. Appl. Phys. 97, 10E302 (2005); A. Giordano, G. Finocchio, L. Torres, M. Carpentieri, and B. Azzerboni, ibid. 111, 07D112 (2012). See also http://www.ctcms.nist.gov/ ∼rdm/mumag.org.html stan- dard problem #4 report by E. Martinez, L. Torres, and L. Lopez-Diaz. 22I. N. Krivorotov, D. V . Berkov, N. L. Gorn, N. C. Emley, J. C.Sankey, D. C. Ralph, and R. A. Buhrman, Phys. Rev. B 76, 024418 (2007). 23Y . Zhou, J. Persson, S. Bonetti, and J. ˚Akerman, Appl. Phys. Lett. 92, 092505 (2008). 24G. Siracusano, G. Finocchio, A. La Corte, G. Consolo, L. Torres, and B. Azzerboni, Phys. Rev. B 79, 104438 (2009). 25A. Pikovsky, M. Rosenbblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge, New York, 2001). 26Z. Zheng, G. 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PhysRevLett.108.057204.pdf	Atomistic Spin Dynamic Method with both Damping and Moment of Inertia Effects Included from First Principles Satadeep Bhattacharjee, Lars Nordstro ¨m, and Jonas Fransson * Department of Physics and Astronomy, Box 516, 75120, Uppsala University, Uppsala, Sweden (Received 14 October 2011; published 31 January 2012) We consider spin dynamics for implementation in an atomistic framework and we address the feasibility of capturing processes in the femtosecond regime by inclusion of moment of inertia. In thespirit of an s-d-like interaction between the magnetization and electron spin, we derive a generalized equation of motion for the magnetization dynamics in the semiclassical limit, which is nonlocal in bothspace and time. Using this result we retain a generalized Landau-Lifshitz-Gilbert equation, also includingthe moment of inertia, and demonstrate how the exchange interaction, damping, and moment of inertia, allcan be calculated from ﬁrst principles. DOI: 10.1103/PhysRevLett.108.057204 PACS numbers: 75.78. /C0n, 71.70.Ej, 72.25.Rb In recent years there has been a huge increase in the interest in fast magnetization processes on a femtosecondscale, which has been initialized by important develop- ments in experimental techniques [ 1–5], as well as poten- tial technological applications [ 6]. From a theoretical side, the otherwise trustworthy spin dynamical (SD) simulationmethod fails to treat this fast dynamics due to the short timeand length scales involved. Attempts have been made togeneralize the mesoscopic SD method to an atomistic SD,in which the dynamics of each individual atomic magneticmoment is treated [ 7,8]. While this approach should in principle be well suited to simulate the fast dynamics observed in experiments, it has not yet reached full pre-dictive power as it has inherited phenomenological pa-rameters, e.g., Gilbert damping, from the mesoscopic SD.The Gilbert damping parameter is well established in thelatter regime but it is not totally clear how it should betransferred to the atomic regime. In addition, very recentlyit was pointed out that the moment of inertia, which typically is neglected, plays an important role for fast processes [ 9]. In this Letter we derive the foundations for an atomistic SD where all the relevant parameters, such asthe exchange coupling, Gilbert damping, and moment ofinertia, can be calculated from ﬁrst-principles electronicstructure methods. Usually the spin dynamics is described by the phenome- nological Landau-Lifshitz-Gilbert (LLG) equation [ 10,11] which is composed of precessional and damping terms driving the dynamics to an equilibrium. By including themoment of inertia, we arrive at a generalized LLG equation _M¼M/C2ð /C0 /C13Bþ^G_Mþ^I€MÞ; (1) where ^Gand ^Iare the Gilbert damping and the moment of inertia tensors, respectively. In this equation the effectiveﬁeldBincludes both the external and internal ﬁelds, of which the latter includes the exchange coupling and an- isotropy effects. Here, we will for convenience include the anisotropy arising from the classical dipole-dipoleinteraction responsible for the shape anisotropy as a part of the external ﬁeld. The damping term in the LLG equa-tion usually consists of a single damping parameter, which essentially means that the time scales of the magnetization variables and the environmental variables are well sepa-rated. This separation naturally brings a limitation to theLLG equation concerning its time scale which is restrictingit to the mesoscopic regime. The addition of a moment of inertia term to the LLG equation can be justiﬁed as follows. A general process of amoment Munder the inﬂuence of a ﬁeld Fis always endowed with inertial effects at higher frequencies [ 12]. The ﬁeld Fand moment Mcan, for example, be stress and strain for mechanical relaxation, electric ﬁeld and electricdipole moment in the case of dielectric relaxation, ormagnetic ﬁeld and magnetic moment in the case ofmagnetic relaxation. In this Letter we focus on the lattercase—the origin of the moment of inertia in SD. Themoment of inertia leads to nutations of the magnetic mo- ments, see Fig. 1. Its wobbling variation of the azimuthal angle has a crucial role in fast SD, such as fast magneti-zation reversal processes. In the case of dielectric relaxation the inertial effects are quite thoroughly mentioned in the literature [ 13,14], espe- cially in the case of ferroelectric relaxors. Coffey et al. [14] have proposed inertia corrected Debye’s theory of dielec-tric relaxation and showed that by including inertial FIG. 1. The three contributions in Eq. ( 1), the bare precession arising from the effective magnetic ﬁeld, and the superimposed effects from the Gilbert damping and the moment of inertia,respectively.PRL 108, 057204 (2012) PHYSICAL REVIEW LETTERSweek ending 3 FEBRUARY 2012 0031-9007 =12=108(5) =057204(5) 057204-1 /C2112012 American Physical Societyeffects, the unphysical high frequency divergence of the absorption coefﬁcient is removed. Very recently Ciornei et al. [9] have extended the LLG equation to include the inertial effects through a magnetic retardation term in addition to precessional and dampingterms. They considered a collection of uniformly magne-tized particles and treated the total angular momentum L as faster variable. They obtained Eq. ( 1) from a Fokker- Plank equation where the number density of magnetizedparticles were calculated by integrating a nonequilibriumdistribution function over faster variables such that faster degrees of freedom appear as parameter in the calculation. The authors showed that at very short time scales the inertial effects become important as the precessional mo-tion of magnetic moment gets superimposed with nutationloops due to inertial effects. It is pointed out that theexistence of inertia driven magnetization dynamics opensup a pathway for ultrafast magnetic switching [ 15] beyond the limitation [ 16] of the precessional switching. In practice, to perform atomistic spin-dynamics simula- tions the knowledge of ^Gand ^Iis necessary. There are recent proposals [ 17,18] of how to calculate the Gilbert damping factor from ﬁrst principles in terms ofKubo-Greenwood–like formulas. Here, we show that simi- lar techniques may by employed to calculate the moment of inertia tensor ^I. Finally, we present a microscopical justiﬁcation of Eq. ( 1), considering a collective magneti- zation density interacting locally with electrons constitut- ing spin moments. Such a description would in principle be consistent with the study of magnetization dynamics wherethe exchange parameters are extracted from ﬁrst-principleselectronic structure calculations, e.g., density functionaltheory (DFT) methods. We ﬁnd that in an atomistic limitEq. ( 1) actually has to be generalized slightly as both the damping and inertia tensors are naturally nonlocal in thesame way as the exchange coupling included in the effec- tive magnetic ﬁeld B. From our study it is clear that both the damping and the moment of inertia effects naturallyarise from the retarded exchange interaction. We begin by considering the magnetic energy E¼M/C1B. Using that its time derivative is _E¼M/C1_Bþ _M/C1Balong with Eq. ( 1), we write _E¼M/C1_Bþ 1 /C13_M/C1ð^G_Mþ^I€MÞ: (2) Relating the rate of change of the total energy to the Hamiltonian H, through _E¼hdH=dti, and expanding Hlinearly around its static magnetization M0, with MðtÞ¼M0þ/C22ðtÞ, we can write H/C25H 0þ/C22ðtÞ/C1 r/C22H 0, where H 0¼HðM0Þ. Then the rate of change of the total energy equals _E¼_/C22/C1h r /C22Hito the ﬁrst order. Following Ref. [ 19] and assuming sufﬁciently slow dynamics such that /C22ðt0Þ¼/C22ðtÞ/C0/C28_/C22ðtÞþ/C282€/C22ðtÞ=2, /C28¼t/C0t0, we can write the rate of change of the magnetic energy as_E¼lim !!0_/C22i½/C31ijð!Þ/C22jþi@!/C31ijð!Þ_/C22j/C0@2!/C31ijð!Þ€/C22j=2/C138: (3) Here, /C31ijð!Þ¼Rð/C0iÞ/C18ð/C28Þh½@iH 0ðtÞ;@jH 0ðt0Þ/C138iei!/C28dt0, /C28¼t/C0t0, is the (generalized) exchange interaction tensor out of which the damping and moments of inertia can beextracted. Summation over repeated indices ( i,j¼x,y,z) is assumed in Eq. ( 3). Equating Eqs. ( 2) and ( 3) results in an internal contribution to the effective ﬁeld about whichthe magnetization precesses B int¼/C22lim!!0/C31ð!Þ, the damping term ^G¼/C13lim!!0i@!/C31ð!Þas well as the mo- ment of inertia ^I¼/C0/C13lim!!0@2!/C31ð!Þ=2. For a simple order of magnitude estimate of the damping and inertial coefﬁcients, ^Gand ^I, respectively, we may assume for a state close to a ferromagnetic state that thespin resolved density of electron states /C26 /C27ð"Þcorrespond- ing to the static magnetization conﬁguration H 0is slowly varying with energy. At low temperatures we, then, ﬁnd ^G/C242/C13/C25sp½h@iH0i/C26h@jH0i/C26/C138"¼"F; (4) in agreement with previous results [ 19]. Here, sp denotes the trace over spin 1=2space. By the same token, the moment of inertia is estimated as ^I/C24/C0 ð /C13=DÞsp½h@iH0i/C26h@jH0i/C26/C138"¼"F; (5) where 2Dis the bandwidth of density of electron states of the host material. Typically, for metallic systems the band-width 2D/C241–10 eV , which sets the time scale of the inertial contribution to the femtosecond (10 /C015s) regime. It, therefore, deﬁnes magnetization dynamics on a timescale that is one or more orders of magnitude shortercompared to, e.g., the precessional dynamics of the mag-netic moment. Next, we consider the physics leading to the LLG equa- tion given in Eq. ( 1). As there is hardly any microscopical derivation of the LLG equation in the literature, we includehere, for completeness the arguments that leads to theequation for the spin-dynamics from a quantum ﬁeld the-ory perspective. In the atomic limit the spin degrees of freedom are deeply intertwined with the electronic degrees of freedom, and hence the main environmental coupling is the one to the electrons. In this study we are mainly concerned with amean ﬁeld description of the electron structure, as in thespirit of the DFT. Then a natural and quite general descrip-tion of the magnetic interaction due to electron-electroninteractions on the atomic site around rwithin the material is captured by the s-d-like model H int¼ /C0RJðr;r0ÞMðr;tÞ/C1sðr0;tÞdrdr0, where Jðr;r0Þrepresents the interaction between the magnetization density Mand the electron spin s. From a DFT perspective the interaction parameter Jðr;r0Þis related to the effective spin dependent exchange-correlation functional Bxc½Mðr0Þ/C138ðrÞ. For gener- ality we assume a fully relativistic treatment of thePRL 108, 057204 (2012) PHYSICAL REVIEW LETTERSweek ending 3 FEBRUARY 2012 057204-2electrons, i.e., including the spin-orbit coupling. In this interaction the dichotomy of the electrons is displayed,they both form the magnetic moments and provide the interaction among them. Owing to the general nonequilibrium conditions in the system, we deﬁne the action variable S¼I CH intdtþSZþSWZWN (6) on the Keldysh contour [ 20–22]. Here, the action SZ¼ /C0/C13H CRBextðr;tÞ/C1Mðr;tÞdtdrrepresents the Zeeman coupling to the external ﬁeld Bextðr;tÞ, whereas the Wess-Zumino-Witten-Novikov (WZWN) term SWZWN ¼RH CR1 0Mðr;t;/C28Þ/C1½@/C28Mðr;t;/C28Þ/C2@tMðr;t;/C28Þ/C138d/C28dtjMðrÞj/C02dr describes the Berry phase accumulated by the magnetization. In order to acquire an effective model for the magneti- zation density Mðr;tÞ, we make a second order [ 23] ex- pansion of the partition function Z½Mðr;tÞ/C138 /C17 trTCeiS, and take the partial trace over the electronic degrees of freedom in the action variable. The effective action /C14SMfor the magnetization dynamics arising from the magnetic inter-actions described in terms of H int, can, thus, be written /C14SM¼/C0IZ Mðr;tÞ/C1Dðr;r0;t; t0Þ/C1Mðr0;t0Þdrdr0dtdt0; (7) where Dðr;r0;t; t0Þ¼RJðr;r1Þð/C0iÞhTsðr1;tÞsðr2;t0Þi /C2 Jðr2;r0Þdr1dr2is a dyadic which describes the electron mediated exchange interaction. Conversion of the Keldysh contour integrations into real time integrals on the interval ( /C01,1) results in S¼Z MðfastÞðr;tÞ/C1½Mðr;tÞ/C2 _Mðr;tÞ/C138dtjMðrÞj/C02dr þZ MðfastÞðr;tÞ/C1Drðr;r0;t; t0Þ/C1Mðr0;t0Þdrdr0dtdt0 /C0/C13Z Bextðr;tÞ/C1MðfastÞðr;tÞdtdr; (8) with MðfastÞðr;tÞ¼Muðr;tÞ/C0Mlðr;tÞandMðr;tÞ¼ ½Muðr;tÞþMlðr;tÞ/C138=2which deﬁne fast and slow varia- bles, respectively. Here, MuðlÞis the magnetization density deﬁned on the upper (lower) branch of the Keldysh con- tour. Notice that upon conversion into the real time do- main, the contour ordered propagator Dis replaced by its retarded counterpart Dr. We obtain the equation of motion for the (slow) magne- tization variable Mðr;tÞin the classical limit by minimiz- ing the action with respect to MðfastÞðr;tÞ, cross multiplying byMðr;tÞunder the assumption that the total moment is kept constant. We, thus, ﬁnd_Mðr;tÞ¼Mðr;tÞ/C2/C18 /C0/C13Bextðr;tÞ þZ Drðr;r0;t; t0Þ/C1Mðr0;t0Þdt0dr0/C19 :(9) Equation ( 9) provides a generalized description of the semiclassical magnetization dynamics compared to theLLG Eq. ( 1) in the sense that it is nonlocal in both time and space. The dynamics of the magnetization at somepointrdepends not only on the magnetization locally at r, but also in a nontrivial way on the surrounding magneti-zation. The coupling of the magnetization at different positions in space is mediated via the electrons in the host material. Moreover, the magnetization dynamics is,in general, a truly nonadiabatic process in which the infor-mation about the past is crucial. However, in order to make a connection to the magne- tization dynamics as described by, e.g., the LLG equationas well as Eq. ( 1) above, we make the following consid- eration. Assuming that the magnetization dynamics is slow compared to the electronic processes involved in the time- nonlocal ﬁeld Dðr;r 0;t; t0Þ, we expand the magnetization in time according to Mðr0;t0Þ/C25Mðr0;tÞ/C0/C28_Mðr0;tÞþ /C282€Mðr0;tÞ=2. Then for the integrand in Eq. ( 9), we get Drðr;r0;t;t0Þ/C1Mðr0;t0Þ ¼Drðr;r0;t;t0Þ/C1/C20 Mðr0;tÞ/C0/C28_Mðr0;tÞþ/C282 2€Mðr0;tÞ/C21 :(10) Here, we observe that as the exchange coupling for the magnetization is nonlocal and mediated through D, this is also true for the damping (second term) and the inertia (third term). In order to obtain an equation of the exact same form as LLG in Eq. ( 1) we further have to assume that the magne- tization is close to a uniform ferromagnetic state, thenwe can justify the approximations _Mðr 0;tÞ/C25 _Mðr;tÞ and €Mðr0;tÞ/C25 €Mðr;tÞ. When Bint¼/C0RDðr;r0;t; t0Þ/C1 Mðr0;tÞdr0dt0=/C13is included in the total effective magnetic ﬁeldB, the tensors of Eq. ( 1)^Gand^Ican be identiﬁed with /C0R/C28Dðr;r0;t; t0Þdr0dt0andR/C282Dðr;r0;t; t0Þdr0dt0=2, re- spectively. From a ﬁrst-principles model of the host mate- rials we have, thus, derived the equation for themagnetization dynamics discussed in Ref. [ 9], where it was considered from purely classical grounds. However,it is clear that for a treatment of atomistic SD that allowsfor all kinds of magnetic orders, not only ferromagnetic, Eq. (1) is not sufﬁcient and the more general LLG equation of Eq. ( 9) together with Eq. ( 10) has to be used. We ﬁnally describe how the parameters of Eq. ( 1) can be calculated from a ﬁrst-principles point of view.Within the conditions deﬁned by the DFT system, theinteraction tensor D ris time local which allows us to write lim"!0i@"Drðr;r0;"Þ¼R/C28Drðr;r0;t; t0Þdt0and lim"!0@2"Drðr;r0;"Þ¼/C0R/C282Drðr;r0;t; t0Þdt0, wherePRL 108, 057204 (2012) PHYSICAL REVIEW LETTERSweek ending 3 FEBRUARY 2012 057204-3Drðr;r0;"Þ¼4spZ Jr/C26J/C260r0fð!Þ/C0fð!0Þ "/C0!þ!0þi/C14 /C2/C27ImGr /C260/C26ð!Þ/C27ImGr /C26/C260ð!0Þd! 2/C25d!0 2/C25d/C26d/C260: (11) Here, Jrr0/C17Jðr;r0Þwhereas Gr rr0ð!Þ/C17Grðr;r0;!Þis the retarded GF, represented as a 2/C22matrix in spin spaces. We notice that the above result presents a general expres-sion for frequency dependent exchange interaction. UsingKramers-Kro ¨nig’s relations in the limit "!0, it is easy to see that Eq. ( 11) leads to D rðr;r0;0Þ¼/C01 /C25sp ImZ Jr/C26J/C260r0fð!Þ/C27Gr /C260/C26ð!Þ /C2/C27Gr /C26/C260ð!Þd!d/C26d/C260; (12) in agreement with, e.g., Ref. [ 24]. We can make connection with previous results, e.g., Refs. [ 25,26], and observe that Eq. ( 11) contains the isotropic Heisenberg, anisotropic Ising, and Dzyaloshinsky-Moriya exchange interactionsbetween the magnetization densities at different points inspace [ 22], as well as the on-site contribution to the mag- netic anisotropy. Using the result in Eq. ( 11), we ﬁnd that the damping tensor is naturally nonlocal and can be reduced to ^Gðr;r 0Þ¼1 /C25spZ Jr/C26J/C260r0f0ð!Þ/C27ImGr /C260/C26ð!Þ /C2/C27ImGr /C26/C260ð!Þd!d/C26d/C260; (13) which besides the nonlocality is in good accordance with the results in Refs. [ 17,25], and is closely connected to the so-called torque-torque correlation model [ 27]. With in- clusion of the spin-orbit coupling in Gr, it has been dem- onstrated that Eq. ( 13) leads to a local Gilbert damping of the correct order of magnitude for the case of ferromag-netic permalloys [ 17]. Another application of Kramers-Kro ¨nig’s relations leads, after some algebra, to the moment of inertia tensor ^Iðr;r 0Þ¼spZ Jr/C26J/C260r0fð!Þ/C27/C20 ImGr /C260/C26ð!Þ /C2/C27@2!ReGr /C26/C260ð!ÞþImGr /C26/C260ð!Þ /C2/C27@2!ReGr /C260/C26ð!Þ/C21d! 2/C25d/C26d/C260; (14) where we notice that the moment of inertia is not simply a Fermi surface effect but depends on the electronic structureas a whole of the host material. Although the structureof this expression is in line with the exchange coupling inEq. ( 12) and the damping of Eq. ( 13), it is a little more cumbersome to compute due the presence of the deriva-tives of the Green’s functions. Note that it is not possible to get completely rid of the derivatives through partial inte- gration. These derivatives also make the moment of inertiavery sensitive to details of the electronic structure, which has a few implications. First, the moment of inertia can take large values for narrow band magnetic materials, suchas strongly correlated electron systems, where these de- rivatives are substantial. For such systems the action of moment of inertia can be important for longer time scales too, as indicated by Eq. ( 5). Second, the moment of inertia may be strongly dependent on the reference magnetic ordering for which it is calculated. It is well known that already the exchange tensor parameters may depend on themagnetic order. It is the task of future studies to determine how transferable the moment of inertia tensor as well as damping tensor are in between different magnetic ordering. In conclusion, we have derived a method for atomistic spin-dynamics which would be applicable for ultrafast (femtosecond) processes. Using a general s-d-like interac- tion between the magnetization density and electron spin, we show that magnetization couples to the surrounding in a nonadiabatic fashion, something which will allow for stud- ies of general magnetic orders on an atomistic level, notonly ferromagnetic. By showing that our method capture previous formulas for the exchange interaction and damp- ing tensor parameter, we also derive a formula for calcu- lating the moment of inertia from ﬁrst principles. In addition our results point out that all parameters are non- local as they enter naturally as bilinear sums in the same fashion as the well established exchange coupling. Ourresults are straightforward to implement in existing atom- istic SD codes, so we look on with anticipation to the ﬁrst applications of the presented theory which would be fully parameter free and hence can take a large step towards simulations with predictive capacity. Support from the Swedish Research Council is acknowl- edged. 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PhysRevB.92.054437.pdf	PHYSICAL REVIEW B 92, 054437 (2015) Spin pumping in YIG /Pt bilayers as a function of layer thickness M. Haertinger and C. H. Back Department of Physics, Universit ¨at Regensburg, 93053 Regensburg, Germany J. Lotze Walter-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany and Physik-Department, Technische Universit ¨at M ¨unchen, 85748 Garching, Germany M. Weiler, S. Gepr ¨ags, and H. Huebl Walter-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany S. T. B. Goennenwein Walter-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany and Nanosystems Initiative Munich (NIM), Schellingstrasse 4, 80799 Munich, Germany G. Woltersdorf* Department of Physics, Universit ¨at Regensburg, 93053 Regensburg, Germany and Institut f ¨ur Physik, Martin-Luther-Universit ¨at Halle-Wittenberg, 06120 Halle, Germany (Received 12 February 2015; revised manuscript received 23 July 2015; published 28 August 2015) We systematically investigate the spin-pumping effect in a series of yttrium-iron-garnet (YIG)–platinum (Pt) bilayers, comparing both broadband ferromagnetic resonance and electrically detected spin-pumpingexperiments. We infer the effective spin-mixing conductance as a function of YIG and Pt layer thicknessfrom ferromagnetic resonance measurements on pure YIG layers as well as on bilayer YIG /Pt samples in a frequency range from 2 to 20 GHz. In bare YIG ﬁlms, we determine a Gilbert damping constant of approximatelyα=0.001. With an additional platinum layer on top of the YIG samples, the damping is signiﬁcantly enhanced. The effective spin-mixing conductance extracted from the ferromagnetic resonance experiments is in agreementwith that determined from independent electrically detected spin-pumping experiments. DOI: 10.1103/PhysRevB.92.054437 PACS number(s): 76 .50.+g,75.76.+j,75.78.−n Spin transport across interfaces between ferromagnetic materials (FM) and normal metals (NMs) has been studiedintensely in recent years [ 1]. Initially such measurements were performed solely on metallic ferromagnetic materials.In addition to spin injection driven by charge currents innanoscale devices based on pillar [ 2] or nonlocal geometries [3,4], the spin-pumping effect was discovered as an efﬁcient method to inject pure spin currents from ferromagnets intonormal metal layers [ 5–7]. In spin-pumping experiments, a pure spin current is emitted at ferromagnetic resonance (FMR)from the ferromagnet into the normal metal layer, leadingto an additional interfacial Gilbert damping contribution tothe magnetization dynamics in the ferromagnet [ 6]. The time-averaged dc component of the pure spin current iscommonly detected using the inverse spin Hall effect (ISHE)in a normal metal layer that is in contact with the ferromagnet[8–11]. Here we refer to these experiments as electrically detected spin pumping (ED-SP). Recently, it was demonstratedexperimentally that the spin-pumping effect is also observedwhen a ferromagnetic insulator is used as a FM [ 12–14]. The most popular insulating ferromagnetic material is yttrium irongarnet (YIG). With its high Curie temperature of 550 K andthe smallest known damping parameter ( α=4×10 −5in bulk single crystals [ 15]), this material is well suited for experiments that involve, for instance, the propagation of spin waves over *georg.woltersdorf@physik.uni-halle.delarge distances. The majority of ED-SP experiments involvingYIG are performed using thick YIG layers that are prepared byliquid phase epitaxy [ 1,16]. In this case, it is difﬁcult to observe the enhanced damping since spin pumping is an interfaceeffect and the magnitude of the corresponding damping scalesinversely with the thickness of the FM. However, recentexperiments performed on well-deﬁned high-quality ultrathinYIG ﬁlms grown by pulsed laser deposition (PLD) withthicknesses of only a few nm clearly identify the enhanceddamping due to spin pumping [ 17,18]. In these experiments, the ultrathin YIG ﬁlms are covered ex situ by additional polycrystalline layers of Au and Fe using thermal evaporationin ultrahigh vacuum [ 17]. The metallic Fe layer acts as a perfect spin sink [ 7,19,20]. In doing so, the spin-mixing conductance was determined from FMR line broadening for YIG /Au interfaces by comparing the FMR linewidth in samples withand without the Fe-layer acting as a spin sink. In a followupexperiment, Burrowes et al. investigated the inﬂuence of surface cleaning by argon bombardment of the YIG ﬁlm priorto the deposition of the metallic cap layers [ 18]. By optimizing the sample treatment, the value of the spin-mixing conductancewas increased by a factor of 3. Recent reports showed thatultrathin YIG ﬁlms with very narrow resonance lines can alsobe prepared by PLD [ 21,22]. In addition, linewidth broadening and voltage generation due to ISHE were observed in a series ofPLD-grown YIG /Pt bilayers [ 23]. The experiments by Yiang et al. indicate the presence of an interface damping effect that is caused by a magnetic proximity polarization of a few 1098-0121/2015/92(5)/054437(6) 054437-1 ©2015 American Physical SocietyM. HAERTINGER et al. PHYSICAL REVIEW B 92, 054437 (2015) FIG. 1. (Color online) (a) Concept of the experiment. By mea- suring the enhanced linewidth in the YIG ﬁlm (left side), we observethe effect of the ac spin current injected into the Pt layer. At the same time, the dc component of the injected spin current (yellow arrow) is converted into a dc voltage via the inverse spin Hall effect. (b) Experimental geometry for the broadband FMR experiments. (c) Experimental geometry for the electrical detection of spinpumping in cavity-based experiments. atomic layers of Pt [ 24]. In sputter-deposited epitaxial YIG ﬁlms capped by platinum and tungsten, Wang et al. observed very large ISHE voltages in the mV range [ 25]. In this article, we quantitatively compare the measurements of the enhanced Gilbert damping and the ED-SP signals inPt/YIG thin-ﬁlm hybrids. The idea is illustrated in Fig. 1(a): spin pumping leads to enhanced damping due to the transfer(pumping) of an ac spin current from YIG into Pt. At thesame time, the dc component of the pumped spin current isconverted in the Pt layer into a dc charge current via ISHE,such that it can be detected by measuring the dc voltage V.I n this way, it becomes possible to quantify both the spin currentgenerated by spin pumping in YIG (from the change in YIGmagnetization damping) and the spin current absorbed in Pt(from the ISHE voltage). Our experimental data show that thespin current generated in YIG reaches the Pt layer. If present atall, a static magnetic proximity effect at the Pt/YIG interfacethus does not contribute signiﬁcantly to the Gilbert dampingin Pt/YIG systems. Our experiments also indicate that the spinmemory loss mechanism that was recently observed at metallicinterfaces with platinum [ 26] is not observed in the YIG /Pt system. Thin epitaxial YIG ﬁlms are grown by pulsed laser deposition on (111)-oriented gadolinium-gallium-garnet(GGG) substrates [ 27]. The deposition is carried out in an oxygen atmosphere at a pressure of 25 μbar and a substrate temperature of 500 ◦C. Metallic capping layers (Pt or Au) are deposited in situ using electron-beam evaporation; more details on the sample fabrication can be found in Refs.[27,28]. Here we focus on plain YIG ﬁlms, as well as YIG /Ptheterostructures, featuring YIG ﬁlm thicknesses 10 /lessorequalslantt YIG/lessorequalslant 200 nm, and Pt thicknesses 1 /lessorequalslanttPt/lessorequalslant20 nm. Furthermore, we also investigate a YIG(15 nm) /Au(8 nm) /Pt(7 nm) trilayer. More speciﬁcally, the magnetization dampingexperiments were performed using four plain YIG ﬁlms withthicknesses 16, 27, 33, and 50 nm, respectively, 4 YIG /Pt bilayers [YIG(12 nm) /Pt(7 nm), YIG(25 nm) /Pt(7 nm), YIG(44 nm) /Pt(7 nm), YIG(53 nm) /Pt(7 nm)], as well as the YIG /Au/Pt trilayer, i.e., in nine samples in total. The ED-SP measurements were performed in all the abovemultilayers, and additionally in 12 YIG /Pt samples fabricated following the same recipe. We ﬁrst address the magnetization damping measurements. The magnetic properties of the samples are characterizedusing FMR measurements. The resonance ﬁeld H FMR and the half-width at half-maximum (HWHM) linewidth /Delta1H are obtained from the FMR spectra by ﬁtting them to Lorentzianline shapes. Exemplary raw data obtained from plain YIGand YIG /Pt at 10 GHz are shown in Fig. 2.W eu s ea broadband FMR setup with a coplanar waveguide and performﬁeld-swept FMR measurements at microwave frequencies(between 2 and 20 GHz) for the external magnetic ﬁeld appliedparallel to the ﬁlm normal [out-of-plane conﬁguration, cf.Fig. 1(b)]. To increase the signal-to-noise ratio, we use ﬁeld modulation and lock-in detection. From FMR measurementsas a function of frequency, the gyromagnetic ratio γ, Gilbert damping parameter α, and effective magnetization M effare determined. By ﬁtting the frequency dependence of HFMRin the perpendicular conﬁguration to the resonance condition,γ=1.7±0.03×10 11 rad Tsis consistently obtained for all samples studied [see Fig. 2(a)]. The value for μ0Meff≈0.2T ﬂuctuates from sample to sample by up to ±10% [see Fig.2(a)]. To avoid micropatterning of the samples as well as complications arising from rectiﬁcation voltages due to spinHall magnetoresistance [ 29,30], the ED-SP measurements are performed in a microwave cavity. In this case, the rectiﬁcationeffects can be easily suppressed since the microwave magneticand electric ﬁelds are spatially separated and the sampleis placed in a node of the electric ﬁeld. In the ED-SPmeasurements, we use a commercial X-band microwave cavitycentered at 9.3 GHz with a Q-factor of approximately 3000.The geometry for these measurements is shown in Fig. 1(c). To ensure that the results from both methods (broadbandCPW and single-frequency cavity) can be compared, we alsoperformed FMR measurements with the X-band cavity [seeFig.2(b)]. For the bare YIG sample with a thickness of 27 nm, the resonance position and linewidth coincide; cf. Figs. 2(a) and2(b). The observation of two FMR lines in the case of the bare YIG ﬁlm in the out-of-plane conﬁguration is dueto spatially inhomogeneous properties of the samples mostlikely caused by ﬂuctuations of saturation magnetization orthe anisotropy ﬁeld. The fact that we observe ﬂuctuations ofthe effective magnetization of up to 20% between the variousYIG ﬁlms studied in this paper [cf. the inset of Fig. 2(a)] indicates that such ﬂuctuations (with a smaller amplitude) mayalso cause the spatially inhomogeneous magnetic propertiesresponsible for the observation of multiple resonance lines.The measurements for the 27 nm YIG ﬁlm in Figs. 2(a) and 2(b) are performed with different pieces from the same wafer. Therefore, it is not surprising that the second line with smaller 054437-2SPIN PUMPING IN YIG /Pt BILAYERS AS A FUNCTION . . . PHYSICAL REVIEW B 92, 054437 (2015) FIG. 2. (Color online) Typical resonance spectra of a bare 27 nm YIG layer and a sample with comparable YIG thickness (25 nm), covered by a 7 nm Pt capping. In (a) the FMR lines are measured in the perpendicular conﬁguration at 10 GHz using the broadband FMR setup [cf.Fig. 1(b)] while the spectra in (b) are obtained with an X-band cavity operating at 9.3 GHz [cf. Fig. 1(b)]. By ﬁtting the resonance spectra to Lorentzian line shapes, the resonance ﬁeld H FMRand the linewidth /Delta1H are extracted. The resonance ﬁeld and linewidth for the case of multiple lines is obtained by two-peak ﬁtting. The black solid lines are the ﬁts while the dotted lines correspond to the individual Lorentzianline shapes. In the further analysis, only the line with the larger amplitude is used. From the frequency dependence of the resonance ﬁeld H FMR, which is shown in the inset of graph (a) for both samples, we ﬁnd a gyromagnetic ratio γ=1.70±0.03×1011rad Ts. The effective magnetization isμ0Meff=0.185±0.001 T and μ0Meff=0.219±0.001 T for the 27-nm-thick YIG ﬁlm and the 25 nm YIG /Pt bilayer, respectively. To compare the resonance lines of the two samples in (a), the spectrum of the 25 nm YIG /Pt bilayer has been shifted to lower ﬁelds by 35 mT. In general, we ﬁnd that the saturation magnetization of the various YIG ﬁlms can vary by up to 20% between the different samples. amplitude in Fig. 2(a) occurs at higher ﬁelds than the main resonance, while it is found at lower ﬁelds in Fig. 2(b).I n both cases, the data can be ﬁtted using two Lorentzian lines(solid lines). The individual resonance lines are shown asdotted lines. All FMR lines of the bare YIG sample that areshown in Fig. 2(10.0 and 9.3 GHz) have a width between 1.3 and 1.6 mT in the perpendicular conﬁguration. Whenincreased damping caused by the Pt capping [see Fig. 2(a)] or by two-magnon scattering in the in-plane conﬁguration [seeFig.2(b)] is present, only one resonance line is visible. In this case, the presence of multiple resonance lines is expected tocause an inhomogeneous broadening of the ﬁtted resonancelinewidth. However, as the amplitude of the second line isfour times smaller than the main resonance, the correspondingsystematic error for the damping constant is small and weestimate it to be less than 5%. This is in line with the fact that noincreased zero-frequency offset is observed for the frequencydependence of the linewidth in the case of the Pt-covered YIGsample compared to the bare YIG sample shown in the inset ofFig.3. Moreover, our analysis is not sensitive to the absolute value of the linewidth. Instead, the slope of the linewidth versusfrequency is evaluated to extract the damping parameter, aswill be discussed below. The FMR linewidth is caused by magnetic relaxation and inhomogeneous line broadening. The different contributionsmay be separated by ﬁtting the frequency dependence ofthe FMR linewidth /Delta1H(ω)[ s e eF i g . 3(b)]. One expects the following behavior [ 31,32]: /Delta1H(ω)=αω μ0γ+/Delta1H(0). (1) In this formula, the intrinsic Gilbert damping parameter α leads to a linear frequency dependence of /Delta1H(ω). Sampleinhomogeneities lead to inhomogeneous FMR line broaden- ing and two-magnon scattering and thereby cause a zero-frequency offset of the linewidth /Delta1H(0). We perform the FIG. 3. (Color online) From the frequency dependence of the FMR linewidth /Delta1H (inset), the damping parameter αcan be determined. The YIG layer without Pt capping shows a small linewidth /Delta1H and a small variation of the Gilbert damping parameter for different thicknesses. For the 27 nm YIG layer, the Gilbert damping is α≈0.0007±0.000 04. With additional 7 nm Pt, the damping parameter is about ﬁve times larger, α≈0.0034±0.0002. The inverse YIG thickness dependence of the damping parameter for samples with 7 nm Pt is consistent with the spin-pumping theory of Eq. ( 2). It can easily be seen that with an additional Au layer inserted between YIG and Pt, spin pumping is reduced. 054437-3M. HAERTINGER et al. PHYSICAL REVIEW B 92, 054437 (2015) frequency-dependent FMR measurements with the magnetic ﬁeld and magnetization oriented along the ﬁlm normal (per-pendicular conﬁguration) in order to avoid the two-magnonscattering contribution [ 31]. The Gilbert damping constant is determined from the slope of /Delta1H versus ω. The inset in Fig. 3 shows /Delta1H obtained in the perpendicular conﬁguration. We ﬁnd a linear frequency dependence of the FMR linewidth witha zero-frequency offset less than μ 0/Delta1H(0)≈2m Tf o rt h e bare YIG ﬁlm with a thickness of 27 nm. When the YIG ﬁlmis capped by an additional Pt layer, one observes a signiﬁcantincrease in the slope of the frequency-dependent linewidthand hence an increased Gilbert damping constant, as shown inFig.3. In bilayers consisting of ferromagnetic and metallic materi- als such as YIG /Pt, the precession of the magnetization in the magnetic layer causes a net ﬂow of spins into the nonmagneticlayer because Pt acts as a spin sink [ 6]. Due to spin pumping, the Gilbert damping parameter for an uncapped YIG sampleα 0is enhanced by the spin-pumping contribution αsp, and one obtains a total damping parameter α=α0+αsp. To determine the Gilbert damping and the contribution arising from spinpumping, the frequency dependence of the FMR linewidth ismeasured in four plain YIG ﬁlms, four YIG /Pt heterostruc- tures, as well as the YIG /Au/Pt trilayer, as mentioned above. A typical result is shown in the inset of Fig. 3. From the difference of the slope of the linewidth versus frequency for both samples,one ﬁnds an additional damping of α sp=0.0024±0.0002 for the 25 nm thick YIG ﬁlm capped by 7 nm Pt. Due toconservation of angular momentum, this additional dampingcan be used to evaluate the Pt/YIG interface spin-mixingconductance. The additional Gilbert damping is related to the effective interface spin-mixing conductance g ↑↓ effby [33] αsp=gμB 4πM Sg↑↓ eff1 tYIG. (2) Here, MSis the saturation magnetization and tYIGis the thickness of the magnetic material. gis thegfactor and μBis the Bohr magneton. The inverse YIG thickness dependenceof the additional damping in Eq. ( 2) is well reproduced experimentally, as shown in Fig. 3. Here four samples with YIG thicknesses between 8 and 50 nm are studied. Additional FMRmeasurements on YIG samples without Pt capping featuringdifferent YIG thicknesses conﬁrm that the intrinsic thicknessdependence of the damping in our YIG ﬁlms is weak; cf. Fig. 3. Using Eq. ( 2), the experimental thickness dependence of the additional damping results in a value of g ↑↓ eff=9.7×1018 1 m2 for the Pt/YIG interface. One should point out that in general the NM layer does not act as a perfect spin sink leading toa partial backﬂow of the spin current. The magnitude of thebackﬂow depends on thickness, spin diffusion length, and theelectron scattering in NM [ 33]: g ↑↓ eff≈g↑↓ 1 1+1 4√ε 3tanh/parenleftBigtPt λsd/parenrightBig. (3) Hereε=τm τsfis the ratio of momentum and spin-ﬂip scattering times, tPtis the Pt-layer thickness, and λsdt h es p i nd i f f u s i o n length in Pt. To quantify the magnitude of the backﬂow for the samples used in the analysis above, a series of samples with differentPt thicknesses (1–20 nm) is investigated [ 14]. Given the very short spin-diffusion length of only 1.5 nm in Pt inferred fromthese experiments, one can expect from Eq. ( 3) a correction to the spin-mixing conductance of less than 5% for a Pt thickness of 7 nm for the effective spin-mixing conductance g↑↓ eff used in ( 2). We also estimate the spin-mixing conductance for the YIG/Au interface (using the YIG /Au/Pt trilayer). As shown above, the properties of our samples are well described with Ptacting as a nearly perfect spin sink. By inserting an Au layerwith a thickness of only 8 nm between Pt and YIG (all metalliclayers are grown in situ ), the additional damping due to spin pumping is limited by the mixing conductance of the YIG /Au interface. As shown in Fig. 3(blue star), we ﬁnd a reduced value of the additional damping. This corresponds to a value ofg ↑↓ eff=6×1018 1 m2for the spin-mixing conductance for the YIG/Au interface. We now compare the mixing conductance of the Pt/YIG interface obtained from damping experiments to the mixingconductance inferred from electrically detected spin-pumpingexperiments, performed using the same set of YIG /Pt samples, as well as YIG /Pt ﬁlms with different respective layer thicknesses, grown with the same recipe. In the electrically detected spin-pumping approach, the dc component of the spin current pumped in ferromagneticresonance is converted into a charge current in the Pt layervia the ISHE, and detected as a dc voltage V[cf. Fig. 1(a)]. More speciﬁcally, the relaxation of the dc component of thepumped spin current leads to the voltage that is recorded in the ISHE-based experiments. The ISHE charge current density is given by j c=αSH−2e /planckover2pi1js×σ, with the spin Hall angle αSH,t h e spin-current propagation direction js, and the spin-current spin polarization σ[34]. The open-circuit ISHE dc voltage recorded in electrically detected spin-pumping experiments thus mustinvert its sign upon magnetization (viz. spin polarization σ) inversion. For the electrically detected spin-pumping measurements, the samples were carefully positioned at the microwavemagnetic ﬁeld antinode of a commercial X-band microwavecavity in order to minimize the inﬂuence of magnetoresistiverectiﬁcation-type effects [ 10,11,35–37]. Typical experimental data from a YIG(58 nm) /Pt(2 nm) bilayer are shown in Figs. 4(a) (FMR absorption) and 4(b) (dc voltage). The FMR signal appears as a peak in the microwave absorptionsignal, at an external magnetic-ﬁeld magnitude of aboutμ 0HFMR≈268 mT for the center frequency ν=9.3 GHz of our microwave cavity [Fig. 4(a)]. The dc voltage Vrecorded simultaneously with the FMR signal [Fig. 4(b)] exhibits a characteristic extremum at HFMR, and it inverts its sign upon inversion of the external magnetic-ﬁeld orientation, asexpected for an ISHE-type spin-pumping voltage. We extract the spin-mixing conductance by solving the established expression for the magnitude of the open-circuitvoltage Vappearing in spin pumping [ 9,33,38–40], V=eg ↑↓ effαSHλSDtanhtPt 2λSDRwνP sin2/Theta1 (4) forg↑↓ eff.Ris the resistance of the Pt ﬁlm of width w, and P≈1 is a correction factor taking into account the ellipticity 054437-4SPIN PUMPING IN YIG /Pt BILAYERS AS A FUNCTION . . . PHYSICAL REVIEW B 92, 054437 (2015) (a) (b) (c) (d) FIG. 4. (Color online) (a) and (b) The electrically detected spin- pumping raw data. The FMR signal of the YIG(58 nm) /Pt(2 nm) bilayer [panel (a)] only depends on the magnitude of the externally applied magnetic ﬁeld H. The dc voltage appearing across the sample in FMR, however, inverts its sign upon magnetic-ﬁeld inversion [panel (b)], as expected for an ISHE-type voltage. Panel (c) shows the effective spin-mixing conductances extracted via Eq. ( 4) from the electrically detected spin-pumping measurements, plotted as a function of the YIG ﬁlm thickness. The full green symbols represent data obtained from the YIG /Pt bilayers also studied in the magnetization damping experiments, while the open circles correspond to data from additional YIG /Pt bilayers. The blue star represents data from the YIG /Au/Pt trilayer. The dashed black and dotted blue lines represent the g↑↓ effvalue inferred from the magnetization damping experiments for YIG /Pt and YIG /Au/Pt, respectively. Panel (d) shows the same data, but now plotted against the Pt ﬁlm thickness. The horizontal black viz. blue dashes again correspond to the g↑↓ effvalues extracted from the broadband damping measurements [cf. panel(c)]. of the magnetization precession trajectory [ 41]. We calculate the magnetization precession cone angle /Theta1=hrf//Delta1H from the microwave magnetic ﬁeld hrf(established in calibration experiments), and /Delta1H. We thus assume dominant Gilbert-likemagnetization damping, validated by the FMR experiments described above. Figure 4(c)shows the g↑↓ effvalues thus obtained from room- temperature electrically detected spin-pumping experimentsfor the series of YIG /Pt heterostructures. Using the values α SH=0.11 and λSD=1.5 nm inferred for our Pt ﬁlms as detailed in Ref. [ 14], we obtain 3 ×1018/lessorequalslantg↑↓ eff/lessorequalslant1.2× 10191 m2for YIG ﬁlm thicknesses smaller than 100 nm, and g↑↓ eff=4.7×10181 m2for the YIG /Au/Pt trilayer, in good agreement with the result determined from the enhancedGilbert damping measurements. For heterostructures with YIG layer thicknesses t YIG/greaterorsimilar100 nm, we ﬁnd larger values g↑↓ eff/lessorequalslant 6×10191 m2. However, in these thicker YIG ﬁlms, standing spin-wave modes become more and more pronounced, suchthat our analysis based on the homogeneous FMR modemight no longer be appropriate. Thus, the calculation of theprecession cone angle from the FMR linewidth becomes lessreliable with increasing YIG ﬁlm thickness. Finally, we wouldalso like to note that the exact value of the microwave cavityquality factor, and the positioning of the sample in the cavity,vary from measurement to measurement. This results in a systematic error in g ↑↓ effin all ED-SP data, which is estimated to be up to a factor of 2. In conclusion, we have quantiﬁed the spin-current induced Gilbert-like damping contribution in YIG /Pt heterostructures. The effective spin-mixing conductance g↑↓ effwe extract from magnetization damping experiments is in quantitative agree- ment with g↑↓ effvalues inferred from electrically detected spin-pumping measurements in the same samples (utilizing theinverse spin Hall effect to convert the spin current pumped intoPt into a voltage). On the one hand, our measurements conﬁrma posteriori the widespread assumption in the literature that g ↑↓ effvalues determined from only magnetization damping, or from only electrically detected spin-pumping measurements,indeed are reliable and can be quantitatively compared. On theother hand, the fact that the two different techniques indeed yield very similar values of g ↑↓ effin one and the same sample suggests that the additional damping in YIG /Pt bilayers is a sole consequence of spin pumping. 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PhysRevApplied.8.034012.pdf	Correlations Between Structural and Magnetic Properties of Co2FeSi Heusler-Alloy Thin Films Weihua Zhu,1Di Wu,1Bingcheng Zhao,1Zhendong Zhu,1Xiaodi Yang,2Zongzhi Zhang,1,and Q. Y. Jin1 1Department of Optical Science and Engineering, Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), and Shanghai Ultra-Precision Optical Manufacturing Engineering Center, Fudan University, Shanghai 200433, China 2Laboratory of Advanced Materials, Fudan University, Shanghai 200438, China (Received 15 May 2017; revised manuscript received 6 August 2017; published 18 September 2017) The structural and magnetic properties are the most important parameters for practical applications of Co-based Heusler alloys. The correlations between the crystallization degree, chemical order, magnetic coercivity, saturation magnetization ( MS), and in-plane magnetic anisotropies are systematically inves- tigated for Co 2FeSi (CFS) films fabricated at different temperatures ( TS). XRD shows that the CFS layer changes progressively from a disordered crystal structure into a chemically disordered A2structure and further into a chemically ordered B2and even L21structures when increasing TSup to 480°C. Meanwhile, the static angular remanence magnetization curves show a clear transition of magnetic anisotropy fromtwofold to fourfold symmetry, due to the competition effect between the uniaxial anisotropy field H Uand biaxial anisotropy field HB. The HUvalue is found to be weakly dependent on TS, while HBshows a continuous enhancement at TS>300°C, implying that the enhancement of the L21ordering degree would not weaken the biaxial anisotropy. The varying trend of HBis similar to MS, which can be respectively attributed to the improved crystal structure and chemical order. The anisotropic fields and their variation behaviors determined by a vibrating sample magnetometer are highly consistent with the results by a time-resolved magneto-optical Kerr effect study. Our findings provide a better understanding of the structural ordering and magnetic anisotropy, which will be helpful for designing advanced spintronic devices. DOI: 10.1103/PhysRevApplied.8.034012 I. INTRODUCTION Full-Heusler alloys (FHAs) are attracting considerable attention due to their great advantages including latticematch with the III-V semiconductors, half-metallic band structure with high spin polarization P, and reduced magnetic damping constant α[1–4]. The compounds are usually described by the formula X 2YZ, where XandYare transition metals and Zis a main group element. Among the various types of FHAs, it has been predicted that theCo-based FHAs, such as Co 2FeSi (CFS) alloy, owns a very high Curie temperature TC∼1100 K and a quite large magnetic moment up to 6μB[5,6]. With these merits, the FHAs as spin source materials are highly appropriate forpractical applications in spintronic devices [7]. The FHA- based magnetic tunnel junctions (MTJs) are very successful and are of great technological interest for developing spin-transfer-torque magnetic random access memory (STT MRAM) devices. Very large tunneling magnetoresistance (TMR) signals up to 429% at room temperature (RT) arereported in MTJs by using the FHA layer as the free and reference magnetic electrodes [8–10]. Moreover, the current –perpendicular-to-plane (CPP) spin-valve devi- ces with a highly spin-polarized electrode of FHAs havealso attracted great attention [11,12] . A rather highgiant MR ratio of 58% at RT is demonstrated in Co 2FexMn 1−xSi=Ag=Co2FexMn 1−xSi CPP devices [13], and the MR ratio and associated resistance-area product satisfy the requirements for a Tbit =in2hard-disk-drive read head. In addition, the half-metallic FHAs allow the injec-tion of highly spin-polarized current into a semiconductorthrough its spin-up channel [14–16], and the injected current is fully spin polarized, which makes them idealcandidates for spin contacts on semiconductors and fordeveloping devices with new functions as well. It is known that developing STT MRAM with high performance requires the ferromagnetic electrode materialwith improved parameters of high spin polarization and lowmagnetic damping for efficient STT switching at a reducedcurrent density and large TMR signal [17,18] . Although high Pand low αhave been theoretically predicted for FHAs due to the lack of minority spin density of states atthe Fermi level, the experimentally measured results are notas good as expected. For the successful implementation of FHA films into next-generation magnetic storage, it is of key importance to optimize the crystal structure andmaterial properties. As we know, the advanced magneticproperties of CFS films are closely related to the chemicalorder [19,20] , which are sensitive to the deposition con- dition, atomic composition, and adjacent layers. The CFSfilm fabricated at low temperatures usually has a disorderedA2phase, with atoms randomly placed in a body-centered zzzhang@fudan.edu.cnPHYSICAL REVIEW APPLIED 8,034012 (2017) 2331-7019 =17=8(3)=034012(10) 034012-1 © 2017 American Physical Societycubic lattice. The formation of chemically ordered B2and L21structures usually requires a high growth temperature and/or postannealing treatment [19,21 –23]. In most cases, multiple structure types may coexist. The existence of disordered phases may worsen the magnetic properties. It isfound that the damping parameter decreases with the increase of atomic ordering [24]. And among these three structure types categorized by occupation of the availableatomic sites, only the chemically ordered structures showed half-metallic behavior [24,25] . Although there are quite a few works performed on the structural and magneticproperties of the FHA films, the exact relationship and underlying mechanism still remain unclear. The saturation magnetization ( M S) and spin polarization of Co-based FHAs increase simultaneously with the increase of the chemical ordering degree [26,27] , while some other researchers observed a reduction of MSwhich may be related to the diffusion of impurity atoms from the bufferlayer [28–30]. Apart from the spin polarization and saturation magnetization, other magnetic properties suchas magnetic anisotropy are also very important and should be carefully characterized and controlled. It is expected that the cubic-crystal structure should present an in-planebiaxial anisotropy for the (001) epitaxial films. However,an additional in-plane uniaxial magnetic anisotropy hasoften been reported for the FHA films grown on varioussingle-crystalline substrates [29,31 –36]. Moreover, quite different and even opposite varying behaviors are observedfor the uniaxial and biaxial magnetic anisotropies as the annealing temperature increases [21,22,37 –39]. A strong reduction of biaxial anisotropy is found and attributed tothe increasing fraction of the L2 1phase [21,22,38,39] . However, the authors do not exclude the origin from anannealing-induced atomic diffusion effect. The magneticanisotropy not only determines the magnetization-switch-ing behaviors but affects the magnetic damping constant and device thermal stability as well. Moreover, the coex- istence of uniaxial and biaxial magnetic anisotropies cancomplicate the magnetization-switching process, whichshould be clarified for practical applications. As we cansee, the performances of spintronic devices rely signifi-cantly on the structural and magnetic properties of the full-Heusler-alloy films. In order to achieve effective control of the material properties for designing appropriate spintronic devices, it is of key importance to acquire a general picture on the correlations between the sample structure and magneticproperties. Considering the chemical order is directly correlated to the atom arrangement, tailoring the lattice site occupation of the CFS alloy provides an efficient wayto control the material properties. Therefore, in this work we perform a detailed study on the CFS thin films grown on MgO(001) single-crystalline substrates. In order to excludethe influence of atomic diffusion on magnetic properties,only the CFS layer is in situ heated at various substrate temperatures ( T S), while the Ta top layer is depositedat RT. The crystal structure and chemical order of CFS layers are characterized by XRD, and the TS-dependent magnetic properties are deduced according to the static anddynamic magnetization measurements. II. EXPERIMENT All the samples, in a structure of CFS ð15nmÞ=Tað7nmÞ, are fabricated on MgO(001) single-crystalline substrates bydc magnetron sputtering under a base pressure better than 3.0×10 −8Torr. The deposition rate is fixed as 0.10Å=s for the CFS layer and 0.52Å=s for the Ta cap layer, respec- tively. The CFS layer is sputtered from an alloy target with a stoichiometric composition of Co 50Fe25Si25under an Ar pressure of 4 mTorr. The top Ta layer is used to protect the CFS layer against oxidation. In order to form differently ordered phases, the CFS layers are deposited at varioussubstrate temperatures of T S¼200, 300, 350, 400, and 480°C, and they are further in situ annealed at the same temperature for 10 min prior to the Ta-layer deposition. Then, the Ta cap layer is sputtered after the CFS layer isnaturally cooled to RT. The film crystallographic texture is examined by a high-resolution four-circle x-ray diffractometer with Cu Kαradiation. A vibrating sample magnetometer (VSM) is utilized to measure the magnetic hysteresis loops andangular remanence magnetization (ARM) curves. The dynamic magnetic properties are acquired by a pump- probe system based on the time-resolved magneto-opticalKerr effect (TRMOKE). The TRMOKE measurements areachieved using a pulsed Ti:sapphire laser with a central wavelength of 800 nm, a pulse duration of 150 fs, and a repetition rate of 1000 Hz [40]. An intense pump pulse beam with a fluence of approximately 7.0mJ=cm 2is used to excite the magnetization precession behavior, and thetransient MOKE signal is detected by a weak probe pulse of approximately 0.6mJ=cm 2which is time delayed with respect to the pump beam. The pump and probe laser beamsare at almost perpendicular incidence, with spot diameters of about 4.0 and 0.2 mm, respectively. During the TRMOKE measurement, an external magnetic field His applied at an angle of θ H¼71° in order to drive the magnetization orientation away from the in-plane easy axis. Note that all the measurements are conducted at RT. III. STRUCTURAL CHARACTERIZATION As we know, the Co 2FeSi Heusler-alloy crystal usually includes three kinds of atomic structure, i.e., L21,B2, and A2with different chemical ordering degrees. Figure 1(a) presents the highly ordered L21structure (space group no. 216: F-43m ), the two Co atoms occupy the (0, 0, 0) and (0.5, 0.5, 0.5) positions, the Fe atom occupies the (0.25,0.25, 0.25) position, and the Si atom occupies the (0.75, 0.75, 0.75) position. The cubic L2 1structure cell consists of four interpenetrating face-centered cubic (fcc) sublattices,ZHU, WU, ZHAO, ZHU, YANG, ZHANG, and JIN PHYS. REV. APPLIED 8,034012 (2017) 034012-2two of which are equally occupied by Co, and the rest two are occupied by Fe and Si. The two Co-site fcc sublattices form a simple cubic sublattice structure, in which theFe and Si atoms are alternatingly located in the center sites. If the Fe and Si atomic sites are randomly occupied, the ordering degree decreases and the L2 1structure changes to theB2structure (space group no. 225: Fm-3m ), as shown in Fig. 1(b). Figure 1(c) displays the completely disordered A2structure, in which all the atomic sites are occupied randomly. Nearly all the magnetic properties of the CFS films are related to the crystalline quality and atomic site occupation [41], which can be characterized by the XRD measurements. The (111) superlattice diffraction peak indicates the presence of ordered L21structure, while the (002) superlattice peak is a main characteristic of B2 structure. Figure 2(a) shows the θ−2θpatterns for CFS films grown at various substrate temperatures. For the samples ofT S¼300°C or lower, apart from the strong (002) peak arising from the MgO substrate, no other diffraction peaks can be detected, which suggests that the samples depositedat low temperatures have a disordered crystal structure, probably due to the presence of defects created in the lattice by energetic atom bombardment during sputtering. A weak(004) peak appears at T S¼350°C, implying that the CFS film with structure disorder starts to crystallize in the chemically disordered A2phase. As TSis further increased up to 400°C and above, the diffraction peaks of both CFS (002) and CFS (004) can be observed, indicating the CFS films are (001) oriented in the out-of-plane direction andhave at least a B2structure. The CFS peak intensity increases considerably with the increase of T S, as the result of reduced lattice defects and hence improved crystal structure. From the diffraction patterns, we can estimate theCFS grain size Dby using the Scherrer equation [42], D¼½ ðkλÞ=βcosθ/C138, where kof approximately 0.89is a dimensionless shape factor, λis the x-ray wavelength of 0.154 nm, βis the full width at half maximum of the diffraction peaks, and θhere is the diffraction angle of the CFS (004) peak (in units of degree). As expected, increas-ing the substrate temperature can lead to obvious crystal- lization and grain growth; the calculated grain sizes areapproximately 7.4, 12.5, and 13.8 nm for T S¼350, 400, and480°C, respectively. In order to clarify the epitaxial relationship and chemical order of CFS films in the atomic level, Φscan measure- ments are performed for the CFS films of TS¼350, 400, and480°C. Figures 2(b)and2(c)show the patterns of CFS h220iobtained at 2θ¼45.48°,ψ¼45.20° and CFS h111i at2θ¼27.36°,ψ¼54.71°, respectively. Here Φcorre- sponds to the in-plane rotation angle and ψthe tilt angle [43], andΦ¼0is defined as the direction of the MgO [110]. Apparently, the four h220idiffraction peaks shown in Fig. 2(b) are separated periodically with an angle dif- ference of 90°, indicating that the CFS film has a fourfold symmetry. From the relative peak positionsof CFS (220) in the Φscan patterns, we can determine FIG. 1. The ordered L21structure (a) and the disorder B2structure (b) andA2structure (c). FIG. 2. XRD patterns of the (a) θ−2θscan, (b) Φscan for the CFS (220) diffraction peaks, and (c) Φscan for CFS (111) diffraction peaks.CORRELATIONS BETWEEN STRUCTURAL AND … PHYS. REV. APPLIED 8,034012 (2017) 034012-3that the CFS films of TS≥350°C are epitaxially grown on the MgO substrates, with an epitaxial relation ofMgO ð001Þ½100/C138∥CFS ð001Þ½110/C138. The peak intensity of all the diffraction lines is decreasing with T S, due to the increased disorder and poor epitaxial growth of the CFScrystal at lower substrate temperatures. Figure 2(c)shows the evolution of CFS h111isuperlattice diffraction peaks, verifying the presence of an atomically ordered L2 1phase atTS>350°C. Note that the corresponding (220) and (111) diffraction peaks are separated by 45°, coincidingwell with the crystal structure of CFS. The parameters of chemical ordering degree, S L21for the L21phase and SB2for the B2phase, can be evaluated by the relative peak intensities according to the extended Webstermodel [19,44] , I200 I400¼S2 B2Ifull-order 200 Ifull-order 400; I111 I220¼/C18 SL21·3−S2 B2 2/C19Ifull-order 111 Ifull-order220; ð3:1Þ where IhklandIfull-order hkl are the experimentally obtained diffraction intensity of the CFS hklplane and the theoreti- cally calculated diffraction intensity of the fully L21 ordered CFS alloys, respectively. According to Ref. [19], the parameter SB2for the full-Heusler X2YZalloy of Co2FeSi represents the random occupation between X andY(orZ) sites. In contrast, the fraction of Yatoms on theZsites is generally used to evaluate the L21ordering degree SL21. As a result, for the completely disordered A2 crystal structure, both order parameters of SB2andSL21are zero. The SB2v a l u ei s1 0 0 %a sl o n ga st h e Xsites are orderly occupied by Co atoms, no matter whether the YandZsites are random or not. It has been pointed out that the SL21value depends on the ordering degree of B2, which increases with SB2[19]. The calculated ordering parameters are listed in Table I. Clearly, SL21andSB2increase with the substrate temperature, and a higher substrate temperature is advanta-geous for the transformation of B2intoL2 1structure. From these results, it can be concluded that the samples of TS¼ 300°C and below have a disordered crystal structure. As TS is350°C, the chemically disordered A2crystal structure forms, which gradually evolves towards a partially chemi-cally ordered B2and completely ordered L2 1structure when further increasing the substrate temperature. As shown in thenext sections, the evolutions of the sample crystal structureand chemical ordering degree are well correlated with thevariation of magnetic properties.IV. INVESTIGATIONS OF MAGNETIC PROPERTIES A. Static magnetic measurements by VSM The magnetic hysteresis loops and ARM curves are measured by VSM for all the samples, with an in-planemagnetic field applied along various orientations. For the convenience of data analyses, we build up a polar coordinate system. As shown in Fig. 3(a), thezaxis is perpendicular to the sample plane, and the xandyaxes are, respectively, parallel to the [110] and ½1¯10/C138crystallographic directions of the CFS layer. The φ M(orφH) is defined as the in-plane angle of magnetization M(or applied magnetic field H) with respect to the xaxis, while θM(orθH) is the corresponding out-of-plane angle between M(orH) and the zaxis. From the VSM measurements, we know that there are two kinds of in-plane magnetic anisotropies for our CFS films: a biaxial magnetic anisotropy with two easy axes along the [110] ( xaxis) and ½1¯10/C138(yaxis) directions and a uniaxial anisotropy with an easy axis along the [110] direction. Based on the polar coordinate system, the total free energy density Eof our CFS films, which includes the Zeeman energy, demagnetizing energy, and anisotropyenergy, can be given as [23,45] E¼E Zeeman þEdemagn þEaniso ¼−MSH½sinθMsinθHcosðφH−φMÞþcosθMcosθH/C138 −2πM2 Ssin2θMþ1 4KBsin2ð2φMÞsin4θM þKUsin2φMsin2θMþK⊥sin2θM; ð4:1Þ where KBrepresents the in-plane biaxial magnetic anisotropy constant, KUthe in-plane uniaxial magnetic anisotropy constant, and K⊥the out-of-plane uniaxial magnetic anisotropy constant. Here, we define HU¼ 2KU=MSandHB¼4KB=MSas the in-plane uniaxial and biaxial magnetic anisotropy fields, respectively, and H⊥¼ 2K⊥=MSas the out-of-plane uniaxial magnetic anisotropy field. According to the Stoner-Wohlfarth coherent rotation model [46], the equilibrium position of magnetization can be determined from the minimum free energy density:TABLE I. The values of part samples for SB2andSL21. TS(°C) SB2(%) SL21(%) 350 0 0 400 73.6 46.7480 82.7 56.9 FIG. 3. The polar coordinate system (a) and its in-planegeometry (b) for magnetic measurement and data analyses.ZHU, WU, ZHAO, ZHU, YANG, ZHANG, and JIN PHYS. REV. APPLIED 8,034012 (2017) 034012-4∂E ∂θM¼−MSH½cosθMsinθHcosðφH−φMÞ−sinθMcosθH/C138 −2πM2 Ssinð2θMÞþKB 2sin2ð2φMÞsin2θMsinð2θMÞ þKUsin2φMsinð2θMÞþK⊥sinð2θMÞ¼0; ð4:2Þ ∂E ∂φM¼−MSH½sinθMsinθHsinðφH−φMÞ/C138 þKB 2sin4θMsinð4φMÞþKUsin2θMsinð2φMÞ¼0: ð4:3Þ As the CFS samples have in-plane magnetic anisotropy and the external magnetic field is applied in plane during the VSM measurement, the magnetization should also stayin the film plane [see the corresponding in-plane geometry depicted in Fig. 3(b)]. In this case, both θ MandθHare equal toπ=2, and the equilibrium position equation of in-plane magnetization can be simplified as KB 2sinð4φMÞþKUsinð2φMÞ¼MSHsinðφH−φMÞ:ð4:4Þ According to this equation, the magnetic anisotropy constants can be deduced by fitting the field dependence of hard-axis magnetization curves. Meanwhile, if we set theapplied field Hin Eq. (4.4) as zero, we can get the angular dependence relation of remanent magnetization: KB 2sinð4φMÞþKUsinð2φMÞ¼0: ð4:5Þ As a result, if the fitted magnetic anisotropy constants are input into Eq. (4.5), the corresponding ARM curves can be simulated. In the following, we show the static magnetic properties and related data analyses based on the above equations of (4.4) and(4.5). Figure 4(a) shows the normalized in-plane magnetic hysteresis loops for CFS films grown at different temper-atures. The external field is applied along the main crystallographic directions of [110], [100], or ½1¯10/C138, i.e., φ H¼0, 45°, and 90°, respectively. Clearly, the loop shape varies considerably with the substrate temperature and field orientation. For the sample deposited at 200°C, from the weak difference in remanent magnetization, we are able to recognize that the [110] and ½1¯10/C138directions are slightly inequivalent. The sample has a weak in-plane uniaxial anisotropy, with the easy axis along the [110] direction and a hard axis along the ½1¯10/C138direction. As TSincreases, the remanences of the [110] and ½1¯10/C138directions both increase; they eventually go to the same value of 1 at TS≥350°C. According to the structural analyses, it infers that a biaxial magnetic anisotropy develops with the improved CFScrystal structure at higher T S, giving rise to the two equivalent easy axes and the transformation of the hard- axis direction from ½1¯10/C138to [100].The saturation magnetization MSand magnetic coerciv- ityHCare derived from the hysteresis loops and summa- rized in Figs. 4(b) and4(c), respectively, as a function of TS. Interestingly, MSandHCdisplay a very similar varying trend. Both of them keep nearly unchanged at TS≤300°C; afterwards, they start to rise considerably. The maximum of MSis1120 emu=cm3, which is slightly lower than the ideal bulk value due to the incompletely chemical ordering of B2 orL21structure [47]. The increasing trend of MSis in good agreement with other studies and is believed to result fromthe improved chemical ordering between Co and Fe sites [37,48] . Note that, at T S<350°C, the HCvalues are rather small and nearly equal for the three different field direc- tions. However, with increasing the substrate temperature, theHCof the [110] and ½1¯10/C138directions exhibits a similar faster rising rate than that of the [100] direction. The observed increase in HCis mainly due to the enhanced biaxial magnetic anisotropy [49]. In addition, the improved crystalline quality may also play some role. As we know from the XRD results, with increasing TSthe crystal structure is improved due to the reduction of lattice defects or imperfections. In this case, the antimagnetized nucleiwould not be easy to form, which will lead to the increase ofH Cas well. Note that in Ref. [20]the authors reported a decreasing behavior of HCwith annealing temperature for their Co 2FeAl film; we consider the opposite variation tendency arises largely from the large film thickness of 55 nm. In order to clearly understand the evolution of magnetic anisotropy and clarify the effect of substrate heating, we measure the angular remanence magnetization curves. The remanence ratio Mr=MSis shown in Fig. 5as a function of angle φHwith respect to the [110] direction. Clearly, the ARM curves of TS¼200 and 300°C show uniaxial FIG. 4. Magnetic properties obtained by VSM measurements. (a) Magnetic hysteresis loops of CFS films grown at different TS, with magnetic field Happlied parallel to the main crystallo- graphic axes of [110], ½1¯10/C138, and [100]. (b) The saturation magnetization MSand (c) magnetic coercivity HCas a function ofTSfor the three kinds of field orientation.CORRELATIONS BETWEEN STRUCTURAL AND … PHYS. REV. APPLIED 8,034012 (2017) 034012-5magnetic anisotropy with twofold symmetry. The maxi- mum remanence ratio at φH¼0and 180° denotes that the uniaxial easy axis is parallel to the [110] direction, and the minimum ratio at φH¼90° implies that the hard-axis direction is along the ½1¯10/C138direction. With the increase ofTS, the curves gradually evolve from a twofold to fourfold rotational symmetry, implying the emergence of a biaxial anisotropy. In addition, the minimum remanenceratio (hard axis) moves towards smaller φ H, i.e., the direction of [100]. The additional remanence peak occurs not at φH¼90° (i.e., the ½1¯10/C138direction) for TS¼350°C, which suggests the easy axes of biaxial anisotropy are not yet strictly along the [110] and ½1¯10/C138directions. This is understandable, since the structural order of the CFS crystal is rather weak at such a low substrate temperature. WhenT Sis over 400°C, the shape of the ARM curves is governed by the enhanced biaxial anisotropy, showing excellent fourfold symmetry with two equivalent easy axes along the [110] and ½1¯10/C138directions. Meanwhile, the minimum remanence occurs at φH¼45°, indicating that the hard magnetic axis shifts to the [100] direction. The quantitative values of uniaxial and biaxial magnetic anisotropy fields, HUandHB, respectively, can be numeri- cally obtained by fitting the reversible part (from saturation field to zero) of the hard-axis magnetization loops, wherethe magnetization reversal can be treated as coherent rotation. Figure 6(a)shows the corresponding experimental data points and fitted reversible curves according toEq.(4.4). The data points are taken from the magnetic loops of the ½1¯10/C138direction for T S≤300°C and of the [100] direction for TS>300°C. Apparently, all the exper- imental data are well fitted by the solid lines, indicating that the hard-axis magnetization curve is basically a coherentrotation process. Figure 6(b)shows the HUandHBvalues obtained from the numerical fitting. By using the fittedanisotropy fields and Eq. (4.5), the corresponding ARM curves can be calculated (the solid red lines in Fig. 5). It is found that, except for the case of 350°C, the calculated ARM curves agree satisfactorily with the experimentaldata, revealing that the fitted H UandHBvalues are appropriate. As for the poor simulation of 350°C, it should come from the fact that the used HUandHBvalues are obtained by fitting the magnetization curve of the [100] direction. However, from Fig. 5, we can see that the actual hard axis apparently deviates away from this orientation,which inevitably gives rise to improper anisotropic fields and hence the observed inconsistent simulation curve. The fitted uniaxial magnetic anisotropy field H Ushown in Fig. 6(b)is rather small, showing a weak nonmonotonic dependence on the substrate temperature. The varyingbehavior is consistent with those of Fe 31Co69grown on (001) Ga [50]and Co 2FeAl on (001) MgO [21].U pt on o w , no clear explanation has been given for the uniaxial anisotropy. It is reported that uniaxial magnetic anisotropyin some soft magnetic thin films may arise, because the process of atoms reaching the substrate is not isotropic during deposition at an oblique angle [51–53]. However, for our CFS films, the uniaxial anisotropy happens not due to this cause. For our sputtering machine, the substrate holder is rotated at a speed of 20 rpm during film depositionto ensure film uniformity, and no external magnetic field is applied in plane. As a result, we consider that the presence FIG. 5. The ARM curves for CFS films grown at different temperatures. The black points represent the experimental data,while the red lines are from the numerical calculation. FIG. 6. (a) The experimental data points and fitted lines for themagnetic curves measured along the corresponding hard-axisorientation. (b) The in-plane uniaxial magnetic anisotropy fieldH Uand biaxial magnetic anisotropy field HBversus TS, calculated according to the Stoner-Wohlfarth model.ZHU, WU, ZHAO, ZHU, YANG, ZHANG, and JIN PHYS. REV. APPLIED 8,034012 (2017) 034012-6of such weak uniaxial anisotropy in this study is most likely related to the symmetry breaking at the MgO =CFS inter- face due to the surface reconstruction or interface bondingeffect [33,54] . Anisotropic relaxation of strains induced by the lattice mismatch may also play a role [32]. The different magnetization-switching behaviors and the transition from twofold to fourfold symmetry in ARMcurves are a result of the competition effect between theuniaxial and biaxial magnetic anisotropies. In contrast toH U, the biaxial magnetocrystalline anisotropy field HB, which arises from the bulk crystal structure, displays a strong dependence on the substrate temperature. It isnegligible at low T Sbut increases rapidly and reaches nearly 60 Oe at TS¼480°C. The increasing tendency of ourHBdata is in accord with the result reported by Gabor et al. for MgO ð001Þ=Co2FeAl samples [37], which is ascribed to the improved crystal structure and enhanced in-plane biaxial strain. Interestingly, the TSdependence of HBis very similar to HCshown in Fig. 4, from which we can get the information that the enhanced biaxialmagnetocrystalline anisotropy governs the change of H C. Accompanying the enhanced biaxial magnetic anisotropy, the ordering parameter SL21increases as well when the substrate temperature is increasing, which suggests that theincreasing amount of the L2 1phase would not give rise to a reduction of HB. As a result, we think that the observed decreasing trend of biaxial anisotropy with TSby others [39] may not be associated with the ordering degree change of L21phase. It is more likely caused by the interfacial diffusion of Cr atoms into the magnetic layer, since theirfull-Heusler films are ex situ annealed simultaneously with the adjacent nonmagnetic Cr buffer layer. One possibleproof is that they observe a continuous reduction in M Swhen increasing the annealing temperature, which is contrary to the increasing behavior of MSof our CFS samples. B. Dynamic magnetic measurements by TRMOKE Ultrafast magnetization dynamics is measured by TRMOKE to verify the influence of TSon the magnetic properties of CFS films. During the dynamic measurement,a magnetic field His applied with a polar angle of θ H¼71° and an azimuth angle φHcorresponding to the magnetic hard-axis direction mentioned above, i.e., φH¼90° for TS≤300°C and 45º for TS>300°C. The typical TRMOKE data points are displayed in Fig. 7(a) for the case of H¼2kOe. Similar to many other magnetic thin films, the dynamic Kerr signal θkdisplays an ultrafast demagneti- zation and a subsequent magnetization precession process. The magnetization precession and damping part can be fittedby using the following sinusoidal equation [22,55] : θ k¼aþbexpðt=t0Þþcsinð2πftþφÞexpð−t=τÞ; ð4:6Þ where the first term is the background signal, the second term corresponds to the slow magnetization recoveryprocess, and the third term describes the uniform magneti- zation precession dynamics. The parameters of c,f,φ, and τ represent the precession amplitude, frequency, initial phase,and decay time of magnetization, respectively. The fittedfrequency is plotted in Fig. 7(b) as a function of external magnetic field Hfor samples grown at different T S, which is seen to increase almost linearly with the magnetic field H. It is known that the varying trend of such a field-dependent precession frequency is determined by the coaction of an external field and effective magnetic anisotropy fields.The field-dependent frequency curves in Fig. 7(b)are fitted by the following Kittel formula of the uniform precession mode [22]: f¼γ 2πf½HcosðφH−φMÞþH1/C138 ×½HcosðφH−φMÞþH2/C138g1=2: ð4:7Þ where γ¼γeg=2is the gyromagnetic ratio of the electron with γe¼1.76×107Hz=Oe and g¼2.06.H1andH2are, respectively, defined as H1¼4πMS−2K⊥ MS−KB MSsin2ð2φMÞ−2KU MSsin2φM; H2¼2KB MScosð4φMÞþ2KU MScosð2φMÞ: ð4:8Þ Note that the experimental data are well fitted by the lines, implying that the magnetization precession behaves in a uniform precession mode. Figure 7(c)shows the in-plane uniaxial and biaxial anisotropy fields obtained from the FIG. 7. (a) The laser-induced magnetization dynamic Kerr signals (symbols) measured at H¼2kOe and the corresponding fitting curves (solid lines) for CFS films grown at different TS. (b) The relationship between precession frequency fand the applied field Hat various TS. (c) The TSdependences of HUand HBvalues obtained by fitting the curves shown in (b).CORRELATIONS BETWEEN STRUCTURAL AND … PHYS. REV. APPLIED 8,034012 (2017) 034012-7curve fitting in Fig. 7(b). Apparently, the values of HUand HB, as well as their varying trends, are very consistent with the results obtained by VSM measurements [Fig. 6(b)], further verifying the existence of a nearly invariable in-planeuniaxial anisotropy and a biaxial magnetic anisotropy whichgrows rapidly with the increase of T S. From the above structural and magnetic results, the evolutions of sample structure and magnetic anisotropies upon thermal heating, as well as their correlations, can be identified. As TSis lower than 300°C, the CFS films show a disordered crystal structure, where the biaxial anisotropyis negligible and the uniaxial magnetic anisotropy domi-nates. As T Sis increased to 350°C, accompanying with the onset of the A2phase, the biaxial anisotropy occurs. With a further increase of TS, both the biaxial anisotropy and saturation magnetization increase significantly, which are,respectively, due to the improved crystal structure andchemical order. The increased biaxial anisotropy plays adominant role in the magnetization-switching behaviors forCFS samples deposited at T S>350°C. V. CONCLUSIONS In this paper, we present correlated analyses on the structure, magnetic coercivity, saturation magnetization,and in-plane magnetic anisotropies for CFS films grownat different T S. XRD reveals that, with the increase of TS, the CFS layer develops first from disordered crystal structure todisordered A2structure and then to chemically ordered B2 andL2 1structures. Accompanying the increased chemical order and saturation magnetization, a gradual transition fromtwofold to fourfold symmetry occurs in the ARM curves,which is ascribed to the competition effect between the in-plane uniaxial and biaxial magnetic anisotropies. The weak uniaxial anisotropy changes slightly with T Sincreasing. In contrast, the biaxial cubic anisotropy that arises from theimproved crystal structure and increased biaxial stress uponthermal heating is negligibly small at T S≤300°C but increases considerably at TS>300°C. Most importantly, we find that the increased L21ordering degree would not reduce the biaxial cubic anisotropy. Our results facilitate thepractical design of spintronic devices based on full-HeuslerCFS thin films with controllable magnetic anisotropy. 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PhysRevApplied.12.044074.pdf	PHYSICAL REVIEW APPLIED 12,044074 (2019) Strain-Enhanced Charge-to-Spin Conversion in Ta /Fe/Pt Multilayers Grown on Flexible Mica Substrate Er Liu ,1,2,T. Fache,2D. Cespedes-Berrocal,2,3Zhi Zhang,1S. Petit-Watelot,2Stéphane Mangin,2 Feng Xu,1,†and J.-C. Rojas-Sánchez2,‡ 1MIIT Key Laboratory of Advanced Metallic and Intermetallic Materials Technology, School of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China 2Université de Lorraine, CNRS UMR 7198, Institut Jean Lamour, F-54011 Nancy, France 3Universidad Nacional de Ingeniería, Rímac 15333, Peru (Received 1 August 2019; revised manuscript received 17 September 2019; published 31 October 2019) Recent demonstration of magnetization manipulation has been focused on the utilization of pure spin current converted by charge current in nonmagnetic materials with strong spin-orbit coupling (SOC), which stimulates intensive studies on the exploration of materials with larger SOC, such as heavy metalsand topological insulators. We suggest an alternative approach for enhancing the eﬀective charge-to- spin conversion eﬃciency by applying strain in Ta /Fe/Pt ﬁlms grown on ﬂexible mica substrates. We experimentally demonstrate a large and tunable charge-to-spin conversion eﬃciency in Ta /Fe/Pt ﬁlms by applying compressive strain within a ﬂexible substrate, and over 50% enhancement of eﬀective spin Hall angle /Theta1 eﬀ SHEof up to approximately 0.2 is achieved in a 6.26‰ strained ﬁlm. Our ﬁndings may spur further work on the integration of ﬂexible electronics and SOC and may potentially lead to the innovation of alternative ﬂexible spintronics devices. DOI: 10.1103/PhysRevApplied.12.044074 I. INTRODUCTION The integration of an electronic system with a ﬂexible substrate has received considerable attention due to the appealing applications in alternative devices such as ﬂex- ible display, electronic skin, and wearable devices [ 1–5]. The fantastic property of such ﬂexible electronics also trig- gers numerous research eﬀorts in the spintronics ﬁeld on the ﬂexible strain eﬀects of spin-dependent phenomena such as exchange bias [ 6,7], interlayer coupling [ 8], and perpendicular magnetic anisotropy (PMA) [ 9]. In recent years, the focus of modern spintronics has been shifted to the generation and manipulation of pure spin currents through spin-orbit coupling (SOC) in nonmagnetic sys- tems [ 10–12], the studies of which even give birth to an emerging ﬁeld, that is, spin orbitronics [ 13]. However, the integration of pure spin currents in ﬂexible electronics remains largely unexplored. As a ﬂow of angular momentum that does not accom- pany a charge ﬂow, pure spin currents carry infor- mation with minimum power dissipation [ 14]. More importantly, similar to spin-polarized current, pure spin ericliu@njust.edu.cn †xufeng@njust.edu.cn ‡juan-carlos.rojas-sanchez@univ-lorraine.frcurrents are capable of manipulating the magnetization of ferromagnetic elements, showing promising applica- tion perspectives in alternative information recording and data processing devices such as spin orbital torque magnetic random access memory (SOT MRAM) [ 15] and spin Hall nano-oscillators [ 16]. In a ferromagnetic- nonmagnetic (FM NM) bilayer system, pure spin cur- rents can be converted by charge current ﬂowing through NM with strong spin orbit coupling (spin Hall eﬀect), then propagate into the adjacent FM, thereby aﬀecting the magnetic ﬂuctuation, magnetization dynamics, and even magnetization switching [ 17–19]. In consideration of the FM-NM interface spin loss during the injection of spin currents, the eﬀective charge-to-spin conversion is quantiﬁed by the eﬀective spin Hall angle /Theta1eﬀ SHE[20, 21]. As such, optimizing the eﬀective spin Hall angle /Theta1eﬀ SHEfor speciﬁc applications is of immense importance and has been achieved by various strategies, including exploring NM with large spin orbital coupling such as heavy metals [ 18,20,22,23], topological insulators [ 19,24] and more recently, the antiferromagnetic material [ 25, 26], engineering the FM-NM interface for greater spin transparency [ 27], or reducing interfacial spin memory loss [ 20]. Here, we report the observation of a large and tunable /Theta1eﬀ SHEin mica //Ta/Fe/Pt by applying mechanical strain in a ﬂexible mica substrate during ﬁlm deposition. 2331-7019/19/12(4)/044074(7) 044074-1 © 2019 American Physical SocietyE. LIU et al. PHYS. REV. APPLIED 12,044074 (2019) II. EXPERIMENTAL PROCEDURES A. Film deposition A series of Ta(4 nm)/Fe (4 nm)/Pt(5 nm) multilayers are deposited on ﬂexible mica substrates, with the 4-nm Ta employed as a buﬀer layer to reduce the roughness of the ﬂexible substrates. Before the deposition, the substratesare cleaned with ethyl alcohol in an ultrasonic cleaner for 10 min and then dried with nitrogen gas. During the deposition, the substrates are bent and ﬁxed on a home- made convex aluminum alloy mold, as shown in Fig. 1(a). Thus, a longitudinal compressive strain is induced in the ﬁlms when ﬂattening the substrate after the deposition and the strain direction is shown in Fig. 1(b). By chang- ing the curvature radii of the mold, Ta /Fe/Pt multilayers with compressive strain εof 0 to approximately 6.26‰ are obtained. The magnitude of such induced strains is estimated by ε=T/2R, where Ris the curvature radii of the mold and Tis the total thickness including both the substrate and multilayers [ 28]. B. Device fabrication The microstrips for measurements are patterned with the long axis parallel to the strain directions by standard UV lithography and have lateral sizes of 20 ´ 90 μm 2.T i/Au electrodes are deposited on the edge of the microstrip by the lift-oﬀ technique to form a ground-signal-ground (GSG) contact that guides the radio frequency (rf) current into the sample. We also prepare a control sample with the long axis perpendicular to the strain direction for the multilayer with a compressive strain of 6.26‰.III. RESULTS AND DISCUSSION A. ST FMR measurements Spin-torque ferromagnetic resonance (ST FMR), [also referred to as spin-orbit ferromagnetic resonance, that is, SO FMR)] [ 22,29] measurements are performed to evalu- ate the eﬃciency of charge-to-spin conversion for all themicrostrip devices, and the schematic of the measurement setup is illustrated in Fig. 1(c). A rf charge current J c with a power of 10 dBm is injected into the microstrip in the presence of an external ﬁeld Hthat is applied in plane at 45° with respect to the microstrip, as shown in Fig.1(d). Note that for all the measurements, the rf current is ﬂowing along the long axis of the microstrip devices with a frequency ranging from 5–20 GHz. A transverse spin current is then converted by the rf current in Pt due to the spin Hall eﬀect, which manipulates the magnetiza- tion procession of Fe by exerting two orthogonal torques, namely, a dampinglike torque ( τD) and a ﬁeldlike torque (τF). Owing to the anisotropic magnetoresistance (AMR) eﬀect, the oscillating magnetization gives rise to a time- dependent resistivity, which mixes with the rf current and results in a rectiﬁed dc voltage. Figure 2(a) shows the dc voltage signal for a Ta /Fe/Pt device with compressive strain of 6.26‰, measured by spin-torque ferromagnetic resonance under a conﬁguration ofJc\|\|ε. It is notable that the resonance ﬁeld shifts toward higher ﬁelds as the rf current frequency increases, which is consistent with the Kittel model [ 30]. Here, the mea- sured dc voltages are a mixture of a symmetric Lorentzian component Vsymdue to the dampinglike torque and an antisymmetric one Vantiarising from the ﬁeldlike torque, (a) (b) (c) (d) Bias tee FIG. 1. (a),(b) Schematic illustration of the strain application process. The ﬁx of mica on a convex aluminum alloy mold during the ﬁlm deposition and the ﬂattening afterward induced a compressive strain εin the ﬁlms with direction shown in (b). (c) Schematic representation of the ST FMR experimental setup. (d) Sketch of the ST FMR experimental conﬁguration showing the spin-transfertorques τ FandτD, magnetization M, and external ﬁeld H. Here, the charge current Jcis parallel to the strain εdirection. 044074-2STRAIN-ENHANCED CHARGE-TO-SPIN CONVERSION . . . PHYS. REV. APPLIED 12,044074 (2019) (a) (b) (c) (d) FIG. 2. ST FMR data for the 6.26‰ strained ﬁlm. (a) The measured ST FMR signals at diﬀerent frequencies, the red lines are the ﬁttings according to Eq. (1). (b) Single ST FMR spectrum measured at 11 GHz. The red solid line represents the ﬁt to the Lorentzian function. The green and purple dash-dotted lines represent symmetric and antisymmetric voltage components, respectively. (c) The ﬁtting of frequency dependence of the resonance ﬁeld from which we can obtain the value of Meﬀ. (d) Eﬀective damping determination from the ﬁtting of frequency dependence of the line width. in a ﬁrst approximation for the level of thicknesses in our samples [ 22,31,32]. We extract the symmetric and antisymmetric voltage contributions by ﬁtting the spin- torque ferromagnetic resonance spectra with the following equation, also considering an oﬀ-set Voﬀ[32] Vdc=Voﬀ+Vsym/Delta1H2 /Delta1H2+(H−Hres)2 +Vanti(H−Hres)/Delta1H /Delta1H2+(H−Hres)2,( 1 ) where H,Hres,a n d /Delta1Hare the applied ﬁeld, the reso- nance ﬁeld, and the resonance line width, respectively. Figure 2(a) shows the ﬁtting result and Fig. 2(b) shows the ﬁtting detail of a ST FMR spectrum measured at a fre-quency of 11 GHz. Both agree well with the experimental data. We extract the resonance ﬁelds and plot them against the resonance frequency f resin Fig. 2(c), in which the reso- nance ﬁeld dependence of the frequency can be well ﬁttedby the Kittel formula fres=γ 2π[(Hres+Huni)(Hres+4πMeﬀ+Huni)]1/2,( 2 ) where γand Huniare the gyromagnetic ratio and the in-plane uniaxial magnetic anisotropy ﬁeld, respectively. 4πMeﬀ=4πMs−Hkis the eﬀective magnetization, Msis the saturation magnetization, and Hkis the perpendicular anisotropy ﬁeld. The value of Meﬀfor all the samples is obtained from the ﬁtting and the strain dependence of Meﬀ and is shown in Fig. 2(c). It is notable that Meﬀslightly increases with the increasing of strain regardless of the strain direction, which probably stems from the increase ofHkdue to an out-of-plane stretch of the lattice by the in-plane compressive strain. B. Strain dependence of magnetic damping The ST FMR measurements also allow us to eval- uate the eﬀective magnetic damping αby ﬁtting the 044074-3E. LIU et al. PHYS. REV. APPLIED 12,044074 (2019) Effective damping a(a) (b)FIG. 3. Strain dependence of Meﬀ(a) and eﬀective damping (b). The dash- dotted lines represent a guide to the eyes.The red circle data point corresponds to the control device where J cis perpendic- ular to ε. We can observe in (a) that Meﬀ is the same in both devices for the higher εwe have measured. This is an indica- tion that the Fe layer is isotropic in theﬁlm plane. We can observe in (b) that the total damping when J cis perpendic- ular to εis lower than the one with Jc\|\|ε. This is an indication of a more eﬃcient spin pumping contribution when Jc\|\|ε. frequency dependence of the resonance line width with equation /Delta1H=/Delta1H0+2πfα/γ, where /Delta1H0is the fre- quency independent contribution due to the inhomogeneity of the ﬁlms. A ﬁtting example is shown in Fig. 2(d) and a linear dependence can be observed. Figure 3(b) shows the strain dependence of α. It is interesting to ﬁnd that rather than the monotonous behavior of Meﬀ,αdepends on the strain direction and it decreases ﬁrst then increases with the increasing of strain for a Jc\|\|εconﬁguration. We will explain this anomalous behavior in the discussion of /Theta1eﬀ SHE below. C. Strain-enhanced charge-to-spin conversion The spin Hall angle /Theta1eﬀ SHEis the ratio of spin current density Jsto the rf current density Jc,, which in the sim- plest model is proportional to the ratio of the symmetric voltage component over the antisymmetric one Vsymm/Vanti. When the resonance ﬁeld is larger than the saturation ﬁeld (Hres>Hsat), the precession of magnetization is consid- ered to be uniform, as such /Theta1eﬀ SHEcan be obtained by the following equation [ 11,22,31,32] /Theta1eﬀ SHE=Js Jc∼=Vsym Vantieμ0MstFetPt /planckover2pi1/bracketleftbigg 1+4πMeﬀ Hres/bracketrightbigg1/2 ×1 1+(HF/HOe),( 3 ) where μ0is the permeability in vacuum and tFeand tPtare the thicknesses of the Fe and Pt layers, respectively. The last factor in Eq. (3)takes into account a possible con- tribution HFdue to ﬁeldlike spin orbit torque and HOe is the Oersted ﬁeld arising from the rf current. Here, weignore the contribution of the Oersted ﬁeld from the Ta layer due to the small current density in Ta, considering the much larger resistivity in Ta (200 µW cm) than that in Pt (24 µW cm). We have estimated the value of the Oer- sted ﬁeld ( HOe∼1.5 Oe) and the ﬁeld due to the ﬁeldlike spin orbit torque ( HF∼0.15 Oe) in our samples accord- ing to the method given by Refs. [ 31]a n d[ 33]. Thus, we can neglect the factor HF/HOein Eq. (3). Figure 4(a) shows the eﬀective spin Hall angles /Theta1eﬀ SHEf o rP ti nt h e Ta/Fe/Pt system without and with a strain of 6.26‰ at varying Jcfrequencies from 6 to 20 GHz, from which a signiﬁcant increase of /Theta1eﬀ SHEby strain is noted. To fur- ther conﬁrm the strain eﬀect on /Theta1eﬀ SHE, we also show the /Theta1eﬀ SHEcomparison derived from the ST FMR measurements with Jcﬂowing along and perpendicular to the strain direc- tion in Fig. 4(b). It is worth noting that in both cases, Figs. 4(a) and4(b), the value of /Theta1eﬀ SHE is increased when Jc\|\|εand when the magnitude of εincreases. The strain dependence of /Theta1eﬀ SHEis shown in Fig. 4(c), indicating the increase of /Theta1eﬀ SHEfrom 0.12 ±0.02 to 0.20 ±0.02 by the increase of strain when Jc\|\|ε. On the contrary, /Theta1eﬀ SHEis almost constant ( /Theta1eﬀ SHE=0.12±0.02 and 0.11 ±0.02 for Jc⊥εandε=0, respectively) when Jcis not ﬂowing along strain direction. It should be mentioned that the mea- sured/Theta1eﬀ SHEis larger compared to that reported previously (around 0.05) [ 20,22,32], and we ascribe this to the pres- ence of the Ta buﬀer layer, which has a negative spin Hall angle [ 17,18] thereby also contributing to the spin current density Js. Even if we do not consider the thickness of Ta, the calculated “eﬀective” value involves the contribu- tion of Ta. Thus, it explains the large value, approximately 0.12, we obtain in our devices without any strain. Similar enhancements of eﬀective values in trilayers have also 044074-4STRAIN-ENHANCED CHARGE-TO-SPIN CONVERSION . . . PHYS. REV. APPLIED 12,044074 (2019) (a) (c) (b)FIG. 4. (a),(b) Determination of the eﬀective spin Hall angle according to Eq. (3), which is reliable only when Hres>Hsat[32]. (c) Strain dependence of the eﬀectivespin Hall angle. been observed in W /(Co, Fe )B/Pt [31]a n dP t /Co/Ta [34] systems. The strain dependence of /Theta1eﬀ SHEcan also be veriﬁed by the anomalous behavior of eﬀective damping αas we mentioned in Fig. 3(b). Here, we consider two damp- ing mechanisms contributing to the eﬀective damping α: one is the intrinsic damping of magnetic layer, which decreases with increasing compressive strain, as reported previously [ 6]. The other is an additional damping due to spin pumping into the heavy metal, Pt and Ta, layers. Note, however, that the enhancement of damping due to the Ta layer is much smaller than that of Pt [ 18]. Thus, the clear increase of /Theta1eﬀ SHEsuggests more eﬃcient spin pumping into the heavy metal (HM) layers, thus a larger αfor the Fe layer, which agrees with the damping measurements for the two devices with a compressive strain of 6.26‰. Despite the diﬀerent rf current orientation, the Fe layers in both devices undergo the same magnitude of strain (6.26‰) and magnetic ﬁeld conﬁguration ( His applied 45° in plane to the strain direction), thus the intrinsic damping of Fe is identical. Consequently, the eﬀective damping is mainly determined by the additional damping due to spin pump- ing. In contrast, for the devices with diﬀerent strain, α decreases ﬁrst then increases with the increase of strain. This can be explained by the competition of strain eﬀect on the intrinsic damping of Fe and the enhancement of spin pumping due to the larger /Theta1eﬀ SHE. Finally, we discuss the possible mechanism accounting for the observed strain-dependent /Theta1eﬀ SHE. It is well known that a magnetoelastic anisotropy will be induced due to strain in the magnetic layer, and in our case, the easy axis of the magnetoelastic anisotropy is in-plane perpendicularto the compressive strain direction [ 6,28]. However, the observed strain-dependent /Theta1eﬀ SHEis unlikely due to the strain-induced anisotropy, since diﬀerent /Theta1eﬀ SHEare still obtained in the 6.26‰ strained devices in which the same orientation (45°) between the magnetic ﬁeld and strain direction is given. We note that the variation of 4 πMeﬀ/Hres [factor in Eq. (3)] due to strain is less than 1%, thus it can- not account for the 50% variation of the eﬀective spin Hall angle. Moreover, magnetic anisotropy independent /Theta1eﬀ SHE is also observed in epitaxial Fe /Pt [32] and an exchange- biased NiFe /IrMn system [ 35]. The increase of SOC in Pt(Ta)is another reason to be ruled out since SOC is dominated by the atomic number of Pt (Ta), which cannot be changed by the strain. A possibility for the strain- dependent /Theta1eﬀ SHEis the decrease of spin-current dissipation at the interface due to the ﬂexible substrate-induced strain, which may improve the interfacial disorder. More detailed theoretical and experimental investigations are needed to explore the exact microscopic mechanism of the enhanced /Theta1eﬀ SHEgiven the complex nature of the system. IV . CONCLUSION With the ST FMR technique, we show a large and tun- able charge-to-spin conversion eﬃciency in compressive strained Ta /Fe/Pt ﬁlms grown on ﬂexible mica substrates, and the eﬀective spin Hall angle /Theta1eﬀ SHEis signiﬁcantly enhanced over 50% from 0.12 ±0.02 to 0.20 ±0.02 by the ﬂexible substrate-induced strain of 6.26‰. The demon- strated strong strain dependence of charge-to-spin conver- sion eﬃciency paves an alternative way for the manipu- lation of spin current, and we anticipate the results can be 044074-5E. LIU et al. PHYS. REV. APPLIED 12,044074 (2019) used for the application of alternative spin memory or logic devices as well as the development of ﬂexible electronics. ACKNOWLEDGMENTS This work was sponsored by the National Natural Science Foundation of China (Grants No. 51601093, No. 51571121, No. 11604148, and No. 61427812), Fun- damental Research Funds for the Central Universities (Grant No. 30916011345), the Natural Science Founda- tion of Jiangsu Province (Grants No. BK20160833 and No. BK20160831), the China Postdoctoral Science Foun- dation Funded Project (Grants No. 2015M571285, No. 2016M601811, and No. 2016M591851), the Postdoctoral Science Foundation Funded Project of Jiangsu Province (Grant No. 1601268C), the Key Research & Development Program of Jiangsu Province (Grant No. BE2017102), and Special fund for the transformation of scientiﬁc and tech- nological achievements in Jiangsu Province (Grant No. BA2017121). The authors also acknowledge and thank LUE Graduated program internship 2019 from Lorraine Université d’Excellence, the Feder and Region Grand Est for its support in Projet Chercheur d’Avenir, Plate- forme Nano-terraHertz and Plus. 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PhysRevA.86.023622.pdf	PHYSICAL REVIEW A 86, 023622 (2012) Excitation transport through a domain wall in a Bose-Einstein condensate Shohei Watabe,1,2,Yusuke Kato,3and Yoji Ohashi1,2 1Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan 2CREST(JST), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan 3Department of Basic Science, The University of Tokyo, Tokyo 153-8902, Japan (Received 12 May 2012; published 20 August 2012) We investigate the tunneling properties of collective excitations through a domain wall in the ferromagnetic phase of a spin-1 spinor Bose-Einstein condensate. Within the mean-ﬁeld theory at T=0, we show that the transverse spin wave undergoes perfect reﬂection in the low-energy limit. This reﬂection property differsconsiderably from that of a domain wall in a Heisenberg ferromagnet where spin-wave excitations exhibitperfect transmission at arbitrary energy. When the Bogoliubov mode is scattered from this domain wall soliton,the transmission and reﬂection coefﬁcients exhibit pronounced nonmonotonicity. In particular, we ﬁnd perfectreﬂection of the Bogoliubov mode at energies where bound states appear. This is in stark contrast to the perfecttransmission of the Bogoliubov mode with arbitrary energy through a dark soliton in a scalar Bose-Einsteincondensate. DOI: 10.1103/PhysRevA.86.023622 PACS number(s): 67 .85.−d, 03.75.Lm, 03 .75.Mn, 05 .60.Gg I. INTRODUCTION Bloch and N ´eel walls in ferromagnetic materials are domain walls with well-known structures in condensed matter, wherespin conﬁgurations are twisted in a magnetic domain regionwith an angular displacement of 180 ◦[1]. It is desirable to use domain wall motions in ferromagnets in the aim of theirapplication to memory devices [ 2–13]. For example, current- driven domain wall motion associated with spin transfer andmomentum transfer from electrons to the domain wall hasbeen theoretically discussed [ 7] and has been experimentally observed [ 8]. Domain walls are also found in two-component Bose- Einstein condensates and spin-1 Bose-Einstein condensates.For spin-1 Bose-Einstein condensates, spin domain structures[14,15], tunneling from metastable spin domains [ 16], and the formation of spin textures and domains [ 17,18] have been experimentally investigated. An earlier theoretical paper [ 19] reported that ferromagnetic domain walls in a spin-1 Bose-Einstein condensate spread out over the entire spatial regionof the system. However, a domain wall may exhibit solitarybehavior with a size comparable with the healing length ofthe Bose-Einstein condensate, which is much smaller than thesystem size. In addition, a uniform particle density around thedomain wall was reported in Ref. [ 19], which is analogous to that of the Heisenberg model, where the magnitude of the spinvector is spatially uniform. However, gaseous ferromagneticspin-1 Bose-Einstein condensates may be free from such aconstraint, so that their particle density, as well as spin density,may become nonuniform in the domain wall region. A searchfor such a domain is anticipated to reveal a solitary domainwall in spin-1 Bose-Einstein condensates. Furthermore, transport phenomena in gaseous ferromag- netic Bose-Einstein condensates are much less understoodthan those in solid materials. Scattering between low-lying Present address: Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan.modes and a domain wall in a Bose-Einstein condensate will be interesting, if it exhibits different characters from conventional ferromagnets. In contrast to the Heisenberg model case, thespin density vector magnitude may be spatially nonuniform inspinor Bose-Einstein condensates. This character is expectedto provide interesting insights into ferromagnets and spinorBose-Einstein condensates. In addition, since these scatteringproblems can be expressed in terms of Nambu-Goldstonemodes and topological excitations, this concept is not speciﬁcto cold atomic gases, but can also be applicable to othersymmetry-broken states. We investigate the tunneling properties of collective exci- tations through a ferromagnetic domain wall in a spin-1 Bose-Einstein condensate. We start by reconsidering the domainstructure of the ferromagnetic Bose-Einstein condensate witha given boundary condition. For simplicity, we consider athree-dimensional system with a planar domain wall whichspatially varies in the xdirection. We study one-dimensional tunneling at T=0, as shown in Fig. 1, where A and B correspond to different hyperﬁne states S z(=±1) of the spin-1 Bose-Einstein condensate. The main results of this paper are summarized as follows. We ﬁnd that the domain wall structure of the spin-1 Bose-Einstein condensate differs considerably from that in theferromagnetic Heisenberg model. Its transverse spin wavealso exhibits strikingly different tunneling properties. Thetransverse spin wave in ferromagnets transmits perfectlythrough a domain wall [ 20]. In contrast, the transverse spin wave of the present Bose system is perfectly reﬂected bythe ferromagnetic domain wall in the low-energy limit. Forthis perfect reﬂection, the wave function does not vanishin the low-energy limit. This contrasts with a conventionalsingle particle in quantum mechanics, which undergoes perfectreﬂection in the low-energy limit usually due to the absenceof the wave function. The quadrupolar spin wave in a spin-1 Bose-Einstein condensate exhibits conventional reﬂectionproperties in the long-wavelength limit for which perfectreﬂection occurs due to the absence of the wave function inthe long-wavelength limit. The coexistence of both perfect 023622-1 1050-2947/2012/86(2)/023622(9) ©2012 American Physical SocietySHOHEI WATABE, YUSUKE KATO, AND YOJI OHASHI PHYSICAL REVIEW A 86, 023622 (2012) FIG. 1. (Color online) One-dimensional tunneling of excitations through a domain wall of a Bose-Einstein condensate. For a ferromagnetic spin-1 Bose-Einstein condensate, A and B representdifferent hyperﬁne states of the condensate (A corresponds to the S z=−1 state and B to Sz=+1). A right-moving incident excitation is scattered from the domain wall. reﬂection and a nonvanishing wave function in the low-energy limit is due to the existence of the zero mode and that of adamping mode whose damping length becomes inﬁnite in thelow-energy limit. Spin-1 Bose-Einstein condensates have another Nambu- Goldstone mode: the Bogoliubov excitation. In the low-energylimit, the Bogoliubov mode does not undergo perfect reﬂectionwhen it is incident on the domain wall. This is similar to thesituation for a scalar Bose-Einstein condensate with a darksoliton. However, energy-dependent nonmonotonic tunnelingproperties are observed at a ﬁnite energy. In particular, theperfect reﬂection occurs when the bound state appears at thedomain. This differs from the dark soliton case in the scalarbosons, where the transmission coefﬁcient is independent ofenergy. This paper is organized as follows. In Sec. II, we compare the domain wall in a ferromagnetic spin-1 Bose-Einsteincondensate with that in the ferromagnetic Heisenberg model.In Sec. III, we examine the tunneling properties of excitations through the ferromagnetic domain wall. II. COMPARISON WITH DOMAIN WALL OF HEISENBERG MODEL AtT=0 in a three-dimensional system, the domain structure of a spin-1 Bose-Einstein condensate is describedby the Gross-Pitaevskii equation [ 21,22] for the condensate wave function ˆ/Phi1=(/Phi1 +1,/Phi10,/Phi1−1)T(where the subscripts ±1,0 represent hyperﬁne states in the S=1s p i ns t a t e ) , i¯h∂ˆ/Phi1(r,t) ∂t=⎛ ⎜⎝h+(r,t)c1√ 2F− 0 c1√ 2F+h(r,t)c1√ 2F− 0c1√ 2F+h−(r,t)⎞ ⎟⎠ˆ/Phi1(r,t).(1) Here, we have introduced h(r,t)≡−¯h2∇2/(2m)+c0ρ(r,t), h±(r,t)≡h(r,t)±c1Fz(r,t), and F±≡Fx±iFy, where m is the atomic mass, ρ(r,t)=ˆ/Phi1†(r,t)ˆ/Phi1(r,t) is the particle density, and F=ˆ/Phi1†(r,t)Sˆ/Phi1(r,t) is the spin density. The spin quantization axis of S=1 spin matrices S=(Sx,Sy,Sz)i s chosen to be parallel to the zaxis. The two coupling constants c0=4π¯h2(a0+2a2)/(3m) andc1=4π¯h2(a2−a0)/(3m)a r e for spin-independent and spin-dependent interactions, respec-tively [ 22], where a Sis the s-wave scattering length for the total spin S=0 or 2 channel. We consider a ferromagnetic spin-dependent interaction (c1<0) and a planar ferromagnetic domain wall with only5 0 50.00.20.40.60.81.01.2 5 0 50.00.20.40.60.81.01.2 5 0 51.00.50.00.51.0Φ±1/√ρ ρ(x)/ρ Fz/ρΦ+1 Φ−1 x/ξ x/ξ x/ξ(a) (b) (c) FIG. 2. Calculated (a) condensate wave functions, (b) particle density, and (c) spin density of Fzas a function of x./Phi10is absent everywhere. ξis the healing length and is given by ξ=¯h/√mc+ρ. anxdependence (i.e., Fz(x=± ∞ )=±ρ, where ρis the density at \|x\|=∞ ). The condensate wave function for the stationary state is determined from Eq. (1)by setting ˆ/Phi1(x,t)= e−iμt/ ¯hˆ/Phi1(x), where μis the chemical potential, under the boundary condition ˆ/Phi1(x=− ∞ )=(0,0,√ρ)Tand ˆ/Phi1(x= +∞)=(√ρ,0,0)T. Figures 2(a)–2(c), respectively, show the spatial proﬁle of the condensate wave function, the particledensity, and the spin density of F z, which were obtained by solving the Gross-Pitaevskii equation with the chemicalpotential μ=(c 0+c1)ρ. The results of Figs. 2(a)–2(c) are characterized by /Phi10=0. The domain wall shown in Fig. 2(a)is quite different from that in a ferromagnet described by the Heisenberg model. Inthe continuum limit, this spin model is described by H spin=/integraldisplayd3x a3/bracketleftbiggJ 2(∇σ)2−K 2σ2 z/bracketrightbigg , (2) where the easy axis is taken to be parallel to the zaxis. σ=(σx,σy,σz)Tis the spin vector, ais the lattice con- stant, and J(>0) and K(>0) are, respectively, the ex- change coupling and easy axis anisotropy constants. Adomain wall solution to Eq. (2)is given by ( σ x,σy,σz)= (0,σ/cosh(x/λ),σtanh(x/λ)), where λ≡√J/K [12,23]. The magnitude of σis homogeneous and the spin direction continuously changes around the magnetic wall. The presentferromagnetic domain wall for the spin-1 Bose-Einsteincondensate differs markedly from this domain wall in thefollowing two ways: (i) F x,y(∝\|/Phi10\|) is absent because /Phi10=0 everywhere, even near the domain wall, and (ii) the magnitudeof the spin density is not homogeneous and it vanishes at thecenter of the wall [see Fig. 2(c)]. A domain wall structure with F x,y/negationslash=0 has been also discussed in the system we are considering [ 19]. We ﬁnd that energy of a domain wall in Ref. [ 19] with a nonzero value of /Phi10is higher than that shown in Fig. 2. The different domain wall structures between the Heisen- berg model and this spin-1 Bose-Einstein condensate originatefrom the following causes. For the spin-1 Bose-Einsteincondensate, the spin-density interacts with itself [i.e., F 2(x)] and the spatial structure is governed by the kinetic term of thecondensate wave function. On the other hand, for the spatialstructure of the domain wall in the Heisenberg model, thenearest-neighbor spin exchange interaction [i.e., ( ∇σ) 2for the continuous approximation] in Eq. (2)is important. From the results in Fig. 2, the condensate wave function can be simply expressed by ˆ/Phi1(x)=(/Phi1+1(x),0,/Phi1−1(x))T,s o 023622-2EXCITATION TRANSPORT THROUGH A DOMAIN WALL IN ... PHYSICAL REVIEW A 86, 023622 (2012) that the Gross-Pitaevskii equation can be simpliﬁed as /bracketleftbigg −¯h2 2m∂2 x−μ+c+\|/Phi1±1\|2+c−\|/Phi1∓1\|2/bracketrightbigg /Phi1±1=0,(3) where c±=c0±c1. In this case, Eq. (3)can be regarded as the two-component Gross-Pitaevskii equation, if we regardc +andc−as the interaction parameters for the same and different species, respectively. In the ferromagnetic case, wehavec −>c+, which corresponds to the condition for phase separation of the two-component Bose-Einstein condensate[24]. Figure 2(a) clearly shows the phase-separated domains of the two hyperﬁne states. Even if we add the quadratic Zeeman term to the Gross- Pitaevskii equation to introduce the easy axis as the zaxis by reference to Eq. (2), the situation remains the same. For this quadratic Zeeman effect, the absence of /Phi1 0is energetically preferable [ 25,26]. That is, the ferromagnetic spin-1 Bose- Einstein condensate has a domain wall solution even whenno easy axis is introduced. In contrast, a ferromagnet in theHeisenberg model has a domain wall with ﬁnite thickness onlywhen it has an easy axis (i.e., K/negationslash=0) [1]. III. TUNNELING OF EXCITATIONS THROUGH DOMAIN WALL The tunneling properties of excitations can be de- termined by considering ˆ/Phi1(x,t)=exp (−iμt/ ¯h)[ˆ/Phi1(x)+ ˆφ(x,t)], where ˆφ=(φ+1,φ0,φ−1)Tdescribes ﬂuctuations of the condensate wave function about the mean-ﬁeld value, andretaining terms to O[ˆφ(x)] in the Gross–Pitaevskii equation (1). The ferromagnetic spin-1 Bose-Einstein condensate with S z=1 is known to have three kinds of collective modes [21,22]. Among them, the Bogoliubov mode is associated with phase ﬂuctuations of the order parameter, where spindegrees of freedom are not crucial. The remaining twospin-wave excitations are associated with spin ﬂuctuationsin the S z=0 and Sz=−1 channels. While the former is called the transverse spin wave, characterized by δF±= δFx±iδFy=√ 2(/Phi1∗ ±1φ0+/Phi1∓1φ∗ 0), the latter is called the quadrupolar spin mode, characterized by δQ+=2/Phi1∗ +1φ−1 andδQ−=2/Phi1+1φ∗ −1). We brieﬂy note that the quadrupolar spin density is given by Q±≡ˆ/Phi1†S2 ±ˆ/Phi1=2/Phi1∗ ±1/Phi1∓1. After applying the Bogoliubov transformation, φ±1,0= u±1,0e−iEt/ ¯h−v∗ ±1,0e+iEt/ ¯h, we can determine the reﬂection and transmission coefﬁcients ( randτ, respectively) by solving the Bogoliubov equation under the boundary condition for thetunneling problem (see Appendix A). Here, r(τ) is deﬁned as the magnitude of the ratio of the ﬂux density of the reﬂected(transmitted) wave to that of the incident wave. Here, the ﬂuxdensity Jis given by J=/summationtext j=±1,0Im[u∗ j∂xuj+v∗ j∂xvj]/m and is proportional to the time-averaged energy ﬂux [ 27]. An excitation with Sz=0, corresponding to the transverse spin wave, is decoupled from other spin states (i.e., φ±1), so that the transmission and reﬂection coefﬁcients are obtainedby solving the Bogoliubov equation for S z=0. Figure 3shows the transmission coefﬁcient τof the transverse spin wave as a function of momentum. The excitation cannot pass through thedomain wall in the low-momentum limit and the transmissioncoefﬁcient increases monotonically with increasing momen-0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0 kξ0.0 0.0τ c1=−0.1c+ c1=−c+ c1=−3c+ FIG. 3. Calculated transmission probability τthrough the domain wall as a function of the incident momentum when the transverse spin mode is incident on the domain wall. tum. (Perfect reﬂection in the low-momentum limit is proved in Appendix B.) This perfect reﬂection contrasts with the case of the Heisenberg model (2). In the latter case, the eigenfunction describing the spin wave in the presence of the domain wall isproportional to [ −ikλ+tanh(x/λ)] exp ( ikx)[12,28]. That is, the incident wave is perfectly transmitted through the domainwall for all k. This phenomenon in a magnetic nanowire has recently been studied by solving the Landau-Lifshitz-Gilbertequation [ 20]. The absence and presence of the amplitude of the order parameter at the domain wall are suggestive of the differenttunneling properties of the transverse spin wave for theHeisenberg model and the spin-1 Bose-Einstein condensates.However, perfect reﬂection does not originate from a zerovalue of the order parameter at the domain wall. Let usconsider the system where lim x→±∞(/Phi1+1,/Phi1−1)=(√ρ,0) holds, but which has a region where ( /Phi1+1,/Phi1−1)=(0,√ρ) holds around x=0. In that system, we can ﬁnd points where /Phi1+1=/Phi1−1, which leads to \|F\|=0. The absence of the spin density represents the same situation as that shown in Fig. 2. In this case, however, the transverse spin wave exhibits perfecttransmission in the low-energy limit (as proved in AppendixB). In these two cases, the excitation wave functions in the low-energy limit are given by the same form of the condensatewave functions. In Bose-Einstein condensates, this propertyusually gives rise to perfect transmission of an excitationthrough a potential barrier in the low-energy limit [ 29,30]. For the present transverse spin wave, the damping length of theexcitation solution becomes inﬁnite in the low-energy limit.This makes it possible for the damping mode of the excitationwave function to smoothly connect with the zero-energy modecorresponding to the condensate wave function in the low-energy limit. Consequently, depending on the conﬁguration ofthe condensate wave function, the damping mode can remainalone in the transmission region where the condensate wavefunction is ﬁnite. See Appendix Bfor more details. We now determine how the Bogoliubov mode and the quadrupolar spin mode are scattered from this domain wall.Hyperﬁne states S z=±1 are coupled as in Eq. (3); however, forx=− ∞ , the Bose-Einstein condensate of Sz=−1 only 023622-3SHOHEI WATABE, YUSUKE KATO, AND YOJI OHASHI PHYSICAL REVIEW A 86, 023622 (2012) 0 2 4 6 80.00.20.40.60.81.0 0 2 4 6 80.00.20.40.60.81.0 0 2 4 6 80.00.20.40.60.81.0 0 2 4 6 80.00.20.40.60.81.0 4 2 0 2 4051015202530 4 20 2 41.00.50.00.51.01.52.0 x/ξ Re[u+1(x)]/√ρ A BAB 0.0 0.0(a) BM (Sz=−1)(c) BM ( Sz=+1) (b) QSM ( Sz=+1) (d) QSM ( Sz=−1) )f( )e(E/(c+ρ) E/(c+ρ)τr x/ξF(x)/ρ2total Sz=+1 Sz=−1 FIG. 4. Transmission and reﬂection properties of excitations through the domain wall when the Bogoliubov mode is incident on the domain wall. Reﬂection probabilities of (a) the Bogoliubovmode (BM) ( S z=−1) and (b) the quadrupolar spin mode (QSM) (Sz=+1). Transmission probabilities of (c) the BM ( Sz=+1) and (d) the QSM ( Sz=−1). (e) A matrix element of the density spectral function in the low-energy regime ( E=10−2c+ρ). (f) The real part of the wave function u+1, where the Bogoliubov mode is perfectly reﬂected. The coupling constant is chosen to be c1=−3c+. exists, so that an excitation of Sz=−1 corresponds to the gapless Bogoliubov mode with E=[ε(ε+2c+ρ)]1/2and an excitation of Sz=+1 to the quadrupolar spin mode with an excitation gap that has the spectrum E=ε+2\|c1\|ρ, where ε=¯h2k2/(2m)[22]. For x=∞ , the Bose-Einstein condensate of Sz=+1 only exists, so that the roles for the two hyperﬁne states in the excitations are interchanged. Figure 4shows the reﬂection and transmission coefﬁcients, randτ, when the Bogoliubov excitation is scattered from the domain wall. Intriguingly, reﬂection does not occur in thelow-energy limit. Excitations in Bose-Einstein condensates areknown to be perfectly transmitted through potential barriersin the low-energy limit; this phenomenon is referred to asanomalous tunneling [ 29]. The Bogoliubov mode in the low- energy limit is a phase excitation of a Bose-Einstein condensate(i.e., a Nambu-Goldstone mode), which plays an essentialrole in anomalous tunneling [ 31,32]. However, the absence of reﬂection in the present case differs from anomalous tunneling.Figure 4(e) shows the matrix element of the local density spectral function F(x)≡\|/Phi1 ∗u−/Phi1v\|2[33] in the low-energy limit. In contrast to anomalous tunneling, the density modesofS z=±1 states localized at the domain wall appear even in the low-energy limit. (See Appendix Cfor more details, where these localized modes are discussed.)With respect to scattering between a topological excita- tion and the Nambu-Goldstone mode, a dark soliton in ascalar Bose-Einstein condensate is known to be completelytransparent to the Bogoliubov mode for all energies [ 34]. In this case, the localized density mode can also be found inthe low-energy limit, given by F(x)∝tanh 2(x)/cosh4(x)i n the dimensionless form [ 34], which is similar to the present case. However, in stark contrast to this dark soliton case,the present transmission and reﬂection coefﬁcients exhibitpronounced nonmonotonicity. In particular, the Bogoliubovmode is perfectly reﬂected at some energy points [A andBi nF i g . 4(a)]. This perfect reﬂection is strongly related to the bound state of the S z=+1 state at the domain wall [see Fig. 4(f)]. This appears to be similar to resonance scattering, where the bound state increases the scattering cross section[35]. However, the present bound state is speciﬁc to the ferromagnetic spin-1 Bose-Einstein condensate. For x< 0, (u +1,v+1) decays when the incident energy is lower than the energy gap of the quadrupolar spin mode. For x> 0, the wave functions ( u+1,v+1) behave as the Bogoliubov mode whose damping mode is given by exp( −κBx) for a positive energy E(>0) (see also Appendix Aand Ref. [ 36]). The damping structures on both sides of the domain wall are different. ForE< 2\|c1\|ρ, the plane wave solution of the quadrupolar spin mode does not exist and the reﬂection and transmissionof the quadrupolar spin mode are both forbidden [Figs. 4(b) and4(d)]. When the incident Bogoliubov energy exceeds the energy gap of the quadrupolar spin mode (i.e., E> 2\|c 1\|ρ), the transmission coefﬁcient of the quadrupolar spin mode(S z=−1) suddenly increases [see Fig. 4(d)]. Figure 5shows the reﬂection and transmission coefﬁcients when the quadrupolar spin mode is incident on the domainwall. Perfect reﬂection occurs in the low-momentum limit, butexcitations with ﬁnite momentum pass through the domainwall. At higher energies, the Bogoliubov mode of S z=+1 accounts for a substantial fraction of the transmitted ﬂux. For 0.00.51.01.52.02.53.00.00.20.40.60.81.0 0.00.51.01.52.02.53.00.00.20.40.60.81.0 0.00.51.01.52.02.53.00.00.20.40.60.81.0 0.00.51.01.52.02.53.00.00.20.40.60.81.0 0.00.0 0.00.0 0.0 0.0c1=−0.1c+ c1=−c+ c1=−3c+(a) QSM ( Sz MSQ)c( )1+= (Sz=−1) (b) BM ( Sz=−1) (d) BM (Sz=+1) kξ kξ τr FIG. 5. Transmission and reﬂection properties of excitations at a domain wall when the quadrupolar spin mode is incident on the domain wall. Reﬂection probabilities of (a) quadrupolar spinmode (QSM) ( S z=+1) and (b) Bogoliubov mode (BM) ( Sz=−1). Transmission probabilities of (c) QSM ( Sz=−1) and (d) BM (Sz=+1). 023622-4EXCITATION TRANSPORT THROUGH A DOMAIN WALL IN ... PHYSICAL REVIEW A 86, 023622 (2012) perfect reﬂection in the low-momentum limit, we conﬁrmed that the wave function vanishes in this limit. This is aconventional reﬂection property in the long-wavelength limit. The momentum transfer can be used to drive the ferro- magnetic domain wall. From the viewpoint of the momentumtransfer that occurs when excitations are reﬂected, the spinwave in ferromagnets does not drive the ferromagnetic domainwall. This is because this spin wave exhibits perfect trans-mission that is independent of the energy [ 20]. On the other hand, in the ferromagnetic spin-1 Bose-Einstein condensate,perfect reﬂection occurs for all three excitations: the transversespin wave, the quadrupolar spin mode, and the Bogoliubovmode. The ﬁrst two excitations do not play a major role indriving the domain wall because they exhibit perfect reﬂectionin the low-momentum limit so that they involve very-low-momentum transfer. In contrast, the Bogoliubov mode exhibitsperfect reﬂection outside of the low-momentum limit (A andBi nF i g . 4). This relatively high-momentum transfer of this mode is useful for driving the domain wall. Spin-transfertorque can also drive a ferromagnetic domain wall [ 7,20]. However, the formalism used in this paper cannot directlydemonstrate excitation-driven domain wall motion due to spintransfer as well as momentum transfer, because it involvesonly a small ﬂuctuation about the condensate wave function.The subject of future studies is to directly demonstrate howexcitation tunneling affects domain wall motion in the spin-1Bose-Einstein condensate by including the nonlinear effect asstudied in Ref. [ 20]. The simple domain wall shown in Figs. 1and2can be mapped onto a domain wall soliton (phase separation) in binarymixtures of Bose-Einstein condensates [ 24][ s e ea l s oE q . (3)]. In this regard, the phase separation can be experimentallycontrolled by changing the scattering length between sameor different components through the Feshbach resonance[37,38]. The term “quadrupolar spin mode” is speciﬁc to the spin-1 Bose-Einstein condensate; however, transmissionand reﬂection properties equivalent to those shown in Figs. 4 and5are easier to obtain experimentally by employing an immiscible two-component Bose-Einstein condensate (e.g., 85Rb-87Rb Bose-Einstein condensates [ 37] and Bose-Einstein condensates of87Rb with two internal spin states [ 38]). To create the localized Bogoliubov mode and localized spin wave excitations, Bragg scattering and Raman scattering,whose light beams are shined over a razor edge, would beuseful, respectively. After an excitation is scattered by thedomain wall, if reﬂection occurs, excitations may propagate inthe direction opposite to the incident excitation. Stern-Gerlachmeasurements can be used to determine their spin degrees offreedom. On the other hand, the movement and deformation ofthe domain wall after scattering can be observed by imagingthe density because the total particle density is low at thedomain wall [Fig. 2(b)]. In this paper, to extract the essential physics, we considered a planar domain wall and one-dimensional scattering in a three-dimensional system. Nambu-Goldstone modes of a Bose-Einstein condensate are known to be reﬂected and refractedin the presence of a potential step [ 39,40]. For junctions in Bose-Einstein condensates with equal densities, reﬂection andrefraction are absent for both the Bogoliubov mode and atransverse spin wave. For junctions in the (anti)ferromagneticHeisenberg model, spin waves also transmit perfectly [ 41]. It will be interesting to extend the present one-dimensionalscattering problem to a three-dimensional one and to comparereﬂection and refraction properties with those reported inRefs. [ 39,40] and for (anti)ferromagnets. The ﬁndings in this study will be helpful for determining the interface physics ofsuperﬂuids and scattering processes through domain walls thathave more complex structures than a planar wall. For scatteringbetween low-lying modes and topological excitations, it is alsoa future problem to study Nambu-Goldstone mode scatteringfrom a (half-)quantized vortex and a skyrmion in spinorBose-Einstein condensates. IV . SUMMARY We studied scattering between low-lying modes and a topological defect in a spin-1 Bose-Einstein condensate. Inparticular, we report the structure of a ferromagnetic domainwall and the transmission properties of excitations through thisdomain wall. We found that the domain wall and the tunnelingproperties of the transverse spin wave differ from those for aplanar domain wall in the ferromagnetic Heisenberg model.We also found that the transmission and reﬂection propertiesare strongly nonmonotonic when the Bogoliubov mode isscattered from the domain wall. ACKNOWLEDGMENTS The authors thank D. Takahashi for useful discussions on zero modes in Bose-Einstein condensates. S.W. thanks J. Iedafor discussions and pointing out Refs. [ 20,28]. This work was supported by Grants-in-Aid for Scientiﬁc Research (GrantsNo. 20500044, No. 21540352, No. 22540412, No. 23104723,and No. 23500056) from JSPS and MEXT, Japan. APPENDIX A: HOW TO DETERMINE THE TRANSMISSION AND REFLECTION COEFFICIENTS Considering ﬂuctuations in the condensate wave function ˆ/Phi1(x,t)=exp (−iμt/ ¯h)[ˆ/Phi1(x)+ˆφ(x,t)] and retaining terms toO[ˆφ(x)] in the Gross-Pitaevskii equation (1), we obtain equations for determining the tunneling properties of exci-tations. This appendix employs /Phi1 0=0, which is consistent with the system we are studying in this paper. For a transversespin wave (excitation with S z=0 state), after applying the Bogoliubov transformation φ0=u0e−iEt/ ¯h−v∗ 0e+iEt/ ¯h,w e obtain the following equation: E/parenleftbiggu0 v0/parenrightbigg =/parenleftbiggh0 −2c1/Phi1+1/Phi1−1 2c1/Phi1∗ +1/Phi1∗ −1 −h0/parenrightbigg/parenleftbiggu0 v0/parenrightbigg ,(A1) where h0=−¯h2 2m∂2 x−μ+c+(\|/Phi1+1\|2+\|/Phi1−1\|2). Amplitude transmission and reﬂection coefﬁcients, T0andR0, are ob- tained, by solving Eq. (A1) with the following boundary con- ditions: u0=e1e+ikx+R0e1e−ikx+A0e2ekx,f o rx=− ∞ , andu0=T0e1e+ikx+B0e2e−kx,f o rx=+ ∞ , where ¯ hk= (2mE)1/2,u0≡(u0,v0),e1≡(1,0), and e2≡(0,1).A0and B0are unknown coefﬁcients of the damping solutions, which should be determined along with T0andR0. On the other hand, as for the excitations with Sz=±1 states, the usual Bogoliubov transformation 023622-5SHOHEI WATABE, YUSUKE KATO, AND YOJI OHASHI PHYSICAL REVIEW A 86, 023622 (2012) φ±1=u±1e−iEt/ ¯h−v∗ ±1e+iEt/ ¯hleads to E⎛ ⎜⎜⎜⎝u +1 v+1 u−1 v−1⎞ ⎟⎟⎟⎠=⎛ ⎜⎜⎜⎝h +−a+b+−c a∗ +−h+c∗−b∗ + b−−ch −−a− c∗−b∗ −a∗ −−h−⎞ ⎟⎟⎟⎠⎛ ⎜⎜⎜⎝u +1 v+1 u−1 v−1⎞ ⎟⎟⎟⎠, (A2) where h ±=−¯h2 2m∂2 x−μ+2c+\|/Phi1±1\|2+c−\|/Phi1∓1\|2,a±= c+/Phi1±1/Phi1±1,b±=c−/Phi1±1/Phi1∗ ∓1, andc=c−/Phi1+1/Phi1−1. To deter- mine the transmission and reﬂection coefﬁcients, we imposethe following boundary condition: /parenleftbiggu +1 u−1/parenrightbigg =/parenleftbigguin +1 uin −1/parenrightbigg +R+1/parenleftbigge1 0/parenrightbigg e−ikQx+A+1/parenleftbigge2 0/parenrightbigg eκQx +R−1/parenleftbigg0 α/parenrightbigg e−ikBx+A−1/parenleftbigg0 β/parenrightbigg eκBx(x=− ∞ ), (A3) /parenleftbiggu+1 u−1/parenrightbigg =T+1/parenleftbiggα 0/parenrightbigg e+ikBx+B+1/parenleftbiggβ 0/parenrightbigg e−κBx +T−1/parenleftbigg0 e1/parenrightbigg e+ikQx+B−1/parenleftbigg0 e2/parenrightbigg e−κQx(x=∞ ), (A4) where uT ±1≡(u±1,v±1);kQandκQare, respectively, the wave vectors of the quadrupolar spin mode and its dampingsolution; and k BandκBare, respectively, wave vectors of the Bogoliubov mode and its damping solution [ 36].T±1andR±1 are, respectively, the amplitude transmission and reﬂection coefﬁcients for propagating modes, and A±1andB±1are coefﬁcients for damping modes, which should be determinedin this problem. For the Bogoliubov mode incident to thedomain wall, we set ( u in +1,uin −1)=(0,α)e x p(ikBx). On the other hand, for the incident quadrupolar spin mode, we set(u in +1,uin −1)=(e1,0)e x p(ikQx). APPENDIX B: PROOF OF PERFECT REFLECTION OF TRANSVERSE SPIN WA VE In this appendix, we prove and discuss perfect reﬂection of the transverse spin wave mode in the low-energy limit. Weuse the dimensionless form, where the wave function, energy,and length are scaled by√ ρ,c+ρandξ=¯h/√mc+ρ.T w o functions S±≡u0±v0obey the following equation: h(±) 0S∓∓i2c1Im(/Phi1+1/Phi1−1)S±=ES±, (B1) where h(±) 0≡h0±2c1Re(/Phi1+1/Phi1−1). Since /Phi1±1are real in this study, we have h(±) 0S∓=ES±. (B2) The ﬂux density, which is independent of x, is given by J∝Im[S∗ +∂xS++S∗ −∂xS−]. For the tunneling problem, the boundary condition for x=− ∞ is given by S±=eikx+ R0e−ikx±A0ekx, and that for x=+ ∞ is given by S±= T0eikx±B0e−kx. The transmission and reﬂection coefﬁcients τandrare now determined as τ=\|T0\|2andr=\|R0\|2, respectively.To study the low-energy properties, we use the following expansion: (T0,R0,A0,B0)=/summationdisplay n=0kn/parenleftbig T(n) 0,R(n) 0,A(n) 0,B(n) 0/parenrightbig .(B3) Fork\|x\|/lessmuch1, the boundary condition for x/lessmuch− 1i sg i v e n byS±=1+R(0) 0±A(0) 0+k(R(1) 0±A(1) 0)+kx(i−iR(0) 0± A(0) 0)+··· , and that for x/greatermuch1 is given by S±=T(0) 0±B(0) 0+ k(T(1) 0±B(1) 0)+kx(iT(0) 0∓B(0) 0)+··· . We also determine S±from Eq. (B2). We expand the functions S±asS±=/summationtext n=0knS(n) ±. Then, h(∓) 0S(0,1) ±=0f o l - lows. Since one of the zero mode solutions for Eq. (A1) is given by ( u0,v0)=(/Phi1+1,/Phi1∗ −1),S(0,1) ±,I=/Phi1+1±/Phi1−1≡φ± follow, where we take /Phi1±1to be real. The other solu- tions are obtained as S(0,1) ±,II(x)=φ±(x)/integraltextx 0dx/primeφ−2 ±(x/prime). The behaviors for \|x\|/greatermuch1a r eg i v e nb y( S(0,1) +,I,S(0,1) −,I)=(1,sgn(x)) and (S(0,1) +,II,S(0,1) −,II)=(sgn(x)γ++x,γ−+\|x\|), where γ±≡/integraltext∞ 0dx(φ−2 ±−1). Here, we used φ±(−x)=±φ±(x). We examine solutions linear in k, which are given by S±(x)= C(0) ±,IS(0) ±,I+C(0) ±,IIS(0) ±,II+k(C(1) ±,IS(1) ±,I+C(1) ±,IIS(1) ±,II). By compar- ing the coefﬁcients of kandx, we obtain /parenleftbig R(0) 0,T(0) 0,A(0)0,B(0) 0/parenrightbig =(−i,0,0,1−i), (B4) ±C(0) ±,I=∓C(1) ±,II=1−i, (B5) C(0) ±,II=0. (B6) As a result, the excitation shows perfect reﬂection in the low- energy limit (i.e., \|R(0) 0\|2=1). We also note that the wave functions show lim E→0S±=±(1−i)φ±, that is, lim E→0(u0,v0)/(1−i)=(/Phi1−1,/Phi1+1). (B7) The spin density ﬂuctuation δF, deﬁned here as the spin density F=ˆ/Phi1†Sˆ/Phi1within the ﬁrst order of φ0, is given by δFx,y∝Fz, where Fz=/Phi12 +1−/Phi12 −1. As for the usual perfect reﬂection in quantum mechanics, the phase of the amplitude reﬂection coefﬁcient R0in the low-energy limit leads to πand the amplitude of the wave function vanishes everywhere. However, in this case, the phaseof the amplitude reﬂection coefﬁcient R 0in the low-energy limit is 3 π/2 and the excitation wave function remains in the low-energy limit when perfect reﬂection occurs. Perfectreﬂection and a nonvanishing wave function coexist in thelow-energy limit for the following reason. The transverse spinmode in this system has a damping solution whose dampinglength is 1 /k. In the low-energy limit, this damping length becomes inﬁnite and the damping mode can smoothly connectwith the condensate wave function, which is uniform for \|x\|/greatermuch 1. Consequently, the propagating wave mode can vanish in thislimit. The coexistence of perfect reﬂection and a nonvanishing wave function is also observed in the scalar Bose-Einsteincondensate. In particular, we ﬁnd perfect reﬂection in the low-energy limit when the Bogoliubov excitation is incident onthe region where the condensate density is absent due to thepresence of a potential step that exceeds the chemical potential[42,43]. If we regard the excitation in the low-energy limit 023622-6EXCITATION TRANSPORT THROUGH A DOMAIN WALL IN ... PHYSICAL REVIEW A 86, 023622 (2012) as the phase mode of the condensate, perfect reﬂection in the low-energy limit is natural because a region with no condensatedensity cannot support the phase mode. On the other hand, inthe present case, the spin wave mode in the low-energy limit,which is related to the rotation symmetry of the spin space,can propagate on both sides of the domain wall, so that perfectreﬂection of this mode originates from different causes. A zerovalue of the order parameter at the domain wall is speciﬁc tothe present system and is suggestive of perfect reﬂection in thelow-energy limit. However, perfect reﬂection of this spin mode does not orig- inate from the zero value of the spin density at the domain wall.To demonstrate this, we assume that lim x→±∞ (/Phi1+1,/Phi1−1)= (1,0); however, the system has a region where ( /Phi1+1,/Phi1−1)= (0,1) around x=0. For the tunneling problem, the boundary conditions for excitations are unaltered. Solutions S(0,1) ±,Iand S(0,1) ±,IIare formally given in the same form as those derived above. However, they have different behaviors for \|x\|/greatermuch1, which are given by S(0,1) ±,I=1 and S(0,1) ±,II=sgn(x)γ±+x, where γ±≡/integraltext∞ 0dx(φ−2 ±−1). Following the same procedures shown above, we obtain /parenleftbig R(0) 0,T(0) 0,A(0)0,B(0) 0/parenrightbig =(0,1,0,0), (B8) /parenleftbig C(0) ±,I,C(0) ±,II,C(1) ±,II/parenrightbig =(1,0,i). (B9) As a result, we have perfect transmission in the low-energy limit (i.e., \|T(0) 0\|2=1), and lim E→0(u0,v0)=(/Phi1+1,/Phi1−1). The spin density ﬂuctuation is given by ( δFx,δFy)∝(Fz,0). In the present case, we can ﬁnd points where /Phi1+1=/Phi1−1, which leads to Fz=0. However, the absence of the amplitude of the order parameter at the domain walls is irrelevant tothe tunneling properties in the low-energy limit. In a scalarBose-Einstein condensate, the Bogoliubov excitation can passthrough the dark soliton [ 34]. This is another example in which perfect transmission occurs in a system where the amplitudeof the order parameter is absent in the soliton. APPENDIX C: DETAILS OF ZERO-MODE TUNNELING THROUGH A SOLITON In this appendix, we discuss the effect of zero-energy modes on the tunneling properties by focusing on the phase anddensity ﬂuctuations. First, by comparing Eq. (A2) forE=0 with Eq. (3)and its derivative with respect to x, we ﬁnd two simple zero-energy- mode solutions for Eq. (A2) , which are given by (u +1,v+1,u−1,v−1)=(/Phi1+1,/Phi1∗ +1,/Phi1−1,/Phi1∗ −1), (C1) (u+1,v+1,u−1,v−1)=∂x(/Phi1+1,−/Phi1∗ +1,/Phi1−1,−/Phi1∗ −1),(C2) where we omit the normalization factors. If we add an external potential term to Eqs. (3)and(A2) by replacing μwithμ− V(x), the ﬁrst solution, Eq. (C1), holds, whereas the second solution, Eq. (C2), does not. Equation (C2) yields a zero mode solution to Eq. (A2) only when the system is uniform. This is easy to understand when we consider (spontaneously) brokensymmetries. As for the broken gauge symmetry, when the con-densate wave function is given by /Phi1(x)=A(x)e x p[iθ(x)], the function A(x)e x p{i[θ(x)+dθ]}/similarequal/Phi1(x)+i/Phi1(x)dθisanother solution of Eq. (3). The factor proportional to dθleads to Eq. (C1). (Here, we omit the subscript ±1; the discussion in the present paragraph also holds for a scalar Bose-Einsteincondensate.) Thus, the ﬁrst solution, Eq. (C1), is the zero mode related to the broken gauge symmetry and it does notcontribute the density ﬂuctuation. (When the condensate wavefunction /Phi1ﬂuctuates as /Phi1+φ, the density ﬂuctuation is given by\|/Phi1+φ\| 2−\|/Phi1\|2=2Re[/Phi1∗φ]+O(φ2). In the low-energy limit,φ=u−v∗and the solution (C1) is not related to the density mode.) On the other hand, in the absence of an externalpotential, /Phi1(x+dx)/similarequal/Phi1(x)+∂ x/Phi1(x)dxis another solution of Eq. (3). The term proportional to dxleads to Eq. (C2) [44]. Thus, the second solution, Eq. (C2), is the zero mode related to the broken translational symmetry due to the presenceof the soliton. This gives a ﬁnite density ﬂuctuation andthe matrix element of the density spectral function is givenbyF(x)∝[∂ xρ(x)]2, where ρ(x)=\|/Phi1(x)\|2. The proﬁle in Fig. 4(e) pertains to the present discussion. To summarize, when the condensate exhibits solitary behavior, the densityﬂuctuation does appear in the soliton. We here discuss the absence of reﬂection of the Bogoliubov mode in the low-energy limit from the domain wall shownin Fig. 4. Starting with a scalar Bose-Einstein condensate is instructive for understanding the characteristic properties ofexcitation tunneling through the domain wall. In the usualmanner and using the usual notation, the condensate wavefunction is obtained by solving the Gross-Pitaevskii equation: /bracketleftbigg −¯h 2 2m∂2 x+V(x)−μ+g\|/Phi1(x)\|2/bracketrightbigg /Phi1(x)=0.(C3) The Bogoliubov excitation is described by E/parenleftbiggu v/parenrightbigg =/parenleftbiggh −g/Phi12 g(/Phi1∗)2−h/parenrightbigg/parenleftbiggu v/parenrightbigg , (C4) where h=−¯h2/(2m)∂2 x+V(x)−μ+2g\|/Phi1(x)\|2. The chem- ical potential μand the boundary condition u≡(u,v) for the tunneling problem are, respectively, given byμ=gρ≡g\|/Phi1(\|x\|→∞ )\| 2and u=αe+ikBx+Rαe−ikBx+AβeκBx(x=− ∞ ),(C5) u=Tαe+ikBx+Bβe−κBx(x=∞ ), (C6) where kB,κB,α, and βare given in Ref. [ 36] (where c+ is replaced by g).TandRare, respectively, the amplitude transmission and reﬂection coefﬁcients for the propagatingmodes and AandBare the respective coefﬁcients for the damping modes, which should be determined in thisproblem. We consider three cases below. The ﬁrst case isthe ground state in the presence of the barrier V/negationslash=0, where the boundary condition of the condensate wave function isgiven by /Phi1(x→± ∞ )=+√ ρ. This case was studied as an anomalous tunneling problem [ 29]. The results are shown by the solid lines in each panel of Fig. 6. In this case, the Bogoliubov mode perfectly tunnels through the barrier in thelow-energy limit [Fig. 6(b)] and the density mode is suppressed [Fig. 6(c)]. The second case is the πstate in the absence of a barrier ( V=0) where the boundary condition of the condensate wave function is given by /Phi1(x→± ∞ )=±√ ρ. 023622-7SHOHEI WATABE, YUSUKE KATO, AND YOJI OHASHI PHYSICAL REVIEW A 86, 023622 (2012) 5 0 51.00.50.00.51.0 0.0 0.2 0.4 0.6 0.80.00.20.40.60.81.01.2 5 0 502468Φ/√ρ τ F(x)/ρ2 ground state ( V=0) π-state ( V=0) π-state ( V=0) 0.00.0 x/ξs kξs x/ξs(a) (b) (c) FIG. 6. Transmission and reﬂection properties of the Bogoliubov excitation in a scalar Bose-Einstein condensate. We consider three cases: (i) the ground state in the presence of the barrier ( V/negationslash=0) (solid lines in each panel), (ii) the πstate in the absence of the barrier ( V=0) (dotted lines in each panel), and (iii) the πstate in the presence of the barrier ( V/negationslash=0) (dashed lines in each panel). (a) The condensate wave function as a function of x, (b) momentum- dependent transmission probability τ=\|T\|2, and (c) the matrix element of the density spectral function in the low-energy regime kξs=10−2.ξsis the healing length, which is given by ξs=¯h/√mgρ. For cases (i) and (iii), we used V(x)=2gρexp(−x2/ξ2 s). This case was considered in Ref. [ 34]. The results are shown by the dotted lines in each panel of Fig. 6. In this case, the Bogoliubov mode perfectly tunnels through the soliton,independent of the energy [Fig. 6(b)]. In contrast to the previous case, density ﬂuctuation appears [Fig. 6(c)]. We here examine whether the Bogoliubov mode transmits through thebarrier in the low-energy limit when the system is in the π state. The results are shown by the dashed lines in each panelof Fig. 6. In the low-energy limit, the Bogoliubov excitation in theπstate perfectly tunnels through the barrier at the center of the soliton. In this case, the potential barrier suppressesthe zero mode of the density. From these three cases, we ﬁndthat the density ﬂuctuation is irrelevant to perfect tunnelingthrough a soliton in a scalar Bose-Einstein condensate. Sincethe presence of the zero mode of the phase is related to thephase coherence of the condensate, this coherence is importantfor perfect tunneling of the low-energy Bogoliubov mode in ascalar Bose-Einstein condensate. We now examine the case for the domain wall of a spin-1 Bose-Einstein condensate. We add an external potential to thecenter of the domain wall, which suppresses the zero modeof the density, to investigate whether the perfect conversionof energy ﬂux from S z=−1 Bogoliubov mode to Sz=+1 Bogoliubov mode occurs only through the zero mode of thephase. We add an external potential term to Eqs. (3)and(A2) by replacing μwithμ−V(x). The results are shown in Fig. 7.0 2 4 6 80.00.20.40.60.81.0 0 2 4 6 80.00.20.40.60.81.0 0 2 4 6 80.00.20.40.60.81.0 0 2 4 6 80.00.20.40.60.81.00.0 0.0(a) BM (Sz=−1)(c) BM ( Sz=+1) (b) QSM ( Sz=+1) (d) QSM (Sz=−1) E/(c+ρ) E/(c+ρ)τr FIG. 7. Transmission and reﬂection properties of excitations through the domain wall deformed by the potential barrier V(x) when the Bogoliubov mode is incident. Reﬂection probabilities of (a) Bogoliubov mode (BM) ( Sz=−1) and (b) quadrupolar spin mode (QSM) ( Sz=+1). Transmission probabilities of (c) BM (Sz=+1) and (d) QSM ( Sz=−1). The coupling constant is chosen asc1=−3c+and we used V(x)=2c+ρexp(−x2/ξ2). We here took the center of the domain wall (i.e., x=0) as the center of the potential barrier. We ﬁnd that perfect reﬂection occurs in the low-energy limit when the Bogoliubov mode is injected into the domain walldeformed by the barrier. This result contrasts with that in theπstate of a scalar Bose-Einstein condensate. We conﬁrmed that in the presence of the barrier, the matrix element ofthe density spectral function near the domain vanishes asthe energy decreases. This result suggests that the localizeddensity mode plays an important role in converting the energyﬂux of a low-energy Bogoliubov mode propagating in oneBose-Einstein condensate into that in the other condensate. We also conﬁrmed that reﬂection is absent in the low-energy limit when the barrier is located far from the interface betweencondensates. In this case, the potential barrier breaks thetranslational invariance so that the zero mode solution is notgiven by ∂ x/Phi1±1. However, we ﬁnd the same behavior of the matrix element of the density spectral function as shown inFig. 4(e). The breaking of the translational invariance by a potential barrier far from a domain wall is irrelevant to theappearance of the localized density mode near the domainwall and to the absence of reﬂection. [1] C. 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Sakurai, Modern Quantum Mechanics (Addison-Wesley, New York, 1985). [36] Parameters are given by ¯ hk Q≡[2m(E+2c1ρ)]1/2,¯hκQ≡ [2m(E−2c1ρ)]1/2,¯hkB≡[2m(WE−c+ρ)]1/2,¯hκB≡ [2m(WE+c+ρ)]1/2,α≡[(W+1)/2]1/2,a n d β≡ [(W−1)/2]1/2,w h e r e W≡[1+(c+ρ/E)2]1/2. We also deﬁne α≡(α,β)a n d β≡(β,−α). [37] S. B. Papp, J. M. Pino, and C. E. Wieman, Phys. Rev. Lett. 101, 040402 (2008). [38] S. Tojo, Y . Taguchi, Y . Masuyama, T. Hayashi, H. Saito, and T. Hirano, P h y s .R e v .A 82, 033609 (2010). [39] S. Watabe and Y . Kato, P h y s .R e v .A 78, 063611 (2008). [40] S. Watabe and Y . Kato, arXiv: 1012.5618 . [41] Y . Kato, S. Watabe, and Y . Ohashi (in preparation).[42] S. Tsuchiya and Y . Ohashi, P h y s .R e v .A 79, 063619 (2009). [43] D. Takahashi, arXiv: 0909.1068 . [44] This type of zero mode is also found in the ferromag- netic Heisenberg model. As for the domain wall solution toEq.(2)g i v e nb y( σ x,σy,σz)=(0,σ/cosh(x/λ),σtanh(x/λ))≡ σ(0,cosθ0(x),sinθ0(x)), the zero mode solution ϕ0(x)= 1/cosh(x/λ) is related to θ0(x) through ϕ0(x)=−λ∂xθ0(x)[12]. 023622-9
PhysRevLett.126.056801.pdf	Current-Driven Magnetization Reversal in Orbital Chern Insulators Chunli Huang , Nemin Wei, and Allan H. MacDonald Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA (Received 20 July 2020; revised 18 December 2020; accepted 5 January 2021; published 2 February 2021) Graphene multilayers with flat moir´ e minibands can exhibit the quantized anomalous Hall effect due to the combined influence of spontaneous valley polarization and topologically nontrivial valley-projectedbands. The sign of the Hall effect in these Chern insulators can be reversed either by applying an external magnetic field, or by driving a transport current through the system. We propose a current-driven mechanism whereby reversal occurs along lines in the (current I, magnetic-field B) control parameter space with slope dI=dB ¼ðe=hÞMA Mð1−γ2Þ=γ, where Mis the magnetization, AMis the moir´ e unit cell area, andγ<1is the ratio of the chemical potential difference between valleys along a domain wall to the electrical bias eV. DOI: 10.1103/PhysRevLett.126.056801 Introduction. —Magnetism in solid state systems is pro- duced by both spin and orbital electronic angular momen- tum, but the two constituents normally have a decidedly asymmetric relationship in which spins order spontaneouslyand orbital magnetism is induced parasitically by spin-orbit interactions. Current control of ordered spins is now routine in spintronics [1–4]. The recent discovery [5,6] of sponta- neous orbital order manifested by a quantum anomalous Hall effect in graphene moir´ e superlattice systems, and of current driven magnetization reversal in those systems, is the firstdemonstration of an influence of a transport current on orbital magnetism. In this Letter, we propose an experi- mentally testable explanation for this effect. The quantum anomalous Hall effect, a property of insulators whose occupied bands carry a net Chern number,is common in graphene moir´ e superlattice systems [5–9] when the minibands are flat and the moir´ e band filling factor ν¼n eAMis close to an odd integer. (Here, neis the carrier density and AMis the moir´ e unit cell area.) In magic angle twisted bilayer graphene [10] (MATBG), for exam- ple, the intriguing family of strongly correlated states in the−4<ν<4flat-band regime includes superconductors and Mott insulators [11–14]and also a Chern insulator state with a Hall resistance close [5,6] to the von Klitzing constant. The quantized Hall conductance appears at ν¼3when the graphene bilayer is aligned with an adjacent hexagonal boron nitride layer but, unlike the caseof magnetized topological insulators [15–17], cannot be a consequence of spin-order plus spin-orbit coupling since the latter is negligible in pristine graphene. Instead, theChern insulator is thought to be a combined consequence of the nontrivial topology of moir´ e minibands in graphene multilayers [18–26]and momentum-space condensation [27–29]in the form of spontaneous valley polarization. Indeed, Hartree-Fock calculations [25,30] predict that odd integer νinsulators in graphene multilayers are very oftenChern insulators. We refer to these states as orbital Chern insulators (OCIs), although they break time reversal sym-metry in both spin and orbital degrees of freedom, because the main observable —the anomalous Hall effect —is of orbital origin, and because spin-order cannot be maintainedat finite temperature when spin-orbit interactions are negligible. Therefore, we drop the spin degree of freedom from the following discussion. The properties of OCIs arequite distinct [31] from those of spin Chern insulators [17]. From a statistical physics point of view, an OCI is an Ising ferromagnet in which the total Chern number of theoccupied bands C /C6¼/C6Ccan be viewed as an order parameter. Experiments have shown that the Hall conductance of an OCI can be switched between þCe2=hand−Ce2=h, signal- ing a complete reversal of orbital magnetization [5,6],b y applying either an external magnetic field Band/or an electrical bias voltage V. The magnetization reversal mechanisms in conventional spin ferromagnets are relatively well established [32–34]and involve a combination of Stoner-Wohlfarth single domain switching and domain-wall depinning, driven by a combination of spin-transfer torques, spin-orbit torques, and magnetic fields. However, consensushas not yet been reached on the microscopic mechanism of orbital-magnetization reversal, although some interesting proposals have been put forward [6,35,36] . Here, we analyze the case of current driven reversal in an OCI with a bulk that isperfectly insulating so that gapless charge excitations are present only at the sample edge and along domain walls. We find that both magnetic fields Band transport bias voltages V apply pressure to domain walls and predict that switching occurs along a line in the (current I, magnetic field B)c o n t r o l parameter space with slope dI=dB ¼ðe=hÞMA Mð1−γ2Þ=γ, where Mis the magnetization, AMis the moir´ e unit cell area, andγ<1is the ratio of the chemical potential difference between valleys along a domain wall to the electrical biasPHYSICAL REVIEW LETTERS 126, 056801 (2021) 0031-9007 =21=126(5) =056801(6) 056801-1 © 2021 American Physical Societyvoltage. In the following, first, we argue that moir´ es u p e r - lattice OCIs are described by an Oð3Þf i e l dt h e o r yi nw h i c h the vector order parameter characterizes the local valleypolarization direction. This property allows domain pinningto be analyzed using conventional Landau-Lifshitz equations. Valley pseudospins in MATBG. —The valley-projected π bands of twisted bilayer graphene are described by a low-energy continuum model [10]in which isolated layer Dirac cones are coupled by an interlayer tunneling term that has the periodicity of the moir´ e pattern H τ 0¼−iℏvFðτσx∂xþσy∂yÞ−ΔBN 2σzj1ih1j þX3 j¼1½TðτÞ je−iτqj·rj1ih2jþH:c:/C138; ð1Þ Tτ j¼w0σ0þw1σxeð2πi=3Þðj−1Þτσz; ð2Þ where τ¼/C6 is the valley label, j1ih2jaccounts for tunneling between layers labeled 1 and 2, the σ Pauli matrices act on the sublattice degree of freedomwithin each layer, q 1¼ð4π=3aMÞð0;−1Þ,q2;3¼ð4π=3aMÞ ð/C6ﬃﬃﬃ 3p =2;1=2Þ, and aMis the moir´ e lattice constant, equal to 13.4 nm at the magic angle θ¼1.05°. In Eq. (2).w1¼ 110meV and w0¼0.8w1are tunneling energy parameters. Since Hτ 0is a periodic function of position for each valley τ, it has a set of Bloch bands Hτ 0junτki¼EnτðkÞjunτki that satisfy the time-reversal symmetry propertyE n−ð−kÞ¼EnþðkÞ, guaranteeing that the densities of states of the two valleys are identical. The OCI ground state at ν¼3empties the conduction band of one valley, chosen spontaneously. Mean-field (MF)calculations [25] have shown that the energy scale Iof single-particle valley-flip excitations of the OCI state is∼10meV, whereas the energy scale Kof long-wavelengthcollective valley reorientation excitations [37]∼0.1meV. This contrast in energy scales is familiar from the properties of the conventional itinerant electron ferromagnets heavily employed in spintronics, although less extreme in the OCIcase, if we identify valley in OCIs with spin in conventionalferromagnets. (In ferromagnetic Ni, for example, I∼ 0.3eV and K∼3μeV[38].) Therefore, we follow the approach used in metal spintronics to address magnetiza-tion reversal by assuming that we can focus on thedynamics of the low-energy collective degrees of freedom,which are described at long wavelengths by the pheno- menological micromagnetic [39] energy density E½n/C138¼Að∇nÞ 2−K AMn2zþK⊥ AMsin2ðθÞsin2ðϕ−ϕpÞ;ð3Þ where the moir´ e unit cell area AM¼ﬃﬃﬃ 3p a2 M=2and ˆn¼ ðsinθcosϕ;sinθsinϕ;cosθÞis the Bloch sphere unit vector that characterizes the local collective valley spinor jΨi∼cos/C18θ 2/C19 jτ¼þ iþ eiϕsin/C18θ 2/C19 jτ¼−i:ð4Þ Equation (3)is parametrized by three parameters (with dimension of energy) A; K; K ⊥>0which arise naturally from the following considerations: Ais a stiffness para- meter that expresses an energetic preference for uniformvalley polarized states, Kis a valley anisotropy constant that favors complete polarization in jτ¼/C6 i in the OCI ground state, and K ⊥is an azimuthal anisotropy constant that accounts for processes that violate valley conservation.Since K ⊥¼0is a consequence of momentum conservation in perfect crystals, we anticipate that K⊥≠0only near sample edges. In Fig. 1, we plot the MF quasiparticle energy bands of an OCI for two different valley orientations ˆnby adding (a) (b) (c) FIG. 1. Local quasiparticle bands [Eq. (5)] for a valley-exchange field pointing along (a) ˆn¼ˆzand (b) ˆn¼ˆx. Occupied (unoccupied) states at ν¼neAM¼3are drawn in black (red) so that (a) is an insulator while (b) is a metal. For better comparison, the bands in (a) have been folded into the irreducible Brillouin zone of (b). Because the interlayer tunneling terms are different in the two valleys(c) the area of the irreducible Brillouin zone (red) for general valley orientation (spanned by q 1andq2)i s1=3of the single-particle valley-projected moir´ e Brillouin zones (black-solid) area (spanned by q2−q1and q3−q1). These bands were calculated using spontaneous valley splitting I¼12meV, hexagonal boron nitride ( h-BN)-induced mass gap ΔBN¼10meV, twist angle θ¼1.05°, and Fermi velocity vF¼9.5×105m=s.PHYSICAL REVIEW LETTERS 126, 056801 (2021) 056801-2an exchange field with Stoner interaction constant I¼12meV [40] to Eq. (2) HMF¼Hþ 0þH− 0 2þτzHþ 0−H− 0 2−I 2τ·n; ð5Þ where the τPauli matrices act on the valley degree of freedom. The choice of an exchange effective magnetic field that is aligned with the valley orientation is motivatedby the observation that the dominant Coulomb interactionsin graphene multilayers are valley-independent, just as theCoulomb interactions in a magnetic metal are spin inde-pendent. The OCI band-structure calculation has three important messages. First, the band structure is indepen- dent of ϕas a result of total valley number conservation in Eq.(5). This band model result is consistent with the expectation that K ⊥¼0in perfect periodic lattice. Second, asˆngoes from the pole [Fig. 1(a)] to the equator [Fig. 1(b)], the bandwidth decreases and total energy increases,suggesting that the easy direction of valley polarization is the polar axis in agreement with experiment. Because the exchange field couples valleys, and the tunnelingHamiltonians in the two valleys are not identical, the moir´ e Hamiltonian unit cell area is increased (by a factor of 3)when ˆn×ˆz≠0, as illustrated in Fig. 1(c) [41] . Third, we found that the local band structure is metallic when ˆnis close to the equator, a property that will have important implications for domain wall dynamics. Domain wall dynamics. —We can calculate magnetiza- tion dynamics from Eq. (3)by recognizing that the two components of valley pseudospin perpendicular to ˆn(when suitably normalized) are canonical conjugate variable. The Euler-Lagrange equation corresponding to Eq. (3)is, therefore, ℏ 2AM∂tn¼n×δE δnþℏ 2AMn×ðˇα·∂tnÞ: ð6Þ Equation (6)is known in spintronics as the Landau- Liftshitz-Gilbert equation and includes a damping tensor ˇαthat accounts for coupling between collective magnetic degrees of freedom and other low-energy degrees of free-dom, including phonons and gapless quasiparticle excita-tions if these are present. Applying Eq. (6)to Eq. (3)and linearizing around n z¼1yields the valley-wave collective mode energies EðqÞ¼4Kþ4AAMq2. By fitting to micro- scopic bulk collective mode calculations [37], we estimate thatK∼0.04meV and AAM∼0.13meVa2 M. A domain wall, like the one illustrated schematically in Fig. 2, is a real-space topological defect obtained by minimizing the energy functional Eq. (3)with the con- straint that nz→/C61forx→/C6∞. This yields ϕ¼ϕpand θ¼2arctan exp ðx−X=λÞ, where Xis the domain wall center, λ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ AAM=Kp is half-width of the domain wall. Using the values for KandAquoted above yields λ¼1.8aM. In order to describe wall dynamics, we use ageneralization of Slonczekswi ’s[42] ansatz by letting the domain wall position and azimuthal phase X¼Xðy; tÞ; ϕ¼ϕðy; tÞ; ð7Þ depend on time and the coordinate along the wall. This dynamics focuses on excitation of the soft mode of adomain wall associated with its invariance under a shift in Xin the absence of pinning. In practice, domain walls are invariably pinned by sample inhomogeneities in realdevices, and this pinning is responsible for hysteresis.For definiteness, we assume that the domain wall is pinned atX¼0by some extrinsic pinning potential E pinwhich can arise from, e.g., a twist-angle extremum at which thecondensation energy of the ordered state is minimized, or a local minimum in the width Wof the sample. There is an energy penalty dE pinto shift Xaway from X¼0. For simplicity, we take it to be specified by a harmonic potential dEpin¼EpinðXÞ−EpinðX¼0Þ¼kW 2X2; ð8Þ up to a maximum jXj<X maxbeyond which the pinning energy is constant. The pinning strength k>0has units of energy per length. Our main results do not depend on the details of Epin. When an external magnetic field Bis present, we must also account for the dependence of itsinteraction with the spontaneous orbital magnetization on domain wall position dW B /C6¼½−MþX−M−ð−XÞ/C138WdB ¼−2MWXdB: ð9Þ Here, M/C6is the net orbital moment per area of the /C6 valley states, and we used time reversal symmetry FIG. 2. Magnetization reversal in an orbital Chern insulator: A domain wall (vertical dashed line) separates the OCI into regionswith opposite signs of the Hall conductances. Domain walls canbe shifted (from 0 to X) and eventually depinned by a valley- dependent chemical potential eðV 1−V3Þ, or by a magnetic field. Inset shows bending of a domain wall close to a hot spot [51].PHYSICAL REVIEW LETTERS 126, 056801 (2021) 056801-3(Mþ¼−M−≡M) in the second equation. Introducing Eqs. (8)and(9)into Eq. (3)a n di n t e g r a t i n gE q . (6)overx yields _ϕ¼4AAM ℏλX00þ2AMMB ℏ−AMkX ℏ−αϕ_X λ;ð10Þ _X λ¼−2AAMπ ℏϕ00þ2K⊥ ℏsin½2ðϕ−ϕpÞ/C138 þ αX_ϕ:ð11Þ Equation (10) equates the precession frequency of the valley pseudospin to the wall pressure generated by the sum of wall curvature, magnetic field, pinning forces, anddamping forces. Note, we distinguished α Xfrom αϕsince it requires processes that change overall valley polariza- tion, and therefore, we expect it to be much smaller. Indeed, αϕhas a substantial electronic contribution since [43] the Chern number change upon valley polarization reversal requires that the quasiparticle gap vanishes in the interior of the domain wall, see Fig. 1(b). Equation (11) is a continuity equation (with K⊥→0) for valley polariza- tion expressed in collective coordinates: the damping term proportional to αXis a valley-transfer torque that accounts for the valley-pumping quasiparticle currents generated by _ϕ[44–50]. So far, we have not directly invoked the unusual physics of OCIs, except by allowing the valley polarization order parameter, which is important for identifying conjugatecoordinates and, therefore, collective coordinate dynamics,and magnetization, which characterizes the strength of interactions with the external magnetic field, to be inde- pendent. For spin-magnets, these two quantities have auniversal relationship characterized by the gyromagnetic ratio. The simple way in which transport currents influence domain wall dynamics, which we now explain, is, however,a very specific consequence of the topological character of OCIs. The pinned domain wall in Fig. 2separates orbital Chern insulator domains with opposite total Chern num- bers. Therefore, the domain wall supports two copropa-gating edge channels that are sourced entirely from different electrical contacts when tunneling between channels is negligible. We i dentify the local chemical potential difference between valleys on the domain wallwith ℏ_ϕvia the Josephson-like voltage-frequency rela- tionship [52] ℏ_ϕ¼δμ: ð12Þ This fundamental relationship allows topological edge states to electrically control the properties of OCIs. Anticipating that substantial equilibration occurs in thehot-spot regions [53–55]indicated in Fig. 2where the valley edge state channel s meet near the sample boun- dary so that momentum is not conserved, we setδμ¼γeV,w h e r e γ<1is a fractional equilibration parameter that depends on the microscopic details of the hot spot. In particular, γshould have very different values for a smooth (gate-defined) and sharp (physical) edge. Quasistatic wall. —Equation (10) has a quasistatic solution with ϕandXindependent of y, andXindependent of time X eq¼2MB−A−1 MγeV k: ð13Þ Equation (13) has the following thermodynamic interpre- tation. The chemical potential is the energy to add an electron to the system. In an ordinary insulator, chemical potentials within the gap are undefined because the systemis incompressible; no states are available to add electrons within the gap. In a Chern insulator, electrons can be added at energies within the gap but only at an edge or a domainwall and only by expanding the area of the system so that it holds one more electron. When a domain wall moves, it adds electrons to one Chern insulator and removes it fromthe other. Equation (13) places the domain wall at the position where the energy change for moving a domain wall by A M=W, adding an electron to one domain and removing it from the other, is the chemical potentialdifference ℏ_ϕ. Reversal occurs at the depinning threshold X eq¼Xmax. According to Eq. (13), the slope of the Xeq¼Xmaxline in theðδμ;BÞparameter space is dδμ dB¼2MA M: ð14Þ To relate reversal to the transport current, we note that, since two hot spots have been traversed, the difference inlocal chemical potentials between top left and top right of the Hall bar in Fig. 2isγ 2eV. It follows that the net current flowing from source to drain is I¼Ve2 h1−γ2 2; ð15Þ and therefore, that reversal occurs along a line in control parameter space with slope dI dB¼MA Me h1−γ2 γ: ð16Þ This is the central result of the Letter. Since MA M∼μBin graphene multilayer OCIs [8] and eμB=ℏ∼1.4× 10−8A=T is 2 orders of magnitude smaller than the experimental result reported in Ref. [6], this mechanism can apply to the samples studied experimentally only if γ∼10−2, i.e., the edge channels are substantially equilibrated in the hot spot regions. This property is, in fact, consistent with reported observations [6].O u r theory of reversal can be tested quantitatively byPHYSICAL REVIEW LETTERS 126, 056801 (2021) 056801-4measuring [56–58]the longitudinal resistance along the upper edge of the Hall bar in Fig. 2to determine a value forγ R¼V2−V4 I¼2h e2γ2 1−γ2: ð17Þ Discussions. —When edge states arrive at a hot spot (cf. Fig. 2) with a valley (momentum) flux perpendicular to the domain wall, they can exert a force on the domain wall. If so, the wall will bend with a vertical profile satisfyingEqs. (10)and(11) [51] . Illustrative wall profiles are shown in Fig. 2. Observation of such domain wall bending would support our proposed reversal mechanism. In Ref. [51], we perform a self-consistent mean field calculation to simulate domain wall magnetization reversalin a quantum Hall valley ferromagnet. We are able to numerically verify Eq. (13) forB¼0. Such analysis lends credence to our proposed magnetization reversal mecha-nism in OCI. Two interesting mechanisms for current reversal of orbital magnetization have recently been proposed in Refs. [35] and[36]. Their theories appeal to finite dis- sipation in the bulk ( σ xx≠0) and do not apply in the quantum anomalous Hall effect regime considered here.The theoretical analysis in Ref. [6]identifies an I 3con- tribution (where Iis the current) to the edge state free energy of conduction edge states and associated reversalwith it becoming comparable to bulk magnetostatic energy. This reversal mechanism does not rely on wall dynamics. Our theory provides an alternative current-reversal mecha-nism based on the depinning of valley domain walls via topological edge states and it is most relevant in devices with well defined orbital Chern insulators that show thequantum anomalous Hall effect, like those imaged recently in Ref. [9]. 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PhysRevB.82.100415.pdf	Experimental evidence for an unidirectional Gilbert damping parameter Chantal Le Graët,1David Spenato,1Souren P. Pogossian,1David T. Dekadjevi,1,2and Jamal Ben Youssef1,* 1Laboratoire de Magnétisme de Bretagne, CNRS–Université de Bretagne Occidentale, 6 Avenue le Gorgeu, 29285 Brest Cedex, France 2Department of Physics, University of Johannesburg, P .O. Box 524, Auckland Park, Johannesburg 2006, South Africa /H20849Received 26 August 2010; published 27 September 2010; publisher error corrected 4 November 2010 /H20850 In magnetization dynamics, the relaxation is driven by the damping. Here, we demonstrate that damping in exchange-coupled systems is not only anisotropic but also unidirectional evidenced by an asymmetry in thedamping by inversion of the magnetic ﬁeld polarity. Our study reveals that this asymmetry in the damping isenhanced by the increase in the exchange bias ﬁeld. Finally, we introduce a modiﬁed relaxation term in theequation of motion. DOI: 10.1103/PhysRevB.82.100415 PACS number /H20849s/H20850: 75.70. /H11002i, 76.50. /H11001g, 75.50.Ee Among the huge variety of scientiﬁc ﬁelds involving dy- namical processes, spin dynamics is one of the most popularbecause of its key role in high volume information-storage orprocessing devices based on metallic, insulators, or semicon-ductors materials. 1In magnetic storage devices, the key point is the speed at which magnetic data-storage elements can bemanipulated with either a magnetic ﬁeld, a spin-polarizedelectric current 2or with an electric ﬁeld.3In these research ﬁelds, commonly accepted theoretical approaches are basedon a well-known phenomenological model applied to mag-netic ﬁeld-induced precession motion of the magnetization 4 or current-induced motion.5In this vast amount of research, the relaxation mechanism is introduced through the use ofthe so-called damping parameter, /H9251. Thus, it governs the re- laxation of the magnetization toward equilibrium. Despitethe general and widespread use of these equations of motion,the physics concerning the phenomenological damping pa-rameter is a matter of controversy. Indeed, the experimentalresults are often interpreted in terms of a scalar isotropic damping parameter 6whereas some authors claim that it should be of a more complicated form that would rely on adamping matrix. This matrix description would then imple-ment the breathing Fermi-surface model 7or crystalline sym- metry considerations.8Then, the damping scalar /H9251should be replaced by an anisotropic term that depends on the orienta-tion of the magnetization. Up to now, the experimental stud-ies that demonstrate the anisotropy of the damping, focus onthe role of the magnetocristalline anisotropy 9and have not focused on exchange-coupled systems despite the fact thatthey seem to be promising for high-frequency applications. 10 In this Rapid Communication, experimental evidence for anunidirectional damping parameter is provided for exchange-coupled ferromagnetic /H20849F/H20850/antiferromagnetic /H20849AF/H20850bilayers. The exchange coupling induces a strong anisotropic relax-ation process which depends on the direction of magnetiza-tion. A frequency-dependent study of the relaxation demon-strates the anisotropic nature of the intrinsic part of therelaxation, i.e., of the damping . The anisotropy of the damp- ing parameter is found to be unidirectional and directly re-lated to the exchange ﬁeld amplitude. Finally, a modiﬁedrelaxation term in the equation of motion is implemented toreproduce these experimental evidences. The investigated NiO /H20849t AF/H20850/Py/H2084920 nm /H20850/H20849where Py denotes Ni81Fe19/H20850bilayers have been grown on Si /H20849100 /H20850substrate by conventional rf diode sputtering, under a static magnetic ﬁeld/H20849HD/H20850of 300 Oe to induce an uniaxial anisotropy in the Py layer. The base pressure prior to the ﬁlm deposition wastypically 10 −7mbar. The NiO thicknesses were tAF=0, 20, 30, 50, 67.5, and 75 nm. A shifted hysteresis loop, at room temperature, along the depositing ﬁeld axis, is observed fort AFhigher than 20 nm. In order to precisely determine the relaxation parameters, the frequency dependence of the fer-romagnetic resonance /H20849FMR /H20850spectra was investigated within a range of microwave frequencies between 6 and 12.5 GHzusing a wideband resonance spectrometer with a nonresonantmicrostrip line. 11The FMR is measured via the derivative of the microwave power absorption /H20849dP /dH/H20850using a small rf exciting ﬁeld. Resonance spectra were recorded with the ap-plied static magnetic ﬁeld oriented in plane at an angle /H9272 relative to the depositing ﬁeld. Moreover our experimentaldevice allows the use of positive and negative polarities ofﬁeld sweeps, /H20849H +,H−/H20850. Static magnetic properties were mea- sured at room temperature by vibrating sample magnetome-ter/H20849VSM /H20850. Figure 1shows the FMR absorption derivative and ab- sorption proﬁles for the Py thin ﬁlm and the NiO /H2084967.5 nm/H20850/Py bilayer measured at 9 and 5 GHz. The presence of FIG. 1. /H20849Color online /H20850An example of /H20849a/H20850FMR absorption de- rivative spectrum and /H20849b/H20850integrated spectrum for a Si /H20849100 /H20850/Py /H2084920nm /H20850ﬁlm /H20851green /H20849light /H20850line /H20852a n daS i /H20849100 /H20850/NiO /H2084967.5 nm /H20850/Py /H2084920 nm/H20850bilayer /H20851blue /H20849dark /H20850line /H20852. Measurements are performed with positive /H20849H+/H20850and negative /H20849H−/H20850electromagnet polarity.PHYSICAL REVIEW B 82, 100415 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 1098-0121/2010/82 /H2084910/H20850/100415 /H208494/H20850 ©2010 The American Physical Society 100415-1the unidirectional anisotropy induces displacement of the resonance ﬁeld /H20849Hres/H20850and an enhancement of the linewidth /H20849/H9004Hpp/H20850, as expected in exchange-coupled systems.12What is more unusual is that the width and intensity of the bilayerresonance spectra depend on the ﬁeld polarity, i.e., there is astrong asymmetry of the resonance line with respect to H =0 Oe. Such an asymmetry with respect to the ﬁeld sweep-ing polarity has previously been reported but not discussed. 13 To achieve an understanding of this pronounced asymmetry,we have systematically measured the FMR spectra for differ-ent applied ﬁeld directions /H20849 /H9272/H20850. For each angular position within the plane and for positive and negative ﬁeld polarities,the following fundamental FMR quantities were extracted:H res,/H9004Hppand the maximum of the absorption intensity, I, as shown in /H20851Figs. 2/H20849a/H20850–2/H20849c/H20850/H20852. The Hresangular dependence reﬂects the symmetry of our exchange bias system. For eachpolarity, the shapes are symmetric with respect to the ex-change bias ﬁeld direction /H20849i.e., /H9272=180° /H20850with two minima and two maxima, as previously expected in unidirectionalsystems 14/H20851Fig. 2/H20849a/H20850/H20852. They present a similar angular depen- dence, i.e., symmetry axes along the exchange bias ﬁeld di-rection. In order to understand Iand/H9004H ppproperties, oneshould consider the origin of the absorbed microwave power in a RFM experiment. For a given azimuthal angle, the ab-sorbed microwave power is proportional to the imaginarypart of the susceptibility tensor component along the exciting ﬁeld: P/H11011 /H9273h/H11033.15We quantitatively examine the asymmetry with respect to the ﬁeld polarity using an asymmetry param-eter /H20849AS/H20850deﬁned as follow: AS =I +−I− I++I−=/H9273h max/H11033+−/H9273h max/H11033− /H9273h max/H11033++/H9273hm a x/H11033−, /H208491/H20850 where I+andI−are the microwave power absorption maxi- mal values at resonance ﬁelds for positive and negative val-ues of resonance ﬁelds as shown in Fig. 1. For a single Py ﬁlm and for all azimuthal angles, the FMR spectra are sym-metric with respect to the applied ﬁeld polarity, i.e., I +=I− and AS=0 /H20849not shown in Fig. 2/H20850. In the presence of the exchange bias, the angular dependence of AS presents acomplex behavior as shown in Fig. 2/H20849d/H20850. In the following, we study this behavior considering the magnetic susceptibilitywhich may be calculated with Landau-Lifshitz-Gilbert/H20849LLG /H20850equation /H20851Eq. /H208495.3/H20850in Ref. 6/H20852where the effective ﬁeld is deﬁned as H eff=/H11509F//H11509M. The free-energy functional F/H20849M/H20850 consists, in our case, of the Zeeman energy, the unidirec-tional anisotropy of the AF layer, the demagnetizing ﬁeldenergy, the saturation magnetization, and the uniaxial aniso-tropy of the F layer. 14In addition one can assume that the precession around the Heffis of small amplitude so the LLG equation may be linearized.16This allows us to calculate the theoretical FMR spectra and thus the theoretical AS param-eter. In a ﬁrst step we have ﬁtted the resonance ﬁeld angulardependence as shown in Fig. 2/H20849a/H20850, it allows us to determine the values of the exchange ﬁeld /H20849H e/H20850, F anisotropy ﬁeld /H20849HK/H20850, and the saturation magnetization /H20849MS/H20850/H20849i.e., He =30 Oe, HK=34 Oe, MS=10900 G, and the gyromagnetic ratio/H9253=1.843 /H11003107s−1Oe−1/H20850. The value of Heobtained from the ﬁt is about 30% lower than the one obtained byVSM for all our exchange biased samples as previously ob-served in such systems. 14In second step, we have simulated the angular dependence of AS directly by supposing that theGilbert damping coefﬁcient /H9251has a constant value and does not depend on the angle between the magnetization and theexciting ﬁeld as generally admitted in azimuthalmeasurements. 17The simulated curves reproduce the general shape of AS /H20851Fig. 2/H20849e/H20850/H20852but not the experimental features along the exchange bias ﬁeld axis, i.e., for /H9272=0° and /H9272 =180°. Following the linearized LLG model including anisotropic damping, the asymmetry should not be presentalong this axis. Therefore, we have considered a nonisotropicdamping for reproducing the AS in the following manner. According to Steiauf and Fähnle, 7in the presence of spin- orbit interaction the damping parameter is described by amatrix and will depend on the direction of the magnetization.Thus, the spin-orbit coupling makes the spin degree of free-dom respond to its orbital environment. The authors showthat there are at most two eigenvalues for the damping ma-trix: /H9251/H20849M/H20850, which depends rather sensitively on the orienta- tion of Min the crystal. In our case the exchange bias breaks the symmetry of uniaxial anisotropy with respect to the an-FIG. 2. /H20849Color online /H20850Azimuthal angular dependencies for posi- tive /H20849red open circles /H20850and negative /H20849black open squares /H20850applied ﬁeld polarity of /H20849a/H20850the experimental and simulated /H20849solid line /H20850FMR resonance ﬁeld /H20849b/H20850the measured FMR linewidth and /H20849c/H20850the mea- sured magnitude of the resonance curve with the integer form, at9.5 GHz for sample Si /H20849100 /H20850/NiO /H2084967.5 nm /H20850/Py. /H20849d/H20850Experimental measurements of the asymmetry ratio, AS, deﬁned in Eq. /H208491/H20850for a Si/H20849100 /H20850/NiO /H2084967.5 nm /H20850/Py bilayer /H20849open triangles /H20850. Solid line simu- lation of AS with /H9264=0.18 /H20851Eq. /H208492/H20850/H20852 /H20849e/H20850Simulation of the asymmetry ration AS for /H9264=0 /H20849dash line /H20850and/H9264/HS110050/H20849solid line /H20850.LE GRAËT et al. PHYSICAL REVIEW B 82, 100415 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 100415-2isotropy axis, and therefore our measurements show different damping in directions of /H9272=0° and /H9272=180° thus supporting the theoretical prediction of the authors mentioned above. Toreproduce the experimental features of the resonance absorp-tion, we modify the LLG equation by assuming a magneti-zation direction-dependent damping parameter that lies be-tween two eigenvalues of damping parameter given by /H9251+ =/H9251/H208491−/H9264/H20850and/H9251−=/H9251/H208491+/H9264/H20850. Thus, dM dt=−/H9253/H20849M/H11003Heff/H20850+1 MM/H11003/H9251/H11569dM dt, /H208492/H20850 where /H9251/H11569=/H9251/H208751−/H9264He.M HeM/H20876. /H208493/H20850 The ﬁrst term on the right-hand side of Eq. /H208492/H20850describes the precession of Maround Heffand the second term represents the damping torque and is the key parameter that governs therelaxation toward equilibrium. The term /H9251/H11569represents the anisotropic damping that depends on the direction of themagnetization retained /H20849withheld /H20850by AF spins via exchange coupling /H20849H e/H20850. In Eq. /H208493/H20850/H9251is the mean Gilbert phenomeno- logical damping parameter and the term /H9264is the difference between the measured damping parameters along the unidi-rectional anisotropy for the positive /H20849 /H9251+/H20850and negative /H20849/H9251−/H20850 polarities /H20851i.e.,/H9264=/H20849/H9251−−/H9251+/H20850//H20849/H9251−+/H9251+/H20850/H20852. The best ﬁt to the ex- perimental data was obtained for /H9264=0.18. The good agree- ment between theoretical and experimental values /H20851Fig.2/H20849d/H20850/H20852 conﬁrms our assumption, given by Eq. /H208493/H20850, reﬂecting the damping parameter dependence on the magnetization direc-tion. As relaxation mechanisms may include intrinsic and ex- trinsic contributions, 18one needs to separate these effects to show unambiguously that the origin of the AS is intrinsic. Inthe parallel FMR conﬁguration, with the magnetization par-allel to the applied ﬁeld, the time derivative /H11509M //H11509tGilbert term in the equation of motion produces a FMR linewidthlinear with the microwave frequency f. However, in many magnetic systems, while a linear behavior is observed, thelinewidth fails to extrapolate to zero with vanishing fre-quency. This zero-frequency contribution /H9004H 0to the line-width is related to extrinsic mechanism and reﬂects the effect of inhomogeneity on the linewidth. Thus, frequency-dependent studies provide intrinsic and extrinsic contribu-tions to the relaxation. The ﬁeld-swept linewidth, in a givendirection, may be written as 8 /H9004Hpp=/H9004H0+2 /H208813/H9251 /H92532/H9266f, /H208494/H20850 where /H9004H0is the inhomogeneous broadening and fis the microwave frequency. The second term is the Gilbert contri-bution that represents intrinsic contribution. In our study, itshould be noted the damping parameter /H9251should be replaced by/H9251/H11569in Eq. /H208494/H20850. Figure 3shows the dependence of /H9004Hppon the micro- wave frequency for both angle /H9272values 0° and 90° with positive and negative polarities. The FMR linewidth is lin-early dependent on the microwave frequency for both orien-tations, as previously observed in such systems 19and as ex- pected from Eq. /H208494/H20850. The inversion of the ﬁeld polarity along the exchange ﬁeld axis /H20849/H9272=0° /H20850induces a change in the slope of the FMR linewidth. It conﬁrms the dependence of theintrinsic contribution toward the ﬁeld polarity as well as theintrinsic origin of the AS. Furthermore, the intercept at theorigin of the FMR linewidth slopes, i.e., /H9004H 0give the same value by inversion of the polarity for /H9272=0°. This shows un- ambiguously that the polarity inversion of the applied mag-netic ﬁeld only modiﬁes the intrinsic contribution to the line-width. It should also be noted that the /H9004H ppfrequency dependence does not depend on the polarity along /H9272=90°. To illuminate the inﬂuence of the exchange coupling on the anisotropic nature of the damping, Py ﬁlms were depos-FIG. 3. Evolution of the peak-to-peak linewidth /H9004Hppas a func- tion of frequency for sample NiO /H2084967.5 nm /H20850/Py with the static ﬁeld is applied with angle /H9272=90° /H20849circles /H20850and 0° /H20849squares /H20850with positive /H20849full symbols /H20850and negative polarity /H20849open symbols /H20850. FIG. 4. /H20849Color online /H20850Exchange bias dependence, in Si/H20849100 /H20850/NiO /H20849tAF/H20850/Py/H2084920 nm /H20850with tAF=20, 30, 50, 67.5, and 75 nm, of /H20849a/H20850the asymmetry AS of the susceptibility intensity /H20849b/H20850the zero-frequency peak-to-peak linewidth /H9004H0with negative /H20849full black squares /H20850and positive /H20849red open circles /H20850polarity /H20849c/H20850the pa- rameter /H9264/H20851see text and Eq. /H208493/H20850/H20852. All the parameters were measured along the F easy axis /H20849/H9272=0° /H20850EXPERIMENTAL EVIDENCE FOR AN UNIDIRECTIONAL … PHYSICAL REVIEW B 82, 100415 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 100415-3ited on NiO underlayers with different thicknesses resulting in a large variety of exchange bias ﬁeld values from 0 to 60Oe. Figure 4/H20849a/H20850shows the inﬂuence of the exchange ﬁeld value on the asymmetry of the linewidth measured along theF easy axis. It shows clearly that the condition for the ap-pearance of AS requires a NiO ﬁlm thicker than a criticalthickness, above which H e/HS110050.20The asymmetry is then pro- portional to He. To conﬁrm the intrinsic origin of AS, intrin- sic and extrinsic contributions were obtained for eachsamples. Figure 4/H20849b/H20850shows the dependence of /H9004H 0on the He and for both polarities. The presence of a thin NiO layer increases /H9004H0relatively to the value obtained for the Py single layer. However, /H9004H0shows no signiﬁcant dependence onHeand does not depend on ﬁeld polarity. Therefore, the extrinsic contribution does not contribute to the AS mecha-nism. The inﬂuence of H eon the intrinsic damping aniso- tropy is depicted in Fig. 4/H20849c/H20850. Under the critical thickness, the value of /H9251/H20849i.e.,/H9264=0/H20850remains unchanged by inversion of thepolarity. However, /H9264increases with He, for a greater NiO thickness. It shows unambiguously that the anisotropy in theintrinsic Gilbert damping originates from the pinned AFspins and is proportional to H e. In conclusion, we report on experimental evidence for an anisotropic unidirectional intrinsic Gilbert damping term inexchange-coupled systems. A modiﬁed damping term in theLLG equation of motion is introduced to reproduce this an- isotropy. It depends on the direction between the magnetiza-tion and the unidirectional anisotropy. We also show that themagnitude of the damping anisotropy is directly related totheH emagnitude. This anisotropy should be considered in the fundamental understanding of the dynamic of exchange-coupled systems and would be great importance in low-energy storage applications or high-frequencies applications. This work was partly supported by Region Bretagne ARED under Grant No. 3596. *jamal.ben-youssef@univ-brest.fr 1D. D. Awschalom and M. E. Flatté, Nat. Phys. 3, 153 /H208492007 /H20850. 2J. A. Katine, F. J. Albert, R. A. 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Rezende, Spin Dynamics in Conﬁned Magnetic Structures II /H20849Springer, New York, 2003 /H20850, V ol. 87, p. 27; C. Vittoria, S. Yoon, and A. Widom, Phys. Rev. B 81, 014412 /H208492010 /H20850. 9W. Platow, A. Anisimov, G. Dunifer, M. Farle, and K. Baber- schke, Phys. Rev. B 58, 5611 /H208491998 /H20850; C. Vittoria, R. Barker, and A. Yelon, Phys. Rev. Lett. 19, 792 /H208491967 /H20850. 10J. McCord, R. Mattheis, and D. Elefant, Phys. Rev. B 70, 094420 /H208492004 /H20850; D. Spenato, S. P. Pogossian, D. T. Dekadjevi, and J. Ben Youssef, J. Phys. D 40, 3306 /H208492007 /H20850.11J. Ben Youssef, N. Vukadinovic, D. Billet, and M. Labrune, Phys. Rev. B 69, 174402 /H208492004 /H20850. 12W. Stoecklein, S. S. P Parkin, and J. C Scott, Phys. Rev. B 38, 6847 /H208491988 /H20850. 13L. Wee, R. L. Stamps, L. Malkinski, and Z. Celinski, Phys. Rev. B69, 134426 /H208492004 /H20850; A. Punnoose, E. Morales, Y . Wang, D. Lederman, and M. Seehra, J. Appl. Phys. 93, 771 /H208492003 /H20850. 14R. D. McMichael, M. D. Stiles, P. J. Chen, and W. F. Egelhoff, Jr.,Phys. Rev. B 58, 8605 /H208491998 /H20850; C. Le Graët, D. Spenato, S. Pogossian, D. Dekadjevi, and J. Ben Youssef, Appl. Phys. Lett. 94, 262502 /H208492009 /H20850. 15A. Jander, J. Moreland, and P. Kabos, Appl. Phys. Lett. 78, 2348 /H208492001 /H20850. 16A. G. Gurevich and G. Melkov, Magnetization Oscillations and Waves /H20849CRC Press, Boca Raton, 1996 /H20850, Chap 2. 17K. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock, M. Farle, U. von Hörsten, H. Wende, W. Keune, J. Rocker, S. S. Kalarickal,K. Lenz, W. Kuch, K. Baberschke, and Z. Frait, Phys. Rev. B 76, 104416 /H208492007 /H20850. 18R. Urban, B. Heinrich, G. Woltersdorf, K. Ajdari, K. Myrtle, J. F. Cochran, and E. Rozenberg, Phys. Rev. B 65, 020402 /H208492001 /H20850. 19D. Twisselmann and R. McMichael, J. Appl. Phys. 93, 6903 /H208492003 /H20850. 20D. T. Dekadjevi, A. Suvorova, S. Pogossian, D. Spenato, and J. Ben Youssef, Phys. Rev. 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PhysRevB.102.174428.pdf	PHYSICAL REVIEW B 102, 174428 (2020) From chaotic spin dynamics to noncollinear spin textures in YIG nanoﬁlms by spin-current injection Henning Ulrichs* I. Physical Institute, Georg-August University of Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany (Received 10 July 2020; revised 21 September 2020; accepted 3 November 2020; published 16 November 2020) In this paper I report on a numerical investigation of nonlinear spin dynamics in a magnetic thin ﬁlm made of yttrium iron garnet (YIG). This ﬁlm is exposed to a small in-plane oriented magnetic ﬁeld and strong spincurrents. The rich variety of ﬁndings encompasses dynamic regimes hosting localized, nonpropagating solitons,as well as a turbulent chaotic regime, which condenses into a quasistatic phase featuring a noncollinear spintexture. Eventually, at the highest spin current, a homogeneously switched state is established. DOI: 10.1103/PhysRevB.102.174428 I. INTRODUCTION Recent advances [ 1–5] in the art of thin-ﬁlm growth al- low us nowadays to prepare yttrium iron garnet (YIG) ﬁlmswith nanometer thickness. These ﬁlms in particular featuremagnetic losses comparable to or lower than metallic ferro-magnets like the widely used permalloy or amorphous CoFeBalloys. Such YIG nanoﬁlms are of great interest to implementfunctionalities based on wave interference in magnon spin-tronic applications [ 6–9]. Being electrically insulating, YIG allows us to completely disentangle spin and charge currentrelated physics. This makes this material in particular attrac-tive for studies on spin-related transport phenomena [ 10–16]. In addition to its appearance in these topical research ﬁelds,YIG has been, since its discovery, a great medium to studyhighly nonlinear spin dynamics. Turbulence [ 17], parametric instabilities [ 18–21], and even Bose-Einstein condensation (BEC) [ 21–23] have been studied in YIG for quite a few decades. But so far many of these intriguing phenomenacould be realized only on rather macroscopic scales, render-ing them less attractive for practical applications. To addressthese effects, YIG samples are usually exposed to strong,monochromatic microwave radiation, whose magnetic partcan directly drive magnetization dynamics. On the other hand, if single-frequency excitation is not a prerequisite, spin currents can be considered a convenientmethod for realizing broadband excitation. Spin currents canbe generated by a charge current when they are led througha spin-Hall material [ 24–27]. These are, for example, the very common heavy metals platinum and tungsten. In a sim-ple picture, one can relate the appearance of spin currentsin patterned ﬁlms consisting of these materials to spin-orbital coupling (SOC). If a lateral charge current carried byap r i o r i non-spin-polarized electrons experiences scattering with SOC, this scattering gives rise to a vertical spin imbal-ance. This imbalance builds up between the top and bottomsurfaces of the conducting ﬁlm. When deposited on top ofa YIG nanoﬁlm, the spin accumulation at the interface caninteract with the magnetic moments in the YIG. This in *hulrich@gwdg.departicular can result in an effective reduction of magneticlosses of magnons. A critical current can, in this context, bedeﬁned as the magnitude at which the mode with the lowestlosses reaches the point of full damping compensation. Thespin-current-induced instability of a particular mode is theessential mechanism behind spin-Hall oscillators, which havebeen realized with metallic permalloy [ 28,29], as well as with insulating YIG [ 30,31], as active magnetic media. The ﬁndings presented in the following in particular shed light on the question of what happens if one exceeds theinstability threshold in a situation when the injection of spincurrents is conﬁned to one lateral dimension or even not atall. Thus, the investigations presented here complement recentﬁndings [ 12,14,15] about magnon transport phenomena in YIG nanoﬁlms and theoretical investigations that predict BECin such an experimental situation [ 32,33]. The micromag- netic approach applied here provides a view inside the ﬁlm,circumventing spatial and temporal resolution limitations en-countered in common experimental approaches like Brillouinlight scattering [ 34], which is used to image magnetization dynamics. When conﬁning the spin current, I have foundﬁrst the nucleation of so-called spin-wave bullets, whose den-sity quickly increases, leading to a chaotic regime. At evenlarger spin current a novel quasistatic phase condenses fromthese turbulent ﬂuctuations. This phase is characterized by astripelike, noncollinear magnetization texture. At higher cur-rent, this texture gradually disappears, and a fully switched,homogeneous magnetic state is established. The rest of this paper is organized as follows. First, I provide details about the numerical method. In particular, Iexplain how I take temperature-related effects into account.Then, I present results obtained for the case of conﬁned spin-current injection. After that, I present the ﬁndings for the caseof unrestricted injection. In the ﬁnal discussion, I explain themagnitude of the numerically found threshold current density,and I explain why turbulence arises. Finally, I present a tenta-tive interpretation for the emerging quasistatic texture. II. EXPERIMENTAL DETAILS To simulate the spin-current injection into a YIG nanoﬁlm with a thickness of tYIG=20 nm, the micromagnetic 2469-9950/2020/102(17)/174428(8) 174428-1 ©2020 American Physical SocietyHENNING ULRICHS PHYSICAL REVIEW B 102, 174428 (2020) simulation code MUMAX 3[35] was used. In this ﬁnite- difference numerical code, the magnetic ﬁlm is divided intorectangular cells that are 5 ×5×20 nm 3. Each cell hosts a magnetic moment with a ﬁxed vectorial length, interacting bymicromagnetic exchange and dipolar ﬁelds with its surround-ings. A total lateral area of 2560 ×2560 nm 2was considered. For the YIG ﬁlm at 285 K, a saturation magnetization ofM 0=0.11 MA /m, an exchange constant of A=3.7p J/m2,a gyromagnetic ratio of γ=1.7588×10111/Ts, and a Gilbert damping constant of α=0.001 were assumed. The spin torque generated by the spin-Hall effect in a tPt=3.5 nm thick Pt layer was taken into account by adding the Slonczewskitorque term [ 36,37] to the equation of motion of the mag- netization. As a conversion factor between charge and spincurrents, a spin-Hall angle of θ SHE=0.11 and an interface transparency of about τi=0.47 were employed. These mate- rial parameters resemble typical experimental values, as usedin [15]. The MUMAX 3 script ﬁle in the Supplemental Mate- rial [ 38] provides all information to reproduce the simulations. Note that I did not consider the Oersted ﬁeld created by the charge current. In the Supplemental Material [ 38], I show that, due to its small magnitude, the Oersted ﬁeld does not inﬂuencethe dynamics. In contrast, the inﬂuence of sample tempera-ture on the magnetization and exchange, enhanced by Jouleheating, is taken into account. For simplicity, I have assumedhomogeneous heating. Laterally inhomogeneous temperatureproﬁles do not affect the dynamics, as discussed in the Sup-plemental Material [ 38]. In the simulation, a static reduction of the magnetization and temperature-driven ﬂuctuations im-plemented by means of a ﬂuctuating thermal ﬁeld [ 35]a r e taken into account. The method is described in what follows.I assume the temperature dependence of the magnetizationshown in Fig. 1(a), which was published in [ 39]. Note that these data are well described by a phenomenological powerlaw with an exponent of 0.511(5) (red curve). Figure 1(b) shows experimental temperature calibration data from [ 15], which extrapolate quadratically (red curve) to T C=560 K at about j=8×1011A/m2. Combining both data sets and ﬁtting curves, I have constructed the current dependence ofthe magnetization shown in Fig. 1(c). This curve is taken for rescaling the effective magnetization at a given current andtemperature in the simulation. This means that in practicethe length of the magnetization vector in each simulated cellis adjusted accordingly. Note that in the simulation long-range, low-frequency ﬂuctuations are included, stochasticallyexcited by the thermal ﬁeld. Such ﬂuctuations further reducethe effective magnetization. Across the whole temperaturerange (285 to 560 K) valid here, I have found it necessary toincrease the magnetization by about 1% to take the additionalreduction of the effective magnetization by such ﬂuctuationsinto account. For the exchange constant, I have assumed theclassical micromagnetic expectation A(T)∝M 0(T)2[40,41]. The resulting current dependence is shown in Fig. 1(d). Figure 2shows the experimental sample designs consid- ered in this work. In Fig. 2(a), the case of a spatially conﬁned spin-current injection is depicted, realized by patterning thecharge current carrying Pt layer to a stripe with a width ofw=500 nm. In the simulation, the Pt stripe is considered only implicitly by enabling the Slonczewski torque only in theinjection region beneath the conductor. Absorbing boundary FIG. 1. Inﬂuence of Joule heating on static and dynamic mag- netization. (a) Temperature dependence of magnetization accordingto [39]; the solid line is a power law ﬁt. The dashed vertical line marks the Curie temperature T C. (b) Current density dependence of the temperature underneath the Pt stripe according to [ 15]; the solid line is a quadratic ﬁt. The dashed horizontal line marks the Curie temperature TC. (c) Derived current density dependence of magnetization M0. (d) Derived current density dependence of the exchange constant A. conditions were applied to the edges parallel to the wire, and periodic boundary conditions (PBCs) were applied tothe perpendicular edges. Therefore, an inﬁnitely long wirewas simulated. The external ﬁeld had a magnitude of μ 0H= 50 mT, and it was oriented in the ﬁlm plane, perpendicularto the wire. The detection stripe included in Fig. 2(a) is de- picted to graphically deﬁne the region underneath the YIGﬁlm. This region is used for probing dynamics outside theactively excited region. In Fig. 2(g) the case of homogeneous spin-current injection is depicted. Here, the PBCs are appliedto all edges. III. RESULTS Figure 2depicts snapshots of the dynamics obtained for the two cases of conﬁned and unrestricted spin-current injections,at current densities above and below a certain critical thresh-oldj th. Note that I quantify this threshold later from the data and use it to deﬁne the overcriticality /Gamma1by /Gamma1=j jth−1. (1) A. Conﬁned spin-current injection Let us begin the inspection of the results by analyz- ing the case of spin-current injection conﬁned to a stripe. 174428-2FROM CHAOTIC SPIN DYNAMICS TO NONCOLLINEAR … PHYSICAL REVIEW B 102, 174428 (2020) FIG. 2. Sketch of the experimental situations and snapshots of simulated magnetization dynamics in terms of the normalized magnetic vector ﬁeld m(x,y). (a) Sketch for conﬁned spin-current generation and injection. (g) Sketch for homogeneous spin-current generation and injection. Edges marked by PBC and ABC refer to periodic and absorbing boundary conditions. Note that both sketches include color-coded maps of m, referring to a current density below the onset of bullet formation, at /Gamma1=− 0.27, as indicated. (b)–(f) Snapshots of mfor increasing /Gamma1as indicated for the case of conﬁned spin-current injection. Dashed lines mark the boundaries of the Pt stripe. (h)–(l) Analogous snapshots ofmfor unconﬁned spin-current injection. Figures 2(a) to2(f) show snapshots of the magnetization after dynamic equilibrium has been established. For a cur-rent density below a certain threshold j<j th[/Gamma1< 0, see Fig.2(a)], no dynamic response can be seen. When the current is increased to j>jth(/Gamma1> 0), this situation changes. Now, the simulation features localized hot spots, where the ﬁlm isstrongly excited [see Fig. 2(b)]. The normalized magnetization component m i y=/angbracketleftMy/angbracketrightinjection M0averaged across the injection area provides quantitative access to these dynamics. A representative time series obtained at/Gamma1=7 is shown in Fig. 3(b). The Fourier transform power spectrum shown in Fig. 3(c) is dominated by a strong peak at frequency f b=2.4 GHz. Note that this value is lower than the bottom of the linear spin-wave spectrum at about f0= 2.7 GHz [dark blue dashed line in Fig. 3(c)]. Both spectral and spatial features are typical for so-called spin-wave bul-let modes [ 42]. Such a bullet is a nonlinear, nonpropagating solitonic solution of the gyromagnetic equation of motion. Onthe other hand, the dynamics in the detection area captured by m d y=/angbracketleftMy/angbracketrightdetection M0[see Figs. 3(b) and3(c)] show oscillations at a frequency close to the frequency of ferromagnetic resonance(FMR): ω 0=2πf0=/radicalbig ωH[ωH+ωM(j)], (2) where ωH=γμ 0HandωM(j)=γμ 0M0(j)[43]. When in- creasing the current density, the number of simultaneouslyexisting bullets in the injection area increases, as Fig. 2(c) illustrates. Simultaneously, their frequency f bdecreases, as shown in Fig. 3(c). This downshift in frequency is well known for bullets in in-plane magnetized magnetic ﬁlms. In the de-tection area, the frequency f 0of the dominating FMR mode follows the thermally driven decrease of the magnetizationdue to Joule heating [for a plot of Eq. ( 2), see the green dashed line in Fig. 3(d)]. When reversing the current polarity, the dynamics in the injection and in the detection area areprogressively suppressed and dominated by the FMR mode, asdemonstrated by the good agreement of the spectral maximawith the calculated dependence of the FMR frequency f 0onj shown in Figs. 3(c) and3(d). The emergence of the bullets in the injection area can be characterized by an order parameter /Psi1=1−mi x 2, (3) where mi x=/angbracketleftMx/angbracketrighti M0. The order parameter /Psi1in essence captures how far the magnetization deviates from the equilibrium ori-entation in the absence of a spin current, when M/bardblH. Figure 4 shows a plot of the dependence of /Psi1onj. One can see a quick initial growth, followed by an intermediate slowing down,which then speeds up again to reach values /Psi1> 0.5. Let us take a closer look at the initial growth. For a continuous phasetransition one can expect, according to Landau [ 44], a generic dependence /Psi1=/parenleftbiggj jth−1/parenrightbiggε =/Gamma1ε. (4) Indeed, ﬁtting Eq. ( 4) to the data yields a critical expo- nent of ε=0.72(3) and a threshold current density of jth= 0.17(1)×1011A/m2(see also the inset in Fig. 4). Figure 4 clearly shows that, at around /Gamma1=32, further evolution of the order parameter deviates from Eq. ( 4). Indeed, the or- der parameter soon exceeds /Psi1=0.5, which implies that, on average, the magnetization is aligned antiparallel to theexternal ﬁeld. Before this switching is completely achieved, 174428-3HENNING ULRICHS PHYSICAL REVIEW B 102, 174428 (2020) FIG. 3. Spectral characterization of dynamics in the injection and detection area. (a) Part of the typical transient dynamics in terms ofthe magnetic component m i y(md y), spatially averaged over the injec- tion (detection) area, obtained at /Gamma1=7. (b) Corresponding Fourier power spectra calculated from a 50-ns-long transient. It features adominant peak at frequency f b, marked by the vertical light blue dashed line (close to the frequency of ferromagnetic resonance f0, marked by the vertical dark blue dashed line). (c) and (d) Depen- dency of power spectra in the injection and detection areas on the current density. The dark blue dashed line marks the calculated f0(j). The light blue dashed line serves as a guide to the eye for fb(j). Blue and orange arrows indicate the spectra shown in (b). a quasistatic magnetic texture emerges [see Fig. 2(e)]. The spin-torque-induced magnon emission from the injection areacan be captured by /Sigma1(j)=/angbracketleftM x(js=0)/angbracketright2 d−/angbracketleftMx(j)/angbracketright2 d, (5) where the spatial average across the detection area /angbracketleftMx(js= 0)/angbracketrightdrefers to a simulation conducted at ﬁnite temperature T(j), as caused by Joule heating, but without taking into account the spin current jsﬂowing from the Pt stripe into the YIG ﬁlm. In contrast, /angbracketleftMx(j)/angbracketrightdrefers to a simulation including the action of the spin current. By construction, /Sigma1 is proportional to the number of magnons emitted from theinjection area, which are caused by only the spin injection,without compromising the thermal background. The currentdependence /Sigma1(j) is included in Fig. 4. It displays a quick initial growth, followed by a saturation around /Gamma1=30. There- after,/Sigma1quickly decreases to zero emission. B. Unrestricted spin-current injection In this section, the situation sketched in Fig. 2(g) is an- alyzed, where no spatial restrictions are imposed on thespin-current injection [see Figs. 2(g) to2(l)for typical snap- shots]. Also here, spin-wave bullets appear, although chaos FIG. 4. Dependence of the order parameter /Psi1(blue circles) and of the magnon emission /Sigma1(orange rectangles) on the current density j. The red dashed line is a ﬁt of Eq. ( 3). The inset magniﬁes the behavior close to the threshold current density. The vertical green and blue dashed lines mark the onset of spin-wave bullet formation and the emergence of the quasistatic texture, respectively. The orangedashed line is a guide to the eye. sets in earlier. The motivation for this experiment is to an- alyze and better understand the transition from bullets tothe emergence of the quasistatic stripelike texture. The evo-lution of this transition is elucidated in Fig. 5in terms of two-dimensional (2D) spatial and spatiotemporal fast Fouriertransform (FFT) power maps P FFT(kx,ky) and PFFT(kx,f)ky=0 ofmz. In the left panel of Fig. 5(a) one can see PFFT(kx,ky) of an already chaotic state, obtained at /Gamma1=1. The mag- netization displays no clear structure, as the quite isotropicFourier spectrum demonstrates. The agreement between thecomputed dispersion of plane spin waves [ 45] with the max- ima of the spatiotemporal Fourier spectrum P FFT(kx,f)ky=0 depicted in the right panel shows that the ﬂuctuations here still correspond mainly to linear spin waves. At /Gamma1=6.4 [Fig. 5(b)], short-wavelength ﬂuctuations strongly increase. Second, one can see a signature of the bullets appearingin the spatiotemporal Fourier spectrum. That is, the largestspectral weight appears around k x=0 at frequencies below the computed spin-wave dispersion (dashed line). This devi-ation is even more pronounced at /Gamma1=21 [see Fig. 5(c)]. At /Gamma1=43, the short-wavelength ﬂuctuations are suppressed, and the spectrum displays a peculiar anisotropy, corresponding tothe stripelike magnetic texture shown in Fig. 2(k). The static behavior of this state is reﬂected by the spatiotemporal Fourierspectrum, which shows two maxima at frequency f=0. Now, these maxima cannot be related to linear spin waves at all(dashed white curve). IV . DISCUSSION A. Bullet dynamics In the simulations that consider a spatially restricted spin-current injection, one sees the appearance of localizedmodes above a current density of j th=0.17×1011A/m2. 174428-4FROM CHAOTIC SPIN DYNAMICS TO NONCOLLINEAR … PHYSICAL REVIEW B 102, 174428 (2020) FIG. 5. Spatiotemporal spectral characterization of spin dynam- ics in the case of unconﬁned spin injection. The left panels show 2D spatial FFT power maps PFFT{mz(x,y)}(kx,ky) of snapshots of the magnetization component mz. The right panels show spatiotemporal FFT power maps PFFT(kx,f) along kxforky=0. (a)–(d) refer to speciﬁc overcriticalites /Gamma1, as indicated. This number can be compared with a simple expectation. In the case of YIG nanoﬁlms, the mode with lowest lossesis the FMR mode. Without spin currents, its relaxation ratereads [ 46] ω R=α(ωH+0.5ωM). (6) The spin torque pumps energy into the magnetic oscilla- tions at a rate [ 36,37] β=jγ¯h 2eM 0tYIG/Theta1SHEτi. (7) Exact compensation, that is, ωR=β, leads to a theoret- ical critical current density of 0 .16×1011A/m2in the Pt stripe. Only when it exceeds this value can the magnetizationbecome unstable. Indeed, the observed threshold almost exactly coincides with this theoretical expectation. All prop-erties derived from inspecting the current dependency of thedynamics comply with the interpretation that the unstablemode is a spin-wave bullet [ 42]. B. Turbulence As more and more bullets appear with increasing current, the dynamics quickly becomes chaotic. Note that this chaosis deterministically driven by the spin-current injection. Asa signature of deterministic chaos, I ﬁnd that, in all spectradiscussed in this paper, the phases are random, and they reactsensitively to small perturbations of the initial state. This sen-sitivity is maintained when excluding the thermal ﬂuctuationﬁeld. In the Supplemental Material [ 38], an analysis of spectral properties of this chaotic state is shown. Chaos appears be-cause, with increasing current, for a larger and larger part ofthe spin-wave spectrum, losses are compensated. Therefore,dissipation can occur only when three-magnon or higher-order scattering pushes energy into higher-frequency modes,whose losses are not yet compensated by the injected spincurrent. These nonlinear processes inevitably set in whenthe unstable modes have achieved large enough amplitudes.Such an energy cascade is, indeed, prototypical for turbu-lence [ 47,48]: energy is injected into the low wave number, low-frequency part of the spectrum, and energy is dissi-pated as it reaches the large wave number, high-frequencypart. Furthermore, there is an interesting connection to classical pipe ﬂow experiments. There, so-called puffs appear as pre-cursors to turbulence [ 49,50]. At ﬁrst glance, puffs and bullets seem to have a lot in common, as both appear prior to the onsetof turbulence and both dynamics are nonlinear and localized.Similar to the puffs in pipes, the bullets have a ﬁnite lifetime.How far does the analogy hold? I would like to emphasizethat, in contrast to puffs, the bullets do not move. They remainstationary inside the injection region. Note that this reﬂectsa quite different experimental situation. In pipe ﬂow experi-ments, one induces turbulence locally by placing objects inthe ﬂow or by a nozzle. Here, my focus is on a spatiallyextended injection region for the spin-current injection, givingrise to chaotic dynamics in this region. The data presented inFig. 3show that, outside this region, the magnetic ﬁlms be- have mainly like a normal, thermally excited system. Second,with increasing spin-current injection, turbulence evolves, andthe lifetime of the bullets decreases. At even higher currentdensity, the turbulence disappears again, in favor of a qua-sistatic texture. In contrast, puffs moving downstream have anincreasing lifetime as a function of the Reynolds number. AsI explain in the Supplemental Material [ 38], the latter can be regarded as being effectively controlled by the spin-currentinjection. To further investigate similarities and differences,one could envisage a different sample design, in which the Ptinjection stripe consists of two adjacent sections with largeand small widths and with a metallic ferromagnetic ﬁlm be-low. Then, one can locally induce bullets ( =puffs) below the small-width part ( =reservoir under pressure). In addition, one may be able to push the bullets into the large-width part 174428-5HENNING ULRICHS PHYSICAL REVIEW B 102, 174428 (2020) FIG. 6. Field dependence of quasistatic texture. The colored map in the background shows the ﬁeld and wave number dependence ofEq. ( 8). The characteristic wave numbers k 0(open circles) lie on the isocontour ζω=0(red line). (=pipe) by means of the spin torque from the current ﬂow inside the ferromagnet, similar to a moving domain wall. C. Noncollinear spin texture At larger overcriticality, the progressive softening of the bullet mode culminates in a quasistatic pattern. Note that,besides softening, the local switching of the magnetizationalso drives the condensation into the stripe pattern: whereverM/bardbl−H, the injected spin exerts a dampinglike torque. Only at small overcriticality does M/bardblHstill hold on average, and the torque is antidampinglike. Regarding the quasistatic texture, one may recall that Bender et al. [32] proposed in 2014 that Bose-Einstein condensation of magnons should set in under spin-current in-jection. In their theory, a phase diagram was derived under theassumption of small-angle dynamics. Here, I emphasize that,in the case of a strongly excited YIG nanoﬁlm, the nonlinearspin-wave bullets must be considered dynamic modes un-dergoing condensation. Their local oscillation angle is large.Therefore, the theory of Ref. [ 32] cannot be applied directly. To further understand the classical condensation phenomenonobserved in this micromagnetic simulation work, I suggestﬁrst considering the dispersion of bullets, which I here ap-proximate by ω b(k)=/radicalbig (ωH−aωMk2)(ωH−aωMk2+ωM), (8) where a=2A μ0M2 0. Note that in this expression, the wave num- berk∝1 dbcharacterizes the diameter of the nonpropagating bullet [ 42]. Comparing the maxima of the Fourier power in Fig. 5(d) with the overlaid dispersion curve Eq. ( 8) (dashed green line), I ﬁnd an intersection approximately at the point ofvanishing frequency. To rule this out as a mere coincidence,I have repeated the simulations for external ﬁelds betweenμ 0H=25 mT and 400 mT. At all ﬁelds, I have found at a current density of j=7.5×1011A/m2(corresponding to /Gamma1=43) the stripe texture, and I determined the corresponding characteristic wave number k0. The ﬁeld dependence of k0is plotted in Fig. 6, on top of a colored map encoding the ﬁeld and wave number dependence of ωb(H,k). For all selectedﬁelds H, the characteristic wave numbers k0lie approximately on the isocontour ζω=0of vanishing frequency. This reﬂects the ﬁnding that the emerging texture is a quasistatic feature. Why should this particular mode be chosen? Recall that the conventional dissipation argument for spin-wave instabilitiesimplies that the mode with the smallest losses is selected [ 17]. For a magnon BEC, this is also the mode with the lowestfrequency. Here, this argument fails because the spin torquecompensates for the direct dissipative losses. By pushing thebullets as far away from the linear spin-wave spectrum aspossible, the system minimizes nonlinear losses that occurbecause of multiple-magnon scattering. Such processes redis-tribute energy from the bullets into high-frequency magnonswhose losses are not compensated by the spin current. Thisindirect route remains an active dissipation channel as long asthe bullet frequency does not vanish. V . SUMMARY The overall picture for spin-current-induced magnetization dynamics in YIG nanoﬁlms obtained from micromagneticsimulation is this: when the threshold current density j th= 0.17×1011A/m2is exceeded, ﬁrst, single spin-wave bullets appear, whose number quickly increases with increasing cur-rent. The bullets then give rise to deterministic chaos. Thisturbulent state eventually freezes out, in favor of a quasistatic,noncollinear magnetic texture, which ﬁnally gradually turnsinto a completely switched state. Note that combining mate-rials with large spin-Hall angles like β-tungsten [ 51], with optimally grown YIG nanoﬁlms, displaying Gilbert damp-ing constants as small as only 7 ×10 −5[2,4], opens up a realistic and fruitful perspective for studying samples withlarge active areas ( w/greatermuchk −1 b). Then, one might be able to observe turbulent dynamics, as well as the novel, quasistatictexture. While, so far, no experimental reports about the emer-gence of such a texture exist, the possibility of establishinga connection to experimental work (see the SupplementalMaterial [ 38]) further supports this chance. In addition to such experimental opportunities, the ﬁndings presented in thispaper also open an interesting perspective for the applicationof spin hydrodynamic theory [ 52–55]. In particular, at large overcriticalities, the emerging texture breathes at its bound-aries, radiating large-amplitude waves (see the video in theSupplemental Material [ 38]). This process bears similarities to the appearance of dissipative exchange ﬂows discussedin [52]. Finally, I would like to emphasize that the dynamics dis- cussed here are, in particular, rather independent of the actualmagnetic material. Qualitatively similar ﬁndings can be ob-tained for metallic ferromagnets like permalloy. 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PhysRevApplied.15.044049.pdf	PHYSICAL REVIEW APPLIED 15,044049 (2021) Editors’ Suggestion Featured in Physics Double-Free-Layer Magnetic Tunnel Junctions for Probabilistic Bits K e r e mY .C a m s a r i ,1,Mustafa Mert Torunbalci ,2William A. Borders ,3Hideo Ohno ,3,4,5,6,7and Shunsuke Fukami3,4,5,6,7 1Department of Electrical and Computer Engineering, University of California, Santa Barbara, California 93106, USA 2Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907 USA 3Laboratory for Nanoelectronics and Spintronics, Research Institute of Electrical Communication, Tohoku University, Japan 4Center for Spintronics Research Network, Tohoku University, Japan 5Center for Science and Innovation in Spintronics, Tohoku University, Japan 6WPI Advanced Institute for Materials Research, Tohoku University, Japan 7Center for Innovative Integrated Electronic Systems, Tohoku University, Japan (Received 13 December 2020; revised 3 March 2021; accepted 25 March 2021; published 29 April 2021) Naturally random devices that exploit ambient thermal noise have recently attracted attention as hard- ware primitives for accelerating probabilistic computing applications. One such approach is to use a lowbarrier nanomagnet as the free layer of a magnetic tunnel junction (MTJ), the magnetic ﬂuctuations of which are converted to resistance ﬂuctuations in the presence of a stable ﬁxed layer. Here, we propose and theoretically analyze a MTJ with no ﬁxed layers but two free layers that are circularly shaped diskmagnets. We use an experimentally benchmarked model that accounts for ﬁnite-temperature magnetiza- tion dynamics, bias-dependent charge, and spin-polarized currents as well as the dipolar coupling between the free layers. We obtain analytical results for statistical averages of ﬂuctuations that are in good agree-ment with the numerical model. We ﬁnd that the free layers with low diameters ﬂuctuate to randomize the resistance of the MTJ in an approximately bias-independent manner. We show how such MTJs can be used to build a binary stochastic neuron (or a p-bit) in hardware. Unlike earlier stochastic MTJs that need to operate at a speciﬁc bias point to produce random ﬂuctuations, the proposed design can be random for a wide range of bias values, independent of spin-transfer-torque pinning. Moreover, in the absenceof a carefully optimized stabled ﬁxed layer, the symmetric double-free-layer stack can be manufactured using present-day magnetoresistive random-access memory (MRAM) technology by minimal changes to the fabrication process. Such devices can be used as hardware accelerators in energy-eﬃcient computingschemes that require a large throughput of tunably random bits. DOI: 10.1103/PhysRevApplied.15.044049 I. INTRODUCTION Intrinsic randomness in nanodevices can be harnessed to do useful computational tasks, especially when the nat- ural physics of a device map to a useful functionality, a principle sometimes expressed as “let physics do the computing” [ 1]. One such example is the physics of low- barrier magnets, which can produce random ﬂuctuations in magnetization that can be turned into ﬂuctuations in resistances in magnetic tunnel junctions (MTJs). Stochas- tic MTJs (SMTJs) and the stochastic behavior of MTJs have attracted a lot of recent attention, both theoretically and experimentally [ 2–12,12–21]. It has been observed camsari@ece.ucsb.eduthat even with the relatively modest tunneling magnetore- sistance of present-day MTJs, where the parallel to antipar- allel resistance ratios are small, ﬂuctuations in resistances can be converted to electrical ﬂuctuations that can be sensed by inverters or ampliﬁers [ 14,22]. Such devices can be useful as compact energy-eﬃcient tunable true random number generators that can be interconnected to accelerate a wide range of computational tasks such as sampling and optimization [ 23]. Both theory and available experimental data suggest that when SMTJs are designed out of perpendicular easy- axis MTJs (PMTJs), where both the ﬁxed and the free layer have perpendicular anisotropy, the ﬂuctuations tend to be slow, since even at the zero-barrier limit E/kBT→0 (where Eis the energy barrier of the magnet, kBis the Boltzmann constant, and Tis the temperature) the ﬂuctua- tions ( τ) are of the order of τ−1≈αγHth(where γis the 2331-7019/21/15(4)/044049(10) 044049-1 © 2021 American Physical SocietyKEREM Y. CAMSARI et al. PHYS. REV. APPLIED 15,044049 (2021) gyromagnetic ratio of the electron, αis the damping coef- ﬁcient, and Hthis an eﬀective thermal noise ﬁeld, given by Hth=kBT/MsV,MsVbeing the total magnetic moment) [24]. Even for a small magnet—for example, with half a million spins— τ−1is limited to frequencies around 1.5–15 MHz for α=0.01–0.1. An alternative is to use circular- disk magnets with no intrinsic anisotropy, since by virtue of their large demagnetizing ﬁeld, these magnets tend to produce much faster ﬂuctuations [ 22,25] with a preces- sional mechanism that has been theoretically analyzed [26–28] and recently observed in experiment [ 20,29]. All focus on SMTJs, whether with in-plane or perpendicular easy-axis magnets, however, has been on magnetic stacks, where there is a stable ﬁxed layer and an unstable free layer that ﬂuctuates in the presence of thermal noise. The current standard in spin-transfer-torque magnetoresistive random access memory (STT MRAM) technology is to use PMTJs [30] and switching to such in-plane easy-axis magnets with at least one stable ﬁxed layer to get fast ﬂuctuations is challenging from an industry standpoint, especially for miniaturized MTJs down to a few tens of nanometers, where techniques to fabricate ﬁxed layers have not been established. In this paper, we propose and evaluate the possibility of using a double-free-layer MTJ where both layers are designed as circular in-plane easy-axis magnets. Such a conﬁguration can easily be achieved by starting from a typical STT MRAM material-stack structure comprised of Co-Fe-B/MgO/Co-Fe-B MTJs by making both the free- and ﬁxed-layer magnets thicker such that their easy-axis orients in the plane of the magnet (Fig. 1). In the rest of this paper, we analyze the behavior of this double- free-layer MTJ device. One advantage of this device comes from its simplicity: it is a completely symmet- ric device with two free layers that are in an in-plane conﬁguration in equilibrium and this does not require a highly optimized magnetic stack design as it is based on the same Co-Fe-B/MgO/Co-Fe-B structure of standard PMTJs. Another key feature of this device is its bias inde- pendence over a wide range of voltages, which can be useful for designing devices to be used in probabilistic computing applications where a large throughput of tun- able random bit streams are needed. We note that double- free-layer structures similar to those shown in Fig. 1 have been discussed in the context of spin-torque nano- oscillators (see, for example, Refs. [ 31–35]); however, our focus in this paper is on fully circular magnets with no intrinsic anisotropy that are in the superparamagnetic regime. The rest of the paper is organized as follows. In Sec. II, we develop a model to describe the dipolar interaction between the layers, starting from Maxwell’s equations in the magnetostatics regime. In Sec. III,w e describe the ﬁnite-temperature coupled-macrospin model that describes the magnetization dynamics. In Sec. IV,Double-free-layer MTJ FreeFixedStandard PMTJ Free Free(a) (b) FIG. 1. The proposed device. (a) The standard MTJ of the MRAM technology using perpendicular easy-axis MTJs(PMTJs) with ﬁxed and free layers. (b) We consider an MTJ with two free layers and no ﬁxed layer. The magnetization of the free layers ﬂuctuates in the plane in the presence of thermal noise that is turned into resistance ﬂuctuations through tunneling mag- netoresistance (TMR). One way to build the proposed device isto start from the standard PMA-MTJs and make both the free and the ﬁxed layers thicker so that their magnetizations fall into the plane. We note that the PMTJ stack sketch shown here is forillustrative purposes and that industrial PMTJ stacks have many more additional layers that account for canceling dipolar ﬁelds, ensuring ﬁxed-layer stability and other eﬀects. we analyze the zero-bias behavior of the double-free- layer MTJ with analytical benchmarks that are obtained from equilibrium statistical mechanics. In Sec. V,w e describe the fully voltage-dependent model that considers bias-dependent spin-polarized currents that inﬂuence the free layers. Finally, in Sec. VI, we show how the pro- posed device can be combined with modern transistors in a 1 transistor (1T)-1MTJ circuit topology to deliver tunable randomness with fast ﬂuctuations. II. MAGNETOSTATICS In the absence of any external magnetic ﬁeld and intrin- sic anisotropies, the energy of the two-magnet system is fully speciﬁed by magnetostatics and is given by [ 36] E=−2πM2 sV⎛ ⎜⎜⎝2/summationdisplay i=1ˆmT iNiiˆmi+2/summationdisplay i,j i/negationslash=jˆmT iDijˆmj⎞ ⎟⎟⎠,( 1 ) where Msis the magnetic moment per volume, Vis the volume, the ˆmiare the three-component magnetiza- tion vectors, and Niiand Dijare the demagnetization and the dipolar tensors, respectively (we adopt cgs units for magnetic models throughout). We implicitly assume the volume and the Msto be the same for both magnets, which is true for all cases considered in this paper. We adopt a macrospin approach where the chosen volume corresponds to the volume of the magnet and the ˆmiare described as three-component vectors. We numerically solve for Dand 044049-2DOUBLE-FREE-LAYER MAGNETIC... PHYS. REV. APPLIED 15,044049 (2021) Nthat are in general position dependent but we average them within the volume of the target magnet to reduce these to single numbers for a given geometry. Starting from the magnetostatics conditions where /vector∇× /vectorH=0, we can deﬁne a magnetic potential /Phi1such that −∇/Phi1=/vectorHand since /vector∇·/vectorB=0a n d /vectorB=(/vectorH+4π/vectorM) always hold, we obtain ∇2/Phi1=4π/vector∇·/vectorM, which is mathe- matically equivalent to the Poisson equation of electrostat- ics. To solve this equation, we ﬁrst introduce a Green’s function, G(/vectorr,/vectorr/prime), deﬁned as the potential at /vectorrdue to a “unit” charge at source /vectorr/prime.Gcan be readily identiﬁed from the deﬁnition of the Dirac delta function [ 37],∇2(−1/\|/vectorr− /vectorr/prime\|)≡4πδ(/vectorr−/vectorr/prime). Once this Green’s function is known, using the linearity of the potential [ 38], we can write the general solution for the magnetic potential as /Phi1(/vectorr)=/integraldisplay d/vectorr/primeG(r,/vectorr/prime)ρM(r/prime),( 2 ) where we have deﬁned a magnetic source density, ρM(r/prime)=4π/vector∇·/vectorM, which is only nonzero at the bound- aries of the magnetic volume, assuming a uniformly mag- netized body. The solution to the magnetic potential at position /vectorrthen becomes /Phi1(/vectorr)=/integraldisplay V−/vector∇·/vectorM \|/vectorr−/vectorr/prime\|d/Omega1,( 3 ) where Vis the volume of the source magnet. Now let us consider the speciﬁc case shown in Fig. 2(a), where we have a cylindrical in-plane magnet, ˆM=Msˆx. The magnetic source density can be expressed as −/vector∇· /vectorM=Msδ(R−r/prime)cosφ, since the magnetization abruptly becomes zero right outside the magnetic boundary. This allows us to write the magnetic potential as follows: /Phi1(/vectorr)=δ/integraldisplay −δ2π/integraldisplay 0dφdz/primecos(φ)MsR/radicalbig (x−Rcosφ)2+(x−Rsinφ)2+(z−z/prime)2, (4) where we introduce δ=t/2,tbeing the thickness of the magnetic layer (source). We do not ﬁnd a closed-form solution of this integral, though it can be partly integrated just along z/primeafter taking derivatives of /Phi1to obtain ﬁeld expressions. These are not necessarily informative, so we do not repeat them here but we use these partial integrals to ease the numerical integration of Eq. (4)(for another treatment, see, e.g., [ 39]). Figure 2(b) shows a typical position-dependent vector plot of this numerical integra-tion. In Fig. 2(c), we show typical results where the ﬁeld strength increases for magnets with lower diameters, also observed in experiments with perpendicular MTJs [ 40] [also, see the inset in Fig. 4(a) for averaged out D xx=(a) (b) Offset Offset (nm) Offset (nm) (c) 10 nm 20 nm 60 nm 100 nm10 nm 20 nm 60 nm 100 nm(d) –40 –20 20 40 –40–20204060 –40 –20 20 40 –100–5050100 FIG. 2. The dipolar model. (a) The geometry and parameters in the calculation of dipolar ﬁelds between the two free lay- ers. (b) An illustrative vector plot of the dipolar ﬁeld due to the bottom layer (dashed box) at the y=0 plane. (c) The dipolar ﬁeld HXdue to a source magnet (bottom, +xpolarized) with varying diameters (2 R=10, 20, 60, 100 nm) with a ﬁlm thick- ness of t=1 nm at a distance d=1 nm (typical MgO thickness in MTJs) measured from the top of the source magnet with Ms=800 emu/cc ≈1 T. At zero oﬀset (circled points), the ﬁelds can be analytically calculated from Eq. (5).( d )T h e z-directed ﬁelds along the oﬀset direction. In our model, these ﬁelds average out to zero when summed over the target magnetic volume. D0values]. This correspondence between PMTJs and the double-free-layer system considered here can be seen from the properties of the dipolar tensor, i.e., tr[ D]=0[41,42]. As mentioned earlier, our approach of calculating dipo- lar tensor components is by averaging the ﬁeld over the target magnetic body; for example, to compute Dxx,w e ﬁrst compute HXover the target magnet volume when the source is magnetized along +xand take an average of this ﬁeld to obtain a single value for Dxx. Equation (4)can eas- ily be integrated on the cylindrical axis ( x=0,y=0) after taking the derivative HZ=−∂/Phi1/∂ zand this results in HZ=πMs/parenleftBigg z−δ/radicalbig R2+(z−δ)2−z+δ/radicalbig R2+(z+δ)2/parenrightBigg .( 5 ) Figure 2(c) shows the application of this formula at zero oﬀset, which is deﬁned as the dashed line in Fig. 2(a) that passes along the y=0 line, and this result matches the numerical integration. It might be tempting to use Eq. (5) to approximate the dipolar tensor coeﬃcients analytically to obtain a single number but we ﬁnd that this tends to sig- niﬁcantly diﬀer from the value we obtain after averaging 044049-3KEREM Y. CAMSARI et al. PHYS. REV. APPLIED 15,044049 (2021) over the volume, especially at larger diameters. With this method of calculating dipolar coeﬃcients, we ﬁnd that Dxy, Dyx,Dzx,a n d Dzyall average to zero, which only leaves the diagonal components [see Fig. 2(d)]. Moreover, the cylindrical symmetry of the problem ensures that Dxx= Dyy=D0and from the symmetry of the dipolar tensor Dzz=−2D0, leaving only one tensor coeﬃcient to com- pute. Similarly, the diameter (2 R) to thickness ( t) ratios of all the magnets analyzed in this paper ensure that t/R/lessmuch1, and the demagnetization tensor always has one component, Nzz≈−1, which is what we use in the rest of the paper. III. MAGNETIZATION DYNAMICS Next, we describe the coupled magnetization dynamics model that considers two coupled Landau-Lifshitz-Gilbert (LLG) equations at ﬁnite temperature [ 43–45]: (1+α2)dˆmi dt=− \|γ\|ˆmi×/vectorHi−α\|γ\|(ˆmi×ˆmi×/vectorHi) +1 qN[ˆmi×/vectorISi(V)×ˆmi] +/braceleftbiggα qN[ˆmi×/vectorISi(V)]/bracerightbigg ,( 6 ) where αis the damping coeﬃcient, γis the gyromagnetic ratio of the electron, qis the electron charge, and iis the magnet index, i∈{1, 2}. The eﬀective ﬁeld for each mag- net is calculated according to Eq. (1),Hi≡− ∇ ˆmE/(MsV) and N=(MsV)/μ B,a n dμBis the Bohr magneton. The eﬀective ﬁeld Hiincludes uncorrelated Gaussian noise (Hn x,y,z) at each direction (x,y,z)with the following statis- tical properties: /angbracketleftHn(t)/angbracketright=0a n d /angbracketleftHn(t)Hn(t/prime)/angbracketright= Dδ(t− t/prime), where D=(2αkBT)/(γ MsV). In our model, this set of equations is solved self-consistently with a transport model for the MTJ, which provides the bias-dependent spin-polarized current, /vectorIS(V)at a given bias, which in turn depends on the instantaneous magnetizations, mi(Fig. 3). An important consideration when modeling circular- disk nanomagnets in the macrospin approximation is the formation of vortex states [ 46]. However, for the highly reduced diameters ( ≤100 nm) and thicknesses ( ≤1n m ) considered in this paper, both detailed micromagnetic sim- ulations [ 47–49] and available experiments [ 50–52] indi- cate that the macrospin-modeling approach in the parame- ter ranges considered should be reasonably informative. We solve the stochastic LLG equation using the transient-noise function of HSPICE . We benchmark this model extensively by comparing its time-dependent statis-tical behavior with respect to the Fokker-Planck equation [53] and also by comparing it against our own implemen- tation that solves the stochastic LLG equation using the Stratonovitch convention [ 54].–IS(V) MTJLLG LLG IS(V)V+ V–m1 m2R(m1,m2,V) H21H12m1 m2Dipolar FIG. 3. The self-consistent model for the magnetization dynamics and transport. The coupled LLG equations provide instantaneous magnetizations to the MTJ model, which in turnproduces a bias-dependent spin-polarized current, /vectorI S(V)in the channel. +/vectorIS(V)is incident to one ferromagnetic interface and −/vectorIS(V)is incident to the other ferromagnetic interface. The dipo- lar model couples the two LLG solvers through ﬁelds that depend on instantaneous ˆmi. IV . ZERO-BIAS BEHA VIOR It is instructive to analyze the zero-bias behavior ( V= 0→/vectorIS=0) of the two free layers before considering their interaction with spin-polarized currents. Starting from Eq. (1)and making use of the symmetry results described at the end of Sec. II, we can write E=−4πM2 sV/bracketleftbigg −(mz 1)2 2−(mz 2)2 2 +(D0)mx 1mx2+(D0)my 1my2−(2D0)mz 1mz2/bracketrightbig .( 7 ) Since both free layers are low-barrier magnets that ﬂuc- tuate in the presence of thermal noise, it is instructive to calculate the cosine of the average angle between the free layers (cos θ1,2) at zero bias, as this determines the resistance of the MTJ. This average can be written down from the Boltzmann distribution (switching to spherical coordinates): /angbracketleftcosθ1,2/angbracketright=1 Z/integraldisplay ˆm1 ×ˆm2exp/bracketleftbigg−E(θ,φ,η,χ) kBT/bracketrightbigg dθdφdηdχ,( 8 ) where (θ,φ)and(η,χ)are spherical coordinate pairs for the two magnets and Zis a normalization constant that ensures that the total probability is 1. We can- not ﬁnd a closed-form expression for this integral butwe can make progress by approximations. Introducing h d≡4πM2 sV/kBTand d0=D0hd, we note that for typ- ical parameters ( Ms=800 emu/cc and 2 R=10−100 nm), hd/greatermuch1, indicating that both magnets always roughly 044049-4DOUBLE-FREE-LAYER MAGNETIC... PHYS. REV. APPLIED 15,044049 (2021) Analytical Eq. (9)Numerical Eq. (6) (b)(a) 20 40 60 80 100 –0.03–0.02–0.01 0.5 1.0 1.5 2.0 2.5 –1.0–0.50.51.0 0.2 0.4 0.6 0.8 1.00.20.40.60.81.0 FIG. 4. The zero-bias behavior. (a) The average cos θ1,2(θ1,2is the angle between magnetizations vectors) at zero bias. Equation (9)is compared with a ﬁnite-temperature LLG simulation at dif- ferent diameters (measured in nanometers). The average is taken over 5 μs for all examples with a time step of /Delta1t=1p s .T h e inset shows the average dipolar tensor component Dxx=Dyy= D0as a function of the diameter, calculated from Eq. (4)assum- ing the same geometric and magnetic parameters that are used in Fig. 2and damping coeﬃcient α=0.01. (b) The normalized autocorrelation of mxfor both layers and cos (θ1,2)are shown for 2R=10 nm as an example, where the total simulation period is 5μs. The inset shows the time dependence of the mxcomponents for a short period. remain in the x-yplane in equilibrium. Expressing this assumption mathematically, we expand the integrand in Eq.(8)at(θ,η)→(π/2,π/2)and keeping the leading order terms, we obtain /angbracketleftcosθ1,2/angbracketright≈I1(d0) I0(d0),( 9 ) where Inis the modiﬁed Bessel function of the ﬁrst kind. This simple expression determines the degree of cou- pling between the two layers in terms of the cosine of the angle between their magnetization and is entirely depen- dent on geometric and material parameters. Figure 4(a) shows a comparison of Eq. (9)with numerical simulationof Eq. (6)and we observe that the analytical expres- sion reproduces the numerically observed average with high accuracy, especially at higher diameters, where our assumption of hd/greatermuch1 becomes more accurate. The slight deviation at low diameters (10, 20)nm is due to this assumption becoming inaccurate. Equation (9)is an impor- tant result of this paper since, as we see in Sec. V, the aver- age angle behaves in a bias-independent manner; therefore, Eq.(9)is also approximately valid in the nonequilibrium condition. By providing the average degree of coupling between the layers in terms of material and geometric parameters, Eq. (9)could be useful in the design process of double-free-layer MTJs. We observe from Fig. 4(a) (inset) that even though the average dipolar interaction strength D0decreases at high diameters, the coupling strength observed through the average angle between the magnetizations increases [Fig. 4(a)]. The reason for this is that increasing the diam- eter increases the volume and the dipolar coupling energy as∝R2compared to the thermal energy kBT, while the dipolar interaction scales as ∝R−1. In other words, the decrease in the dipolar coupling at high diameters is not enough to compensate for the rapidly increasing dipolar energy. Another interesting observation is made when we observe the autocorrelation, C(τ)=1 Tp/integraldisplay mx(t)mx(t+τ)dt, (10) of two in-plane components mx[Fig. 4(b)], where Tpis the simulation period. The mxcomponents of each layer lose memory very rapidly in around 100 ps but the cosine of the angle between the magnets takes about 1 ns to become completely uncorrelated. Since this is the param-eter that determines the resistance of the MTJ, it is the more relevant time scale to consider to generate random signals. Note that in our analysis, we do not include the eﬀects of exchange interaction between the layers that might be present in real MTJs, as we assume that MgO pro- duces a low degree of exchange coupling compared to the dipolar coupling [ 55], but this may require further investi- gation. We also ignore the eﬀect of an existing interfacial anisotropy in our energy model. The existence of a strong interfacial anisotropy may be detrimental to the speed of ﬂuctuations [ 26]; hence strategies to reduce it might be useful. V . VOLTAGE-DEPENDENT BEHA VIOR In order to describe the full bias dependence of the double-free-layer MTJ, we ﬁrst introduce our combined transport and magnetization dynamics model (Fig. 3). We describe the MTJ as a bias-dependent resistor that provides a bias-dependent spin-polarized current to two separate 044049-5KEREM Y. CAMSARI et al. PHYS. REV. APPLIED 15,044049 (2021) ﬁnite-temperature LLG solvers described by Eq. (6).T h e LLG solvers provide magnetization vectors that change the resistance of the MTJ. This self-consistency between the resistance that depends on magnetization and magne- tization that depends on resistance is well deﬁned since magnetization dynamics are far slower than the electronic time scales; as such, at each time point, the resistance of the MTJ can be taken as a lumped model that provides spin-polarized currents to the LLG solvers. Similarly, the dipolar coupling acts “instantaneously,” providing updated ﬁelds that are fed back into the LLG solver at each time step. We model the bias-dependent conductance ( GMTJ≡ 1/RMTJ) based on two voltage-dependent interface polar- izations [ 53,57,58]: GMTJ(V)=G0/bracketleftbig 1+P1(V)P2(V)cosθ1,2/bracketrightbig , (11) where Vis the bias voltage across the MTJ, G0is the con- ductance measured when cos θ1,2=π/2, and the Pi(V)are the voltage-dependent polarizations of the two interfaces. By a physically motivated choice of interface polariza- tions, this model reproduces the bias dependence of the resistance, as well as the asymmetric bias dependence of the spin-polarized current, /vectorIS(V)[57]. A reasonable model for the polarization is [ 53] P(V)=1 1+P0exp(−V/V0), (12) where P0is a parameter that is determined by the low-bias magnetoresistance and V0is determined by the high-bias features of the MTJ. This model is motivated by the observation that at higher voltages, the polarization of the injected currents becomes weaker considering parabolic ferromagnetic bands in the contacts [ 57]. For simplicity, we assume a symmetric junction where P1(V)=P2(−V)=P(V)is satisﬁed and we drop the sub- script to denote only one polarization function, deﬁned by Eq.(12). We also assume a weak dependence of the polar- ization with respect to the voltage by choosing a large roll-oﬀ parameter ( V0) compared to the applied biases of interest (approximately ±0.5 V) for the p-bit considered in Sec. VI. In actual experiments, the bias dependence of the torque can be controlled by the roll-oﬀ parameter that can introduce asymmetries in larger biases depending on the bias asymmetry of the spin-polarized currents, similar to what is observed in standard MTJs [ 57]. With the bias-dependent polarizations and conductance of the MTJ deﬁned, we can deﬁne the magnetization- dependent spin-polarized current in the channel as [ 57,59] /vectorIS(V)=G0V[P(V)ˆm1+P(−V)ˆm2], (13) where the total spin-polarized current is the vectorial sum of two components proportional to the magnetization ofeach layer. In typical descriptions of spin-transfer-torque in MTJs, only one of these terms appear since the ﬁxed layer is assumed to be inert. In the double-free-layer sys- tem, however, both magnets are active and they respond to spin-polarized currents that are polarized in the direc- tion of the other magnets; therefore, we need to consider the total spin-polarized current. A key point to note is that Eq.(13) describes the total spin-polarized current in the channel. For one free layer, +/vectorIsis incident to the ferro- magnetic interface and for the other layer −/vectorIsis incident to the interface. Therefore, what is supplied to the LLG equations diﬀers by a minus sign (Fig. 3). This can also be intuitively understood by considering one free layer as the instantaneously ﬁxed reference layer of the other one: A +V(−V) bias that would make the layers parallel (antipar- allel) would switch sign if we imagine the other magnet as the reference layer. There is also another term that is along a direction that is orthogonal to both ˆm1andˆm2, the so-called ﬁeldlike torque, but it is typically small compared to the main terms and we ignore it in this paper [ 60–62]. The form of Eq.(13) can be justiﬁed by microscopic quantum trans- port models based on the nonequilibrium Green’s function (NEGF) formalism that is able to reproduce the bias depen- dence of torque and resistance values in experiments [ 57]. Even though there are two terms in Eq. (13), the individ- ual magnetization dynamics of each free layer only pick up a torque from the transverse component of the other free layer [ 63], since the form of Eq. (6)ensures that ˆmi×/vectorIS×ˆmicancels out components of /vectorISalong ˆmi.W e put together all the ingredients discussed so far, the dipo- lar tensors based on Eq. (4), the ﬁnite-temperature LLG dynamics based on Eq. (6), and the transport equations of MTJ described by Eqs. (11)–(13), in a modular circuit environment [ 53] that is simulated in HSPICE (Fig. 3). Figure 5shows representative voltage-dependent char- acteristics of a double-free-layer MTJ. A striking ﬁnding is that the bias dependence of the resistance is approx- imately independent of the applied voltage, even in the presence of the full eﬀect of spin-transfer-torque between the layers. The bias dependence also shows symmetry with respect to voltage, a result we intuitively expect since the device is completely symmetric with two identical free lay- ers. Figure 5(a) illustrates the dynamics of the free layers at diﬀerent time instances. We observe that the free lay- ers ﬂuctuate close to the x-yplane but also occasionally pick up a zcomponent. These ﬂuctuations are reminiscent of the fast precessional ﬂuctuations of easy-plane mag- nets that have been examined in Refs. [ 20,26,27] but all with stable reference layers, unlike the case considered here. Figure 5(b) shows the Rversus Vcharacteristics of a 10-nm double-free-layer MTJ where the resistance keeps ﬂuctuating approximately uniformly between RPand RAP values at all bias voltages. This bias independence of the 044049-6DOUBLE-FREE-LAYER MAGNETIC... PHYS. REV. APPLIED 15,044049 (2021) (a) (b) (c) FIG. 5. The Rversus Vcharacteristics of a double-free-layer MTJ. (a) Representative ﬂuctuations at diﬀerent time instances for 2 R=20 nm free layers at t=0n s , t=0.5 ns, and t=20 ns. (b) The Rversus Vcharacteristics for a (2R)=10 nm MTJ, where we assume a low-bias TMR of 115% [ P0=0.65 and V=0i nE q . (12)] and a roll-oﬀ constant V0=50 V, ensuring a symmetric bias dependence for the spin-polarized currents. We assume an RA product of 9 /Omega1-μm2[56] to obtain G0for all MTJs in this paper. The plot is obtained by sweeping the voltage from(−2V ,2V )i n2 μs with 1-ps time steps; the dots correspond to 1-μs averages taken at that bias. (c) The average cos (θ 1,2) taken over 1 μs at diﬀerent bias points for diﬀerent diameters. All diameters roughly show bias-independent average angles or resistances. ﬂuctuations is a signiﬁcant advantage of the double-free- layer MTJ, since it can provide a ﬂuctuating resistance over a wide range of values without getting pinned, unlike MTJs with ﬁxed layers [ 14,20], where the random ﬂuctu- ation point needs to be identiﬁed carefully in an eventual device implementation. We also note from Fig. 5(c) that above 60–100 nm, the average of the angle (and the resistance of the MTJ), even though random, is largely stuck around the RAPvalue where the increasing dipo- lar energy with magnetic volume overcomes the thermal noise. For this reason, strategies to use scaled dimensions or low-magnetic-moment ( Ms) materials will be useful. VI. TUNABLE RANDOMNESS WITH DOUBLE-FREE-LAYER MTJ In this section, we show how the double-free-layer MTJ can be used to deliver a hardware binary-stochastic-neuron(BSN) functionality in a 1T-1MTJ circuit plus an inverter circuit (Fig. 6) based on Ref. [ 22], although we attach a source resistance R Sto be able to shift the overall char- acteristics to the left or to the right, similar to what has (b) (a) VOUT (V) VIN(V)(c)VD (V) VOUT VIN+VDD Free FreeRMTJ RS14 nm FinFETVDVIN(V) FIG. 6. A BSN with a double-free-layer MTJ. (a) The double- free-layer MTJ (2 R=20 nm with all the same parameters that are used in previous ﬁgures) in a 1T-1MTJ circuit, where a RS= 5k/Omega1resistance is used to shift the overall characteristics. (b) The drain voltage is measured while the input is swept from 0 V to 0.8V over 0.5 μs. The circled points are averages over 250 ns at each bias point. (c) The output of the inverter, which shows the BSN characteristics with the same measurement times as reportedin (b). been carried out in Ref. [ 14]. We use a high-performance 14-nm FinFET model from the PREDICTIVE TECHNOLOGY MODEL (PTM)[64] to model the NMOS and combine the NMOS model with our circuit model simulated in HSPICE . We choose the 20-nm MTJ (with the same parameters that are used in this paper) to illustrate the circuit operation (Fig. 6), since its average resistance (1 /G0) approximately matches the transistor resistance plus the source resistance (RS) when the input to the NMOS is 0.4 V. Our purpose is not to provide a comprehensive circuit design of the BSN but simply to illustrate how the proposed MTJ, which includes eﬀects of thermal noise, dipolar coupling, and bias-dependent spin and charge currents, can be used to build a viable hardware BSN. Figure 6shows the output of the inverter while the input gate voltage of the NMOS is swept from 0 to VDD, the aver- age of which shows the familiar sigmoidal behavior of the binary stochastic neuron (BSN). One potential challenge in the design of double-free-layer MTJs will undoubtedly be to design the average P /AP ratio of the ﬂuctuations. Indeed, as can be observed from Fig. 5(c), the average resistance of the MTJ at diﬀerent diameters is not in the middle of RPand RAPvalues but closer to RAPdue to the dipolar coupling of the in-plane free layers. While at very high diameters >60 nm, this ratio may mostly be skewed toward RAP, the skew at 20 nm can be mitigated by a source resistance ( RS) to center the sigmoid of the hardware BSN, as shown in Fig. 6. We observe that changing this resis- tance value causes a shift of the overall characteristics (with higher values causing a rightward shift). 044049-7KEREM Y. CAMSARI et al. PHYS. REV. APPLIED 15,044049 (2021) Unlike previous stochastic MTJs in which the bias point where ﬂuctuations between RAPand RPcan vary signif- icantly between diﬀerent devices, the weak bias depen- dence of the proposed MTJ can be globally centered by a ﬁxed source resistance. This diﬀerence between having to align each p-bit precisely at its midpoint by a diﬀer- ent bias current and obtaining an approximately uniform randomness at all relevant bias voltages constitutes an important advantage of the proposed design that can be exploited at the system level. Secondary variations arising from diﬀerences in MTJ resistances and transistor process variations can further be dealt with at the “synaptic level,” where weighted inputs of p-bits can be modiﬁed by con- stant biases to counter these variations. Further, probabilis- tic computations are generally robust to small variations [65,66] and strategies that may counter such variations by adjusting the interconnection weights of probabilistic devices (as has been done in Ref. [ 14]) can be useful. Moreover, ultrascaled MTJs beyond 20 nm can become truly random as predicted by Eq. (9)provided that we can match the resistance of the NMOS to that of the MTJ, either by transistor design or by an additional source resis- tance, as shown in Fig. 6. Device-level simulations have shown that BSN characteristics similar to those shown in Fig.6can be used as building blocks to design probabilis- tic circuits to solve optimization [ 22,67,68] and sampling problems to train neural networks [ 69]. Therefore, the pro- posed double-free-layer MTJ can function in similar ways to be useful for these applications. It is important to note that even though we present the double-free-layer MTJ in terms of circular-disk magnets that can ﬂuctuate in nanosecond time scales, recent exper- imental [ 20,29] and theoretical work [ 26–28] have now ﬁrmly established that elliptical in-plane easy-axis mag- nets can also ﬂuctuate in similar time scales. While the exact details of our model might be diﬀerent, the double- free-layer concept as a bias-dependent building block to build probabilistic bits applies to such structures with qualitatively similar results. VII. CONCLUSION We propose and analyze a MTJ with two free layers to generate random ﬂuctuations using a comprehensive model where the dipolar interaction, thermal noise, and bias-dependent spin and charge currents are considered. Our ﬁndings reveal approximately bias-independent mag- netization ﬂuctuations that can produce random resistance values over a wide range of bias values, which can simplify circuit design with such MTJs. Another key advantage of the proposed magnetic stack is in its simplicity: the twofree layers can be completely symmetric and their in-plane magnetization can easily be achieved by modifying the existing STT MRAM technology by increasing the thick- ness of both the free and the ﬁxed layers. RepurposingSTT MRAM technology with double-free-layer MTJs can lead to massively parallel tunable random number gen- erators that can ﬁnd useful applications in probabilistic computing. ACKNOWLEDGMENTS Use was made of computational facilities purchased with funds from the National Science Foundation (Grant No. CNS-1725797) and administered by the Center for Scientiﬁc Computing (CSC). The CSC is supported by the California NanoSystems Institute and the Materials Research Science and Engineering Center (MRSEC; NSF DMR 1720256) at University of California Santa Barbara. 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PhysRevB.96.094428.pdf	PHYSICAL REVIEW B 96, 094428 (2017) Micromagnetic simulations of spin-torque driven magnetization dynamics with spatially resolved spin transport and magnetization texture Simone Borlenghi,1M. R. Mahani,2Hans Fangohr,3,4M. Franchin,3Anna Delin,1,2,5and Jonas Fransson1 1Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden 2Department of Applied Physics, School of Engineering Sciences, KTH Royal Institute of Technology, Electrum 229, SE-16440 Kista, Sweden 3Department of Engineering and the Environment, University of Southampton, Southampton, SO171BJ, United Kingdom 4European XFEL GmbH, Holzkoppel 4, 22869 Schenefeld, Germany 5Swedish e-Science Research Center (SeRC), KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden (Received 26 March 2017; revised manuscript received 14 July 2017; published 21 September 2017) We present a simple and fast method to simulate spin-torque driven magnetization dynamics in nanopillar spin-valve structures. The approach is based on the coupling between a spin transport code based on random matrixtheory and a micromagnetics ﬁnite-elements software. In this way the spatial dependence of both spin transportand magnetization dynamics is properly taken into account. Our results are compared with experiments. Theexcitation of the spin-wave modes, including the threshold current for steady-state magnetization precession andthe nonlinear frequency shift of the modes are reproduced correctly. The giant magneto resistance effect and themagnetization switching also agree with experiment. The similarities with recently described spin-caloritronicsdevices are also discussed. DOI: 10.1103/PhysRevB.96.094428 I. INTRODUCTION The orientation of the magnetization in a magnetic ﬁlm can be inﬂuenced using a spin-polarized current. Consequently, adirect current can transfer spin angular momentum betweenmagnetic layers, separated by either a normal metal or athysichin insulating layer. This effect is called spin-transfertorque (STT) and was ﬁrst discussed in the 1970s in the contextof moving magnetic domain walls [ 1] and fully understood in the 1990s [ 2,3]. STT has been of profound importance for the development of spintronic devices such as read-heads based onthe giant magnetoresitive (GMR) effect [ 4,5], the spin-transfer torque magnetic random-access memory (STT-MRAM) [ 6], and spin-torque nano-oscillators (STNO) [ 7]. Until very recently [ 8] the approaches to theoretically describe the magnetization dynamics induced by a spin torqueusually greatly simpliﬁed or neglected the description ofeither the spatial inhomogeneity of the spin torque or thethree-dimensional magnetization texture [ 9–11]. In the present work, we go beyond such approaches by coupling a ﬁnite element micromagnetic method [ 12]t oa numerical solver for spin transport, based on continuousrandom matrix theory (CRMT) [ 13,14]. In this way the effect of spin torque is described, with the transport and magneticdegrees of freedom treated on an equal footing. The spatialinhomogeneity of both spin transport and magnetizationdynamics is thus explicitly included. In our implementation,CRMT is parametrized by the same set of experimentallyaccessible parameters as in Valet-Fert theory [ 15], so that our numerical simulations contain no free parameters. We demonstrate the capabilities of our computational method by addressing theoretically the effect of spin torque onthe magnetization dynamics in the perpendicularly magnetizedcircular spin-valve nanopillar experimentally investigated inRef. [ 16]. This conﬁguration is obtained by saturating the device with a large applied ﬁeld perpendicular to the layers.The device setup is kept very simple and as such serves as aprototype for spin-valve structures, which ﬁnd application also in the emerging ﬁeld of spin-caloritronics [ 17–20]. In partic- ular, our conﬁguration corresponds to a circular precession of the magnetization, allowing for a precise identiﬁcation of the spin wave (SW) modes. This is crucial if one wants to couplethe system to an external radio frequency signal, since only asignal with the same symmetries of the SW modes can excitethe magnetization. Breaking the axial symmetry results in amore complicated conﬁguration where modes with differentsymmetries mix up [ 16]. Moreover, the system can be described using the language of coupled oscillators [ 21] and in particular of the discrete nonlinear Schrödinger equation [ 17] (DNLS). Indeed, this setup corresponds to the simplest realization of the DNLS, con-taining only two elements. Here spin-transfer torque physicallycorresponds to a magnon chemical potential [ 18] that controls the propagation of the energy and magnetization currents be- tween the two layers. The DNLS appears in many branches of Physics, including Bose-Einstein condensates, photonicswave guides, and photosynthetic reactions. Understanding thedynamics in our setup can therefore shed light on a very generaloscillator model. The remainder of this paper is organized as follows: in Sec. IIwe describe the geometry of the nanopillar and also brieﬂy review the classiﬁcation of SW modes in a perpendicularly magnetized nanopillar. Those modes are then identiﬁed by means of micromagnetic simulations at zerocurrent (thus without spin torque). This study was previouslyperformed in Ref. [ 16], to which we refer for a thorough discussion. Section IIIreviews key concepts of scattering approach and CRMT to describe transport in magneticmultilayers, and constitutes an extension of the material presented in Refs. [ 13,22]. In Sec. IVwe describe how to couple the CRMT transport code to Nmag [ 12], to simulate the effect of spatially inhomogeneous spin-transfer torqueand magnetization dynamics on the same footing. Section V contains our micromagnetic simulations of current driven 2469-9950/2017/96(9)/094428(13) 094428-1 ©2017 American Physical SocietySIMONE BORLENGHI et al. PHYSICAL REVIEW B 96, 094428 (2017) spin dynamics. Here we identify the SW modes excited by spin-transfer torque, the critical current for auto oscillations,and the frequency shift beyond the critical threshold [ 21]. Using CRMT we provide a precise characterization of themagnetoresistance. Our simulations are then compared withthe experimental results found in Ref. [ 16]. Finally, in the conclusion we summarize the main results of this paper and point out further possible developments. II. PHYSICAL SYSTEM AND MODEL A. Spin valve structure The nanopillar studied here is displayed in Fig. 1(a). It consists of a trilayer structure made of two Permalloy(Ni 80Fe20alloy) disks Pyaand Pybseparated by a 10-nm Cu spacer. The disks have diameter d=200 nm and thicknesses ta=4 nm (upper disk) and tb=15 nm (lower disk). The upper disk is connected to a 25-nm Au contact and thelower disk to a 60-nm Cu contact. In the experiment [ 16], the sample was mounted inside a magnetic resonance forcemicroscope (MRFM) and the whole apparatus was placedinside a vacuum chamber operated at room temperature.The external magnetic ﬁeld H ext, was oriented along the pillar axis z, which corresponds to the precession axis of the magnetization. The MRFM consists of an ultrasoft cantilever with a 100-nm-diameter magnetic sphere glued to its tip. The sphereis positioned precisely above the center of the nanopillar,so as to retain the axial symmetry. The mechanical-FMR FIG. 1. (a) Sketch of the nanopillar under study, with the arrows representing the magnetization vectors. In the steady state, the spin-transfer torque M×Scompensates the damping torque M×d and the local magnetization vector Msprecesses regularly at the Larmor frequency along a circular orbit. (b) The SW modes in each disk are Bessel functions J/lscriptm, with ( /lscript)(m) the azimuthal (radial) index. Modes with /lscript=0 correspond to a uniform in-phase precession of the magnetization, while modes with /lscript=1 correspond to a nonuniform precession, with the magnetization vector rotating along the azimuthal direction. (c) In the case where two disks arecoupled through the dipolar interaction, the SW J B/A /lscriptmmodes separate into binding (B) and antibinding (A), which correspond, respectively, to an antiphase precession (occurring mainly in the thick layer) andin-phase precession (occurring mainly in the thin layer).spectroscopy consists in recording, by optical means, the vibration amplitude of the cantilever as a function of thebias out-of-plane magnetic ﬁeld in the presence of a RF ﬁeldwith ﬁxed frequency excitation [ 16]. The dynamics can be excited also by injecting a dc current along z, which excites the dynamics in the thin layer due to spin-transfer torque.In the steady state, the combined torques exerted by the RFexcitations and spin transfer compensate the damping, sothat the local magnetization vector precesses regularly at theLarmor frequency along a circular orbit; see Fig. 1(a). The instantaneous magnetization Mcan be decomposed into a large static component M /bardbland a small oscillating component M⊥, where M/bardblis parallel to the local precession axis (i.e., the direction of the external magnetic ﬁeld), and M⊥ is perpendicular to that axis. An essential feature of our system is that different SW modes have different spatial distribution of the phase of M⊥ inside the magnetic disks. Those modes can be excited only by RF ﬁelds with the same rotational symmetries, giving selection rules for the excitation of the SW modes [ 16]. The dipolar force acting on the cantilever is proportional to the spatially averaged value of the longitudinal (static)component of the magnetization inside the whole nanopillar, /angbracketleftM z/angbracketright=1 V/integraldisplay VMz(r)d3r. (1) The latter is not subject to any selection rule, so that the the mechanical-FMR setup detects all possible SW modes.Experimentally, it has been observed that the presence of thecantilever introduces a shift of +0.57 GHz in the SW spectrum. This has be taken into account in our simulations. The dynamics of the magnetization can be excited also by means of a radio frequency current ﬂowing along the axisof the pillar, which generates an radio frequency orthoradialOersted ﬁeld, and by mean of an homogeneous in-plane RFmagnetic ﬁeld. We remark that, although in experiments the dynamics driven by a combination of RF ﬁelds and STT, in oursimulations the dynamics is excited by STT only. The effectof the RF ﬁeld is modeled by setting different initial conditionfor the magnetization. In our circuit, a positive current corresponds to a ﬂow of electrons from the bottom Py bthick layer to the top Pyathin layer, and stabilizes the parallel conﬁguration due to the spin-transfer effect. Vice versa, a negative current stabilizes thethick layer and destabilizes the thin one. At low current, thickand thin layer are thus the ﬁx and free layer, correspondingly.However, at high enough current, both layers precess due tothe repulsive dipolar interaction, as will be discussed later. B. Magnetization dynamics In this section, we brieﬂy review the magnetization dynam- ics in our system. For a more comprehensive discussion, seeRef. [ 16]. The local dynamics of the magnetization M j, which depends continuously on the position rjin the layer j=(a,b), is described by the Landau-Lifshitz-Gilbert equation [ 23,24]: 1 γ˙Mj=Mj×Hj eff+αMj×˙Mj+Mj×Sj.(2) 094428-2MICROMAGNETIC SIMULATIONS OF SPIN-TORQUE . . . PHYSICAL REVIEW B 96, 094428 (2017) Here, γ< 0 is the gyromagnetic ratio in the magnetic layer. The ﬁrst term on the right-hand side of Eq. ( 2) describes the adiabatic torque, which accounts for the precession of themagnetization vector around the local equilibrium direction.This precession axis is deﬁned by the effective magnetic ﬁeld experienced locally by the magnetization, H j eff=−∂F ∂Mj, which contains all the static contributions to the free energy F of the layers [ 25]. In particular, the effective ﬁeld contains con- tributions from applied ﬁeld, exchange interaction, and dipolarinteraction between the layers. We refer to Refs. [ 14,16,20,25] for the explicit expressions. The second term on the right-hand side of Eq. ( 2)i st h e damping torque, d j=αj˙Mj, (3) proportional to the Gilbert damping parameter αj.W ei n t r o - duce here the notation Mj≡Mj smj, withMj sthe norm of the magnetization (a constant of the motion) and mjthe unit vector along the magnetization direction. The dissipative term djis responsible for the ﬁnite line width (full width at half height)of the resonance peaks, /Delta1H=2d. For a normally magnetized nanopillar, with circular precession of the magnetization, thesimple relation α=\|γ\|/Delta1H/ (2ω) holds [ 21]. If a charge current I dcis ﬂowing between two layers jand j/prime, the Slonczewski-Berger [ 2,3] spin-transfer torque reads Sj=Idc 2πλ[mj×mj/prime]. (4) The latter depends on the relative angle between the magneti- zation mjin the layer jand the spin polarization of the current, which coincides with the direction of the magnetization mj/prime of the polarizer (here the thick layer). The term λj=2eMj sV ηh(5) has the dimension of a distance. Here, his the Planck constant, ethe absolute value of the electron charge. ηis the spin polarization of the current and Vthe volume of the thin layer. Since djandSjare collinear, spin-transfer torque can compensate the damping torque, as shown in Fig. 1. When the dc current through the nanopillar reaches the thresholdcurrent I th=− 2πλαH eff, the thin layer starts auto-oscillating. Combining Eqs. ( 3) and ( 4), it is possible to deﬁne an effective damping for the thin layer, d=α(1−Idc/Ith), (6) which depends linearly on the spin polarized dc current [26,27]. The critical threshold corresponds to the value of the current Idcat which the effective damping vanishes and the system starts auto-oscillating. C. Coupled oscillator model and classiﬁcation of the SW modes Since our layers are thinner than 15 nm, one can assume that the magnetization dynamics is uniform along the thickness. Inthis approximation, the linearized LLG equation simpliﬁesto two equations describing the circular precession of the transverse magnetization projections M j xandMj yaround the z axis, which depends only on the two spatial variables ( x,y)i nthe layer j[16,25]. The two real equations of each layer can be rewritten as one complex equation for the dimensionlessspin-wave amplitude, c j=Mj x+iMj y/radicalBig 2Mj s/parenleftbig Mj s+Mj z/parenrightbig, (7) which depends on the polar coordinates ( rj,φj)o fd i s k j. The dynamics of the two disks, written in terms of the cjs, is described by the equations [ 16,17,21] ˙ca=iωaca−[/Gamma1a−−/Gamma1a+]ca+ihabcb, (8) ˙cb=iωbcb−[/Gamma1b−−/Gamma1b+]cb+ihbaca, (9) which are the equations of motion of two coupled nonlinear oscillators with resonance frequency ωj(pj) and damping rates /Gamma1j(pj). Both depend on the SW power pj=\|cj\|2, which describes the amplitude of the oscillations in each disk. Fromhere on, to keep the notation simple we do not write explicitly the dependence on p j. The frequencies ωj=γ\|Hj eff\|are proportional to the local magnetic ﬁeld, while the dampingrates/Gamma1 j−are proportional to αjωj. Both can be therefore controlled by means of the applied ﬁeld along z. The terms /Gamma1j+, proportional to Idcare due to spin-transfer torque, which can compensate the damping and lead to autooscillations of the layers. In the present case, those terms donot have the same sign. At positive current, /Gamma1 a+is positive, while/Gamma1b+is negative, favoring the auto oscillations in the thin layeraand stabilizing the thick layer b. The dipolar coupling strength hjj/primeis an effective term obtained by averaging the dipolar ﬁeld over the volumes ofthe samples; see Ref. [ 16] for the explicit expression. Equations ( 8) and ( 9) describe the dynamics of a nonlinear Schrödinger dimer, the simplest realization of the discretenonlinear Schrödinger equation (DNLS) [ 17,18,28]. Upon multiplying Eqs. ( 8) and ( 9), respectively, by c ∗ aandc∗ band summing them with their complex conjugate equations, onehas the following continuity equation for the SW power: ˙p a=− 2(/Gamma1a−−/Gamma1a+)pa+jp ab, (10) and a similar equation for pb. The magnetization current jp ab=2Im[habcac∗ b] describes the transfer of Mzbetween the two layers and is essentially the SW current written fora discrete systems with only two spins [ 17]. Upon writing c a=√pa(t)eiφa(t), the current reads jp ab=2hab√papbsin[φa(t)−φb(t)+β]. (11) The quantity βcomes from the condition of dissipative coupling between the oscillators [ 19,28]. When the two oscillators are synchronized, φa≈φband the magnetization current approaches the constant value jp ab∝sinβ. On the other hand, if the oscillators are not synchronized, the magnetizationcurrent oscillates around zero and vanishes in average. Withinthis DNLS formulation, the spin-transfer torque that appears inEq. ( 10) plays the role of a magnon chemical potential, which by controlling the lifetime of the excitations, controls also theSW current between them. The diagonalization of the LLG equation in a conﬁned geometry leads to a discrete series of normal modes having 094428-3SIMONE BORLENGHI et al. PHYSICAL REVIEW B 96, 094428 (2017) each a different eigen-value, ω/(2π), the so-called Larmor precession frequency. The normal modes of the system arenumbered according to the number of half waves in thevibration. In the case of a 2D axially symmetric structure,the normal modes are identiﬁed by two integers: /lscriptandm, respectively, the mode number in the azimuthal and radialdirections. The analytical expression of the normal modes ofa perpendicularly magnetized disk is found in Refs. [ 16,29], c /lscript,m(r,φ,t )=J/lscript(k/lscript,mr)e+i/lscriptφe−iω/lscriptmt, (12) where J/lscriptare the Bessel functions of the ﬁrst kind and k/lscript,mis the modulus of the in-plane SW wave vector, which dependson the boundary conditions. The above labeling can be extended to the case of two different magnetic disks coupled by dipolar interaction. In theperpendicular geometry, the strength of the dynamical dipolarcoupling is attractive (lower in energy) when both layersvibrate in antiphase, because the dynamical dipolar chargesin each layer are alternate [ 30,31]. Thus, the binding state B corresponds to a collective motion where the two layers vibrateantisymmetrically and the antibinding state Ato a collective motion where the two layers vibrate symmetrically. The B modes correspond to a precession amplitude that is larger inthe thick layer b, while the Amodes correspond to a precession amplitude that is larger in the thin layer a[16]; see Fig. 1for an illustration. The dynamical dipolar coupling does not modify the nature of the modes; hence, to describe the dynamics of the bi-layersystem, we shall just add a new index BorA, indicating if the precession occurs in antiphase (mostly in the thick layer)or in phase (mostly in the thin layer), respectively. Thereare thus three indices to label the observed eigen-modes: theusual azimuthal and radial indices for a single disk ( /lscript,m), plus an additional index referring to the symmetrical orantisymmetrical ( AorB) coupling between both layers. The identiﬁcation of the SW modes and their symmetry is essential to couple the oscillator to an external source,since SW modes can couple only to a source with thesame symmetry. Here the /lscriptindex determines the rotational symmetry of the SW mode. The /lscript=0 modes correspond to SW that do not rotate in the x-yplane and can be excited only by a spatially uniform in-plane RF ﬁeld, while the /lscript=1 modes correspond to SW that rotate around the disk in the samedirection as the Larmor precession and can be excited only byan RF Oersted ﬁeld with orthoradial symmetry. Thus, excitingthe system with these different means gives two differentspectra. III. MICROMAGNETIC SIMULATIONS AT ZERO CURRENT In this section we describe the SW spectra by means of micromagnetic simulations without spin-transfer torque.Those simulations were performed with the NMAG micro-magnetic software [ 12], where the sample is described by ﬁnite element tetrahedral mesh. The latter has a maximumintersite distance of 6 nm, of the order of the Py exchangelength. The micromagnetic parameters are the same used inRef. [ 16] and are reported in Table Ifor convenience. The dynamics at each of the i=1,...,N nodes of the mesh of diskTABLE I. Parameters of thin (a) and thick (b) layer used in micromagnetics simulations. 4πMa(G) αa 4πMb(G) αb γ(rad.s−1.G) 8.2×1031.5×10−29.6×1039×10−31.87×107 j=(a,b) is described by the following LLG equation for the unit magnetization vector, mj i=Mj i/Ms: ˙mj i=−γ/parenleftbig mi×Hj effi/parenrightbig +α Ms/parenleftbig mj i×˙mj i/parenrightbig . (13) The integration of the LLG equation at each mesh site is performed by the Sundials ODE solver [ 32], which is based on variable steps multistep methods. The ﬁeld Hj i at each mesh node has contributions from applied ﬁeld, ﬁrst neighbor exchange interaction and long-range dipolarinteraction, responsible for the coupling between the layers.The quantity of interest is the space-averaged magnetization/angbracketleftm j(t)/angbracketright=1 Vj/integraltext Vjmj(rj,t)d3r, which for our ﬁnite-elements mesh reduces to mj=1 NN/summationdisplay i=1mj i. (14) From this quantity, the SW amplitudes cj(t) are calculated. The power spectrum, shown in Fig. 2, is given by the Fourier transform of the time series of the collective SW amplitudeaveraged over the sample thicknesses: c=(c ata+cbtb)/(ta+ tb). The modes with /lscript=0 (displayed in blue tones) are excited starting from an initial condition where the magnetizationuniformly tilted 8 ◦in the x direction with respect to the pre- cession axis z. Instead, the the modes with /lscript=+ 1 (displayed in red tones) are excited by applying to the magnetizationaligned with the zaxis the orthoradial vector ﬁeld perturbation θ(r,z)=/epsilon1ˆz×ˆρ.H e r e /epsilon1=0.01 and ˆρis the unit vector in FIG. 2. SW power spectrum for the /lscript=0 (a) and /lscript=1( b ) modes, obtained from the volume-averaged magnetization. The B (respectively, A) modes corresponds to antiphase (respectively, in phase) precession, whose amplitude is larger in the thick (respectively,thin) layer. 094428-4MICROMAGNETIC SIMULATIONS OF SPIN-TORQUE . . . PHYSICAL REVIEW B 96, 094428 (2017) TABLE II. Comparative table of the resonance ﬁelds of the /lscript= 0 SW modes. Top are the peak locations measured experimentally in Ref. [ 16]. Bottom are the eigen-frequencies extracted from the simulation at Hext=1T . Exp.f(GHz) 6.08 8.95 9.82 11.98 Sim.f(GHz) 6.08 8.94 9.83 12.00 SW modes B00 B01 A00 A01 the radial direction. Starting from these conditions, we have computed the time evolution of the system for 50 ns, withan integration time step of 5 ps. The frequencies of the SWmodes are displayed in Tables IIand IIIand compared with the experimental values. IV . CONTINUOUS RANDOM MATRIX THEORY (CRMT) FOR SPIN TRANSPORT This section contains a thorough review of the CRMT semiclassical theory of spin-dependent transport in magneticmultilayers. We follow closely the material presented inRefs. [ 13,14,22] and we extend it by providing an explicit formula for spin torque, that will be used in micromagneticssimulations of next section. A. Scattering matrix approach Within the scattering matrix formulation developed by Landauer and Buttiker [ 33], a sample is deﬁned by the scattering matrix S, which expresses the outgoing propagating modes in term of the incoming ones. The incoming modes areﬁlled according to the Fermi-Dirac distribution of the leads towhich they are connected. From the elements of the scatteringmatrix, various physical quantities can be calculated, such asthe conductance G, the spin currents J, and charge current I. The system contains N ch/greatermuch1 propagating modes (or channels) per spin. In particular, one has Nch≈A/λ2 F, where Ais the transverse area of the electrode and λFis the Fermi wavelength. The amplitude of the wave function on thedifferent modes is given by the vector ψ i±withNchelements, ψi±=/parenleftbiggψi±↑ ψi±↓/parenrightbigg . (15) The latter contains the amplitudes for the right ( +) and left (−) moving electron direction with spin σ=↑,↓along the zaxis in region i=0,2 of the multilayer; see Fig. 3for an illustration. The Smatrix is a 4 Nch×4Nchunitary matrix that TABLE III. Comparative table of the resonance ﬁelds of the /lscript= +1 SW modes. Top are the peak locations measured experimentally in Ref. [ 16]. The mode A11is not visible in experimental data. Bottom are the eigen-frequencies extracted from the simulation at Hext=1T . Exp.f(GHz) 7.44 10.47 10.85 — Sim.f(GHz) 7.46 10.46 10.90 13.85 SW modes B10 B11 A10 A11 FIG. 3. (a) Scattering matrix relating incoming and outgoing modes ψnin region n=0,1. The +and−sign represent, respectively, right and left propagating modes. (b) Illustration of the “hat” matrix approach. We deﬁne the regions 0, 1, and 2, respectively, on the left, middle, and right of the two samples SaandSb. In each region the four vectors Pi±σrepresents the probability to ﬁnd right ( +) propagating and left ( −) propagating electrons. (c) The CRMT equations allow one to calculate the hat matrices as a function of the length Lof the sample, using the sum law for hat matrices. relates the outgoing modes to the ingoing ones: /parenleftbigg ψ0− ψ1+/parenrightbigg =S/parenleftbigg ψ0+ ψ1−/parenrightbigg ; (16) see Fig. 3(a) for a schematic of the system. The scattering matrix consist of 2 Nch×2Nchtransmission t,t/primeand reﬂection r,r/primesub-blocks, S=/parenleftbigg r/primet t/primer/parenrightbigg . (17) Here the ( r,t) and ( r/prime,t/prime) describe reﬂection and transmission, respectively, from left to right and from right to left. Thetransmission and reﬂection matrices have an internal spinstructure: t=/parenleftbigg t ↑↑t↑↓ t↓↑t↓↓/parenrightbigg , (18) where tσσ/primeareNch×Nchmatrices containing amplitudes for transmission between σ/primeandσspin states, capturing both spin preserving and spin-ﬂip phenomena. The conductance of the system is given by the Landauer formula [ 33], G=e2 hTr[t†t], (19) while the spin current in region 0 reads ∂/vectorJ0 ∂μ=1 4πTr[t/vectorσt†], (20) μbeing the difference of chemical potential between the two electrodes [ 22]. 094428-5SIMONE BORLENGHI et al. PHYSICAL REVIEW B 96, 094428 (2017) B. Random matrix theory (RMT) The scattering matrix approach is fully quantum and contains interference effects such as weak localization oruniversal conductance ﬂuctuations [ 22,34]. The system studied in this paper is a nanopillar of 250 nm of diameter connectedto top and bottom electrodes. Those contain ≈10 4–105prop- agative channels. Here the scattering is not perfectly ballistic(mismatch at the interfaces, surface roughness, or impurityscattering) so that channels get mixed up. Random matrixtheory (RMT) [ 22,35,36] assumes that this mixing is ergodic : an electron entering the system in a given mode will leave it inan arbitrary mode, acquiring a random phase in the process. In this case, the transmission and reﬂection probabilities of an electron are well characterized by their average over thepropagative channels. This average is obtained by taking thetrace over the N chof the original reﬂection and transmission matrices. For instance, the hat matrix ˆt[13,14] is deﬁned as ˆtση,σ/primeη/prime=1 NchTrNch[tσσ/primet† ηη/prime]. (21) Explicitly, this reads ˆt=1 NchTrNch⎛ ⎜⎜⎜⎜⎝t ↑↑t† ↑↑t↑↑t† ↑↓t↑↓t† ↑↑t↑↓t† ↑↓ t↑↑t† ↓↑t↑↑t† ↓↓t↑↓t† ↓↑t↑↓t† ↓↓ t↓↑t† ↑↑t↓↑t† ↑↓t↓↓t† ↑↑t↓↓t† ↑↓ t↓↑t† ↓↑t↓↑t† ↓↓t↓↓t† ↓↑t↓↓t† ↓↓⎞ ⎟⎟⎟⎟⎠,(22) with the same structure for the reﬂection hat matrix. The elements of this matrix correspond to the proba- bility for an electron with a given spin to be transmit-ted ( ˆt) or reﬂected ( ˆr) by the system. In particular, the terms (1 /N ch)Tr Nch(t↑↑t† ↑↑)≡T↑↑and (1 /Nch)Tr Nch(t↓↓t† ↓↓)≡ T↓↓correspond to the probability to transmit an electron with up and down spin correspondingly. The terms at the corners, (1 /Nch)Tr Nch(t↑↓t† ↓↑)≡T↑↓andT↓↑, correspond to transmission probabilities with spin ﬂip. The so-called “mixing transmission”, (1 /Nch)Tr Nch(t↑↑t† ↓↓)=Tmxis a complex num- ber whose amplitude measures how much of a spin transverseto the magnetic layer can be transmitted through the system,while its phase amounts for the corresponding precession. T mx decays exponentially with the size of the ferromagnet and accounts for spin-transfer effect [ 14,22]. The other off diagonal elements in the hat matrix Eq. ( 21) can be ignored [ 14,22]. In the basis parallel to the magnetiza- tion, Eq. ( 22) therefore becomes ˆt=⎛ ⎜⎝T↑↑ 00 T↑↓ 0Tmx 00 00 T∗ mx 0 T↓↑ 00 T↓↓⎞ ⎟⎠, (23) with the same structure for the reﬂection matrix ˆr. The hat- matrix ˆShas a form similar to Eq. ( 17), ˆS=/parenleftbigg ˆr/primeˆt ˆt/primeˆr/parenrightbigg . (24) To describe transport in noncollinear multilayers, where the orientation of the magnetization changes inside the system, one needs to rotate the original Smatrix as ˜S=Rθ,/vectornSR† θ,/vectornin thechosen working basis. Here the matrix Rθ,/vectorn=exp(−i/vectorσ·/vectornθ/ 2) (25) is the rotation matrix of angle θaround the unit vector /vectornthat brings the magnetization onto the z-axis of the working basis.In terms of hat matrices, this translates directly into ˆ˜S=ˆR θ,/vectornˆSˆR† θ,/vectorn, (26) with ˆRση,σ/primeη/prime=Rσσ/primeR∗ ηη/primea unitary matrix. From Eq. ( 23), the conductance is given by G=1 Rsh(T↑↑+T↑↓+T↓↑+T↓↓), (27) where the Sharvin resistance Rsh=h Nche2,i sam a t e r i a l property that can be experimentally measured, related tothe number N chof transverse propagative channels for the electrons crossing the system. Equation ( 27) is analogous to the Landauer formula Eq. ( 19) and consists of the sum of all the possible transmission processes (spin preserving and spinﬂipping) for an electron. In analogy with the modes ψ i±σfor the scattering matrix, the four vectors Pi±are introduced: Pi±=⎛ ⎜⎝Pi±,↑ Pi±,mx P∗ i±,mx Pi±,↓⎞ ⎟⎠. (28) The components o Pi±↑,↓have interpretation in term of probabilities for an electron to propagate in the region i of the system. The “mixing” components, Pmxare complex numbers which correspond to probability to ﬁnd the electronwith spin along the x(real part) or y(imaginary part) axis. Inside magnetic layers where the zaxis will correspond to the direction of the magnetization, they will correspond tothe probability for the spin to have a part transverse to themagnetization. For the following discussion, it is convenient to consider a system made of two conductors connected in series, describedby the two hat matrices ˆS aand ˆSb; see Fig. 3(b). The space is thus divided into three regions: region 0 and 2, respectively, atthe leftmost and rightmost part of the system, and region 1 inbetween the two hat matrices. In analogy with Eq. ( 17), which expresses the amplitudes of the outgoing modes in term of theincoming ones, in subsystems aandbone has, respectively, /parenleftbigg P 0− P1+/parenrightbigg =ˆSa/parenleftbigg P0+ P1−/parenrightbigg , (29) /parenleftbigg P1− P2+/parenrightbigg =ˆSb/parenleftbigg P1+ P2−/parenrightbigg , (30) while for the total system a+b, /parenleftbigg P0− P2+/parenrightbigg =ˆSa+b/parenleftbigg P0+ P2−/parenrightbigg . (31) Here we have used the addition law of hat matrices, a fundamental property that will be useful in the derivation ofspin currents and spin torque inside the system. According to 094428-6MICROMAGNETIC SIMULATIONS OF SPIN-TORQUE . . . PHYSICAL REVIEW B 96, 094428 (2017) this rule, given the hat matrices for separate systems aandb, the hat matrices of the composed system a+bread [ 13,14] ˆta+b=ˆta1 ˆ1−ˆr/prime bˆraˆtb, (32) ˆra+b=ˆrb+ˆt/prime b1 ˆ1−ˆraˆr/prime bˆraˆtb, (33) with ˆ1t h e4 ×4 identity matrix. Similar expression holds for r/prime a+bandt/prime a+b. The main result of Refs. [ 13,14] is that the spin current in region iof the system can be expressed in terms of probability vectors as follows: Ji=Nch 4π/vectorσ·(Pi+−Pi−), (34) where /vectorσ=(/vectorσ↑↑,/vectorσ↑↓,/vectorσ↓↑,/vectorσ↓↓) is the vector of components of Pauli matrices. To express the spin current in region 1 inbetween the two conductors as a function of the hat matrices,we use Eqs. ( 29)–(33) to eliminate P 2+andP2−and obtain /parenleftbigg P1+ P1−/parenrightbigg =⎛ ⎝1 ˆ1−ˆraˆr/prime bˆt/prime a1 ˆ1−ˆraˆr/prime bˆraˆtb 1 ˆ1−ˆr/prime bˆraˆr/prime bˆt/prime a1 ˆ1−ˆr/prime bˆraˆtb⎞ ⎠/parenleftbigg P0+ P2−/parenrightbigg . (35) The theory is completed imposing boundary conditions on the incoming electrons on both sides of the system. For normalelectrodes, one has P 0+=⎛ ⎜⎝μ0 0 0μ 0⎞ ⎟⎠,P2−=⎛ ⎜⎝μ2 0 0μ 2⎞ ⎟⎠, (36) where μ0andμ2are the respective chemical potentials of the two electrodes. The generalization to magnetic electrodesis done imposing different chemical potentials for majorityand minority electrons in the leads. By taking as boundaryconditions μ 0=eU, with Uthe potential difference between the two sides and μ2=0. The spin current ﬁnally reads J1=Nch 4πJeU, (37) where the spin current per channel and per unit of potential difference reads J=/vectorσ·1−ˆr/prime b ˆ1−ˆraˆr/prime bˆta. (38) Equation ( 37) allows one to compute the spin current in the region between two bulk materials of arbitrary thicknesses.In the multilayer considered here, the currents J a/b±δare calculated at positions δ=± 1 nm before and after the two magnetic layers, as shown in Fig. 4. The torque is the spin current absorbed by each layer, i.e., the quantity τj=Nch 4πfjeU, (39) withfj=Jj−δ−Jj+δ,j=a,b. C. From scattering matrices to CRMT To calculate the current in different regions of the multi- layer, one needs to calculate the matrices ˆS(L)a saf u n c t i o n FIG. 4. Illustration of the nanopillar simulated with CRMT. Electrons enter from the left and are spin-polarized along the direction ofMb. When they enter the thin layer, they change their polarization along the direction of Ma. Because of the conservation of angular momentum, the transverse component of the spin-polarisation is transferred to the thin and thick layers as a spin torque. of the position Linside the system. The main result of Refs. [ 13,14] is that the matrix ˆS(L+δL) for an inﬁnitesimal increment of the position δLis entirely characterized by two matrices /Lambda1tand/Lambda1r, deﬁned as ˆt(δL)=1−/Lambda1tδL, ˆr(δL)=/Lambda1rδL. (40) Once ˆS(δL) is known, one can make use of the addition law Eqs. ( 32) and ( 33) to obtain a differential equation to compute ˆS(L+δL). By taking the limit δL→0 one gets the two CRMT differential equations that describe hat matrices as afunction of the length of the system: ∂ˆr ∂L=/Lambda1r−/Lambda1tˆr−ˆr/Lambda1t+ˆr/Lambda1rˆr, (41) ∂ˆt ∂L=−/Lambda1tˆt+ˆr/Lambda1rt. (42) A bulk magnetic material is characterized by four independent parameters /Gamma1↑,/Gamma1↓,/Gamma1sf, and/Gamma1mx, which describe spin preserv- ing and spin ﬂip phenomena. With this parametrization, onehas for the transmission /Lambda1 t=⎛ ⎜⎝/Gamma1↑+/Gamma1sf 00 −/Gamma1sf 0 /Gamma1mx 00 00 /Gamma1∗ mx 0 −/Gamma1sf 00 /Gamma1↓+/Gamma1sf⎞ ⎟⎠, (43) and for the reﬂection /Lambda1r=⎛ ⎜⎝/Gamma1↑−/Gamma1sf 00 /Gamma1sf 00 00 00 00 /Gamma1sf 00 /Gamma1↓−/Gamma1sf⎞ ⎟⎠. (44) These four parameters correspond in turn to ﬁve different lengths. The two most important one are the mean freepaths for majority ( σ=↑) and minority ( σ=↓) electrons deﬁned as /lscript σ=1//Gamma1σ. Next comes the spin diffusion length lsf=[4/Gamma1sf(/Gamma1↑+/Gamma1↓)]−1/2. Last come the complex number /Gamma1mx=1/l⊥+i/lL, where l⊥is the penetration length of transverse spin current inside the magnet while lLis the Larmor precession length. Upon integrating Eq. ( 42), one obtains for the mixing transmission. Tmx(L)=e−L/l⊥−iL/l L, (45) which shows the exponential decay of the transverse spin current, absorbed by the layer, giving the phenomenon of 094428-7SIMONE BORLENGHI et al. PHYSICAL REVIEW B 96, 094428 (2017) spin-transfer torque. The two lengths l⊥andlLare of taken of 3 nm [ 14] so that spin-transfer torque is a phenomenon that occurs close to the interface. For bulk materials, anotherimportant quantity is the current polarization, P σ=T↑σ−T↓σ T↑σ+T↓σ. The latter decays exponentially as a function of the distance in the material, with spin-ﬂip length scale of the order of15 nm for Py, and reaches a constant value. The behavior ofspin and charge currents in a multilayer for different magneticconﬁgurations have been extensively studied in Ref. [ 14], to which we refer for a thorough discussion. The CRMT parameters ( /Gamma1 ↑,/Gamma1↓, and /Gamma1sf)a r ei n1 - 1 correspondence with the Valet-Fert (VF) parameters [ 15][ρ↑, ρ↓(resistivities for majority and minority electrons) and lsf (spin-ﬂip diffusion length)]. Using the standard notations for the average resistivity ρ∗and polarization β[ρ↑(↓)= 2ρ∗(1∓β)] one has [ 13] 1 lsf=2/radicalbig /Gamma1sf/radicalbig /Gamma1↑+/Gamma1↓, (46) β=/Gamma1↓−/Gamma1↑ /Gamma1↑+/Gamma1↓, (47) ρ∗ Rsh=(/Gamma1↑+/Gamma1↓)/4. (48) This parametrization does not ﬁx the mixing coefﬁcients /Gamma1mx, which only play a role in noncollinear conﬁgurations [ 22]. In the VF theory, the interfaces are characterized by the average resistance rb∗and polarization γwithr↑,↓= 2rb∗(1±γ). In the CRMT formalism, interfaces are modeled as a virtual material with their own transmission and reﬂectionhat matrices. Those are related to the VF parameters as follows: T ↑↑=(1+e−δ)/2 1+2(rb∗/Rsh)(1−γ), T↓↑=(1−e−δ)/2 1+2(rb∗/Rsh)(1−γ), T↑↓=(1+e−δ)/2 1+2(rb∗/Rsh)(1+γ), T↓↓=(1+e−δ)/2 1+2(rb∗/Rsh)(1+γ), (49) while for the reﬂection coefﬁcient one has R↑↑=1−1 1+2(rb∗/Rsh)(1−γ), R↓↓=1−1 1+2(rb∗/Rsh)(1+γ), (50) andR↑↓=R↓↑=0. At this point, the parametrization of CRMT is complete. Each material and interface are simulated separately, andthe whole system is recovered by applying the addition lawEqs. ( 32) and ( 33). Equations ( 41) and and ( 42) can be integrated analytically in some simple cases, to obtain the scattering matrix of anarbitrary bulk part [ 14]. For numerical purposes, instead of integrating numerically Eqs. ( 41) and ( 42) we have adopted a more efﬁcient procedure, by using directly Eqs. ( 32) and ( 33). In practice, one starts with an extremely small piece of material FIG. 5. Calculation of the angular dependence of spin torque (a) and resistance (b) obtained keeping mbﬁxed and rotating ma. Magnetoresistance hysteresis curve of the nanopillar for an in-plane magnetic ﬁeld. The dark (respectively, light) symbols indicate themagnetic ﬁeld being ramped up (respectively, down). described by Eqs. ( 40) and recursively doubles its size using the addition law Eqs. ( 32) and ( 33) until the full length Lof the layer is obtained. This leads to an extremely fast integrationtime, proportional to log L. To calculate magnetoresistance and spin torque as a function of the magnetic conﬁguration, we simulate transportthe nanopillar depicted in Fig. 4by keeping the magnetization M bﬁxed and rotating the magnetization Maof an angle θin spin space. Such rotation is obtained by applying to the hatmatrices of the thin layer the transformation Eq. ( 26). Figures 5(a) and5(b) show, respectively, the angular depen- dence of spin torque and resistance inside the nanopillar for asingle propagative channel and with eU=1. The resistance difference between parallel and antiparallel conﬁguration is27 m/Omega1. We remark that this calculation, performed without adjustable parameters, reproduces experimental data withinthe 10%. We note also that in our system, the current is spin-polarized by the thick layer and impinges the thin layer, exerting atorque that destabilizes its magnetization. Because of multiplereﬂections of spin polarized electrons between the two Pylayers, STT tends to stabilize the thick layer, increasing itseffective damping. Thus, at low current, one can consider thethick layer as “ﬁxed” and the thin layer as “free”. However,at high current both layers undergo a coupled precession (seenext section). It is instructive to plot the CRMT computation of the spin current proﬁles across the sample. Figure 6shows thezcomponent of the spin current j zalong the zaxis of the nanopillar. black circles correspond to the collinearconﬁguration, while other symbols correspond to differentrotation angles of the magnetization of the thin layer in thex-yplane. One can see that, in the collinear conﬁguration, j zdecreases continuously across the thin layer. However, in noncollinear conﬁgurations the current drops abruptly, sincethe transverse spin is absorbed by the thin Py layer throughspin-transfer torque. The highest jump occurs at an angle of 094428-8MICROMAGNETIC SIMULATIONS OF SPIN-TORQUE . . . PHYSICAL REVIEW B 96, 094428 (2017) FIG. 6. zcomponent of the spin current along the zaxis of the nanopillar. Black circles correspond to the collinear conﬁguration, while other symbols correspond to different rotation angles of themagnetization of the thin layer in the x-yplane. θ=0.7π. As one can see from Fig. 5, this corresponds to the maximum of spin-transfer torque, as expected. In Fig. 7we plot the current polarization Pσupon sending σ=↑,↓electrons with respect to the zaxis, for different magnetic conﬁgurations. In terms of transmission coefﬁcients,the polarization is deﬁned as [ 14] Pσ=T↑σ−T↓σ T↓σ+T↑σ. (51) The behavior of the current polarization is similar to that observed in Ref. [ 38]. When injected into the sample, the polarization is ±1, respectively, for ↑and↓electrons, and remain essentially the same through the non magnetic layers.When entering Py, the polarization rapidly changes and decaysexponentially to the bulk value. The typical decay length for Pyis around 20 nm. Across the thinner Py layer, the polarizationchanges abruptly in noncollinear conﬁgurations, due to thefact that the current is absorbed by the layer. The largest jumpof the spin polarization occurs for θ=0.7π, corresponding to the peak of spin-transfer torque. FIG. 7. Spin polarization for ↑(a) and ↓(b) electrons.TABLE IV . Bulk VF parameters for the materials which consti- tute the multilayer. The VF bulk resistivity ρ∗is expressed in units 10−9/Omega1m and the spin-ﬂip lsflength in nm. Material ρ∗ β 1/lsf Cu 5 0 0.002 Au 20 0 0.033 Py 291 0.76 0.182 V . SIMULATIONS OF CURRENT DRIVEN DYNAMICS This section contains the main results of the paper. Here solve simultaneously the transport equations coupled to theLLG equation with spin-transfer torque, by coupling CRMTto Nmag. The approach described here is valid for one-dimensional systems, where the magnetization varies onlyalong the direction of propagation of electrons. In the presentcase, the magnetization is uniform along zinside the material, but its dynamics is different in each layer. A generalization ofCRMT for a fully three-dimensional spin transport has beenrecently developed [ 37]. However, for the case considered here the one-dimensional CRMT approach captures the physicswell and is extremely fast. The other advantage of CRMTis that it is parametrized by the same set of experimentallyaccessible parameters as the VF theory (reported in Tables IV and V), so that no free parameter is needed to characterize realistic systems and materials. To include the effect of spin-transfer torque into Nmag, the LLG Eq. ( 13) at node iof disk j=a,bneeds to be modiﬁed as follows: ˙m j i=−γ/parenleftbig mj i×Hj effi/parenrightbig −αj Mj s/parenleftbig mj i×˙mj i/parenrightbig +gμB ¯hCM sτj iˆwj i. (52) Heregis the Landé factor, μBis the Bohr magneton, and Cis the volume associated to each site of the mesh. τj iis the torque given by Eq. ( 39). For our relatively homogeneous mesh, we have taken this volume as the total volume Vjof disk j=a,b divided by the total number of sites Nj. For simplicity, we have deﬁned the vector ˆwj i=mj i×(mj i×mj/prime i). To include STT in our micromagnetic simulations, we adopt the following self-consistent loop: (1) At time tand site iin disk j, Nmag computes the vector mj i. (2) The magnetic conﬁguration is passed to the CRMT solver which computes the torque τj i. TABLE V . VF parameters for the interfaces between magnetic and nonmagnetic material in our system. The VF interface resistivity is expressed in units of 10−15/Omega1m2. Interface r∗ b γ Cu\|Py 0.5 0.7 Au\|Py 0.5 0.77 094428-9SIMONE BORLENGHI et al. PHYSICAL REVIEW B 96, 094428 (2017) FIG. 8. Schematic of the method adopted to couple CRMT to Nmag, with the nanopillar viewed from proﬁle and the current ﬂowing along the zaxis. (a) One considers the sites lying on the inner surfaces of disks aandbthat face each other. Then one selects a site of surface 1 (thin layer, red dots) and looks for its nearest neighbor among the sitesof surface 2 (thick layer, blue dots). (b) To each couple of such sites, one associates a column k, which corresponds to a CRMT system, with magnetization m k 1andmk 2. The system is then represented as an assembly of CRMT columns connected in parallel. From the magnetic conﬁguration in each column k, STT and conductance are computed. (c) Illustration of the system divided into CRMT pillars, view fromtop. (3) Then the quantity mj i(t)+τj i(t)/Delta1tis set as new initial condition for the Nmag solver, which performs the timeintegration of the LLG equation between times tandt+/Delta1t. At this point, spin torque is recalculated as a function of the new magnetic conﬁguration and the loop starts again. Theintegration time step /Delta1tis of the order of the ps. Since Nmag uses a ﬁnite element mesh while CRMT a ﬁnite difference discretization of space, to implement the coupling,we divide the mesh into columns of sites, each column k representing a CRMT pillar with cross section S k. The whole system is then considered as an assembly of CRMT pillarsconnected in parallel, as shown in Fig. 8. Each column contains two sites, lying on surfaces of thin (a) and thick ( b) layer facing each other, with magnetizations m ak(t), and mbk(t), correspondingly. From the angle θk between these two sites, one obtains the torques τj k(θ) that act on the magnetization vectors mj k, and the resistance Rk(θk). This procedure takes into account the three-dimensional texture of the magnetization, while electronic transport is along zonly, without considering the lateral diffusion of spins. Since the system is an assembly of columns connected in parallel, the current Ikﬂowing in each column is given by the total current Idcdivided by the number of columns Nk, which in our system corresponds to the number of sites lying at the surface of each disk. Each column has FIG. 9. Time evolution of Mzfor different values of the dc current. (a) Subcritical regime, where both magnetization vectors are aligned. (b) Slightly higher than critical current, where the thin layer precesses and the thick layer stand still. (c) High current, with reversal of thin layer and coupled precession. a cross section Sk≈S/N k. The current Ikis related to the potential difference between the conductors via Ohm’slaw:eU=R k(θ)Idc/Nk, where the resistance Rkdepends on the transmission probability Tk(θ) for an electron to cross column k: RK=Rsh Tk(θ)Sk. (53) Inserting Eq. ( 53) into Eq. ( 39), and recalling that Rsh= h/(e2Nch), the torque ﬁnally reads τjk=gμB 2eVjNj NkIdcfjk(θ) Tk(θ)ˆwj. (54) We remark that Eq. ( 54) is quite general and can be applied to systems with arbitrary geometry, provided that electronictransport is one-dimensional only. VI. RESULTS AND DISCUSSION We turn now to the discussion of the coupled Nmag- CRMT simulations. A qualitative description of the differentdynamical regimes is given by the time evolution of M z, displayed in Fig. 9. Figure 9(a) shows the region beyond critical current, where both the magnetisations are ﬁxed andaligned with the zaxis. Figure 9(b) shows the region slightly beyond critical current, where the thin layer precesses andthe thick one remains ﬁxed. Finally, Fig. 9(c) displays the high current regime, where the thin layer is reversed and both layers undergo coupled precession. This behavior is due to therepulsive character of dipolar interaction: as the magnetizationof the thin layer reverses, it repels the magnetization of thethick one, causing it to precess. Figure 10shows the resistance as a function of the dc current, for values of the applied ﬁeld between 1T and1.3T. One can recognize clearly the three dynamical regions discussed above. In particular, the resistance starts increasingaround −6 mA, indicating the region of critical currents. The resistance increases with the current until it reaches a peak 094428-10MICROMAGNETIC SIMULATIONS OF SPIN-TORQUE . . . PHYSICAL REVIEW B 96, 094428 (2017) FIG. 10. Resistance difference /Delta1R between parallel and antipar- allel conﬁguration as a function of the dc current Idc, calculated for different values of the applied ﬁeld Hext. between −13 and −16 mA, corresponding to the reversal of the magnetization maof the thin layer. Note that at increasing ﬁeld the resistivity graph moves rightwards towards higher currents. This is due to the factthat the precession frequency increases proportionally tothe applied ﬁeld, so that the damping /Gamma1 a+=≈ 2αaωaalso increases and a higher current is needed to compensate it. The power spectrum of the system, with the modes /lscript= 0 and +1, is shown, respectively, in Figs. 11(a) and 11(b) in logarithmic scale. The modes A00andA10, corresponding to the dynamics of the thin layer, increase with the currentand dominate the spectrum around I dc=− 5 mA. Beyond the critical threshold, their frequency increases with the ﬁeld. Onthe other hand, the modes B 00andB10, which dominate the spectrum at zero currents, decrease until they almost disappearnear the critical threshold. Figure 12shows the line width and frequencies of the modes A 00andA10as a function of the dc current. In Fig. 12(a) , one FIG. 11. SW power spectrum of the /lscript=0 (a) and /lscript=+ 1 (b) modes, computed for different values of the dc current. The yellow squares (respectively, circles) indicate the positions of theB(respectively, A) modes. FIG. 12. Line widths (a) and frequencies (b) of the modes A00 andA10as a function of the dc current near the critical threshold. The symbols are numerical calculations, while dashed lines are linear ﬁts. can see that the line widths of both modes decrease linearly of an order of magnitude at increasing current, and they vanish atthe critical threshold, where the damping is compensated andthe system begins to auto oscillate. Note that the two modeshave slightly different critical currents, of about 5 ( /lscript=0) and 5.5 (/lscript=1) mA. The behavior of the spectrum and the critical currents agree with the experimental result of Refs. [ 16,38]. The frequencies of the excited modes remain constant untilthe critical threshold and then starts increasing linearly witha rate around 2 GHz per mA. This behavior also agrees withexperimental data. We conclude this section by noting that the physics discussed here is very similar to that of the spin-caloritronicsdiode [ 17]. In Fig. 13, one can see that increasing the electrical dc current leads to an increase of the magnon current betweenthe two disks. This current describes the transfer of magneticmoment M zbetween the two disks and corresponds to the usual SW current written for a system of only two spins[17–20]. The increase of the SW current with the dc current is due to the fact that spin-transfer torque excites only onemode, which eventually dominates the spectrum, inducing a FIG. 13. Magnetization current from thick to thin layer as a function of dc current Idc, computed for different values of the applied ﬁeld. 094428-11SIMONE BORLENGHI et al. PHYSICAL REVIEW B 96, 094428 (2017) phase locking between the two disks. In this kind of discrete system, the magnon current is indeed a measure of the phasesynchronization of the system. VII. CONCLUSIONS By coupling CRMT to micromagnetic simulations with the Nmag code we allow description, on an equal footingand without free parameters, of both transport and magneticdegrees of freedom. The results of this work have importantconsequences regarding the characterization and the optimiza-tion of the performance of STNOs. Using the developed method, we have identiﬁed the nature of the modes that auto-oscillate when the current exceeds thecritical threshold. In particular, we have predicted differentcritical currents for the modes A 00andA10, and a nonlinear frequency shift of the order of 2 GHz per mA. The precisedetermination of the SW mode symmetry of the auto-oscillating mode is important for the phase synchronizationof a STNO to an external source. In fact, it will be successfulonly if the latter can couple efﬁciently to the spin-transferdriven auto-oscillation modes, i.e., if it has the appropriatesymmetry. The ﬂexibility of CRMT and of the procedure to couple it with Nmag allows one to use our numerical tool to simulatedifferent geometries and materials. Further development andinvestigations are possible. A multiscale approach which com-bines systematically CRMT with a fully quantum approachhas been implemented [ 14]. This should allow to compute current driven dynamics in a large variety of systems, includingmultiterminal devices and tunnel junctions.We remark that in our simulations we have not taken into account the lateral diffusion of spins, since the system isdescribed using one dimensional CRMT columns, whereelectrons propagate only along the zdirection. This 1D model of transport is effective to describe se- lection rules into a perpendicularly magnetized nanopillarwith magnetic ﬁeld applied along e z, but more complicated conﬁgurations (such as magnetic vortices and multiterminalspin valves) may require a fully three-dimensional description. Finally, further investigation is necessary to understand the behavior of the system at high current, where nonlineareffects (such as the dependence of the Gilbert damping oncurrent [ 21]) may play an important role. The present work can be considered as an intermediate step toward a fully3D description of transport and magnetization dynamics inrealistic systems. ACKNOWLEDGMENTS We thank R. Lassalle-Balier and J. Dubois for fruitful discussions and X. Waintal, O. Klein, and G. de Loubensfor useful comments and support in the analysis of theexperiments. Financial support from Vetenskapsrådet (VR),Carl Tryggers Stiftelse (CTS), Stiftelsen Olle Engqvist Byg-gmästare and Swedish Energy Agency (STEM) is grate-fully acknowledged. The computations were performed onresources provided by the Swedish National Infrastructurefor Computing (SNIC) at the Swedish Super computingCenter (NSC), Linköping University, the PDC Centre for HighPerformance Computing (PDC-HPC), KTH, and the HighPerformance Computing Center North (HPC2N), UmeåUni-versity. [1] L. Berger, J. Appl. Phys. 49,2156 (1978 ). [2] L. Berger, Phys. Rev. B 54,9353 (1996 ). [3] J. C. Slonczewski, J. Magn. Magn. Mater. 159,L1(1996 ). [4] M. N. 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PhysRevB.82.100413.pdf	Nonlinear magnetization relaxation of superparamagnetic nanoparticles in superimposed ac and dc magnetic bias ﬁelds Serguey V . Titov,1Pierre-Michel Déjardin,2Halim El Mrabti,2and Yuri P. Kalmykov2 1Institute of Radio Engineering and Electronics, Russian Academy of Sciences, 1, Vvedenskii Square, Fryazino 141190, Russia 2LAMPS, Université de Perpignan Via Domitia, 52, Avenue de Paul Alduy, 66860 Perpignan Cedex, France /H20849Received 9 July 2010; revised manuscript received 9 August 2010; published 24 September 2010 /H20850 The nonlinear ac response of the magnetization M/H20849t/H20850of a uniaxially anisotropic superparamagnetic nano- particle subjected to both ac and dc bias magnetic ﬁelds of arbitrary strengths and orientations is determined byaveraging Gilbert’s equation augmented by a random ﬁeld with Gaussian white-noise properties in order tocalculate exactly the relevant statistical averages. It is shown that the magnetization dynamics of the uniaxialparticle driven by a strong ac ﬁeld applied at an angle to the easy axis of the particle /H20849so that the axial symmetry is broken /H20850alters drastically leading to different nonlinear effects due to coupling of the thermally activated magnetization reversal mode with the precessional modes of M/H20849t/H20850via the driving ac ﬁeld. DOI: 10.1103/PhysRevB.82.100413 PACS number /H20849s/H20850: 75.40.Gb, 05.40.Jc, 75.50.Tt, 76.20. /H11001q Fine ferromagnetic particles are characterized by thermal instability of their magnetization M/H20849t/H20850resulting in spontane- ous change in their orientation from one metastable state toanother by surmounting energy barriers, giving rise to super-paramagnetism which is very important in information stor-age and rock magnetism as well as in biomedicalapplications. 1,2Due to the large magnitude of the magnetic dipole moment /H20849/H11011104–105/H9262B/H20850giving rise to a relatively large Zeeman energy even in moderate external magneticﬁelds, the magnetization reversal process has a strong-ﬁelddependence causing nonlinear effects in the dynamic suscep-tibility and ﬁeld-induced birefringence, 2,3stochastic resonance,4–6dynamic hysteresis,7,8etc. However, nonlinear response to an external ﬁeld represents an extremely difﬁculttask even for dilute systems because it always depends onthe precise nature of the stimulus. Thus no unique response function valid for all stimuli exists unlike in linear response. 9 The nonlinear magnetic response of an individual super- paramagnetic nanoparticle in the presence of the thermal agi-tation can be evaluated 2,3,9by calculating the relevant statis- tical averages from Gilbert’s /H20849or Landau-Lifshitz /H20850equation augmented by a random ﬁeld h/H20849t/H20850with Gaussian white-noise properties, accounting for thermal ﬂuctuations of M/H20849t/H20850due to the heat bath, viz.,10 /H11509tM/H20849t/H20850=/H9253/H20853M/H20849t/H20850/H11003/H20851−/H11509MV/H20849t/H20850−/H9257M˙/H20849t/H20850+h/H20849t/H20850/H20852/H20854. /H208491/H20850 Here/H9253is the gyromagnetic ratio, /H9257is the damping param- eter, and V/H20849M,t/H20850is the free energy per unit volume. This is made up of the nonseparable Hamiltonian of the magnetic anisotropy U/H20849M/H20850and Zeeman energy densities, the latter arising from external magnetic dc and ac ﬁeldsH 0+Hcos/H9275tof arbitrary strengths and orientations. Now the nonlinear ac stationary response has hitherto been calcu-lated for uniaxial superparamagnets either /H20849i/H20850by assuming the energy of a particle in external ﬁelds is much less thanthe thermal energy kTso that the response may be evaluated viaperturbation theory /H20849e.g., Refs. 3and11/H20850or/H20849ii/H20850by as- suming that strong external ﬁelds are directed along the easyaxis of the particle so that axial symmetry is preserved /H20849e.g., Refs. 2and12/H20850. Thus the results are very restricted. In par- ticular, the conventional assumption of axial symmetry ishardly realizable in nanoparticle systems under experimental conditions because the easy axes of the particles are randomly oriented in space. Furthermore, many interestingnonlinear phenomena /H20849such as damping dependence of the response and interplay between precession andthermoactivation 3/H20850cannot be included because in axial sym- metry no dynamical coupling between the longitudinal andtransverse /H20849or precessional /H20850modes of motion exists. In con- trast, discarding the above assumptions we shall now presentan exact nonperturbative method for the nonlinear magneti- zation relaxation of superparamagnetic particles with anarbitrary anisotropy potential Uin a strong ac driving ﬁeld superimposed on a strong dc bias ﬁeld of arbitrary orienta-tions. Moreover, taking as example uniaxial superparamag- nets, we shall demonstrate that for arbitrary orientations of ac and dc bias ﬁelds /H20849so breaking the axial symmetry /H20850, the mag- netization dynamics changes substantially leading to differ-ent nonlinear effects which cannot be treated via perturbationtheory. We remark in passing that nonlinear effects in relax-ation processes of superparamagnetic nanoparticles areclosely related to those in nonlinear dielectric relaxation andthe dynamic Kerr effect in molecular liquids and liquidcrystals, 11,13harmonic mixing in a cosine potential,14the nonlinear impedance of Josephson junctions,15ac-driven vor- tices in superconductors,16etc. Thus our approach can also be applied to nonlinear effects in these. When the magnitude of the ac ﬁeld H/H20849t/H20850is so large that the Zeeman energy of a particle is comparable to or higherthan kT, one is faced with an intrinsically nonlinear problem which of course cannot be treated by perturbation theory andwhich we solve as follows. First we transform the stochasticGilbert Eq. /H208491/H20850to an inﬁnite hierarchy of stochastic differential-recurrence relations which on averaging overtheir realizations using the properties of white noise yielddifferential-recurrence relations for the statistical moments/H20855Y l,m/H20856/H20849t/H20850/H20849the expectation values of the spherical harmonics Yl,m/H20850, viz.,9,17 /H9270N/H11509t/H20855Yl,m/H20856/H20849t/H20850=/H20858 s,rel,m,l+r,m+s/H20849t/H20850/H20855Yl+r,m+s/H20856/H20849t/H20850, /H208492/H20850 where /H9270N=/H92700/H20849/H9251+/H9251−1/H20850,/H9251=/H9253/H9257Msis a dimensionless damping constant, /H92700=/H9252MS//H208492/H9253/H20850is the free-rotational diffusion timePHYSICAL REVIEW B 82, 100413 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 1098-0121/2010/82 /H2084910/H20850/100413 /H208494/H20850 ©2010 The American Physical Society 100413-1ofM/H20849t/H20850,MSis the saturation magnetization, /H9252=v//H20849kT/H20850, and vis the volume of the particle. Equation /H208492/H20850was derived in Ref. 16and we may now solve it exactly as follows. By introducing column vectors cn/H20849t/H20850/H20849n=1,2,3,... /H20850with c0=/H20855Y00/H20856=1 //H208814/H9266, which are formed from the statistical mo- ments cl,m/H20849t/H20850=/H20855Yl,m/H20856/H20849t/H20850, Eq. /H208492/H20850becomes the matrix recur- rence equation /H9270N/H11509tcn/H20849t/H20850=qn−cn−1/H20849t/H20850+qncn/H20849t/H20850+qn+cn+1/H20849t/H20850 +/H20851pn−cn−1/H20849t/H20850+pncn/H20849t/H20850+pn+cn+1/H20849t/H20850/H20852/H20849ei/H9275t+e−i/H9275t/H20850, /H208493/H20850 where the supermatrices qn,qn/H11006and pn,pn/H11006are generated from the coefﬁcients el,m,l+r,m+s/H20849t/H20850. We remark in passing that the explicit form of the vectors cn/H20849t/H20850depends on the type of the free-energy density V/H20849/H9277,/H9272/H20850; for uniaxial superparamag- nets, cn/H20849t/H20850are given by Eq. /H208497/H20850below. Since we are solely concerned with the stationary ac response, which is indepen-dent of the initial conditions, we require the steady-state so-lution of Eq. /H208493/H20850only. In the steady-state response, symmetry under time translation is retained under the discrete timetransformation t→t+2 /H9266//H9275. Thus we may seek all the cn/H20849t/H20850 in the form of the time Fourier series cn/H20849t/H20850=/H20858k=−/H11009/H11009cnk/H20849/H9275/H20850eik/H9275t. On substituting this series into Eq. /H208493/H20850, we have the recur- rence relations for the Fourier amplitudes cnk/H20849/H9275/H20850, namely, qnk/H20849k/H9275/H20850cnk/H20849/H9275/H20850+qn+cn+1k/H20849/H9275/H20850+qn−cn−1k/H20849/H9275/H20850 +pn−/H20851cn−1k−1/H20849/H9275/H20850+cn−1k+1/H20849/H9275/H20850/H20852+pn/H20851cnk−1/H20849/H9275/H20850+cnk+1/H20849/H9275/H20850/H20852 +pn+/H20851cn+1k−1/H20849/H9275/H20850+cn+1k+1/H20849/H9275/H20850/H20852=0 , /H208494/H20850 where qn/H20849k/H9275/H20850=−ik/H9270N/H9275I+qnand Iis the identity matrix. Now Eq. /H208494/H20850can be transformed into the matrix recurrence relations Q1C1+Q1+C2=R, Qn−Cn−1+QnCn+Qn+Cn+1=0 /H20849n/H110221/H20850, /H208495/H20850 where the column vectors RandCnand the tridiagonal su- permatrices QnandQn/H11006are deﬁned as Cn/H20849/H9275/H20850=/H20898] cn−2/H20849/H9275/H20850 cn−1/H20849/H9275/H20850 cn0/H20849/H9275/H20850 cn1/H20849/H9275/H20850 cn2/H20849/H9275/H20850 ]/H20899,R=−1 /H208814/H9266/H20898] 0 p1− q1− p1− 0 ]/H20899, /H20851Qn/H20852l,m=/H9254l−1,mpn+/H9254l,mqn/H20849m/H9275/H20850+/H9254l+1,mpn, /H20851Qn/H11006/H20852l,m=/H9254l−1,mpn/H11006+/H9254l,mqn/H11006+/H9254l+1,mpn/H11006. The exact solution of Eq. /H208495/H20850forC1can be now given using matrix continued fractions, viz.,C1=S1·R, /H208496/H20850 where the matrix continued fraction S1is deﬁned by the re- currence equation Sn=− /H20851Qn+Qn+Sn+1Qn+1−/H20852−1. The vector C1contains all the Fourier amplitudes required for the nonlinear stationary response. We emphasize that sofar our matrix continued fraction solution, Eq. /H208496/H20850, is valid for an arbitrary anisotropy potential U. Next we shall apply the above general method to the par- ticular case of uniaxial superparamagnets subjected to the acand dc bias ac ﬁelds H 0+Hcos/H9275tapplied in arbitrary direc- tions, where the free energy can be written in dimensionlessform as /H9252V=/H9268sin2/H9258 −/H92640/H20849/H92531sin/H9277cos/H9272+/H92532sin/H9277sin/H9272+/H92533cos/H9277/H20850 −/H9264cos/H9275t/H20849/H92531/H11032sin/H9277cos/H9272+/H92532/H11032sin/H9277sin/H9272+/H92533/H11032cos/H9277/H20850. Here/H92531,/H92532,/H92533and/H92531/H11032,/H92532/H11032,/H92533/H11032are the direction cosines of the vectors H0andH, respectively, Kis the anisotropy constant, /H9268=/H9252K,/H92640=/H9252H0MS, and /H9264=/H9252HM S. Now the vectors cn/H20849t/H20850 and the supermatrices qn,qn/H11006andpn,pn/H11006are given by cn/H20849t/H20850=/H20898/H20855Y2n,−2n/H20856/H20849t/H20850 ] /H20855Y2n,2n/H20856/H20849t/H20850 /H20855Y2n−1,−2 n+1/H20856/H20849t/H20850 ] /H20855Y2n−1,2 n−1/H20856/H20849t/H20850/H20899, pn−=/H20873oo b2n−1o/H20874,pn=/H20873a2n b2n d2n−1a2n−1/H20874, pn+=/H20873od 2n oo/H20874,qn−=/H20873V2n o W2n−1V2n−1/H20874, qn+=/H20873Z2nY2n oZ 2n−1/H20874,qn=/H20873X2nW2n Y2n−1X2n−1/H20874. /H208497/H20850 Here oand0are zero matrices and vectors of appropriate dimensions, respectively. The tridiagonal submatricesa l,bl, and dlhave the dimensions /H208492l+1/H20850/H11003/H208492l+1/H20850, /H208492l+1/H20850/H11003/H208492l+3/H20850, and /H208492l+1/H20850/H11003/H208492l−1/H20850, respectively. Their matrix elements are given by /H20849al/H20850n,m=/H9254n−1,mal,−l+m−+/H9254n,mal,−l+m−1+/H9254n+1,mal,−l+m−2+, /H20849bl/H20850n,m=/H9254n,mbl,−l+m−1−+/H9254n+1,mbl,−l+m−2+/H9254n+2,mbl,−l+m−3+, /H20849dl/H20850n,m=/H9254n−2,mdl,−l+m+1−+/H9254n−1,mdl,−l+m+/H9254n,mdl,−l+m−1+, where an,m=−im/H9264/H92533/H11032 4/H9251,bn,m=−/H9264/H92533/H11032n 4/H20881/H20849n+1/H208502−m2 /H208492n+1/H20850/H208492n+3/H20850,TITOV et al. PHYSICAL REVIEW B 82, 100413 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 100413-2an,m+=−i/H9264/H20849/H92531/H11032−i/H92532/H11032/H20850 8/H9251/H20881/H20849n+m+1/H20850/H20849n−m/H20850, bn,m+=/H9264/H20849/H92531/H11032−i/H92532/H11032/H20850n 8/H20881/H20849n+m+1/H20850/H20849n+m+2/H20850 /H208492n+1/H20850/H208492n+3/H20850, dn,m=/H9264/H92533/H11032/H20849n+1/H20850 4/H20881n2−m2 /H208492n+1/H20850/H208492n−1/H20850, dn,m+=/H9264/H20849/H92531/H11032−i/H92532/H11032/H20850/H20849n+1/H20850 8/H20881/H20849n−m/H20850/H20849n−m−1/H20850 /H208492n+1/H20850/H208492n−1/H20850 an,m−=−/H20849an,−m+/H20850/H11569,bn,m−=−/H20849bn,−m+/H20850/H11569, and dn,m−=−/H20849dn,−m+/H20850/H11569. The ma- trices qn,qn/H11006consist of the ﬁve submatrices Vl,Wl,Xl,Yl, andZl/H20849they also appear in the linear response and are de- ﬁned explicitly, e.g., in Ref. 9, Chap. 9 /H20850. Having determined the amplitudes cl,mk/H20849/H9275/H20850from Eq. /H208496/H20850, we can evaluate the magnetization MH/H20849t/H20850=vMS/H20858k=1/H11009Re/H20851m1k/H20849/H9275/H20850eik/H9275t/H20852, where m1k/H20849/H9275/H20850=4/H20881/H9266 3/H20875/H92533/H11032c10k/H20849/H9275/H20850 +/H20849/H92531/H11032+i/H92532/H11032/H20850c1−1k/H20849/H9275/H20850−/H20849/H92531/H11032−i/H92532/H11032/H20850c11k/H20849/H9275/H20850 /H208812 /H20876. /H208498/H20850 Here we shall assume from now on that the vectors H0and Hare parallel and they lie in the XZplane of the laboratory coordinate system so that /H92531=/H92531/H11032=sin/H9274,/H92532=/H92532/H11032=0, and /H92533=/H92533/H11032=cos/H9274, where /H9274is the angle between H0and the Z axis is taken as the easy axis of the particle. For nonparallelac and dc ﬁelds, results will be presented elsewhere. For a weak ac ﬁeld, /H9264→0,/H927311/H20849/H9275/H20850=3m11/H20849/H9275/H20850//H9264deﬁnes the normalized linear dynamic susceptibility and our resultsagree in all respects with the benchmark linear-responsesolution. 18The plots of the magnetic loss spectrum −Im /H20851/H927311/H20849/H9275/H20850/H20852vs/H9275/H9270Nare shown in Fig. 1. Here the low- frequency behavior of /H927311/H20849/H9275/H20850can be described by a single Lorentzian, viz., /H927311/H20849/H9275/H20850 /H927311/H208490/H20850/H110151−/H9254 1+i/H9275/H9270+/H9254, /H208499/H20850 where /H9270is the longest relaxation time in the absence of an ac external ﬁeld and /H9254is a parameter /H20849here/H9270is associated with the reversal time of the magnetization;19simple analytic equations for /H9270and/H9254are given in Refs. 9and20/H20850. Equation /H208499/H20850is also plotted in Fig. 1. Now /H9270is related to the charac- teristic frequency /H9275max, where −Im /H20851/H927311/H20849/H9275/H20850/H20852reaches a maxi- mum, and/or the half-width /H9004/H9275of Re /H20851/H927311/H20849/H9275/H20850/H20852of the Lorent- zian as /H9270/H11015/H9275max−1/H11015/H9004/H9275−1. /H2084910/H20850 For zero dc bias ﬁeld, /H92640=0,/H9270is independent of the angle /H9274 while in a strong dc ﬁeld, e.g., for /H92640=3,/H9270substantially depends on /H9274/H20851Fig. 1/H20849b/H20850/H20852. In addition, a far weaker second relaxation peak appears at high frequencies. This relaxationband is due to the “intrawell” modes which are virtuallyindistinguishable in the frequency spectrum appearing as asingle high-frequency Lorentzian band. The third or ferro- magnetic resonance /H20849FMR /H20850peak /H20849due to excitation of trans- verse modes with frequencies close to the precession fre-quency /H9275prof the magnetization /H20850appears only at low damping and strongly manifest itself at high frequencies.Moreover, for /H9274=0, when the axial symmetry is restored, the FMR peak disappears /H20851curves 1 in Fig. 1/H20849a/H20850/H20852because the transverse modes no longer take part in the relaxation pro-cess so that this peak is a signature of the symmetry-breakingaction of the applied ﬁeld. In strong ac ﬁelds, /H9264/H110221, pronounced nonlinear effects occur /H20849see Fig. 2/H20850. In particular, the low-frequency band of −Im /H20851/H927311/H20849/H9275/H20850/H20852can no longer be approximated by a single Lorentzian. Nevertheless, Eq. /H2084910/H20850may still be used in order to estimate an effective reversal time of the magnetization /H9270. /H20849We remark that /H9270may also be evaluated from the spectra of the higher order harmonics12because the low-frequency parts of their spectra are also dominated by the magnetiza- tion reversal /H20850. The behavior of /H9275max /H20849and, hence, /H9270/H20850as func- tions of the ac ﬁeld amplitude depends on whether or not adc ﬁeld is applied. For a strong dc bias, /H92640/H110221, the low- frequency peak shifts to lower frequencies reaching a maxi-mum at /H9264/H11011/H92640thereafter decreasing exponentially with in- creasing /H9264. In other words, as the dc ﬁeld increases, the reversal time of the magnetization initially increases and then having attained its maximum at some critical value /H9264/H11011/H92640decreases exponentially /H20849see Fig. 2/H20850. For weak dc bias 0/H11349/H92640/H110211, the low-frequency peak shifts monotonically to higher frequencies. As seen in Fig. 2/H20849a/H20850, as the ac ﬁeld am-10−410−310−210−1100 10−510−310−110110310−410−310−21 5432 1σ=10 α=0.01 ξ0=0ξ=0.01−Im[χ1 1(ω)]1:ψ=0 2:ψ=π/6 3:ψ=π/4 4:ψ=π/3 5:ψ=π/2(a) 1(b) 543 2 1σ=10 α=0 . 0 1 ξ0=3 ξ=0.01 ωτN FIG. 1. /H20849Color online /H20850−Im /H20851/H927311/H20849/H9275/H20850/H20852vs/H9275/H9270Nfor various values of the angle /H9274and /H20849a/H20850/H92640=0 /H20849no dc bias ﬁeld /H20850and /H20849b/H20850/H92640=3 /H20849strong dc bias ﬁeld /H20850. Solid lines: matrix continued fraction solution; ﬁlled circles: Eq. /H208499/H20850with/H9270calculated using the method of Ref. 20. 10−410−310−210−1 10−510−310−110110310−410−310−2ξ0=3ξ=1σ=10 ψ=π/443 2−Im[χ1 1(ω)]1:ξ=0,01 2:ξ=1 3:ξ=3 4:ξ=5α=0.01 σ=10 ξ0=3ψ=π/4 1(a) (b) 42 3 1:α=0.01 2:α=0.1 3:α=1 4:α=10 ωτΝ1 FIG. 2. /H20849Color online /H20850−Im /H20851/H927311/H20849/H9275/H20850/H20852vs/H9275/H9270N/H20849a/H20850for various values of the ac ﬁeld parameter /H9264and /H20849b/H20850for various values of damping /H9251.NONLINEAR MAGNETIZATION RELAXATION OF … PHYSICAL REVIEW B 82, 100413 /H20849R/H20850/H208492010 /H20850RAPID COMMUNICATIONS 100413-3plitude increases, the FMR peak decreases and also broadens showing pronounced nonlinear saturation effects characteris-tic for a soft spring. This effect is very similar to that alreadyknown in atomic and molecular spectroscopy. 21Thus we see that the intrinsic damping dependence of the ac nonlinearresponse for the oblique ﬁeld conﬁguration /H20851see Fig. 2/H20849b/H20850/H20852 serves as a signature of the coupling between the longitudi-nal and precessional modes of the magnetization. Hence, itshould be possible to determine the damping coefﬁcient /H9251 from measurements of nonlinear response characteristics,2 e.g., by ﬁtting the theory to the experimental dependence of /H927311/H20849/H9275/H20850on the angle /H9274and the ac and dc bias ﬁeld strengths, so that the sole ﬁtting parameter is /H9251, which can be deter- mined at different temperatures T, yielding its temperature dependence. This is important because a knowledge of /H9251and itsTdependence allows for the separation of the various relaxation mechanisms.9 We now estimate the parameter range, where the nonlin- ear effects appear. For cobalt nanoparticles with mean diam-eter a/H1101110 nm and saturation magnetization M S/H110111460 G, the ﬁeld parameter /H9264forT/H1101130 K is on the order of unity for H/H110116kT //H20849/H9266a3MS/H20850/H110115 Oe. Furthermore, an ac ﬁeld of this order of magnitude is easily attained in measurements of thenonlinear response of magnetic nanoparticles; e.g., Bitoh et al. 22have measured the nonlinear susceptibility of cobalt nanoparticles in the temperature range 4.2–280 K using an acmagnetic ﬁeld /H110115–30 Oe. As far the characteristic time /H9270N is concerned, for /H9253/H110152/H11003107rad /Oe s and /H9251/H110110.1, we have at room temperature /H9270N/H1101110−8s. To conclude we have developed a nonperturbative ap- proach in terms of matrix continued fractions for the nonlin-ear relaxation of a uniaxial superparamagnetic particle forarbitrary strengths and orientations of the dc bias and ac driving ﬁelds. We have shown that the nonlinear ac station-ary response to a strong ac ﬁeld applied at an angle to theeasy axis of the particle /H20849so that the axial symmetry is bro-ken /H20850is very sensitive to both the ac ﬁeld orientation and amplitude owing to the coupling induced by the symmetry breaking driving ﬁeld between the precession of the magne-tization and its thermally activate reversal over the saddlepoint. In particular, the pronounced damping and ac ﬁeld dependence of the nonlinear response /H927311/H20849/H9275/H20850can be used to determine the damping coefﬁcient /H9251just as for higher har- monic responses.3We emphasize that these nonlinear effects in/H927311/H20849/H9275/H20850cannot be treated via perturbation theory. Our cal- culations, since they are valid for ac ﬁelds of arbitrarystrengths and orientations, allow one both to predict and in-terpret quantitatively nonlinear phenomena in magneticnanoparticles such as nonlinear magnetic susceptibility, non-linear stochastic resonance and dynamic hysteresis, nonlinearac ﬁeld effects on the switching ﬁeld curves, etc., whereperturbation theory and the assumption that axial symmetryis preserved are no longer valid /H20849these results will be pub- lished elsewhere /H20850. For practical applications /H20849e.g., in mag- netic nanoparticle hyperthermia 1/H20850, in order to account for the polydispersity of the particles of a real sample and the factthe easy axes of particles are randomly distributed in space,one must also average the nonlinear response functions m 1k/H20849/H9275/H20850over appropriate distribution functions2/H20849averaging of m1k/H20849/H9275/H20850over particle volumes and orientations can be readily accomplished numerically using Gaussian quadraturs23/H20850. Here, only uniaxial superparamagnetic particles have beentreated. Particles with nonaxially symmetric anisotropies /H20849cu- bic, biaxial, etc. /H20850can be considered in like manner. Finally, our results can be adapted to other nonlinear phenomenasuch as nonlinear dielectric relaxation and the dynamic Kerreffect in molecular liquids and liquid crystals. 11,13 The support of the work by the Agence Nationale de la Recherche, France /H20849Project No. ANR-08-P147-36 /H20850and by EGIDE, France /H20849Program ECO-NET, Project No. 21394NH /H20850 is gratefully acknowledged. 1Q. A. Pankhurst et al. ,J. Phys. D 42, 224001 /H208492009 /H20850. 2Yu. L. Raikher and V . I. Stepanov, Adv. Chem. Phys. 129, 419 /H208492004 /H20850. 3J. L. García-Palacios and P. Svedlindh, Phys. Rev. Lett. 85, 3724 /H208492000 /H20850. 4L. Gammaitoni et al. ,Rev. Mod. Phys. 70, 223 /H208491998 /H20850. 5Yu. L. Raikher et al. ,Phys. Rev. E 56, 6400 /H208491997 /H20850. 6Yu. L. Raikher and V . I. Stepanov, Phys. Rev. Lett. 86, 1923 /H208492001 /H20850; Yu. L. Raikher et al. ,J. Magn. Magn. Mater. 258-259 , 369 /H208492003 /H20850. 7I. Klik and Y . D. Yao, J. Appl. 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PhysRevB.74.054411.pdf	Giant nonlocal damping by spin-wave emission: Micromagnetic simulations Gerrit Eilers, Mario Lüttich, and Markus Münzenberg * IV . Physikalisches Institut, Universität Göttingen, 37077 Göttingen, Germany /H20849Received 21 March 2006; revised manuscript received 12 May 2006; published 9 August 2006 /H20850 A micromagnetic simulation scheme is employed to evaluate the damping processes occurring in all-optical pump-probe experiments. We reveal that small wavelength and high-energy, nanometer-sized domains formduring the ﬁrst stage of the relaxation process. The emission of spin waves propagating out of the excitationspot area dominates the energy dissipation process. This process becomes dominant for small excitation spotareas and increases the macroscopically observed damping constant /H9251drastically. We ﬁnd that the damping time can be increased by a factor of more than 100, depending on the geometry, and must be considered in theinterpretation of all-optical pump probe experiments with spot diameters less than 1 /H9262m. DOI: 10.1103/PhysRevB.74.054411 PACS number /H20849s/H20850: 72.25.Rb, 75.30.Fv, 75.75. /H11001a I. INTRODUCTION Evaluating the origin of different damping processes is to date one of the most challenging subjects in magnetizationdynamics. All-optical pump-probe experiments are a valu-able tool to study these processes. 1Typical time scales in ultrafast magnetization dynamics range from 100 fs /H20849all- optical pump-probe experiments /H20850to 100 ps /H20849precession ex- cited by ultrashort magnetic ﬁeld pulses /H20850.1–5Using femtosec- ond lasers, where the pulse length of the ultrafast lasersystem /H20849typically 50 fs to 100 fs /H20850determines the time reso- lution, so far provides the tool with the highest temporalresolution to probe the evolution of a nonequilibrium spinconﬁguration. Hereby the light ﬁeld of the ultrashort laserpulse acts onto the electron system. The development of aprecise description and understanding of these experiments isan important aspect of the studies of ultrafast magnetizationdynamics and allows identiﬁcation of the blocks building thecollective processes as a magnetic coherent precession of amacrospin. 6Two time scales generally determine the charac- teristics of the subsequent relaxation processes of the excitedsystem. On the ultrashort time scale, the thermalization ofthe electron system dominates the dynamics. The excess en-ergy is distributed to the lattice within the time scale of a fewphonon oscillation periods, involving also the spin system.The mechanisms that determine the time scale and the degreeof demagnetization are still widely under discussion and areprobably a sum of more than one elementary relaxation step.In the following, the details of the demagnetization processare neglected. The demagnetization originates most probablyfrom various scattering events, strongly connected with theelectron relaxation. On the large time scale from5 ps to 1 ns, three processes govern the dynamics: energytransfer by heat diffusion, excitation of stress pulses, andspin-wave excitations. The purpose in the following is toexamine the magnetization dynamics and spin-wave excita-tions generated by the ultrafast demagnetization. Spin wavesexcited by homogeneous pumping and the energy dissipationinvolved have been described already in Ref. 7. Recently especially local magnetic excitations in nanometer-sizedpoint contacts by spin currents have gained strong interestand reveal strong similarities. 8Then a relaxation time is ex- tracted and details of the relaxation mechanism are discussedfor various simulation patterns, from small structures starting from a few hundred nanometers to larger structures 20 /H9262mi n length. II. MICROMAGNETIC MODEL In the following, the detailed nature of the different de- magnetization processes described in a unifying theory byKoopmans 9is neglected. The electronic nature of the spin- split electronic states and the energy dissipation processesenter by the size of the exchange energy and the dampinginto the micromagnetic model. For a 5 ps to nanosecondstime scale the relaxation processes are known to be welldescribed by the physical processes included within theLandau-Lifshitz-Gilbert /H20849LLG/H20850equation. The typical energy scales, as there are exchange and dipolar energies, determinethe dynamics of the relaxation processes /H20849precessional fre- quency and angular motion /H20850by the strength of the effective ﬁeld from all neighboring cells adding at a cell site chosen. Itis important to notice that both interactions have intrinsiclength and time scales. While the exchange interaction domi-nates within a few nanometers, it can be neglected for largerdistances. By tilting the magnetization of a neighboring cell,the energy deposited within the exchange interaction can betransferred and distributed on a larger length scale. As a re-sult, spin waves are excited, owning a characteristic energygiven by the tilt angles between the cells cooperatively mov-ing as a spin-wave packet with a characteristic length ofexcitation. A frozen image after electron thermalization and subse- quent complete demagnetization within the laser spot area isthe starting point for our simulation. The demagnetized areais transcribed within a discretized model by rotating the mag-netization of each cell by a random angle. This increases theenergy locally by exchange energy between neighboringcells and an additional demagnetization ﬁeld arising from thespot area and results in a quenched magnetization. There areseveral approaches to implement temperature-excited ﬂuc-tuations and spin disorder. 10–12One common way to take into account a nonzero temperature is by adding a randomly ﬂuc-tuating ﬁeld contributing to the effective ﬁeld term in theLLG equation for each time step. Certainly, this descriptionis an approximation with limitations since the elementaryPHYSICAL REVIEW B 74, 054411 /H208492006/H20850 1098-0121/2006/74 /H208495/H20850/054411 /H208495/H20850 ©2006 The American Physical Society 054411-1processes happen on a single spin unit: in a micromagnetic simulation using a rectangular grid, the magnetic modes thatcan be excited are cut off by the wave vector k=2 /H9266/Lcell. Lcellis given by the cell size in the direction considered. However, it gives a valuable insight into the modular pro-cesses resulting in a restoration of the magnetization. We willdemonstrate in the following that it is possible to study thebasic relaxation processes in the exchange-energy or dipole-ﬁeld limit within this simpliﬁed micromagnetic toy model.Important conclusions for the interpretation of all-opticalpump-probe experiments, as well as ultrafast magnetic re-cording that involves bit sizes as small as a few 100 nm 2, are derived. The homogeneous magnetization within a domain serves as a target region for the laser spot; therefore, the demagne-tized spot is inserted into a previously relaxed Landau orvortex state. The dynamic micromagnetic simulations withinthe LLG equation were conducted using the O BJECT ORI- ENTED MICROMAGNETIC FRAMEWORK /H20849OOMMF /H20850code de- veloped at the National Institute of Standards and Technol-ogy/H20849NIST /H20850. 13The program approximates the continuum micromagnetic theory for a thin ﬁlm by a two-dimensionalgrid of square cells. Within each cell, a three-dimensionalmagnetization vector represents the magnetization. The ex-change energy is calculated using an eight-neighbor bilinearscheme. The Landau-Lifshitz-Gilbert equation is solved byintegration until a speciﬁed simulation time has elapsed tomaintain equal time steps. For the micromagnetic simulation,a damping constant of /H9251=0.01 and a Permalloy /H20849Py/H20850thin ﬁlm with zero crystalline anisotropy was used. The diameterof the excitation area is always 1/8 of the total length of thestructure. A limitation of the micromagnetic simulation isthat for large structures in the size range of a few microme-ters the cell size has to be increased to adapt the problem tothe computational capabilities. Therefore, the transition re-gion from small toward large structures was studied withspecial care and a cell size not larger than 20 nm was used. 14 For a review of pitfalls in magnetic modeling, see Refs. 13 and15. III. RESULTS AND DISCUSSION In Fig. 1the evolution of the remagnetization process is shown in the time range from 0 ps to 300 ps for a 1 /H9262m /H110030.5/H9262m structure with a Py thickness of 10 nm and a cell size of 2 nm. The starting conﬁguration is a vortex structureconﬁguration owning the lowest energy. 16As a color scale, theMXcomponent is plotted. On the right, the effective ﬁeld /H20849sum of exchange and dipolar energy /H20850is shown. After 35 ps, the outer region of the excitation spot is already signiﬁcantlyrelaxed by the emission of short wavelength spin waves. For300 ps the excited region completely disappeared. The shortwavelength excitations are consecutively damped within themagnetic nanostructure. The extension of the spin-wave pat-tern is seen best in the total effective ﬁeld plotted at thecorresponding time on the right of Fig. 1. Since selected wavelengths are dominating, the interference of the spinwaves emitted can be observed. In Fig. 2a horizontal cut through the middle of the structure is shown for 30 ps, 70 ps,and 300 ps. The excitation spot extends from y =125 to 200 nm. The Néel wall separating the two domainsis located at y=250 nm, where the M xcomponent changes from positive to negative direction. The spin waves emittedfrom the excitation spot area have reached the position y =344 nm for 30 ps. Passing through the domain wall, thespin waves show an altered sense of rotation. While beforepassing the domain wall the M zcomponent is /H9266/2 ahead of theMycomponent, the phase difference is − /H9266/2 after pass- ing through the wall. The inﬂuence on the phase of spinwave interacting with a domain wall has been described inRef. 17. Comparing the situation for 30 ps, 70 ps, and 300 ps one can observe that for longer time scales the higher fre-quency excitations have been strongly damped. This obser-vation is the key to discover the details of the remagnetiza-tion process. Within the excited area small domains areforming with a size of a few nanometers. Because of theirsmall size, the magnetic excitations own a high exchangeenergy that is distributed in the following by short wave-length propagating spin waves. They relax by dissipating en-ergy consecutively decaying into the lower order eigenmodesof the structure. These eigenmodes form by constructive anddeconstructive interference of the spin waves after multiplereﬂections at the verge of the structure. The number two tofour reﬂections for 300 ps after excitation, determined by thespin-wave velocity and the structure size, results already in a FIG. 1. /H20849Color online /H20850Micromagnetic simulation for a 0.5 /H9262m /H110031/H9262m Permalloy ﬁlm structure with a 125 nm demagnetized spot diameter with 10 nm thickness. On the left the evolution of thespin-wave emission from the excited area is shown. On the right,the total effective ﬁeld reﬂects the energy located within the domainwalls and spin waves excited. The color code “red-white-blue”/H20849white to black /H20850indicates “positive-zero-negative” /H20849the absolute /H20850 value of the xcomponent.EILERS, LÜTTICH, AND MÜNZENBERG PHYSICAL REVIEW B 74, 054411 /H208492006/H20850 054411-2characteristic pattern of higher order eigenmodes superim- posed onto each other /H20849Fig. 1, Ref. 18/H20850. The characteristic wavelength decreases from 1/40 of the pattern width for35 ps, to 1/20 and 1/12 for 75 ps and 300 ps, respectively. The emission of spin waves out of the excitation spot areahas an important consequence. Because of the nature of theemission along the boundary of the excited-nonexcited area,a strong inﬂuence on the damping is expected with the varia-tion of the excited area. In Fig. 3, the relaxation patterns for the 1 /H9262m large structure are compared to the relaxation pat- tern of a much larger structure 20 /H9262m in width. The situation shortly after excitation is shown for 0 ps and at a time of30 ps for the smaller structure and 700 ps for the largerstructure. The corresponding effective ﬁeld, including ex-change and dipolar contributions, is plotted on the right side.At 30 ps and 700 ps, respectively, the spin-wave relaxationpattern has moved by a similar length compared to the totalsize of the pattern in both examples. For 700 ps in the caseof the 1 /H9262m large structure, no excitation would be visible anymore in the spot region. The ﬁrst conclusion is that thetime scales for the relaxation of the patterns are differentwhen changing the size of the system as expected. Second,we observe a different nonsymmetric relaxation pattern inthe case of the larger structure. This points to a dominance ofthe dipolar ﬁelds. For the larger length scales they becomedominant over the exchange ﬁeld that is negligible then. Thedipole ﬁeld generated by a demagnetized hole in a continu-ous domain can be seen nicely for 0 ps in both structures.The transition between both relaxation mechanisms has beencarefully examined and appears at around 2 /H9262m in length of the structure. For the small structure a very different situationis found: the spin-wave emission pattern is centrosymmetricand is not determined by the long-range dipolar interaction.On these length scales, the driving force is the exchangeinteraction. The characteristic dynamic behavior of the sys- tem shows a crossover from exchange dominated to dipolardynamic relaxation modes. This is analogous and within thesame length scale range where the crossover is observed inthe static case. We deﬁne a relaxation time when the random orientation of the cells within the excitation area has com-pletely disappeared. From there we calculated a dampingtime /H9270damping , which is shown as a function of the excitation area diameter in Fig. 4. As a reference, we included the re- laxation time /H92700of a demagnetized circular nanostructure with a diameter of the laser spot. Clearly, the relaxation timeof a demagnetized circular nanostructure alone has no radia-tive damping contribution. The observed damping time /H9270damping deviates drastically from the intrinsic value for Per- malloy for a small angle oscillation that is in the order of afew nanoseconds. A damping time of 20 ps only is found forexcitation spots approaching 20 nm diameter. Also, for thereference structure, the damping time decreases below 1 /H9262m diameter. High-energy eigenmodes result in a high preces-sion frequency /H9275/H11011k2and thus stronger damping, but still the difference, marked as the shaded region, amounts to /H9270damping //H92700/H110151/40 for a 20 nm spot size. Concluding from the previous paragraph, the emission of spin waves observed plays an important role in the dampingprocess. The additional energy introduced by the magneticdisorder within the laser spot region is dissipated by the in-trinsic, local damping parameter /H9251within the LLG equation on one hand. This is analogous to the precessional motionand subsequent damping of a macrospin in every single cell.The ground state forms again if all excess energy is dissi-pated. The domain conﬁguration is “relaxed”. Miltat has FIG. 2. /H20849Color online /H20850Fort=30 ps, 70 ps, and 300 ps the mag- netization components for a cut along the ydirection are shown. The width of the averaging is 2 nm. The lower spectra are verticallyshifted for clarity. FIG. 3. /H20849Color online /H20850Micromagnetic simulation for a 0.5 /H9262m /H110031/H9262ma n da1 0 /H9262m/H1100320/H9262m Permalloy ﬁlm with 10 nm thick- ness. The demagnetized spot diameter is 125 nm and 1.25 /H9262m, re- spectively. In the upper part, the evolution of the spin-wave emis-sion from the excited area is shown. In the lower part, the totaleffective ﬁeld reﬂects the energy located within the domain wallsand spin waves excited. The color code “red-white-blue” /H20849white to black/H20850indicates “positive-zero-negative” /H20849the absolute /H20850value of the xcomponent.GIANT NONLOCAL DAMPING BY SPIN-WA VE ¼ PHYSICAL REVIEW B 74, 054411 /H208492006/H20850 054411-3shown19that as a general rule for a macroscopic observation the observed will always be larger than the intrinsic that onewould observe microscopically. This plays an important role,especially for strong deviations from a simple macrospinmodel and quasichaotic excitations /H20849e.g., for high and ul- trashort ﬁeld pulses applied /H20850. This additional damping con- tribution is called local because the energy dissipation hap-pens in the same place where the precession is observed.There are also damping mechanisms by spin currents, dis-covered very recently, that act as a nonlocal damping. 20–23 Here the additional damping is a nonlocal damping process by spin waves. It was already proposed in Ref. 6that these excitations might play a role in all-optical pump-probe ex-periments. The energy is dissipated over the magneticallynondisturbed sample and consecutively damped. This helpsto distribute the energy all over the sample and acts as anadditional sink for the energy deposited within the laser spotarea. By decreasing the excitation area, the ratio area to outerboundary increases. In magnetic nanocontacts with diametersless than 100 nm, the radiative damping determines thethreshold of a spin-wave excitation by spin currents. Thecontribution of radiative damping was estimated in a simplemodel and was found to depend on the spot size Ras /H9270=1//H20849c1+c2/R2/H20850. An analysis using c1=3/H1100310−4ps and c2 =2/H1100310−4ps//H9262m2is given as a reference in Fig. 4. Thus, in the limit of the larger diameters than a few micrometers, theemission of spin waves contributes to the intrinsic dampingwith less than a few percent. In conventional all-opticalpump-probe geometry with typical pump-spot diameters ofmore than 10 /H9262m, the contribution can be neglected. In the limit of diameters approaching a single spin exitation, therelaxation time approaches a few hundred femtoseconds.This corresponds to the time scale observed in all-opticalpump-probe experiments for ferromagnetic thin ﬁlms. Herethe exchange ﬁeld dominates the effective ﬁeld. 9Precession frequencies of /H9275/2/H9266=100 THz can be reached. With an in- trinsic Gilbert damping of /H9251=0.01, one can approximate the resulting relaxation time to /H9270/H110111//H9251/H9275=150 fs.9The energy is distributed on the surrounding cells. Since these time scalesare beyond the range where the approximations used to de-rive the LLG equation for the description of a spin ensembleare valid, these values have to be regarded as a rather coarseapproach to describe the processes ongoing on a 100 fs timescale. IV. CONCLUSION In conclusion, we discussed the magnetization dynamics after instantaneous demagnetization of a ferromagnetic struc-ture by a laser pulse, accessible by the Landau-Lifshitz-Gilbert equation within a micromagnetic simulation. Withinthe text we discussed the dipolar interactions dominating inmagnetic structures above some micrometer length scale.The long-range dipolar interaction gives rise to a character-istic spin-wave emission pattern. Within the excited area,small magnetic high-energy ﬂuctuations form and continueto increase in size as the energy is consecutively dissipated.The emission of spin waves from the excitation spot contrib-utes signiﬁcantly to the relaxation of the magnetizationwithin the spot region and acts as an additional nonlocaldamping term. The energy dissipation by spin waves emittedfrom the excitation spot and results in a strong decrease ofthe relaxation times to up to 20 ps for the smallest spot sizesin the range of 20–200 nm. Spin-wave generation will there-fore have an effect on the thermally assisted writing processin the future generation of magnetic hard disks. ACKNOWLEDGMENT Support by the Deutsche Forschungsgemeinschaft within the priority program SPP 1133 is gratefully acknowledged. *Corresponding author. Email address: mmuenze@gwdg.de 1M. Djordjevic, M. Lüttich, P. Moschkau, P. Guderian, T. Kamp- frath, R. G. Ulbrich, M. Münzenberg, W. Felsch, and J. S.Moodera, Phys. 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The blue /H20849dark gray /H20850line through the data is an analysis using the formula given in the text.EILERS, LÜTTICH, AND MÜNZENBERG PHYSICAL REVIEW B 74, 054411 /H208492006/H20850 054411-45I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann, J. Stöhr, G. Ju, B. Lu, and D. Weller, Nature /H20849London /H20850428, 831 /H208492004/H20850; C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, D. Weller, E. L. Garwin, and H. C. Siegmann, Science 285, 864 /H208491999/H20850. 6M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W. J. M. de Jonge, and B. Koopmans, Phys. Rev. Lett. 88, 227201 /H208492002/H20850. 7V . S. L’vov, Wave Turbulence Under Parametric Excitation /H20849Springer-Verlag, Berlin, 1994 /H20850, Chap. 6. 8J. Slonczewski, J. Magn. Magn. Mater. 195, L261 /H208491999/H20850;A . Slavin and P. Kabos, IEEE Trans. Magn. 41, 1264 /H208492005/H20850;A . Slavin and V . Tiberkevich, Phys. Rev. Lett. 95, 237201 /H208492005/H20850. 9B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, and W. J. M. de Jonge, Phys. Rev. Lett. 95, 267207 /H208492005/H20850. 10D. V . Berkov and N. L. Gorn, Phys. Rev. B 57, 14332 /H208491998/H20850. 11J. Fidler and T. Schreﬂ, J. Phys. D 33, R135 /H208492000/H20850. 12U. Nowak, O. N. Mryasov, R. Wieser, K. Guslienko, and R. W. Chantrell, Phys. Rev. B 72, 172410 /H208492005/H20850. 13M. Donahue and B. McMichael, http://math.nist.gov/oommf/; M. J. Donahue and R. D. McMichael, Physic aB&C 233, 272 /H208491997/H20850. 14We want to state here that micromagnetic problems are not scal- able because of the different length scales intrinsic to the ex-change and dipolar interactions involved. Therefore, to studylarger structures and length scales, the use of larger cell sizes hasto be taken into account. On the other hand, for large lengthscales the rotation angles between the cells are very small and dipolar interactions dominate the total energy. Thus, even forlarger cell structures and a suited micromagnetic problem, theuse of larger cell sizes will give a reasonable result. 15E. Della Torre, Physica B & C 343,1/H208492004/H20850. 16The V ortex conﬁguration has with 2 kJ/m3the lowest total en- ergy, followed by the Landau state with a cross tie wall with3k J / m 3and the sstate with 3.8 kJ/m3. 17R. Hertel, W. Wulfhekel, and J. Kirschner, Phys. Rev. Lett. 93, 257202 /H208492004/H20850. 18M. Bolte, G. Meier, and C. Bayer, Phys. Rev. B 73, 052406 /H208492006/H20850. 19J. Miltat, in Spin Dynamics in Conﬁned Magnetic Structures I , edited by B. Hillebrands and K. Ounadjela /H20849Springer-Verlag, Berlin, 2001 /H20850. 20Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 /H208492002/H20850. 21S. Mizukami, Y . Ando, and T. Miyazaki, Jpn. J. Appl. Phys., Part 140, 580 /H208492001/H20850; S. Mizukami, Y . Ando, and T. Miyazaki, J. Magn. Magn. Mater. 226, 1640 /H208492001/H20850. 22J. Foros, G. Woltersdorf, B. 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PhysRevLett.122.247202.pdf	Spin Pinning and Spin-Wave Dispersion in Nanoscopic Ferromagnetic Waveguides Q. Wang,1,B. Heinz,1,2,R. Verba,3M. Kewenig,1P. Pirro,1M. Schneider,1T. Meyer,1,4B. Lägel,5 C. Dubs,6T. Brächer,1and A. V . Chumak1,† 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern, D-67663 Kaiserslautern, Germany 2Graduate School Materials Science in Mainz, Staudingerweg 9, 55128 Mainz, Germany 3Institute of Magnetism, Kyiv 03680, Ukraine 4THATec Innovation GmbH, Augustaanlage 23, 68165 Mannheim, Germany 5Nano Structuring Center, Technische Universität Kaiserslautern, D-67663 Kaiserslautern, Germany 6INNOVENT e.V., Technologieentwicklung, Prüssingstraße 27B, 07745 Jena, Germany (Received 9 July 2018; revised manuscript received 5 March 2019; published 19 June 2019) Spin waves are investigated in yttrium iron garnet waveguides with a thickness of 39 nm and widths ranging down to 50 nm, i.e., with an aspect ratio thickness over width approaching unity, using Brillouin light scattering spectroscopy. The experimental results are verified by a semianalytical theory andmicromagnetic simulations. A critical width is found, below which the exchange interaction suppresses the dipolar pinning phenomenon. This changes the quantization criterion for the spin-wave eigenmodes and results in a pronounced modification of the spin-wave characteristics. The presented semianalytical theoryallows for the calculation of spin-wave mode profiles and dispersion relations in nanostructures. DOI: 10.1103/PhysRevLett.122.247202 Spin waves and their quanta, magnons, typically feature frequencies in the gigahertz to terahertz range and wave-lengths in the micrometer to nanometer range. They areenvisioned for the design of faster and smaller next genera-tional information processing devices where information is carried by magnons instead of electrons [1–9]. In the past, spin-wave modes in thin films or rather planar waveguideswith thickness-to-width aspect ratios a r¼h=w≪1have been studied. Such thin waveguides demonstrate the effectof“dipolar pinning ”at the lateral edges, and for its theoretical description the thin strip approximation was developed, in which only pinning of the much-larger-in- amplitude dynamic in-plane magnetization component istaken into account [10–15]. The recent progress in fabrica- tion technology leads to the development of nanoscopicmagnetic devices in which the width wand the thickness h become comparable [16–23]. The description of such wave- guides is beyond the thin strip model of effective pinning,because the scale of nonuniformity of the dynamic dipolarfields, which is described as “effective dipolar boundary conditions, ”becomes comparable to the waveguide width. Additionally, both in-plane and out-of-plane dynamic mag- netization components become involved in the effectivedipolar pinning as they become of comparable amplitude.Thus, a more general model should be developed andverified experimentally. In addition, such nanoscopic fea-ture sizes imply that the spin-wave modes bear a strong exchange character, since the widths of the structures are now comparable to the exchange length [24]. A proper description of the spin-wave eigenmodes in nanoscopicstrips which considers the influence of the exchange interaction, as well as the shape of the structure, is funda-mental for the field of magnonics. In this Letter, we discuss the evolution of the frequencies and profiles of the spin-wave modes in nanoscopic wave- guides where the aspect ratio a revolves from the thin film casear→0to a rectangular bar with ar→1. Yttrium iron garnet (YIG) waveguides with a thickness of 39 nm andwidths ranging down to 50 nm are fabricated and thequasiferromagnetic resonance (quasi-FMR) frequencies within them are measured using microfocused Brillouin light scattering (BLS) spectroscopy. The experimentalresults are verified by a semianalytical theory and micro-magnetic simulations. It is shown that a critical waveguidewidth exists, below which the profiles of the spin-wavemodes are essentially uniform across the width of the waveguide. This is fundamentally different from the profiles in state-of-the-art waveguides of micrometer [16–19]or millimeter sizes [25,26] , where the profiles are nonuniform and pinned at the waveguide edges due to the dipolarinteraction. In nanoscopic waveguides, the exchange inter-action suppresses this pinning due to its dominance over the dipolar interaction and, consequently, the exchange inter- action defines the profiles of the spin-wave modes as well asthe corresponding spin-wave dispersion characteristics. In the experiment and the theoretical studies, we con- sider rectangular magnetic waveguides as shown schemati- cally in Fig. 1(a). In the experiment, a spin-wave mode is excited by a stripline that provides a homogeneous exci-tation field over the sample containing various waveguidesPHYSICAL REVIEW LETTERS 122, 247202 (2019) 0031-9007 =19=122(24) =247202(6) 247202-1 © 2019 American Physical Societyetched from a h¼39nm thick YIG film grown by liquid phase epitaxy [27] on gadolinium gallium garnet. The widths of the waveguides range from w¼50nm to w¼ 1μm with a length of 60μm. The waveguides are patterned by Arþion beam etching using an electron-beam litho- graphically defined Cr =Ti hard mask and are well separated on the sample in order to avoid dipolar coupling betweenthem [9]. The waveguides are uniformly magnetized along their long axis by an external field B(xdirection). Figures 1(b) and1(c)show scanning electron microscopy (SEM) micrographs of the largest and the narrowestwaveguide studied in the experiment. The intensity ofthe magnetization precession is measured by microfocusedBLS spectroscopy [28] (see Supplemental Material S3 [29]) as shown in Fig. 1(a). Black and red lines in Fig. 1(d) show the frequency spectra for a 1μm and a 50 nm wide waveguide, respectively. No standing modesacross the thickness were observed in our experiment, astheir frequencies lie higher than 20 GHz due to the smallthickness. The quasi-FMR frequency is 5.007 GHz for the1μm wide waveguide. This frequency is comparable to 5.029 GHz, the value predicted by the classical theoreticalmodel using the thin strip approximation [12–14,34] .I n contrast, the quasi-FMR frequency is 5.35 GHz for a 50 nmwide waveguide which is much smaller than the value of7.687 GHz predicted by the same model. The reason is thatthe thin strip approximation overestimates the effect ofdipolar pinning in waveguides with aspect ratio a rclose to one, for which the nonuniformity of the dynamic dipolarfields is not well localized at the waveguide edges.Additionally, in such nanoscale waveguides, the dynamicmagnetization components become of the same order of magnitude and both affect the effective mode pinning, in contrast to thin waveguides, in which the in-plane mag-netization component is dominant. In order to accurately describe the spin-wave character- istic in nanoscopic longitudinally magnetized waveguides,we provide a more general semianalytical theory which goes beyond the thin strip approximation. Please note that the theory is not applicable in transversely magnetizedwaveguides due to their more involved internal field landscape [16]. The lateral spin-wave mode profile m kxðyÞand frequency can be found from [35,36] −iωkxmkxðyÞ¼μ×½ˆΩkx·mkxðyÞ/C138; ð1Þ with appropriate exchange boundary conditions, which take into account the surface anisotropy at the edges (see Supplemental Material S1 [29]). Here, μis the unit vector in the static magnetization direction and ˆΩkxis a tensorial Hamilton operator, which is given by ˆΩkx·mkxðyÞ¼/C20 ωHþωMλ2/C18 k2x−d2 dy2/C19/C21 mkxðyÞ þωMZ ˆGkxðy−y0Þ·mkxðy0Þdy0: ð2Þ Here, ωH¼γB,Bis the static internal magnetic field that is considered to be equal to the external field due to the negligible demagnetization along the xdirection, ωM¼γμ0Ms,γis the gyromagnetic ratio. ˆGkxðyÞis the Green ’s function (see Supplemental Material S1 [29]). A numerical solution of Eq. (1)gives both the spin-wave profiles mkxand frequency ωkx. In the following, we will regard the out-of-plane component mzðyÞto show the mode profiles, representatively. The profiles of the spin-wavemodes can be well approximated by sine and cosine functions. In the past, it was demonstrated that in micro- scopic waveguides, that the fundamental mode is well fittedby the function m zðyÞ¼A0cosðπy=w effÞwith the amplitude A0and the effective width weff[12,13] . This mode, as well as the higher modes, are referred to as “partially pinned. ” Pinning hereby refers to the fact that the amplitude of the modes at the edges of the waveguides is reduced. In that case, the effective width weffdetermines where the amplitude of the modes would vanish outside the waveguide [9,12,23] . With this effective width, the spin-wave dispersion relation can also be calculated by the analytical formula [9] ω0ðkxÞ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ½ωHþωMðλ2K2þFyy kxÞ/C138½ωHþωMðλ2K2þFzz kxÞ/C138q ; ð3Þ where K¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ k2xþκ2p andκ¼π=weff. The tensor ˆFkx¼ ð1=2πÞR∞ −∞ðjσkj2=˜wÞˆNkdkyaccounts for the dynamic FIG. 1. (a) Sketch of the sample and the experimental con- figuration: a set of YIG waveguides is placed on a microstrip lineto excite the quasi-FMR in the waveguides. BLS spectroscopy isused to measure the local spin-wave dynamics. (b) and (c) SEMmicrograph of a 1μm and a 50 nm wide YIG waveguide of 39 nm thickness. (d) Frequency spectra for 1μm and 50 nm wide waveguides measured for a respective microwave power of 6and 15 dBm.PHYSICAL REVIEW LETTERS 122, 247202 (2019) 247202-2demagnetization, σk¼Rw=2 −w=2mðyÞe−ikyydyis the Fourier transform of the spin-wave profile across the width of the waveguide, ˜w¼Rw=2 −w=2mðyÞ2dyis the normalization of the mode profile mzðyÞ. In the following, the experiment is compared to the theory and to micromagnetic simulations. The simulations are performed using MUMAX3[37]. The structure is schemati- cally shown in Fig. 1(a). The following parameters were used: the saturation magnetization Ms¼1.37×105A=m and the Gilbert damping α¼6.41×10−4were extracted from the plain film via ferromagnetic resonance spectros-copy before patterning [38]. Moreover, a gyromagnetic ratio γ¼175.86rad=ðns T Þand an exchange constant A¼ 3.5pJ=m for a standard YIG film were assumed. An external field B¼108.9mT is applied along the waveguide long axis (see Supplemental Material S2 [29]). The central panel of Fig. 2shows the spin-wave mode profile of the fundamental mode for k x¼0, which corre- sponds to the quasi-FMR, in a 1μm (a2) and 50 nm (b2) wide waveguide which has been obtained by micromag- netic simulations (red dots) and by solving Eq. (1)numeri- cally (black lines) (higher width modes are discussed inSupplemental Material S6 [29]). The top panels (a1) and (b1) illustrate the mode profile and the local precession amplitude in the waveguide. As it can be seen, the two waveguides feature quite different profiles of their funda-mental modes: in the 1μm wide waveguide, the spins arepartially pinned and the amplitude of m zat the edges of the waveguide is reduced. This still resembles the cosinelike profile of the lowest width mode n¼0that has been well established in investigations of spin-wave dynamics in waveguides on the micron scale [19,23,39] and that can be well described by the simple introduction of a finite effective width weff>w (weff¼wfor the case of full pinning). In contrast, the spins at the edges of the narrow waveguide are completely unpinned and the amplitude of the dynamic magnetization mzof the lowest mode n¼0is almost constant across the width of the waveguide, result- ing in weff→∞. To understand the nature of this depinning, it is instruc- tive to consider the spin-wave energy as a function of the geometric width of the waveguide normalized by the effective width w=w eff. This ratio corresponds to some kind of pinning parameter taking values in between 1 for the fully pinned case and 0 for the fully unpinned case. The system will choose the mode profile which minimizes the total energy. This is equivalent to a variational minimiza-tion of the spin-wave eigenfrequencies as a function of w=w eff. To illustrate this, the lower panels of Figs. 2(a3) and 2(b3) show the normalized square of the spin-wave eigenfrequencies ω02=ω2 Mfor the two different widths as a function of w=w eff. Here, ω02refers to a frequency square, not taking into account the Zeeman contribution (ω2 HþωHωM), which only leads to an offset in frequency. The minimum of ω02is equivalent to the solution with the lowest energy corresponding to the effective width weff.I n addition to the total ω02(black), also the individual contributions from the dipolar term (red) and the exchange term (blue) are shown, which can only be separatedconveniently from each other if the square of Eq. (3)is considered for k x¼0. The dipolar contribution is non- monotonic and features a minimum at a finite effective width weff, which can clearly be observed for w¼1μm. The appearance of this minimum, which leads to the effect known as “effective dipolar pinning ”[13,14] , is a result of the interplay of two tendencies: (i) an increase of thevolume contribution with increasing w=w eff, as for common Damon-Eshbach spin waves, and (ii) a decrease of the edge contribution when the spin-wave amplitude at the edges vanishes ( w=w eff→1). This minimum is also present in the case of a 50 nm wide waveguide (red line), even though this is hardly perceivable in Fig. 2(b3) due to the scale. In contrast, the exchange leads to a monotonicincrease of frequency as a function of w=w eff, which is minimal for the unpinned case, i.e., w=w eff¼0implying weff→∞, when all spins are parallel. In the case of the 50 nm waveguide, the smaller width and the corresponding much larger quantized wave number in the case of pinned spins would lead to a much larger exchange contribution than this is the case for the 1μm wide waveguide (please note the vertical scales). Consequently, the system avoids pinning and the solution with lowest energy is situated at FIG. 2. Schematic of the precessing spins and simulated precession trajectories (ellipses in the second panel) and spin-wave profile m zðyÞof the quasi-FMR. The profiles have been obtained by micromagnetic simulations (red dots) and by thequasianalytical approach (black lines) for an (a) 1μm and a (b) 50 nm wide waveguide. Bottom panel: Normalized square of the spin-wave eigenfrequency ω 02=ω2 Mas a function of w=w eff and the relative dipolar and exchange contributions.PHYSICAL REVIEW LETTERS 122, 247202 (2019) 247202-3w=w eff¼0. In contrast, in the 1μm wide waveguide, the interplay of dipolar and exchange energy implies that energyis minimized at a finite w=w eff. The top panel of Fig. 2(b1) shows an additional feature of the narrow waveguide: as the aspect ratio of the waveguides approaches unity, the ellip- ticity of precession, a well-known feature of micron-sizedwaveguides which still resemble a thin film [23,40] , vanishes and the precession becomes nearly circular. Also, in nano- scale waveguides, the ellipticity is constant across the width, while in the 1μm wide waveguide it can be different at the waveguide center and near its edges. Please note that thepinning phenomena and ellipticity of precession also influ-ence the spin-wave lifetime which is described in the Supplemental Material S5 [29]. As it is evident from the lower panel of Fig. 2, the pinning and the corresponding effective width have a large influence on the spin-wave frequency. This allows for an experimental verification of the presented theory, since thefrequency of partially pinned spin-wave modes would besignificantly higher than in the unpinned case. Blacksquares in Fig. 3(a) summarize the dependence of the frequency of the quasi-FMR measured for different widths of the YIG waveguides. The magenta line shows theexpected frequencies assuming pinned spins, the blue(dashed) line gives the resonance frequencies extrapolating the formula conventionally used for micron-sized wave- guides [34] to the nanoscopic scenario, and the red line gives the result of the theory presented here, together withsimulation results (green dashed line). As it can be seen, the experimentally observed frequencies can be well repro- duced if the real pinning conditions are taken into account. As has been discussed along with Fig. 2, the unpinning occurs when the exchange interaction contributionbecomes so large that it compensates the minimum inthe dipolar contribution to the spin-wave energy. Since the energy contributions and the demagnetization tensor change with the thickness of the investigated waveguide,the critical width below which the spins become unpinned is different for different waveguide thicknesses. This is shown in Fig. 3(b), where the inverse effective width w=w eff is shown for different waveguide thicknesses. Symbols are the results of micromagnetic simulations, lines are calcu- lated semianalytically. As can be seen from the figure, thecritical width linearly increases with increasing thickness. This is summarized in the inset, which shows the critical width (i.e., the maximum width for which w=w eff¼0)a sa function of thickness. The critical widths for YIG, Permalloy, CoFeB, and Heusler compound (Co 2Mn 0.6Fe0.4Si) with different thicknesses are given in the SupplementalMaterial S9 [29]. A simple empirical linear formula is found by fitting the critical widths for different materials in a wide range of thicknesses: w crit¼2.2hþ6.7λ; ð4Þ where his the thickness of the waveguide and λis the exchange length. Please note that additional simulations with rough edges and a more realistic trapezoidal cross section ofthe waveguides are also provided in the Supplemental Material S7, S8 [29]. The results show that these effects have a small impact on the critical width. Up to now, the discussion was limited to the special case ofk x¼0. In the following, the influence of a finite wave vector will be addressed. The spin-wave dispersion relationof the fundamental ( n¼0) mode obtained from micro- magnetic simulations (color code) together with the semianalytical solution (white dashed line) are shown inFig.3(c)for the YIG waveguide of w¼50nm width. The figure also shows the low-wave-number part of the dispersion of the first width mode ( n¼1), which is pushed up significantly in frequency due to its large exchange contribution. Both modes are described accurately by the quasianalytical theory. As it is described above, the spinsare fully unpinned in this particular case. In order to demonstrate the influence of the pinning conditions on the spin-wave dispersion, a hypothetic dispersion relationfor the case of partial pinning is shown in the figure with adash-dotted white line (the case of w=w eff¼0.63is considered that would result from the usage of the thin strip approximation [12]). One can clearly see that the spin- wave frequencies in this case are considerably higher. Figure 3(d) shows the inverse effective width w=w effas a function of the wave number kxfor three exemplary FIG. 3. (a) Experimentally determined resonance frequencies (black squares) together with theoretical predictions and micro-magnetic simulations. (b) Inverse effective width w=w effas a function of the waveguide width. The inset shows the criticalwidth ( w crit) as a function of thickness h. (c) Spin-wave dispersion relation of the first two width modes from micro-magnetic simulations (color code) and theory (dashed lines).(d) Inverse effective width w=w effas a function of the spin-wave wave number kxfor different thicknesses and waveguide widths, respectively.PHYSICAL REVIEW LETTERS 122, 247202 (2019) 247202-4waveguide widths of w¼50, 300, and 500 nm. As it can be seen, the effective width and, consequently, the ratio w=w eff shows only a weak nonmonotonic dependence on the spin- wave wave number in the propagation direction. This dependence is a result of an increase of the inhomogeneityof the dipolar fields near the edges for larger k x, which increases pinning [14], and of the simultaneous decrease of the overall strength of dynamic dipolar fields for shorterspin waves. Please note that the mode profiles are not onlyimportant for the spin-wave dispersion. The unpinnedmode profiles will also greatly improve the couplingefficiency between two adjacent waveguides [9,41 –43]. In conclusion, the quasi-FMR of individual wires with widths ranging from 1μm down to 50 nm are studied by means of BLS spectroscopy. A difference in the quasi-FMRfrequency between experiment and the prediction by theclassical thin strip theory is found for 50 nm wide wave-guides. A semianalytical theory accounting for the non- uniformity of both in-plane and out-of-plane dynamic demagnetization fields is presented and is employedtogether with micromagnetic simulations to investigatethe spin-wave eigenmodes in nanoscopic waveguides withaspect ratio a rapproaching unity. It is found that the exchange interaction is getting dominant with respect to thedipolar interaction, which is responsible for the phenome-non of dipolar pinning. This mediates an unpinning of thespin-wave modes if the width of the waveguide becomessmaller than a certain critical value. This exchange unpin- ning results in a quasiuniform spin-wave mode profile in nanoscopic waveguides in contrast to the cosinelike pro-files in waveguides of micrometer widths and in a decreaseof the total energy and, thus, frequency, in comparison tothe fully or the partially pinned case. Our theory allows usto calculate the mode profiles as well as the spin-wavedispersion, and to identify a critical width below whichfully unpinned spins need to be considered. The presentedresults provide valuable guidelines for applications innanomagnonics where spin waves propagate in nanoscopicwaveguides with aspect ratios close to one and lateral sizes comparable to the sizes of modern CMOS technology. The authors thank Burkard Hillebrands and Andrei Slavin for valuable discussions. This research has been supportedby ERC Starting Grant No. 678309 MagnonCircuits and bythe DFG through the Collaborative Research Center SFB/TRR-173 “Spin þX”(Projects B01) and through the Project No. DU 1427/2-1. R. V. acknowledges support fromthe Ministry of Education and Science of Ukraine, ProjectNo. 0118U004007. *These authors have contributed equally to this work. †Corresponding author. chumak@physik.uni-kl.de [1] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. 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PhysRevB.91.041408.pdf	RAPID COMMUNICATIONS PHYSICAL REVIEW B 91, 041408(R) (2015) Creation and ampliﬁcation of electromagnon solitons by electric ﬁeld in nanostructured multiferroics R. Khomeriki,1,2L. Chotorlishvili,1B. A. Malomed,3and J. Berakdar1 1Institut f ¨ur Physik, Martin-Luther Universit ¨at Halle-Wittenberg, D-06120 Halle/Saale, Germany 2Physics Department, Tbilisi State University, 0128 Tbilisi, Georgia 3Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel (Received 6 October 2014; revised manuscript received 8 January 2015; published 20 January 2015) We develop a theoretical description of electromagnon solitons in a coupled ferroelectric-ferromagnetic heterostructure. The solitons are considered in the weakly nonlinear limit as a modulation of plane wavescorresponding to two electriclike and magneticlike branches in the spectrum. Emphasis is put on magneticlikeenvelope solitons that can be created by an alternating electric ﬁeld. It is shown also that the magnetic pulses canbe ampliﬁed by an electric ﬁeld with a frequency close to the band edge of the magnetic branch. DOI: 10.1103/PhysRevB.91.041408 PACS number(s): 85 .80.Jm,75.78.−n,77.80.Fm Multiferroic materials, i.e., materials exhibiting coupled order parameters, are in the focus of current research. Thesesystems not only offer new opportunities for applicationsbut also provide a test ground for addressing fundamentalissues regarding the interplay between electronic correlations,symmetry, and the interrelation between magnetism andferroelectricity [ 1–3]. Here we address magnetoelectrics which possess a simultaneous ferroelectric-magnetic response. Aninteresting aspect is the nonlinear nature of the magneto-electric excitation dynamics, which hints at the potentialof these systems for exploring nonlinear wave-localizationphenomena, such as multicomponent solitons [ 4,5], nonlinear band-gap transmission [ 6,7], and the interplay between the nonlinearity and Anderson localization [ 8]. In this Rapid Communication we aim at exciting robust magnetic signalsby means of electric ﬁelds. Particularly, we consider amultiferroic nanoheterostructure consisting of a ferromagnetic(FM) part deposited onto a ferroelectric (FE) substrate. Asdemonstrated experimentally, under favorable conditions, acoupling between the ferroelectric and the ferromagnetic orderparameters may emerge (this coupling is referred to as the mag-netoelectric coupling), thus allowing one to control magnetism(ferroelectricity) by means of electric (magnetic) ﬁelds. Herewe consider the case when the multiferroic structure is drivenby an electric ﬁeld with a frequency located simultaneouslywithin the band gap of the FE branch and in the band ofthe magnetic-excitation branch. With a proper choice of the electric-ﬁeld frequency (that follows from the electromagnon soliton theory developed below) it is thus possible to excitepropagating magnetic solitons. In addition, we point out apossibility for the ampliﬁcation of weak magnetic signals,which suggests the design of a digital magnetic transistor,where the role of the pump is played by the electric ﬁeld. Examples of two-phase multiferroics under study [ 9–12] are BaTiO 3/CoFe 2O4or PbZr 1−xTixO3/ferrites. The devel- oped model will be applied to a system where the FE and FMregions are coupled at an interface with a weak magnetoelectricinteraction. The theory is, however, more general and can, inprinciple, be applied to single-phase magnetoelectrics [ 13–15]. For the creation of electromagnon solitons, which is the subjectof the present work, a two-phase multiferroic structure ismore appropriate, as it allows one to generate and manipulateisolated FE or FM signals away from the interface.Both single- and two-phase multiferroics may be modeled by a ladder consisting of two weakly coupled chains: Onechain is ferroelectric (FE), built out of unidimensional electricdipole moments P n. The second chain is ferromagnetic (FM), composed of classical three-dimensional magnetic moments /vectorSn, where nnumbers the site in the lattice. Each chain is characterized by an intrinsic nearest-neighbor coupling, and eachPnis coupled to /vectorSnvia interchain weak magnetoelectric coupling. For a discussion of the microscopic nature of thiscoupling, we refer to Ref. [ 16]. We assume the direction of FE dipoles at some arbitrary angle with respect to the FManisotropy axis ξ, as depicted in Fig. 1(a). The magnetoelectric coupling will cause a rearrangement of magnetic moments. Leta new ground-state ordering direction of FM be the axis z, and φis the angle between zand anisotropy axis ξ. The magnetic ﬁeld/vectorh(t) is applied along z, and θis an angle between z and FE moments [see Fig. 1(a)].S 0(P0) stands for the FM (FE) equilibrium conﬁguration. We will consider perturbationsaround the equilibrium. Deﬁning the scaled dipolar deviationsp n≡(Pn−P0)/P0and the scaled magnetic variables /vectorsn≡ /vectorSn/S0, the Hamiltonian is written as H=HP+HS+HSP,H SP=− ˜gN/summationdisplay n=1pnsx n, HP=N/summationdisplay n=1˜α0 2/parenleftbiggdpn dt/parenrightbigg2 +˜α 2p2 n+˜β 4p4 n+˜αJ 2(pn+1−pn)2, HS=N/summationdisplay n=1/bracketleftbig −˜J/vectorsn/vectorsn+1+˜D1/parenleftbig sx n/parenrightbig2+˜D2/parenleftbig sy n/parenrightbig2/bracketrightbig , (1) where HSPstands for the linearized interfacial magnetoelectric coupling between the FM and the FE chain [ 16].HPis the FE part of the energy functional for N-interacting FE dipole moments [ 17,18]. Further, ˜ α0is a kinetic coefﬁcient; ˜ αJis the nearest-neighbor coupling constant; ˜ αand ˜βare second- and forth-order expansion coefﬁcients of the Ginzburg-Landau-Devonshire (GLD) potential [ 17,19] near the equilibrium state P 0.HSstands for the ferromagnetic contribution [ 20], where Jis the nearest-neighbor exchange coupling in the FM part. ˜D1=˜DS 0cos2φand ˜D2=˜DS 0are anisotropy constants, 1098-0121/2015/91(4)/041408(5) 041408-1 ©2015 American Physical SocietyRAPID COMMUNICATIONS KHOMERIKI, CHOTORLISHVILI, MALOMED, AND BERAKDAR PHYSICAL REVIEW B 91, 041408(R) (2015) 0 1 2 301234 WavenumberFrequencyz (b)ξ (a) ωxyθφS0 P0 (c) FIG. 1. (Color online) (a) A schematic of the multiferroic ladder built of FE and FM chains. Arrows indicate the directions and themagnitudes of the electric dipole moments and magnetic moments in the course of the soliton advancement along the ladder. (b) Mutual orientations of FE and FM ground-state vectors, in the frameworkof Hamiltonian ( 1). (c) Dispersion relations for FE (upper) and FM (lower) branches of the multiferroic composite. The arrow indicates the selected carrier frequency. andDis the uniaxial anisotropy constant along axis ξ[see Fig. 1(b)].We operate with dimensionless quantities by using the scaling t→ω0twithω0=√αJ/α0(for the examples shown below, ω0∼1012rad/s). The other parameters of the model, ˜g, ˜α,˜β,˜J,˜D1,˜D2, are scaled with ω0. The scaled quantities are indicated by omitting the tilde superscript. The time evolutionis governed by ∂s x n ∂t=−J/bracketleftbig sy n/parenleftbig sz n−1+sz n+1/parenrightbig −sz n/parenleftbig sy n−1+sy n+1/parenrightbig/bracketrightbig −2D2sy nsz n, ∂sy n ∂t=J/bracketleftbig sx n/parenleftbig sz n−1+sz n+1/parenrightbig −sz n/parenleftbig sx n−1+sx n+1/parenrightbig/bracketrightbig +2D1sx nsz n−gpnsz n, d2pn dt2=−αpn−βp3 n+(pn−1−2pn+pn+1)+gsx n.(2) As we are interested in small perturbations, sx n,sy n, andpn are much less than unity and the approximate equality sz n= 1−(sx n)2/2−(sy n)2/2 is justiﬁed. We seek weakly nonlinear harmonic solutions to Eq. ( 2), with a frequency ωand a wave number k, in the form of a column vector ( sx n,sy n,pn)=Rexp[i(ωt−kn)]+c.c., where Ris a set of complex amplitudes R≡(a,b,c ), and c .c.stands for the complex conjugate. Neglecting higher harmonics inEq. ( 2), we ﬁnd the set of nonlinear algebraic equations ˆW∗R=Q nl, (3) where the matrix and source are, respectively, ˆW=⎛ ⎜⎝iω J sin2(k/2)+2D2 0 −[Jsin2(k/2)+2D2] iω g g 0 ω2−α−sin2(k/2)⎞ ⎟⎠, Qnl=⎛ ⎜⎝b(\|b\|2+\|a\|2)(J−Jcosk+D2)−b∗(b2+a2)(Jcosk−Jcos 2k−D2) −a(\|b\|2+\|a\|2)(J−Jcosk+D1)+a∗(b2+a2)(Jcosk−Jcos 2k−D1) −3β\|c\|2c⎞ ⎟⎠. (4) The linear limit amounts to the set of linear homogeneous algebraic equations ˆW∗R=0. The solvability condition Det( ˆW)=0leads to two branches of the dispersion relation ω(k) which are shown in Fig. 1(c), with the corresponding amplitude set R=(a,b,c ), where bandcare expressed via the arbitrary constant a:b=−iaω/ [Jsin2(k/2)+D2], c=ga/[α+sin2(k/2)−ω]. We call a dispersion branch ferroelectric (deﬁning its frequency ωEand labeling the amplitude with index E) if it has \|c\|>\|a\|[the red curve in Fig. 1(c)], while a ferromagnetic branch ( ωM) is deﬁned by the relation \|c\|<\|a\|(the blue curve in the same ﬁgure). Of particular interest is the case when the system is excited at an edge (at the left one, for the sake of deﬁniteness)with a frequency ω swhich falls into the band gap of the FE mode and, simultaneously, the propagation band of theFM one, as shown by the arrow in Fig. 1(c). The dispersion relation with the ﬁxed frequency ω=ω sthen becomes a cubic equation for sin2(k/2). For ωsbelonging to the band of the FM mode and a band gap of an FE one, the cubicequation yields two complex wave numbers, associated with the FE and FM modes, and a real one, corresponding tothe FM mode. These three wave numbers determine a setof three orthogonal eigenvectors, R E≡(aE,bE,cE),R− M≡ (a− M,b− M,c− M), and R+ M≡(a+ M,b+ M,c+ M), where the ﬁrst two correspond to complex FE and FM wave numbers kEand k− M, respectively, while the last one is related to the real wave number k+ M. In linear systems, the solutions with the complex wave numbers are evanescent waves localized at the left edgeof the multiferroic chain. Thus, the solution for the vectorfunction F n=(sx n,sy n,pn)i s FE n=A(t)REeiωst−\|kE\|n+c.c., (5) FM− n=B(t)R− Meiωst−\|k− M\|n+c.c., where the amplitudes A(t) andB(t) may vary slowly in time. As mentioned above, the solutions corresponding to the complex wave numbers are localized at the boundary. To 041408-2RAPID COMMUNICATIONS CREATION AND AMPLIFICATION OF ELECTROMAGNON . . . PHYSICAL REVIEW B 91, 041408(R) (2015) examine the possibility of a solitonic self-localization of another solution with a real wave number (cf. Refs. [ 21–23] for similar solutions in multiferroic models), we consider thenonlinear frequency shift produced by the small terms Q nl in (3). Assuming a shifted frequency, ω+δωinstead of ω, the matrix ˆWis substituted by a modiﬁed one, ˆW+δˆW, with the diagonal matrix δˆW≡iδωDiag(1 ,1,2ω). We also deﬁne a row vector L=(a/prime,b/prime,c/prime) which solves for the equation L∗ˆW=0. Then, multiplying both sides of Eq. ( 3)o n L, we obtain the nonlinear plane-wave frequency shift: δω(k,\|a\|2)=−i(L∗Qnl)/(L∗δˆW∗R). (6) From these results, operating with the envelope function ϕ(n,t) deﬁned from FM+ n=ϕ(n,t)ei(ωst−ik+ Mn)(a+ M,b+ M,c+ M), one can derive the nonlinear Schr ¨odinger equation (NLS) (cf. Refs. [ 24,25]) in the form 2i/parenleftbigg∂ϕ ∂t+v∂ϕ ∂n/parenrightbigg +ω/prime/prime∂2ϕ ∂n2+/Delta1\|ϕ\|2ϕ=0, (7) which gives rise to the respective envelope-soliton solution with the FM-like localized mode being written as FM+ n=ei(ωst−k+ Mn) cosh/bracketleftbiga+ M 2/radicalBig /Delta1 2ω/prime/prime(n−vt)/bracketrightbig⎛ ⎝a+ M b+ M c+ M⎞ ⎠+c.c.. (8) The velocity, dispersion, and the nonlinearity coefﬁcients are v=∂ωM ∂k,ω/prime/prime=−∂2ωM ∂k2,/Delta1=∂[δωM(k,\|a\|2)] ∂[\|a\|2].(9) The carrier frequency of this soliton is deﬁned by the dispersion relation [the lower blue curve in Fig. 1(c)] ωs=ωM(k)+δωM(k,\|a\|2)/2. (10) Thus, one can generate both the FE evanescent ( 5) and FM solitonic ( 8) modes, driving the left edge of the chain at the same frequency ωs. It is possible to produce a combination of these solutions, too. Generally, in nonlinear systems, linearcombinations of particular solutions is not another solution, butif the solutions are widely separated, which makes interactionsbetween them negligible, the linear combination F n=f1FE n+f2FM− n+FM+ n (11) is still a solution of the nonlinear problem. In the weakly nonlinear limit it is even possible to construct a solution for thecase when particular modes overlap (i.e., the magnetic solitonis located near the edge), adding a time-dependent phase toeach term ( 5) and ( 8)i nt h es u m[ 26]. For instance, one can consider an approximate solution at the left edge of the ladder,n=0, in the form of F 0=f1FE 0ei/Psi1E(t)+f2FM− 0ei/Psi1M−(t)+FM+ 0ei/Psi1M+(t),(12) where, in the weakly nonlinear limit, the phases /Psi1are proportional to the wave amplitudes. Hence the waves do notgain signiﬁcant phase shifts due to interaction effects, if theirrelative group velocity is not negligible. In this case, all phaseshifts may be neglected. Our particular aim is to create an FM soliton by exciting only the FE degree of freedom at the edge, i.e., s x 0=sy 0=0. To this end, we choose A(t)=B(t)=sech[a+ Mvt√/Delta1/8ω/prime/prime],seeking to impose the following vector relation at the edge, n=0: (0,0,p0)=(f1RE+f2R− M+R+ M)eiωst cosh[a+ Mvt√/Delta1/8ω/prime/prime]+c.c. (13) Now using the orthogonality of eigenvectors R, we readily get the appropriate expression for p0: p0(t)=\|R+ M\|2 (c+ M)∗eiωst cosh[a+ Mvt√/Delta1/8ω/prime/prime]+c.c. (14) Further, it is possible to compute the coefﬁcients f1andf2 using the same orthogonality property: f1=\|R+ M\|2(cE)∗ \|RE\|2(c+ M)∗,f 2=\|R+ M\|2(c− M)∗ \|R− M\|2(c+ M)∗. (15) In numerical simulations, if we make p0a function of time as per Eq. ( 14), keeping the magnetic moments pinned at the boundary, it is possible to excite the FE and FM evanescentwaves ( 5), and also the propagating FM soliton ( 8). These simulations correspond to an experimental setup with pinnedboundary conditions at both FE and FM edges, and to theapplication of the electric ﬁeld E(t)=p 0(t) according to Eq. ( 14) at the ﬁrst cell of the FE chain. In this way, one can realize the excitation of magnetic solitons in the FM chainof the multiferroic ladder via an electric (rather than magnetic)ﬁeld by virtue of the magnetoelectric coupling. For an assessment of the above, we performed full numer- ical simulations with the following values of the normalizedparameters: α=0.2,β=0.1,J=1,D 1=0.1, D2=0.2,g=0.1. (16) These values correspond to BaTiO 3/Fe [27,28]. Furthermore, we assume for the FE second- and fourth-order poten-tial coefﬁcients ˜ α 1/(a3 FE)=2.77×107Vm/C, ˜ α2/(a3 FE)= 1.7×108Vm5/C3, and for the FE coupling coefﬁcient ˜αJ/(a3 FE)=1.3×108Vm/C, the equilibrium polarization P0=0.265C/m2, and the coarse-grained FE cell size aFE= 1 nm. The FM exchange interaction strength is ˜J=3.15× 10−20J, the FM anisotropy constant is ˜D=6.75×10−21J, and the ME coupling strength is ˜g0≈10−21Vm2. We drive the left boundary of FE according to ( 14) with a driving frequency ωs=0.4 and an amplitude a+ M=0.07 and apply pinned boundary conditions for FM, sx 0=sy 0=0. The results are displayed in Fig. 2, where the comparisons of numerical simulations and approximate analytical solu-tion ( 11) are shown at different moments of time [Figs. 2(a) and2(b)]. Next, we consider the possibility of amplifying magnetic pulses. To this end, apply a continuous electric signal withthe frequency ω swhich is slightly below the FM band boundary ωM(0), keeping FM moments pinned at the edge. In this setting, and for small driving amplitudes, no energyis transmitted through the chain. Both FM and FE modes areevanescent and described by the solutions ( 5). A propagating FM soliton emerges only if the electric-ﬁeld amplitude attainsthe band-gap transmission threshold [ 6,29]: This happens if the amplitude is large enough so that a solution of the nonlineardispersion relation ( 10) for a real wave number kexits. Then, 041408-3RAPID COMMUNICATIONS KHOMERIKI, CHOTORLISHVILI, MALOMED, AND BERAKDAR PHYSICAL REVIEW B 91, 041408(R) (2015) FIG. 2. (Color online) Numerical simulations of the creation of a ferromagnetic soliton by the electric ﬁeld, using Eq. ( 2) with parameters ( 16). (a) and (b) show a comparison between the analytical (solid lines) and the numerical (points) results for the soliton’s spatial proﬁle at different moments of time. The analytical solutionis taken according to Eq. ( 11) with the coefﬁcients ( 15). (c), (d) The propagation of the FM soliton in the ferroelectric and the ferromagnetic layers, respectively. if one keeps the amplitude of the electric ﬁeld just slightly below this band-gap threshold, a small-amplitude in-phasemagnetic signal coupled to the FM chain allows one to passthe threshold. This gives rise to a large-amplitude FM solitonpropagating through the multiferroic (see Fig. 3). In this way, one can realize an ampliﬁcation of the magnetic pulses byelectric ﬁeld. It may happen that almost all the energy of theelectric ﬁeld will be transferred to the ferromagnet chain, andthe corresponding ampliﬁcation rate may achieve values asmuch as ∼100. For instance, in the simulations presented in Fig. 3, we choose the driving frequency and the amplitude of the electric ﬁeld ω s=0.229 (that is, ≈30 GHz in real units) and ( p0)max=0.414, while the FM signal amplitude is (sx 0)max=0.02. In this Rapid Communication we do not address dissipation effects which, in principle, could be taken into account by in-troducing the conventional Landau-Lifshitz-Gilbert dampingterm in the magnetic part of the evolution equations ( 2), as well FIG. 3. (Color online) Results of numerical simulations of the ampliﬁcation of a magnetic signal in the multiferroic chain (a). Thespace-time evolution of excitations in the ferroelectric part. (b) The ampliﬁed propagation of the magnetic soliton in the ferromagnetic part. The arrow indicates the moment of the injection of the magneticsignal. Inset (c) presents the time dependence of the energy of the input magnetic signal at the FM edge, n=0, while the energy of output signal is measured at site n=200. as damping terms in the electric part. Here, we assume that dissipation has no qualitative effects for the considered lengthand on the time scales comparable with the magnetic/electricsignal transmission (that is, 10 −9s) and thus do not consider the respective terms in the evolution equations. Concluding, an electromagnon soliton theory is developed and the results are applied for electric-ﬁeld-induced magneticsoliton generation. A proper choice of parameters of the pumpelectric ﬁeld enables the ampliﬁcation of magnetic signals.In the amplifying regime the total (pump+signal) amplitudeovercomes the band-gap transmission threshold and the energyof the electric ﬁeld is completely transferred to the magneticsoliton. As we have shown above, substantial (more than 100times) ampliﬁcation of the magnetic input/output signals couldbe realized. L.C. and J.B. are supported by DFG through SFB 762. R.K. is supported by a DAAD fellowship and Grant No. 30/12from SRNSF and Grant No. 6084 from STCU. 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PhysRevB.104.024415.pdf	PHYSICAL REVIEW B 104, 024415 (2021) Theory of spin-Hall magnetoresistance in the ac terahertz regime David A. Reiss ,1Tobias Kampfrath,2,3and Piet W. Brouwer1 1Dahlem Center for Complex Quantum Systems and Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany 2Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany 3Department of Physical Chemistry, Fritz-Haber Institut, Faradayweg 4–6, 14195 Berlin, Germany (Received 15 April 2021; revised 25 June 2021; accepted 28 June 2021; published 12 July 2021) In bilayers consisting of a normal metal (N) with spin-orbit coupling and a ferromagnet (F), the combination of the spin-Hall effect, the spin-transfer torque, and the inverse spin-Hall effect gives a small correction to thein-plane conductivity of N, which is referred to as spin-Hall magnetoresistance (SMR). We here present a theoryof the SMR and the associated off-diagonal conductivity corrections for frequencies up to the terahertz regime.We show that the SMR signal has pronounced singularities at the spin-wave frequencies of F, which identiﬁes itas a potential tool for all-electric spectroscopy of magnon modes. A systematic change of the magnitude of theSMR at lower frequencies is associated with the onset of a longitudinal magnonic contribution to spin transportacross the F-N interface. DOI: 10.1103/PhysRevB.104.024415 I. INTRODUCTION In recent years, it was experimentally shown that basic spintronic effects not only operate in the DC and GHz regime but also in the ultrafast (THz) regime [ 1]. Examples include the spin-Hall effect (SHE) and its inverse (ISHE) [ 2,3]i n metals with strong spin-orbit coupling, spin pumping, andspin-transfer torque (STT) [ 4,5] at interfaces of nonmagnetic metals and ferromagnets, and magnon generation [ 6]. Fur- thermore, ultrafast versions of the spin-Seebeck effect [ 7,8] and the giant magnetoresistance in ferromagnet \|normal-metal multilayers [ 9] were demonstrated. Magnetoresistance effects are important for eventual applications, because their utiliza-tion on ultrafast time scales has the potential of increasingspeeds for the electrical readout of magnetic memories. In this paper, we consider the ultrafast version of the “spin-Hall magnetoresistance” (SMR) [ 10–15]. The SMR is observed in current-in-plane experiments on bilayers of an insulating or metallic ferromagnet F and a nonmagnetic metal N with strong spin-orbit coupling, as shown in Fig. 1.I ti s a small correction δσ xxto the diagonal in-plane conductivity (i.e., for currents parallel to an applied electric ﬁeld) thatresults from a combination of SHE, ISHE, and STT, and thatis sensitive to the magnetization direction in the ferromagneticlayer. Measurements of the SMR in the DC regime were foundto be in good agreement with theoretical predictions [ 16,17]. Lotze et al. measured the ﬁnite-frequency SMR in YIG \|Pt bilayers up to 3 GHz and observed no frequency dependenceofδσ xxwithin their measurement accuracy [ 18]. With the present advances in the availability of THz sources [ 19], the experimental investigation of the SMR in the ultrafast regime becomes a realistic possibility. At zero frequency, the SMR involves a combination of key spintronic phenomena [ 16,17,20], shown schematically in Fig. 1: (i) When an electric ﬁeld Eis applied to N, the SHE generates a spin accumulation at the F-N interface. (ii) Via thespin-transfer torque or, in the case of a metallic ferromagnet,via the ﬂow of a spin-polarized electron current, spin angular momentum is transferred between the spin accumulation atthe interface and the F layer. (iii) The ISHE converts thespin current through the F-N interface into a charge currentin N that ﬂows parallel to the interface. The component ofthis induced current parallel to the applied electric ﬁeld corre-sponds to a change of the diagonal in-plane conductivity δσ xx, whereas the perpendicular current component gives a contri-bution δσ xyto the off-diagonal conductivity. The conductivity corrections depend on the magnetization direction, becausethe amount of angular momentum transferred to F depends onit. The SMR can be distinguished from the anisotropic mag-netoresistance in a proximity-induced magnetic N layer bymeasuring the magnetoresistance for out-of-plane directionsof the magnetization [ 17]. In order to adequately describe the frequency dependence of the SMR, we must decompose the spin transport acrossthe F-N interface into “longitudinal” and “transverse” con-tributions, polarized collinear with and perpendicular to themagnetization direction m, respectively [ 20]. The difference of the transverse and longitudinal contributions determinesthe size of the SMR [ 16,17]. As we show in detail in this paper, both contributions to the SMR depend on frequencybut in different ways. On one hand, the coherent excitationof spin waves (magnons) causes pronounced singular featuresin the frequency dependence of the transverse contribution tothe SMR, which, with sufﬁciently high frequency resolution,identiﬁes the SMR as an all-electric spectroscopic probe ofmagnon modes in F. Collinear spin transport, on the otherhand, occurs via the incoherent excitation or annihilation ofthermal magnons via spin-ﬂip scattering at the F-N interface[20–24] and, in metallic ferromagnets, via spin-dependent transport of conduction electrons [ 25]. For thin insulating F layers with a long magnon lifetime, longitudinal spin transportis strongly suppressed at zero frequency, as the buildup ofan excess density of magnons in F causes a backﬂow ofspin angular momentum from F into N. This compensation 2469-9950/2021/104(2)/024415(20) 024415-1 ©2021 American Physical SocietyREISS, KAMPFRATH, AND BROUWER PHYSICAL REVIEW B 104, 024415 (2021) FIG. 1. Explanation of the SMR in a ferromagnet \|normal-metal (F\|N) bilayer: Via the spin-Hall effect, an electric ﬁeld applied to N creates a spin accumulation at its boundaries. The spin accumulation at the F-N boundary drives a spin current from N into F. This spin current is converted into a charge current in N via the inverse spin-Hall effect. For a time-dependent driving ﬁeld, additionally magnons (spin waves) can be excited in F. The possibility to coherently excite magnons is what distinguishes the THz SMR from its DCcounterpart. mechanism ceases to be effective at ﬁnite frequencies, how- ever, which leads to an appreciable decrease of the SMR fortypical YIG \|Pt devices for frequencies in the GHz range and above. Previous work on the ﬁnite-frequency SMR by Chiba, Bauer, and Takahashi [ 26] theoretically considered transverse spin transport across the F-N interface in a current-in-planeversion of the current-induced spin-torque ferromagnetic res-onance (FMR) [ 27–29] and the nonlinear spin-torque diode effect [ 30]. Chiba et al. considered frequencies close to the FMR frequency ω 0, for which only the uniform mode of the magnetization in the F layer is excited (see also Ref. [ 31]). The mechanisms governing transverse spin transport acrossthe F-N interface—spin torque and spin pumping—are essen-tially the same in the THz regime as they are in the lower-GHzregime considered in Ref. [ 26], which is why this part of our theoretical analysis closely follows Ref. [ 26]. The main dif- ference to Ref. [ 26] is that for driving frequencies in the THz regime the magnetization mode excited by the SHE-inducedspin torque is no longer the uniform ferromagnetic-resonancemode but an acoustic magnon mode, whose wavelength ismuch shorter than the typical thickness d Fof the F layer. Such current-induced coherent magnon excitation was consid-ered theoretically by Sluka [ 32] and Johansen, Skarsvåg, and Brataas [ 33] for antiferromagnetic layers, in which magnon frequencies are typically higher. We restrict our theory to thelinear-response regime and do not consider nonlinear rectiﬁ-cation effects responsible for a DC response of driven F \|N bilayers [ 26,34,35]. This is appropriate for the THz regime, because the ﬁeld amplitudes used in standard THz time-domain spectroscopy are usually too small to induce nonlineareffects [ 9]. A theory for the zero-frequency limit that includes both the transverse and the longitudinal contribution to the SMRwas considered by Zhang, Bergeret, and Golovach [ 20]b u t without considering the backﬂow of spin current resultingfrom a nonequilibrium population of thermal magnons in F.As we show in this paper, it is this backﬂow term that causesa systematic frequency dependence of the SMR in bilayersinvolving a ferromagnetic insulator with long magnon life-time. Although, as we show below, the difference betweenzero-frequency and high-frequency limits does not depend on interface properties and device parameters, the character-istic frequency separating low-frequency and high-frequencyregimes depends on these details. The authors of Ref. [ 18] measured the SMR in YIG \|Pt bilayers in the GHz regime and did not observe an appreciable frequency dependence of theSMR up to approximately 3 GHz. We attribute the apparentabsence of a frequency dependence in this experiment to thepresence of the large applied magnetic ﬁeld, which effectivelypinned the magnetization direction, thus shifting the charac-teristic frequency to a value outside the range accessible inthe experiment of Ref. [ 18]. Whereas the magnetic ﬁeld generated by the applied AC current played a signiﬁcant role if the applied frequenciesare close to the ferromagnetic-resonance frequency [ 26], the Oersted ﬁeld only has a minimal effect on the SMR in the THzregime. The reason is that the Oersted ﬁeld is approximatelyhomogeneous in F such that it can not effectively excitemagnon modes at the frequency of the driving ﬁeld. The sameapplies to the magnetic ﬁeld of the electromagnetic wave thatdrives the SMR at high frequencies. The spin-transfer torque,on the other hand, acts locally at the F-N interface, so that itcouples to magnon modes of all wavelengths. For this reason,we will not consider the Oersted ﬁeld in the main text and,instead, discuss its effect in the Appendix. This paper is organized as follows: In Sec. IIwe describe the system relevant for the SMR in an F \|N bilayer geometry and introduce the necessary notation. In Sec. III, charge and spin current densities driven by an applied time-dependentelectric ﬁeld are calculated for an F \|N bilayer with a thickness d Nof the N layer much larger than the spin-relaxation length λN. This is the geometry relevant for the existing experiments in the low-frequency regime. Following the idea of a “mag-netoelectric circuit theory,” the result is formulated in termsof “spin impedances” characterizing the N layer, the F layer,and the F-N interface. Separate sets of impedances describethe transverse and longitudinal contributions to the SMRand the associated off-diagonal conductivity corrections. Theimpedances are calculated from elementary electronic andmagnetic transport equations in Sec. IV. Speciﬁc predictions for the SMR in bilayers of YIG \|Pt and Fe \|Au (as proto- types for insulating and metallic ferromagnets) are discussedSec. V. We conclude in Sec. VI. A discussion of the effect of the Oersted ﬁeld, of F \|N bilayers with ﬁnite thickness d N/lessorsimilarλNand of F \|N\|F trilayers, as well as a theory of the lon- gitudinal magnonic spin transport through the F-N interfacewith ballistic magnons in F can be found in the appendices. II. SYSTEM AND NOTATION We consider the SMR in an F \|N bilayer geometry, shown schematically in Fig. 2. (A discussion of the SMR in an F \|N\|F trilayer geometry can be found in Appendix C.) Following Ref. [ 16], we choose coordinates such that the zdirection is perpendicular to the thin ﬁlms, the normal metal N ofthickness d Nis located at 0 <z<dN, and the magnet F at−dF<z<0. A spatially uniform time-dependent electric ﬁeld E(t)=E(t)exis applied in the xdirection. We assume that the thickness dNof the N layer is much larger than the spin-relaxation length λN. In this limit, the 024415-2THEORY OF SPIN-HALL MAGNETORESISTANCE IN THE … PHYSICAL REVIEW B 104, 024415 (2021) FIG. 2. F \|N bilayer with a normal metal of thickness dNand a ferromagnet of thickness dF. Coordinates are chosen such that the xy plane is the interface between the F and N layers. The electric ﬁeld Eis applied in the xdirection. small corrections δσxxandδσxyto the conductivity of the N layer from the combination of SHE and ISHE are the sumof contributions from the F-N interface at z=0 and the N- vacuum interface at z=d N. Since the latter does not depend on the magnetization direction m, for a theory of the SMR it is sufﬁcient to consider the contribution from the F-N interfaceatz=0 only. The case d N/similarequalλNis discussed in Appendix C. The relevant variables for the charge and spin currents in the normal metal N are the charge current densities jx,y c(z,t) in the xand ydirections, the spin current density jz s(z,t) ﬂowing in the zdirection, and the spin accumulation μs(z,t) (0<z<dN) as deﬁned in the following. Throughout we use superscripts to denote spatial directions and subscripts forcomponents associated with the spin direction. We deﬁne thespin current density as j z sz=(¯h/2e)[jz c↑−jz c↓], where jz c↑,↓ is the charge current density carried by electrons with spin ↑,↓projected onto the zaxis, respectively. Similarly, the spin accumulation μsz=μ↑−μ↓, where μ↑,↓is the (elec- tro)chemical potential for electrons with spin ↑,↓projected onto the zaxis. We use analogous deﬁnitions for the compo- nents jz sx,jz sy,μsx, andμsy. The equilibrium direction e/bardblof the magnetization of the magnet F is meq≡e/bardbl=mxex+myey+mzez, (1) see Fig. 2. To parametrize the direction perpendicular to e/bardbl, we combine two orthogonal vectors spanning the plane per-pendicular to e /bardblas the real and imaginary part of a complex unit vector e⊥with the property e⊥×e/bardbl=+ie⊥, (2) which deﬁnes e⊥up to a phase factor. Anticipating that the y direction plays a special role, as discussed in the next section,a convenient choice is e ⊥=1/radicalBig 2/parenleftbig m2x+m2z/parenrightbig/bracketleftbig/parenleftbig m2 x+m2 z/parenrightbig ey−(mzmy−imx)ez −(mxmy+imz)ex/bracketrightbig . (3) (The phase factor of the vector ( 3) is not deﬁned if e/bardbl= ey, which does not affect the ﬁnal results.) To account for the different response to spin excitations collinear with andperpendicular to e /bardbl, we separate the spin accumulation μs and the spin current jz sinto “longitudinal” and “transverse” components with respect to the equilibrium magnetizationdirection e/bardbl, μs(z,t)=μs/bardbl(z,t)e/bardbl+μs⊥(z,t)e⊥+μ∗ s⊥(z,t)e∗ ⊥, jz s(z,t)=jz s/bardbl(z,t)e/bardbl+jz s⊥(z,t)e⊥+jz∗ s⊥(z,t)e∗ ⊥, (4) where μs⊥(z,t) and jz s⊥(z,t) are complex variables. The relevant dynamical variables for the magnet F are the magnetization direction m(z,t), the spin current density jz s(z,t)ﬂ o w i n gi nt h e zdirection, and the spin accumulation μs(z,t), where −dF<z<0. We consider small deviations ofmfrom the equilibrium direction e/bardblonly, which we parametrize by the complex amplitude m⊥(z,t), m(z,t)=e/bardbl+m⊥(z,t)e⊥+m∗ ⊥(z,t)e∗ ⊥. (5) As the exchange ﬁeld is large, a metallic ferromagnet F can sustain a longitudinal component of the spin accumulationonly, μ s(z,t)=μs/bardbl(z,t)e/bardbl,−dF<z<0, (6) where we neglect the effect of the SHE in F, because the spin- Hall conductance of common metallic ferromagnets such asFe, Co, and Ni is smaller than that for Pt and Au [ 36,37] and does not lead to a frequency dependence. Performing a Fourier transform to time we write E(t)=1 2π/integraldisplay+∞ −∞dωE(ω)e−iωt, (7) where E(−ω)=E∗(ω). The same Fourier representation will be used for all time-dependent quantities introduced above.Note that the transverse amplitudes j s⊥,μs⊥, and m⊥at frequencies ωand−ωneed not be complex conjugates of each other because these amplitudes are complex in the timedomain. III. SMR FOR SINGLE F-N INTERFACE In this section, we state the relations between charge cur- rents, spin currents, and spin accumulations in both N and Flayers and across the F-N interface. In these relations, “spinimpedances” appear in a natural way and the results can beformulated as equivalent magnetoelectronic circuit diagrams,similar to electric circuit analysis. In Sec. IV, explicit expres- sions for the impedances are derived. To linear order in the applied ﬁeld and the induced potential gradients, the charge current densities j x,y cand the spin current density jz sin the normal metal N satisfy the characteristic response equations of the SHE and ISHE [ 38–41], jx c(z,ω)=σNE(ω)−θSHσN 2e∂ ∂zμsy(z,ω), (8) jy c(z,ω)=θSHσN 2e∂ ∂zμsx(z,ω), (9) jz s(z,ω)=−¯hσN 4e2∂ ∂zμs(z,ω)−θSH¯hσN 2eE(ω)ey.(10) Here 0 <z<dN,θSHis the spin-Hall angle and σNthe con- ductivity of the N layer. Since the thickness dNof the normal metal is assumed to be much larger than its spin-relaxationlength λ N, the spin accumulation near the F-N interface at z=0 does not lead to a spin current for zsufﬁciently far away from the interface. Averaging Eqs. ( 8) and ( 9) over z,w e 024415-3REISS, KAMPFRATH, AND BROUWER PHYSICAL REVIEW B 104, 024415 (2021) may then express the ISHE corrections δ¯jx,y cto the (effective) current densities associated with the F-N interface: In termsof the spin accumulation μ satz=0 they read as δ¯jx c(ω)=θSHσN 2edNμsy(z↓0,ω), (11) δ¯jy c(ω)=−θSHσN 2edNμsx(z↓0,ω). (12) To solve for the spin accumulation μs(z↓0,ω), we ob- serve that, within linear response, Eq. ( 10)i m p l i e st h a t μs(z↓ 0,ω) must be proportional to jz s(0,ω)+θSH¯hσNE(ω)ey/2e. The proportionality relation may be written as μs(z↓0,ω)=ZN(ω)/bracketleftbigg jz s(0,ω)+θSH¯hσN 2eE(ω)ey/bracketrightbigg ,(13) where the “spin impedance” ZN(ω) has the dimension of “area.” We have written jz s(0,ω) instead of jz s(z↓0,ω) be- cause the spin current density is conserved across the F-Ninterface. In Eq. ( 13) we neglect a contribution from the spin-Hanle effect to the magnetoresistance [ 42,43], because the direct effect of the applied magnetic ﬁeld on the spin accumulationin N is typically weak compared to that of the coupling to theferromagnet F. Inclusion of the spin-Hanle effect into our the-ory would require the introduction of a spin impedance Z N(ω) that differs for directions collinear with and perpendicular tothe applied magnetic ﬁeld but without an additional frequencydependence. Equation ( 13) completely determines the spin accumula- tionμ s(z↓0,ω) at the interface to a nonmagnetic insulator, since jz s(0,ω)=0 in that case. To ﬁnd μs(z↓0,ω) for an interface with an insulating or metallic magnet F, as is ap-propriate for the geometry of Fig. 2, we now consider the spin currents through an F-N interface and in F. In the fer-romagnet, the spin current j z s=jz se+jz smhas contributions from magnons and conduction electrons. The spin currentj z se=jz se/bardble/bardblcarried by conduction electrons has a longitudi- nal component only. Again, within linear response there isa simple proportionality to the (electron) spin accumulationμ s/bardbl(z↑0,ω) at the magnetic side of the interface, which can be written in a form similar to Eq. ( 13), μs/bardbl(z↑0,ω)=− Ze F/bardbl(ω)jz se/bardbl(0,ω). (14) An additional equation for μs/bardbl(z↑0) is found by considering the boundary conditions at the F-N interface, which also takethe simple form of a proportionality [ 44–46], μ s/bardbl(z↓0,ω)−μs/bardbl(z↑0,ω)=− Ze FN/bardbl(ω)jz se/bardbl(0,ω).(15) As in Eq. ( 13), the proportionality constants Ze F/bardblandZe FN/bardblhave the dimension of “area.” The magnonic spin current jz smhas a coherent transverse component related to the magnetization dynamics as wellas an incoherent longitudinal component carried by thermalmagnons. In linear response, the equations for the longitudinalmagnonic spin current are analogous to Eqs. ( 14) and ( 15)f o r the electronic spin current [ 21,23], μ m(0,ω)=− Zm F/bardbl(ω)jz sm/bardbl(0,ω), (16) μs/bardbl(z↓0,ω)−μm(0,ω)=− Zm FN/bardbl(ω)jz sm/bardbl(0,ω),(17)where μm(0,ω) is the chemical potential describing the distribution of thermal magnons in F [ 24]. The transverse component jz sm⊥is proportional to the time derivative of the magnetization, −¯hωm⊥(0,ω)=− ZF⊥(ω)jz sm⊥(0,ω), (18) and satisﬁes the boundary condition [ 47,48] μs⊥(z↓0,ω)+¯hωm⊥(0,ω)=− ZFN⊥(ω)jz sm⊥(0,ω) (19) at the interface. Again, the proportionality constants Zm FN/bardbl,Zm F/bardbl, ZF⊥, and ZFN⊥have the dimension of “area.” The interface impedances Ze FN/bardbl,Zm FN/bardbl, and ZFN⊥may be expressed in terms of the spin-dependent interface conductances g↑↑,g↓↓and the spin-mixing conductance g↑↓that are used in the theory of Refs. [ 16,17], see Secs. IV C ,IV E , and IV F. In the next section we show that of the seven “spin impedances” deﬁned in Eqs. ( 13)–(19) only ZF⊥(ω) and Zm F/bardbl(ω)—associated with the coherent magnon excitation and the nonequilibrium accumulation of magnons, respectively—have a non-negligible frequency dependence in the THzregime and below. Anticipating this result, we retain the fre-quency argument for Z F⊥(ω) and Zm F/bardbl(ω) but drop it for the ﬁve other spin impedances. Solving the coupled equations ( 13)–(19) for the longitudi- nal and transverse components of the spin current density jz sis straightforward and one ﬁnds jz s/bardbl(0,ω)=−ZN Z/bardbl(ω)θSH¯hσN 2eE(ω)e/bardbl·ey, jz s⊥(0,ω)=−ZN Z⊥(ω)θSH¯hσN 2eE(ω)e∗ ⊥·ey, (20) where we deﬁned Z/bardbl(ω)=ZN+/bracketleftbigg1 Zm FN/bardbl+Zm F/bardbl(ω)+1 Ze FN/bardbl+Ze F/bardbl/bracketrightbigg−1 , Z⊥(ω)=ZN+ZFN⊥+ZF⊥(ω). (21) Using Eq. ( 13) to calculate μs(z↓0,ω), Eqs. ( 1) and ( 3) for the unit vectors e/bardblande⊥, and Eqs. ( 11) and ( 12), one can calculate the SMR corrections δjx candδjy cto the current densities parallel and perpendicular to the applied electricﬁeld as δ¯j x c(ω)=δσxx(ω)E(ω), δ¯jy c(ω)=δσxy(ω)E(ω), (22) with δσxx(ω)=θ2 SH¯hσ2 N 4e2dNZN/braceleftbigg 1−m2 yZN Z/bardbl(ω) −1−m2 y 2/bracketleftbiggZN Z⊥(ω)+ZN Z∗ ⊥(−ω)/bracketrightbigg/bracerightbigg , (23) δσxy(ω)=θ2 SH¯hσ2 N 4e2dNZN ×/braceleftbigg mxmy/bracketleftbiggZN Z/bardbl(ω)−ZN 2Z⊥(ω)−ZN 2Z∗ ⊥(−ω)/bracketrightbigg −imz 2/bracketleftbiggZN Z⊥(ω)−ZN Z∗ ⊥(−ω)/bracketrightbigg/bracerightbigg . (24) 024415-4THEORY OF SPIN-HALL MAGNETORESISTANCE IN THE … PHYSICAL REVIEW B 104, 024415 (2021) These two equations are the central results of this paper. They describe the longitudinal and transverse contributionsto the SMR. The term “1” between the curly brackets inEq. ( 23), which does not depend on the interface properties, is the ISHE correction to the diagonal conductivity from asingle nonmagnetic insulating interface. The remaining terms,which depend on the properties of the F-N interface and onthe magnetization direction m eq, describe the change of the conductivity corrections δσxxandδσxydue to the presence of the magnet F. The contribution to the off-diagonal conductiv-ityδσ xyin Eq. ( 24) proportional to mxmycan be identiﬁed with a spin-Hall version of the planar Hall effect (PHE),which is symmetric under magnetization reversal. The termsproportional to m zcorrespond to a spin-Hall version of the anomalous Hall effect (AHE), which is antisymmetric undermagnetization reversal. The interface to the vacuum at z=d N, which is not con- sidered here, gives an additional correction to σxx. In the limit of large dNthis correction is given by the term “1” between the curly brackets in Eq. ( 23). The particular combination of impedances appearing in Eq. ( 23) can be illustrated with the equivalent magnetoelec- tronic circuit diagrams in Fig. 3, which shows the corrections toδσxxfrom transverse and longitudinal spin currents and spin accumulations on the left and right, respectively. The currentsand spin accumulations in the magnetoelectronic circuits ofFig. 3are related via spin versions of Ohm’s law and Kir- choff’s circuit laws and thus fulﬁll Eqs. ( 11)–(19)( c f .a l s o Ref. [ 49]). In Sec. IVwe ﬁnd that in the DC limit ω=0 one has Z F⊥(0)=0. Additionally, for a typical thickness dFof the F layer, Zm F/bardbl(0) is much larger than all other impedances, so that it is a very good approximation to not consider the contribu-tion from the longitudinal magnon spin current. This is theapproach taken in Refs. [ 15–17]. In this limit the contributions inversely proportional to Z ⊥in Eqs. ( 23) and ( 24) agree with the DC theory of the SMR in an F \|N bilayer for a ferromag- netic insulator of Refs. [ 15–17]. The contributions inversely proportional to Z/bardblare consistent with previous results for an F\|N bilayer with a metallic ferromagnet [ 25]. IV . IMPEDANCES As shown below, the spin impedances deﬁned in Eqs. ( 13)– (19) characterize different physical processes that affect spin transport: (i)ZNthe relaxation of the diffusing spin accumulation in N (due to magnetic and nonmagnetic impurities, phonons viaspin-orbit coupling, and other spin-ﬂip mechanisms); (ii)Z e F/bardblthe same as ZN, but for a ferromagnetic metal F; (iii) Ze FN/bardblthe spin current carried by electrons transported across an F-N interface; (iv) ZF⊥the transport and relaxation of spin currents by coherent magnons in F; (v)ZFN⊥the spin-transfer torque and spin pumping at an F-N interface; (vi) Zm FN/bardblthe incoherent creation and annihilation of ther- mal magnons in F by spin-ﬂip scattering of conductionelectrons at an F-N interface;σNE δjx c⊥dN /planckover2pi1 jz SH⊥ZNjz sN ZFN⊥ ZF⊥ jz s⊥(0)μs⊥(z↓0) −/planckover2pi1ωm⊥(0)σSH⊥ σSH⊥σN δjx c/bardbldN /planckover2pi1 jz SH/bardblZNjz sN Ze FN/bardbl Ze F/bardbl jz se/bardbl(0)Zm FN/bardbl Zm F/bardbl jz sm/bardbl(0)E μs/bardbl(z↓0) μs/bardbl(z↑0) μm(0)σSH/bardbl σSH/bardbl FIG. 3. Equivalent AC magnetoelectronic circuit diagrams for δσxx. The left circuit shows the correction from the transverse spin accumulation at the F-N interface; the right one shows the correction from the longitudinal spin accumulation. The dashed lines in the upper part indicate the transport of charge, while the solid linesin the lower part indicate the transport of spin angular momentum. Spin currents and spin accumulations are related via (spin versions of) Ohm’s law and Kirchoff’s circuit laws, see Eqs. ( 11)–(19). The impedances with labels σ SHand arrows indicate the conversion between charge and spin currents due to the SHE and ISHE, see Eqs. ( 8)–(10). The magnetization direction enters via the projection on longitudinal and transverse components, σSH/bardbl=θSH¯hσNmy/2e andσSH⊥=θSH¯hσN(1−m2 y)1/2/2e. The spin currents jz SH/bardblandjz SH⊥ generated via the SHE are given by jz SH/bardbl=θSH¯hσNEey·e/bardbl/2e= σSH/bardblEand jz SH⊥=θSH¯hσNEey·e⊥/2e=σSH⊥E. The fact that spin is not a conserved quantity—neither in N due to ﬂips of electron spins, nor in F due to Gilbert damping—is reﬂected by the presenceof “spin sinks” in N and F, represented as dissipation channels to the “ground” in the circuit diagram. (vii) Zm F/bardblthe buildup and relaxation of a chemical potential for magnons in F (a “magnon capacitance” effect). In the following we describe the calculation of each of these spin impedances separately. A. Normal metal The continuity equation for spin currents and accumula- tions, including spin ﬂips at a rate 1 /τsf,Ndue to spin-orbit coupling and magnetic impurities, reads 2 ¯h∂ ∂zjz s+νN˙μs=−νNμs τsf,N, (25) where νNis the density of states per spin direction. Combining this equation with the transport equation ( 10) and Fourier transforming to time, it follows that the spin accumulation μs satisﬁes the spin-diffusion equation, ∂2 ∂z2μs(z,ω)=μs(z,ω) λ2 N. (26) 024415-5REISS, KAMPFRATH, AND BROUWER PHYSICAL REVIEW B 104, 024415 (2021) The spin-relaxation length λNis given by λ2 N=σNτsf,N/2e2νN 1−iωτsf,N. (27) The frequency dependence of λNcan be neglected up to fre- quencies in the THz regime, because spin-ﬂip rates 1 /τsf,Nin heavy metals with signiﬁcant spin-Hall angle θSHare typically much higher due to the strong spin-orbit coupling. For exam-ple, in Pt one has 1 /τ sf,N∼102THz, see Ref. [ 50]. FordN/greatermuchλN, the solution of Eq. ( 26) decays exponentially away from the F-N interface at z=0. Integration over zim- mediately yields Eq. ( 13), with ZN=4e2λN ¯hσN. (28) The impedance ZNis the resistance (normalized to cross- sectional area) for a normal-metal slab of thickness λN. B. Ferromagnet: Longitudinal contribution from electrons In a metallic ferromagnet, conduction electrons can carry a spin current parallel to the magnetization direction e/bardbl. The calculation of this spin current component proceedsin the same manner as the calculation of the spin currentdensity in the normal metal. Hereto, we introduce (elec-tro)chemical potentials μ ↑,↓and charge current densities jz c↑,↓ carried by electrons with spin ↑,↓, where now the arrows “↑” and “ ↓” indicate spin polarization parallel or antiparal- lel to e/bardbl, respectively. From the transport equations jz c↑,↓= −(σ↑,↓/e)∂μ↑,↓/∂zfor a metallic ferromagnet we derive that the charge current density jz cand spin current density jz se/bardblare jz c=−σc e∂ ∂zμc−σs 2e∂ ∂zμs/bardbl, (29) jz se/bardbl=−¯hσc 4e2∂ ∂zμs/bardbl−¯hσs 2e2∂ ∂zμc, (30) where μc=(1/2)(μ↑+μ↓),σc=σ↑+σ↓, and σs=σ↑− σ↓. Since charge density ﬂuctuations are strongly suppressed by the long-range Coulomb interactions, we require that jz c= 0, so that jz se/bardbl=−¯hσF 4e2∂ ∂zμs/bardbl,σ F=σ2 c−σ2 s σc. (31) The continuity equations for charge and spin read 1 e∂ ∂zjz c+2νc˙μc+νs 2˙μs/bardbl=0, (32) 2 ¯h∂ ∂zjz se/bardbl+νc˙μs/bardbl+νs˙μc=−νF τsf,Fμs/bardbl, (33) where τsf,Fis the phenomenological spin-ﬂip time in the fer- romagnet; νc=(ν↑+ν↓)/2 andνs=ν↑−ν↓are densities of states, and νF=4ν2 c−ν2 s 4νc. (34) Again, using the absence of charge currents in the zdirection we eliminate ˙ μcfrom these equations and ﬁnd 2 ¯h∂ ∂zjz se/bardbl+νF˙μs/bardbl=−νF τsf,Fμs/bardbl. (35)Combining Eqs. ( 35) and ( 31) and performing a Fourier trans- form to frequency, we obtain the spin-diffusion equation ∂2 ∂z2μs/bardbl(z,ω)=μs/bardbl(z,ω) λ2 F, (36) with the spin-relaxation length λFof F given by λ2 F=σFτsf,F/2e2νF 1−iωτsf,F. (37) As in the case of the normal metal N, spin ﬂip rates 1 /τsf,F in metallic ferromagnets are assumed to be large, so that we may safely ignore the frequency dependence of λFfor the frequencies of interest in the THz range and below. Solving Eq. ( 37) with the boundary condition jz se/bardbl(−dF,ω)=0, one ﬁnds μs/bardbl(z,ω)=μs/bardbl(z↑0,ω)cosh[( z+dF)/λF] cosh( dF/λF). (38) The spin current jz se/bardblat the F-N interface is jz se/bardbl(0,ω)=−¯hσF 4e2λFtanh/parenleftbiggdF λF/parenrightbigg μs/bardbl(z↑0,ω). (39) Comparing with Eq. ( 14) we conclude that Ze F/bardbl=4e2λF ¯hσFcoth/parenleftbiggdF λF/parenrightbigg . (40) C. F-N interface: Longitudinal contribution from electrons The linearized charge and spin currents through the F-N interface at z=0 collinear with mare given by the equations [46] jz c(0,ω)=−e hgc/Delta1μ c(ω)−e 2hgs/Delta1μ s/bardbl(ω), (41) jz se/bardbl(0,ω)=−gs 4π/Delta1μ c(ω)−gc 8π/Delta1μ s/bardbl(ω), (42) where /Delta1μ c(ω)=μc(z↓0,ω)−μc(z↑0,ω),/Delta1μ s/bardbl(ω)= μs/bardbl(z↓0,ω)−μs/bardbl(z↑0,ω) are the drops of potential and spin accumulation over the F-N interface, gc=g↑↑+g↓↓is the total dimensionless interface conductance per unit area,and g s=g↑↑−g↓↓. For a ferromagnetic insulator g↑↑= g↓↓=0 and, hence, gc=gs=0. As before, we require that there be no charge current through the interface, which gives jse/bardbl(0,ω)=−gFN 8π/Delta1μ s/bardbl(ω), (43) with gFN=g2 c−g2 s gc. (44) Comparing Eq. ( 43) with Eq. ( 15), we conclude that Ze FN/bardbl=8π gFN. (45) Since the reﬂection at the F-N interface is effectively in- stantaneous, the frequency dependence Ze FN/bardblm a yb es a f e l y neglected for frequencies in the THz range and below. 024415-6THEORY OF SPIN-HALL MAGNETORESISTANCE IN THE … PHYSICAL REVIEW B 104, 024415 (2021) D. Ferromagnet: Transverse contribution For both insulating and metallic ferromagnets, to linear order in the applied ﬁelds, spin currents transverse to themagnetization direction e /bardblare carried by magnons. Up the THz frequency range, the resulting magnetization dynamicsmay be calculated from the Landau-Lifshitz-Gilbert equation.This approximation is valid as long as the frequency of theapplied ﬁelds is far below the excitation threshold of opticaland zone-boundary magnons, which, for YIG, holds for fre-quencies up to ω/2π≈5T H z[ 51]. The Landau-Lifshitz-Gilbert equation reads [ 52,53] ˙m=ω 0e/bardbl×m+αm×˙m−Dexm×∂2m ∂z2. (46) Hereω0is the ferromagnetic-resonance frequency, which includes effects of static external magnetic ﬁelds, demagne-tization ﬁeld, and anisotropies; αis the bulk Gilbert damping coefﬁcient, and D exthe spin stiffness. The spin current density from the magnetization dynamics is [ 54] jz s=−DexMs γm×∂m ∂z, (47) where Msis the magnetic moment per unit volume and γ= μBg/¯hthe gyromagnetic ratio. Inserting the parametrization ( 5) and keeping terms to linear order in m⊥only, one gets the linearized Landau- Lifshitz-Gilbert equation −Dex∂2 ∂z2m⊥(z,ω)=(ω+iαω−ω0)m⊥(z,ω). (48) With the boundary condition that the spin currents must vanish at the boundary of the F layer at z=−dF, the solution of this equation is m⊥(z,ω)=cos[K(ω)(z+dF)] cos[K(ω)dF]m⊥(0,ω), (49) where K(ω) is the solution of ω0+DexK2=ω(1+iα). (50) For the transverse spin current through the F-N interface at z=0 we then ﬁnd jz s⊥(0,ω)=−iDexMs γK(ω)t a n [ K(ω)dF]m⊥(0,ω),(51) so that ZF⊥(ω)=i¯hγω DexMsK(ω)cot[K(ω)dF]. (52) To further analyze this expression, we separate real and imaginary parts of the complex wave number K(ω), K(ω)=k(ω)+iκ(ω). (53) For frequencies ω>ω 0magnon modes exist in F. For small Gilbert damping α, the complex wave number K(ω)i sc l o s e to being real if ω>ω 0, with small imaginary part κ(ω)≈αω v(ω),ω/greaterorsimilarω0, (54)where v(ω)=dω/dkis the magnon velocity. For ω<ω 0, which includes negative frequencies, the complex wave num-berK(ω) is close to being purely imaginary, with imaginary part κ(ω)≈/radicalbigg ω0−ω Dex,ω/lessorsimilarω0, (55) reﬂecting the absence of magnon modes at these frequencies. In the limit of large dF, such that κ(ω)dF/greatermuch1, one has cot[k(ω)dF]→− i, so that ZF⊥(ω)→Z∞ F⊥(ω)≡¯hγω DexMsK(ω). (56) For\|ω\|/greatermuchω0, the limiting impedance Z∞ F⊥(ω) may be approx- imated as Z∞ F⊥(ω)≈¯hγ Ms/radicalBigg \|ω\| Dex×/braceleftbigg 1i f ω/greatermuchω0, −iifω/lessmuch−ω0.(57) Upon going to smaller thicknesses dFof the ferromagnetic layer, the approximation κ(ω)dF/greatermuch1 ﬁrst breaks down for frequencies ω/greaterorsimilarω0, because the imaginary part κ(ω) is small- est in that case, see Eqs. ( 54) and ( 55). To analyze the impedance ZF⊥(ω) in the regime κ(ω)dF/lessmuch1 for frequencies ω/greaterorsimilarω0, we note that ZF⊥(ω) Z∞ F⊥(ω)≈icot[k(ω)dF] +/summationdisplay nκ(ω)dF (k(ω)dF−nπ)2+κ(ω)2d2 F, (58) where the summation is over the integers n. The real part of ZF⊥(ω)/Z∞ F⊥(ω) exhibits a resonance structure with resonance spacing /Delta1ω≈πv/dFand a Lorentzian line shape with height 1/κ(ω)dF≈v/αωdFand full width at half maximum ≈2αω. It averages to one if averaged over a frequency window ofwidth much larger than the resonance spacing /Delta1ωbut smaller thanω−ω 0. The imaginary part of ZF⊥(ω)/Z∞ F⊥(ω)s h o w s large oscillations with period /Delta1ω that average to zero if av- eraged over frequency. The magnetic ﬁeld of the driving ﬁeld and the (Oersted) ﬁeld of the alternating charge current give an additional cor-rection to the transverse impedance discussed here and, hence,to the SMR. Since these ﬁelds are spatially uniform, theymainly couple to the uniform precession mode. Their effecton the SMR is strongest for frequencies in the vicinity ofthe ferromagnetic-resonance frequency, but it is negligible forother frequencies. We refer to Appendix Bfor a more detailed discussion. E. F-N interface: Transverse contribution The transverse spin current across the F-N interface cou- ples to the magnetization dynamics via the spin transfer torqueand spin pumping [ 46], j z s⊥(0,ω)=−g↑↓ 4π[μs⊥(z↓0,ω)+¯hωm⊥(0,ω)],(59) where g↑↓is the dimensionless spin-mixing conductance per unit area. Using the Landauer-Büttiker approach, g↑↓is 024415-7REISS, KAMPFRATH, AND BROUWER PHYSICAL REVIEW B 104, 024415 (2021) deﬁned as [ 44] gσσ/prime=1 Atr (1−rσr† σ/prime),σ , σ/prime=↑,↓, (60) with Athe total area of the F-N interface and r↑andr↓the reﬂection matrix for majority and minority electrons, respec-tively. Comparing Eqs. ( 59) with Eq. ( 19), we obtain Z FN⊥=4π g↑↓. (61) F. F-N interface: Longitudinal magnon contribution As described in Sec. IV D , a spin wave at frequency /Omega1/2π carries an alternating spin current jz s⊥with spin polarization perpendicular to the magnetization direction e/bardbl. The mag- nitude of this transverse spin current is proportional to theamplitude m ⊥of the spin wave, see Eq. ( 47). Additionally, a spin wave carries a nonoscillating (i.e., steady-state) spincurrent j z sm/bardblwith spin polarization collinear with the mag- netization direction. The magnitude of this longitudinal spincurrent is proportional to the difference of the densities of magnons moving in the positive and negative zdirection, which is quadratic in m ⊥. The net longitudinal magnon current across the F-N interface is nonzero if the F and N layers arebrought out of equilibrium, which occurs, e.g., if the spinaccumulation μ s/bardbl(z↓0) in N is nonzero [ 21]. In Refs. [ 23,24] the longitudinal spin current is calculated to leading order inthe spin-mixing conductance g ↑↓of the F-N interface (see also Refs. [ 55,56]) jz sm/bardbl(0)=−¯hγ πMsRe(g↑↓)/integraldisplay d/Omega1ν m(/Omega1)/Omega1/parenleftbigg −df0 d/Omega1/parenrightbigg ×[μs/bardbl(z↓0)−μm(0)], (62) where f0(/Omega1)=1/(e¯h/Omega1/kBT−1) is the Planck function, i.e„ a Bose-Einstein distribution with zero chemical potential, andνm(/Omega1)=(/Omega1−ω0)1/2/(4π2D3/2 ex) the magnon density of states. Equation ( 62) assumes that the temperatures on both sides of the F-N interface are the same. Since in our system the spin accumulation μs/bardbl(z↓0) oscil- lates at frequency ω/2πdue to the applied alternating electric ﬁeld and the SHE, the associated longitudinal magnon cur-rent j z sm/bardblin F oscillates at the same frequency. For driving frequencies ω/lessorsimilarkBT/¯h, this alternating longitudinal magnon spin current can be obtained from the steady-state result ( 62). For the spin impedance Zm FN/bardblit follows that 1 Zm FN/bardbl=¯hγ πMsRe(g↑↓)/integraldisplay d/Omega1ν m(/Omega1)/Omega1/parenleftbigg −df0 d/Omega1/parenrightbigg . (63) Corrections to Eq. ( 63) from the breakdown of the adia- batic approximation will become relevant at frequencies ω/greaterorsimilar kBT/¯h. At room temperature this is at ω/2π/greaterorsimilar6T H z . Equation ( 63) may be further simpliﬁed in the limit ¯ hω0/lessmuch kBT, which is applicable at room temperature. In this limit one ﬁnds [ 24] 1 Zm FN/bardbl≈3¯hγζ(3/2) 16Msπ5/2k3 TReg↑↓, (64)where kT=√kBT/¯hDexis the thermal magnon wave number andζ(3/2)≈2.61. G. Ferromagnet: Longitudinal magnon contribution If the thickness dFof the F layer is so small and the magnon lifetime in F so long that the excess magnons excited at the F-N interface cannot be transported away from the interface anddissipated in the bulk F efﬁciently enough, a ﬁnite magnonchemical potential μ m(ω) in F builds up—the magnet acts like a “magnon capacitor.” We here describe this effect in the limitof small d F, assuming that relaxation processes conserving the magnon number are fast enough that the magnon distributioncan be characterized by a uniform magnon chemical potentialacross the F layer. (The opposite limit, in which magnonspropagate ballistically in F and do not relax, is discussed inAppendix A.) Balancing the inﬂux of magnons through the F-N inter- face and the decay of excess magnons with lifetime τ(/Omega1)= 1/2α/Omega1 due to the phenomenological Gilbert damping, we ﬁnd, to linear order in μ m, jz sm/bardbl(0)=− dF/integraldisplay∞ ω0d/Omega1ν m(/Omega1)/parenleftbigg −df0(/Omega1) d/Omega1/parenrightbigg/bracketleftbigg ˙μm−μm τ(/Omega1)/bracketrightbigg . (65) Fourier transforming and comparing to Eq. ( 16) results in an impedance of the form Zm F/bardbl(ω)=1 Cm(−iω+1/τm). (66) In the limit ¯ hω0/lessmuchkBT, the expressions for the “magnon capacitance” Cmper unit area and the effective magnon life timeτmare Cm=dF 8π√ω0Dexk2 T, (67) τm=√π 3ζ(3/2)αkT√ω0Dex, (68) where the thermal magnon wave number kTis deﬁned below Eq. ( 64). The effective magnon lifetime τmis signiﬁcant in the low-frequency regime ωτm/lessorsimilar1 only and may be effectively set to inﬁnity for frequencies in the THz regime. H. Numerical estimates for the impedances To obtain an understanding of the order of magnitude of the spin impedances ZN,Ze F/bardbl,Zm F/bardbl,ZF⊥,Ze FN/bardbl,Zm FN/bardbl, and ZFN⊥, we calculate numerical values using typical parameters for anF\|N bilayer consisting of the ferromagnetic insulator YIG and the normal metal Pt, as well as for a bilayer consisting of theferromagnetic metal Fe and the normal metal Au. Numericalvalues for the relevant material and device parameters arecollected in Table I, together with experimental references. (We note, however, that there is a large variation in litera-ture values for the spin-Hall angle [ 12–15,36,57–63]θ SH,t h e spin-relaxation lengths [ 12,14,15,57–59,64,65]λN,F, and the interface conductances [ 12–15,57–59]gσσ/prime. Different values for these quantities lead to different quantitative predictionsbut do not affect our qualitative conclusions.) 024415-8THEORY OF SPIN-HALL MAGNETORESISTANCE IN THE … PHYSICAL REVIEW B 104, 024415 (2021) TABLE I. Typical values for the relevant material and device parameters of the F \|N bilayers considered in this paper. The last column states the references used for our estimates. Material and device parameters Ref. Pt σN=9×106/Omega1−1m−1,[ 66] θSH=0.1,λN=2×10−9m, [ 15,18,67] dN=4×10−9m[ 18] Au σN=4×107/Omega1−1m−1,[ 68] θSH=0.08,λN=6×10−8m, [ 58] dN=6×10−8m[ 69] YIG ω0/2π=8×109Hz,α=2×10−4,[ 13] Dex=8×10−6m2s−1,Ms=1×105Am−1,[ 67] dF=6×10−8m[ 18] Fe ω0/2π=8×109Hz,α=5×10−3,a[69] Dex=4×10−6m2s−1,Ms=2×106Am−1,[ 70] σF=1×107/Omega1−1m−1,[ 71] λF=9×10−9m, [ 65] dF=2×10−8m[ 69] YIG\|Pt ( e2/h)Reg↑↓=6×1013/Omega1−1m−2,[ 13,59] YIG\|Pt ( e2/h)Img↑↓=0.3×1013/Omega1−1m−2 b Fe\|Au ( e2/h)Reg↑↓=1×1014/Omega1−1m−2[58] (clean) ( e2/h)Img↑↓=0.05×1014/Omega1−1m−2,[ 73] (e2/h)g↑↑=4×1014/Omega1−1m−2,[ 73] (e2/h)g↓↓=0.8×1014/Omega1−1m−2[73] aThe resonance frequency ω0depends strongly on the external mag- netic ﬁeld applied in experiments. To make a comparison possible,we assume a magnetic ﬁeld which results in the same resonance frequency as for the YIG \|Pt bilayers of Ref. [ 13]. bAccording to Refs. [ 15,72], Im g↑↓/Reg↑↓≈0.05, which is also used here to estimate Im g↑↓. Estimates for the frequency-independent impedances as well as the “magnon capacitance” Cmand the effective magnon lifetime τmobtained this way can be found in Table II. Since the spin-relaxation lengths λNandλFare smaller than typical layer thicknesses dNand dFused in experiments, we list ZNand Ze F/bardblfor the limit of large dN TABLE II. Estimates for the impedances of the F \|N bilayers considered in this paper. Parameter values are taken from Table I.A l l impedances are given in 10−12/Omega1m2; the “magnon capacitance” Cm is in 103Fm−2and the effective magnon lifetime in 10−9s. The in- terface impedance Zm FN/bardbl,Cm,a n dτmare evaluated at T=300 K. The estimates for Zm F/bardblandZ∞ F⊥are atω/2π=1 GHz and ω/2π=1T H z , respectively. The full frequency dependence of ZF⊥(ω)a n d Zm F/bardbl(ω)i s shown in Fig. 4for YIG \|Pt. F\|NY I G \|Pt Fe \|Au (h/e2)ZN 0.0055 0.038 (h/e2)ZFN⊥ 0.21–0.01i 0.13–0.006i (h/e2)Zm FN/bardbl 0.30 1.3 (h/e2)Ze FN/bardbl 0.096 (h/e2)Ze F/bardbl 0.023 (e2/h)Cm F/bardbl 0.72 0.68 τm 0.81 0.032 (h/e2)Zm F/bardbl(2πGHz) 0 .043+0.21i 0.045+0.009i (h/e2)Z∞ F⊥(2πTHz) 0 .0043–4 .3×10−7i 0.0003–7 .6×10−7iFIG. 4. Numerical estimates for the spin impedances of a YIG \|Pt bilayer as a function of the frequency ω/2π. For the impedance ZNthe large- dNlimit is shown; for \|ZF⊥(ω)\|both the large- dFlimit (solid red line) and the ﬁnite- dFcase (dotted red line) are shown, as discussed in the main text. Parameter values are taken from Table I. The inset shows the same impedances on a logarithmic scale in the GHz frequency regime. The horizontal lines indicate ZN(blue dashes, main panel and inset), ZFN⊥(green dot-dashes, inset), and Zm FN/bardbl(light-purple dots, inset). and dF, respectively. For the impedances ZF⊥and Zm F/bardbl,a s well as Cm, we take values for dFtypical for recent experi- ments, see Table I. Figure 4shows the frequency-dependent spin impedances ZF⊥(ω) and Zm F/bardbl(ω)f o raY I G \|Pt bilayer. For a comparison of numerical values, the transverse spinimpedance Z ∞ F⊥(ω)a tω/2π=1 THz and the longitudinal spin impedance Zm F/bardbl(ω)f o rω/2π=1 GHz are also included in Table II. The impedance ZF⊥(ω)f o raF e \|Au bilayer has a similar frequency dependence as in Fig. 4but a value that is a factor ∼10 smaller (not shown). The smallness of ZF⊥(ω) in comparison to the transverse interface impedance ZFN⊥for YIG\|Pt and Fe \|Au bilayers means that in both YIG and Fe spin angular momentum is efﬁciently transported away fromthe F-N interface for frequencies well into the THz regime. For both YIG \|Pt and Fe \|Au, the longitudinal and trans- verse interface impedances are of comparable magnitude atT=300 K. Since \|Z F⊥(ω)\|is typically much smaller than \|ZFN⊥\|, except in the immediate vicinity of resonances, the transverse spin current through the interface is dominated bythe interfacial impedance. The same applies to the longitu-dinal spin current carried by conduction electrons if F is ametallic ferromagnet. The situation is different for the lon-gitudinal magnonic spin current which depends strongly onfrequency. In the zero-frequency limit, the longitudinal spinimpedance Z m F/bardblis much larger than the corresponding inter- facial spin impedance, so that the longitudinal spin currentcarried by magnons is strongly suppressed. At high frequen-cies, Z m F/bardblbecomes small, and magnons can carry a sizable longitudinal spin current. The crossover between these tworegimes depends on the thickness d Fof the F layer and the ferromagnetic-resonance frequency ω0. For the parameters 024415-9REISS, KAMPFRATH, AND BROUWER PHYSICAL REVIEW B 104, 024415 (2021) listed in Table I, the longitudinal magnon current sets in at ω/2π/greaterorsimilar1 GHz for YIG \|Pt. Taking typical values for ω0,α, and Dexfrom Table I, one ﬁnds that at a frequency ω/2π∼1 THz the asymptotic large- dFregime κ(ω)dF/greatermuch1 for the transverse spin impedance ZF⊥(ω)s e t si na t dF/greaterorsimilar1×10−5m for YIG and dF/greaterorsimilar3× 10−7m for Fe. (The difference is caused by the smallness of the Gilbert damping in YIG.) Both values are larger than thetypical thickness d F∼10−8m of thin magnetic ﬁlms used in experiments, see Table I. At these smaller thicknesses, ZF⊥(ω) develops a resonance structure, which may be measurable inexperiments, if variations in d Fand frequency resolution are small enough. For example, at a thickness dF∼6×10−8m of a YIG layer, the resonance spacing /Delta1ω/ 2πat frequency ω/2π∼1T H zi s /Delta1ω/ 2π∼1×10−1THz, and for dF∼2× 10−8m of an Fe layer /Delta1ω/ 2π∼2×10−1THz, whereas the typical frequency resolution of THz time-domain experimentsis larger than 100 GHz, owing to the <10 ps wide time window that is sampled [ 9]. V . NUMERICAL ESTIMATES OF THE SMR The conductivity corrections δσxxandδσxydepend on the magnetization direction meq=e/bardbl,s e eE q .( 1). We char- acterize the conductivity corrections δσxx(e/bardbl) and δσxy(e/bardbl) corresponding to the spin-Hall magnetoresistance (SMR), pla-nar Hall effect (PHE), and anomalous Hall effect (AHE) usingthe three complex dimensionless quantities /Delta1 SMR=[δσxx(ey)−δσxx(ex)]/σN, /Delta1PHE=−δσxy(exy)/σN, /Delta1AHE=−δσxy(ez)/σN, (69) where we abbreviated exy=(ex+ey)/√ 2. The planar Hall effect and spin-Hall magnetoresistance characteristics are re-lated, see Eqs. ( 23) and ( 24), /Delta1 PHE=1 2/Delta1SMR. (70) Experimentally, the real part of /Delta1SMR is a magnetization direction-dependent correction to the magnitude of the currentdensity ¯j x c(ω) averaged over the thickness dNof the N layer, whereas the imaginary part of /Delta1SMR is the magnetization direction-dependent part of the phase shift between ¯jx c(ω) and the applied electric ﬁeld E(ω). The modulus and phase of the complex coefﬁcients /Delta1PHEand/Delta1AHEdescribe magnitude and phase of the transverse current ¯jy c(ω). Below we discuss the full frequency-resolved characteris- tics/Delta1SMR and/Delta1AHE, which contain a contribution from the sharp magnon resonances in F, as well as from the asymp-totic characteristics ¯/Delta1 SMR and ¯/Delta1AHE, which are obtained by replacing the transverse spin impedance ZF⊥(ω) with Z∞ F⊥(ω) of Eq. ( 56). For deﬁniteness, we take experiment-motivated sample thicknesses dNanddFas listed in Table I. Numerical estimates for /Delta1SMR and/Delta1AHEare shown in Figs. 5and6for YIG\|Pt and Fe \|Au bilayers, respectively. No separate results are shown for /Delta1PHE, since /Delta1PHE=(1/2)/Delta1SMR,s e eE q .( 70). In both ﬁgures the frequency dependence of /Delta1SMR at low frequencies ( ω/2π/lessorsimilar1 GHz) mainly results from the fre- quency dependence of Zm F/bardbl(ω), while above ω0it stems from the resonance structure of ZF⊥(ω). Comparing Re /Delta1SMR(ω)FIG. 5. Real and imaginary parts of the dimensionless charac- teristics /Delta1SMR of the spin-Hall magnetoresistance (top) and /Delta1AHE of the anomalous Hall effect (bottom) for a YIG \|Pt bilayer. The thick solid lines use the asymptotic value Z∞ F⊥(ω)o fE q .( 56); thin dashed lines use the full expression for ZF⊥(ω), which includes the effect of spin-wave resonances in the YIG layer. Material and deviceparameters are taken from Table I. atω=0 and ω/2π=1 THz, there is a decrease of the size of the SMR by 68% for YIG \|Pt and an increase by ca. 3% for Fe \|Au, which mainly has its origin in the frequency range below 1 GHz. The net frequency dependence of /Delta1SMR for Fe \|Au is weaker than for YIG \|Pt, because for Fe \|Au the frequency-independent electronic spin impedances Ze FN/bardbl+ Ze F/bardblshunt the frequency-dependent magnon spin impedances Zm FN/bardbl+Zm F/bardbl(ω)( c f .F i g . 3). For Fe \|Au,/Delta1SMR is negative, as the longitudinal contributions dominate over the transverseone. In Fig. 5, the imaginary part Im /Delta1 SMR exhibits a peak for frequencies ω/2πin the GHz range, where the (almost purely imaginary) longitudinal spin impedance Zm F/bardblmatches the corresponding interface impedance Zm FN/bardbl. The discussion of Z∞ F⊥(ω) in Sec. IV D implies that the asymptotic characteristic ¯/Delta1SMR describes the SMR for the case that the frequency resolution is larger than thespacing between spin-wave resonances in F. Although the 024415-10THEORY OF SPIN-HALL MAGNETORESISTANCE IN THE … PHYSICAL REVIEW B 104, 024415 (2021) FIG. 6. Same as Fig. 5but for an Fe \|Au bilayer. Parameter values are taken from Table I. spin impedance Z∞ F⊥(ω) depends strongly on frequency, see Sec. IV D , for both material combinations the transverse con- tribution to ¯/Delta1SMR depends only very weakly on frequency, as the spin impedance Z∞ F⊥(ω) appears in series with the much larger interface spin impedance ZFN⊥, which is frequency independent. The same applies to the PHE and AHE correc-tions to the conductivity, characterized by ¯/Delta1 PHEand ¯/Delta1AHE, respectively. If the effect of a ﬁnite thickness dFis taken into ac- count, the SMR response acquires sharp resonances, reﬂectingthe resonance structure of Z F⊥(ω). The real part of /Delta1SMR shows narrow symmetric features for frequencies around the standing spin-wave modes in F. For a quantitative discussionwe notice that the transverse contribution to the conductiv-ity correction δσ xx(ω) is a sum of contributions involving impedances at frequencies ωand−ω. Of these, it is only the positive-frequency contribution that is affected by theresonances in Z F⊥(ω). Neglecting the small imaginary part of ZFN⊥and taking into account that \|Z∞ F⊥(ω)\|/lessmuch\| ZFN⊥\|for both material combinations we consider, one ﬁnds that at the res-onance center the transverse contribution to /Delta1 SMR is reduced by a factor ≈[1−dc F/2(dF+dc F)], with dc F=2γ¯h/αMs(ZN+ZFN⊥). The full width at half maximum of the resonant fea- tures in Re /Delta1SMR isδω≈2αω(1+dc F/dF). The crossover scale dc Fseparates the low- dFregime [ 47,48], in which the life- time of spin waves is limited by decay into N, and the large- dF regime, in which intrinsic Gilbert damping determines the magnon lifetime. Taking material parameters from Table I, we ﬁnd that dc F≈0.2μmf o rY I G \|Pt and dc F≈0.6n m f o r Fe\|Au. The imaginary part of /Delta1SMR has abrupt jumps at the spin-wave frequencies and is a smooth function of frequencyotherwise. The same discussion applies to /Delta1 PHEand to /Delta1AHE but with the roles of real and imaginary parts reversed for the latter. Lotze et al. [18] performed a measurement of the spin- Hall magnetoresistance of a YIG \|Pt bilayer for frequencies up toω/2π=3 GHz. The magnetic ﬁeld in this experiment was 0.6 T, resulting in a ferromagnetic-resonance frequency ω0/2π≈17 GHz, which is an order of magnitude larger than what we used for our numerical evaluation. The in-creased value of ω 0shifts the lowest magnon resonances to higher frequencies and it lowers the “magnon capacitance” Cm,s e eE q .( 67). As a result, the low-frequency regime, in which Zm F/bardbl(ω) dominates the longitudinal spin current, is extended to higher frequencies. Setting ω0/2π=17 GHz and taking the other material parameters from Table I,w e ﬁnd that Re /Delta1SMR decreases by approximately 10% between ω=0 and ω/2π=3 GHz, consistent with the observation of Ref. [ 18], who found that the magnitude of the spin-Hall magnetoresistance does not change within the experimentaluncertainty of ∼10%. VI. CONCLUSION AND OUTLOOK The main difference between the zero-frequency and ﬁnite- frequency versions of the spin-Hall magnetoresistance (SMR)results from the onset of spin transport by incoherent magnonsand the resonant coherent excitation of magnetization modesin F for frequencies in the GHz range and above. Extend-ing the work of Chiba et al. [26], who developed a theory of the ﬁnite-frequency SMR that accounted for the uniformprecession mode at the ferromagnetic-resonance frequency,we here present a theory that includes acoustic spin waves ofarbitrary wavelength, accounting for their role in incoherentlongitudinal as well as coherent transverse spin transport, thusallowing for an extension to the GHz and THz frequencyrange. Our theory is a general linear response theory, in whichthe relevant response functions of the normal metal, the fer-romagnet, and the interface are lumped together into “spinimpedances,” see Fig. 3. These spin impedances relate spin current and spin accumulation in the same way as standardimpedances relate charge current and voltage. This allows foran efﬁcient description of F \|N bilayers and F \|N\|F trilayers, in which the thickness d Nexceeds the spin-relaxation length λN, so that the SMR is the sum of contributions from the two interfaces of the normal-metal layer, as well as of bilayers ofa smaller thickness. The SMR corrections δσ xxandδσxyto the diagonal and off-diagonal conductivity—see Eqs. ( 23) and ( 24) and Figs. 5, 6—are the difference of contributions associated with the ﬂow of spin angular momentum collinear (“longitudinal”) and per-pendicular (“transverse”) to the magnetization direction. The 024415-11REISS, KAMPFRATH, AND BROUWER PHYSICAL REVIEW B 104, 024415 (2021) longitudinal contribution is carried by conduction electrons and thermal magnons inside the ferromagnet. The magnoniclongitudinal contribution to the SMR, which is the sole lon-gitudinal contribution if F is insulating, has a systematicfrequency dependence, with a characteristic frequency scalein the GHz range for magnetic layer thickness d F∼10 nm. The origin of this frequency dependence is the buildup of amagnon chemical potential in F, which effectively blocks alongitudinal magnonic spin current in the DC limit but notfor high frequencies. The transverse contribution, which iscarried by coherent magnons, features sharp resonances at thespin-wave frequencies of the F \|N bilayer but otherwise has only a small systematic frequency dependence. With sufﬁcientfrequency resolution the THz-SMR may be an interesting toolfor all-electric spectroscopy of magnon modes. Furthermore,our analysis has shown that the excitation and propagation ofcoherent magnons is as efﬁcient at THz frequencies as in theGHz range. The combination of the longitudinal and transverse con- tributions leads to a signiﬁcant decrease of the SMR forYIG\|Pt by 68% between ω=0t oω/2π=1 THz. The SMR decreases so strongly, because for bilayers involving YIGas a ferromagnetic insulator, the transverse and longitudinalcontributions to the SMR are of comparable magnitude athigh frequencies at room temperature, with the transversecontribution dominating, whereas the longitudinal contribu-tion is suppressed at low frequencies. The thickness d Fof the ferromagnetic layer, the applied magnetic ﬁeld, and thechoice of the normal metal affect the characteristic crossoverfrequency but not the overall change of the SMR between thelow and high frequency limits. Since the ratio of longitudinalto transverse spin impedances of the F-N interface system-atically decreases with temperature [ 23,24,74,75], we expect that the longitudinal and transverse contributions for bilayersinvolving YIG as a ferromagnetic insulator eventually canceleach other and, hence, that the SMR vanishes, upon raising thetemperature above room temperature. Further, whereas for thethicknesses typically used in experiments the characteristicfrequency for the onset of the longitudinal contribution to theSMR is comparable to the ferromagnetic-resonance frequencyω 0, the two frequency scales can be pushed apart by consider- ing smaller or larger layer thicknesses. In particular, for larged Fwe predict that the frequency-dependent suppression of the SMR can set in signiﬁcantly below 1 GHz, so that it is mea-surable by conventional high-frequency electronic techniques. An upper limit for the applicability of our theory is the maximum frequency ω maxof acoustic magnons. For the mag- nets we consider here, one has ωmax/2π∼5T H z[ 51,76]. At this frequency, both the description of the F-N interface as amere “boundary condition” and the description of magnetiza-tion dynamics by the Landau-Lifshitz-Gilbert equation ( 46) cease to be valid. Approximately the same frequency restrictsthe use of the quasiadiabatic approximation for longitudinalspin transport by thermal magnons at room temperature. Bothfrequency limits are below frequency scales at which otherassumptions of our theory cease to be valid, such as neglectingfrequency-dependent corrections to the Drude scattering ratesin N and F (valid for ω/2π/lessorsimilar10 THz). Apart from the appearance of sharp resonances at spin wave frequencies, the transverse contribution to the SMRhas no appreciable systematic frequency dependence for the material combinations YIG \|Pt and Fe \|Au for frequencies well into the THz regime. Within our formalism, the reason is thatthe transverse contribution to the SMR is dominated by thefrequency-independent spin impedance Z FN⊥of the interface, which obscures the strong frequency dependence of ZF⊥(ω) in the THz regime. We have also considered other materialsthat are used in spintronics experiments, such as Cu, Co,CoFe(B), Py (permalloy), GdFe(Co), GdIG (gadolinium irongarnet), Fe 3O4(magnetite), NiFe 2O4,E u S[ 77], and EuO [ 78] in combination with different nonmagnetic heavy metals as Nlayers and arrived at the same qualitative conclusion: In allcases, the transverse impedance Z ∞ F⊥(ω), which describes the frequency-averaged part of magnon-mediated spin transportin F, is much smaller than Z FN⊥, ruling out a substantiative effect of the magnon spin impedance Z∞ F⊥on the SMR. This leads to the question, whether there are other, less- explored material combinations, for which the transversecontribution to the SMR has a stronger systematic frequencydependence. To address this question, we ﬁnd it instructiveto consider Z F⊥(ω) at the highest frequency ωmaxfor which our long-wavelength theory is valid, which is at the boundaryof the (magnetic) Brillouin zone. Using M s/¯hγ=S/ax Say Saz S, where Sandax,y,z Sare the spin and linear dimensions of the magnetic unit cell, respectively, and ωmax≈Dex(π/az)2,w e estimate that /vextendsingle/vextendsingleZ∞ F⊥(ω)/vextendsingle/vextendsingle/lessorequalslant\|ZF⊥(ωmax)\|∼πax Say S S. (71) A lower limit for the interface impedance ZFN⊥is obtained using the Sharvin resistance of the F-N interface [ 44,73], ZFN⊥/greaterorsimilar4πλ2 e, (72) where λeis the Fermi wavelength in N. The fact that for most material combinations λeandax,y Sare comparable, whereas S is of order unity or larger, explains why material combinationswith Z F⊥(ω) larger than ZFN⊥are hard to ﬁnd. As a guiding principle for the search for material combina- tions in which ZF⊥(ω) is large and ZFN⊥is small, Eqs. ( 71) and (72) suggest to consider materials with a large magnetic unit cell, small S, and a spin-mixing conductance that approaches the Sharvin limit as closely as possible. The spin stiffnessD exdoes not directly enter into the comparison of ZF⊥(ωmax) andZFN⊥,b u tas m a l l Dexlowers the frequency scale ωmax, making it easier to reach this frequency scale experimentally.A promising class of materials in this regard are half-metallic(fully) compensated ferrimagnetic Heusler compounds, suchas Mn 3Al or Mn 1.5FeV 0.5Al. The former compound has a band gap ∼0.5 eV for minority carriers [ 79] and a magne- tization Ms≈2×104Am−1[80], which is almost an order of magnitude below the corresponding value for YIG. Thesmall value of M sshould result in a relatively large value of ZF⊥, consistent with the expectation that acoustic magnons do not efﬁciently transport spin angular momentum in an almostcompensated ferrimagnet, whereas the half-metallic charac-ter ensures a large spin-mixing conductance. Indeed, therelated half-metallic ferromagnetic Heusler compounds suchas Co 2MnSi [ 81–85], Co 2FeAl [ 86], and Co 2Fe0.4Mn 0.6Si [87] are reported to have spin-mixing conductances around or above the spin-mixing conductance of YIG \|Pt. For a 024415-12THEORY OF SPIN-HALL MAGNETORESISTANCE IN THE … PHYSICAL REVIEW B 104, 024415 (2021) more complete answer to this question, however, a detailed calculation of the spin-mixing conductances for the almostcompensated ferrimagnetic compounds is necessary, as wellas a description of the F-N interface that accounts for theeffect of the ﬁnite penetration of minority electrons into F onthe coupling to short-wavelength acoustic magnons. ACKNOWLEDGMENTS We thank Y . Acremann, G. Bierhance, S. T. B. Gönnen- wein, L. Nádvornik, M. A. Popp, I. Radu, U. Ritzmann, S.M. Rouzegar, and R. Schlitz for stimulating discussions. Thiswork was funded by the German Research Foundation (DFG)via the collaborative research center SFB-TRR 227 “UltrafastSpin Dynamics” (projects A05, B02, and B03). APPENDIX A: LONGITUDINAL SPIN CURRENT FOR BALLISTIC MAGNON DYNAMICS In this Appendix we describe the longitudinal magnonic spin current through the F-N interface if the magnon dynamicsin F is ballistic, with specular reﬂection at the interface of theF layer to the vacuum at z=−d F. Magnons at the F-N inter- face are described by their distribution function f(/Omega1,kx,ky), explicitly accounting for the wave-vector components kxand kyparallel to the F-N interface. We distinguish the distribution functions fin(/Omega1,kx,ky) of magnons incident on the F-N inter- face and fout(/Omega1,kx,ky) of magnons moving away from the interface. With ballistic magnon transport in F and specularreﬂection of magnons at the ferromagnet-vacuum interface atz=−d F, we obtain the relation fin(/Omega1,kx,ky;t)=fout(/Omega1,Kx,Ky;t−2dF/vz), (A1) where vz(/Omega1,kx,ky)=2Dexkz(/Omega1,kx,ky) is the magnon veloc- ity perpendicular to the interface. Magnons incident on the F-N interface are reﬂected into F with probability Rm(/Omega1,kx,ky). The difference Tm=1−Rm is the probability that a magnon is annihilated at the interface and transfers its angular momentum to the conduction elec-trons in N. It is [ 56,88] T m(/Omega1,kx,ky)=MsDexkz/Omega1/π ¯hγ \|MsDexkz/¯hγ+/Omega1g↑↓/4π\|2Reg↑↓,(A2) where kz=/radicalBigg /Omega1−ω0 Dex−k2x−k2y. (A3) The magnon distribution at the interface then satisﬁes the additional boundary condition [ 88] fout(/Omega1,kx,ky;t)=Rm(/Omega1,kx,ky)fin(/Omega1,kx,ky;t) +Tm(/Omega1,kx,ky)f0(/Omega1−μs/bardbl(z↓0,t)/¯h), (A4) where f0is the Planck distribution. The second term in Eq. ( A4) ensures that fout=finif the temperatures in F and N are the same and the magnon chemical potential in F equalsthe spin accumulation in N [ 24].FIG. 7. Real part (top) and imaginary part (bottom) of the com- bined impedance Zm /bardbl(ω) from Eq. ( A7) for a ballistic F layer (dashed and dot-dashed curve) and of the impedance sum Zm FN/bardbl+Zm F/bardbl(ω) (solid curves) of Secs. IIIandIV F,IV G . Material and device pa- rameters are taken from Table I. In linear response, the distribution functions finand fout may be expanded as fin,out(/Omega1,kx,ky;t) =f0(/Omega1)+/parenleftbigg −df0(/Omega1) d/Omega1/parenrightbigg φin,out(/Omega1,kx,ky;t). (A5) Setting μs/bardbl(t)=μs/bardbl(ω)e−iωt+μs/bardbl(−ω)e+iωtand solving for the linear-response corrections φin,out(/Omega1,kx,ky;t) to the dis- tribution functions, we ﬁnd that the spin current through theF-N interface is j sm/bardbl(0,ω)=−1 Zm /bardbl(ω)μs/bardbl(z↓0,ω), (A6) with 1 Zm /bardbl(ω)=1 (2π)3/integraldisplay dkxdky/integraldisplay d/Omega1/parenleftbigg −df0 d/Omega1/parenrightbigg Tm(/Omega1,kx,ky) ×/bracketleftbigg 1−Tm(/Omega1,kx,ky)e2iωdF/vz 1−Rm(/Omega1,kx,ky)e2iωdF/vz/bracketrightbigg . (A7) The ﬁrst term between the square brackets is identical to the (inverse) longitudinal interfacial spin impedance ZFN/bardbl.T h e correction term accounts for the magnon accumulation in F. Inthe limit of zero frequency, the correction term imposes that j sm/bardbl→0, i.e., Zm /bardbl→∞ . For frequencies large enough such thatωdF/greatermuchvzfor thermal magnons, the integrand in the cor- rection term is a fast-oscillating function of frequency, so thatZ m /bardbl(ω)→Zm FN/bardbl. Figure 7compares the ballistic impedance Zm /bardbl(ω)o fE q .( A7) with the corresponding sum impedance Zm FN/bardbl+Zm F/bardbl(ω)o faY I G \|Pt interface calculated using the ef- fective magnetoelectronic circuit theory of the main text. Theplot illustrates this limiting behavior of Z m /bardbl(ω). 024415-13REISS, KAMPFRATH, AND BROUWER PHYSICAL REVIEW B 104, 024415 (2021) APPENDIX B: RESPONSE TO OERSTED FIELD The (Oersted) magnetic ﬁeld of the alternating charge cur- rent jc(t) drives a magnetization precession which, via spin pumping and the ISHE, results in an additional correctionto the charge current. Since the SMR correction δj cto the charge current density in N is small, we may neglect it whencalculating the magnetic ﬁeld B(t) in F and set [ 89] B(t)=B y(t)ey,By(t)=μ0E(t)σNdN 2. (B1) The magnetization response is found by inclusion of the mag- netic ﬁeld into the Landau-Lifshitz-Gilbert equation ( 46), ˙m=ω0e/bardbl×m+αm×˙m−Dexm×∂ ∂z2m+γm×B. (B2) Linearizing Eq. ( B2) and performing a Fourier transformation to frequency as in Sec. IV D , we ﬁnd that only the transverse component B⊥(ω)=e∗ ⊥·B(ω)( B 3 ) of the Oersted ﬁeld affects the magnetization dynamics, −Dex∂2 ∂z2m⊥(z,ω)=(ω+iαω−ω0)m⊥(z,ω)−γB⊥(ω). (B4) With the boundary condition that the spin currents must vanish at the boundary of the F layer at z=−dF, the solution of this equation is m⊥(z,ω)=m⊥(0,ω)cos[K(ω)(z+dF)] cos[K(ω)dF]+χ⊥(ω)B⊥(ω), (B5) where χ⊥(ω)=γ ω+iαω−ω0. (B6) The uniform magnetization precession driven by the Oersted ﬁeld does not carry a spin current inside F, but it does lead toan additional contribution to the spin current through the F-Ninterface via spin pumping. Hence, Eqs. ( 13) and ( 14) remain unchanged, whereas the transverse part of Eqs. ( 19) has to be modiﬁed, μ s⊥(z↓0,ω)+¯hωm⊥(0,ω) =−ZFN⊥(ω)jz s⊥(0,ω)−χ⊥(ω)¯hωB⊥(ω). (B7) Repeating the calculations of Sec. III, we then ﬁnd that the corrections δσxxandδσxyto the in-plane conductivity are still given by Eqs. ( 23) and ( 24) but with the replacement 1 Z⊥(ω)→1 Z⊥(ω)/bracketleftbigg 1+eωμ 0χ⊥(ω)dN ZNθSH/bracketrightbigg . (B8) For the material and device parameters used in Secs. IV H andV, see Table I, inclusion of the Oersted ﬁeld only affects the conductivity correction for frequencies in the immediatevicinity of the ferromagnetic-resonance frequency ω 0, where the transverse susceptibility χ⊥(ω) is maximal. This is illus- trated in Fig. 8for the case of a YIG \|Pt bilayer. InclusionFIG. 8. Comparison of the characteristics /Delta1SMR(dashed and dot- dashed curve) and ¯/Delta1SMR(solid curves) with and without inclusion of the Oersted ﬁeld. Thick (colored) curves are for the case with Oersted ﬁeld; thin (black) curves are for the case without it. The left and rightpanels show real and imaginary parts, respectively. Parameter values are taken from Table I. of the Oersted ﬁeld has a much larger effect on the SMR in the limit of large dF(solid curves in Fig. 8) than in the limit of small dF(dashed and dash-dotted curves). Also, upon inclusion of the Oersted ﬁeld, the form of the lineshape ofthe resonance near ω 0is switched between imaginary and real parts of /Delta1SMR—something that can be understood noting that the correction factor in Eq. ( B8) is almost purely imaginary forω=ω0. Since the Oersted ﬁeld only drives a uniform precession mode of the magnetization, it does not couple tothe magnon modes at higher frequencies. APPENDIX C: SMR FOR F \|N BILAYERS AND F \|N\|F TRILAYERS The results of Sec. IIIdescribe the corrections to the charge current densities jx cand jy cassociated with a single interface for the case that the normal layer has a thickness dNmuch larger than the spin-relaxation length λN. In this limit, the total correction to the current density from the combination of theSHE and ISHE is the sum of the corrections associated withthe F-N interface at z=0 and the N-vacuum interface at z= d N, which is independent of the magnetization direction m.I n this section we consider F \|N bilayers and F \|N\|F trilayers with dN/lessorsimilarλN, for which the contributions from the two interfaces no longer add up. For a normal layer of ﬁnite thickness dN, the ISHE cor- rection δ¯jcto the (effective) current densities follows by integration of Eqs. ( 11) and ( 12) over the entire cross section of the normal metal, δ¯jc(ω)=θSHσN 2edNez×[μs(z↑dN,ω)−μs(z↓0,ω)].(C1) 024415-14THEORY OF SPIN-HALL MAGNETORESISTANCE IN THE … PHYSICAL REVIEW B 104, 024415 (2021) Thezdependence of the spin accumulation μsis found from the solution of the spin-diffusion equation ( 26), μs(z)=1 sinh( dN/λN)/bracketleftBigg μs(z↓0) sinh/parenleftbiggdN−z λN/parenrightbigg +μs(z↑dN)s i n h/parenleftbiggz λN/parenrightbigg/bracketrightBigg . (C2) For the spin current densities jz s(z) at the normal-metal inter- faces at z=0 and z=dNone thus obtains, see Eq. ( 10), jz s(0)=1 ZN/bracketleftbiggμs(z↓0) tanh( dN/λN)−μs(z↑dN) sinh( dN/λN)/bracketrightbigg −θSH¯hσN 2eEey, jz s(dN)=1 ZN/bracketleftbiggμs(z↓0) sinh( dN/λN)−μs(z↑dN) tanh( dN/λN)/bracketrightbigg −θSH¯hσN 2eEey, (C3) where the impedance ZNis deﬁned in Eq. ( 28). In the limit dN/greatermuchλN, the terms proportional to 1/sinh( dN/λN) can be neglected and the spin accumulations at the two interfaces at z=0 and z=dNcan be calculated separately. The resulting current correction is the sum ofcontributions from the two interfaces, as discussed above.To calculate the spin accumulations for ﬁnite d N,w en o w consider the cases of an F \|N bilayer and an F \|N\|F trilayer separately. 1. SMR for F \|N bilayer The boundary condition for the insulating interface at z= dNis that of zero spin current density, jz s(dN)=0. (C4) With the help of this boundary condition, the spin accumula- tionμs(z↑dN) at the insulating surface may be eliminated from Eq. ( C3). This allows us to express the spin current density jz s(0) at the ferromagnet interface at z=0i nt e r m s of the spin accumulation μs(z↓0) at that interface, jz s(0)=μs(z↓0) ZNtanh( dN/λN) −θSH¯hσN 2eEey/bracketleftbigg 1−1 cosh( dN/λN)/bracketrightbigg . (C5) This is the same expression as Eq. ( 13) but with the replace- ments ZN→˜ZN≡ZNcoth( dN/λN), E→˜E=E/bracketleftbigg 1−1 cosh( dN/λN)/bracketrightbigg . (C6) When these replacements are made, the calculation of μs(z↓ 0) then follows that of Sec. III. Compared to Sec. III,t h e calculation of the charge currents involves the replacement ofthe spin accumulation μ s(z↓0) at the F-N interface by the difference μs(z↓0)−μs(z↑dN), see Eq. ( C1). Using that jz s(dN)=0, the spin accumulation μs(z↑dN) at the insulat- ing surface can be easily obtained from Eq. ( C3) and one ﬁnds that, up to a constant term that does not depend on the magne-tization orientation, μ s(z↓0)−μs(z↑dN)=(˜E/E)μs(z↓ 0). As a consequence, the resulting charge currents δ¯jx candδ¯jy care still given by Eqs. ( 23) and ( 24), respectively, but with the substitution ( C6)f o r ZN, the substitution E→˜E2/E forE, and with the corresponding substitutions Z/bardbl→˜Z/bardbland Z⊥→˜Z⊥for the sum impedances ˜Z/bardbland ˜Z⊥,s e eE q .( 21). In the limit dN/greatermuchλN, the results of Sec. IIIare recovered, because then ˜ZN→ZNand ˜E→E. The leading correction term is an overall factor (1 −4e−dN/λN) for the charge current δ¯jc, which comes from the substitution E→˜E2/E. In the op- posite limit dN/lessmuchλN, we note that the prefactor ˜ZN˜E2/dNE= (ZNE/dN) tanh2(dN/2λN) tanh( dN/λN)i nE q s .( 23) and ( 24) is proportional to ( dN/λN)2, so that the SMR is strongly sup- pressed. A further suppression of the SMR occurs once dN is so small that both sum impedances Z/bardblandZ⊥of Eq. ( 21) are dominated by ˜ZN. In this limit longitudinal and trans- verse contributions cancel and no observable SMR remains.This effect is not relevant for YIG \|Pt, since it would require an unrealistically small layer thickness d Nfor that material combination, but is relevant for Fe \|Au, where this asymptotic small- dNbehavior sets in for dN/lessorsimilar6×10−8m. 2. SMR for F \|N\|Ft r i l a y e r To describe the SMR in an F \|N\|F trilayer, we introduce two sets of spin impedances Ze(j) F/bardbl,Zm(j) F/bardbl,Z(j) F⊥,Ze(j) FN/bardbl,Zm(j) FN/bardbl, and Z(j) FN⊥with j=1,2, where the superscripts j=1 and j=2 refer to the ferromagnet and the F-N interface at z=0 and z=dN, respectively, and the corresponding sum impedances Z(j) /bardblandZ(j) ⊥are deﬁned as in Eq. ( 21). The longitudinal and transverse components of the spin impedances are deﬁnedwith respect to the unit vectors e (1) /bardblande(2) /bardblpointing along the magnetization direction in the two magnets. Restoring thevector notation for the spin current density j z s(z) and the spin accumulation μs(z), the boundary conditions ( 14) and ( 19)f o r spin currents and spin accumulations at the interfaces at x=0 andx=dNcan be summarized as μs(z↓0)=− (Z(1)−ZN)jz s(0), μs(z↑dN)=+ (Z(2)−ZN)jz s(dN), (C7) where, for j=1,2, Z(j)=Z(j) /bardble(j) /bardble(j)T /bardbl+Z(j) ⊥(ω)e(j) ⊥e(j)T∗ ⊥+Z(j)∗ ⊥(−ω)e(j)∗ ⊥e(j)T ⊥, (C8) with T denoting the transpose vector and ∗complex conju- gation. The system of equations is closed by the boundarycondition ( C3). Solving for the spin accumulations μ s(z↓0) andμs(z↑dN), one ﬁnds the charge current densities δ¯jx,y c from Eq. ( C1), δ¯jc(ω)=−θ2 SH¯hσ2 N 4e2dNE(ω)Z2 Ntanh2dN 2λN ×2/summationdisplay j=1[ez×Z(j)(ω)−1C(ω)−1ey]. (C9) Here ˆZ(j)=ZNcothdN 2λNI+Z(j),j=1,2, (C10) 024415-15REISS, KAMPFRATH, AND BROUWER PHYSICAL REVIEW B 104, 024415 (2021) with Ithe identity matrix, and C=1 2/summationdisplay jˇZ(j)ˆZ(j)−1, (C11) with ˇZ(j)=ZNtanhdN 2λNI+Z(j),j=1,2. (C12) In Eq. ( C9) we subtracted a constant term that does not depend on the magnetization direction. One veriﬁes that C(ω)→1i n the limit dN/greatermuchλN, so that the result for the charge current correction δ¯jcis the sum of contributions from the two in- terfaces separately. The case of an F \|N bilayer discussed in Sec. C1can be obtained from Eq. ( C9) upon taking the limit Z(2) FN→∞ . In the limit dN/greatermuchλN, one may approximate C−1≈I+2ZNe−dN/λN2/summationdisplay j=1ˆZ(j)−1. (C13) In this limit, we ﬁnd that Eq. ( C9)g i v e s δ¯jcas the sum of separate contributions from the two F layers in the form ofEqs. ( 23) and ( 24) and a weak interaction between the current corrections from the two interfaces, δ¯j c(ω)=/bracketleftbig δ¯j(1) c(ω)+δ¯j(2) c(ω)/bracketrightbig (1−4e−dN/λN) −θ2 SH¯hσ2 N 2e2dNE(ω)Z3 NedN/λNez ×[ˆZ(1)−1(ω)ˆZ(2)−1(ω)+ˆZ(2)−1(ω)ˆZ(1)−1(ω)]ey, (C14) where δ¯j(j) cis the correction to the charge current for a single F-N interface, j=1,2, and the factor (1 −4e−dN/λN)i st h e leading correction factor for a ﬁnite-width interface, see thediscussion below Eq. ( C6). In the opposite limit d N/lessmuchλN, one has C−1≈4λNZN dNZ−1 /Sigma1−2ZNZ−1 /Sigma1(2ZN−Z/prime /Sigma1)Z−1 /Sigma1, (C15) where we abbreviated Z/Sigma1=2/summationdisplay j=1/parenleftbig Z(j) FN+Z(j) F/parenrightbig ,Z/prime /Sigma1=1 ZN2/summationdisplay j=1/parenleftbig Z(j) FN+Z(j) F/parenrightbig2. (C16) In the same manner, the sum of inverses /summationdisplay jZ(j)−1≈dN λNZNI−d2 N 4λ2 NZ2 NZ/Sigma1. (C17) Combining these results, we ﬁnd that for small dN/λNone has δ¯jc(ω)=4e2θ2 SHdN ¯hE(ω)ez ×/bracketleftbigg Z−1 /Sigma1−dN 2λNZ−1 /Sigma1(2ZN−Z/prime /Sigma1)Z−1 /Sigma1/bracketrightbigg ey,(C18) where we omitted a constant term that does not depend on the magnetization direction. In the limit of a small layer width dN, the charge current correction δ¯jcof Eq. ( C18) for an F \|N\|F trilayer is a factor(dN/λN)2smaller than the SMR for two independent F-N interfaces. To understand why, note that for small dN/λNthe coupled equations ( C3) imply that both the spin currents jz s and the spin accumulations at the interfaces are proportional todN/λN. This reﬂects the fact that the interface spin accumu- lation generated by the spin-Hall effect is proportional to thelayer thickness d Nfor small dN. However, to this order in the layer thickness dN, the difference μs(z↓0)−μs(z↑dN)o f the spin accumulations, which is what determines the chargecurrent correction, see Eq. ( C1), depends on the driving ﬁeld Eand the spin-Hall angle θ SHbut not on the orientations of the magnetizations of the two F layers. This can be seen, e.g., bytaking the sum of the two equations in Eq. ( C3), which gives μ s(z↓0)−μs(z↑dN)=2edNθSHEeyto leading (ﬁrst) order indN. The dependence on the magnetization direction, which is what constitutes the SMR, occurs to subleading (second)order in d N/λN, which explains the smallness of the charge current by a factor ( dN/λN)2in the small- dNlimit. In spite of its smallness, the charge current correction δjc for an F \|N\|F trilayer is a factor ∼λN/dNlarger than the charge current correction for an F \|N bilayer, which was discussed in Sec. C1. The reason is that for the F \|N bilayer the spin current jz s(dN) is strictly zero, whereas for the F \|N\|F trilayer one ferro- magnet can serve as a “sink” for the spin current generated bythe other ferromagnet and vice versa, so that both spin currentsj z s(0) and jz s(dN) can be nonzero. This observation was made by Chen et al. for the DC limit [ 16]. Our calculation shows that the same mechanism also applies at ﬁnite driving frequency. Figure 9shows the characteristic ¯/Delta1SMR of the spin-Hall magnetoresistance in an Fe \|Au bilayer and an Fe \|Au\|Fe FIG. 9. Dimensionless characteristic ¯/Delta1SMR,s e eE q .( 69), of a Fe\|Au\|Fe trilayer (dashed blue line) and a Fe \|Au bilayer (solid red line) as function of the thickness of the Au layer. The Fe \|Au\|Fe trilayer has identical F layers with parallel magnetization directions and a thickness dN, while the thickness of the Au layer in the Fe \|Au bilayer is dN/2. (This ensures that the magnetic interface /volume ratio in both geometries is same.) Parameter values are taken from Tables IandII. The two curves are for ω/2π=0 Hz. For compari- son, d(0) N=6×10−8m is the thickness of the N layer of Ref. [ 69], which is also the reference thickness used for the numerical estimates of Sec. V. 024415-16THEORY OF SPIN-HALL MAGNETORESISTANCE IN THE … PHYSICAL REVIEW B 104, 024415 (2021) trilayer with Au layers of thickness dN/2 and dN, respectively, as a function of dNforω=0. The magnetization directions e(1) /bardblande(2) /bardblin the F \|N\|F trilayer are parallel. In the large- dN limit, ¯/Delta1SMR is the same for both conﬁgurations, because in this limit the two F layers of the F \|N\|F trilayer are inde- pendent and their identical contributions simply add up. Forlarge d N, the characteristic /Delta1∞ SMR is inversely proportional to dN, since the magnetization-direction dependent contribution to the conductivity exists on top of a large background cur-rent independent of the magnetization direction. For smalld N,¯/Delta1SMR differs for a bilayer and a trilayer because of the interplay between the two ferromagnetic layers; ¯/Delta1SMR∝d2 N for the F \|N bilayer, which is the asymptotic dependencediscussed in Sec. C1, whereas ¯/Delta1SMR∝dNfor the F \|N\|F trilayer. 3. F \|N\|F trilayer with collinear magnetization directions If the magnetization directions e(1) /bardblande(2) /bardblare collinear, i.e.,e(1) /bardbl=±e(2) /bardbl, there exists a basis in which the matrices Z(1)andZ(2)can be diagonalized simultaneously. In this limit, it is possible to directly express the charge current com- ponents δ¯jx candδ¯jy cin terms of the scalar impedances Z(j) N, Z(j) FN, and Z(j) F,j=1,2. For the case of parallel magnetization directions, e(1) /bardbl=e(2) /bardbl, one ﬁnds (up to constant terms that do not depend on the magnetization direction) δ¯jx c(ω)=−θ2 SH¯hσ2 N 2e2dNE(ω)Z2 Ntanh2dN 2λN/bracketleftbigg1 2/parenleftbig 1−m2 y/parenrightbig ˆZ(1)∗ ⊥(−ω)+ˆZ(2)∗ ⊥(−ω) ˇZ(1)∗ ⊥(−ω)ˆZ(2)∗ ⊥(−ω)+ˇZ(2)∗ ⊥(−ω)ˆZ(1)∗ ⊥(−ω) +1 2/parenleftbig 1−m2 y/parenrightbig ˆZ(1) ⊥(ω)+ˆZ(2) ⊥(ω) ˇZ(1) ⊥(ω)ˆZ(2) ⊥(ω)+ˇZ(2) ⊥(ω)ˆZ(1) ⊥(ω)+m2 yˆZ(1) /bardbl+ˆZ(2) /bardbl ˇZ(1) /bardblˆZ(2) /bardbl+ˇZ(2) /bardblˆZ(1) /bardbl/bracketrightBigg , (C19) δ¯jy c(ω)=−θ2 SH¯hσ2 N 2e2dNE(ω)Z2 Ntanh2dN 2λN/bracketleftbigg1 2(mxmy−imz)ˆZ(1)∗ ⊥(−ω)+ˆZ(2)∗ ⊥(−ω) ˇZ(1)∗ ⊥(−ω)ˆZ(2)∗ ⊥(−ω)+ˇZ(2)∗ ⊥(−ω)ˆZ(1)∗ ⊥(−ω) +1 2(mxmy+imz)ˆZ(1) ⊥(ω)+ˆZ(2) ⊥(ω) ˇZ(1) ⊥(ω)ˆZ(2) ⊥(ω)+ˇZ(2) ⊥(ω)ˆZ(1) ⊥(ω)−mxmyˆZ(1) /bardbl+ˆZ(2) /bardbl ˇZ(1) /bardblˆZ(2) /bardbl+ˇZ(2) /bardblˆZ(1) /bardbl/bracketrightBigg . (C20) Here ˆZ(j) ⊥,/bardbl(ω) and ˇZ(j) ⊥,/bardbl(ω) are the sum impedances of layer j,j=1,2, see Eq. ( 21), with ZNreplaced by ZNcoth( dN/2λN) andZNtanh( dN/2λN), respectively [compare with Eqs. ( C10) and ( C12)]. For the antiparallel conﬁguration e(1) /bardbl=−e(2) /bardblthe charge current correction δ¯jx,y ccan be obtained from Eqs. ( C19) and ( C20) by exchanging the impedances ˇZ(2) ⊥(ω) and ˆZ(2) ⊥(ω) byˇZ(2)∗ ⊥(−ω) and ˆZ(2)∗ ⊥(−ω), respectively, and vice versa. This substitution rule follows from the observation that the roles of the vectors e(2) ⊥ande(2)∗ ⊥are interchanged upon inverting the magnetization direction e(2) /bardbl. In the DC limit and neglecting the imaginary part of the spin-mixing conductances at the F-N interfaces, the current correction δ¯jcin an F \|N\|F trilayer is the same for the parallel and antiparallel magnetization conﬁguration [ 16]. At ﬁnite frequencies, the resonant features of δ¯jcat the magnon frequencies are different for the parallel and antiparallel magnetization conﬁguration: Upon inverting the magnetization direction of one of the magnets, the polarization vector of a propagating magnon mode in thatmagnet changes from e ⊥toe∗ ⊥or vice versa, which affects the interaction term involving both magnets. (Mathematically, the difference between parallel and antiparallel magnetization conﬁgurations at large frequencies follows, because ZF⊥(ω)s h o w s resonant features for positive frequencies but not for negative frequencies.) 4. SMR in the perpendicular F \|N\|F conﬁguration When the spin valve is in the perpendicular conﬁguration, e(1) /bardbl·e(2) /bardbl=0, a simultaneous diagonalization of the matrices Z(1) andZ(2)is possible only in the DC limit ω→0 and neglecting the imaginary part of the spin-mixing conductance at the F-N interfaces. In this limit, all impedances are real and one ﬁnds δ¯jx c(0)=−θ2 SH¯hσ2 N 2e2dNE(ω)Z2 Ntanh2dN 2λN/bracketleftBigg m(1)2 yˆZ(1) /bardbl+ˆZ(2) ⊥ ˇZ(1) /bardblˆZ(2) ⊥+ˇZ(2) ⊥ˆZ(1) /bardbl+m(2)2 yˆZ(1) ⊥+ˆZ(2) /bardbl ˇZ(1) ⊥ˆZ(2) /bardbl+ˇZ(2) /bardblˆZ(1) ⊥ +/parenleftbig 1−m(1)2 y−m(2)2 y/parenrightbig ˆZ(1) ⊥+ˆZ(2) ⊥ ˇZ(1) ⊥ˆZ(2) ⊥+ˇZ(2) ⊥ˆZ(1) ⊥/bracketrightbigg , (C21) δ¯jy c(0)=θ2 SH¯hσ2 N 2e2dNE(ω)Z2 Ntanh2dN 2λN/bracketleftBigg m(1) xm(1) yˆZ(1) /bardbl+ˆZ(2) ⊥ ˇZ(1) /bardblˆZ(2) ⊥+ˇZ(2) ⊥ˆZ(1) /bardbl+m(2) xm(2) yˆZ(1) ⊥+ˆZ(2) /bardbl ˇZ(1) ⊥ˆZ(2) /bardbl+ˇZ(2) /bardblˆZ(1) ⊥ −/parenleftbig m(1) xm(1) y+m(2) xm(2) y/parenrightbig ˆZ(1) ⊥+ˆZ(2) ⊥ ˇZ(1) ⊥ˆZ(2) ⊥+ˇZ(2) ⊥ˆZ(1) ⊥/bracketrightbigg . (C22) 024415-17REISS, KAMPFRATH, AND BROUWER PHYSICAL REVIEW B 104, 024415 (2021) This result generalizes an expression obtained in Ref. [ 16], which considers the case of two insulating ferromagnets and aperpendicular magnetization conﬁguration with both magne-tizations in the xyplane. In this case the third terms between the square brackets in Eqs. ( C21) and ( C22) are independent of the magnetization direction. Moreover, since Z (j) /bardbl(ω)/greatermuch Z(j) ⊥(ω) for insulating ferromagnets at frequency ω→0, each of the two remaining terms in the expressions for δ¯jx candδ¯jy c depends on properties of a single magnet only. Indeed, if the magnetization directions e(1) /bardblande(2) /bardblare both in the xyplane and if the spin-mixing conductance of the F-N interface is real,there is nothing in the system that rotates spin currents j z sout of the xyplane in the DC limit ω=0. Since each ferromagnet drives only spin currents that are transverse to its magnetiza-tion direction, spin currents driven by one magnet are longi-tudinal for the other magnet and vice versa. Longitudinal spin currents are fully reﬂected by insulating ferromagnets in theDC limit, which explains why there is no “interaction” con-tribution to δj cin this case. Our full result of Eqs. 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In contrast, for the direct magnetic ﬁeld of the driving radiation one has By/E=1/c≈3×10−9sm−1.I n - clusion of the direct magnetic ﬁeld into the considerations ofAppendix Bis straightforward, if desired. 024415-20
PhysRevB.85.024416.pdf	PHYSICAL REVIEW B 85, 024416 (2012) Spin-transfer torque in a thick N ´eel domain wall P. Bal ´aˇz,1V . K. Dugaev,2and J. Barna ´s1,3 1Department of Physics, Adam Mickiewicz University, ulica Umultowska 85, PL-61-614 Pozna ´n, Poland 2Department of Physics, Rzesz ´ow University of Technology, Aleja Powsta ´nc´ow Warszawy 6, PL-35-959 Rzesz ´ow, Poland, and Department of Physics and CFIF , Instituto Superior T ´ecnico, TU Lisbon, avenida Rovisco Pais, PT-1049-001 Lisbon, Portugal 3Institute of Molecular Physics, Polish Academy of Sciences, ulica M. Smoluchowskiego 17, PL-60-179 Pozna ´n, Poland (Received 24 August 2011; published 23 January 2012) Current-induced spin transfer torque in relatively thick N ´eel domain walls in ferromagnetic metals /semiconductors is considered theoretically in the linear response regime. Using quasiclassical approxi- mation and unitary transformation we calculate the nonequilibrium (current-induced) spin accumulation in thedomain wall. The current-induced spin density is then used to determine adiabatic and nonadiabatic componentsof the spin transfer torque. The results show that the spin torque as well as its nonadiabaticity can be enhancedby spin-conserving momentum scattering processes inside the wall. DOI: 10.1103/PhysRevB.85.024416 PACS number(s): 75 .60.Ch, 75 .70.Cn I. INTRODUCTION Many recent experiments have demonstrated that the domain wall motion can be effectively controlled not onlyby an external magnetic ﬁeld but also by an electric current. 1–5 This possibility appears owing to exchange coupling between conduction electrons and local magnetic moments, which leadsto transfer of spin angular momentum and gives rise to a torqueacting on the localized magnetic moments. The broad interestin the current-induced dynamics of domain walls in magneticnanostructures is stimulated by perspective applications innovel spintronic devices as well as in modern magneticmemory elements. 6The existence of spin transfer torque was theoretically predicted by Slonczewski7and Berger,8 and later observed experimentally9,10in multilayered metallic structures.11–13However, the spin transfer occurs in any system with nonuniform magnetic texture, including alsodomain walls. 14,15In the latter case the spin torque can lead effectively to a domain wall displacement when the currentdensity exceeds a certain critical value. Experiments as wellas numerical simulations have shown that the spin torque notonly moves the domain wall but also modiﬁes its shape. Thelatter effect generally reduces the spin torque, and the wallusually stops after passing a certain distance. The domain-wall motion can be described by a classical Landau-Lifshitz-Gilbert equation, which additionally includesthe current-induced spin torque. Solution to this equationrelies usually on micromagnetic numerical simulations. Animportant issue is the problem of calculating the appropriatespin transfer torque. Generally, the current-induced torquehas two components—one in the plane in which the domainmagnetization changes (usually called adiabatic component),and the second one (nonadiabatic) normal to this plane. Bothcomponents have been calculated within various models ofdomain walls and also for various physical situations. The adiabatic component of the spin torque has been studied theoretically in many papers, and some consensus has beenreached on the theoretical side. 16–19However, the experiments with current-induced domain wall motion cannot be explainedby the adiabatic torque only. More speciﬁcally, it has beenshown that the adiabatic torque, in the absence of externalmagnetic ﬁeld, is unable to maintain a constant domain wallmotion observed in some experiments. 2Therefore, it has been proposed to include a nonadiabatic term20–22attributed to spin relaxation processes due to mistracking of the conductionelectrons’ spins and localized moments. The main problem isthat the calculated nonadiabaticity of the torque, described bythe parameter β, has been found rather small. 23Then, the role of spin-conserving relaxation process has been raised,20,24–26 which can contribute to the constant βand also can be signiﬁcant even in the case of a domain wall with a smoothmagnetization proﬁle. More recently it has been shown 27,28 that in the case of relatively thick (in comparison to the Larmor precession length) domain walls the parameter βmay be increased up to the order of β∼10−1or even 1. In addition, it has been shown29–31that the nonadiabaticity parameter βmay oscillate with the domain wall width. However, the oscillationsare damped with increasing thickness of the wall and β becomes constant for large domain wall widths. Generally,the problem of nonadiabatic torque and especially of themagnitude of constant βis still a matter of discussions. 15,26,32 In this paper we calculate the spin transfer torque in a relatively thick N ´eel domain wall and show that not only spin-ﬂip scattering processes but also spin-conserving momentum relaxation ones contribute to the spin torque, in agreement with Ref. 26. The presented approach is based on a unitary transformation method.24,33,34Even though we assume that variation of the magnetization is smooth, we ﬁnd it more convenient to perform calculations without expandingin small gauge potential. Using the linear response theory and Green function formalism, we calculate the current- induced spin density and spin torque exerted on the domainwall. We identify both in-plane and out-of-plane spin torque components. Moreover, as the out-of-plane component is fully nonadiabatic, the in-plane one includes also a nonadiabaticterm in addition to the usual adiabatic one. From numerical calculations for realistic material parameters we have found that the nonadiabatic component is generally smaller than theadiabatic one, but in a certain range of exchange coupling it can be remarkable. In Sec. IIwe brieﬂy describe the model and theoreti- cal method. Nonequilibrium spin accumulation (induced bycurrent) is calculated in Sec. III. The current-induced spin 024416-1 1098-0121/2012/85(2)/024416(7) ©2012 American Physical SocietyP. BAL ´AˇZ, V . K. DUGAEV , AND J. BARNA ´S PHYSICAL REVIEW B 85, 024416 (2012) accumulation is then used to calculate spin torque components, as described and discussed in Sec. IV, where we also present some numerical results. Summary and ﬁnal conclusions are inSec. V. II. MODEL We consider a ferromagnetic metal with nonuniform mag- netization corresponding to a single domain wall described bythe magnetization proﬁle M(r). Assuming that \|M(r)\|≡M= const, one can write M(r)=Mn(r), where n(r) is a unit vector ﬁeld corresponding to M(r). The single-particle Hamiltonian describing conduction electrons which are locally exchange coupled to the magneti-zation M(r) can be written in the form H 0=−¯h2 2mψ† α∂2 ∂r2ψα−Jψ† ασακ·n(r)ψκ, (1) where Jis the exchange parameter, ψαandψ† αare the spinor ﬁeld operators of electrons, while σ=(σx,σy,σz) represents the Pauli matrices. Our considerations are restricted to a speciﬁc domain wall shape. First, we assume the wall is translationally invariant in thex-yplane, so n(r)→n(z). Second, we consider a N ´eel domain wall, where M(z) rotates always staying within the x-zplane. Thus, one can parametrize the vector n(z)a s n(z)=[sinϕ(z),0,cosϕ(z)], (2) where the phase ϕ(z) describes shape of the domain wall. For instance, if we assume the domain wall in the form of a kink,then ϕ(z)=−π 2tanh (z/L). (3) where Lis the domain wall width. The schematic magne- tization proﬁle associated with the studied domain wall ispresented in Fig. 1. The ﬁrst step of our analysis is to perform a unitary trans- formation to a local frame, ψ→T(r)ψwithT †(r)T(r)=1, which removes the inhomogeneity of n(r).33In other words, T(r) transforms the second term in Eq. ( 1)a s T†(r)σ·n(r)T(r)=σz. (4) The explicit form of the transformation corresponding to the wall described by Eq. ( 2)i s T(z)=1√ 2/parenleftbigg/radicalbig 1+cosϕ(z)−iσysinϕ(z)√1+cosϕ(z)/parenrightbigg .(5) FIG. 1. (Color online) Schematic picture of the magnetization proﬁle in a N ´eel domain wall.The transformed Hamiltonian can be then presented in the following form: H0=−¯h2 2m∂2 ∂r2−Jσz +¯h2/bracketleftbiggmκ2(z) 2+iσyκ/prime(z) 2+iσyκ(z)∂ ∂z/bracketrightbigg ,(6) where κ(z)=ϕ/prime(z) 2m, (7) andκ/prime(z)≡∂κ(z)/∂z, while ϕ/prime(z)≡∂ϕ(z)/∂z.F o ras l o w l y varying smooth function ϕ(z) (thick domain wall centered at z=0), the perturbation due to the domain wall is weak and localized close to the center of the wall,35\|z\|/lessmuchL. When the domain wall is in the form of a kink [see Eq. ( 3)] the parameter κ(z) takes the form κ(z)=−π 4mLcosh2(z/L). (8) The term proportional to κ/primein Hamiltonian (6) is generally smaller than the others and may be ignored in furthercalculations. Then, in the adiabatic approximation, which isvalid for thick domain walls, L/greatermuchλ(where λis the electron wavelength), one can write the Hamiltonian (6) in the basis ofsemiclassical functions as H 0=¯h2k2 2m−Jσz+¯h2/bracketleftbiggmκ2(z) 2+iσyκ/prime(z) 2−σyκ(z)kz/bracketrightbigg . (9) The domain wall gives rise to excess spin density within the wall. The equilibrium (in the absence of external electricﬁeld) spin density of conduction electrons in the local framecan be calculated as S=−iTr/integraldisplaydε 2πd3k (2π)3σGε(k), (10) where Gε(k) is the Green function corresponding to the Hamiltonian (9). The equilibrium spin density within thedomain wall was calculated and thoroughly analyzed inRef. 34. In this paper, however, we are interested in the current-induced part of the spin density, which is responsiblefor the current-induced spin transfer torque exerted on thedomain wall. III. SPIN ACCUMULATION IN THE LINEAR RESPONSE REGIME We assume now that the system is in an exter- nal ﬁeld described by the time-dependent vector po-tential A(t)=A ωexp(−iωt). The electric ﬁeld is then E(t)=−(1/c)[∂A(t)/∂t]=(iω/c )Aωexp(−iωt), or equiva- lently Eω=(iω/c )Aω. Hamiltonian of the system in the vector ﬁeldA(t) can be obtained by replacing −i¯h(∂/∂r)i nE q .( 6) by−i¯h(∂/∂r)−(e/c)A(t). The total Hamiltonian takes then the form H=H0+HA, (11) 024416-2SPIN-TRANSFER TORQUE IN A THICK N ´EEL DOMAIN WALL PHYSICAL REVIEW B 85, 024416 (2012) where HAis the perturbation due to the external ﬁeld. The linear in A(t) term contributing to Eq. ( 11) takes then the following form: HA(k,ω)=−¯hek·Aω mc+¯heAz ω cκσy. (12) Now we calculate the electron spin density in the local reference frame, induced by the external ﬁeld A(t) in the linear approximation with respect to A(t). Furthermore, since our considerations are based on the adiabatic approximation [validfor small values of κ(z)], we will consider only the terms linear in the parameter κ(z), too. The ﬁeld-induced spin density S can be calculated from the formula S=−ReiTr/integraldisplaydε 2πd3k (2π)3σGAε(k), (13) where the Green’s function GAε(k) should be taken in the linear approximation with respect to the perturbation given byEq. ( 12), G Aε(k)→Gε+¯hω(k)HA(k,ω)Gε(k). (14) Since we are interested in the result linear in κ(z), we may approximate the Green function Gε(k)a s Gε(k)=ε−εk−Jσz−¯h2kzκσy+μ (ε−εk↑+μ+iδ↑sgnε)(ε−εk↓+μ+iδ↓sgnε), (15) where εk=¯h2k2/2m,εk↑(↓)=εk∓J,μdenotes the chemical potential, and δ↑(↓)=¯h/2τ↑(↓), with τ↑andτ↓denoting the relaxation times for the majority and minority conductionelectrons, respectively. Since there are two different terms in H A[see Eq. ( 12)] we will calculate separately the corresponding contributionsto the induced spin density. The contribution of the ﬁrst termin Eq. ( 12)t oScan be then written as S (1)=¯heEω mωRe Tr/integraldisplaydε 2πd3k (2π)3kzσGε+¯hω(k)Gε(k).(16) Taking into account Eqs. ( 15) and ( 16), and calculating the trace one ﬁnds the two components of the ﬁeld-induced spindensity in the form S (1) x=0, (17a) S(1) y=−2e¯h3Eωκ mωRe/integraldisplaydε 2πd3k (2π)3k2 z2(ε−εk+μ)+¯hω F(ε)F(ε+¯hω), (17b) where F(ε) is deﬁned as F(ε)=(ε−εk↑+μ+iδ↑sgnε)(ε−εk↓+μ+iδ↓sgnε). (18) Calculating the integrals in Eq. ( 17b) one obtains S(1) y=e¯hEωκ 6π2J/parenleftbig τ↑k3 F↑−τ↓k3 F↓/parenrightbig +e¯h2Eωκ 24J2π2ω/parenleftbigg1 τ↑−1 τ↓/parenrightbigg/parenleftbig k3 F↑−k3 F↓/parenrightbig ,(19) where kF↑andkF↓are the Fermi wave vectors corresponding to the spin majority and spin minority electron subbands,respectively. We note that kF↑andkF↓are related via the formula 2 J=(¯h2/2m)(k2 F↑−k2 F↓). Now we calculate the contribution due to the second term in Eq. ( 12). This contribution can be written as S(2)=−¯heEωκ ωRe Tr/integraldisplaydε 2πd3k (2π)3σGε+¯hω(k)σyGε(k). (20) Since we intend to ﬁnd this contribution in the linear approx- imation with respect to κ, and the above formula includes already a prefactor κ, we can take the Green functions in the limit of κ=0. As a result one obtains S(2) x=−e¯hEωκ ωRe/integraldisplaydε 2πd3k (2π)3 ×i/bracketleftbig G0↓ ε+¯hω(k)G0↑ ε(k)−G0↑ ε+¯hω(k)G0↓ ε(k)/bracketrightbig ,(21a) S(2) y=−e¯hEωκ ωRe/integraldisplaydε 2πd3k (2π)3 ×/bracketleftbig G0↓ ε+¯hω(k)G0↑ ε(k)+G0↑ ε+¯hω(k)G0↓ ε(k)/bracketrightbig ,(21b) where G0↑(↓) ε=1/[ε−εk↑+μ+iδ↑(↓)sgnε] are the ele- ments of the Green function in the zeroth order with respect toκ(Green functions for spin-up and spin-down electrons in the absence of the wall). Calculating the integrals one ﬁnds S (2) x=−e¯h2Eωκ 12J2π2/parenleftbig k3 F↑−k3 F↓/parenrightbig , (22a) S(2) y=−e¯hEωmκ 8π2J2τ↑kF↑+τ↓kF↓ τ↑τ↓ −e¯h2Eωκ 24π2J2ω/parenleftbigg1 τ↑−1 τ↓/parenrightbigg/parenleftbig k3 F↑−k3 F↓/parenrightbig .(22b) The total spin density components are Sx=S(1) x+S(2) xand Sy=S(1) y+S(2) y. Note that the singular terms in S(1) yandS(2) y, proportional to ω−1, cancel each other, so we can take the static limit ω→0. Thus the ﬁeld-induced spin density is given ﬁnally by the formulas Sx=−e¯h2Eκ 12J2π2/parenleftbig k3 F↑−k3 F↓/parenrightbig , (23a) Sy=e¯hEκ 6π2J/parenleftbig τ↑k3 F↑−τ↓k3 F↓/parenrightbig −e¯hEmκ 8π2J2τ↑kF↑+τ↓kF↓ τ↑τ↓, (23b) where Eis the static electric ﬁeld that drives the current. The above expressions for spin density will be now used tocalculate spin torque exerted on the domain wall. IV . SPIN TORQUE According to Eq. ( 1), exchange interaction between the local magnetic moments and the current-induced spin densitycan be written as H int=−JS·n(r), (24) 024416-3P. BAL ´AˇZ, V . K. DUGAEV , AND J. BARNA ´S PHYSICAL REVIEW B 85, 024416 (2012) where Sis transformed back to the original reference frame. Equation of motion for the localized magnetization takes thenthe form dn dt=−\|γ\|J M0n×S, (25) so the torque exerted on the localized magnetization is T=J M0M×S. (26) Thus, the in-plane and out-of-plane components of the spin torque in the cylindrical coordinates take the form T/bardbl=−JSy, (27a) T⊥=JSx, (27b) where SxandSyare given by Eqs. ( 23a) and ( 23b), respectively. It is convenient to express both the in-plane and out-of-plane components of the spin torque in the following form: T/bardbl=˜an×/parenleftbigg n×∂n ∂z/parenrightbigg , (28a) T⊥=˜bn×∂n ∂z, (28b) where ˜aand˜bare the respective coefﬁcients which are linear inEand depend on the material parameters as ˜a=e¯hE/bracketleftBigg τ↑kF↑+τ↓kF↓ 16π2Jτ↑τ↓−τ↑k3 F↑−τ↓k3 F↓ 12π2m/bracketrightBigg ,(29a) ˜b=−e¯h2 24π2JmE/parenleftbig k3 F↑−k3 F↓/parenrightbig . (29b) The spin torque components are usually related to current density and not to electric ﬁeld. Using the Drude formula forelectric conductivity of two independent spin channels oneﬁnds the relation E=6π 2m e21 τ↑k3 F↑+τ↓k3 F↓I, (30) where Istands for current density. From this ˜aand ˜bcan be rewritten in the form ˜a=¯h 2eI/bracketleftBigg 3m 4Jτ↑τ↓τ↑kF↑+τ↓kF↓ τ↑k3 F↑+τ↓k3 F↓−τ↑k3 F↑−τ↓k3 F↓ τ↑k3 F↑+τ↓k3 F↓/bracketrightBigg , (31a) ˜b=−¯h 2eI/bracketleftBigg ¯h 2Jk3 F↑−k3 F↓ τ↑k3 F↑+τ↓k3 F↓/bracketrightBigg . (31b) The spin torque components, Eqs. ( 28a) and ( 28b), can be thus rewritten in the form T/bardbl=aIn×/parenleftbigg n×∂n ∂z/parenrightbigg , (32a) T⊥=bIn×∂n ∂z, (32b)where aandbare given as a=−¯h 2e/bracketleftbigg P−1 η/parenleftbiggτex τ↑/parenrightbigg23 43π2(1−x2)(1+ηx) 1+ηx3/bracketrightbigg ,(33a) b=−¯h 2e/bracketleftbigg1 4π/parenleftbiggτex τ↑/parenrightbigg1−x3 1+ηx3/bracketrightbigg , (33b) withτex=2π¯h/J andη=τ↓/τ↑. The parameter xis deﬁned as x=kF↓ kF↑=/parenleftbigg1−J/μ 1+J/μ/parenrightbigg1/2 . (34) When deriving the above equations we have also taken into account the fact that 2 J=(¯h2/2m)(kF↑2−kF↓2)( s e ea l s ot h e previous section). In the expression (33a) Pis the current polarization, deﬁned as P=σ↑−σ↓ σ↑+σ↓=τ↑k3 F↑−τ↓k3 F↓ τ↑k3 F↑+τ↓k3 F↓, (35) where σ↑(↓)is the conductivity of spin- ↑(spin-↓) channel. The current polarization might be rewritten using dimensionlessparameters xandηas P=1−ηx 3 1+ηx3, (36) or alternatively using Jandμas P=(1+J/μ)3/2−η(1−J/μ)3/2 (1+J/μ)3/2+η(1−J/μ)3/2. (37) The relaxation times τ↑(↓)include spin-conserving as well as spin-ﬂip contributions,25 1 τ↑(↓)=1 τ0 ↑(↓)+1 τsf ↑(↓), (38) where τ0 ↑(↓)andτsf ↑(↓)are the relaxation times due to spin- conserving and spin-ﬂip scattering processes for the spinmajority (minority) channel, respectively. Deﬁning τ sfandτ∗ as 1 τsf=1 2/parenleftBigg 1 τsf ↑+1 τsf ↓/parenrightBigg , (39) 1 τ∗=1 2/parenleftBigg 1 τ0 ↑+1 τ0 ↓/parenrightBigg , (40) and assuming for simplicity the same spin asymmetry of the spin-conserving and spin-ﬂip relaxation times, τ0 ↓/τ0 ↑= τsf ↓/τsf ↑=η=τ↓/τ↑, one may write τex τ↑=2η 1+η/parenleftbiggτex τsf+τex τ∗/parenrightbigg . (41) Thus, Eqs. ( 33) can be rewritten as a=−¯h 2e/bracketleftbigg P−/parenleftbiggτex τsf+τex τ∗/parenrightbigg2η (1+η)2 ×3 42π2(1−x2)(1+ηx) 1+ηx3/bracketrightbigg , (42a) b=−¯h 2e/bracketleftbigg1 2π/parenleftbiggτex τsf+τex τ∗/parenrightbiggη 1+η1−x3 1+ηx3/bracketrightbigg .(42b) 024416-4SPIN-TRANSFER TORQUE IN A THICK N ´EEL DOMAIN WALL PHYSICAL REVIEW B 85, 024416 (2012) The above expressions for the current-induced spin torque components constitute our ﬁnal results. These formula sim-plify in some limiting cases. For instance, when η=1 one ﬁnds a=−¯h 2e/bracketleftBigg P−3 43π2/parenleftbiggτex τsf+τex τ∗/parenrightbigg2(x−1)(x+1)2 1+x3/bracketrightBigg , (43a) b=−¯h 2e/bracketleftbigg1 4π/parenleftbiggτex τsf+τex τ∗/parenrightbigg1−x3 1+x3/bracketrightbigg , (43b) where the polarization reduces now to P=k3 F↑−k3 F↓ k3 F↑+k3 F↓=1−x3 1+x3. (44) V . NUMERICAL RESULTS AND DISCUSSION Let us analyze now the derived results. The ﬁrst term of Eq. ( 42a) is proportional to the current polarization Pand is analogical to the adiabatic spin torque term proposed by Li andZhang. 19Following them we shall call this term as adiabatic . In turn, all the other spin torque components which differ fromthis term and depend on the relaxation processes shall be calledasnonadiabatic torque. Thus, the nonadiabatic part of the spin torque includes the second term of Eq. ( 42a) and also the whole out-of-plane torque corresponding to Eq. ( 42b). Hence, one can write a=a 0+asct, where a0∝Pis the amplitude of the adiabatic spin torque, while asctis the nonadiabatic contribu- tion due to electron scattering. To express the nonadiabaticityof the in-plane torque we deﬁne the in-plane nonadiabaticity γasγ=a sct/a0. Similarly, we can deﬁne the out-of-plane nonadiabaticity asβ=b/a 0, which is the standard deﬁnition of nonadiabaticity as introduced by Zhang and Li.21,22Follow- ing this, one can write the spin torque as a sum of the adia-batic T adand nonadiabatic Tnacontributions; T=Tad+Tna, where Tad=a0In×/parenleftbigg n×∂n ∂z/parenrightbigg , (45a) Tna=a0γIn×/parenleftbigg n×∂n ∂z/parenrightbigg +a0βIn×∂n ∂z.(45b) Since the properties of the adiabatic torque are well known,18,19,21,22,29,36let us now focus on the nonadiabatic terms. In the theory of Zhang and Li21the out-of-plane nonadiabaticity is proportional to the ratio τex/τsf. Equation (42a) shows that the nonadiabaticity can be enhanced by the second term ( τex/τ∗) in the bracket, which arises from the momentum relaxation processes. When τsf/greatermuchτ∗, this term may increase the out-of-plane torque component by one ortwo orders of magnitude. This clearly shows that momentumscattering can enhance the spin torque, in agreement withRef. 26. This is due to the fact that when electron momentum scattering is strong, electron spends more time within thedomain wall and therefore its coupling to the wall is effectivelyenhanced and also its chance to suffer a spin-ﬂip scatteringincreases as well.Similarly, one can split b[Eq. ( 42b)] into two parts, b= b sf+bsc. The ﬁrst term, bsf∝τex/τsf, is related to the spin-ﬂip processes and is analogical to the nonadiabatic spin torque termobtained by Zhang and Li. 21The second one, bsc∝τex/τ∗, is related to the spin-conserving scattering. The out-of-planenonadiabaticity can be then written as β=β sf+βsc, where βsf=bsf/a0is the spin-ﬂip contribution, and βsc=bsc/a0 is the spin-conserving contribution to the nonadiabaticity. Similar two contributions to the out-of-plane nonadiabaticitywere originally proposed by Tatara et al. 20,24As the ﬁst term βsf has the same origin, the second one βscis different. In the works of Tatara et al.20,24βscarises from the momentum transfer when electron is reﬂected from a thin domain wall and it is claimedto be negligibly small in the case of adiabatic (thick) domainwall. However, in the present model, β scdescribes the effect of momentum scattering inside the domain wall, which enhancesthe spin transfer between the localized magnetic moments andconduction electrons. Let us now present some numerical results illustrating physical consequences of the equations derived above. In thecorresponding numerical calculations we assumed μ=10 eV. Typically, the momentum relaxation time is smaller than spin-ﬂip one. Here we assumed τ ∗=10−14s andτsf=10−12s.21 To specify the applicability range of the description, we have to take into account the fact that the adiabatic approximationis limited to thick domain walls. Following Xiao et al. , 18 one can deﬁne a characteristic length of the material, which in our notation reads /lscript=(¯h2kF)/(2mJ). Then the adiabatic approximation is valid if L/greatermuch/lscript, which leads to the following condition for the exchange coupling parameter J: J/greatermuch/radicalbiggμ 2m¯h L. (46) Thus, for an adiabatic domain wall of width L=100 nm one obtains J/greatermuch10−3μ, that is, appropriate values for our calculations are J/μ> 10−2. First, let us analyze variation of the spin torque amplitudes with the basic parameters of themodel. Figure 2shows the dependence of the coefﬁcients a andbon the exchange parameter J. As one could expect, ais linear in Jin the relevant range of the parameter J, where the approach based on the quasiclassical approximation is valid.The parameter bdepends on Jalso almost linearly, although now deviations from the linear dependence are stronger than inthe case of a. Apart from this, its magnitude is smaller than that ofa. Moreover, while aincreases with decreasing ηand also with increasing J, the parameter bdecreases with decreasing ηand also with increasing J. In other words, when aincreases, the corresponding bdecreases. The reason of such behavior follows from the fact that the stronger is the adiabatic spintorque, the smaller is the mistracking of conduction electrons’spins and localized moments. As a consequence, the parameterbdecreases with increasing J. In turn, the increase of awith ηdeviating from 1 is a consequence of increasing current spin polarization [see Eq. ( 37)]. Variation of the in-plane and out-of-plane spin torque components along the zaxis is shown in Fig. 3. Both torque components reach the corresponding maxima in the center ofthe wall and decay toward the wall boundaries. Similarly tothe coefﬁcients aandb,T /bardblincreases while T⊥decreases with decreasing η. In addition, both spin torque components scale 024416-5P. BAL ´AˇZ, V . K. DUGAEV , AND J. BARNA ´S PHYSICAL REVIEW B 85, 024416 (2012) (a) (b) FIG. 2. (Color online) Dependence of the spin torque coefﬁcients (a)aand (b) bon the exchange parameter Jfor indicated values of the parameter ηdeﬁned in the main text. with the domain wall width Lwhich stems from the relation κ∝1/L. In Fig. 3we show the nonadiabaticity parameters βand γ. Experimentally, the parameter βappears to be small (a) (b) FIG. 3. (Color online) Variation of the amplitudes of the (a) in- plane and (b) out-of-plane spin torque components along the zaxis calculated for J/μ=0.1 and indicated values of η. The center of the domain wall is located at z=0.(β∼10−2),21which corresponds to our results when ηis markedly smaller than 1 and/or the exchange coupling is strongenough. Oppositely, for η∼1 and small J, one observes an increase of β. This enhancement of βis connected with a weak coupling of conduction electrons to the localized magneticmoments and/or small current polarization, which leads tomisalignment of the conduction and localized spins and givesrise to the nonadiabatic spin torque component T ⊥. Following Zhang and Li,19,21the initial domain wall velocity (at t=0) in the absence of external magnetic ﬁeld is mainly controlled by the in-plane torque component, thatis, the initial velocity is proportional to a.I nR e f . 21the in-plane torque component was entirely adiabatic. In our casehowever, the in-plane torque includes a small nonadiabaticcontribution proportional to γ. This contribution however, is rather small [see Fig. 4(b)] in comparison to the adiabatic term, and therefore it does not have any important effect on the initialdomain wall velocity. On the other hand, the terminal wallvelocity (for t→∞ ) is determined by the ratio of b/α, where αis the Gilbert damping parameter. In the model of Zhang and Li, 21the out-of-plane torque appeared due to the spin-ﬂip process. Here we have shown that this velocity depends alsoon the spin-conserving momentum relaxation processes. Our description shows, in agreement with other models, 20,24 that the overall value of βmight be larger than that predicted by Li and Zhang. This value might be especially enhanced inmagnetic materials with small values of Jand/or small current polarization. The high values of βhave been also observed experimentally 27,28in magnetic multilayered wires with do- main walls obeying the assumption of adiabatic spin transport.These experiments have shown that the nonadiabaticity might (a) (b) FIG. 4. (Color online) (a) The nonadiabaticity parameter βand (b) the nonadiabaticity parameter γas a function of the exchange parameter J, calculated for indicated values of η. 024416-6SPIN-TRANSFER TORQUE IN A THICK N ´EEL DOMAIN WALL PHYSICAL REVIEW B 85, 024416 (2012) reach values of the order of β∼10−1–100. By a thorough analysis of different contributions to the domain wall motion,the authors arrived at the conclusion that the spin-conservingscattering of the linear momentum might be responsible for thehigh nonadiabaticity of the domain wall, which is in agreementwith our description. VI. CONCLUSIONS Using the Green function formalism and linear response theory we have calculated the current-induced spin torqueexerted on a N ´eel domain wall. The considerations are applicable to relatively thick domain walls, when the adiabaticapproximation is justiﬁed. Such a situation takes place, forexample, in metallic systems, especially in the bulk limit.Although the unitary transformation [Eq. ( 5)] is applicable to N´eel walls, the method can be used for other nonuniformly magnetized systems including also Bloch domain walls. In thelatter case the results should be qualitatively similar, althoughsome formal derivation and formulas may be different. We have derived analytical formulas for both adiabatic and nonadiabatic components of the torque, and analyzedtheir dependence on basic parameters of the domain wall and ferromagnetic materials. The derived results can be usedin the analysis of current-induced domain wall dynamicsby including the spin torque into the quasiclassical Landau-Lifshitz-Gilbert equation. The earlier models have shown that the nonadiabatic spin torque component is proportional to τ ex/τsfwithτsf being the spin-ﬂip relaxation time. Here we have shown that the spin torque incudes a term proportional to τex/τ∗, withτ∗describing spin-conserving momentum relaxation processes. This shows that spin-conserving scattering canenhance the spin transfer torque, which is also in agree-ment with some experimental 27,28as well theoretical20,24–26 suggestions. ACKNOWLEDGMENTS This work is partly supported by FCT Grant PTDC/FIS/70843/2006 in Portugal (VKD) and by the PolishMinistry of Science and Higher Education as a research projectin years 2011–2013 (VKD) and a research project in years2010–2011 (PB). 1J. Grollier, P. Boulenc, V . Cros, A. Hamzi ´c, A. Vaur `es, A. Fert, and G. Faini, Appl. Phys. Lett. 83, 509 (2003). 2A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, P h y s .R e v .L e t t . 92, 077205 (2004). 3M. Yamanouchi, D. Chiba, F. Matsukura, and H. 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PhysRevB.85.140408.pdf	RAPID COMMUNICATIONS PHYSICAL REVIEW B 85, 140408(R) (2012) Autonomous and forced dynamics in a spin-transfer nano-oscillator: Quantitative magnetic-resonance force microscopy A. Hamadeh,1G. de Loubens,1,V. V. N a l e t ov ,1,2J. Grollier,3C. Ulysse,4V. C r o s ,3and O. Klein1,† 1Service de Physique de l’ ´Etat Condens ´e (CNRS URA 2464), CEA Saclay, 91191 Gif-sur-Yvette, France 2Physics Department, Kazan Federal University, Kazan 420008, Russian Federation 3Unit ´e Mixte de Physique CNRS/Thales and Universit ´e Paris Sud 11, RD 128, 91767 Palaiseau, France 4Laboratoire de Photonique et de Nanostructures, Route de Nozay 91460 Marcoussis, France (Received 30 January 2012; published 20 April 2012) Using a magnetic-resonance force microscope (MRFM), the power emitted by a spin-transfer nano-oscillator consisting of a normally magnetized Py \|Cu\|Py circular nanopillar is measured both in the autonomous and forced regimes. From the power behavior in the subcritical region of the autonomous dynamics, one obtains aquantitative measurement of the threshold current and of the noise level. Their ﬁeld dependence directly yieldsboth the spin torque efﬁciency acting on the thin layer and the nature of the mode which ﬁrst auto-oscillates:the lowest energy, spatially most uniform spin-wave mode. From the MRFM behavior in the forced dynamics,it is then demonstrated that in order to phase lock this auto-oscillating mode, the external source must have thesame spatial symmetry as the mode proﬁle, i.e., a uniform microwave ﬁeld must be used rather than a microwavecurrent ﬂowing through the nanopillar. DOI: 10.1103/PhysRevB.85.140408 PACS number(s): 76 .50.+g, 75.30.Ds, 78 .47.−p, 85.75.−d Recent progress in spin electronics have demonstrated that, owing to the spin-transfer torque (STT),1,2biasing magnetic hybrid nanostructures by a direct current can lead to microwaveemission. These spin-transfer nano-oscillators (STNOs) 3–5 offer decisive advantages compared to existing technology in tunability, agility, compactness, and integrability. In view of their applications in high-frequency technologies, a promisingstrategy to improve the coherence and increase the emittedmicrowave power of these devices is to mutually synchronizeseveral of them. 6–10 The synchronization of the STNO oscillations to an external source has already been demonstrated.11,12In particular, it has been shown that symmetric perturbations to the STNOtrajectory favor even synchronization indices (ratio of theexternal frequency to the STNO frequency r=2,4,6,...), while antisymmetric perturbations favor odd synchronizationindices. 13,14But, so far, the inﬂuence of the spatial symmetry of the spin-wave (SW) mode which auto-oscillates on thesynchronization rules has not been elucidated. To address this open question, the spectroscopic identiﬁ- cation of the auto-oscillating mode is crucial. It is usually a challenge, as a large variety of dynamic modes can be excitedin STNOs, and their nature can change depending on thegeometry, magnetic parameters, and bias conditions. In thiswork, we study a STNO in the most simple conﬁguration:a circular nanopillar saturated by a strong magnetic ﬁeldapplied along its normal. It corresponds to an optimum conﬁguration for synchronization, since it has a maximal nonlinear frequency shift, which provides a large ability forthe STNO to lock its phase to an external source. 8Moreover, the perpendicular conﬁguration coincides with the universaloscillator model, for which an exact analytical theory can bederived. 15Last, but not least, this highly symmetric case allows for a simpliﬁed classiﬁcation of the SW eigenmodes inside the STNO.16 We shall use here a magnetic-resonance force microscope (MRFM) to monitor directly the power emitted by thisarchetype STNO vs the bias dc current and perpendicular magnetic ﬁeld. In the autonomous regime, these quantitativemeasurements allow us to demonstrate that the mode whichauto-oscillates just above the threshold current is the funda-mental, spatially most uniform SW mode. By studying theforced regime, we then show that this mode synchronizesonly to an external source sharing the same spatial symmetry,namely, a uniform microwave magnetic ﬁeld, and notthe common microwave current passing through the device. For this study, we use a circular nanopillar of nominal diam- eter 200 nm patterned from a (Cu60 \|Py B15\|Cu10\|PyA4\|Au25) stack,16where thicknesses are in nm and Py =Ni80Fe20.Ad c current Idcand a microwave current irfcan be injected through the STNO using the bottom Cu and top Au electrodes. Apositive current corresponds to electrons ﬂowing from the thickPy Bto the thin Py Alayer. This STNO device is insulated and an external antenna is patterned on top to generate a spatiallyuniform microwave magnetic ﬁeld h rforiented in the plane of the magnetic layers. The bias magnetic ﬁeld Hext, ranging between 8.5 and 11 kOe, is applied at θH=0◦from the normal to the sample plane. The room temperature MRFM setup17consists of a spheri- cal magnetic probe attached at the end of a very soft cantilever,coupled dipolarly to the buried nanopillar (see inset of Fig. 1) and positioned 1 .5μm above its center. This mechanical detection scheme 18,19sensitively measures the variation of the longitudinal magnetization /Delta1Mzover the whole volume of the magnetic body,20a quantity directly proportional to the normalized power pemitted by the STNO:15 p=/Delta1Mz 2Ms, (1) where Msis the saturation magnetization of the precessing layer. First, we measure the phase diagram of the STNO au- tonomous dynamics as a function of IdcandHext; see Fig. 1. In this experiment, Idcis fully modulated at the cantilever 140408-1 1098-0121/2012/85(14)/140408(4) ©2012 American Physical SocietyRAPID COMMUNICATIONS A. HAMADEH et al. PHYSICAL REVIEW B 85, 140408(R) (2012) FIG. 1. (Color online) Phase diagram of the STNO autonomous dynamics measured by MRFM. frequency, fc≈12 kHz, and the mechanical signal represents /Delta1Mzsynchronous with the injection of Idcthrough the STNO. This quantitative measurement21is displayed using the color scale indicated on the right of Fig. 1. Three different regions can be distinguished in this phase diagram. At low negative or positive current (region a/circlecopyrt),/Delta1Mz is negligible, because in the subcritical region the STT is not sufﬁcient to destabilize the magnetization in the thin or thicklayer away from the perpendicular applied ﬁeld direction. AsI dcis reaching a threshold negative value (from −3t o−7m A asHextincreases from 8 .5t o1 0 .7 kOe; see pink solid line in Fig. 1), the MRFM signal starts to smoothly increase in region b/circlecopyrt. It corresponds to the onset of spin-transfer driven oscillations in the thin layer, which will be analyzed in detailbelow. As I dcis further decreased toward more negative values, the angle of precession increases in the thin layer, until iteventually reaches 90 ◦: at the boundary between regions b/circlecopyrtand c/circlecopyrt(see black dashed line) 4 π/Delta1M zequals the full saturation magnetization in the thin layer, 4 πMs=8k G .22Let us now concentrate on the spin-transfer dynamics in the thin layer at Idc<0. We ﬁrst turn to the quantitative analysis of the subcritical region a/circlecopyrt. We introduce N=VMs/(gμB), the number of spins in the thin layer ( Vis its volume, gthe Land ´e factor, and μBthe Bohr magneton). The averaged normalized power pin the subcritical regime ( \|Idc\|<I th) is evaluated in the stochastic nonlinear oscillator model described in Sec. VIIof Ref. 15. Under the assumption that only one SW mode dominates the STNO autonomous dynamics, Eq. (1)follows the simple relationship /Delta1M z 2Ms=kBT N¯hων1 1−Idc/Ith, (2) where Ith=2αωνNe//epsilon1 is the threshold current for auto- oscillation of the SW mode νwith frequency ων(αis the Gilbert damping constant in the thin layer, ethe electron charge, and /epsilon1the spin torque efﬁciency). In Eq. (2),t h e prefactor η≡kBT N¯hων(3) is the noise power: the ratio between the thermal energy ( kB is the Boltzmann constant and Tthe temperature) and the maximal energy stored in the SW mode ν(¯his the Planck constant over 2 π). From Eq. (2), the inverse power is linear with the bias current Idcin the subcritical region. A sample measurement at Hext=10 kOe (along the white dashed line in Fig. 1)i ss h o w n in Fig. 2(a). From a linear ﬁt, one can thus obtain the threshold current Ithand the noise power ηat this particular ﬁeld. The dependencies of Ithandηon the perpendicular magnetic ﬁeld are plotted in Figs. 2(b) and2(c), respectively. The parameters V,Ms,g(hence N/similarequal6.3×106) and α=0.014 of the thin layer have been determined from an extensive MRFM spectroscopic study performed at Idc=0 on the same sample and published in Ref. 16. This study also yields the dispersion relations ων=γ(Hext−Hν)o ft h e thin layer SW modes ( γ=gμB/¯h=1.87×107rad s−1G−1 is the gyromagnetic ratio and Hνthe so-called Kittel ﬁeld associated to the mode ν). By injecting ωνin the expression of the threshold current, it is found that the latter depends linearly FIG. 2. (Color online) (a) Determination of the threshold current Ithand noise power ηatHext=10 kOe, from the inverse MRFM signal in the subcritical regime. Dependencies of the threshold current (b) and noise power (c) on the perpendicular magnetic ﬁeld. 140408-2RAPID COMMUNICATIONS AUTONOMOUS AND FORCED DYNAMICS IN A SPIN- ... PHYSICAL REVIEW B 85, 140408(R) (2012) FIG. 3. (Color online) MRFM measurement of the STNO dynam- ics forced by (a) the uniform ﬁeld hrfat 8.1 GHz and (b) the orthoradial Oersted ﬁeld produced by irfat 9.2 GHz, as a function of IdcandHext. The black traces show the MRFM signal vs IdcatHext=8.8 kOe. The pink solid lines show the location of the threshold current determined in Fig. 2(b). The dashed lines are guides to the eye. on the perpendicular bias ﬁeld: Ith=2αNe /epsilon1γ(Hext−Hν), (4) as observed in Fig. 2(b). The linear ﬁt of IthvsHextusing Eq.(4)yields Hν=6.80±0.15 kOe and /epsilon1=0.30±0.005. The importance of the analysis of Fig. 2(b) is that, ﬁrst, it provides an accurate determination of the spin torqueefﬁciency, found to be in agreement with the accepted valuein similar STNO stacks. 23Second, a comparison with the SW modes of the thin layer [see black symbols extracted fromRef. 16and mode proﬁles in Fig. 2(b)] shows that the ﬁtted value of H νprecisely corresponds to the Kittel ﬁeld of the (/lscript,n)=(0,0) mode, /lscriptandnbeing respectively the azimuthal and radial mode indices. It thus allows us to conclude aboutthe nature of the mode that ﬁrst auto-oscillates at I dc<0a s being the fundamental, most uniform precession mode of thethin layer. To gain further insight in our analysis of the subcritical regime, we compare in Fig. 2(c) the noise power determined as a function of H extwith the prediction of Eq. (3),i n which the dispersion relation of the ν=(0,0) SW mode is used. It is found that the ﬂuctuations of the STNO powerare well accounted for by those of the previously identiﬁedauto-oscillating mode, which conﬁrms that the single mode assumption made to derive Eq. (2)is a good approximation. Using two different microwave circuits, we shall now compare the ability of the auto-oscillating SW mode to phaselock either to the uniform microwave ﬁeld h rfgenerated by the external antenna, or to the microwave current irfﬂowing through the nanopillar. We know from previous studies thatin the exact perpendicular conﬁguration, the SW spectrumcritically depends on the method of excitation: 16hrfexcites only the axially symmetric modes having azimuthal index/lscript=0, whereas due to the orthoradial symmetry of the induced microwave Oersted ﬁeld, i rfexcites only the modes having azimuthal index /lscript=+ 1. The dependencies on IdcandHext of the STNO dynamics forced respectively by hrfandirfare presented in Figs. 3(a) and3(b). The plotted quantity is /Delta1Mz synchronous with the full modulation of the external source power: hrf=1.9 Oe (a) and irf=140μA (b). Although the /lscript=0 and/lscript=+ 1 spectra are in principle shifted by 1.1 GHz from each other, a direct comparison of the phase diagrams (a)and (b) can be made by using different excitation frequenciesforh rf(8.1 GHz) and irf(9.2 GHz). Below the threshold current (indicated by the pink lines in Fig.3), the observed behaviors of the /lscript=0 and/lscript=+ 1 modes are alike: a small negative dc current slightly attenuates theSW modes B /lscriptnof the thick Py Blayer, while it promotes quite rapidly the SW modes A/lscriptnof the thin Py Alayer, in agreement with the expected symmetry of the STT.16On the contrary, there is a clear qualitative difference between the modes A00 andA10beyond Ith. Although both peaks similarly shift toward lower ﬁeld as Idcis decreased toward lower negative values, A00gets strongly distorted, with the appearance of a negative dip on its high ﬁeld side, in contrast to A10, which remains a positive peak. The negative MRFM signal observed in Fig. 3(a) in the region of spin-transfer driven oscillations in the thin layer isstriking, because it means that the precession angle can bereduced in the presence of the microwave excitation h rf.A sa matter of fact, this distortion of the peak A00is associated to the synchronization of the auto-oscillating mode to the externalsignal. Figure 4(a) illustrates the distortion of the STNO emission frequency induced by this phenomenon. These datawere obtained by monitoring the ﬂuctuating voltage across FIG. 4. (Color online) (a) Magnetic ﬁeld dependence of the STNO frequency in the free and forced regimes (the external source at 8.1 GHz ishrf). (b) Comparison between the STNO frequency shift deduced from (a) and the MRFM signal. 140408-3RAPID COMMUNICATIONS A. HAMADEH et al. PHYSICAL REVIEW B 85, 140408(R) (2012) the nanopillar at Idc=− 7 mA with a spectrum analyzer as a function of the applied magnetic ﬁeld.24The frequency shift of the forced oscillations with respect to the free runningoscillations is plotted in Fig. 4(b), along with the MRFM signal. This demonstrates that, in the so-called phase-lockingrange, the STNO amplitude adapts ( /Delta1M z>0: increases; /Delta1Mz<0: decreases), so as to maintain its frequency equal to the frequency of the source, here ﬁxed at 8.1 GHz. Thiscomparison also allows one to estimate the phase-lockingbandwidth, found to be as large as 0.4 GHz despite the smallamplitude of the external signal. The nonlinear frequency shiftis indeed the largest in the perpendicular conﬁguration, N= 4γM s/similarequal48 GHz;15therefore, a small change of the power emitted by the STNO is sufﬁcient to change its frequency bya substantial amount. Such a signature of synchronization of the auto-oscillating mode is not observed in Fig. 3(b), where the external source is the microwave current. This highlights the crucial importanceof the symmetry associated to the SW mode driven by STT: inthe exact perpendicular conﬁguration, i rfcan only excite /lscript= +1 SW modes, therefore, it has the wrong symmetry to couple to the auto-oscillating mode, which was shown in Fig. 2to bare the azimuthal index /lscript=0. We add that in our exact axiallysymmetrical case, no phase-locking behavior is observed with the even synchronization index r=2, neither with irf, nor with hrf, which is due to the perfectly circular STNO trajectory. To conclude, based on the quantitative analysis of both the critical current and the noise power in the subcriticalregime, we have unambiguously identiﬁed the auto-oscillatingmode in the perpendicular conﬁguration of a nanopillar. Thiscase is particularly interesting due to its large ability tosynchronize to an external source. But we have shown thatin addition to the symmetry of the perturbation with respectto the STNO trajectory, 14the overlap integral between the external source and the auto-oscillating mode proﬁle is crucialto synchronization rules. Due to symmetry reasons, only theuniform microwave ﬁeld applied perpendicularly to the biasﬁeld and with the synchronization index r=1 is efﬁcient to phase lock the STNO dynamics in the present work. We believethat this ﬁnding might be important for future strategies tosynchronize large STNOs arrays. We thank A. N. Slavin for useful discussions and his support. This research was supported by the European GrantMaster (NMP-FP7 212257) and by the French Grant V oice(ANR-09-NANO-006-01). Corresponding author: gregoire.deloubens@cea.fr †olivier.klein@cea.fr 1J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 2L. Berger, P h y s .R e v .B 54, 9353 (1996). 3S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature (London) 425, 380 (2003). 4W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, P h y s .R e v .L e t t . 92, 027201 (2004). 5D. Houssameddine, U. Ebels, B. Delat, B. Rodmacq, I. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-P. Michel, L. Prejbeanu-Buda, M.-C. Cyrille, O. Redon, and B. Dieny, Nat. Mater. 6, 447 (2007). 6S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, andJ. A. Katine, Nature (London) 437, 389 (2005). 7F. B. Mancoff, N. D. Rizzo, B. N. Engel, and S. Tehrani, Nature (London) 437, 393 (2005). 8A. N. Slavin and V . S. Tiberkevich, Phys. Rev. B 72, 092407 (2005). 9J. Grollier, V . Cros, and A. Fert, P h y s .R e v .B 73, 060409 (2006). 10A. Ruotolo, V . Cros, B. Georges, A. Dussaux, J. Grollier, C. Deranlot, R. Guillemet, K. Bouzehouane, S. Fusil, and A. Fert,Nature Nanotech. 4, 528 (2009). 11W. H. Rippard, M. R. Pufall, S. Kaka, T. J. Silva, S. E. Russek, and J. A. Katine, P h y s .R e v .L e t t . 95, 067203 (2005). 12B. Georges, J. Grollier, M. Darques, V . Cros, C. Deranlot, B. Marcilhac, G. Faini, and A. Fert, P h y s .R e v .L e t t . 101, 017201 (2008). 13M. Quinsat, J. F. Sierra, I. Firastrau, V . Tiberkevich, A. Slavin,D. Gusakova, L. D. Buda-Prejbeanu, M. Zarudniev, J.-P. Michel,U. Ebels, B. Dieny, M.-C. Cyrille, J. A. Katine, D. Mauri, and A. Zeltser, Appl. Phys. Lett. 98, 182503 (2011). 14S. Urazhdin, P. Tabor, V . Tiberkevich, and A. Slavin, Phys. Rev. Lett. 105, 104101 (2010). 15A. Slavin and V . Tiberkevich, IEEE Trans. Magn. 45, 1875 (2009). 16V . V . Naletov, G. de Loubens, G. Albuquerque, S. Borlenghi, V . Cros, G. Faini, J. Grollier, H. Hurdequint, N. Locatelli, B. Pigeau,A. N. Slavin, V . S. Tiberkevich, C. Ulysse, T. Valet, and O. Klein,Phys. Rev. B 84, 224423 (2011). 17O. Klein, G. de Loubens, V . V . Naletov, F. Boust, T. Guillet, H. Hurdequint, A. Leksikov, A. N. Slavin, V . S. Tiberkevich, andN. Vukadinovic, Phys. Rev. B 78, 144410 (2008). 18G. de Loubens, V . V . Naletov, O. Klein, J. Ben Youssef, F. Boust, and N. Vukadinovic, P h y s .R e v .L e t t . 98, 127601 (2007). 19B. Pigeau, G. de Loubens, O. Klein, A. Riegler, F. Lochner, G. Schmidt, and L. W. Molenkamp, Nature Phys. 7, 26 (2011). 20G. de Loubens, V . V . Naletov, and O. Klein, P h y s .R e v .B 71, 180411 (2005). 21V . V . Naletov, V . Charbois, O. Klein, and C. Fermon, Appl. Phys. Lett. 83, 3132 (2003). 22This is corroborated by transport measurements, as the increase of the dc resistance measured at this boundary equals 11 m /Omega1, i.e., half of the full GMR of the spin-valve nanopillar. 23V . S. Rychkov, S. Borlenghi, H. Jaffres, A. Fert, and X. Waintal,Phys. Rev. Lett. 103, 066602 (2009). 24Here, a slight tilt of the angle θH=2◦is required. Indeed, no oscillatory voltage is produced in the exact perpendicularconﬁguration, due to the perfect axial symmetry of the STNOtrajectory. 140408-4
PhysRevB.93.214405.pdf	PHYSICAL REVIEW B 93, 214405 (2016) Cycloidal versus skyrmionic states in mesoscopic chiral magnets Jeroen Mulkers,1,2,Milorad V . Milo ˇsevi´c,1and Bartel Van Waeyenberge2 1Department of Physics, Antwerp University, Antwerp, Belgium 2DyNaMat Lab, Department of Solid State Sciences, Ghent University, Ghent, Belgium (Received 20 April 2016; published 6 June 2016) When subjected to the interfacially induced Dzyaloshinskii-Moriya interaction, the ground state in thin ferromagnetic ﬁlms with high perpendicular anisotropy is cycloidal. The period of this cycloidal state dependson the strength of the Dzyaloshinskii-Moriya interaction. In this work, we have studied the effect of conﬁnementon the magnetic ground state and excited states, and we determined the phase diagram of thin strips and thinsquare platelets by means of micromagnetic calculations. We show that multiple cycloidal states with differentperiods can be stable in laterally conﬁned ﬁlms, where the period of the cycloids does not depend solely onthe Dzyaloshinskii-Moriya interaction strength but also on the dimensions of the ﬁlm. The more complex statescomprising skyrmions are also found to be stable, though with higher energy. DOI: 10.1103/PhysRevB.93.214405 I. INTRODUCTION Magnets can be chiral due to the Dzyaloshinskii-Moriya interaction (DMI). In bulk materials, this interaction is causedby a lack of inversion symmetry in the crystal structure [ 1–3], but in thin ﬁlms DMI can also be induced by symmetrybreaking at interfaces [ 4]. Bogdanov et al. gave a micro- magnetic description of the basic chiral spin states—helices,cycloids, skyrmions—in ferromagnetic materials subject tothe DMI [ 5–8]. Bode et al. imaged the cycloidal state in a single atomic layer of manganese in 2007 [ 9], whereas the existence of skyrmions and skyrmion lattices was onlyconﬁrmed experimentally in 2009 [ 10–12]. More exotic chiral magnetic structures have been observed in a Sc-doped bariumhexaferrite thin ﬁlm by Yu et al. [13]. Skyrmionics, and the study of related chiral spin states, has gained a lot of interest since the ﬁrst experimental evidence forthe existence of magnetic skyrmions. In particular, the effectof the interfacially induced DMI on the magnetization of thinﬁlms with perpendicular magnetic anisotropy (PMA) becamea prominent subject in micromagnetism. As one prominenteffect, DMI makes N ´eel walls energetically favorable in thin PMA ﬁlms. Consequently, the magnetic ground stateis no longer homogeneous but cycloidal (in the absence ofan external magnetic ﬁeld) or a N ´eel skyrmion lattice (in a perpendicular ﬁeld) [ 14,15]. Many applications based on the DMI in thin PMA ﬁlms have already been proposed:skyrmion writer [ 16], racetrack memory [ 17–21], skyrmion- based transistor [ 22], storage [ 23] and logic gates [ 24]. The DMI strength can be tuned experimentally by using differentsubstrates, different ﬁlm thicknesses, or by using a Ta bufferlayer [ 25–28]. To cover a broad range of possible materials, we take the DMI strength as a variable parameter. For prospective applications to compete with existing information technologies, the device dimensions should besmall [ 29]. When entering the mesoscopic regime, i.e., when the dimensions of the system become comparable to the typicallength scales of the magnetic state, one has to take into accountthe effect of the conﬁnement on the magnetic structures. jeroen.mulkers@uantwerpen.beKeesman et al. have studied the effect of conﬁnement on skyrmionic ground states in thin PMA strips using Monte Carlosimulations [ 30]. Stabilization of skyrmions in nanowires, as a function of sample parameters, has been considered inRef. [ 31] by solving the Landau-Lifshitz-Gilbert equation. The skyrmionic state in a mesoscopic disk was studied by Rohartet al. [32] and is to date the only analytic consideration of a conﬁned chiral state. In this paper, we contribute to the understanding of the phase diagram of PMA strips and square platelets in thepresence of DMI. First, we present an analytic derivationof the cycloidal state in an inﬁnitely long strip of ﬁnitewidth, where demagnetization is approximated by an effectiveanisotropy, and the effect of the boundaries is carefullydiscussed. Subsequently, the analytic results are comparedwith micromagnetic simulations, where we do not use anapproximation for the demagnetization energy. This enables usto check the validity of the effective anisotropy approximationfor ﬁnite-size thin PMA ﬁlms. Finally, we determine theground state and excited states of thin square PMA platelets ofdifferent sizes, and we present a complete equilibrium phasediagram and its governing rules. The paper is organized as follows. Section IIpresents the micromagnetic framework and numerical algorithms.Section IIIpresents our results for cycloidal states in long mesoscopic strips, considered analytically (Sec. III A ) and numerically (Sec. III B ). In Sec. IV, we ﬁnally present the complete diagram of cycloidal and skyrmionic phases for asquare platelet of varied size and DMI. A summary is given inSec. V. II. METHODS In this section, we recapitulate the micromagnetic descrip- t i o no ft h i nP M Aﬁ l m s[ 33]. The quantity of interest is the magnetization ﬁeld /vectorM(x,y)=Msat/vectorm(x,y) with magnetization modulus \|/vectorM\|=Msatand magnetization direction /vectorm(x,y). The dynamics of the magnetization is governed by the Landau-Lifshitz-Gilbert equation (LLG) /vectorm t=γLL 1+α2(/vectorm×/vectorHeff+α[/vectorm×(/vectorm×/vectorHeff)]) (1) 2469-9950/2016/93(21)/214405(8) 214405-1 ©2016 American Physical SocietyMULKERS, MILO ˇSEVI ´C, AND V AN W AEYENBERGE PHYSICAL REVIEW B 93, 214405 (2016) with damping factor αand the gyromagnetic ratio γLL.T h e effective magnetic ﬁeld is the derivative of the magnetic energy density ε:/vectorHeff=−∂ε/∂/vectorm. When studying very thin ﬁlms with saturation magnetization Msatand anisotropy constant K, one can approximate the demagnetization energy by using an effective anisotropy Keff=K−1/2μ0M2 sat[34]. The three remaining energy terms of interest in this paper are related toexchange, DMI, and magnetic anisotropy, respectively, ε ex=A/bracketleftbigg/parenleftbigg∂/vectorm ∂x/parenrightbigg2 +/parenleftbigg∂/vectorm ∂y/parenrightbigg2/bracketrightbigg , (2) εDMI=D/bracketleftbigg mx∂mz ∂x−mz∂mx ∂x+my∂mz ∂y−mz∂my ∂y/bracketrightbigg ,(3) εanis=−Keffm2 z. (4) The total magnetic energy E(/vectorm)=/integraltext ε(/vectorm)dV of a thin ﬁlm is a functional of the magnetization direction /vectorm(x,y). Minimizing the total energy yields the magnetic ground state(global minimum) and the excited states (local minima). Inour work, we use both analytic and numerical techniques tominimize the energy. a. Analytic approach. The cycloidal state in inﬁnite PMA ﬁlms and skyrmionic states in circular PMA disks have beencalculated analytically with variational calculus in Ref. [ 32]. We have used the same approach to derive the conﬁnedcycloidal state in thin PMA strips. b. Numerical methods. Using the LLG equation, we fol- low the magnetization converging to a stable state at a localenergy minimum [ 33]. Subtracting the Larmor precession term from the LLG equation speeds up the computation ofrelaxed states. Different initial conditions can be used to ﬁnddifferent stable magnetic states. In our simulations, we startedfrom random magnetic states, V oronoi-like domains, and smartinitial guesses to probe the equilibrium phase diagram. We use the ﬁnite-difference-based simulation package MUMAX3 for the micromagnetic simulations presented in this paper [ 35]. In these simulations, we calculate the demagneti- zation ﬁeld and do not use the thin-ﬁlm approximation withthe effective anisotropy. We employ the boundary conditions d/vectorm dn=D 2A(/vectorez×/vectorn)×/vectorm (5) at an edge with normal /vectorn[32,35]. The origin of these boundary conditions will become clear in Sec. III A . We use material parameters corresponding to Pt/Co ﬁlms, as used in Ref. [ 36]:Msat=580 kA /m,A=15 pJ/m,K= 0.8M J/m3, and Keff=0.59 MJ /m3. The thickness of the ﬁlmtis 0.4 nm (a single layer). The used cell size in all our simulations is 1 nm ×1n m×0.4 nm. This cell size guarantees a maximal angular variation of the magnetizationin neighboring cells below 20 ◦while preserving a reasonable computation time. III. CYCLOIDAL STATE IN A MAGNETIC STRIP A. Analytic considerations The ﬁrst steps in the derivation of the cycloidal state in thin strips are analogous to the derivation of the cycloidal state in0 wxz FIG. 1. Cycloidal state in a strip of width w. inﬁnite ﬁlms presented in Ref. [ 32]. The cycloidal state in a thin PMA strip (with inﬁnite length but with ﬁnite width w) can be calculated analytically after assuming that the magnetizationof the cycloidal state rotates in the ( x,z) plane and changes only along the width of the strip ( xdirection) and is thus constant along the length ( ydirection) and the height ( zdirection). This means that the magnetization mis now fully described by a single angle θ(x):/vectorm=(sin(θ),0,cos(θ)). Figure 1sketches the magnetization in a thin PMA strip. One can make the total energy functional, given in Sec. II, more explicit by using these assumptions about the magneti-zation of the cycloidal state. In this paper, we will work withthe average energy density functional E, which is proportional to the total energy E: E[θ]=1 w/integraldisplayw 0/bracketleftbigg A/parenleftbiggdθ dx/parenrightbigg2 −Ddθ dx−Keffcos2θ/bracketrightbigg dx. (6) Here we assume free boundary conditions at x=0 and x=w. Using variational calculus, we minimize (or maximize) the energy functional E[θ]. This gives the Euler-Lagrange equation d2θ dx2=Keff Asinθcosθfor 0<x<w . (7) The free boundary conditions become Dirichlet boundary conditions after the minimization: dθ dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=0=dθ dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=w=D 2A. (8) Taking the indeﬁnite integral and subsequently the square roots of both sides yields dθ dx=±/radicalbigg Keff A/radicalbig C+sin2θ. (9) Later on, we will use the integration constant Cas the tuning parameter for meeting the boundary conditions. Equation ( 9) tells us that the angle θ(x) is a monotonic function. When looking at the energy functional E, especially at the sign of the DMI term, one can conclude that the angle θ(x)i sa monotonically increasing function for a magnetic state witha local energy minimum. This is why we will only considerthe positive square root of Eq. ( 9). Inverting and integrating Eq. ( 9) from 0 to xyields /radicalbigg Keff Ax=/integraldisplayθ(x) θ01√ C+sin2θdθ, (10) with the yet unknown initial angle θ0:=θ(0). This is an implicit expression for the angle θ(x). One can calculate the quarter period of the cycloidal state by integrating the integrandin Eq. ( 10)f r o m0t o π/2. The period of the ground state in an inﬁnite ﬁlm can be found by altering the integration constantCuntil the energy density of the corresponding magnetic state θ(x) reaches the energy minimum. 214405-2CYCLOIDAL VERSUS SKYRMIONIC STATES IN . . . PHYSICAL REVIEW B 93, 214405 (2016) 01020304050 Position x(nm)00.5ππ1.5π2πAngle θ(x) φ+ 0φ− 1φ+1φ−2φ+ 2 φ− 0(a) 0.0 0.2 0.4 0.6 0.8 1.0 10C/C max−0.70−0.65−0.60−0.55E(MJ/m3) φ+ 0φ− 1 φ+ 1φ− 2 φ+ 2(b)-101(c) θ(w)=φ+ 2 -101(d) θ(w)=φ− 2 -101 Magnetization mxmz (e) θ(w)=φ+ 1 -101(f) θ(w)=φ− 1 01020304050 Position x(nm)-101(g) θ(w)=φ+ 0 FIG. 2. (a) θ(x) proﬁles of the stable cycloidal states with initial angleθ0=φ− 0in a 50-nm-wide PMA strip with DMI strength D= 4mJ/m2. The magnetic energy density is given as a function of the free parameter Cin panel (b). The magnetizations mz=cosθand mx=sinθof the stable and metastable states are shown in (c)–(g). The stable and metastable states are denoted by their boundary angleθ(w)∈φ ± n. In contrast to the work of Rohart and Thiaville in Ref. [ 32], we will focus on the cycloidal state in thin strips where theboundary conditions need special treatment. Using Eqs. ( 8) and ( 9), we conclude that the angles at the boundaries have to be in the set φ ± n=± arcsin/parenleftbigg/radicalBigg D2 4AK eff−C/parenrightbigg +nπ withn∈Z(11) in order to meet the Dirichlet boundary conditions. From this set, we choose the initial angle θ0(left boundary) to be φ− 0. The integration constant Ccan now be tuned in order to meet the boundary condition at x=w, i.e., until θ(w)∈φ± n.W ed o this by scanning Cfrom 0 to Cmax=D2/(4AK eff). For every C, we calculate the magnetic state θ(x) and the corresponding energy and check if θ(w)∈φ± n. As a representative example, we discuss a full sweep of Cfor a strip of width w=50 nm and DMI strength D=4m J/m2. The results are shown in Fig. 2. There are ﬁve different values of Cthat yield a correct boundary angle θ(w)∈φ± nfor the given example. Their corresponding energies are local energy extrema, as is expectedfor the Euler-Lagrange equations. The three stable states have aborder angle θ(w)=φ + nwithn=0,1,2. From Figs. 2(c)–2(g) and symmetry arguments, we can conclude that for stablestates in general, the left boundary angle θ 0∈φ− nand the rightboundary angle θ(w)∈φ+ n. This conﬁrms the correctness of our initial guess for the initial angle θ0=φ− 0. For the given example in Fig. 2, it is easy to check that the three stable states are the only stable states: choosing a different initial angleθ 0∈φ− nyields equivalent solutions due to the periodicity of θ(x). From here on, we will label the stable states with the given integer n∈N. The total rotation of the magnetization of state nisnπplus a small correction in order to satisfy the boundary conditions: φ+ n−φ− 0=nπ+2a r c s i n/parenleftbigg/radicalBigg D2 4AK eff−C/parenrightbigg . (12) We can repeat our calculations to obtain the stable states in strips of different widths wand for varying DMI strengths D. The energy densities of the stable states are shown in Figs. 3(a) and3(b) for two different DMI strengths D. With these energy plots, one determines if a cycloidal state is stable in a strip ofwidth w. After ﬁnding the lowest energy state for each Dand w, we obtain the phase diagram of the ground state shown in Fig. 3(c). If the DMI strength Dis below the critical DMI strength D c=4√AK eff/π, then the ground state is the cycloidal state n=0 (the quasiuniform state). This is similar for inﬁnite ﬁlms [32]. IfD>D c, then the ground state depends on the width w of the strip: the larger the width or the stronger the DMI D,t h e larger is the nof the ground state. This behavior is consistent with the results for inﬁnite ﬁlms [ 32]. The effect of the DMI and the width of the strip on a cycloidal state is shown in Fig. 4. Making the strip narrower compresses the state. It is interesting to note that \|mz\|→ 1a t the boundaries when narrowing the width wof the strip. If we conﬁne the strip even further, then the state becomes unstable. −0.60−0.55−0.50E(MJ/m3)(a) D=3.5mJ/m2state n:0123456 −0.65−0.60−0.55E(MJ/m3)(b) D=4.0mJ/m2 20 40 60 80 100 120 Width w(nm)3.63.84.04.24.44.64.85.0D(mJ/m2) Dc(c) FIG. 3. The energy density of the stable states in a thin PMA strip is shown for DMI strength D=3.5m J/m2(a) and D=4.0m J/m2 (b). The ground state as a function of the DMI interaction strength D and the width wis presented in panel (c). 214405-3MULKERS, MILO ˇSEVI ´C, AND V AN W AEYENBERGE PHYSICAL REVIEW B 93, 214405 (2016) −1.0−0.50.00.51.0mz(a) Eﬀect of w 35nm 70nm 100nm −40−2002040 x(nm)−1.0−0.50.00.51.0mx(b) Eﬀect of D 2mJ/m24mJ/m2 −30−20−100102030 x(nm) FIG. 4. The zandxcomponents of the magnetization in the cycloidal stable state n=2 in a PMA strip with width wand DMI strength D. Panel (a) illustrates the inﬂuence of the width wwith D=4m J/m2, and panel (b) illustrates the inﬂuence of the DMI strength Din a 60-nm-wide strip. The periodicity of the cycloidal state in inﬁnite ﬁlms depends strongly on the DMI strength D. For a strip of a given width, the periodicity of a cycloidal state is practically ﬁxed as a resultof the conﬁnement. Still, some effect of the DMI strength D is visible since it alters the boundary condition [see Eq. ( 8)]. B. Micromagnetic simulations In this subsection, we investigate possible deformations of the cycloidal state in a PMA strip when we drop theassumptions that the magnetic state is constant along the lengthof the strip and that the magnetization direction lies in the(x,z) plane. Analytic calculations are no longer possible, and we resort to micromagnetic simulations. Periodic boundaryconditions and a large simulation box (500 nm) are usedin the ydirection in order to simulate an inﬁnitely long strip. The initial states in these simulations are cycloidalwith a small amount of random noise in order to triggerpossible deformations. The resulting energy densities afterrelaxation of the cycloidal states n=0,1,2,3 for DMI strength D=4m J/m 2are shown in Fig. 5. Examples of the obtained magnetic states are shown in Fig. 6. The results for the quasiuniform state ( n=0) correspond exactly with the analytic results. The same is true for thecycloidal state n=1 if the width of the strip is small, i.e., w< 80 nm. If the width of the strip is larger ( w> 80 nm), we observe buckling in the domain wall, which somewhatlowers the energy density. The magnetization is no longerconstant along the ydirection. If the width of the strip is taken even larger, w> 125 nm, the energy density drops drastically after a complex deformation of the initial state. Note that forw=140 nm, the typical domain width in the relaxed n=1 state is similar to the domain widths in the cycloidal staten=3 (ground state). Further increasing the width of the strip will yield similar results for the cycloidal states n> 1. For example, note the buckled domains in the relaxed cycloidalstaten=2 in a 140-nm-wide strip in Fig. 6. The relaxation of perfect cycloidal states (without buckling) can result in magnetic states that are no longer perfect cycloids.60 80 100 120 140 Width w(nm)−0.650−0.645−0.640−0.635−0.630−0.625−0.620−0.615E(MJ/m3) n=0 n=1 n=2 n=3 FIG. 5. The energy densities of the numerically relaxed cycloidal statesn=0,1,2,3 in thin strips with DMI strength D=4m J/m2are shown by dots. Lines show the analytical results, previously plottedin Fig. 3. However, all the observed ground states are perfect cycloids, and are thus analytically calculable. The analytic results agreeperfectly with the numerical results of the uniform state and ofthe cycloidal states without buckling (see Fig. 5). This justiﬁes the approximation of the demagnetization of thin strips inanalytic calculations by using the effective anisotropy K eff. This is not surprising since the thickness of the strip is muchsmaller than its width. IV . SQUARE PLATELETS In this section, we study the ground state and the excited states of square mesoscopic PMA platelets as a function of theside length land the DMI strength D. Relaxing a randomly magnetized sample using the LLG equation gives one ofthe stable states. Repeating this process for different initialmagnetic states, sizes l, and DMI strengths Dwill reveal the full phase diagram of square PMA platelets. The number ofstable states in mesoscopic samples with a low DMI strengthor a small side length ( l<60 nm) will turn out to be limited, which facilitates identifying the ground state as well as allexcited states. This is done in Sec. IV A . If the platelet is large ( l>60 nm) and the DMI is strong, then the number of possible states can be very large, making it difﬁcult to identifyall stable states. However, it is still possible to determine theground state. This is detailed in Sec. IV B . w(nm) n=1 n=2 n=3 80 100 120 140 FIG. 6. The cycloidal states n=1,2,3 in thin strips with D= 4m J/m2after relaxation using micromagnetic simulations. The direction of the magnetization is depicted by colors shown in thecolor wheel. 214405-4CYCLOIDAL VERSUS SKYRMIONIC STATES IN . . . PHYSICAL REVIEW B 93, 214405 (2016) A. Excited states in small platelets We identiﬁed all stable states in square platelets with a size length below 60 nm for DMI strengths D=3m J/m2< DcandD=5m J/m2>D c. To convince ourselves that we identiﬁed every possible state, we used 10 000 initialconﬁgurations for each set of parameters, while a few hundredinitial conﬁgurations are usually sufﬁcient to ﬁnd all statesin such small platelets. This brute force method yields manyequivalent states, where we took a single representative statefor each set of equivalent states using a comparison algorithm.The states are compared pairwise, taking into account the D 4h symmetry of the sample. The representative states and their energies are shown in Fig. 7forD=3m J/m2and in Fig. 8 forD=5m J/m2. Some of the representative states are labeled for convenient referral. Figure 7shows that a stable magnetic state in a platelet of certain size is not necessarily stable in smaller platelets. Forexample, the excited state C08 shown in Fig. 7is unstable in square platelets with a side length smaller than 47 nm, as oneof the three skyrmions will be pushed out of the sample. 30 35 40 45 50 55 60 Side length l(nm)−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.1E(MJ/m3) 40 nm A00 A01 A02 A03 Label 0.000 0.085 0.179 0.225 ΔE(MJ/m3) 50 nm B00 B01 B02 B03 B04 B05 Label 0.000 0.061 0.108 0.151 0.187 0.232 ΔE(MJ/m3) C00 C01 C02 C03 C04 C05 C06 60 nm 0.000 0.048 0.073 0.111 0.129 0.131 0.154 C07 C08 Label 0.215 0.237 ΔE(MJ/m3) FIG. 7. The energy densities Eof the magnetic states of a square l×lplatelet with DMI strength D=3m J/m2<D c. The magnetic states for platelets with side lengths 40, 50, and 60 nm are shown separately and labeled in order of their energies. The energy differencewith the quasiuniform state, shown below the label, is given in MJ /m 3.20 25 30 35 40 45 50 Side length l(nm)−0.95−0.90−0.85−0.80−0.75−0.70−0.65−0.60E(MJ/m3) 30 nm D00 D01 D02 D03 D04 D05 40 nm E00 E01 E02 E03 E04 E05 E06 E07 E08 E09 F00 F01 F02 F03 F04 F05 F06 50 nm F07 F08 F09 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F0 0F 01 F0 2F 03 F0 4F 05 F0 6 F0 7F 08 F0 9F 10 F1 1F 12 F1 3 F1 4F 15 F1 6F 17 F1 8F 19 F2 0 F21 F22 F23 F24 FIG. 8. The energy densities Eof the magnetic states of a square l×lplatelet with DMI strength D=5m J/m2>D c: the uniform state (black), the parallel cycloidal states (green), the diagonalcycloidal states (red), the single skyrmion state (blue), and other states (gray). The magnetic states for platelets with side lengths 30, 40, and 50 nm are shown separately and labeled in the order of theirincreasing energies. For weak DMI strengths D<D c(D=3m J/m2in our case), the ground state in platelets of arbitrary size is uniform.Furthermore, the sequence of the excited states ordered bytheir energies does not depend on the size of the platelet.The excited states contain distinct features such as skyrmionsand domain walls, which can be considered as particlelikeexcitations. The creation of a domain wall or skyrmion willgenerally increase the energy. However, this energy differenceis not trivial. For instance, the energy difference betweenthe double skyrmion state C06 and the uniform state C00is not twice as large as the energy difference between thesingle skyrmion state C02 and the uniform state C00. Thesame holds true for states with domain walls or with the 214405-5MULKERS, MILO ˇSEVI ´C, AND V AN W AEYENBERGE PHYSICAL REVIEW B 93, 214405 (2016) combination of skyrmions and domain walls. We thus infer that an important ingredient is the repulsion energy betweenskyrmions, domains, and boundaries. Further, a special kindof domain wall is identiﬁed in C04. This domain wall is anordinary N ´eel wall except at the center, where the in-plane magnetization makes a full rotation. The topological charge ofthis state is one due to this rotation, just as in a skyrmion. The phase diagram of the platelets becomes more complex for increasing DMI strengths D. Figure 8shows that the number of possible excited states can be very large for DMIstrengths larger than D c.F o rD=5m J/m2, we identiﬁed 25 and 77 different stable states in square platelets with,respectively, a side length of l=50 and 60 nm. Identifying all possible stable states for larger ﬁlms is a very laborioustask. Furthermore, the sequence of the magnetic states orderedby their energies does depend on the size of the platelet, whichalso contributes to the complexity of the phase diagram. The ground states of square platelets shown in Fig. 8are cycloids parallel with an edge. The number of domain walls inthe cycloid depends on the size of the platelet. For example, forl=40 nm there is a single domain wall in the ground state E00, while for l=50 nm there are two domain walls in the ground state F00. This is consistent with our analytical calculations.In general, the low-energy states have a cycloidal character.For instance, state E02 can be considered as a cycloidal stateparallel with the diagonal of the square platelet, and state F02 can be considered as a slightly deformed cycloid. However, not every cycloidal state has a relatively low energy. Note that,e.g., cycloidal state F20 is a high-energy state due to its smallperiod. Other high-energy states, such as F21–F24, containskyrmions. These states are stable since the skyrmions aretopologically protected. The effects of the square shape and the boundaries of the studied platelets are visible in Fig. 8. For example, the only difference between states F00 and F02 is caused by the upperright corner. If the effect of the boundary would be weakeror the shape more round, then state F02 would transforminto ground state F00 during relaxation without the need tojump over an energy barrier. Other sets of states in which thiseffect is visible are {F11,F12,F13,F15 }and{F08,F09 }.T h e energy differences between the states within one set are smallcompared to the energy differences between states of differentsets. B. Ground state of square platelets Using random initial magnetizations in large samples with strong DMI will predominantly yield high-energy states.The reason is that the randomness of the magnetizationcauses the formation of many small skyrmions, which inturn stabilizes the high-energy state since skyrmions aretopologically protected. We constructed a coarser randomdistribution of magnetization by using uniformly magnetizedV oronoi domains in order to avoid the formation of smallskyrmions. Varying the size of the V oronoi domains yields amultitude of stable states with disperse energies. Besides usingthis coarse random initial magnetization, we also identiﬁedsome smart choices for the initial state in order to ﬁnd thelow-energy states in large samples. One can imagine thatcycloidal states discussed in Sec. IIIare good candidates as Parallel Diagonal Circular Voronoi Random n=4 n=4 n=2ltyp= 20nm ⇓⇓⇓⇓⇓ FIG. 9. An example of different types of initial magnetization and the resulting relaxed state in square platelets. low-energy states. We thus initialize the calculation from the cycloidal state parallel with an edge of the square platelet(parallel state ) or parallel with the diagonal ( diagonal state ). We also consider the radially symmetric cycloidal states(circular state ), which are actually skyrmionic. These different types of initial conﬁgurations and some typical results areshown in Fig. 9. After selecting the lowest-energy state for different side lengths land DMI strengths Dof the samples, we obtain the phase diagram shown in Fig. 10. The ground state is always, as already suspected, a parallel, diagonal, or circular state. In mostcases, the ground state is a parallel state, which is very similarto the cycloidal states in an inﬁnite strip. Consequently, thephase diagram shares general trends with the phase diagram ofthe magnetic state in an inﬁnitely long strip shown in Fig. 3(c). However, there are two important differences. First, the groundstate in large samples is circular (skyrmionic) in the vicinity ofthe critical DMI strength D c. Second, there are regions in the phase diagram where the ground state is diagonal. This can beexplained by pointing out that, in comparison with the parallelstate, the period of a diagonal state in one of these regions iscloser to the period of the cycloid in an inﬁnite ﬁlm. We end this discussion by mentioning that the skyrmion (or double-wall skyrmion) in the circular ground state aroundthe critical DMI strength D cis deformed to a rounded square 50 100 150 200 Side length l(nm)3.54.04.55.05.56.0D(mJ/m2) 01234567891011 n FIG. 10. The ground states of a square l×lplatelet. Different cycloidal states are represented by n(as deﬁned in Sec. III A ). The states are diagonal inside the red borders, circular inside the magenta border, and parallel elsewhere. The stepwise character of the delimiting lines is a side effect of the ﬁnite resolution of the phasediagram (2 nm ×0.2m J/m 2). 214405-6CYCLOIDAL VERSUS SKYRMIONIC STATES IN . . . PHYSICAL REVIEW B 93, 214405 (2016) in large platelets. This gives the state a cycloidal character in both directions of the sample symmetry, and the periodsof the cycloids are maximized. For large ﬁlms, the energycontribution of the relatively small rounded corners becomesnegligible. This explains why the circular state is the groundstate in large platelets with a DMI strength close to D c. V . CONCLUSIONS We have investigated in depth the magnetic phase diagram of thin strips with perpendicular magnetic anisotropy and inthe presence of Dzyaloshinskii-Moriya interaction (DMI). Wehave started the analysis by showing how the cycloidal states insuch mesoscopic strips can be calculated analytically, and howconﬁnement promotes hysteretic effects and excited magneticstates. We further resort to micromagnetic simulations toshow that numerical results agree very well with the analyticmodel for a nonstretched cycloidal state. On the other hand, astretched cycloidal state is shown to buckle in the numericalexperiments, and will deform drastically in order to minimizethe energy. To address further the conﬁnement effects on the magnetic state in chiral mesoscopic magnets, we reported the detailedphase diagram for square platelets. We show that the excitedmagnetic states in square samples with a weak DMI ( D< D c=4√AK eff/π) consist of well-deﬁned skyrmions and domain walls. We ﬁnd that the energy of a domain wallacross the sample is lower than the energy of a skyrmion independently of the sample size, and that stable states withincreasing energy sequentially comprise one added wall orskyrmion, all of which interact repulsively to form a stableconﬁguration. In the case of a strong DMI ( D>D c), the phase diagram is very complex. Besides the known skyrmion-skyrmion andskyrmion-edge interactions, we point out the interaction of thedomain wall with sample edges (connecting the adjacent or thefacing edges of the sample), while interacting with the presentskyrmion(s) as well. As a general rule, the cycloidal stateswith domain walls parallel to the sample edge have the lowestenergy, followed by the cycloidal states with diagonal domains,and then those comprising skyrmions. 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PhysRevLett.116.186601.pdf	Enhancement of Thermally Injected Spin Current through an Antiferromagnetic Insulator Weiwei Lin,1,Kai Chen,2Shufeng Zhang,2and C. L. Chien1,† 1Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA 2Department of Physics, University of Arizona, Tucson, Arizona 85721, USA (Received 15 October 2015; revised manuscript received 8 February 2016; published 5 May 2016) We report a large enhancement of thermally injected spin current in normal metal (NM)/antiferromagnet (AF)/yttrium iron garnet (YIG), where a thin AF insulating layer of NiO or CoO can enhance the spincurrent from YIG to a NM by up to a factor of 10. The spin current enhancement in NM =AF=YIG, with a pronounced maximum near the Néel temperature of the thin AF layer, has been found to scale linearly with the spin-mixing conductance at the NM =YIG interface for NM ¼3d,4d, and 5dmetals. Calculations of spin current enhancement and spin mixing conductance are qualitatively consistent with the experimental results. DOI: 10.1103/PhysRevLett.116.186601 Pure spin current phenomena and devices are new advents in spin electronics [1,2]. A pure spin current has the unique attribute of delivering spin angular momentum without anet charge current thus with higher energy efficiency. A purespin current can be generated by several mechanisms, including the spin Hall effect [1–3], lateral spin valves [4,5], spin pumping [6,7], and longitudinal spin Seebeck effect (LSSE) [8,9]. The inverse spin Hall effect (ISHE) in a metal can detect a pure spin current by converting it into acharge current with a resultant charge accumulation [3,10] . Inevitably, a spin current decays as it traverses through a material on the scale of the spin diffusion length λ SF, which depends on the strength of the intrinsic spin orbit interactionand the quality of the material [5]. The transmission of a spin current across an interface between two materials, such as a ferromagnet and a nonmagnetic material, is further limited by the spin-mixing conductance at the interface [7]. The rapidly diminishing spin current has severely hampered its exploi-tation. It is highly desirable to explore ways to enhance purespin current. Pure spin current phenomena and devices have employed ferromagnetic (F) metals [3–5,10] , F insulators [8,9],a n d normal metals (NMs) [3,8–10], where the F magnetization sets the spin index of the spin current injected from the F material, light NM (e.g., Cu) and heavy NM (e.g., Pt), respectively, transmits and detects the spin current. Veryrecently, spin current exploration involves antiferromagnetic(AF) materials [11–18]. The employment of antiferromagnets in spintronic devices is particularly attractive for terahertz (THz) devices [19]. Recently, spin pumping experiment in Pt=YIG (where YIG ¼Y 3Fe5O12) shows enhanced spin transport through an intervening AF NiO layer betweenYIG and Pt at room temperature [13,14] . It was suggested that the spin transport through the AF insulators is related to AF magnons and spin fluctuations [13,14] , where the AF spins, strongly coupled to the precessing YIG magnetization,transport the spin current [13,14] .H o w e v e r ,t h u sf a r ,s p i ntransport through AF insulators has only employed ferro- magnetic resonance measurements (FMR) at the GHz fre-quency range [11,13 –15,18] , which is far less than the characteristic frequencies (up to 1 THz) of the AF NiO. The excitation and transmission of spin current, including amplification, through AF are far from clear. Coherent Néel dynamics employed to explain the spin transport and enhancement in such systems at room temperature [16] implies a more prevalent spin transport enhancement at T≪T N. With the absence of the key experimental results, the mechanism for the large spin current enhancementobserved at room temperature remains elusive [14]. The spin current amplification phenomena have thus far been observed in Pt=NiO=YIG and only at FMR frequencies. To unlock the underlying physics, it is essential to employ a different spin current injection method, different AF materials, and a variety of metals other than Pt, and perform measurements over a wide temperature range. The comprehensive experimental studies would constrain the theory that accounts for the results. In this Letter, we report enhanced spin current through AF (AF ¼NiO and CoO) generated by the LSSE in the layer structure of NM =AF=YIG over a wide temperature range. The pure spin current injected from YIG, trans- porting through the AF layer, is detected by the ISHE in various 3d,4d, and 5dNMs. In contrast to spin pumping, LSSE is a dc injection method without coherent resonance excitations at high frequencies. We show that the trans- mitted spin current detected in the NM has a maximum near theT Nof the AF layer of a specific thickness, indicating the dominant roles of magnons and spin fluctuation in the AF on the spin transport, rather than the collective AF ordering dynamics. Equally important, we also demonstrate in various NMs that the spin current enhancement scales linearly with the spin-mixing conductance at the NM =YIG interface. Theoretical calculations of the spin current enhancement and the spin mixing conductance in suchPRL 116, 186601 (2016) PHYSICAL REVIEW LETTERSweek ending 6 MAY 2016 0031-9007 =16=116(18) =186601(6) 186601-1 © 2016 American Physical Societylayer geometry are qualitatively consistent with the exper- imental results. NiO is a well-known AF insulator with a face-centered cubic rock salt structure and a bulk Néel temperature ofT N¼525K[20]. We used magnetron sputtering to fab- ricate polycrystalline multilayers onto polished polycrystal-line YIG substrates 0.5 mm thick via dc Ar sputtering forthe NMs, reactive (Ar þO 2) sputtering for NiO, and rf Ar sputtering for CoO at ambient temperature. The samples are denoted as Pt ð3Þ=NiOð1Þ=YIG, where the numbers in parentheses are the thickness in nm. The lateral sizes ofall the rectangular samples are 7mm × 2mm. As shown in Fig.1(a), the sample is thermally linked to a copper holder as a heat sink with its temperature measured by a thermocouple, while a heater is at the top of the sample surface with its temperature T NMmeasured via its electrical resistance. Between the heater and the heat sink, weestablished an out-of-plane temperature gradient ∇T, for which most of temperature drop occurs in the thicker YIGand injects a pure spin current J Sinto the multilayer. The direction of ∇Tdictates that of JS. A small magnetic field aligns the YIG magnetization along the short direction ofthe sample that sets the spin index σof the pure spin current. The ISHE in the NM generates an electric field inthe direction of σ×J Swith a voltage VISHEalong the long direction of the sample. In this open circuit dc measure- ment, there is no high frequency coherent excitations. The applied magnetic field, less than 100 mT in magnitude,only aligns the YIG magnetization and does not alterappreciably the AF ordering in NiO. The measured ISHE voltage V ISHE in NM =YIG and NM=NiO=YIG are shown in Figs. 1(b) and 1(c) as a function of the applied magnetic field. All the results havebeen obtained at T NM¼303K in samples of the same size and same ∇T¼10K=mm. In Fig. 1(b), the results of Ptð3Þ=YIG (red curve) are similar to those previously observed by spin pumping [8,9]. With 1 nm thick NiO inserted between Pt(3) and YIG, VISHEof Pt ð3Þ=Nið1Þ=YIG (blue curve) dramatically increases. The null result of VISHEin the Pt =NiO=SiOx=Si (black curve) shows that NiO itself does not generate any spin current at all. The large enhancement of VISHEdue to the insertion of NiO also occurs in Ta =NiO=YIG [Fig. 1(c)], but that the polarity of the enhanced VISHEin YIG =NiO=Ta is reversed due to its spin Hall angle of the opposite sign. In both cases, the inserted NiO layer greatly increases VISHEwhile preserving the spin index. The enhancement of pure spin current due to thepresence of the thin NiO spacer layer is clearly established. Since V ISHE is proportional to the separation Lof the voltage leads and the temperature gradient ΔT=t YIGwithin the YIG thickness tYIG, we use the normalized parameter S¼ðVISHE=LÞ=ðΔT=t YIGÞ, also known as the transverse thermopower, that allows comparison of results taken under different experimental conditions. We use the ratioSðt NiOÞ=Sð0Þ, where SðtNiOÞwith, and Sð0Þwithout, the presence of the NiO layer of thickness tNiO.A ss h o w ni n Fig. 2(a), both Pt =NiO=YIG and Ta =NiO=YIG, SðtNiOÞ= Sð0Þincreases sharply from 1, reaching a maximum at tNiO≈1nm before decreasing exponentially as shown in the inset of Fig. 2(a). The maximal value of SðtNiOÞ=Sð0Þfor Ta=NiO=YIG at tNiO≈1nm is higher, but its decay length λðTaÞNiO¼1.3nm is considerably smaller than λðPtÞNiO¼ 2.5nm for Pt =NiO=YIG. Similar behavior has also been observed in another AF insulator of CoO inserted betweenYIG and NM. Figure 2(b) shows in Ta =CoO=YIG, Sðt CoOÞ=Sð0Þhas a maximum at tCoO≈2nm. With the CoO results, we show that the spin current enhancement phenomenon is not exclusive to NiO. To illustrate the uniquefeature of the intervening AF layer, we have also inserted AlO xbetween YIG and Pt. As shown in Fig. 2(c), SðtAlOxÞ=Sð0Þin Pt ð3Þ=AlOxðtAlOxÞ=YIG at TPt¼303K exhibits the expected exponential decay, monotonicallydecreasing with a very short decay length of λðPtÞ AlOx ¼ 0.23nm, without enhancement at all. In Fig. 3(a), we show the temperature dependences of theSvalue from about 10 K to room temperature in Ptð3Þ=NiOðtNiOÞ=YIG for tNiO¼0,0 . 6 ,1 . 2 ,a n d2n m , highlighting the strong temperature dependence and FIG. 1. (a) Schematic of thermal spin transport measurement. Inverse spin Hall voltage Vas a function of the applied field Hin (b) Pt ð3Þ=YIG, Pt ð3Þ=NiOð1Þ=YIG, and Pt ð3Þ=NiOð1Þ=SiO x=Si, (c) Ta ð3Þ=YIG and Ta ð3Þ=NiOð1Þ=YIG. The temperature of the metal layer is about 303 K, and the out-of-plane temperature gradient cross the YIG is about 10K=mm. The number in the layered structure denotes thickness in nm.PRL 116, 186601 (2016) PHYSICAL REVIEW LETTERSweek ending 6 MAY 2016 186601-2sensitivity to the NiO layer thickness. Without the NiO layer, theSvalue of Pt ð3Þ=Y I G( l a b e l e da s0n m )i ss m a l l ,h a r d l y varying for TPtbetween 65 and 300 K. However, with the insertion of the NiO layer, the large Svalue of Pt =NiO=YIG acquires a very different temperature dependence exhibitinga well-defined broad peak. For t NiO¼0.6,1 . 2 ,a n d2n m , the peak temperature progressively increases, whereas the peak height changes sharply and nonmonotonically from4.3μV=Kt o 6μV=Ka n dt o 1.2μV=K, respectively. The spin current injected into the NM layer is J S¼ ftNMσNM=½ΘSHλNMtanhðtNM=2λNMÞ/C138gðVISHE=LÞ, where σNM,tNM,ΘSH,λNMare the conductivity, the thickness, the spin Hall angle, and the spin diffusion length of the NMlayer, respectively [21]. Then, we have J S¼ftNMσNM= ½ΘSHλNMtanhðtNM=2λNMÞ/C138gðSΔT=t YIGÞ. Since the injected pure spin current JSin NM is proportional to the parameter SðtNiOÞ, the ratio SðtNiOÞ=Sð0Þgives JSðtNiOÞ=JSð0Þfor a temperature gradient in YIG, the amplification of pure spin current due to the presence of NiO. The results in Fig. 3(b) ofJSðtNiOÞ=JSð0Þfor Pt ð3Þ=NiOðtNiOÞ=YIG appears sim- ilar to those shown in Fig. 3(a)because Sð0Þfor Pt =YIG without NiO varies little except at low temperatures. As shown in Fig. 3(b), the presence of the intervening NiO layer greatly enhances the spin current, up to a factor of11.6 for the 1.2 nm thick NiO. The enhancement of J Shas a well-defined peak at Tpeak, whose values of 142 K, 191 K, and 263 K depend stronglywith t NiO¼0.6nm, 1.2 nm, and 2 nm, respectively. As shown in Fig. 3(c), theTpeakincreases linearly with the tNiO astNiO<2nm. Similar behavior has been also observed recently in IrMn =Cu=NiFe by spin pumping [18]. The Tpeak is near the reduced intrinsic Néel temperature TNðtNiOÞof the isolated thin NiO layer due to finite size effects [22,23] . The value of TNðtNiOÞof NiO thin film can be estimated by the blocking temperature at which exchange bias of aferromagnetic layer exchange coupled to the NiO vanishes [24]. The blocking temperature is close to and usually slightly lower than T N[22]. As shown in the inset ofFig.3(d), the magnetic hysteresis loop of a NiO ð1Þ=Coð3Þ film shifts to a negative field at T¼90K after cooling from 330 K under a 0.5 T field, due to the exchange bias [24,25] . From the temperature dependence of the exchange bias field shown in Fig. 3(d), the blocking temperature of 1 nm thick NiO layer is around 170 K, which agrees with Ref. [25]. The Svalues in all cases, with or without NiO, as shown in Fig. 3(a), decrease towards zero as TPt approaches 0 K due to the lack of thermal excitations of magnons in YIG at low temperatures [26,27] . To address the physics of the observed behavior, we calculated the spin current transmission under an out-of- plane temperature gradient in NM =AF=F. In contrast to FIG. 3. Temperature dependences of (a) Sand (b) JSðtNiOÞ= JSð0Þin Pt ð3Þ=NiOðtNiOÞ=YIG for various NiO thicknesses tNiO. In (b), JSðtNiOÞ=JSð0ÞfortNiO≠0has a peak at the peak temperature Tpeak, whereas the dashed line denotes JSð0Þ= JSð0Þ¼1. (c) Peak temperature Tpeakas a function of tNiO. (d) Temperature dependence of exchange bias field in aNiOð1Þ=Coð3Þfilm. Inset shows the magnetic hysteresis loop of the NiO ð1Þ=Coð3Þfilm at T¼90K after the field cooling. FIG. 2. (a) Transverse thermopower SðtNiOÞnormalized by Sð0Þ, the transverse thermopower without NiO, of Pt ð3Þ=NiOðtNiOÞ=YIG and Tað3Þ=NiOðtNiOÞ=YIG as a function of the NiO thickness tNiOatTNM¼303K. Inset shows SðtNiOÞ=Sð0Þin the logarithmic scale as a function of tNiO. (b)SðtCoOÞ=Sð0Þas a function of CoO thickness tCoOin Ta ð3Þ=CoO ðtCoOÞ=YIG at TTa¼303K. (c) SðtAlOxÞ=Sð0Þas a function of AlO xthickness tAlOxin Pt ð3Þ=AlO xðtAlOxÞ=YIG at TPt¼303K. Inset shows SðtAlOxÞ=Sð0Þin the logarithmic scale as a function of tAlOx.PRL 116, 186601 (2016) PHYSICAL REVIEW LETTERSweek ending 6 MAY 2016 186601-3coherent zero-wave number magnons for spin pumping, the spatial dependent nonequilibrium thermal magnons have abroad spectrum distribution [27], and thus it is possible to transfer one F magnon to one AF magnon via interface exchange interaction. The spin currents in the NM forNM=F and NM =AF=F can be expressed as J NM=F S ¼κ∇Te−ðx=λNMÞ 1þGF/C18 1 GNM=Fþ1 GNM/C19 ð1Þ JNM=AF=F S ¼κ∇Te−ðx=λNMÞ 1þGF/C18 1 GNM=AFþ1 GNM/C191 cosh ðtAF λAFÞþδsinhðtAF λAFÞ; ð2Þ where κis the spin current coefficient due to the temper- ature gradient, GNM,GF,GAF,GNM=F, and GNM=AFare the spin current conductance of bulk NM, bulk F, bulk AF, the NM =F interface, and the NM =AF interface, respec- tively. λNMis the spin diffusion length of the NM, λAF the magnon decay length of the AF, tAFthe AF thickness, and δ¼GAF½ð1=GFÞþð1=GAF=FÞ/C138. Then, the spin current ratio is JNM=AF=F S JNM=F S¼/C20 1þða−1ÞGNM GNM=AFþGNM/C211 cosh ðtAF λAFÞþδsinhðtAF λAFÞ; ð3Þ where a¼ðGNM=AFÞ=ðGNM=FÞ. In the spin wave approxi- mation, GNM=AFscales as ðJNM=AFÞ2ðT=T NÞ2andGNM=F scales as ðJNM=FÞ2ðT=T CÞ3=2[27], where JNM=AFand JNM=Fares−dexchange constants of the NM =AF inter- face and the NM =F interface, respectively, and TCCurie temperature of the F. Then, ðGNM=AFÞ=ðGNM=FÞ¼ bðJNM=AF=JNM=FÞ2ðT=T NÞ2×ðT=T CÞ−3=2(bis a numeri- cal constant of order of 1). The TNof NiO is lower than the TCof YIG. Thus, GNM=AFincreases much faster with T.A t the high temperature, GNM=AFis larger than GNM=F. This is primary due to enhanced AF magnons or enhanced spin fluctuation in NiO. Thus, the significant enhancement of spin current occurs near TNin agreement with experiments. The enhancement of spin current decreases but still pronounced in a large temperature range above TNin the absence of long range AF ordering. This indicates the prominent roles of spin fluctuation and short range spin c o r r e l a t i o ni nA Fo ns p i nt r a n s p o r t [17,18,28] .N o t et h a t short range spin correlation in AF still exists at temperaturesmuch higher than T N, as revealed by neutron scattering [29]. Above the TN, the magnons whose wavelength is shorter than the spin correlation length remain. The spin correlationlength of NiO is ξ¼l½ðT=T NÞ−1/C138−0.64, where l¼0.5nm [29]. The number of magnons participating the spin transport decreases due to the loss of the magnons whose wavelengthis longer than the spin correlation length. Thus, the spin current enhancement is maximum near the TN. For thicker NiO layers with tNiO>3.5nm, there is no appreciable enhancement because of the drastic decay of spin current. Overall, the largest enhancement occurs near tNiO≈1nm, as shown in Fig. 2(a). As discussed above, the observed enhancement of spin transport in the NM =AF=YIG is attributed to the large spin conductance at both the NM =AF and the AF =YIG inter- faces. We experimentally explore this essential feature in spin current enhancement in NM =NiO=YIG with various NMs in addition to Pt, the only metal studied to date. Wedetermine J SðtNiOÞ=JSð0ÞwithtNiO¼1nmat TNM¼303K for3d(Cr, Mn), 4d(Pd), and 5d(Ta, W, Pt, Au) metals. The spin-mixing conductances at the NM =YIG interfaces have been measured from the FMR linewidth in spin pumping [30–33].F i g u r e 4(a) shows our measured values of JSð1Þ=JSð0Þfor various NMs vs the spin-mixing conductance G↑↓ NM=YIGreported by spin pumping at room temperature in NM=YIG[30–33].TheG↑↓ NM=YIGvaluesinunitsof 1018m−2in ascending order for NM ¼Cr, Pd, W, Au, Pt, Mn, Ta are 0.83, 1.1, 1.2, 2.7, 3.9, 4.5, and 5.4, respectively [30–33].M o s t remarkably, JSð1Þ=JSð0Þisproportional toG↑↓ NM=YIG,i . e . , FIG. 4. (a) Our measured spin current enhancement in NMð3Þ=NiOð1Þ=YIG for various NM at TNM¼303K vs the measured spin-mixing conductance in NM =YIG (from Refs. [30–33]). (b) Calculated spin current enhancement in NM=NiOð1Þ=YIG as a function of the spin mixing conductance at the NM =YIG interface, in the case of JNM=NiO¼2JNM=YIG, λNiO¼2.5nm,TC¼560K,TN¼190K, and T¼300K.PRL 116, 186601 (2016) PHYSICAL REVIEW LETTERSweek ending 6 MAY 2016 186601-4JSð1Þ=JSð0Þ¼CG↑↓ NM=YIG, where C¼8.5×10−19m2.I n NM=YIG, the spin current transmission is dictated by the spin-mixing conductance at the NM =YIG. With the insertion of a NiO layer in NM =NiO=YIG, the spin fluctuation in the thin AF NiO layer amplifies the spin current transmission. The ratio of spin current in the NM between the NM =F and the NM =AF=F can be calculated from Eq. (3).A s shown in Fig. 4(b), the calculated spin current enhancement in NM =NiOð1Þ=YIG increases with the spin-mixing con- ductance in NM =YIG. The calculated spin current enhance- ment is consistent with the linear correlation observedexperimentally at room temperature. It may be noted that spin Hall angle Θ SH, an important property of the NM in converting pure spin current, doesnotplay a role in spin current enhancement. In particular, while Cr, Ta, W, and Pt have large Θ SHvalues [30,32] , only Ta and Pt have large JSð1Þ=JSð0Þabove 3, whereas those of Cr and W have small values of less than 1, i.e.,only reduction. The linear behavior of J Sð1Þ=JSð0Þwith G↑↓ NM=YIGprovides an essential criterion for selecting materials for large spin current enhancement. We also notethat such spin current enhancement is observed with 1 nmthick paramagnetic NiO at room temperature (above T N), and thus not related to coherent AF ordering dynamics. In conclusion, we have observed spin current enhance- ment through AF by dc thermal injection in a broad temperature range in various metals. The spin conductance can be enhanced in NM =AF=YIG due to the magnons and spin fluctuation in the thin AF layer. The degree of enhance-ment increases with the spin-mixing conductance at the NM=YIG interface. These key results provide the criteria for selecting materials with effective spin current enhancement. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Science, underAward Grant No. DE-SC0009390. W. L. was supported in part by C-SPIN, one of six centers of STARnet, a SRC program sponsored by MARCO and DARPA. K. C. andS. Z. were supported by National Science Foundation underGrant No. ECCS-1404542. W. L. thanks Ssu-Yen Huang from National Taiwan University for fruitful discussions. Note added. —Recently, Ref. [34] accounts for the AF insulator thickness dependence of spin current in NM = AF=F by diffusive thermal AF magnons. 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PhysRevB.101.054422.pdf	PHYSICAL REVIEW B 101, 054422 (2020) Electric ﬁeld control of magnetic susceptibility in laminate magnetostrictive/piezoelectric composites: Phase-ﬁeld simulation and theoretical model Liwei D. Geng ,1Yongke Yan,2Shashank Priya,2and Yu U. Wang1,* 1Department of Materials Science and Engineering, Michigan Technological University, Houghton, Michigan 49931, USA 2Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, USA (Received 29 August 2019; accepted 30 January 2020; published 18 February 2020) Electric ﬁeld control of magnetic susceptibility in laminate magnetostrictive/piezoelectric composites promises to create a new class of magnetoelectric elements, voltage tunable inductors. To elucidate the underly-ing mechanism of electric ﬁeld modulated magnetic susceptibility at the domain level, phase-ﬁeld modeling,and computer simulation are employed to systematically study the laminate magnetoelectric composites ofTerfenol-D and PZT, where polycrystalline Terfenol-D can provide a giant magnetoelectric coupling thatis important for high-tunability voltage tunable inductors. The simulations focus on the interplay betweenmagnetocrystalline anisotropy and stress-induced anisotropy that is induced by electric ﬁeld and reveal threeregimes of magnetic susceptibility behaviors: constant (regime I), fast-varying (regime II), and reciprocal linear(regime III), where regimes II and III can give rise to a high tunability. Such three regimes are attributedto different magnetization distribution and evolution mechanisms that are modulated by the stress-inducedanisotropy. To further characterize the electric ﬁeld control of magnetic susceptibility behaviors, a generaltheoretical model of laminate magnetoelectric (ME) composites based on polycrystalline magnetostrictivematerials is developed, which reproduces the three regimes of susceptibility behaviors for polycrystallineTerfenol-D material. The general theoretical model for this speciﬁc system can also be extended to other laminatepolycrystalline ME composites. DOI: 10.1103/PhysRevB.101.054422 I. INTRODUCTION Magnetoelectric (ME) coupling effect that arises from the cross-linking between magnetostrictive and piezoelectricproperties via the interface in multiferroic materials is offundamental and technical importance in creating novel elec-tronic and spintronic devices [ 1–5]. While the direct ME effect (the appearance of an electric polarization upon applyinga magnetic ﬁeld) has been extensively studied [ 6–14], the converse ME effect, especially the modulation of magneticsusceptibility or permeability by electric ﬁeld, was much lessinvestigated. In particular, multiferroic laminate compositesmade of magnetostrictive and piezoelectric phases take ad-vantage of strain-mediated interaction that allows control ofmagnetic susceptibility or permeability through electric ﬁeldor voltage via the converse ME coupling effect, and promiseto develop a new class of ME components, voltage tunableinductors (VTIs) [ 15–18], which is very important for enhanc- ing the efﬁciency of power electronics as well as reducingthe number of passives by actively changing the magnitudeof inductance. Thus, an understanding of electric ﬁeld controlof magnetic susceptibility in laminate ME composites at thedomain level and even further, from a phenomenologicaltheoretical model, is required. Bichurin, Petrov, and Srinivasan have developed var- ious phenomenological theoretical models to elucidate *Author to whom correspondence should be addressed: wangyu@mtu.eduthe ME coupling mechanisms in laminate magnetostric- tive/piezoelectric composites of various forms and condi-tions [ 11–14]. While most of those theoretical works were related to the direct ME effect, the theoretical model forelectric ﬁeld control of magnetic susceptibility in laminateME composites is still lacking. Through the strain-mediatedinteraction, applying an electric ﬁeld on the piezoelectric layercan exert a stress on the magnetostrictive layer, and thus, it isactually the interplay between magnetocrystalline anisotropyand stress-induced anisotropy of the magnetostrictive layerthat determines the susceptibility behaviors modulated byelectric ﬁeld, which means that susceptibility behaviors ex-hibited by different magnetostrictive materials must be dif-ferent due to their different magnetocrystalline anisotropies.However, since polycrystalline magnetostrictive materials aremore commonly used in ME composites, whose grain orien-tations are randomly distributed, the effect of individual mag-netocrystalline anisotropy could be “averaged” in a certainmanner so as to exhibit a more general effective magneticanisotropy eventually, which allows the existence of a generalbehavior of magnetic susceptibility modulated by electricﬁeld. Thus, we will develop a theoretical model to describethis general behavior. Among various magnetostrictive materials, Terfenol-D, a rare-earth-iron alloy, is the most widely used giant mag-netostrictive alloy that exhibits a very high magnetostric-tion (over 1000 ppm) as well as a large magnetocrystallineanisotropy, and the composites made of Terfenol-D andpiezoelectric materials can manifest a strong ME couplingeffect. Such a giant ME coupling as well as the underlying 2469-9950/2020/101(5)/054422(11) 054422-1 ©2020 American Physical SocietyGENG, Y AN, PRIY A, AND W ANG PHYSICAL REVIEW B 101, 054422 (2020) mechanism was discussed in detail by Nan et al. i nar e - view on Terfenol-D-based ME composites [ 2]. Since a giant ME coupling usually leads to a large tunability of magneticsusceptibility that is an essential factor of VTIs, we adoptpolycrystalline Terfenol-D as the magnetostrictive materialfor our laminate ME composite system in this work to sys-tematically study the electric ﬁeld control of magnetic sus-ceptibility behaviors by employing domain-level phase-ﬁeldmodeling and computer simulation. The simulations focuson the interplay between magnetocrystalline anisotropy andstress-induced anisotropy. Such a stress-induced anisotropy ismainly introduced by two types of strain, namely, electric ﬁeldtunable strain and internal bias strain. The simulations revealthree regimes of susceptibility, which are further characterizedby a theoretical model developed for laminate ME compositesystems and good agreement is obtained. Such a generaltheoretical model can be also extended to other laminate MEcomposites, for example, similar susceptibility or permeabil-ity behaviors were also observed in our recent experimen-tal studies of coﬁred [ 19] and bonded [ 18] polycrystalline ferrite/Pb(Mg 1/3Nb2/3)O3-PbTiO 3(PMN-PT) ME compos- ites. In fact, because of the polycrystalline microstructure ofTerfenol-D, this work exhibits a different magnetization evo-lution mechanism and thus a different magnetic susceptibilitybehavior from the previous amorphous Metglas/PZT modelsystem [ 20] which does not exhibit the polycrystalline natureand hence signiﬁcantly reduces the complexity to develop a theoretical model. II. PHASE-FIELD MODELING In ME composites consisting of magnetostrictive and ferroelectric phases, the evolution of magnetic and electricdomains is coupled with their secondary elastic domainsdue to magnetostriction and electrostriction. The phase-ﬁeldME composite model integrates two stand-alone phase-ﬁeldmodels for magnetostrictive materials [ 21] and ferroelectric materials [ 22] that are developed in our previous works, which treats domain processes, grain structures, and phase mor-phology in two-phase polycrystalline composites. This modelallows the strain-mediated domain-level coupling betweenmagnetization and polarization, and the electrical conductivityin magnetostrictive phase is also considered. While this sec-tion will only brieﬂy describe this ME composite model, moredetailed description can be found in our previous publications[20,23]. The state of a ME composite can be described by ﬁelds of magnetization M(r), polarization P(r), and free charge density ρ(r). The total system free energy under exter- nally applied magnetic ﬁeld H exand electric ﬁeld Eexis [21,22] F=/integraldisplay [(1−η)fM(RijMj)+ηfE(RijPj)+βM\|∇M\|2+βE\|∇P\|2−μ0Hex·M−Eex·P]d3r +/integraldisplayd3k (2π)3/bracketleftbiggμ0 2\|n·˜M\|2+1 2ε0/vextendsingle/vextendsingle/vextendsingle/vextendsingle˜ρ k−in·˜P/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +1 2Kijkl˜εij˜ε∗ kl/bracketrightbigg , (1) where fM(RijMj) and fE(RijPj) are the local free-energy density of magnetostrictive and ferroelectric phases, respec-tively. Both M(r) and P(r) are deﬁned in a global coor- dinate system of the composite. The operations R ijMjand RijPjin the functions fM(RijMj) and fE(RijPj) transform M(r) and P(r) from the global system to the local crystal- lographic system in each grain. Figure 1shows the grain structure of the two-phase ME composite used in our sim-ulations, where the two-phase morphology of the compos-ite is characterized by a phase-ﬁeld variable η(r) that dis- tinguishes magnetostrictive grains ( η=0) and ferroelectricgrains ( η=1). In the local crystallographic system, f M(M)i s formulated as the magnetocrystalline anisotropy energy [ 24]: fM(M)=K1/parenleftbig m2 1m2 2+m2 2m2 3+m2 3m2 1/parenrightbig +K2m2 1m2 2m2 3,(2) where m=M/Mis the magnetization direction, K1and K2are the magnetic anisotropy constants, and fE(P)i s formulated by the Landau-Ginzburg-Devonshire (LGD)polynomial energy [ 25]: fE(P)=α1/parenleftbig P2 1+P2 2+P2 3/parenrightbig +α11/parenleftbig P4 1+P4 2+P4 3/parenrightbig +α12/parenleftbig P2 1P2 2+P2 2P2 3+P2 3P2 1/parenrightbig +α111/parenleftbig P6 1+P6 2+P6 3/parenrightbig +α112/bracketleftbig P4 1/parenleftbig P2 2+P2 3/parenrightbig +P4 2/parenleftbig P2 3+P2 1/parenrightbig +P4 3/parenleftbig P2 1+P2 2/parenrightbig/bracketrightbig +α123P2 1P2 2P2 3, (3) where α’s are the expansion coefﬁcients. The two gradient terms in Eq. ( 1) characterize the energy contributions from the magnetization gradient (exchange energy) andpolarization gradient, respectively. The k-space integral terms characterize the domain conﬁguration-dependent energies ofthe long-range magnetostatic, electrostatic, and elastostaticinteractions, where ˜M(k),˜P(k), ˜ρ(k), and ˜ε(k) are the Fouriertransforms of the respective ﬁeld variables M(r),P(r), ρ(r), and ε(r),K ijkl=Cijkl−nmCijm n/Omega1npCklpqnq,/Omega1ik= (Cijklnjnl)−1,Cijklis the elastic stiffness tensor, and n=k/k. The spontaneous strain εresults from magnetostriction and electrostriction, εij=λijklmkml+QijklPkPl, where λijkland Qijkl are magnetostrictive and electrostrictive coefﬁcient tensors, respectively. 054422-2ELECTRIC FIELD CONTROL OF MAGNETIC … PHYSICAL REVIEW B 101, 054422 (2020) FIG. 1. Grain structure and phase morphology of laminate ME composite. Blue color represents magnetostrictive phase and red color represents piezoelectric phase. The domain evolution described by magnetization M(r,t) and polarization P(r,t) are, respectively, governed by the Landau-Lifshitz-Gilbert equation [ 26] and the time-dependent Ginzburg-Landau equation [ 27]: ∂M(r,t) ∂t=γM×δF δM(r,t)+αM×/bracketleftbigg M×δF δM(r,t)/bracketrightbigg , (4) ∂P(r,t) ∂t=−LδF δP(r,t), (5) where γandαare gyromagnetic ratio and damping parameter, respectively, for magnetization evolution, and Lis kinetic coefﬁcient for polarization evolution. The evolution of freecharge density ﬁeld ρ(r,t) is governed by charge conservation and microscopic Ohm’s law [ 28]: ∂ρ(r,t) ∂t=− ∇· j(r,t), (6) ji=σikEk, (7) where σik(r) is the electrical conductivity distribution in the ME composite. The local electric ﬁeld E(r) that is generated by free charge density distribution ρ(r) and polarization dis- tribution P(r) as well as externally applied electric ﬁeld Eexis given by E(r)=Eex−1 ε0/integraldisplayd3k (2π)3/bracketleftbigg n·˜P(k)+i˜ρ(k) k/bracketrightbigg neik·r.(8) In this computational study of laminate ME composites, TbxDy1−xFe2(Terfenol-D) [ 24] and Pb(Zr 1−xTix)O3(PZT) [25] are considered as model magnetostrictive and ferroelec- tric materials. The material-speciﬁc input parameters used inthis work are magnetocrystalline anisotropy constants K 1= −6×104J/m3,K2=− 2×105J/m3, saturation magnetiza- tion Ms=8×105A/m, magnetostrictive constants λ111= 1.64×10−3,λ100=1×10−4for Terfenol-D [ 24]; andLGD coefﬁcients α1=− 2.67×107m/F,α11=− 1.43× 107m5/C2F,α12=1.57×107m5/C2F,α111=1.34× 108m9/C4F,α112=1.17×109m9/C4F,α123=− 4.77× 109m9/C4F, electrostrictive constants Q11=0.0966 m4/C2, Q12=− 0.046 m4/C2,Q44=0.0819 m4/C2for Pb(Zr 0.5 Ti0.5)O3[25,29]. While these material-speciﬁc parameters are used in this work, the obtained simulation results reﬂectgeneral behaviors of laminate ME composite made ofpolycrystalline magnetostrictive and piezoelectric materials. III. SIMULATION RESULTS AND DISCUSSION In this computational study, we focus on the interplay between magnetocrystalline anisotropy and stress-inducedanisotropy on laminate Terfenol-D/PZT ME composites. Sucha stress-induced anisotropy is mainly introduced by two typesof strain, namely, electric ﬁeld tunable strain and internalbias strain. The electric ﬁeld tunable strain is induced byelectric ﬁeld. When the electric ﬁeld is applied perpendicu-lar to the PZT layer (i.e., along the zaxis), a compressive strain will be transferred to the magnetostrictive Terfenol-Dlayer and exert a compressive stress on it due to the straincoupling. Such a compressive stress can induce an effectiveuniaxial magnetic anisotropy on Terfenol-D layer, i.e., thestress-induced anisotropy, to control its magnetic properties.Unlike the electric ﬁeld tunable strain, the internal bias strainintroduced by speciﬁc means does not change with the tunableelectric ﬁeld, so the resultant stress-induced anisotropy isconstant and independent of the electric ﬁeld. Although internal bias strain can be introduced in different ways, two types of bias strain commonly exist in ME compos-ites, namely, the preexisting strain and the thermal mismatchstrain. The preexisting strain can be caused by electricallypoling the piezoelectric layer, while the thermal mismatchstrain that usually exists in coﬁred laminate composites arisesfrom different thermal expansion coefﬁcients of the layersof different phases. These two types of bias strain producesimilar effect on laminate ME composite due to the plane-stress condition, that is, the resultant internal stress is zeroalong the layer thickness direction ( zaxis) and thus only the in-plane bias stress is produced in the laminate composite. Inthis work, the poling-generated bias strain in the PZT layeris treated explicitly through polarization domain evolutionduring simulated poling procedure, while the remaining partof the bias strain is treated in terms of an in-plane bias strainεin the Terfenol-D layer. To systematically study the strain effect on magnetic susceptibility and its tunability, variousmagnitudes of the bias strain εare considered in the computer simulations and for each ε, tunable electric ﬁeld within a speciﬁc range is applied. To simulate the magnetic susceptibility of Terfenol-D/PZT ME composites, the same following procedure is adopted inall simulation cases. Firstly, the in-plane ( x-yplane) bias strain εof a given value is exerted on the PZT layer. Secondly, a strong out-of-plane ( z-axis) electric ﬁeld is applied on the PZT layer for a full electrical poling, after which the ferroelectricdomains are fully relaxed upon removal of the poling ﬁeld.Thirdly, a series of electric ﬁeld within a certain tuning rangeis applied perpendicular to the PZT layer to induce a tunablestress on Terfenol-D layer via strain-coupling effect. Finally, 054422-3GENG, Y AN, PRIY A, AND W ANG PHYSICAL REVIEW B 101, 054422 (2020) FIG. 2. In-plane stress σin Terfenol-D layer induced by bias strain εin PZT layer, and resultant domain structures of PZT and Terfenol-D layers at points A and B, which correspond to zero bias strain ( ε=0) and zero induced stress ( σ=0), respectively. Black and white arrows represent polarization and magnetization vectors, respectively. To distinguish the composite structure, polarization domain structure in PZT layer is visualized by color map with red, green, blue (RGB) components proportional to Px,Py,Pz, and magnetic domain structure in Terfenol-D layer is visualized by color contour proportional to Mz. a small magnetic ﬁeld /Delta1Halong the yaxis is applied on the Terfenol-D layer to induce a small magnetization response/Delta1M, from which the simulated magnetic susceptibility can be obtained by χ=/Delta1M//Delta1H. In this systematical study, a series of internal bias strain εfrom−2.4×10 −3to 1.5 ×10−3is exerted in the PZT layer, which will induce a stress in the Terfenol-D layer. Sinceelectrical poling of the PZT layer also exerts a stress on theTerfenol-D layer, the total stress that determines initial mag-netic domain structures and initial susceptibility magnitudesshould include both contributions. Figure 2shows the total induced stress σin the Terfenol-D layer as a function of the bias strain εin the PZT layer, which, as expected, exhibits a linear interdependence within a certain range of the biasstrain. It is worth noting that there are two points of particularinterest in Fig. 2: point A with zero bias strain ( ε=0) and point B with zero induced stress ( σ=0). Point A of zero bias strain corresponds to a ME composite where the unpoled PZTlayer is bonded with the Terfenol-D layer and subsequentlypoled; thus, a compressive stress is generated in the Terfenol-D layer caused by the poling procedure, while point B of zeroinduced stress corresponds to a ME composite where the PZTlayer is poled before bonded with the Terfenol-D layer, thusno internal stress is generated. According to the simulationresults, the poling generates a bias strain of −1.8×10 −3in the PZT layer, and thus use of a bias strain ε=− 1.8×10−3 in the Terfenol-D layer can completely cancel the in-plane stress induced by electrical poling of the PZT layer, leadingto point B in Fig. 2. Domain structures of ME composites corresponding to points A and B are also shown in Fig. 2. As expected for the electrically poled PZT layer, almost allpolarization vectors point toward the poling direction (positivezaxis) corresponding to both points A and B. However, the magnetic domain structure manifests a big difference:corresponding to point A, almost all magnetization vectorsare oriented along positive or negative zaxis forming 180 ◦ domain walls due to the stress-induced uniaxial anisotropy, while corresponding to point B, the magnetization vectorsare oriented with signiﬁcant components within the x-yplane due to lack of stress-induced anisotropy and large demagne-tization ﬁeld. Starting from those initial domain structures, once a tunable electric ﬁeld is applied, the domain evolutiontakes place and hence the magnetic susceptibility is tuned.Figure 3and Fig. 4show the respective simulated results for case A and case B (starting from points A and B, respectively).Since the domain structure of poled PZT layer does notchange signiﬁcantly with the tuning electric ﬁeld, hereafteronly the magnetic domain structures of the Terfenol-D layerare visualized for clarity. Under zero internal bias strain ( ε=0), the electrical poling of the PZT layer induces an in-plane compressive stress(−36 MPa) on the Terfenol-D layer, which is equivalent to a large magnetic uniaxial anisotropy with the easy axisperpendicular to the layer (i.e., along the zaxis). Such a large stress-induced anisotropy leads to the out-of-plane align-ment of magnetization vectors and the formation of multipleantiparallel domains in the Terfenol-D layer, as shown inFig. 3(b). To simulate the susceptibility, a small magnetic ﬁeld/Delta1H=3 Oe along the yaxis is applied which induces a small magnetization response /Delta1Min the same direction. The simulated susceptibility is χ=/Delta1M//Delta1H=8.2f o rt h e ME composite at E=0 under zero bias strain. Since the small magnetic ﬁeld is almost perpendicular to all magneticdomains and domain walls, the obtained susceptibility ismainly contributed by the domain-rotation process rather thanthe domain-wall motion process. When an electric ﬁeld isapplied, the susceptibility is tuned due to the change in thestress-induced anisotropy. Figure 3(a) shows the simulated susceptibility as a function of the tunable electric ﬁeld. Asthe electric ﬁeld is tuned from E=− 15 to 40 kV /cm, the susceptibility decreases from χ=10.0 to 4.8. Actually, such a decrease in susceptibility arises from the increasedcompressive stress ( σ=− 28→− 57 MPa) and the resultant increased stress-induced anisotropy. Larger anisotropy makesthe magnetization rotation more difﬁcult and thus reduces thesusceptibility. However, larger electric ﬁeld or stress-inducedanisotropy does not change the domain structure signiﬁcantly,and only better align the magnetization vectors along thezaxis, as shown in Fig. 3(c) atE=40 kV /cm. Note that the susceptibility almost changes linearly with electric ﬁeld 054422-4ELECTRIC FIELD CONTROL OF MAGNETIC … PHYSICAL REVIEW B 101, 054422 (2020) FIG. 3. (a) Simulated magnetic susceptibility χas a function of electric ﬁeld E(or induced stress σ) for Terfenol-D/PZT ME composites under zero internal bias strain ( ε=0). Red line represents linear ﬁtting curve with slope dχ/dE=− 0.087 cm /kV. Magnetic domain structures of Terfenol-D layer under (b) E=0k V/cm and (c) E=40 kV /cm. Black arrows represent magnetization vectors. Magnetic domain structures are visualized by color map with RGB components proportional to Mx,My,Mz. in this range of tuning electric ﬁeld with a slope of dχ/dE= −0.087 cm /kV as shown in Fig. 3(a). As will be discussed later, such a nearly linear χ-Erelation with a negative slope dχ/dEis characteristic of the magnetic susceptibility behav- ior in regime III. ME composites under zero internal bias stress ( σ=0 achieved with internal bias strain ε=− 1.8×10−3as dis- cussed above) manifest quite different susceptibility and do-main behaviors from the above case. Due to the absenceof stress-induced anisotropy, the demagnetization ﬁeld effectprefers the magnetization vectors to stay in the x-yplane, as shown in Fig. 4(b). Since the demagnetization ﬁeld is not uniform due to varying domain structures and uneven layerinterfaces, the intrinsic magnetocrystalline anisotropy of thegrains competes against the inhomogeneous demagnetizationﬁeld and thus makes the in-plane magnetization alignmentimperfect, and some magnetization vectors orient partiallyout of plane. At zero electric ﬁeld, the susceptibility is χ= 2.8, and such a small value actually results from the mostly in-plane domain structure. Figure 4(a) shows the simulated susceptibility as a function of tunable electric ﬁeld. As theelectric ﬁeld is tuned from E=− 20 to 40 kV /cm, the suscep-tibility increases from 2.52 to 4.80, with an average slope of dχ/dE=0.037 cm /kV, which is positive in contrast to the case of zero bias strain shown in Fig. 3. In this case, increasing the electric ﬁeld leads to an increased compressive stress(σ=7→− 12 MPa) and thus an increased stress-induced anisotropy in the Terfenol-D layer, which results in progres-sively more out-of-plane magnetization alignment to enhancethe domain-rotation process and gradually increased suscepti-bility. Figure 4(c) shows the corresponding domain structure atE=40 kV /cm with more out-of-plane magnetizations. A closer inspection of the χ-Erelation shown in Fig. 4(a)reveals that the simulated susceptibility indeed falls into two differ-ent regimes: regime I for E<10 kV /cm (or σ>−3M P a ) , where the magnetizations mostly stay in the x-yplane and further reducing the electric ﬁeld does not signiﬁcantly changethe domain structure and the susceptibility; and regime II forE>10 kV /cm (or σ<−3 MPa), where further increasing the electric ﬁeld gives rise to more out-of-plane magnetiza-tions and thus effectively increases the susceptibility. Whileboth regime I and II are characterized by positive slopedχ/dEin contrast to the negative slope dχ/dEin regime III, the susceptibility in regime I is relatively insensitive to the FIG. 4. (a) Simulated magnetic susceptibility χas a function of electric ﬁeld E(or induced stress σ) for Terfenol-D/PZT ME composites under zero internal bias stress ( σ=0). Red line represents linear ﬁtting curve with slope dχ/dE=0.037 cm /kV. Blue dashed lines represent the linear ﬁtting in regimes I and II. Magnetic domain structure of Terfenol-D layer under (b) E=0k V/cm and (c) E=40 kV /cm. 054422-5GENG, Y AN, PRIY A, AND W ANG PHYSICAL REVIEW B 101, 054422 (2020) FIG. 5. (a) Simulated susceptibility χatE=0 as a function of bias strain εfrom−2.4t o1.5×10−3. Three susceptibility regimes (I, II, and III) are divided by two critical bias strain values, εc1≈− 1.6×10−3andεc2≈− 0.6×10−3. (b) The linear ﬁtting slope dχ/dEas a function of bias strain εin the three regimes. tuning electric ﬁeld (for instance, the χ-Eslope in regime II is about 6 times that in regime I). Further increasing electricﬁeld beyond regime II will induce greater compressive stressand greater out-of-plane anisotropy, resulting in decreasedsusceptibility, entering regime III as discussed above. While different domain structures and susceptibility be- haviors are demonstrated for the two speciﬁc cases shown inFig. 3and Fig. 4, to investigate the whole-range susceptibility and tunability behaviors of the ME composite, more caseswith different internal bias strain magnitudes are systemati-cally simulated. For each case with a given bias strain, tunableelectric ﬁeld is applied to simulate the magnetic susceptibilityand its tunability. Figure 5(a) shows the simulated suscepti- bility at E=0 as a function of bias strain εwithin the range of−2.4t o1.5×10 −3. Note that the magnetic susceptibility reaches its maximum at ε=− 0.6×10−3, while deviation from such a critical bias strain leads to continuous decreasein susceptibility. According to the domain structures andsusceptibility behaviors discussed above for the two speciﬁccases shown in Fig. 3and Fig. 4, three regimes can be deﬁned for the simulated susceptibility behavior and eachregime is dominated by a different mechanism. Regime I andregime II are divided by ε c1≈− 1.6×10−3corresponding to E=10 kV /cm or σ=− 3 MPa in Fig. 4(a), while regime II and regime III are divided by εc2≈− 0.6×10−3corre- sponding to the magnetic susceptibility maximum, as shownin Fig. 5(a). In regime I, almost all the magnetization vectors in the Terfenol-D layer lie in the x-yplane, and decreasing the bias strain does not increase the in-plane magnetization align-ment, leading to small susceptibility and small tunability. Inregime II, increasing the bias strain increases the compres-sive bias stress, leading to more out-of-plane magnetizationalignment and increased susceptibility. It is worth noting thatregime II is quite narrow, in which domain structures aresensitive to the bias strain, internal stress, and tuning electricﬁeld, leading to fast-increasing susceptibility and hence largetunability. In regime III, almost all the magnetization vectorsare aligned out of plane, and increasing the bias strain andcompressive stress makes magnetization rotation more difﬁ- cult, leading to decreased susceptibility. As mentioned above, the induced stress exerted on the Terfenol-D layer produces a uniaxial anisotropy along thezaxis, which controls the magnetic domain structure and susceptibility. Figure 6shows three representative magnetic domain structures under different bias strain to illustrate thestress-induced domain evolution process. Figure 6(a) corre- sponds to a tension stress exerted on the Terfenol-D layer,where most magnetization vectors lie in the x-yplane due to the stress-induced negative uniaxial anisotropy. Figure 6(b) illustrates the domain structure caused by a compressive stressand the resultant positive uniaxial anisotropy, where antipar-allel domains start to form with most magnetization vectorsaligned out of plane, corresponding to the maximum magneticsusceptibility at ε=− 0.6×10 −3shown in Fig. 5(a). Figure 6(c) shows the domain structure under a large compressive stress, where almost all magnetization vectors are alreadyaligned out of plane, and thus further increasing the stress-induced anisotropy does not signiﬁcantly change the domainstructure. According to Fig. 5(a), large tunability can be pre- dicted to occur in regimes II and III rather than regime I.Figure 5(b) shows the χ-Eslopes to characterize the tunability as a function of bias strain ε. As expected, two tunability peaks emerge: one at ε=− 1.2×10 −3in regime II with a positive slope of 0.120 kV /cm, while the other at ε=0 in regime III with a negative slope of −0.087 kV /cm. It is worth noting that the latter in regime III just corresponds to the aforementionedcase with zero bias strain shown in Fig. 3, while the former in regime II exhibits an even greater tunability. To reveal moredetails of the former case with ε=− 1.2×10 −3,F i g . 7shows the simulated susceptibility and the corresponding domainstructures. At zero electric ﬁeld, the susceptibility is χ=5.0. Figure 7(a) shows the simulated susceptibility as a function of tunable electric ﬁeld. As the electric ﬁeld increases fromE=− 20 to 40 kV /cm, the compressive stress is tuned from σ=− 5t o−22 MPa, which falls in regime II and exhibits the 054422-6ELECTRIC FIELD CONTROL OF MAGNETIC … PHYSICAL REVIEW B 101, 054422 (2020) FIG. 6. Magnetic domain structures of Terfenol-D layer under representative bias strain of (a) ε=− 2.4×10−3,( b )−0.6×10−3,a n d (c) 1.5×10−3. largest tunability. Within this tuning range, the susceptibility increases from 3.2 to 9.6 with a positive slope of dχ/dE= 0.120 cm /kV, which is much larger than any other case, and such a large tunability is attributed to the stress-sensitivedomain structures. When the stress-induced anisotropy onthe Terfenol-D layer is continuously increased, more out-of-plane magnetization alignment is induced that enhancesthe domain-rotation process, as shown in Figs. 7(b) and 7(c), effectively increasing the susceptibility and eventually producing a large tunability. Although the above-simulatedresults and discussions are based on the ME composites thatutilize Terfenol-D as the magnetostrictive inductor material,the general behaviors of susceptibility and tunability revealedby this study are also applicable to ME inductors made ofother polycrystalline magnetostrictive materials. IV . THEORETICAL MODEL To further characterize the simulated susceptibility behav- iors of the laminate Terfenol-D/PZT ME composites dis-cussed in Sec. III, a general theoretical model is developed based on the underlying domain-level mechanisms revealedin the above computer simulations. Although a few modelshave been proposed to study the stress-dependent magnetic susceptibility [ 22–24], they mainly focus on amorphous mag- netostrictive materials in traditional inductors, thus a theo-retical model treating polycrystalline magnetostrictive mate-rials in VTIs is still lacking. In this section, we propose atheoretical model that takes into account the magnetocrys-talline anisotropy in randomly oriented polycrystal grainsand describes the interplay between effective magnetocrys-talline anisotropy and stress-induced anisotropy as well asthe resultant susceptibility behaviors of Terfenol-D/PZT MEcomposites over the three regimes. It also provides a generalunderstanding of the stress-modulated susceptibility behav-iors exhibited by VTIs built from different polycrystallinemagnetostrictive materials. In this theoretical model, the magnetization direction is described by M=M s(sinθcosϕ,sinθsinϕ,cosθ), as illustrated in Fig. 8(a). For polycrystalline magnetostric- tive materials, a unit vector p=(sinθ0cosϕ0,sinθ0sinϕ0, cosθ0) characterizes the local magnetization easy axis that is closest to the magnetization direction m=M/Ms= (sinθcosϕ,sinθsinϕ,cosθ) in a grain. The magnetocrys- talline anisotropy energy in Eq. ( 2) can be approximately expanded around the easy axis pin a form of uniaxial FIG. 7. (a) Simulated magnetic susceptibility χas a function of electric ﬁeld E(or induced stress σ) for Terfenol-D/PZT ME composites under internal bias strain ε=− 1.2×10−3. Red line represents linear ﬁtting curve with slope dχ/dE=0.120 cm /kV. Magnetic domain structures of Terfenol-D layer under (b) E=0k V/cm and (c) E=40 kV /cm. 054422-7GENG, Y AN, PRIY A, AND W ANG PHYSICAL REVIEW B 101, 054422 (2020) FIG. 8. (a) Schematic illustration of laminate ME composite and coordinate system. Calculated (b) susceptibility χand (c) χ−1as a function of stress σbased on the theoretical model for polycrystalline magnetostrictive materials. anisotropy fM=K0[1−(m·p)2], where K0=−2 3K1−2 9K2 (K0=8.44×104J/m3for Terfenol-D). The effective mag- netocrystalline anisotropy in the polycrystal will be ob-tained by averaging over θ 0andϕ0for randomly distributed grain orientations. In addition to the effective magnetocrys-talline anisotropy, the magnetic domain structure and sus-ceptibility in a laminate ME composite system are also con-trolled by stress-induced anisotropy and shape anisotropy.The stress-induced anisotropy is characterized by an energyterm K σ=−3 2λs[σxx(m2 x−1 3)+σyy(m2 y−1 3)], where λs= 3 5λ111+2 5λ100is saturation magnetostriction constant ( λs= 1.0×10−3for Terfenol-D), and σxxandσyyare the in-plane stress components from both internal bias stress and tunablestress induced by electric ﬁeld. The shape anisotropy is de-scribed by demagnetization energy term in the magnetostric-tive layer, which depends on the magnetic domain structureand thus its magnitude is usually signiﬁcantly smaller thanK d0=0.5μ0M2 s[26]. It is the competition between these three anisotropy terms that determines the susceptibility be-havior. As revealed in above computer simulations, in regimeI under large tensile stress where a negative K σdominates, the magnetization vector lies inside the x-yplane ( θ→π/2); on the other hand, in regime III under large compressive stresswhere a positive K σdominates, the magnetization vector lies out of the x-yplane along the stress-induced easy zaxis (θ→ 0o rπ). Between regime I and III, there exists a transition regime II, whose range is determined by the inhomogeneousdomain structures and the resultant demagnetization energyK das well as the effective magnetocrystalline anisotropy, which usually leads to a sharp increase in susceptibility as thecompressive stress increases and thus yields a large tunability. Based on above simplifying assumptions, the total free energy under induced in-plane stress σand small external magnetic ﬁeld /Delta1Happlied along the yaxis is given by F=K 0[1−(m·p)2]+Kdm2 z−3 2λs/bracketleftbigg σxx/parenleftbigg m2 x−1 3/parenrightbigg +σyy/parenleftbigg m2 y−1 3/parenrightbigg/bracketrightbigg −μ0Ms/Delta1Hm y. (9) The second term is the demagnetization energy, where Kd is usually much smaller than the maximum value Kd0= 0.5μ0M2 sas discussed above. For a magnetostrictive layerof in-plane dimensions much greater than its thickness, the in-plane components of induced stress are equiaxial σxx= σyy=σ. As such, Eq. ( 9) can be reformulated as F=K0[1−(sinθcosϕsinθ0cosϕ0+sinθsinϕsinθ0sinϕ0 +cosθcosθ0)2]+(Kd−Kσ)cos2θ+1 3Kσ −μ0Ms/Delta1Hsinθsinϕ, (10) where Kσ=− 3λsσ/2 is the stress-induced anisotropy (note thatσ< 0 for compressive stress). In regime I under large tensile stress, a negative Kσdomi- nates so that the magnetization vector lies inside the x-yplane (θ=π/2), and Eq. ( 10) becomes FI=K0[1−sin2θ0cos2(ϕ−ϕ0)]+1 3Kσ−μ0Ms/Delta1Hsinϕ. (11) Since the magnetization vector stays close to the directionof easy axis p,ϕ−ϕ 0=/Delta1ϕ is a small angle. Solving ∂FI/∂/Delta1ϕ =0 gives magnetization response /Delta1Mto magnetic ﬁeld/Delta1H, yielding the magnetic susceptibility χI(θ0,ϕ0)= dM/dHas χI(θ0,ϕ0)=μ0M2 scos2ϕ0 2K0sin2θ0, (12) which depends on the direction of the easy axis pas speciﬁed byθ0andϕ0in each grain. Upon orientation average over θ0∈(/Psi1,π−/Psi1) and ϕ0∈(−π,π ) for randomly oriented /angbracketleft111/angbracketrighteasy axes in polycrystalline Terfenol-D, where /Psi1= sin−1(1/√ 3), the effective susceptibility in regime I becomes χI=/angbracketleftχI(θ0,ϕ0)/angbracketright=μ0M2 s 4KI, (13) where /angbracketleftcos2ϕ0/angbracketright=1/2, and KI=K0/angbracketleftsin2θ0/angbracketright=K0[1+ sin 2/Psi1/(π−2/Psi1)]/2=0.75K0=6.3×104J/m3is the effective magnetocrystalline anisotropy in regime I. Equation(13) shows that the effective susceptibility in regime I is a constant that depends only on the effective magnetocrystallineanisotropy K I. In regime III under large compressive stress, a positive Kσdominates so that the magnetization vector lies out of thex-yplane along the stress-induced easy zaxis (θ=0o r π, which are equivalent with respect to the magnetic ﬁeld /Delta1Happlied along the yaxis; for convenience, θ=0i s considered). While ϕis undeﬁned in such a “ground state” of 054422-8ELECTRIC FIELD CONTROL OF MAGNETIC … PHYSICAL REVIEW B 101, 054422 (2020) FIG. 9. (a) Simulated magnetic susceptibility χas a function of induced stress σ. Three susceptibility regimes (I, II, and III) are deﬁned by two critical induced stress values, σc1≈− 3M P aa n d σc2≈− 20 MPa. (b) Plot of χ−1as a function of induced stress σ. Red line represents linear ﬁtting in regime III: χ−1 III=− 0.0035σ+0.011. regime III, application of magnetic ﬁeld /Delta1Halong the yaxis induces magnetization response /Delta1Min the same direction, thusθis a small angle and ϕ=π/2 can be assumed under /Delta1H, and Eq. ( 10) becomes FIII=K0[1−(sinθsinθ0sinϕ0+cosθcosθ0)2] +(Kd−Kσ)cos2θ+1 3Kσ−μ0Ms/Delta1Hsinθ.(14) Solving ∂FIII/∂θ=0 gives magnetization response /Delta1Mto magnetic ﬁeld /Delta1H, yielding the magnetic susceptibility χIII(θ0,ϕ0)=dM/dHas χIII(θ0,ϕ0)=μ0M2 s 2K0(cos2θ0−sin2θ0sin2ϕ0)−2(Kd−Kσ), (15) which depends on the direction of the easy axis pas spec- iﬁed by θ0andϕ0in each grain. Upon orientation average overθ0∈(0,/Theta1) and ϕ0∈(−π,π ) for randomly oriented /angbracketleft111/angbracketrighteasy axes in polycrystalline Terfenol-D, where /Theta1= sin−1(√ 2/√ 3), the effective susceptibility in Regime III be- comes χIII=/angbracketleftχIII(θ0,ϕ0)/angbracketright=μ0M2 s 2(KIII−Kd+Kσ), (16) where /angbracketleftcos2θ0/angbracketright=1/2+sin 2/Theta1/4/Theta1=0.75,/angbracketleftsin2θ0/angbracketright=1/2− sin 2/Theta1/4/Theta1=0.25,/angbracketleftsin2ϕ0/angbracketright=1/2, and KIII=K0(/angbracketleftcos2θ0/angbracketright− /angbracketleftsin2θ0/angbracketright/angbracketleftsin2ϕ0/angbracketright)=0.62K0=5.2×104J/m3is the effective magnetocrystalline anisotropy in regime III. It is worth notingthat Eq. ( 16) reveals a linear relationship between the recip- rocal susceptibility χ −1 IIIand stress σin regime III, which is more evident upon reformulation: χ−1 III=2(KIII−Kd) μ0M2s−3λs μ0M2sσ. (17) Therefore, it is expected that a large tunability with a desir- able linear dependence on tuning stress (and voltage) can beachieved in regime III if the stress is tuned in a large range. To compare the theoretical predictions and the simulation results, the data points of the phase-ﬁeld simulated suscep-tibility χand its reciprocal χ −1are plotted as a function of stress σover the three regimes in Fig. 9. Figure 9(a) shows χI≈2 in regime I, which according to Eq. ( 13) cor- responds to an effective magnetocrystalline anisotropy KI≈ 1.0×105J/m3. This value is about 60% greater than the theoretical value KI=6.3×104J/m3. This discrepancy is attributed to three major factors: ( 1) the uniaxial crystallo- graphic anisotropy approximation adopted in the theoretical model is accurate only for magnetization direction close to the easy axis while in regimes I and III the deviation angle of local magnetization vectors from easy axes in some grains may be not small due to random grain orientations; ( 2)t h e orientation averaging procedure performed in Eqs. ( 13) and (16) treats θ0andϕ0as uniformly distributed in respective ranges while such simpliﬁcation does not accurately describe random grain orientation distribution with uniform probabil-ity; and ( 3) the phase-ﬁeld simulation considers only a ﬁnite number of grains in the Terfenol-D layer which is insufﬁcient to sample random grain orientation. Despite these theoretical simpliﬁcations and computational limitations, the obtained values of the effective magnetocrystalline anisotropy K Iin regime I are of the same order of magnitude, indicating that the theoretical model captures the main mechanisms of magnetic susceptibility behavior. Figure 9(b) shows a linear relationship of χ−1∼σin regime III with a slope −0.0035 MPa−1, in good agreement with the theoretical slope −3λs/μ0M2 s=− 0.0038 MPa−1according to Eq. ( 17). Also, according to Eq. ( 17) and Fig. 9(b),2 (KIII−Kd)/μ0M2 s= 0.011 gives KIII−Kd=4.4×103J/m3.U s i n g KIII=5.2× 104J/m3yields Kd=4.76×104J/m3, which is signiﬁ- cantly smaller than Kd0=0.5μ0M2 s=4.02×105J/m3as expected. It is worth noting that such a linear relationship χ−1∼σwas also observed in amorphous inductor materials when the stress is large enough to orient magnetization along the stress-induced easy axis [ 30–33]. Regime II is a smooth transition between regimes I and III, which is associated with the gradual reorientation ofmagnetization from in-plane alignment under tensile stress inregime I to out-of-plane alignment under compressive stressin regime III caused by the change in the stress. Figure 9(a) 054422-9GENG, Y AN, PRIY A, AND W ANG PHYSICAL REVIEW B 101, 054422 (2020) plots the three regimes of the simulated susceptibility that are deﬁned by two critical stress values, σc1andσc2. In contrast to regimes I and III, regime II is narrower and the susceptibilitychanges signiﬁcantly, thus providing higher tunability perunit stress (or voltage). During the gradual reorientation ofmagnetization caused by the change in stress in regime II, themagnetic domain structure also gradually evolves as shownin Fig. 6, leading to change in the demagnetization energy K d. To capture such an effect and reproduce the transition regime II, a Gaussian distribution of the demagnetizationenergy K daround Kd=4.76×104J/m3with a standard deviation σK=9×103J/m3is assumed in the theoretical model. The susceptibility χandχ−1as a function of stress σ calculated from this theoretical model are plotted in Figs. 8(b) and8(c), respectively, which show good agreement with the simulated results shown in Fig. 9. In regime I the susceptibility approaches a constant value χI≈2, and in regime III the linear relationship between χ−1andσis reproduced with the theoretical slope consistent with the simulated one. The effects of the domain-level mechanisms responsible for the three susceptibility regimes as revealed by the phase-ﬁeld simulations are captured by above theoretical modelof the polycrystalline Terfenol-D/PZT ME composites. Thismodel predicts three susceptibility regimes, which is differentfrom the previous model of amorphous Metglas/PZT MEcomposites [ 20] where only two susceptibility regimes exist due to different magnetic anisotropy. While the direct andconverse ME effects in laminate Terfenol-D/PZT compos-ites have been extensively studied [ 34–37], the electric ﬁeld control of magnetic susceptibility in those Terfenol-D/PZTcomposites was much less investigated; nevertheless, the ex-istence of regime III was experimentally evidenced by theobserved stress-induced permeability behavior of Terfenol-Dmagnetostrictive transducers [ 38,39]. It is worth noting that similar susceptibility behaviors exist in other polycrystallinemagnetostrictive inductor materials. For example, regime IIIis experimentally observed in VTIs made of coﬁred polycrys-talline ferrite/PMN-PT ME composites [ 19], while regimes I and II are observed in bonded polycrystalline ferrite/PMN-PT ME composites [ 18,40]. Among the three regimes, large tunability can be achieved in either regime II or regime III. Inregime III, the reciprocal susceptibility χ −1changes linearly with stress σ, thus a large tunability can be obtained if the tun- ing range in stress is wide, which requires a large piezoelectricconstant d 31and/or large dielectric breakdown strength for piezoelectric layer (a large magnetostriction coefﬁcient λs for magnetostrictive layer is also desired). In regime II, the susceptibility changes signiﬁcantly within a narrow tuningrange, thus a high tunability could be achieved without therequirements for large piezoelectric constant, high dielectricbreakdown strength, or large magnetostriction coefﬁcient.It is worth noting that the tunability in regime II exhibitsa strong dependency on the magnetocrystalline anisotropy, that is, a smaller magnetocrystalline anisotropy leads to anarrower regime II and a higher tunability per unit stress. Asreported in our recent work for polycrystalline ferrite/PMN-PT VTIs [ 18,40], reducing the magnetocrystalline anisotropy effectively enhances the tunability in regime II and a colossaltunability up to 750% is obtained in VTIs with an almostdiminished magnetocrystalline anisotropy. V . CONCLUSION Electric ﬁeld control of magnetic susceptibility in laminate magnetostrictive/piezoelectric composites is of great impor-tance in creating a new class of magnetoelectric elements,voltage tunable inductors. To elucidate the mechanism ofelectric ﬁeld modulated magnetic susceptibility at the domainlevel, phase-ﬁeld modeling and computer simulation are em-ployed to systematically study the laminate Terfenol-D/PZTmagnetoelectric composites. Polycrystalline Terfenol-D canprovide a giant magnetoelectric coupling because of its largemagnetostriction, which is important for high-tunability volt-age tunable inductors. The simulations mainly focus on theinterplay between magnetocrystalline anisotropy and stress-induced anisotropy. The stress-induced anisotropy is intro-duced by two types of strain, namely, electric ﬁeld tunablestrain and internal bias strain. The simulations reveal threeregimes of magnetic susceptibility: Regime I corresponds toa nearly constant susceptibility, regime III exhibits a lin-ear behavior of the reciprocal susceptibility, while regimeII is a fast transition between regimes I and III. Actually,such three regimes are attributed to different magnetizationdistribution and evolution mechanisms that are modulated bythe stress-induced anisotropy. The simulated results indicatethat high tunability can occur in regime II or III. To furthercharacterize the electric ﬁeld control of magnetic suscepti-bility behaviors, a general theoretical model of laminate MEcomposites based on polycrystalline magnetostrictive materi-als is developed, which reproduces the three regimes of sus-ceptibility behaviors for polycrystalline Terfenol-D material,in good agreement with the simulation ﬁndings. Such threeregimes commonly exist in polycrystalline magnetostrictiveinductor materials, such as in ferrite/PMN-PT ME composites[18,19,40]. This general theoretical model presented here for Terfenol-D model system can be also extended to otherlaminate magnetoelectric composites based on polycrystallinemagnetostrictive materials. ACKNOWLEDGMENTS Financial support from DARPA MATRIX Program is ac- knowledged. The parallel computer simulations were per-formed on XSEDE supercomputers. [1] J. Ma, J. Hu, Z. Li, and C.-W. Nan, Adv. Mater. 23,1062 (2011 ). [2] C.-W. Nan, M. I. Bichurin, S. Dong, D. Viehland, and G. Srinivasan, J. Appl. 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PhysRevLett.112.187203.pdf	ac Current Generation in Chiral Magnetic Insulators and Skyrmion Motion induced by the Spin Seebeck Effect Shi-Zeng Lin, Cristian D. Batista, Charles Reichhardt, and Avadh Saxena Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Received 12 August 2013; revised manuscript received 23 February 2014; published 9 May 2014) We show that a temperature gradient induces an ac electric current in multiferroic insulators when the sample is embedded in a circuit. We also show that a thermal gradient can be used to move magneticSkyrmions in insulating chiral magnets: the induced magnon flow from the hot to the cold region drives theSkyrmions in the opposite direction via a magnonic spin transfer torque. Both results are combined tocompute the effect of Skyrmion motion on the ac current generation and demonstrate that Skyrmions ininsulators are a promising route for spin caloritronics applications. DOI: 10.1103/PhysRevLett.112.187203 PACS numbers: 75.70.Kw, 75.10.Hk, 75.70.Ak, 75.85.+t The possibility of inducing magnon currents with thermal gradients in magnetically ordered insulators is attracting considerable attention for its promise in spin caloritronics[1]. For finite magnetoelectric coupling, the magnon current creates oscillations in the electric polarization, implying thatan oscillating current is generated if the sample is embeddedin an electric circuit. This simple observation suggests that the ability to induce spin currents with thermal gradients can also be exploited for other applications. The recent discov-ery of stable spin textures called Skyrmions in magneticinsulators without inversion symmetry (e.g., Cu 2OSeO 3) [2,3] posed a challenge for moving and manipulating these textures in insulating materials. Because Cu 2OSeO 3exhib- its a finite magnetoelectric coupling, it is important to study how Skyrmions respond to a thermal gradient and the effectof Skyrmion motion on the electric polarization. Skyrmions were predicted to be stable in chiral magnets without inversion symmetry [4–6]. The triangular Skyrmion crystal was first observed experimentally in metallic bulk [7]and thin films [8,9]. In a metal, conduction electrons interact with the local magnetic moments anddrive the Skyrmion via spin transfer torque [10]. The threshold current to make Skyrmions mobile is low[11–13]; thus, Skyrmions are very promising for applica- tions in spintronics. Dissipation due to the conduction electrons is absent in insulators, where the dominant dissipation mechanism is theweak Gilbert damping of the spin precession. Therefore,Skyrmions that emerge in insulating materials are attractivefor applications that require low energy dissipation. Further- more, the finite magnetoelectric coupling that is intrinsic to Skyrmion textures opens the possibility of manipulating thesemagnetic structures with external electric fields [2,14,15] . In analogy with the thermal control of magnetic domain wallsin insulators [1,16 –20], we propose to drive Skyrmions with am a g n o nc u r r e n ti n d u c e db yat h e r m a lg r a d i e n t . We study the dynamics of Skyrmions in insulating chiral magnets under the presence of a thermal gradient and findthat Skyrmions move from the cold to the hot region due tothe magnonic spin transfer torque generated by thetemperature gradient (spin Seebeck effect). We also present a phenomenological description of Skyrmion dynamicswith a monochromatic magnon current. We then computethe dynamics of electric polarization Pthat produces an electric ac current density dP=dtwhen the insulator is embedded in a closed circuit. The role of the Skyrmionmotion on the ac current generation is clarified. Finally,when the temperature in the hot region is high enough toinduce a local paramagnetic state, we find that Skyrmionsare continuously generated by thermal fluctuations andthen diffuse into the cold region. We consider a thin film of insulating chiral magnet with thickness d(Fig. 1), which is described by the Hamiltonian [4–6,21,22] , H¼Z dr 2/C20Jex 2ð∇nÞ2þDn·ð∇×nÞ−Ha·n/C21 ;(1) where Jexis the exchange interaction, Dis the Dzyaloshinskii-Moriya (DM) interaction [23–25], which is generally present in magnets without inversion sym- metry, and nis a unit vector denoting the spin direction and r¼ðx; yÞ. The system is assumed to be uniform along the z direction for thin films. The external field Ha¼Haˆzis perpendicular to the film and its magnitude is fixed atH a¼0.6D2=Jexto stabilize the Skyrmion phase [22]. The spin dynamics is governed by the Landau-Lifshitz-Gilbertequation [26] ∂ tn¼−γn×ðHeffþ~HÞþα∂tn×n; (2) where Heff≡−δH=δn¼Jex∇2n−2D∇×nþHa,γ¼ a3=ðℏsÞ,ais the lattice constant and sthe magnitude of local spins. ~His the fluctuating magnetic field that introduces thermal fluctuations and, consequently, satisfiesthe local fluctuation-dissipation theorem: h~Hi¼ 0and h~H μðr;tÞ~Hνðr0;t0Þi ¼2kBTðrÞα dγδμ;νδðr0−rÞδðt0−tÞ; (3)PRL 112, 187203 (2014) PHYSICAL REVIEW LETTERSweek ending 9 MAY 2014 0031-9007 =14=112(18) =187203(5) 187203-1 © 2014 American Physical Societywhere TðrÞis the local temperature at r,kBis the Boltzmann constant, and μ;ν¼x,y,z. Here we have assumed a local equilibrium for magnons characterized byspatially dependent temperature TðxÞand that the Seebeck effect is driven by magnons [16,27] . Depending on the kinetics of the phonons and magnons, and on theircoupling, other effects may come into play such as the phonon drag spin Seebeck effect [28] and the nonlinear temperature profile along the sample [29,30] . We first study the Skyrmion dynamics in the presence of a finite temperature gradient: Tðx¼0Þ¼ 0and Tðx¼L xÞ¼ΔT. Note that ΔTis not high enough to induce a local paramagnetic state on the hot side, andto excite new Skyrmions thermally. First, we prepare a stationary Skyrmion in the ferromagnetic background state atTðxÞ¼ 0, and then turn on the temperature gradient. We calculate numerically [31,32] Skyrmion position r c¼Rdr2½n·ð∂xn×∂ynÞ/C138r=Rdr2½n·ð∂xn×∂ynÞ/C138, and its velocity v¼_rc. The corresponding trajectory is shown in Fig. 2(a). Surprisingly, the Skyrmion moves from the cold towards the hot region. The average velocity depends linearly on the temperature gradient as shown in Fig. 2(b). Similar behavior is obtained for many Skyrmions [33]. In magnetically ordered insulators, the heat current is carried both by magnons and phonons [18,34] . In the same way a finite temperature gradient induces an electron current in metals and produces the conventional Seebeck effect, it also produces a magnon current in magneticinsulators (magnon flow) that leads to the so-called spinSeebeck effect. This effect has been measured in metals [35], semiconductors [36], and insulators [37]. The magnon current carries a magnetic moment and interacts with theSkyrmion via the magnonic spin transfer torque. Since a Skyrmion is topologically equivalent to a magnetic bubble,we represent the Skyrmion as a bubble domain of down- ward spins in a spin up background to illustrate the basic mechanism [see Fig. 2(c)]. When the magnon passes through the Skyrmion, the magnetic bubble is displaced by one lattice constant in the opposite direction of themagnon current to conserve the magnetic moment, which explains the results in Fig. 2(a). Note that the net entropy still flows from the hot to the cold region, because the density of magnons is higher than the density of Skyrmions. The thermally excited magnon has all frequency com- ponents. We will start by considering the interaction between a monochromatic magnon current and aSkyrmion in the absence of noise ~H¼0[38]. The magnon decay produced by damping and the decay length can be obtained from the magnon dispersion. The initial distance between the Skyrmion and the magnon source is L x=2 to ensure that the magnon current is not fully damped out before interacting with the Skyrmion: Lx< 4ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ Jexγðωm−HaγÞp =ðωmαÞ≈1μm for typical parameters, where ωmis the magnon frequency. The Skyrmion oscil- lates with frequency ωmand drifts in the opposite direction to the magnon current [Fig. 3(a)]. The drift velocity increases with the amplitude Aof the ac driving field [see Fig. 3(b)], and it has a nonmonotonic dependence onωm. It reaches its maximum value when the wavelength of the magnon, 2π=kmwithkmgiven by ωm¼ΩðkmÞ,i so f the order of the Skyrmion size [Fig. 3(c)]. This effect could be used to measure the Skyrmion size. The magnon current is quickly damped for increasing Gilbert damping αand the FIG. 2 (color online). (a) Trajectory of the Skyrmion center of mass in the presence of a temperature gradient. Here,ΔT=ðL xdÞ¼ 0.003 andα¼0.1. (b) Average drift velocity as a function of temperature gradient. The curves are obtained byaveraging over 8 independent runs. (c) Schematic view of themagnonic spin transfer torque and the dynamics of the Skyrmion. FIG. 1 (color online). Schematic view of the motion of aSkyrmion in the presence of a temperature gradient. The temper-ature gradient excites a magnon current flowing from the hotto the cold region, which drives the Skyrmion opposite tothe magnon current direction. The Skyrmion motion changesthe electric polarization, which generates an ac current when theinsulating magnet is embedded in a closed circuit.PRL 112, 187203 (2014) PHYSICAL REVIEW LETTERSweek ending 9 MAY 2014 187203-2drift velocity decreases [Fig. 3(d)]. For thermally excited magnons, the oscillation of the Skyrmion velocity smearsout while the drift motion remains. To understand our numerical results we introduce a phenomenological description of the interaction between Skyrmions and a magnon current. For this purpose, we separate ninto a slow component n s, corresponding to the Skyrmion motion and a fast component ~n, induced by the magnon current n¼nsþ~n. By following the procedure in Ref. [19] and coarse-graining over ~n, Eq. (2)becomes ∂tns¼αns×∂tns−γns×ðJ∇2ns−2D∇×nsþHaÞ −∂μJμþΓ; (4) withΓ∝γDkmj~nj2being the contribution from the DM interaction. Here, the tensor Jμ¼γJex~n×∂μ~nis the spin current density (note that μ¼x,yis the spatial coordinate). The total spin is not conserved, ∂μJμ≠0, because of the presence of DM interaction and external magnetic fields. Γ can be neglected for typical parameters D≪Jex=a, and magnon wavelength much shorter than the Skyrmion size [ jΓj=j∂μJμj≈D=ðJexkmÞ≪1]. We note that jnsj¼1 andns×~n¼0in the linear approximation. In addition, ns×∂μ~n≃0because this term is linear in ~nimplying that it must vanish after coarse graining in time over the fastoscillations of ~n. Therefore, we can rewrite the magnon current as J μ¼Jμnswith Jμ¼γJexð~n×∂μ~nÞ·ns, and the magnon spin transfer torque is given by ∂μJμ≈ Jμ∂μns[19].Equation (4)is equivalent to the case of Skyrmion motion driven by a spin polarized current −Jμin metals. The magnonic spin transfer torque is always adiabatic because the magnetic moment of magnons is antiparallel to the local moment ns. TheΓterm cannot be neglected when the magnon wavelength becomes comparable to the Skyrmion size. In this case, Skyrmions get distorted by the interaction with magnons. This effect can be accounted for by adding a nonadiabatic spin transfer torque term to Eq.(4), ∂tns¼−γns×Heffþαns×∂tns þðvm·∇Þns−βns×ðvm·∇Þns: (5) The equation for drift motion is derived from Eq. (5) [39,40] by treating the Skyrmion as a rigid particle [41]: −ˆz×ðvmþvÞþηðαv−βvmÞ¼ 0; (6) where η¼ημ¼Rdr2ð∂μnsÞ2=ð4πÞis the form factor of a Skyrmion. The effective magnon velocity vmand the parameter βcan be estimated by fitting the Skyrmion trajectory obtained from simulations to Eq. (6). Note that the Skyrmion velocity has a component transverse to the temperature gradient as displayed in Figs. 1and2(a).T o linear order in α≪1, we obtain a Hall angle tan θH≡ v⊥=v∥¼−βη−ðηþβ2η3Þαfor the Skyrmion motion, where v⊥andv∥are the velocity components perpendicular and parallel to the flow direction of the magnon current. The inset of Fig. 3(d)shows the dependence of θHonα.B y fitting this curve, we obtain β≈0.4and η≈1.4, which indicates that the nonadiabatic spin transfer torque plays an important role in the interaction between the magnon current and the Skyrmion when 2π=kmis comparable to the Skyrmion size. We next study the electric polarization induced by the Skyrmion motion. For Cu 2OSeO 3, it was demonstrated experimentally that the mechanism for the generation of electric polarization is d-phybridization [2,14] , which arises from the interaction between a ligand (oxygen) ion and a transition metal (copper) ion with a single magnetic moment [42–44]. Because the electric polarization depends on the direction of the external magnetic field, we introduce a new coordinate frame with the zaxis parallel to Haand thexaxis along the ½¯101/C138direction. We consider the caseHa∥½110/C138. The electric polarization is given by Pdp¼ ðP0=2Þð−2nxny;n2z−n2x;2nynzÞ, where P0≈50μC=m2 for Cu 2OSeO 3[14]. In other potential realizations of Skyrmion lattices, electric polarization could also be induced by the mechanism known as the inverse DM (IDM) effect [45,46] :PIDM ¼P0½ˆex×ðn×∂xnÞþ ˆey× ðn×∂ynÞ/C138. If the sample is embedded in a circuit, the time dependent polarization leads to an ac electric current density Je¼∂tP. For the inverse DM mechanism, this current density can be expressed in terms of the magnon current Jμ,Je;IDMðωÞ¼Im½P0ω γ½ˆex×JxðωÞþ ˆey×JyðωÞ/C138/C138.0.0 0.5 1.0 1.5 2.0-0.008-0.0040.0000.0040.008 02468 1 0-0.002-0.0010.0000.0010.0020.003 0.03 0.06 0.09-0.0020.0000.0020.0041000 1020 1040 1060 1080 1100-0.03-0.02-0.010.000.010.020.03 0.04 0.06 0.08 0.10-0.72-0.68-0.64-0.60vx vy vx vy A[D2/Jex] vx vyVelocity [ D/] ωm[D2/Jex]vx vy αA=1.0 D2/Jex,ωm=1.0 D2/Jexα=0.03, A=1.0 D2/Jexα=0.03, ωm=1.0 D2/JexVelocity [ D/] t[Jex/(D2)]α=0.03, A=1.0 D2/Jex,ωm=1.0 D2/Jex (d) (c)(b) tanH=v⊥/v α(a) FIG. 3 (color online). (a) Oscillation of the Skyrmion velocity with a nonzero dc component in the presence of a magnon current. (b) —(d) Dependence of the Skyrmion velocity on the amplitude Aand frequency ωmof the ac magnetic field applied at the right edge of the sample x¼Lx, which acts as a source of magnon current, and on the Gilbert damping α. The inset in (d) is the Hall angle as a function of α. The line is a fit to tan θHin the main text.PRL 112, 187203 (2014) PHYSICAL REVIEW LETTERSweek ending 9 MAY 2014 187203-3The ac electric current induced by a monochromatic magnon current is shown in Fig. 4. The electric current has the frequency of the magnon current. The time averageof the induced current vanishes and the amplitude of the acelectric current density is about 5×10 3A=m2for typical parameters. For the d-phybridization mechanism, Pdpis finite for collinear spin textures and an ac electric current isstill induced in the absence of Skyrmions. The presence ofSkyrmions produces a small change in the electric current.In contrast, the inverse DM mechanism is absent forcollinear spin textures ( P IDM ¼0) and Skyrmions are then necessary to induce electric polarization. Because theSkyrmion contribution to spin fluctuations is much smallerthan that from magnons, the magnitude of the ac current forthe inverse DM mechanism is much smaller than that for thed-phybridization mechanism [see Fig. 4]. Finally, we increase the temperature in the hot region by ΔTsuch that the hot region is in the paramagnetic phase, where Skyrmions are created and destroyed dynamically bythermal fluctuations. Because of the presence of a temper- ature gradient, some of the created Skyrmions diffuse into the cold region and stabilize there. Skyrmions are driventowards the cold region by the newly created Skyrmions in the hot region because of an effective repulsion between them. Meanwhile, the magnon current drivesthe Skyrmions from the cold to the hot region. However, the repulsive interaction dominates and Skyrmions keep diffusing from the hot to the cold region, as observed inour simulations [33]. Skyrmions can also be driven by a temperature gradient in metallic magnets. In this case, the Skyrmion motioninduces an emergent electric field given by E¼ ℏn·ð∇n×∂ tnÞ=2e[10]; i.e., there is a Skyrmion Hall voltage induced by the spin Seebeck effect. The conven-tional Seebeck effect for electrons also produces a longi-tudinal and a Hall voltage in the presence of an externalmagnetic field. This electronic contribution is dominantbecause the electron carrier density is much higher than thedensity of Skyrmions. In inhomogeneous systems, there is a random pinning potential for Skyrmions. Then, a threshold temperaturegradient or a magnon current density is required to movethe Skyrmions. We can estimate the threshold temperature gradient by using the depinning current density measured for metallic magnets, expecting that the pinning potential is similar for insulators and metals. By using typical parameters and the results shown in Fig. 2, we estimate the velocity to be 0.1m=s for a temperature gradient ΔT=L x≈0.04K=nm and a film thickness of d≈10nm. The required current density to achieve a similar velocity in metallic magnets by a spin-polarized current is J≈109A=m2, i.e., much larger than the typical depinning current J≈106A=m2. Thus, the Skyrmion can be depinned at a temperature gradient larger than ΔT=L x≳4×10−5K=nm. We note that local thermal fluctuations are also helpful for Skyrmions to creep from the pinning sites. To summarize, we have studied the generation of an ac current and the motion of Skyrmions in insulating chiral magnets subject to a temperature gradient. The Skyrmions move from the cold to the hot region because of the magnonic spin transfer torque. When the temperature of the hot region is high enough to induce a local paramagneticstate, Skyrmions are created by thermal fluctuations and diffuse into the cold region. The generation of the ac current by a thermal gradient does not depend on the presence of Skyrmions for the magnetoelectric coupling that arises from the d-phybridization mechanism. However, for the inverse Dzyaloshinskii-Moriya mechanism, an ac current is induced only when Skyrmions are present to render the spin texture noncollinear. Our results indicate that Skyrmions in insulating chiral magnets are promising for spin calori- tronics applications. Computer resources for numerical calculations were supported by the Institutional Computing Program at LANL. This work was carried out under the auspices of the NNSA of the U.S. DOE at LANL under Contract No. DE-AC52-06NA25396, and was supported by the U.S.Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. Note added in proof. —After completion of the present work, we become aware of the similar results on the motion of a Skyrmion in the presence of a temperature gradient in Ref. [50]. However the usual diffusion of Skyrmions and the induced electric current due to the magnetoelectric coupling are not discussed in Ref. [50]. The non-adiabatic magnonic spin transfer torque proposed in Eq. (5)is derived systematically in Ref. [51]. [1] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012) . [2] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science 336, 198 (2012) . [3] T. Adams, A. Chacon, M. Wagner, A. Bauer, G. Brandl, B. Pedersen, H. Berger, P. Lemmens, and C. Pfleiderer,Phys. Rev. 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PhysRevB.93.134427.pdf	PHYSICAL REVIEW B 93, 134427 (2016) Homodyne-detected ferromagnetic resonance of in-plane magnetized nanocontacts: Composite spin-wave resonances and their excitation mechanism Masoumeh Fazlali, Mykola Dvornik, Ezio Iacocca,Philipp D ¨urrenfeld, and Mohammad Haidar Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden Johan ˚Akerman Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden; Materials Physics, School of ICT, KTH Royal Institute of Technology, 164 40 Kista, Sweden; and NanOsc AB, Electrum 205, 164 40 Kista, Sweden Randy K. Dumas† Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden and NanOsc AB, Electrum 205, 164 40 Kista, Sweden (Received 25 January 2016; published 25 April 2016) This work provides a detailed investigation of the measured in-plane ﬁeld-swept homodyne-detected ferromagnetic resonance (FMR) spectra of an extended Co/Cu/NiFe pseudo-spin-valve stack using a nanocontact(NC) geometry. The magnetodynamics are generated by a pulse-modulated microwave current, and the resultingrectiﬁed dc mixing voltage, which appears across the NC at resonance, is detected using a lock-in ampliﬁer. Mostnotably, we ﬁnd that the measured spectra of the NiFe layer are composite in nature and highly asymmetric,consistent with the broadband excitation of multiple modes. Additionally, the data must be ﬁt with two Lorentzianfunctions in order to extract a reasonable value for the Gilbert damping of the NiFe. Aided by micromagneticsimulations, we conclude that (i) for in-plane ﬁelds the rf Oersted ﬁeld in the vicinity of the NC plays thedominant role in generating the observed spectra, (ii) in addition to the FMR mode, exchange-dominated spinwaves are also generated, and (iii) the NC diameter sets the mean wave vector of the exchange-dominated spinwave, in good agreement with the dispersion relation. DOI: 10.1103/PhysRevB.93.134427 I. INTRODUCTION Spin-torque ferromagnetic resonance (ST-FMR) [ 1–9]i sa powerful and versatile tool that enables the characterization ofmagnetodynamics on the nanoscale. Unlike more conventionalFMR measurement techniques, where a resonant cavity orwaveguide is used to generate rf magnetic excitation ﬁelds, theresonant precession in an ST-FMR measurement is assumed tobe primarily a result of the ST from a spin-polarized current.However, ST-FMR represents a speciﬁc type of a more generalhomodyne-detection scheme where the excitation mechanismitself can originate from a variety of physical mechanisms apartfrom or in combination with ST, including, e.g., rf Oerstedﬁelds [ 10] and electric ﬁelds [ 11]. While homodyne-detected FMR studies on magnetic tun- nel junction (MTJ) and all-metallic spin-valve nanopillardevices have dominated the literature [ 2–6], there have been an increasing number of studies utilizing point andnanocontacts (NCs) on extended multilayer ﬁlm stacks [ 7,12– 15]. The NC geometry is particularly promising for high- frequency spin-torque oscillators (NC-STOs) [ 16–22] and for the emerging ﬁeld of ST-based magnonics, [ 23–27], in Present address: Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0526, USA and Department of Physics, Division for Theoretical Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden. †randydumas@gmail.comwhich highly nonlinear auto-oscillatory modes are utilized for operation. In the NC geometry literature [ 7,12–15,28], the observed ST-FMR spectra of the NiFe-based free layers have beenanalyzed as a single resonance, despite a signiﬁcant peakasymmetry hinting at additional contributions. The linewidthof this asymmetric peak has not been understood so far [ 7]. The same studies also note that the typical ﬁeld condition of anin-plane ﬁeld aligning both magnetic layers in parallel shouldnot result in any ST, calling into question the fundamentalexcitation mechanism of the observed spectra. This signiﬁcantdiscrepancy has been tentatively explained as being caused bylocal misalignments due to sample imperfections. However,given how robust ST-FMR measurements are over sets ofdifferent devices, it is rather unsatisfactory to have to referto unknown extrinsic factors for the ST-FMR technique tofunction. It appears that the rf Oersted ﬁeld generated by theinjected microwave current into NC could be at play [ 15]. Therefore, a better fundamental understanding of both thelinear spin-wave (SW) modes in the NC geometry and theirexcitation mechanism is highly desirable. In this work, we show that the observed resonance spectrum in a NiFe NC-STO free layer is composite in nature and can be described as a sum of two distinct resonances withvery different behaviors and origins. Experimentally, the datamust be ﬁt with two Lorentzian functions in order to extract areasonable value for the Gilbert damping of the FMR mode.From these ﬁts, it is also clear that only the width of one of the resonances shows a dampinglike linear dependence 2469-9950/2016/93(13)/134427(7) 134427-1 ©2016 American Physical SocietyMASOUMEH FAZLALI et al. PHYSICAL REVIEW B 93, 134427 (2016) on the frequency, whereas the width of the other is mostly frequency independent. Aided by micromagnetic simulations,we conclude that (i) the composite resonance is a sum ofa FMR mode and an exchange-dominated spin-wave mode,(ii) the NC diameter sets the mean wave vector of theexchange-dominated spin waves, in good agreement withthe dispersion relation, and (iii) for in-plane ﬁelds the rfOersted ﬁeld, not ST, in the vicinity of the NC plays thedominant role in exciting the observed spectra. We argue thathomodyne-detected FMR studies in the NC geometry mustaccount for such additional excitations to accurately extractthe fundamental magnetodynamical properties. II. EXPERIMENT NC-STO fabrication starts with a blanket Pd (8 nm)/Cu (15 nm)/Co (8 nm)/Cu (8 nm)/NiFe (4.5 nm)/Cu (3 nm)/Pd(3 nm) ﬁlm stack deposited by magnetron sputtering on athermally oxidized Si substrate, where the NiFe (Ni 80Fe20) and Co play the role of the free and ﬁxed layers, respectively,a ss h o w ni nF i g . 1(a). The blanket ﬁlm is then patterned into 16×8μm 2spin-valve mesas, and a 30 nm SiO 2layer is deposited by rf magnetron sputtering. Circular NCs of nominaldiameters Dof 90, 160, and 240 nm are deﬁned through the SiO 2insulating layer using electron-beam lithography at the center of the mesa. A ﬁnal photolithographic process thendeﬁnes a coplanar waveguide for electrical connections andefﬁcient microwave signal pickup. FIG. 1. (a) Schematic of the NC-STO and the measurement setup. (b) Plot of the ﬁeld vs frequency of the dominant resonance peak. The resonance ﬁelds can be well ﬁt by the Kittel equation using μ0Ms=0.85±0.02 T for the NiFe layer. Inset: ST-FMR spectra at four different frequencies for the D=160 nm device.All measurements were performed at room temperature in a custom-built probe station utilizing a uniform in-plane magnetic ﬁeld. Our homodyne-detected FMR measurementsutilized both a microwave generator and lock-in ampliﬁer,which were connected to the device using a bias tee, asschematically shown in Fig. 1. The rf power injected into the NC is −14 dBm, which ensures that the excited magne- todynamics are in the linear regime. The resulting dc mixingvoltage [ 3]V mixis measured as a function of the magnetic ﬁeld and at a ﬁxed excitation frequency. The microwave currentwas amplitude modulated at a low (98.76 Hz) modulationfrequency for lock-in detection of V mix. III. EXPERIMENTAL RESULTS The ﬁeld-swept spectra measured for different frequencies, which are vertically offset for clarity, are shown in the inset ofFig. 1(b) for the D=160 nm device. As shown in the main panel of Fig. 1(b), the dominant resonance peak (data points) can be well ﬁt (solid line) with the Kittel equation, whichresults in the saturation magnetization μ 0Ms=0.85±0.02 T and a negligible magnetocrystalline anisotropy. Interestingly, upon closer inspection, it becomes clear that the measured spectra are highly asymmetric, exhibiting asigniﬁcant shoulder on the low-ﬁeld side of the dominantresonance peak. In Fig. 2we show a single representative resonance at f=18 GHz for the D=160 nm device. While it is well known that the mixing voltage can be intrinsicallyasymmetric [ 2,29,30], it is important to point out that we cannot ﬁt our data with a single resonance having bothsymmetric and antisymmetric contributions. Most importantly,the prior theoretical results are virtually independent of the NCdiameter, in direct contrast to our experimental observations.In order to properly ﬁt (red solid line) the entire spectrum wemust instead use twoLorentzian functions, each with its own resonance ﬁeld and linewidth, as shown in Fig. 2(inset). The ﬁt shows a vanishing antisymmetric contribution to the line FIG. 2. Zoom-in of a representative ST-FMR spectrum of the D=160 nm device taken at f=18 GHz and Irf=1.3 mA, together with a ﬁt (red line) based on two Lorentzians as described in the text. The inset shows the two individual contributions of the quasiuniformFMR mode (black) and the spin wave resonance (blue). 134427-2HOMODYNE-DETECTED FERROMAGNETIC RESONANCE OF . . . PHYSICAL REVIEW B 93, 134427 (2016) FIG. 3. The measured (dots) and ﬁtted (solid lines) linewidths of the FMR and SWR modes are shown for the different NC diameters. shape for both of the resonances, Vmix=offset+/summationdisplay i=FMR,SWRslope×B +1 /Delta1Bi/bracketleftbigg Si /Delta1Bi2 4/parenleftbig B−Bires/parenrightbig2+/Delta1Bi2 +Ai/Delta1Bi(B−Bi res) 4/parenleftbig B−Bires/parenrightbig2+/Delta1Bi2/bracketrightbigg , (1) where BandBi resare applied and resonance ﬁelds, respectively, and/Delta1Biis the linewidth of the corresponding peak. Si andAiare amplitudes of its symmetric and antisymmetric components, respectively. As the frequency vs ﬁeld behaviorof the main resonance mode can be well ﬁt with the Kittelequation, Fig. 1(b), we ascribe this peak to the FMR mode of the NiFe layer and the second low-ﬁeld mode with a higher-order spin-wave resonance (SWR), which will be discussed indetail later. The linewidth vs frequency of both the FMR and SWR modes are plotted in Fig. 3for three different NC diameters of 90, 160, and 240 nm. Three signiﬁcant observations can bemade. First, the FMR mode shows a clear linear increase oflinewidth with the frequency, from which the Gilbert dampingαcan be extracted using the following relation: /Delta1B i=4πα γf+/Delta1Bi 0, (2) where γ/2πis the gyromagnetic ratio and /Delta1Bi 0is an inhomogeneous broadening of the corresponding resonance.Our measured values of α, which are all on the order of 0.01, are also consistent with those measured in Ref. [ 31]. This provides further evidence that the dominant resonancemode can indeed be correlated with the usual FMR modeof NiFe. Second, the linewidth of the SWR mode is mostlyindependent of frequency, indicating that the primary originof the linewidth is not damping. Third, the inhomogeneousbroadening is approximately inversely proportional to the NC diameter, which at ﬁrst seems counterintuitive as one would FIG. 4. Measured (dots) and calculated (solid lines) resonance ﬁelds of the FMR and SWR modes for the different NC diameters.The black solid line is a ﬁt to an average of the FMR mode for all three devices. Inset: A plot of the ﬁtted NC diameter D /primevs the nominal diameter D, together with a line indicating D/prime=D. expect a larger NC to sample a larger sample volume and therefore include more inhomogeneities. The origin of thisinteresting effect will be explained later. The frequency versus ﬁeld dependence of the measured FMR and SWR modes are summarized in Fig. 4. The black solid line shows the average behavior of the FMR mode for NCdiameters of 90, 160, and 240 nm and essentially reproduceswhat is shown in Fig. 1. For a ﬁxed frequency, we ﬁnd that the SWR mode shifts to lower ﬁelds as the NC diameter decreases.Assuming that the origin of the SWR mode is the exchangeinteraction, the diameter of the NC, D /prime, can be estimated using the following dispersion relation: f=γ 2π/braceleftbig/bracketleftbig BSWR res+μ0MS(λexk)2/bracketrightbig ×/bracketleftbig BSWR res+μ0MS+μ0MS(λexk)2/bracketrightbig/bracerightbig1/2, (3) where λex=/radicalbig 2A/μ 0M2sandk=π/D/primeare the exchange length and the SWR wave vector, respectively. The room-temperature value of the exchange stiffness is set to A= 11 pJ/m[32]. The estimated sizes of the NCs are in reasonable agreement with the corresponding nominal values, as shownin the inset of Fig. 4. IV . MICROMAGNETIC SIMULATIONS The micromagnetic simulations were performed using the MUMAX3 solver [ 33]. Since the actual spin-valve mesa is too large to be simulated in its entirety in a reasonable time frame,we limited our calculations to a 5.120 μm×2.560μm×4n m volume with periodic boundary conditions tailored to mimicthe lateral aspect ratio of the experimental spin-valve mesa.To break the symmetry of the system, which might otherwisefully forbid any ST-related effects and nonconservative SWscattering, we assume that the applied ﬁeld points 5 ◦out of plane, comparable to the possible error in the experimentalﬁeld alignment. As a ﬁrst step, the evolution of the ground 134427-3MASOUMEH FAZLALI et al. PHYSICAL REVIEW B 93, 134427 (2016) state of the entire Co/Cu/NiFe stack is calculated, conﬁrming that (i) the Co and NiFe layers remain virtually collinear inthe given range of the applied magnetic ﬁelds and (ii) thereare no mutual stray ﬁelds produced between the layers inthe vicinity of the NC. Since there is a signiﬁcant spin-wavedispersion mismatch between Co and NiFe, we do not expectany resonant dynamic magnetic coupling between the layers.Under these three considerations we can conﬁdently simulatethe dynamics of the NiFe free layer alone. In the simulations we replicate the experimental data acquisition routine by performing the ﬁeld sweeps witha harmonic excitation of f=18 GHz. The inﬁnite wire approximation is used to calculate the Oersted ﬁeld producedby the NC [ 34,35]. For every value of the applied ﬁeld we ﬁrst let the system reach the steady state and then samplethe spatial map of the magnetization for the following5 ns at 3.5-ps time intervals with a subsequent pointwise fastFourier transform (FFT) applied and the amplitude and phaseof the magnetization precession extracted at the excitationfrequency. Where applicable, the direction of the spin-currentpolarization is assumed to be collinear with the magnetizationin the nominally ﬁxed Co layer. The implemented saturationmagnetization, gyromagnetic ratio, and damping constant areestimated by ﬁtting a Kittel equation to the experimental data.The room-temperature value of the exchange stiffness is settoA=11 pJ/m[32]. The simulated magnetic responses shown in Fig. 5(b)agree well with the experimentally measured data shown in Fig. 5(a). To identify the origin of the observed peak asymmetry weinvestigate the spatial proﬁles of the magnetization precessionamplitude in the vicinity of the resonance [see Fig. 5(c)]. The snapshots clearly show propagating SWs on the low-ﬁeld sideof the main peak, while no SWs are resolved on the high-ﬁeldside. Looking closer at the phase proﬁles of the correspondingmodes, which essentially depict the wavelength of the excitedmagnons, the following conclusions can be made: (i) Thepropagation of SWs perpendicular to the saturation directionis suppressed, and (ii) the lowest excited mode is not uniformbut antisymmetric with respect to the NC center. V . DISCUSSION If the free layer is magnetized in plane, then both back- ward volume magnetostatic SWs (BVMSSW) and surfacemagnetostatic-exchange SWs (SMSSW) can be excited: f BVMSSW =/braceleftbigg [fB+fM(λexk)2] ×/bracketleftbigg fB+fM(λexk)2+fM/parenleftbigg1−e−kd kd/parenrightbigg/bracketrightbigg/bracerightbigg1/2 , fSMSSW =/braceleftbigg [fB+fM(λexk)2][fB+fM(λexk)2+fM] +f2 M 4(1−e−2kd)/bracerightbigg1/2 , where fB=γ 2πB,fM=γ 2πμ0MS. They are calculated using Eqs. (5.97b) and (5.111b) from Ref. [ 36] for propagation along and perpendicular to the FIG. 5. (a) The normalized measured mixing voltage Vmixand (b) the normalized simulated magnetization precession amplitude for the three NC diameters as a function of the applied in-plane magneticﬁeld. (c) Spatial maps of the magnetization precession amplitude (top row) and phase (bottom row) simulated for a D=160 nm NC diameter taken at the ﬁelds corresponding to the main peak and its 1 2 and1 4heights [as shown by the corresponding black symbols in (b)]. Propagating spin waves are clearly seen for the two lowest ﬁelds. saturation direction, respectively. The exchange contribution is included by substituting B→B+μ0Ms(λexk)2. The corresponding dispersion relations are shown in Fig. 6 for NiFe thicknesses dof 100 nm (green lines) and 4.5 nm (red lines). There is always a region of resonance ﬁelds, 00.10.20.30.40.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14Resonance ﬁeld (T) Wave vector (rad/nm)100 nm 4.5 nm 4.5 nm (pinned) Eq. (3)00.10.20.30.40.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14BFMR k⊥Mk Mmagnetostaticexchange-dominated co-existence of magnetostatic and exchange-dominated SWs FIG. 6. The dispersion relations for the SWs propagating parallel and perpendicular to the saturation direction are shown for different thicknesses of the NiFe layer. The points correspond to the minimumof the SW group velocity. 134427-4HOMODYNE-DETECTED FERROMAGNETIC RESONANCE OF . . . PHYSICAL REVIEW B 93, 134427 (2016) where magnetostatic and exchange-dominated SWs coexist, as highlighted by the shaded area in Fig. 6for a NiFe thickness of 100 nm. Although the band is broad for relatively thicklayers, it only amounts to 1.16 mT for the 4.5-nm NiFe, i.e.,an order of magnitude smaller than the intrinsic linewidth ofthe FMR peak. We therefore conclude that SWs contributingto the low-ﬁeld tail of the FMR peak are exchange-dominated.Note that the calculated dispersion relations differ from what isfound using Eq. ( 3) (thick solid line in Fig. 6). This difference arises as the dispersion relations also strongly depend onthe exact boundary conditions at the free-layer surfaces. Forinstance, if the NiFe ﬁlm is pinned on both surfaces, e.g.,if it is placed in between sufﬁciently thick metallic layers,the dispersion of the exchange-dominated backward volumeSWs is given by the following equation (shown by the squaresymbols in Fig. 6)[37]: f BVMSSW-pinned =/braceleftbigg [fB+fM(λexk)2] ×/parenleftbigg fB+fM(λexk)2+fM 1+(kd/π )2/parenrightbigg/bracerightbigg1/2 . Note that in this case the spectrum of the exchange-dominated surface SWs will be dispersionless and not accessible experi-mentally. Since Eq. ( 3) ﬁts the NC diameter reasonably well, we conclude that (i) the detected mixing voltage is generated bythe exchange-dominated backward volume SWs and (ii) thereis undoubtedly some surface pinning of the NiFe layer. Theexact origin of the pinning and its strength is beyond the scopeof the present study. Due to the collinear free and ﬁxed layers we do not expect any signiﬁcant contribution from the ST to the observedmagnetization dynamics, which is conﬁrmed by comparingmicromagnetic simulations performed with and without STincluded. Correspondingly, for the in-plane applied magneticﬁelds in the NC geometry with no dc bias currents applied,the rf Oersted ﬁeld is the primary excitation mechanismresponsible for the observed dynamics. In a linear approximation the response of the system is essentially determined by the spectrum of the excitation, whichin our case is provided by the rf Oersted ﬁeld. If the excitationhas a ﬁnite amplitude at some point of reciprocal space (i.e.,at the given frequency and wave vector), then, if allowed,the corresponding magnon will be excited. The spatial proﬁleof the Oersted ﬁeld and its spectrum for the D=160 nm NC are shown in Figs. 7(a) and7(b), respectively. We can identify both local and global antisymmetries with respectto the NC center with corresponding periods determined by twice the NC diameter and mesa width L, respectively. Since both spatial components are naturally conﬁned to their unitperiods, the linewidth of the corresponding excitation peak isﬁnite. Therefore, the Oersted ﬁeld most efﬁciently couples tothe SW bands having widths of 2 π/L andπ/D corresponding to the wave vectors of 2 π/L andnπ/D , respectively, where n=1,3,... [see Fig. 7(c) and its inset]. As the NC diameter decreases, the position and width of the former band stayconstant, while the latter one shifts towards lower resonanceﬁelds and increases its width, leading to the observed extensionof the tail in the excitation spectrum. FIG. 7. (a) The spatial distribution and (b) the corresponding 2D FFT of the out-of-plane component of the Oersted ﬁeld by the D=160 nm diameter NC. (c) The Oersted ﬁeld as a function of the wave vector component along the saturation direction is shownfor the D=160 nm diameter NC. The inset shows a zoom-in of the small wave-vector part of the spectrum. (d) The dispersion relation of the exchange-dominated SWs. (e) The experimentallyacquired magnetization dynamics spectrum using a nanocontact of D=160 nm diameter. It is important to mention that the circular NC cannot effectively couple to the uniform FMR. Instead, the mainpeak observed in the experiments and simulations correspondsto the antisymmetric mode with k=2π/L. However, due to the vanishing magnetostatic dispersion, its resonant ﬁeldis virtually indistinguishable from the uniform FMR ( k=0) mode. Considering a typical FMR experiment where the excitation frequency is ﬁxed, according to Eq. ( 3), the wave vector of the generated propagating SW is ultimately determinedby the value of the applied magnetic ﬁeld. As the ﬁeld isswept towards zero past the dominant FMR resonance, theNC continuously excites propagating SWs of increasing wavevectors. Since the excitation amplitude drops rapidly for the low values of the applied ﬁeld (i.e., for short-wavelength SWs), the detected magnetic signal vanishes accordingly, leading tothe appearance of the low-ﬁeld tail [see Figs. 7(c)and7(e)]. By assuming that the extent of the tail is estimated at 1/10 of its peak amplitude, we can project the corresponding experimentally observed applied magnetic ﬁeld to the cutoffwave vector of the excitation spectrum, as schematicallydemonstrated by the shaded rectangles in Figs. 7(c),7(d), and7(e). This gives us the cutoff wave vectors (in units of π/D /prime) of 1.93, 1.95, and 2.34 for the NCs of 90, 160, and 240 nm nominal diameters, respectively. Since these values fall 134427-5MASOUMEH FAZLALI et al. PHYSICAL REVIEW B 93, 134427 (2016) roughly inside the ﬁrst two fundamental SW bands attributed tok=2π/L andk=π/D , the two-peak scheme used to ﬁt the experimental data is fully justiﬁed. It should be noted that the micromagnetic simulations do not reproduce the shoulder as it is observed experimentally forall the NC diameters. According to our model, the shouldershould be inherited from the excitation spectrum. Perhapsthe approximation we used to calculate the Oersted ﬁeld,an inﬁnite wire, is not sufﬁcient to bring out this feature.Nevertheless, this does not change the interpretation of theresults and conclusions of the present study. Finally, the NC size dependence of the FMR and SWR inhomogeneous broadenings shown in Fig. 3can be well understood by assuming that it is inherited from the linewidthof the corresponding excitation peaks. For the SWR mode,the expected extrinsic contribution to the magnonic linewidthis 96, 43, and 15 mT for the NC diameters of 82, 122, and205 nm, respectively, in excellent agreement with the ﬁttedvalues. In contrast, for the FMR mode the contribution isvanishing and should be virtually independent of the NCsize. However, if the NC exhibits shape imperfections, thecorresponding irregularities in the Oersted ﬁeld proﬁle shouldbroaden the excitation peaks and, eventually, the FMR andSWR. As we typically observe a less perfect NC for smallerdiameters, the inhomogeneous broadening of FMR shouldincrease accordingly, consistent with the experimental data. VI. CONCLUSIONS In conclusion, using homodyne-based measurement tech- niques we provide an in-depth study of the magnetodynamicsin a quasiconﬁned system, namely, a NC patterned on anextended pseudo-spin-valve ﬁlm stack. The observed spectra are highly asymmetric and cannot be explained by a singleresonance mode, as has been done in the past [ 7,12]. Instead, each spectrum is ﬁt by a combination of two Lorentziansfrom which we can extract the FMR mode resonance ﬁeldand linewidth. The secondary mode corresponds to thegeneration of exchange-dominated spin waves with a wavevector inversely proportional to the NC diameter. The resultsare reproduced by the micromagnetic simulations that showthe rf Oersted ﬁeld generated by the injected rf current isthe dominant excitation mechanism of the observed magne-tization dynamics. We thereby demonstrate experimentallya highly tunable point source of the propagating SW withthe wave vectors limited only by the resolution of thefabrication process used. This is of paramount importancefor applications of subterahertz and terahertz magnonics andspintronics. ACKNOWLEDGMENTS We would like to thank P. Muduli and M. Ranjbar for useful discussions. 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PhysRevLett.120.027201.pdf	Voltage Control of Rare-Earth Magnetic Moments at the Magnetic-Insulator –Metal Interface Alejandro O. Leon,1Adam B. Cahaya,1and Gerrit E. W. Bauer1,2 1Institute for Materials Research, WPI-AIMR, and CSRN, Tohoku University, Sendai 980-8577, Japan 2Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, The Netherlands (Received 5 September 2017; published 10 January 2018) The large spin-orbit interaction in the lanthanides implies a strong coupling between their internal charge and spin degrees of freedom. We formulate the coupling between the voltage and the local magneticmoments of rare-earth atoms with a partially filled 4fshell at the interface between an insulator and a metal. The rare-earth-mediated torques allow the power-efficient control of spintronic devices by electric- field-induced ferromagnetic resonance and magnetization switching. DOI: 10.1103/PhysRevLett.120.027201 Introduction. —The power demand for magnetization control in magnetic memories is an important design param- eter. The power consumption of voltage-driven magnetiza-tion dynamics can be orders of magnitude lower than the oneof electric-current-induced dynamics [1]. In addition, electric voltages are a more localized driving mechanism compared tomagnetic fields [1]. From the experimental point of view, magnetization reversal [2,3] and ferromagnetic resonance [1,4] driven by electric voltages have been achieved. In those studies, transition-metal films are capped with an insulatingbarrier that prevents the electric current flow. The mainmechanism to couplevoltage and magnetization is the controlof the perpendicular magnetic anisotropy [5].T h em a g n e t i - zation has been manipulated by electric fields for the first time in (Ga,Mn)As [6]and also in magnetoelectric materials [7–9]. The spin-charge coupling that causes the observed phenomena has been modeled by the Rashba [10,11] and Dzyaloshinskii-Moriya interactions [11–13], whose origin is the relativistic magnetic field induced by linear momentum of the electron in a transverse electric field. On the other hand, the spin-orbit interaction in central fields ofsingle atoms can best be expressed in terms of the effectivemagnetic field generated by orbital angular momentum . Here we focus on local magnetic moments in a condensedmatter system for which the second picture of the spin-orbitinteraction is the best starting point. Local magnetic moments in solids are formed by partially filled 3dand4fshells of transition metals and rare earths, respectively. The former are relatively light, and their spin dynamics are dominated by the exchange interaction, with correction by the crystal fields. Rare earths (REs), on theother hand, have their magnetic subshell shielded by outer shells, which decreases the effect of crystal fields and allows the electrons to orbit almost freely in the central Coulombfield of the ionic core with large nuclear charges. The spin- orbit interaction (SOI) of REs is therefore large and free- atomic-like. Since SOI couples the electric and magneticdegrees of freedom, we may expect significant effects of electric fields on the RE magnetization dynamics. Controland switching of RE moments by crystal fields in multi-ferroic systems has been predicted [14]. Here we study the voltage-driven dynamics of rare earths at the interface between a magnetic insulator (or badconductor) and a metal. When one of the layers is magnetic, the presence of REs at the interface strongly couples the magnetization to an applied static and dynamic voltage bythe local spin-orbit interaction. Electric fields, applied by high-frequency signal generators, for example, are constant inside an insulator but nearly vanish in a metal. Thelarge spatial gradients of the electric field at the interfacerenormalize the RE electrostatic interactions with neighbor- ing atoms (crystal fields) and appear as a voltage-modulated magnetic anisotropy and the associated magnetizationtorque that we derive in the following in more detail. Magnetism of rare-earth ions. —In the Russell-Saunders scheme [15], the total spin ( S) and orbital ( L) momenta are the sum of the single-electron momenta of the 4forbitals S¼P jsjandL¼P jlj. The spin-orbit coupling reads HSOI¼ΛS·L, and the coupling parameter Λis positive (negative) for less (more) than a half filled subshell [15,16] . The total angular momentum vector J¼SþLand the angular part of the eigenfunctions can be written as jΨi¼ jS; L; J; J zi, where the quantum numbers are governed by S2jΨi¼ℏ2SðSþ1ÞjΨi,L2jΨi¼ℏ2LðLþ1ÞjΨi,J2jΨi¼ ℏ2JðJþ1ÞjΨi, and ˆJzjΨi¼ℏJzjΨi,ˆJzis the zcomponent of the vector J, andℏis Planck ’s constant divided by 2π. The lowest-energy state of RE ions as governed by Hund ’s rules [15] are listed in Table I. The Wigner-Eckart theorem ensures that within this ground state manifold the angularmomenta are collinear, viz. S¼ðg J−1ÞJandL¼ð2−gJÞJ in terms of the Land´ egfactor gJ. Furthermore, for constant ðS; L; J Þ, the orbital symmetry axis and the spin vector move rigidly together [17,18] , implying that the atomic charge and spin distributions are strongly locked by the spin-orbitPHYSICAL REVIEW LETTERS 120, 027201 (2018) 0031-9007 =18=120(2) =027201(5) 027201-1 © 2018 American Physical Societyinteraction. The electron density of a partially filled 4f subshell can be written as n4fðrÞ¼X3 ml¼−3jR4fðrÞYml 3ðˆrÞj2ðfml↑þfml↓Þ; ð1Þ where r¼rˆris the position vector in spherical coordinates, R4fðrÞis (approximately) the radial part of the 4fatomiclike wave function, and the spherical harmonic Yml 3ðˆrÞdescribe the angular dependence. fml;msis the occupation number of the single-electron state with magnetic quantum numbers of orbital mland spin msangular momenta. The density n4fis normalized to the number of electrons in the 4fshell N4f¼Rn4fðr;tÞdr. The typical 4fradius hri∼0.5Å is much smaller than typical interatomic distances R∼3Å, which motivates the multipole expansion [17,18] n4fðrÞ≈jR4fðrÞj2 4π/C18 N4fþ5Q2 4hr2i½3ðm·ˆrÞ2−1/C138/C19 ;ð2Þ where Q2≡R ð3z2−r2Þn4fðrÞdris the quadrupole moment listed in Table I.m¼−J=jJjis the unit magneti- zation vector that at equilibrium is taken to be ezbut in an excited state may depend on time. The unit position vector in spherical coordinates is ˆr¼sinθ½excosϕþeysinϕ/C138þ ezcosθ, where fex;ey;ezgare the unit vectors along the Cartesian axes. For Q2>0ðQ2<0Þ, the envelope function of the electron density is a pancake or cigarlike (oblate orprolate) ellipsoid, respectively. A local magnetic ion interacts weakly with static electric fields, E¼−∇V, where Vis the voltage or potential energy of a positive probe charge. To leading order, the ions experience the electrostatic energy [19]hψj−eX N4f i¼1VðriÞjψi¼−eZ d3rVðrÞn4fðrÞ; ð3Þ where−eis the electron charge and −eVðriÞis the potential energy of the ith electron. jψiis the 4fmany-electron wave function in the ground state. Again, the leading order in a multipole expansion of the crystal field around the origin r¼0can be parameterized by a quadrupolar term Að0Þ 2: eVðrÞ¼−Að0Þ 2r2ð3cos2θ−1Þ: ð4Þ Inserting Eqs. (2)and (4)into Eq. (3), we arrive at a Hamiltonian that depends on the magnetization direction as Hani¼3 2Q2Að0Þ 2m2z: ð5Þ The crystal symmetry orients here the easy ( Q2Að0Þ 2<0)o r hard ( Q2Að0Þ 2>0) magnetic axis along the zdirection. This crystal field energy accounts for the single rare-earth ion magnetic anisotropy. The parameter Að2Þ 0can be calculated by first principles or to fit to experiments. Typical values areAð2Þ 0¼300Ka−2 0forðREÞ2Fe14B,Að2Þ 0¼34Ka−2 0for ðREÞ2Fe17, and Að2Þ 0¼−358Ka−2 0forðREÞ2Fe17N3[18], where a0¼0.53Å is the Bohr radius. The origin of the strong magnetic anisotropy of REs is their large spin-orbitinteractions. On the other hand, for 3dtransition-metal moments, the anisotropy is usually very small, except atinterfaces, where the orbital motions are partiallyunquenched. In such cases, the anisotropy emerges as the consequence of SOI, the quadrupolar shape of electric potentials at the interface [5], the hybridization of orbitals, and change in the orbital occupation. At an interface between materials with different work functions, the symmetry is reduced. The electric fieldexhibits spatial gradients due to charge accumulationimmediately at the interface that result in a step likepotential. An external voltage difference ΔVdrops over the insulator but is constant in the metal (when the ferromagnet is a bad conductor, the effects are weaker but still exist). The electric field field gradient couples toquadrupoles in the immediate proximity of the interface. Voltage coupling at interfaces. —Let us focus on a magnetic-insulator film with thickness L F. At the surface, the insulator exposes nRErare-earth moments per unit of area. Inside the insulator, the electric field is approximatelyconstant, Eðz<0Þ¼e zΔV=L F, while it vanishes in the metal Eðz>0Þ¼0; see Fig. 1(a). Using Eq. (3), the electric energy of a magnetic moment at the origin is then He¼H0−15 64eΔV LFQ2hri hr2im2z; ð6Þ where H0¼5eE0Q2hri=ð64hr2iÞ þeðΔV=L FÞN4fhri=4 does not depend on the magnetization and hrni≡ N−1 4fRrnn4fðrÞdr.F o r hr2i1=2∼hri∼0.5Å the coupling energy per unit area at equilibrium ( m2z¼1),TABLE I. Ground state ðS; L; J Þbased on Hund ’s rules and shape of the 4fground state electron density [17].Q2is the quadrupole moment calculated using the Wigner-Eckart theorem for the state Jz¼J, and a0¼0.53Å is the Bohr radius. The Wigner-Eckart theorem cannot be applied to Eu because J¼0. Ion 4fnSLJ Shape Q2=a2 0 Ce3þ4f1 1 235 2Oblate −0.686 Pr3þ4f2 1 5 4 Oblate −0.639 Nd3þ4f3 3 2692 Oblate −0.232 Pm3þ4f4 2 6 4 Prolate 0.202 Sm3þ4f5 5 2552 Prolate 0.364 Eu3þ4f6 330 /C1/C1/C1 /C1/C1/C1 Gd3þ4f7 7 2072 Spherical 0 Tb3þ4f8 3 3 6 Oblate −0.505 Dy3þ4f9 5 2515 2Oblate −0.484 Ho3þ4f10 2 6 8 Oblate −0.185 Er3þ4f11 3 2615 2Prolate 0.178 Tm3þ4f12 1 5 6 Prolate 0.427 Yb3þ4f13 1 237 2Prolate 0.409PHYSICAL REVIEW LETTERS 120, 027201 (2018) 027201-2/C12/C12/C12/C12n REðHe−H0ÞLF ΔV/C12/C12/C12/C12¼750fJ VmQ2 10−3nm2nRE nm−2;ð7Þ is one order of magnitude larger than the corresponding coupling in transition metals [20,21] . For electric fields ΔV=L F∼10mV=nm¼100kV=cm, the surface energy density becomes ðHe−H0ÞnRE≈7.5×10−3erg=cm2¼ 7.5μJ=m2. Our electrostatic model requires that the atomic terms of the4fshell in the ground state are stable under applied electric fields, which possibly excludes valence fluctuation compounds. In the Supplemental Material [22] we show that a finite screening length leads to expressions very similar to Eq. (6). In good metals, exchange correlation modifies the screening only slightly and can therefore bedisregarded here. The step field model can also be applied to nonmagnetic insulators andtransition-metal ferromagnets (such as Fe, Co, and Ni, or their alloys) with RE ions at the interface that are antiferromagnetically coupled to the magnetic order[25,26] and facilitate a large coupling of the magnetization to electric fields. Good insulators, such as MgO, can endure very large electric fields (of the order of 300mV=nm, in FeCo jMgO, Ref. [20], for example). Thus, MgO-based magnetic tunnel junctions with rare-earth doping or dusting are promising devices to study and apply electric-field-induced modulations of the magnetization configuration. In magnetic materials, local angular momenta are strongly locked by the exchange interaction. When a sufficiently strong static magnetic field Bis applied, the macrospin model is valid; i.e., the magnetization Mis constant in space. The total magnetic energy H Mper unit area then reads HM μ0M2sLF¼−m·h−βx 2m2xþβz−Γ 2m2z: ð8ÞThe first term on the (dimensionless) right-hand side with h¼B=ðμ0MsÞis the Zeeman energy and Msthe saturation magnetization. The parameters βx(βz) account for the in- plane (out-of-plane) magnetic anisotropy in the absence of applied electric fields, ΔV¼0. The dimensionless cou- pling parameter Γmeasures the relative strength of the electrostatic coupling ∼nREeΔVQ 2=LFthat should be compared with magnetic anisotropies. Γ∼0.06with the following parameters representative for a rare-earth irongarnet thin film such as Tm 3Fe5O12: Γ¼0.06nRE 1=nm2/C18105A=m Ms/C192/C1810nm LF/C192ΔV 0.1VQ2 10−3nm2: ð9Þ Since the Msof 8-nm-thick Tm 3Fe5O12[27] is at room temperature about 10 times smaller than that of even asubnanometer FeCo film [20], the coupling strength Γis 10 times larger for magnetic insulators for the same applied electric field without the need for additional tunnel barriers.Intraband transition and electric breakdown is of no concern as long as eE 0≪ϵ2gap=ðϵFaÞ, where ϵgapis the band gap, ϵF the Fermi level in the metal, and athe lattice constant [28]. Using ϵF∼2eV, and the gap or lattice constant for yttrium iron garnet (YIG) [29,30] ϵgap∼2.85eV=a¼1.2nm, we estimate E0≪2V=nm to be safe. The coupling strength Γ decreases ∼L−2 Ffor a given voltage, so much can be gained by choosing an insulator with a large gap and breakdown voltage that permits working with thin layers. Figure 1(b) shows the stable magnetizations that mini- mize of the energy (8)in the presence of a magnetic fieldh¼h½excosφþezsinφ/C138that is tilted by an angle φ. The parameter are h¼0.01,φ¼5.72°,βx¼0, and βz¼−0.03. The application of a constant voltage allows the transition from the easy-axis (right zone) to the easy-plane (left zone) configuration. The electric field effects in transition-metal devices, as well as the one proposed here, derive from the sametype of magnetic anisotropy, although the microscopic coupling mechanism is different. The phenomenology of electric-field-induced precessional dynamics as observed intransition-metal systems [31] does not differ from the one we expect for RE systems. The advantage of interface REs is the lower power consumption and the possibility of usinga wider range of materials including magnetic insulators, such as YIG. The magnetization dynamics is described by the Landau-Lifshitz-Gilbert equation _m¼−γm×h effþαm×_m; ð10Þ where αis the Gilbert damping constant, γ>0is the (modulus of the) gyromagnetic ratio, _mis the temporal derivative of m, and the effective magnetic field heff satisfieszE-1-0.8-0.6-0.4-0.20.20.40.60.81 -8 -6 -4 -2 0 24 680(a) (b) mj mzup=mzdown mzdownmzupTwo out-of-plane states In-plane state Γ 10-2mx FIG. 1. Spin-charge coupling for an interface local magnetic moment. (a) Electric field at an interface between an insulator(constant electric field) and a metal (vanishing electric field).The magnetic dipole and charge quadrupole at the interfaceare coupled. (b) Ground state magnetization directions m¼ ðm x;my;mzÞas a function of the interface electric field with coupling parameter Γ[Eq. (9)] and a magnetic field tilted from the zaxis by an angle φ∼6°. The system switches from a perpendicular easy-axis to an easy-plane configurationatΓ<0.018.PHYSICAL REVIEW LETTERS 120, 027201 (2018) 027201-3heff μ0Ms≡1 μ0M2sLF∂HM ∂m¼hþβxmxexþ½Γ−βz/C138mzez:ð11Þ The magnetic torque exerted by the electric field is propor- tional to −m×γmzez. Ferromagnetic resonance. —We now turn to an ac electric field that modulates the coupling Γ¼Γ0cosðΩtÞ, with frequency Ωclose to the ferromagnetic resonance (GHz). Since the electric field is normal to thin metallic films <100nm, the induced Oersted-like magnetic field and associated power are negligibly small. In linear response,the model (10) can be solved analytically for β x¼βz¼0. The polar coordinate system is spanned by the unit vectors e1¼excosφþezsinφ,e2¼−exsinφþezcosφ, and e3¼−ey. At the equilibrium state, meq¼e1along the applied magnetic field. Around the equilibrium state, the magnetization is m¼e1þδm, where δm¼δm2e2þ δm3e3is the deviation from meq, with jδmj≪1and δm·meq¼0. To leading order in the coupling ( Γ0) and dissipation ( α), the effective field is ðμ0MsÞ−1heff¼he1þ ΓcosðΩtÞsinφ½e1sinφþe2cosφ/C138and δ_m ωM¼e1×/C20 hδmþαδ_m ωM−Γ0 2cosðΩtÞsinð2φÞe2/C21 ; where ωM≡γμ0Ms. The effective ac magnetic field Bac¼μ0MsΓ0sinð2φÞcosðΩtÞez=2. Then δm¼ðΓ0=4Þsinð2φÞχ0ðΩÞ½cosðΩtÞe2þsinðΩtÞe3/C138 þðΓ0=4Þsinð2φÞχ00ðΩÞ½sinðΩtÞe2−cosðΩtÞe3/C138; where χ0andχ00are the real and imaginary parts, respec- tively, of the dynamics susceptibility χðωÞ≡ωMðω0−ωÞ ðω0−ωÞ2þω2α2þiωMαω ðω0−ωÞ2þω2α2 and the natural frequency is ω0≡ωMh¼γμ0Msh. Figure 2 illustrates δmðtÞ(continuous lines) together with the numeric solution (dots). We see that a large oscillation cone jδmj∼0.15can be achieved by a relatively low voltage forthe aforementioned parameter values and Γ0¼0.01 (orΔV=L F∼1.6mV=nm). Magnetization switching. —Magnetic reversal in tunnel junctions is the key process in magnetic random access memories. An applied voltage can reduce the energy barrier for magnetic-field- and current-induced switching or directlytrigger the magnetization reversal [31]. The latter effect is illustrated by Fig. 3assuming perpendicular magnetization (for in-plane magnetization, see Refs. [2,3]). An equilibrium magnetization along z[either an upordown state in the right zone in Fig. 1(b)] is excited by a steplike voltage pulse into large damped precessions around the in-plane equilibrium[left zone in Fig. 1(b)]. When thevoltage is turned off again at the right time, the magnetization can be fully reverted. Theswitching is observed with a large tolerance in the pulseduration between the pico- and nanosecond scales. In the simulation in Fig. 3, the pulse duration is around 1 ns, while the application of subsequent pulses toggles the magnetiza-tion direction faithfully. Conclusions and remarks. —We report voltage-modu- lated magnetic anisotropies and magnetization dynamics of rare-earth magnetic moments at insulator-metal bilayer interfaces. An applied voltage generates inhomogeneous electric fields at interfaces with a large conductivitymismatch that couple efficiently to rare-earth ions withnonspherical electron distributions, i.e., when the shell isnot half or completely filled. The voltage can then rigidlyprecess the charge and spin distributions of the entire 4f subshell via a stronger and direct coupling to the spin than in transition metals. The localized character of 4felectrons allow us to derive the leading order voltage coupling usingsingle-ion-like and step field models without having to carry out first-principles calculations (that are essential fortransition-metal systems). Adding rare-earth impurities toinsulator-metal bilayers can be used to efficiently switch the magnetization and induce ferromagnetic resonance. Future applications may include rare-earth-dusted magnetic-insulator –normal-metal interfaces, such as YIG jPt, thatmj t (a.u.)0.8 -0.20 mxmy -0.150.15 0.8 0.60 0.7t0t0+300 γMsmx mymz FIG. 2. (a) Magnetization dynamics induced by time-varying voltages. Comparison between analytic (solid line) and (b) nu-meric (dot) precessional (FMR) solutions, obtained forΓ¼Γ 0cosðΩtÞ,Γ0¼0.01,βx¼βz¼0,Ω¼0.08,h¼0.1, φ¼45°, and α¼0.005.10-1-0.8-0.6-0.4-0.200.20.40.60.81 mzmx t × 105γMsmymj 9 8 7 6 5 4 3 2 1 0 10t × 105γMs9 8 7 6 5 4 3 2 1 0Γ(t)0 -0.3-0.4 FIG. 3. Precessional switching for an easy-axis perpendicular magnet ( βz¼−0.03) induced by a voltage box train. The upper panel shows the magnetization components, while the lowerpanel shows the box consisting of a negative voltage with Γ¼ −0.03forΔt¼4000 γM s(∼1ns) followed by Γ¼0. This signal is repeated all 2×105γMs(∼25ns). Other parameters are βx¼0,h¼0.01,φ¼5.72°, and α¼0.005.PHYSICAL REVIEW LETTERS 120, 027201 (2018) 027201-4can efficiently convert an ac voltage into a spin current by spin pumping. We acknowledge the financial support from JSPS KAKENHI Grants No. 25247056, No. 25220910, andNo. 26103006, JSPS Fellowship for Young ScientistsNo. JP15J02585, and CONICYT Becas ChileNo. 74170017. We thank J. Barker for useful discussions.We profited from preparatory research by Dr. MojtabaRahimi. [1] T. Nozaki, Y . Shiota, S. Miwa, S. Murakami, F. Bonell, S. Ishibashi, H. Kubota, K. Yakushiji, T. Saruya, A. Fukushima,S. Yuasa, T. Shinjo, and Y . Suzuki, Electric-field-inducedferromagnetic resonance excitation in an ultrathin ferromag-netic metal layer, Nat. Phys. 8, 491 (2012) . [2] Y. Shiota, T. Maruyama, T. Nozaki, T. Shinjo, M. Shiraishi, and Y. Suzuki, Voltage-assisted magnetization switching inultrathin Fe 80Co20alloy layers, Appl. Phys. Express 2, 063001 (2009) . [3] S. Kanai, M. Yamanouchi, S. 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PhysRevB.83.224418.pdf	PHYSICAL REVIEW B 83, 224418 (2011) Critical thickness investigation of magnetic properties in exchange-coupled bilayers R. L. Rodr ´ıguez-Su ´arez,1L. H. Vilela-Le ˜ao,2T. Bueno,2A. B. Oliveira,3,4J. R. L. de Almeida,2 P. Landeros,5S. M. Rezende,2and A. Azevedo2 1Facultad de F ´ısica, Pontiﬁcia Universidad Cat ´olica de Chile, Casilla 306, Santiago, Chile 2Departamento de F ´ısica, Universidade Federal de Pernambuco, Recife, PE 50670-901, Brazil 3Departamento de F ´ısica Te ´orica e Experimental, Universidade Federal do Rio Grande do Norte, 59078-970 Natal, RN, Brazil 4Escola de Ci ˆencias e Tecnologia, Universidade Federal do Rio Grande do Norte, 59078-970, Natal, RN, Brazil 5Departamento de F ´ısica, Universidad T ´ecnica Federico Santa Mar ´ıa, Avenida Espa ˜na 1680, 2390123 Valpara ´ıso, Chile (Received 7 December 2010; revised manuscript received 30 March 2011; published 24 June 2011) We present a systematic investigation of the magnetic properties of two series of polycrystalline ferromagnetic- antiferromagnetic bilayers (FM-AF) of Ni 81Fe19(10nm)/Ir 20Mn 80(tAF) grown by dc magnetron sputtering. One series was grown at an oblique angle of 50◦and the other one was grown at 0◦. Ferromagnetic resonance (FMR) was used to measure the exchange bias ﬁeld HE, the rotatable anisotropy ﬁeld HRA, and the FMR linewidth /Delta1Has a function of the antiferromagnetic layer thickness tAF. Three relaxation channels due to isotropic Gilbert damping, anisotropic two-magnon scattering, and mosaicity effects are simultaneously distinguished through theangular dependence of the FMR linewidth. In the regime of small IrMn layer thicknesses, not enough to establishthe exchange bias anisotropy, the FMR linewidth shows a sharp peak due to the contribution of the two-magnonscattering mechanism. The results presented here are of general importance for understanding the dynamics ofmagnetization in the FM-AF structures. DOI: 10.1103/PhysRevB.83.224418 PACS number(s): 75 .70.Cn, 76 .50.+g, 75.30.Et I. INTRODUCTION Since the discovery in 1956 by Meiklejohn and Bean1of the exchange bias (EB) phenomenon, a complete theoreticalunderstanding of the magnetic coupling phenomena betweena ferromagnetic (FM) and an antiferromagnetic (AF) materialhas posed one of the most remarkable challenges in the ﬁeldof magnetism. While the phenomenon is observed in a largevariety of systems, 2–4it is in thin-ﬁlm multilayers that it has found important technological applications, as a domainstabilizer of magnetoresistive heads and in spin valve design.Complete references can be found in some recent reviewspublished on this subject. 2–8 Despite the large amount of research reported on the topic, there are still several aspects of the EB mechanismat the FM-AF interface that are not well understood. Forinstance, the complete origin of rotatable anisotropy and thefact that different measurement techniques may yield differentvalues for the FM-AF exchange ﬁeld ( H E) between the layers still lack a satisfactory elucidation.9,10In addition, the mechanism that controls the spin structure at the interface isstill controversy. N ´eel 11was the ﬁrst one to realize that FM-AF coupling involves so many aspects to be well explained by asimple model such as the one initially proposed by Meiklejohnand Bean 1from which the values of HEpredicted are typically two orders of magnitude larger than the experimental results. In order to predict reasonable values for HE, Mauri et al.12 proposed the ﬁrst domain-wall model of EB. Although this model results in more reasonable values for HE, it cannot explain features such as the enhanced coercivity HCof the FM-AF bilayer systems, or the training effect. A model proposed by Stiles and McMichael13to describe the behavior of polycrystalline FM-AF bilayers is based onthe existence of independent AF grains that are coupled tothe FM ﬁlm at the FM-AF interface. Some of the grains arelarge enough in order to stabilize the AF order and thereforeare responsible for the existence of uniaxial anisotropy. On the other hand, the smaller grains, in which the AFmagnetization rotates irreversibly as the FM magnetizationrotates, are responsible for hysteretic behavior observed intorque curve measurements and for the overall shift ofthe resonance ﬁeld observed in ferromagnetic resonance(FMR) measurements. These irreversible transitions set thesystem to lower-energy states, meaning that, on average,the easy direction of these grains goes along with the FMmagnetization direction. This behavior is phenomenologicallydescribed by a rotatable ﬁeld that always points alongthe local FM magnetization vector. This model has beensuccessfully used to explain most of the properties measuredby both FMR and dc magnetization techniques in FM-AFbilayers. Many experimental results have shown that the exchange ﬁeldH Eand coercivity HCin FM-AF bilayers are inversely proportional to the FM layer thickness.14On the other hand, the dependence of the magnetic properties on the AF layerthickness needs a much more elaborate analysis. For instance,there exists a critical thickness of the AF below which the EBvanishes. 15,16In addition to this, it has been observed that the coercivity and the rotatable anisotropy ﬁeld show maximumvalues for different AF layer thicknesses. 15,17–21 FM resonance has been shown to be one of the most successful techniques used to determine the values of theeffective ﬁelds associated with the magnetic anisotropiesin FM thin ﬁlms and multilayers. 22–26In addition, FMR linewidth measurements also give accurate information aboutthe magnetic relaxation mechanisms in these systems. Becausethey are very sensitive to the details of microscopic interactionsand materials microstructure, the FMR linewidths can giverelevant information about the magnetization dynamics ofFM-AF systems. It also provides powerful insight into thenature of the FM-AF interactions that cannot be obtained byother techniques. 224418-1 1098-0121/2011/83(22)/224418(8) ©2011 American Physical SocietyR. L. RODR ´IGUEZ-SU ´AREZ et al. PHYSICAL REVIEW B 83, 224418 (2011) In this paper we present a systematic investigation of magnetic properties in polycrystalline FM-AF bilayers as afunction of the AF layer thickness. The angular dependencesof the FMR resonance ﬁeld and linewidth are analyzed by theLandau-Lifshitz-Gilbert (LLG) equation. It is found that thein-plane angular dependence of the FMR linewidths can bedescribed by a superposition of different mechanisms, such asthe intrinsic Gilbert damping effect, magnetic inhomogeneity,and two-magnon scattering. By investigating the in-planedependence of the FMR linewidth for a series of Py(10nm)/IrMn( t AF), we are able to separate the contribution of the different relaxation mechanisms to the magnetizationdamping. For instance, in the low t AFregime (below the EB onset), the rise in /Delta1His mainly associated with the presence of the two-magnon scattering process. The analysis carriedout in this work is certainly relevant for understanding themagnetization dynamics in EB polycrystalline thin ﬁlms. II. EXPERIMENT The samples investigated here are multilayers of Cu(6 nm)/Ni 81Fe19(10 nm)/Ir 20Mn 80(0–16.4 nm) (labeled series A) and Cu(6 nm)/Ni 81Fe19(10 nm)/Ir 20Mn 80(0–6.1 nm) (labeled series B). Series B was grown by sputter deposition at anoblique angle 50 ◦and series A was grown at 0◦. The ﬁlms were deposited by dc magnetron sputtering system on commerciallyavailable Si(001) after being cleaned in ultrasound baths ofacetone and ethanol for 30 min and dried in nitrogen ﬂow.The base pressure was 2.2 ×10 −7Torr. In order to establish the uniaxial anisotropy of the FM layer, a magnetic ﬁeld of∼50 Oe was applied on the sample surface during deposition. The FMR data were taken with a homemade X-band spectrometer operating at 8.61 GHz, with the sample mountedon the tip of an external goniometer and introduced through ahole in the back wall of the cavity. Thus, it could be rotatedto allow measurements of the in-plane resonance ﬁeld H Rand linewidth /Delta1H as a function of the azimuthal angle, which were determined by ﬁtting the derivative of a Lorentzian lineshape to the measured ﬁeld spectrum and the peak-to-peak ﬁeldspacing, respectively. The dc magnetic ﬁeld was provided bya 9-in. electromagnet and was modulated with a 1.2-kHz accomponent of a few Oersteds using a pair of Helmholtz coilsto allow lock-in detection of the absorption derivative. III. THEORETICAL BACKGROUND The magnetization dynamics in the FM layer can be described by the LLG equation27,28 dMFM dt=−γMFM×Heff+α/parenleftbigg MFM×∂ˆm ∂t/parenrightbigg , (1) where γis the absolute value of the electron gyromagnetic ratio, ˆmis the unit vector in the direction of the FM magnetization vector MFM, andαis the dimensionless Gilbert damping parameter. The ﬁrst term in Eq. ( 1) represents the precessional torque in the effective magnetic ﬁeld Heffand the second term represents the well-known Gilbert dampingtorque. 28 Considering only the ﬁrst term in Eq. ( 1) and following the Smit and Beljers scheme,29the dispersion relation for theFM-AF bilayer can be calculated from the roots of a 4 ×4 matrix.30Then, the resonance condition obtained is /parenleftbiggω2 γ2/parenrightbigg =/bracketleftBigg/parenleftBigg EφFMφFM−E2 φFMφAF EφAFφAF/parenrightBigg/parenleftBigg EθFMθFM−E2 θFMθAF EθAFθAF/parenrightBigg/bracketrightBigg ×1 (tFMMFM)2, (2) where tFMis the FM layer thickness and each term Eijdenotes the second derivative of the free energy per unit surface withrespect to the equilibrium angles of the ferromagnetic ( φ FM) and the antiferromagnetic ( φAF) magnetization. In deriving Eq. ( 2), we consider that the dc magnetic ﬁeld is applied in the plane of the sample, therefore, we can regard that themagnetizations of the both layers are in the plane, i.e., θ FM= θAF=π/2. The behavior of the system can be described by a phe- nomenological model in terms of the magnetic free energy.Here we assume that a magnetization domain wall sets upat the AF layer as the FM magnetization rotates away fromthe equilibrium position. In the framework of this model, themagnetic free energy per unit area of the FM-AF bilayer canbe written as E=/bracketleftbigg −H·M FM+/parenleftbigg 2πM2 FM−KS tFM/parenrightbigg/parenleftbiggMFM·ˆn MFM/parenrightbigg2 −KU/parenleftbiggMFM·ˆuFM MFM/parenrightbigg2/bracketrightbigg tFM−JEMFM·MAF MFMMAF −σWMAF·ˆuAF MAF−(MFM·HRA)tFM, (3) where the ﬁrst three terms in order are the Zeeman, de- magnetizing, and uniaxial anisotropy energies of the FMlayer, with K SandKUthe surface and uniaxial anisotropy constants, respectively. The fourth and ﬁfth terms are theFM-AF exchange coupling and the AF domain-wall energies,respectively. The last term in Eq. ( 3) is the rotatable anisotropy energy, 13where HRAis the corresponding effective ﬁeld which accounts for the spins at the AF part of the FM-AF interfacethat rotate together with the FM magnetization. The unitvectors ˆn,ˆu FM, and ˆuAFrepresent the normal to the ﬁlm’s surface direction, the FM uniaxial anisotropy direction, andthe AF pinning direction, respectively. Figure 1shows the coordinate system used to describe these vectors and theorientation of the magnetizations M FM,MAF,a sw e l la st h e applied magnetic ﬁeld Hdirection. For the resonance ﬁeld values measured in this work, the magnetization and the external ﬁeld are parallel, i.e., φFM∼= φHand from Eqs. ( 2) and ( 3) the FMR frequency is given by ω=γ/radicalbig HYHZ, (4) where HY=H+HUcos 2(φH−η)+HRA+Heff 2 (5) and HZ=H+4πM eff+HUcos2(φH−η)+HRA+Heff 1, (6) 224418-2CRITICAL THICKNESS INVESTIGATION OF MAGNETIC ... PHYSICAL REVIEW B 83, 224418 (2011) where24 Heff 1=HWcosφAFcos(φH−φAF−β)−HEsin2(φH−φAF−β) HW HEcosφAF+cos(φH−φAF−β) (7) and Heff 2=HWcosφAFcos(φH−φAF−β) HW HEcosφAF+cos(φH−φAF−β). (8) Here,His the strength of applied magnetic ﬁeld, HU=2KU/M FMis the FM uniaxial anisotropy ﬁeld, HE=JE/(tFMMFM) is the exchange coupling ﬁeld, HW=σW/(tFMMFM) is the domain-wall effective ﬁeld, and Meffis the effective magnetization deﬁned by 4 πM eff=4πM FM−HS, where HS=2KS/M FMtFMis the surface anisotropy ﬁeld. As shown in Fig. 1, the easy axes of both FM and AF layers are supposed to be oriented along the angles ηandβrespectively, with respect to the reference direction that is the direction of the applied magnetic ﬁeld during the growth. From Eq. ( 4), we can write the resonance ﬁeld HRas function of the azimuthal angle φHas HR=1 2/bracketleftbigg HU(1−3 cos2(φH−η))−4πM eff−2HRA−Heff 1−Heff 2 +/radicalBigg /bracketleftbig HUsin2(φH−η)+4πM eff+Heff 1−Heff 2/bracketrightbig2+4/parenleftbiggω γ/parenrightbigg2/bracketrightbigg . (9) Thus we can determine the anisotropy ﬁelds from the measured angular dependence of HR. As we pointed out, the second term in Eq. ( 1) corresponds to the phenomenological Gilbert damping torque, which leads to a ﬁnite width of theresonance signal in a FMR experiment. However, not all themagnetization damping processes can be written as the Gilbertdamping term. We have to add the so-called extrinsic relaxationprocesses, and their quantitative description depends on thespeciﬁc physical origin of the phenomenon responsible by theenergy loss. xy φ FIG. 1. (Color online) Coordinate system used in the ferromag- netic resonance analysis. All the vectors are in the ﬁlm plane ( xy plane).In this work, the measured FMR linewidths are analyzed taking into consideration the superposition of three differentcontributions, /Delta1H=/Delta1H Gilbert+/Delta1H 2M+/Delta1Hmosaic, (10) where /Delta1H denotes the peak-to-peak linewidth of the FMR signal. Here, the ﬁrst term corresponds to the standard Gilbertcontribution, which is derived directly from the LLG equation(1), leading to the linewidth given by 31 /Delta1H Gilbert=2√ 3α γω. (11) The second term in Eq. ( 10) is the two-magnon contribution to the linewidth, /Delta1H 2M, which is caused by scattering of the uniform ( k=0) precession mode, excited by FMR, into nonuniform modes ( k/negationslash=0magnons) that are degenerate in frequency.32–39As such a process cannot conserve momentum, it requires the presence of defects that serve as scatterersin order to conserve the total momentum. To describe thetwo-magnon contribution to the FMR linewidth in our ﬁlms,we use the Arias-Mills formulation, 32–39in which the angular dependence of the linewidth /Delta1H 2Mis obtained by introducing a scattering matrix of extrinsic nature into the spin-waveHamiltonian, leading to /Delta1H 2M=/Gamma10 (HY+HZ)2/braceleftbigg H2 Y+[HYcos2φH+HZcos (2φH)]2 ×/parenleftBig/angbracketleftBiga c/angbracketrightBig −1/parenrightBig +[HYsin2φH−HZcos (2φH)]2 ×/parenleftbigg/angbracketleftbiggc a/angbracketrightbigg −1/parenrightbigg/bracerightbigg sin−1/radicalBigg HY HZ. (12) 224418-3R. L. RODR ´IGUEZ-SU ´AREZ et al. PHYSICAL REVIEW B 83, 224418 (2011) In this formulation, the surface and interface roughness of the ﬁlms occurring on the short length scales resultsin a variation of the surface anisotropy that can activatetwo-magnon scattering. Here, the defects are supposed to berectangular in shape, with height band lateral dimensions a andcrandomly distributed. It is important to note that Eq. ( 12) is a generalization of Eq. (94) of Ref. 33, in which we include the in-plane uniaxial anisotropy for the case where the externalﬁeldHis applied, making an angle φ Hwith the easy axis.40 The last term in Eq. ( 10) is the line broadening due to the mosaicity ( /Delta1Hmosaic). This term is caused by a small spread of parameters on a very large scale.37,41,42The simplest model of the FMR linewidth and inhomogeneity attributesthe inhomogeneous broadening to parameter variations withinthe sample. This variation can be found in the internalﬁelds, magnetization, surface anisotropy, ﬁlm thickness, andmagnetocrystalline anisotropy. Thus, individual regions ofthe ﬁlm will have slightly different resonance ﬁelds. Here,we will consider ﬂuctuations in the uniaxial anisotropy ﬁelddirections and in the exchange coupling ﬁeld strength H E. Then, the resulting spread of resonance frequencies yields aﬁeld linewidth that is given by 37,41,42 /Delta1Hmosaic=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂H R ∂η/Delta1η/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂H R ∂β/Delta1β/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂H R ∂HE/Delta1HE/vextendsingle/vextendsingle/vextendsingle/vextendsingle,(13) where /Delta1ηand/Delta1βrepresent the average spread of the direction of the easy axes of the FM and the AF ﬁlms, respectively, and/Delta1H Erepresents the strength variation of HEat the FM-AF interface. IV . RESULTS AND DISCUSSION In this section we present the experimental results and interpretation, including a comparison with the theoreticalresults. The effective magnetization value 4 πM eff=9.93 kG was extracted from the ﬁt of the experimental data corre-sponding to the Ni 81Fe19ﬁlm, where γ=17.6 GHz /kOe. Figure 2shows the in-plane dependence of the FMR ﬁeld for three bilayers of Ni 81Fe19(10 nm)/Ir 20Mn 80(tAF)o fs e r i e sA withtAF=1.2, 4.2, and 12 nm. The solid line in each case corresponds to the best ﬁt to the experimental data usingEq. ( 9) from which the values of the exchange and anisotropy ﬁelds were extracted. Notice that when the AF thicknessincreases, the average value of the resonance ﬁeld is overalldownshifted by an amount that corresponds to the rotatableanisotropy ﬁeld H RA, as seen in Eq. ( 9). At small tAF,t h e curves exhibit the symmetry of a typical uniaxial anisotropy,and for large t AFthe curves show a bell-shaped symmetry, which is typical of unidirectional anisotropy as expected forthe case in which H E/greatermuchHU. Figure 3(a)shows the behavior of the EB ﬁeld HE(circles) and the rotatable anisotropy ﬁeld HRA(squares) as a function of the Ir 20Mn 80layer thickness tAFfor samples of series A. Three regions are clearly distinguished, shown in Fig. 3(a) as separated by vertical dashed lines: In region I, wherethe AF layer thickness is smaller than the critical thickness,t AF<t cr=40˚A, the EB HEﬁeld vanishes, and the HRAﬁeld increases sharply with increasing thickness; region II, with40˚A<t AF<80˚A, where the EB reaches the saturation value and the rotatable anisotropy ﬁeld reduces monotonically with0 60 120 180 240 300 3600.40.50.60.70.80.91.0 t = 1.2 nm t = 4.2 nm t = 12 nmH (kOe) φH (deg) FIG. 2. (Color online) In-plane angular dependence of the FMR ﬁeld for a series of NiFe/IrMn bilayers grown at a deposition angle of 0◦. While the FM layer thickness is ﬁxed in 10 nm, we show data for three different AF layer thicknesses tAF=1.2, 4.2, and 12 nm. an increasing thickness value; and region III for tAF>80˚A, where both HEandHRAreach their saturation values. In region I the increase in HRAis attributed to the establishment of the AF order of the Ir 20Mn 80layer. However, since HE=0i nt h i s region, the AF order occurs in the rotatable AF grain regime.13 AttAF∼=40˚A, where the EB anisotropy starts to establish, HRAreaches its maximum value. In region II, opposite trends forHEandHRAindicate that changes at the FM-AF interface occur,15with the increase of HEproduced by the increasing number of the stable AF grains at the expense of a reduction inthe number of the rotatable ones. In region III, for t AF>80˚A, bothHEandHRAattain their saturation values because most of the IrMn grains at the interface are magneticallystable. The FMR linewidth dependence as a function of the AF layer thickness, for the samples of series A, is consistent withthe analysis discussed above. With the increase of the AF layer 0150300450 0 2 04 06 08 0 1 0 0 1 2 0 1 4 0050100150200 (a)HRA, HE (Oe) H ; H (b) ΔH; ΔH ; ΔH +ΔH ΔH (Oe) t (Å) FIG. 3. (Color online) (a) AF layer thickness dependence of the EB ﬁeld HE(circles) and rotatable anisotropy ﬁeld HRA(squares) obtained from the ﬁtting of the FMR data. (b) AF layer thickness dependence of the FMR linewidth (circles). The linewidth value isaveraged over all the in-plane resonance spectra for each sample. The FM-AF bilayers were grown at a deposition angle of 0 ◦. The solid lines in (a) and (b) are guides for the eyes. 224418-4CRITICAL THICKNESS INVESTIGATION OF MAGNETIC ... PHYSICAL REVIEW B 83, 224418 (2011) ΔH (Oe) t = 80 Å ΔH (Oe) Δ Δ Δ Δβ t = 61 Å φH (deg) FIG. 4. (Color online) Angular dependence of the FMR linewidth for the bilayers deposited at an angle of 0◦. The circles are the experimental data obtained at a frequency of 8.61 GHz, whereas the solid lines are ﬁts obtained from Eq. ( 10) with the parameters showed in Table I. The dashed lines correspond to the individual contributions of the three relaxation mechanisms considered. thickness, /Delta1H initially increases, exhibiting a peak value at tAF∼=26˚A, and then decreases until it reaches a saturation value at tAF∼=42˚A. Notice that the abrupt variation of /Delta1H occurs in region I, where HRAalso varies sharply and the EB ﬁeld vanishes. Hence, both behaviors are mostly related tothe same interface phenomenon. In fact, the existence of AFgrains at the interface can contribute to the enhancement ofthe magnetic damping with two different mechanisms: two-magnon scattering and mosaicity, as described in Sec. III. Inorder to determine the contributions from the two relaxationmechanisms, we have investigated the angular dependence ofthe FMR linewidth. Figure 4shows the linewidth data (symbols) and theoretical ﬁts (solid lines) obtained with the model based on Eq. ( 10)f o r two different Py(10 nm)/IrMn( t AF) bilayers of series A, with tAF=61 and 80 ˚A. The parameters used to ﬁt the angular dependence of the FMR linewidth are the Gilbert dampingcoefﬁcient αof Eq. ( 11),/Gamma1 0and/angbracketleftc/a/angbracketrightof the expression corresponding to the two-magnon contribution [Eq. ( 12)], as well as /Delta1η,/Delta1β, and /Delta1HEof Eq. ( 13) (see Table I). The values of 4 πM effand the anisotropy ﬁelds needed forﬁtting the linewidth data were determined by analyzing the angular dependence of FMR ﬁeld as shown in Fig. 2. The ﬁtting parameters /Gamma10and/angbracketleftc/a/angbracketrightof Eq. ( 12) both affect the vertical scale of /Delta1HvsφH,b u t/angbracketleftc/a/angbracketrightalso affects the shape of the curve. Thus, by ﬁtting the theory to the data, we canobtain information about the parameters related to the extrinsictwo-magnon process. Taking into account only one of themechanisms through to Eqs. ( 11), (12), or ( 13), one obtains the dashed curves. This shows that each of them separately is notsufﬁcient to describe the magnetization relaxation of the ﬁlms.The entire angular dependence of the FMR linewidth (solidline) can be explained by adding the two-magnon scatteringand local inhomogeneous effects. In Fig. 3(b) we show the Gilbert (open circles) and the extrinsic ( /Delta1H 2M+/Delta1Hmosaic, open squares) contributions to the linewidth for tAF/greaterorequalslant54˚A. In this region we can clearly identify both extrinsic and intrinsicprocesses. For lower values of t AFthe analysis of the data is not straightforward due to the fact that the angular variationof/Delta1H is less pronounced, and consequently in region I it remains undetermined which of the mechanisms is the mostpronounced. In order to get detailed information on the origin of the pronounced peak exhibited by the linewidth shown in Fig. 3(b), we decided to prepare a series of samples in which the differentrelaxation mechanisms could be extracted. Following this idea,we fabricate a series of Ni 81Fe19(10 nm)/Ir 20Mn 80(tAF) bilayers (named series B) by oblique sputtering at a deposition angleof 50 ◦. It has been known that magnetic ﬁlms fabricated under oblique deposition conditions can exhibit very strong uniaxialanisotropy (see Table II) that depends on the inclination angle of the substrate during the deposition process. 26T h eo r i g i no f this growth-induced magnetic anisotropy has been attributedto columnar grain structures that grow tilted toward the depo-sition direction with respect to the substrate normal. 43–46On the other hand, within the plane of the ﬁlm, the self-shadowingeffect that occurs during the ﬁlm growth is more effective inthe deposition ﬂux direction, producing grains that tend tobe elongated perpendicular to this direction. 47,48These two effects, shape anisotropy and magnetocrystalline anisotropy,both contribute to the observed magnetic anisotropy ofthe ﬁlms. As shown in Ref. 26and emphasized here, the ﬁlms grown by oblique sputter deposition turned out to bea useful prototype for the investigation of FM relaxationmechanisms. The phenomenological parameters used in the calculation areγ=17.6 GHz/kOe and 4 πM eff=9.65 kG, extracted from the ﬁt of the experimental data corresponding to the Ni 81Fe19 ﬁlm grown under oblique angle conditions. The in-planevariations of the FMR ﬁelds for three samples of series B,shown in Fig. 5, exhibit a pronounced uniaxial anisotropy in comparison to the bilayers of series A, shown in Fig. 2. TABLE I. Magnetic relaxation parameters extracted from the ﬁt of the linewidth data. The samples were grown at a deposition angle of 0◦. tAF(˚A) HU(Oe) α/Delta1 H 2M(Oe) /Gamma10(Oe)/angbracketleftbigc a/angbracketrightbig /Delta1HE(Oe) /Delta1β(deg) /Delta1η(deg) 61 40 0.006 43 37 1.003 12 11 0 80 45 0.006 39 34 1.002 16 13 0 224418-5R. L. RODR ´IGUEZ-SU ´AREZ et al. PHYSICAL REVIEW B 83, 224418 (2011) TABLE II. Magnetic relaxation parameters extracted from the ﬁt of the linewidth data showed in Fig. 7. All samples were grown at a deposition angle of 50◦. tAF(˚A) HU(Oe) α/Delta1 H 2M(Oe) /Gamma10(Oe)/angbracketleftbigc a/angbracketrightbig /Delta1HE(Oe) /Delta1β(deg) /Delta1η(deg) 2 90 0.006 17 13 1.019 0 0 0 6 90 0.007 17 13 1.018 0 0 0 19 125 0.008 60 46 1.013 0 0 0 33 130 0.009 34 26 1.012 4 0 045 110 0.008 39 30 1.017 12 0 0 56 119 0.008 39 30 1.016 11 0 0 This is clear from the angular dependence of the resonance ﬁeld that shows two peaks near φH=90◦andφH=270◦. Furthermore, the curves at different AF thicknesses exhibittwo different features. The average values of H Rare shifted along the vertical axis by an amount given by HRAand the unidirectional symmetry becomes evident as tAFincreases. Figure 6(a) shows the exchange ﬁeld HE(circles) and the rotatable anisotropy ﬁeld HRA(squares) as a function of the AF layer thickness for all samples of series B. As can beseen, as the samples of series A, three regions are clearlydistinguished: region I, in which the exchange ﬁeld H Eis null and the HRAvalue increases; region II, where EB starts to establish and HRAreduces; and region III, where all the ﬁelds reach saturation. This shows that the same interplay betweenrotatable and frozen AF grains occurs in the samples grownat an oblique angle of 50 ◦, as was investigated in the samples grown at an oblique angle of 0◦. The solid circles in Fig. 6show the average values of the FMR linewidth as a function of the AF layer thicknessfor the samples of series B. As one can see, following thetrend observed in the samples of series A [see Fig. 3], /Delta1H experiences a nonmonotonic behavior with increasing t AF. First it exhibits a maximum at tAF=17˚A, and then it decreases until it reaches saturation at tAF=20˚A. Notice that this interval corresponds to region I of Fig. 6(a) in which HRAattains its maximum value. As previously explained, 0 60 120 180 240 300 3600.60.70.80.91.0 tAF=2 Å tAF=20 Å tAF=55 ÅHR(kOe) φH (deg) FIG. 5. (Color online) In-plane angular dependence of the FMR resonance ﬁeld for bilayers with three different AF layer thicknesses (the solid lines are ﬁt to the data). The FM-AF bilayers were grown at a deposition angle of 50◦.at low values of tAF, where the AF layers do not exhibit a stable AF ordering with ﬂuctuations of its magnetizationcontributing to the enhancement of the magnetic damping,the origins of the relaxation mechanisms were not obvious.Actually, as is clear from the results shown in Fig. 6(b),t h e dominant contribution to the linewidth in this interval (17 ˚A/lessorequalslant t AF/lessorequalslant22˚A) is due to the two-magnon scattering mechanism. This supports our assumption that the onset of HRAis related to the existence of unstable AF grains. Below tAF=17˚A, the two-magnon scattering and Gilbert relaxation processesare the two main operative mechanisms. For t AF>22˚A we have to also consider the contribution from mosaicity inaddition to two-magnon scattering and Gilbert processes [seeEq. ( 10)]. The angular dependence of the linewidth for some rep- resentative bilayers of series B are shown in Fig. 7.A sw e mentioned, these ﬁlms exhibit a high uniaxial anisotropy,and consequently the angular dependence of /Delta1H shows a pronounced uniaxial symmetry. This is a clear signature ofthe occurrence of the two-magnon scattering mechanism that 0100200300 ΔH; ΔH ÅGilbert; ΔH2M; ΔHmosaic 0 1 02 03 04 05 06 00255075100HE,HRA(Oe)HE HRA(a)ΔH (Oe) tAF()(b) FIG. 6. (Color online) (a) AF layer thickness dependence of the EB ﬁeld HE(circles) and rotatable anisotropy ﬁeld HRA(squares) obtained from the ﬁtting of the FMR data. (b) AF layer thicknessdependence of the FMR linewidth (solid circles). The linewidth value is averaged over all the in-plane resonance spectra for each sample. The open symbols correspond to the intrinsic and extrinsiccontributions to the linewidth. All the samples were grown at a deposition angle of 50 ◦. The solid lines in (a) and (b) are a guide for the eye. 224418-6CRITICAL THICKNESS INVESTIGATION OF MAGNETIC ... PHYSICAL REVIEW B 83, 224418 (2011) 0306090120 0306090120 0306090120 0306090120 0 60 120 180 240 300 3600306090120 0 50 100 150 200 250 300 3500306090120ΔH (Oe)tAF=2 ÅΔH (Oe)tAF=6 Å tAF=19 Å tAF=33 ÅΔH (Oe)tAF=45 Å φH (deg) φH (deg) tAF=56 Å FIG. 7. (Color online) Angular dependence of the FMR linewidths for the bilayers deposited at an angle of 50◦. The squares are the experimental data obtained at a frequency of 8.61 GHz, whereas the solid lines are ﬁts obtained from Eq. ( 10) with the parameters showed in the Table II. The dashed lines correspond to the individual contributions of the three relaxation mechanisms considered ( /Delta1H 2M: dashed line; /Delta1Hmosaic: dashed-dotted line; /Delta1H Gilbert : dashed-dotted-dotted line). becomes quite active in these samples. The dashed curves in Fig.7represent the contributions of each of the three different relaxation mechanisms considered [see Eq. ( 13)]. As we can see, the presence of /Delta1Hmosaicdue to the strong variation of HE at the FM-AF interface ( /Delta1HE, see Table II), fortAF=33, 45, and 56 ˚A, tends to break the uniaxial symmetry coming from the/Delta1H 2Mmechanism. The ﬁts also reveal that the average spread of the direction of the easy axes in the FM and AF ﬁlms(represented by /Delta1ηand/Delta1β, respectively) is negligible. V . CONCLUSIONS In summary, we have studied the AF thickness dependence of the exchange bias, the rotatable anisotropy ﬁeld, and theFMR linewidth in polycrystalline Ni 81Fe19/Ir20Mn 80bilayers. The different trends of the rotatable anisotropy and EB ﬁeldsare correlated with the stability of the AF grains at the interface,conﬁrming the assumption that the isotropic resonance ﬁeldshift is induced by the rotatable anisotropy. We believe thatthis is a general feature of the EB polycrystalline systems. TheFMR linewidth shows a nonmonotonic dependence on the AFthickness. We conﬁrmed that the model of Arias and Mills 33is capable of quantitatively describing the angular dependenceof the measured linewidth. Besides, the Gilbert damping, the two-magnon damping, and the mosaicity effects stronglyaffect its dependence. We showed that the effective magneticdamping can be adjusted over a wide range by changing theAF thickness. In particular, we found that the peak in the FMRlinewidth at certain AF thickness is due to the enhancementof the two-magnon scattering. These results are important forapplications that require the understanding of how tune themagnetic damping, which is directly applied in the dynamicresponse in spintronic devices. ACKNOWLEDGMENTS This work has been supported by the Brazilian agencies CNPq, CAPES, FINEP, FACEPE, by the Fondo Nacionalde Investigaciones Cient ´ıﬁcas y Tecnol ´ogicas (FONDECYT, Chile) under Grants No. 1085229 and No. 11080246, byMillennium Science Nucleus “Basic and Applied Magnetism”Grant N ◦P10-061-F, and by the Financiamiento Basal para Centros Cient ´ıﬁcos y Tecnol ´ogicos de Excelencia CEDENNA FB0807. 224418-7R. L. RODR ´IGUEZ-SU ´AREZ et al. PHYSICAL REVIEW B 83, 224418 (2011) 1W. H. Meiklejohn and C. P. Bean, Phys. Rev. 102, 1413 (1956). 2J. Nogu ´es and I. K. Schuller, J. Magn. Magn. Mater. 192, 203 (1999). 3A. E. Berkowitz and K. Takano, J. Magn. Magn. Mater. 200, 552 (1999). 4J. Nogu ´es, J. Sort, V . Langlais, V . Skumryev, S. Suri ˜nach, J. S. Mu˜n o za n dM .D .B a r ´o,Phys. 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PhysRevMaterials.3.114410.pdf	PHYSICAL REVIEW MATERIALS 3, 114410 (2019) Spin-orbit torque in chemically disordered and L1 1-ordered Cu 100-xPtx Xinyu Shu,1Jing Zhou,1Jinyu Deng,1Weinan Lin,1Jihang Yu,1Liang Liu,1Chenghang Zhou,1 Ping Yang,1,2and Jingsheng Chen1,* 1Department of Materials Science and Engineering, National University of Singapore, Singapore 117576, Singapore 2Singapore Synchrotron Light Source (SSLS), National University of Singapore, 5 Research Link, Singapore 117603, Singapore (Received 22 May 2019; published 18 November 2019) The binary alloys with heavy elements have been considered promising candidates for spin-orbit torque application due to the tunable spin Hall effect. In light of previous studies, the effect of crystalline structureon spin Hall effect in nonmagnetic alloys has not been thoroughly studied. Here, we present a systematicinvestigation of the spin-orbit torques in chemically disordered Cu 100-xPtxand L1 1-Cu 50Pt50by the spin torque ferromagnetic resonance technique. The results indicate that both the atomic concentration and the degree ofthe chemical ordering substantially inﬂuence the spin-orbit torque efﬁciency of the CuPt alloys. In chemicallydisordered Cu 100-xPtx, the primary mechanism of spin Hall effect changes from extrinsic to intrinsic when the Pt concentration is increased to larger than 80%. In L1 1-Cu 50Pt50with weak chemical ordering, the side-jump and intrinsic mechanisms dominate, whereas the skew scattering mechanism dominates for strong chemical ordering.This work provides a perspective to control the spin-orbit torques in alloys. DOI: 10.1103/PhysRevMaterials.3.114410 I. INTRODUCTION The spin-orbit torque (SOT) has attracted increasing atten- tion due to its potential application in energy-efﬁcient devices[1]. The SOT comprises a dampinglike component, which can switch the magnetization of a ferromagnet [ 2–4], and a ﬁeld- like component, which drives the magnetic precession [ 3,4]. In a ferromagnet/heavy metal (FM/HM) heterostructure, it iscommonly believed that the SOT originates from the Rashba-Edelstein effect due to the inversion-symmetry breaking atthe interface [ 5] and/or the spin Hall effect (SHE) arising from the strong spin-orbit coupling (SOC) in nonmagneticHMs [ 2]. The Rashba-Edelstein effect at the interface between nonmagnetic HM and FM mainly exerts a ﬁeldlike torque onmagnetization to promote the precession [ 5,6]. In the scenario of the SOT from the SHE, spin current generated in the HMlayer can be transferred to an adjacent FM layer and exertsa dampinglike torque on the magnetic moment to switch themagnetization of the FM [ 4]. This magnetization switching efﬁciency depends on the SOT efﬁciency ( θ SH), which incor- porates the charge-to-spin conversion efﬁciency in the HMlayer through the SHE and the spin transmission efﬁciency atthe interface. Therefore, the investigation of large and tunableθ SHis of critical importance for magnetization switching. Extensive studies on the SHE in nonmagnetic HMs (such as Ta [ 3,7,8], Au [ 9,10], and Pt [ 11–13]) and alloys [ 1,14– 17] have been carried out. The origin of the SHE has been attributed to three mechanisms: intrinsic, side jump, and skewscattering [ 2]. It is believed that the intrinsic SHE is propor- tional to Z 4, in which Z is atomic weight [ 2,9]. The extrinsic mechanism, including side jump and skew scattering, canbe adjusted using crystalline defects [ 9,11,18] and impurities Corresponding author: msecj@nus.edu.sg[1,14–17]. The introduction of defects and impurities would increase the probability of electron scattering inside the HM.As a result, the electrical resistivity ( ρ o) and thus θSHare enhanced according to their correlation θSH=σSHρo(where σSHis the spin Hall conductivity) [ 2,7,9,14,18]. Based on this knowledge, researchers have succeeded in manipulatingtheθ SHby changing the atomic composition in nonmagnetic alloys [ 1,14–17]. Given the strong correlation among the atomic composition, ρoandθSH, however, the understanding of the intrinsic and the extrinsic contributions to SHE in alloysis still vague. Furthermore, ρ ocan be controlled by changing the degree of chemical ordering for single-crystalline alloys [ 19,20]. It is thus possible to modulate θSHby adjusting the degree of chemical ordering. To date, studies on the chemical-ordering-dependent SOT review that the magnetic structures of FMs[21,22] and antiferromagnets (AFMs) [ 23,24] can affect SOT generation. However, experimental evidence of chemical-ordering-dependent θ SHin nonmagnetic alloys is still lacking. In this work, θSHandρoin the chemically disor- dered Cu 100-xPtxand L1 1-Cu 50Pt50alloys are systematically studied. We used a spin torque ferromagnetic resonance(ST-FMR) technique to evaluate the θ SHof Cu 100-xPtxin Cu100-xPtx/Ni81Fe19[also known as Permalloy (Py)] bilayer. The results show that the θSHof CuPt alloys changes with ρo, which is inﬂuenced by both the atomic concentration and the degree of chemical ordering. Moreover, the mechanismsof SHE in L1 1-Cu 50Pt50and chemically disordered Cu 100-xPtx are interpreted by analyzing the components of ρousing phenomenological models. II. EXPERIMENTAL DETAILS Three types of Cu 100-xPtx/Py bilayer samples were fabri- cated using DC magnetron sputtering, including chemicallydisordered polycrystalline Cu 100-xPtx(11)/Py(11), and 2475-9953/2019/3(11)/114410(7) 114410-1 ©2019 American Physical SocietyXINYU SHU et al. PHYSICAL REVIEW MATERIALS 3, 114410 (2019) 20 30 40 50 20 30 40 50108 104 1001012 109 106 1031012 500 °C (d) (c)(111) STO(222) R (111) Py (111) R 2θ (°)(a) (b) I(111)R /I(222)R2θ (°) Φ(222) R (101) RCounts (arb. units) Counts (arb. units)Counts (arb. units)300 °C 700 °C(111) R(111) STO(222) R (111) Py Cu50Pt50 Cu30Pt70 Pt -180 -90 0 90 180 Φ (°)101102 0 300 600 9000.00.20.40.60.81.0 Tgrowth (°C) FIG. 1. (a) XRD θ-2θresults for Cu 50Pt50/Py in which the Cu 50Pt50samples were prepared at 300◦C, 500◦C, and 700◦C. (b) XRD θ-2θ results for Cu 50Pt50/Py, Cu 30Pt70/Py, and Pt/Py in which the Cu 100-xPtxlayers were prepared at 500◦C. (c) The /Phi1-scan result of L1 1-Cu 50Pt50 deposited at 500◦C. (d) The correlation between the square-rooted integrated intensity ratio of I(111) Rover I(222) RandTgrow. chemically disordered single-crystalline Cu 100-xPtx(11)/ Py(11) and L1 1-Cu 50Pt50(11)/Py(11). Numbers in the parentheses are the thicknesses in nanometers, which weremeasured by x-ray reﬂectometry. Polycrystalline Cu 100-xPtx thin ﬁlms were deposited on the thermally oxidized SiO 2 substrate at room temperature, while single-crystalline Cu100-xPtxﬁlms were epitaxially grown on SrTiO 3(STO) (111) substrate at elevated temperatures. The Py layerwas deposited at room temperature. All samples wereprotected by a 2-nm SiO 2layer. The atomic composition of Cu100-xPtxwas adjusted by changing the sputtering power and measured by energy-dispersive x-ray spectroscopy(EDS). The crystal structure of single-crystalline Cu 100-xPtx was examined by x-ray diffraction (XRD). The devices for ST-FMR measurements were patterned by a combination ofphotolithography and ion-beam etching [ 25]. III. RESULTS AND DISCUSSION Figure 1(a) shows the θ-2θscans of Cu 50Pt50(11)/Py(11) deposited on STO(111) substrates with Cu 50Pt50fabricated at different temperatures ( Tgrow). The peak at 2 θ=20.31◦ corresponds to the (111) plane of the L1 1superlattice in a rhombohedral structure denoted as (111) R, while the peakat 2θ=41.17◦corresponds to the ﬁrst fundamental peak (222) R.F o rC u 50Pt50prepared at 500◦C and 700◦C, evi- dent (111) Rpeaks can be observed. For Cu 50Pt50prepared at 300◦C, the (111) Rpeak disappears, which indicates the chemically disordered phase. There is no shift of (222) Rpeak position among those three Cu 50Pt50samples, which indicates the same atomic composition according to the Vegard’s law.The peaks at 2 θ=44.08 ◦indicate the (111)-textured Py [ 26]. Figure 1(b) shows the θ-2θscan of Cu 100-xPtxdeposited at 500◦C. The shift of the (222) Rpeak is attributed to composi- tion changes. The (111) texture of Py is retained regardlessof the Cu content. Figure 1(c) shows a typical /Phi1scan of Cu 100-xPtxdeposited at temperatures higher than 300◦Ct o conﬁrm the epitaxial growth. A threefold symmetry consistingof (101) R, (011) R, and (110) Rpeaks around the (222) Rpeak is evident. The long-range ordering parameter, which is deﬁned as S=rA−cA cB[19], quantiﬁes the degree of chemical ordering of the L1 1-Cu 50Pt50ﬁlms. Here, rCustands for the fraction of Cu atoms on the correct atomic position to form the L1 1 structure, cCuand cPtare the atomic concentrations of Cu and Pt atoms. For the perfectly ordered L1 1structure, S=1, which means all the Cu and Pt atoms occupy their respectiveright position to form the L1 1structure. In the case of perfectly 114410-2SPIN-ORBIT TORQUE IN CHEMICALLY DISORDERED … PHYSICAL REVIEW MATERIALS 3, 114410 (2019) DC inAC generator Bias Tee SiO2 Py(a) (c) xyz JCHext M 40°ADFL D CA Hrf(b) Ref. in Lock-in ampli erCuPt 010 GHz 400An symmetricData 0.04.0 -4.0 -8.0 Vmix (V)Data Fing curveH (Oe) 20 0406080 0 f (GHz) = 0.0134 Hinh = 1.01 Oe f = 7 GHz-10.0-5.00.05.0Me = 619.32 emu.cc Hk = 10.79 Oe4812Frequency (GHz) 0 0 Hext (Oe)SymmetricFing curve 800 12009 GHz6 GHz5 GHz 7 GHz 8 GHz Hext (Oe)500 1000 1500 2000Data Fing curveVmix (V)0 500 1000 1500 2000 4 8 12 Hres (Oe)(d) (f) (g)(e)HextHigh Freq. Low Freq. FIG. 2. (a)–(c) Illustrations for ST-FMR measurements. (d) Kittel ﬁtting and (e) damping constant ﬁtting of polycrystalline Cu50Pt50(11)/Py(11) for the ST-FMR measurements. (f) ST-FMR curves of polycrystalline Cu 50Pt50(11)/Py(11) at different frequencies. (g) Decomposition of the ST-FMR curve which was measured at 7 GHz. disordered CuPt, S=0, where Cu and Pt atoms are com- pletely random in the crystal lattice. Experimentally, Scan be extracted from XRD θ-2θcurves as S=C/radicalBig Is Io, in which Isis the integrated intensity of the ﬁrst superlattice peak [ I(111) R], andIois the integrated intensity of the ﬁrst fundamental peak [I(222) R].Cis a constant, including the Lorentz polarization factor and the structure factor [ 27]. The dependence of/radicalBig Is Io onTgrow for Cu 50Pt50is summarized in Fig. 1(d). The ratio/radicalBig Is Io, which is linear with S, shows a monotonic increase when Tgrowincreases from 300◦C to 700◦C. Further increas- ingTgrow to 900◦C leads to a slight decrease of chemical ordering.ST-FMR measurements were performed upon the polycrystalline Cu 100-xPtxand L1 1-Cu 50Pt50, both of which were patterned into microstrips [highlighted by a blackcircle in Fig. 2(a)] with dimensions of 30 ×150μm. The external magnetic ﬁeld H extwas applied at 40◦with respect to the direction of microwave in the sample plane [as shownin Fig. 2(b)]. The magnetic precession in the FM layer arises from the combined effects of antidamping torque (τ AD), damping torque ( τD), and ﬁeldlike torque ( τFL) [see Fig. 2(c)]. For the bilayer heterostructure composed by nonmagnetic HM and FM, the traditional ST-FMRmodel attributes τ ADto the SHE and τFLto the Oersted ﬁeld [ 3]. The τFLfrom the Rashba-Edelstein effect is believed to be negligible [ 3,4]. The magnetic precession 114410-3XINYU SHU et al. PHYSICAL REVIEW MATERIALS 3, 114410 (2019) FIG. 3. (a) θSHandρoin polycrystalline Cu 100-xPtx(11)/Py(11), and (b) θSHandρoin L1 1-Cu 50Pt50(11)/Py(11). leads to alternating resistance due to the anisotropic magnetoresistance effect of the FM layer. A rectifying DCvoltage V mixcan arise from mixing the alternating resistance and alternating current. Vmixcan be decomposed into a symmetric component Vs=/Delta1H/[/Delta1H2+(Hext−Hres)2] (corresponding to τAD) and an antisymmetric component VA=/Delta1H(Hext−Hres)/[/Delta1H2+(HextHres)2] (corresponding toτFL)[3,22], where Hres is the resonant ﬁeld and/Delta1His the half-width at half maximum of the ST-FMR curve. Finally, θSH is expressed as θSH= (VS/VA)(eμoMstd/¯h)[1+(4πMeff/Hres)]1/2[3,24], where MsandMeffare saturation and effective magnetization of the FM layer, respectively. ¯ his the reduced Plank constant, and t anddare thicknesses of the FM and HM layers, respectively. It should be noted that the measured θSHcorresponds to the lower boundary of the spin Hall angle due to the loss ofspin current density at the interface during the spin transferprocess. We exemplify a full set of ST-FMR results of polycrys- talline Cu 50Pt50(11)/Py(11) in Figs. 2(d)–2(g). The resonant frequency fis ﬁtted against Hresusing the Kittel equation f= (γ/2π)[(Hres+Hk)(Hres+Hk+4πMeff)]1/2, where γis the gyromagnetic ratio and Hkis the in-plane effective anisotropic ﬁeld [ 28]. In Fig. 2(d),MeffandHkare ﬁtted as 619.32 emu.cc and 10.79 Oe, respectively. The damping constant αcan be ex- tracted through /Delta1H=/Delta1Hinhom+2παf/γ, where /Delta1Hinhom is the inhomogeneous linewidth broadening [ 28], as shown in Fig. 2(e). Figure 2(f) presents Hext-Vmixcurves obtained at different frequencies. A further decomposition of Vmixat f=7 GHz is shown in Fig. 2(g). In this paper, the θSHfor each sample is averaged at four different frequencies (6.5, 7,7.5, and 8 GHz). The effects of atomic concentration and chemical ordering on the θ SHof CuPt are displayed in Fig. 3. Note that the θSHhas been calibrated considering interfacial transparency (see details in the Supplemental Materials, Sec. I[29]). In order to understand the mechanisms of SHE, ρoof all the single-layer CuPt was measured by four-probe technique tostudy its correlation with θ SH[7–13]. In Fig. 3(a),ρoof polycrystalline Cu 100-xPtxcan be well ﬁtted against cPtusing Nordheim’s rule as expressed by Eq. ( 1)[14]. Figure 3(b) presents the dependence of ρoon the degrees of chemicalordering in L1 1-Cu 50Pt50, which is ﬁtted by Rossiter’s model in Eq. ( 2)[19,20]: ρo=AcPtcCu+cPtρPt+cCuρCu, (1) ρo=ρdis,r1−/parenleftbig C/radicalBig Is Io/parenrightbig2 1−B/parenleftbig C/radicalBig Is Io/parenrightbig2+ρT. (2) In Eq. ( 1),Ais the Nordheim coefﬁcient, and ρCuandρPt are the resistivity of Cu and Pt, respectively. The ﬁtting re- sults are A=171.01μ/Omega1cm,ρPt=29.74μ/Omega1cm, and ρCu= 19.69μ/Omega1cm. According to Nordheim’s rule, the residual resistivity ρris conveyed as AcPtcCuand the temperature- dependent ρTis described as cPtρPt+cCuρCu[20]. In Eq. ( 2), Ccan be obtained by ﬁtting in order to calculate S.T h e constant B, which is also ﬁtted, depends on the relative po- sition of the Fermi surface and the superlattice Brillouin-zoneboundaries [ 19].ρ dis,ris the residual resistivity of perfectly disordered Cu 50Pt50(i.e., S=0), and the term ρdis,r1−S2 1−BS2 stands for the ρrof Cu 50Pt50with different degrees of chem- ical ordering. The measured ρoof L1 1-Cu 50Pt50with S= 0.46 deviates from the ﬁtting, which can be attributed to the effect of short-range ordering [ 20]. From ﬁtting, ρdis,r= 44.93μ/Omega1cm,ρT=30.08μ/Omega1cm,C=1.3, and B=0.78. Figure 3exhibits an evident correlation between θSHand ρoin polycrystalline Cu 100-xPtxand L1 1-Cu 50Pt50, regardless of how ρois modulated. In polycrystalline Cu 100-xPtx, both theθSHandρopresent a parabolic trend with the maxima at cPt=50%, where θSH=0.1452 and ρo=67.3μ/Omega1cm. On the other hand, the L1 1-Cu 50Pt50displays negative depen- dence of θSHonS. Unlike the cases in FM [ 21] and AFM materials [ 24], where dramatic changes in θSHare observed with simultaneous changes in crystal and magnetic structures,θ SHof L1 1-Cu 50Pt50can be modiﬁed by tuning the crystal structure alone, as represented by the differed degree of chem-ical ordering. In light of the single-element HMs, the correlation between θ SHandρocan be explained by the intrinsic and extrinsic contributions to spin Hall resistivity ρSH[7–9,18]. However, the correlation between θSHandρoin nonmagnetic binary 114410-4SPIN-ORBIT TORQUE IN CHEMICALLY DISORDERED … PHYSICAL REVIEW MATERIALS 3, 114410 (2019) FIG. 4. (a) Correlation between ρSHand c Ptin polycrystalline Cu 100-xPtxand ﬁtting results by applying Eq. ( 3). (b) Calculated intrinsic and extrinsic spin Hall resistivity including the side-jump and skew scattering contributions. (c) Correlation between ρSHandSin L1 1-Cu 50Pt50 and ﬁtting results by applying Eq. ( 4). The inset in (c) shows the change of θSHwithρo. Two dependences of θSHonρoare visually guided by the red dashed lines. A blue dotted line marks the crossover of these two dependences, which corresponds to S=0.78. alloys is more complicated due to the effects of the atomic concentration and the chemical ordering. Here we developtwo models to understand the correlations between ρ oandθSH in polycrystalline Cu 100-xPtxand L1 1-Cu 50Pt50, respectively: −ρSH=/parenleftbig θint SH,CucCu+θint SH,PtcPt/parenrightbig ρo +σsj SH(cPt)ρ2 r+θss,T SHρT+θss,imp SHρr, (3) −ρSH=σint SH(S)(ρr+ρT)2+σsj SH(S)ρ2 r +θss,T SHρT+θss,imp SHρr. (4) Equation ( 3) describes ρSHof polycrystalline Cu 100-xPtx. Here, the θint SH,Cu=0 and θint SH,Pt=0.07 are the intrinsic SOT efﬁciencies of Cu and Pt, respectively. σsj SH(cPt) stands for the concentration-dependent side-jump spin Hall conductiv- ity.θss,imp SH andθss,T SHare impurity-dependent and temperature- dependent components of skew scattering-induced SOT efﬁ-ciency, respectively [ 30]. The detailed construction of Eq. ( 3) is discussed in the Supplemental Material, Sec. II. Equa- tion ( 4) depicts ρ SHof L1 1-Cu 50Pt50.I nt h eL 1 1-Cu 50Pt50, the degree of chemical ordering is deemed to strongly affectthe electronic band structure [ 31,32]. Therefore, σ int SH(S) and σsj SH(S) stand for the S-dependent intrinsic and side-jump σsj SH, respectively. The effects of atomic concentration and chemical ordering on the ρSHof CuPt are well ﬁtted by Eqs. ( 3) and ( 4), as shown in Fig. 4. Figure 4(a) presents both the ﬁtting results and data of polycrystalline Cu 100-xPtx. The side-jump spin Hall conduc- tivityσsj SH(cPt) is approximated as ( acPt+b)[33]. The ﬁtted parameters from Eq. ( 3)a r eθss,T SH=0.0016, θss,imp SH=0.01, a=4.73×10−5μ/Omega1cm, and b=0.002μ/Omega1cm for the poly- crystalline Cu 100-xPtx. The contributions from different mech- anisms are plotted in Fig. 4(b), where \|ρint SH\|=(θint SH,CucCu+ θint SH,PtcPt)ρo,\|ρSJ SH\|=σsj SH(cPt)ρ2 r, and \|ρSS SH\|=θss,T SHρT+θss,imp SHρr, corresponding to intrinsic, side jump, and skew scattering ρSH, respectively. When cPt<80%, the side-jump contribution dominates the ρSH. When cPt>80%, the intrin- sic contribution overwhelms the extrinsic contribution. Themaximum of ρ SHin Fig. 4(a), as well as the maximum of θSHin Fig. 3(a), can be attributed to the dominant role of the side-jump mechanism, which is also maximum at cPt=50% in Fig. 4(b). Figure 4(c) shows the data of L1 1-Cu 50Pt50and its ﬁt through Eq. ( 4), where we approximately take σint SH(S) and σsj SH(S) as two constants. The well-ﬁtted results verify the validity of Eq. ( 4). In light of the σsj SH(S)ρ2 rin Eq. ( 4), the negative dependence of θSHonSin Fig. 3(b) can be attributed to the decrease of the side-jump contribution in L1 1-Cu 50Pt50 with increasing S. In a perfectly ordered L1 1-Cu 50Pt50,t h e crystalline structure acts as a translationally invariant conﬁgu-ration in which the electron scattering at impurities, as well asρ r, is negligible [ 19,20]. Therefore, it is expected that the ρr, as well as the side-jump contribution, reduces with increasingSin L1 1-Cu 50Pt50. Additionally, when S/greaterorequalslant0.78, the decrease ofρois much faster than the reduction in θSH, where θSHis almost independent of ρo, as shown in the inset in Fig. 4(c). This implies that there is a threshold value of Sat around 0.78 above which the dominant spin Hall mechanism changes fromside jump and intrinsic to skew scattering in L1 1-Cu 50Pt50. IV . CONCLUSION In summary, the SOTs of chemically disordered polycrys- talline Cu 100-xPtxand L1 1-Cu 50Pt50were investigated system- atically in this work. We demonstrate that both the atomiccomposition and the chemical order can strongly inﬂuencetheθ SHin CuPt alloy. In chemically disordered Cu 100-xPtx,t h e primary spin Hall mechanism changes from a side-jump tointrinsic mechanism when c Ptincreases to larger than 80%. The dominance of the side-jump mechanism accounts for 114410-5XINYU SHU et al. PHYSICAL REVIEW MATERIALS 3, 114410 (2019) the maximum of θSHandρSHat Pt concentration of 50%. In L1 1-Cu 50Pt50, there appears to be a threshold value of S at around 0.78, which corresponds to the transition of thedominant mechanism of the SHE, from side jump and intrinsicto skew scattering. Note added. We noticed that the dependence of θ SHon the Pt concentration in our work is different from the results ina previous paper [ 14] in which the θ SHsaturates at certain Pt concentrations. Such a difference may arise from differentconsiderations of the interfacial transparency. As addressedin the Supplemental Material, Sec. I[29], the dependence of θ SHon Pt concentration may be signiﬁcantly inﬂuenced by the interfacial transparency, which is neglected in Ref. [ 14]. Furthermore, the elemental segregation of Pt as mentioned inRef. [ 14] can also lead to a different θ SHfrom that of the homogeneous solid solution. The preferential distribution ofPt can lead to the drop of resistivity and thereby a decreaseofθ SH. These two factors can possibly result in the different dependences of θSHon Pt concentration.ACKNOWLEDGMENTS We gratefully acknowledge support from the Singapore National Research Foundation under CRP Award No. NRF-CRP10-2012-02, and the Singapore Ministry of EducationMOE2018-T2-2-043, AMEIRG18-0022, ASTAR IAF-ICP11801E0036, and MOE Tier 1- FY2018–P23. We thank theSingapore Synchrotron Light Source (SSLS) for providing thenecessary facility. P.Y . was supported by the SSLS via NUSCore Support C-380-003-003-001. X.S. constructed the main part of this project and operated most of the experiments. 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PhysRevB.99.094406.pdf	PHYSICAL REVIEW B 99, 094406 (2019) Nearly isotropic spin-pumping related Gilbert damping in Pt /Ni81Fe19/Pt W. Cao,1,L. Yang,1S. Auffret,2and W. E. Bailey1,2,† 1Materials Science and Engineering, Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA 2SPINTEC, Université Grenoble Alpes /CEA/CNRS, F-38000 Grenoble, France (Received 9 November 2018; revised manuscript received 15 February 2019; published 6 March 2019) A recent theory by Chen and Zhang [ Phys. Rev. Lett. 114,126602 (2015 )] predicts strongly anisotropic damping due to interfacial spin-orbit coupling in ultrathin magnetic ﬁlms. Interfacial Gilbert-type relaxation, dueto the spin pumping effect, is predicted to be signiﬁcantly larger for magnetization oriented parallel to comparedwith perpendicular to the ﬁlm plane. Here, we have measured the anisotropy in the Pt /Ni 81Fe19/Pt system via variable-frequency, swept-ﬁeld ferromagnetic resonance (FMR). We ﬁnd a very small anisotropy of enhancedGilbert damping with sign opposite to the prediction from the Rashba effect at the FM /Pt interface. The results are contrary to the predicted anisotropy and suggest that a mechanism separate from Rashba spin-orbit couplingcauses the rapid onset of spin-current absorption in Pt. DOI: 10.1103/PhysRevB.99.094406 I. INTRODUCTION The spin-transport properties of Pt have been studied in- tensively. Pt exhibits efﬁcient, reciprocal conversion of chargeto spin currents through the spin Hall effect (SHE) [ 1–4]. It is typically used as detection layer for spin current evaluatedin novel conﬁgurations [ 5–7]. Even so, consensus has not yet been reached on the experimental parameters which charac-terize its spin transport. The spin Hall angle of Pt, the spindiffusion length of Pt, and the spin mixing conductance ofPt at different interfaces differ by as much as an order ofmagnitude when evaluated by different techniques [ 2,3,8–12]. Recently, Chen and Zhang [ 13,14] (hereafter CZ) have proposed that interfacial spin-orbit coupling (SOC) is a miss-ing ingredient which can bring the measurements into greateragreement with each other. Measurements of spin-pumping-related damping, particularly, report spin diffusion lengthswhich are much shorter than those estimated through othertechniques [ 15,16]. The introduction of Rashba SOC at the FM/Pt interface leads to interfacial spin-memory loss, with discontinuous loss of spin current incident to the FM /Pt interface. The model suggests that the small saturation lengthof damping enhancement reﬂects an interfacial discontinuity,while the inverse spin Hall effect (ISHE) measurements reﬂectthe bulk absorption in the Pt layer [ 15,16]. The CZ model predicts a strong anisotropy of the enhanced damping due to spin pumping, as measured in ferromag-netic resonance (FMR). The damping enhancement for time-averaged magnetization lying in the ﬁlm plane ( pc-FMR, or parallel condition) is predicted to be signiﬁcantly larger thanthat for magnetization oriented normal to the ﬁlm plane ( nc- FMR, or normal condition). The predicted anisotropy can be wc2476@columbia.edu †web54@columbia.eduas large as 30%, with pc-FMR damping exceeding nc-FMR damping, as will be shown shortly. In this paper, we have measured the anisotropy of the enhanced damping due to the addition of Pt in symmetricPt/Ni 81Fe19(Py)/Pt structures. We ﬁnd that the anisotropy is very weak, less than 5%, and with the opposite sign from thatpredicted in Ref. [ 13]. II. THEORY We ﬁrst quantify the CZ-model prediction for anisotropic damping due to the Rashba effect at the FM /Pt interface. In the theory, the spin-memory loss for spin current polarizedperpendicular to the interfacial plane is always larger thanthat for spin current polarized in the interfacial plane. Thepumped spin polarization σ=m×˙mis always perpendicular to the time-averaged or static magnetization /angbracketleftm/angbracketright t/similarequalm.F o r nc-FMR, the polarization σof pumped spin current is always in the interfacial plane, but for pc-FMR is nearly equally in-plane and out-of-plane. A greater damping enhancementis predicted in the pccondition than in the nccondition, /Delta1α pc>/Delta1 α nc: /Delta1α nc=K/bracketleftbigg1+4ηξ(tPt) 1+ξ(tPt)/bracketrightbigg (1) /Delta1α pc=K/bracketleftbigg1+6ηξ(tPt) 1+ξ(tPt)+η 2[1+ξ(tPt)]2/bracketrightbigg (2) ξ(tPt)=ξ(∞)×coth( tPt/λsd), (3) where the constant of proportionality Kis the same for both conditions and the dimensionless parameters, ηandξ,a r e always real and positive. The Rashba parameter η=(αRkF/EF)2(4) is proportional to the square of the Rashba coefﬁcient αR, deﬁned as the strength of the Rashba potential, V(r)= αRδ(z)(ˆk×ˆz)·σ, where δ(z) is a delta function localizing 2469-9950/2019/99(9)/094406(4) 094406-1 ©2019 American Physical SocietyW. CAO, L. YANG, S. AUFFRET, AND W. E. BAILEY PHYSICAL REVIEW B 99, 094406 (2019) the effect to the interface at z=0 (ﬁlm plane is xy),kFis the Fermi wave number, and EFis the Fermi energy. The backﬂow factor ξis a function of Pt layer thickness, where the backﬂow fraction at inﬁnitely large Pt thickness deﬁnedas/epsilon1=ξ(∞)/[1+ξ(∞)]./epsilon1=0 (1) refers to zero (complete) backﬂow of spin current across the interface. λ sdis the spin diffusion length in the Pt layer. To quantify the anisotropy of the damping, we deﬁne Q: Q≡(/Delta1α pc−/Delta1α nc)//Delta1α nc (5) as an anisotropy factor , the fractional difference between the enhanced damping in pc and nc conditions. PositiveQ(Q>0) is predicted by the CZ model. A spin-memory lossδfactor of 0 .9±0.1, corresponding to nearly complete relaxation of spin current at the interface with Pt, was mea-sured through current perpendicular to plane magnetoresis-tance (CPP-GMR) [ 8]. According to the theory [ 13,14], the spin-memory loss can be related to the Rashba parameter byδ=2η,s ow et a k e η∼0.45. The effect of variable η< 0.45 will be shown in Fig. 3. To evaluate the thickness dependent backﬂow ξ(t Pt), we assume λPt sd=14 nm, which is associated with the absorption of the spin current in the bulk of the Ptlayer, as found from CPP-GMR measurements [ 8] and cited in Ref. [ 13]. Note that this λ Pt sdis longer than that used sometimes to ﬁt FMR data [ 15,16]; Rashba interfacial coupling in the CZ model brings the onset thickness down. The calculatedanisotropy factor Qshould then be as large as 0.3, indicating that/Delta1α pcis 30% greater than /Delta1α nc(see Results for details). III. EXPERIMENT In this paper, we present measurements of the anisotropy of damping in the symmetric Pt( tPt)/Py(5 nm) /Pt(tPt)s y s t e m , where “Py” =Ni81Fe19. Because the Py thickness is much thicker than its spin coherence length [ 17], we expect that spin-pumping-related damping at the two Py /Pt interfaces will sum. The full deposited stack is Ta(5 nm) /Cu(5 nm) / Pt(tPt)/Py(5 nm) /Pt(tPt)/Al2O3(3 nm), tPt=1–10 nm, de- posited via DC magnetron sputtering under computer controlon ion-cleaned Si /SiO 2substrates at ambient temperature. The deposition rates were 0.14 nm /s for Py and 0.07 nm /s for Pt. Heterostructures deposited identically, in the samedeposition chamber, have been shown to exhibit both robustspin pumping effects, as measured through FMR linewidth[18,19], and robust Rashba effects (in Co /Pt), as measured through Kerr microscopy [ 20,21]. The stack without Pt layers was also deposited as the reference sample. The ﬁlms werecharacterized using variable frequency FMR on a coplanarwaveguide (CPW) with center conductor width of 300 μm. The bias magnetic ﬁeld was applied both in the ﬁlm plane(pc) and perpendicular to the plane ( nc), as previously shown in Ref. [ 22]. The nc-FMR measurements require precise alignment of the ﬁeld with respect to the ﬁlm normal. Here,samples were aligned by rotation on two axes to maximizethe resonance ﬁeld at 3 GHz. IV . RESULTS AND ANALYSIS Figure 1shows frequency-dependent half-power linewidth /Delta1H1/2(ω)i npc- and nc-FMR. The measurements were taken FIG. 1. Frequency-dependent half-power FMR linewidth /Delta1H1/2(ω) of the reference sample Py(5 nm) (black) and symmetric trilayer samples Pt(t) /Py(5 nm) /Pt(t) (colored). (a) pc-FMR measurements. (b) nc-FMR measurements. Solid lines are linear ﬁts to extract Gilbert damping α. (Inset): inhomogeneous broadening /Delta1H0inpc-FMR (blue) and nc-FMR (red). at frequencies from 3 GHz to a cutoff frequency above which the signal-to-noise ratio becomes too small for reliable mea-surement of linewidth. The cutoff ranged from 12–14 GHz forthe samples with Pt (linewidth ∼200–300 G) to above 20 GHz fort Pt=0. Solid lines stand for linear regression of the variable-frequency FMR linewidth /Delta1H1/2=/Delta1H0+2αω/γ , where /Delta1H1/2is the full width at half maximum, /Delta1H0is the inhomogeneous broadening, αis the Gilbert damping, ωis the resonance frequency, and γis the gyromagnetic ratio. The ﬁts show good linearity with frequency ω/2πfor all experimental linewidths /Delta1H1/2(ω). The inset summarizes inhomogeneous broadening /Delta1H0inpc- and nc-FMR; its error bar is ∼2O e . In Fig. 2, we plot Pt thickness dependence of damping parameters α(tPt) extracted from the linear ﬁts in Fig. 1,f o r both pc-FMR and nc-FMR measurements. Standard deviation errors in the ﬁts for αare∼3×10−4. The Gilbert damping αsaturates quickly as a function of tPtin both pc and nc conditions, with 90% of the effect realized with Pt(3 nm).The inset shows the damping enhancement /Delta1αdue to the addition of Pt layers /Delta1α=α−α 0, normalized to the Gilbert damping α0of the reference sample without Pt layers. The Pt thickness dependence of /Delta1αmatches our previous study on Py/Pt heterostructures [ 19] reasonably; the saturation value of/Delta1α Pt/Py/Ptis 1.7×larger than that measured for the single interface /Delta1α Py/Pt[19]( 2×expected). The dashed lines in the inset refer to calculated /Delta1α ncusing Eq. ( 1) (assuming λPt sd=14 nm and /epsilon1=10%). η=0.25 shows a threshold of Pt thickness dependence. When η> 0.25, the curvature of /Delta1α(tPt) will have the opposite sign to that observed in exper- iments, so η=0.25 is the maximum which can qualitatively reproduce the Pt thickness dependence of the damping. As shown in Fig. 2inset, the damping enhancement due to the addition of Pt layers is slightly larger in the ncgeometry than in the pcgeometry: /Delta1α nc>/Delta1 α pc. This is opposite to the prediction of the model in Ref. [ 13]. The anisotropy factor Q≡(/Delta1α pc−/Delta1α nc)//Delta1α ncfor the model ( Q>0) and the experiment ( Q<0) are shown together in Figs. 3(a) and 094406-2NEARLY ISOTROPIC SPIN-PUMPING RELATED GILBERT … PHYSICAL REVIEW B 99, 094406 (2019) FIG. 2. Pt thickness dependence of Gilbert damping α=α(tPt) inpc-FMR (blue) and nc-FMR (red). α0refers to the reference sam- ple (tPt=0). (Inset): Damping enhancement /Delta1α(tPt)=α(tPt)−α0 due to the addition of Pt layers in pc-FMR (blue) and nc-FMR (red). Dashed lines refer to calculated /Delta1α ncusing Eq. ( 1) by assuming λPt sd=14 nm and /epsilon1=10%. The red dashed line ( η=0.15) shows a similar curvature with experiments. The black dashed line ( η/greaterorequalslant0.25) shows a curvature with the opposite sign. 3(b). The magnitude of Qfor the experiment is also quite small, with −0.05<Q<0. This very weak anisotropy, or near isotropy, of the spin-pumping damping is contrary to theprediction in Ref. [ 13] and is the central result of our paper. The two panels (a) and (b), which present the same exper- imental data (triangles), consider different model parameters,corresponding to negligible backﬂow [ /epsilon1=0.1, panel (a)] and moderate backﬂow [ /epsilon1=0.4, panel (b)] for a range of Rashba couplings 0 .01/lessorequalslantη/lessorequalslant0.45. A spin diffusion length λ sd=14 nm for Pt [ 8] was assumed in all cases. The choice of backﬂow fraction /epsilon1=0.1 or 0.4 and the choice of spin diffusion length of Pt λsd=14 nm followthe CZ paper [ 13] for better evaluation of their theory. For good spin sinks like Pt, the backﬂow fraction is usually quitesmall. If /epsilon1=0, then there will be no spin backﬂow. In this limit,/Delta1α pc,/Delta1α ncand the Qfactor will be independent of Pt thickness. In the case of a short spin diffusion length of Pt, e.g., λsd= 3 nm, the anisotropy Qas a function of Pt thickness decreases more quickly for ultrathin Pt, closer to our experimentalobservations. However, we note that the CZ theory requiresa long spin diffusion length in order to reconcile differentexperiments, particularly CPP-GMR with spin pumping, andis not relevant to evaluate the theory in this limit. Leaving apart the question of the sign of Q, we can see that the observed absolute magnitude is lower than that predictedforη=0.05 for small backﬂow and 0.01 for moderate back- ﬂow. According to Ref. [ 13], a minimum level for the theory to describe the system with strong interfacial SOC is η=0.3. V . DISCUSSION Here, we discuss extrinsic effects which may result in a discrepancy between the CZ model ( Q∼+0.3) and our experimental result ( −0.05<Q<0). A possible role of two- magnon scattering [ 23,24], known to be an anisotropic contri- bution to linewidth /Delta1H1/2, must be considered. Two-magnon scattering is present for pc-FMR and nearly absent for nc- FMR. This mechanism does not seem to play an importantrole in the results presented. It is difﬁcult to locate a two-magnon scattering contribution to linewidth in the pure Pyﬁlm: Figure 1shows highly linear /Delta1H 1/2(ω), without offset, over the full range to ω/2π=20 GHz, thereby reﬂecting Gilbert-type damping. The damping for this ﬁlm is muchsmaller than that added by the Pt layers. If the introductionof Pt adds some two-magnon linewidth, eventually mistakenfor intrinsic Gilbert damping α, this could only produce a measurement of Q>0, which was not observed. One may also ask whether the samples are appropriate to test the theory. The ﬁrst question regards sample quality. TheRashba Hamiltonian models a very abrupt interface. Samples FIG. 3. Anisotropy factor Qfor spin-pumping enhanced damping, deﬁned in Eq. ( 5). Solid lines are calculations using the CZ theory [ 13], Equations ( 1)–(3), for variable Rashba parameter 0 .01/lessorequalslantη/lessorequalslant0.45.λPt sdis set to be 14 nm. Backﬂow fraction /epsilon1is set to be 10% in (a) and 40% in (b). Black triangles, duplicate in (a) and (b), show the experimental values from Fig. 2. 094406-3W. CAO, L. YANG, S. AUFFRET, AND W. E. BAILEY PHYSICAL REVIEW B 99, 094406 (2019) deposited identically, in the same deposition chamber, have exhibited strong Rashba effects, so we expect the samples tobe generally appropriate in terms of quality. Intermixing of Ptin Ni 81Fe19(Py)/Pt [25] may play a greater role than it does in Co/Pt [26], although defocused TEM images have shown fairly well-deﬁned interfaces for our samples [ 27]. A second question might be about the magnitude of the Rashba parameter ηin the materials systems of interest. Our observation of nearly isotropic damping is consistent with thetheory, within experimental error and apart from the oppositesign, if the Rashba parameter ηis very low and the backﬂow fraction /epsilon1is very low. Ab initio calculations for (epitaxial) Co/Pt in Ref. [ 28] have indicated η=0.02–0.03, lower than the values of η∼0.45 assumed in Refs. [ 13,14] to treat interfacial spin-memory loss. The origin of the small, negative Qobserved here is un- clear. A recent paper has reported that /Delta1α pcis smaller than /Delta1α ncin the YIG /Pt system via single-frequency, variable- angle measurements [ 7], which is contrary to the CZ modelprediction as well. It is also possible that a few monolayers of Pt next to the Py /Pt interfaces are magnetized in the samples [19], and this may have an unknown effect on the sign, not taken into account in the theory. VI. CONCLUSIONS In summary, we have experimentally demonstrated that in Pt/Py/Pt trilayers the interfacial damping attributed to spin pumping is nearly isotropic, with an anisotropy betweenﬁlm-parallel and ﬁlm-normal measurements of <5%. The nearly isotropic character of the effect is more compatiblewith conventional descriptions of spin pumping than with theRashba spin-memory loss model predicted in Ref. [ 13]. ACKNOWLEDGMENTS We acknowledge support from the US NSF-DMR- 1411160 and the Nanosciences Foundation, Grenoble. [1] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett.88,182509 (2006 ). [ 2 ]O .M o s e n d z ,J .E .P e a r s o n ,F .Y .F r a d i n ,G .E .W .B a u e r ,S .D . Bader, and A. Hoffmann, P h y s .R e v .L e t t . 104,046601 (2010 ). [3] L. Liu, T. 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PhysRevB.90.144430.pdf	PHYSICAL REVIEW B 90, 144430 (2014) Thermally driven magnetic precession in spin valves David Luc and Xavier Waintal CEA-INAC/UJF Grenoble 1, SPSMS UMR-E 9001, Grenoble F-38054, France (Received 20 February 2014; revised manuscript received 22 September 2014; published 24 October 2014) We investigate the angular dependence of the spin torque generated when applying a temperature difference across a spin valve. Our study shows the presence of a nontrivial ﬁxed point in this angular dependence. Thisﬁxed point opens the possibility for a temperature gradient to stabilize radio frequency oscillations without theneed for an external magnetic ﬁeld. This so-called “wavy” behavior can already be found upon applying a voltagedifference across a spin valve but we ﬁnd that this effect is much more pronounced with a temperature difference.We ﬁnd that a spin asymmetry of the Seebeck coefﬁcient of the order of 20 μVK −1should be large enough for a temperature gradient of a few degrees to trigger the radio-frequency oscillations. Our semiclassical theory isfully parametrized with experimentally measured(able) parameters and allows one to quantitatively predict theamplitude of the torque. DOI: 10.1103/PhysRevB.90.144430 PACS number(s): 85 .75.−d,72.25.−b I. INTRODUCTION Spin caloritronics [ 1–7] studies the interplay of charge, spin, and heat transport and provides extensions to some of thespintronics concepts. One of interest to us is the spin-transfertorque (STT) [ 8–10], ﬁrst predicted by Slonczewski and Berger in 1996 [ 11,12]. STT is the angular momentum deposited by a spin-polarized current on a ferromagnetic layer. It is atthe origin of interesting out of equilibrium dynamics for themagnetization layer leading to magnetic reversal or sustainedRF oscillations. The later effect, known as the spin-torqueoscillator (STO) [ 13,14], is a promising candidate for agile RF sources. Although most STOs require an external magneticﬁeld, it was also discovered that the angular dependence of theSTT can, in some asymmetric spin valves, vanish at a nontrivialangle ϕ ∗. This peculiar angular dependence, the so-called waviness [ 15–18], can lead to a stable oscillating state in the absence of an external magnetic ﬁeld: while the precessionalangle of most STOs depends on a balance between the torqueand the restoring (e.g., anisotropy) ﬁelds, “wavy” torquesintrinsically favor the precessional angle ϕ ∗. A qualitative understanding of the origin of the wavy behavior and its link tothe asymmetry of the spin valve can be found in a geometricalconstruction of the torque given in [ 17]. In 2007, in one of the ﬁrst articles on “caloritronics,” Bauer et al. considered another route for creating STT via the combination of spintronicswith thermoelectric effects [ 2]: the so-called thermal STT. Spin-dependent thermoelectric effects soon started to attractsome theoretical and experimental interest [ 4–6,19–22]. In this letter, we investigate the angular dependence of the STTinduced by temperature gradients applied across various typesof magnetic spin valves. Our semiclassical theory, carefullytabulated with experimentally measured parameters, showsthat thermally induced STT is naturally “wavy” for a widerange of devices. We predict that RF oscillations can betriggered with temperature differences as low as a few degreesfor reasonable material parameters. II. SEMICLASSICAL DRIFT-DIFFUSION APPROACH Our starting point is a semiclassical approach for magnetic multilayers that treats the charge degrees of freedom at thedrift-diffusion level yet retains all the information about spin degrees of freedom [ 17,23,24]. This approach [called continu- ous random matrix theory (CRMT)] generalizes the Valet-Ferttheory [ 25] to systems with noncollinear magnetization [ 26] and is also equivalent to the so-called (generalized) circuittheory [ 27]. Here we generalize CRMT to include heat ﬂow and thermoelectric effects. In addition to the charge I αand spin Jαcurrent densities, we therefore add the heat current density Qα(α=x,y,z being the direction of propagation). Similarly, in addition to the charge μcand spin μpotentials, we include the temperature θ(θ=kBTwhere Tis the actual temperature in K). We assume that the energy relaxationlength between different spin species is small so that a singletemperature can be used for both majority and minorityelectrons. Allowing for two different temperatures wouldintroduce small corrections to the results given below (ofthe order of 5% or less, i.e., of the GMR in the system).Thermoelectric effects are described by spin-dependent See-beck and Peltier coefﬁcients [ 4,20,28,29]. We note S ↑(S↓)t h e spin-dependent Seebeck coefﬁcients for majority (minority)electrons while the Peltier coefﬁcients are given by Onsagerrelation /Pi1 σ=SσT∗where T∗is the average temperature. We further introduce dimensionless Seebeck coefﬁcients inunits of k B/e≈80μVK−1:s=e(S↑+S↓)/(2kB) and/Delta1s= e(S↑−S↓)/(2kB) characterize, respectively, the average and the polarization of the Seebeck effect. Recent experimentsprovide the ﬁrst spin-resolved values of these quantitiesfor ferromagnetic materials [ 20]:s Co≈− 0.25 and /Delta1sCo≈ −0.02 for cobalt, and sPy≈− 0.21 and /Delta1sPy≈− 0.044 for permalloy. We introduce reduced currents (with unit of energy)as follows, I α=4jc α/slashbig (eRSh), (1) Jα=2/planckover2pi1jα/(e2RSh), (2) Qα=4kBT∗jq α/slashbig (e2RSh), (3) whereRShis the Sharvin resistance for a unit surface (with typical value RSh≈1f/Omega1. m2), and e<0 is the charge of the electron. These variables follow a set of Ohm-like (or 1098-0121/2014/90(14)/144430(8) 144430-1 ©2014 American Physical SocietyDA VID LUC AND XA VIER W AINTAL PHYSICAL REVIEW B 90, 144430 (2014) Fourier-like) equations, −/lscript∗∂αμc=jc α−βjα·m+/lscript∗ /lscriptHs/parenleftbig sjc α+/Delta1sjα·m/parenrightbig −/lscript∗ /lscriptHsjq α (4) −/lscript∗∂αμ=jα−βjc αm+/lscript∗ /lscriptH/Delta1s/parenleftbig sjc αm+/Delta1sjα/parenrightbig −/lscript∗ /lscriptH/Delta1sjq αm+/lscript∗ /lscript⊥(m×jα)×m−/lscript∗ /lscriptL(m×jα) (5) −/lscriptH∂αθ=−sjc α−/Delta1sjα·m+jq α. (6) Equations ( 4)–(6) are the extension of Eqs. (1)–(4) of Ref. [ 26]. The unit vector mis the local direction of the magnetization (bold vectors correspond to spin space whileexplicit components α=x,y,z are used for real space). The parameters involved are the mean free paths for the majority(/lscript ↑) and minority ( /lscript↓) electrons, related to the spin-dependent resistivities ρσas/lscript↑(↓)=RSh/ρ↑(↓). They can be expressed alternatively in term of /lscript∗andβ, the average mean free path and the asymmetry of the spin-resolved resistivity (witha deﬁnition identical to the usual Valet-Fert parameter),deﬁned by 1 /lscript∗=1 /lscript↑+1 /lscript↓, (7) β=/lscript↑−/lscript↓ /lscript↑+/lscript↓. (8) Two length scales characterize the behavior of a spin perpen- dicular to the magnetization: the Larmor precession length /lscriptL and the transverse penetration length /lscript⊥(see [ 26]). Finally, /lscriptHis the heat diffusion length deﬁned as /lscriptH=/lscript∗(L− s2+2βs/Delta1s +/Delta1s2)/(1−β2) where L=e2κ/(σk2 BT∗), is the dimensionless Lorenz number. If the only contribution tothe thermal conductivity comes from the transport electrons(no phonon contribution), Wiedemann-Franz law implies L= π 2/3. A second set of equations expresses the conservation (or lack thereof) of the different currents, /summationdisplay α∂αjc α=0, (9) /summationdisplay α∂αjq α=0, (10) /summationdisplay α∂αjα=−/lscript∗ /lscript2 sfμ−1 /lscript⊥(m×μ)×m+1 /lscriptL(m×μ),(11) where /lscriptsfis the spin diffusion length. Similarly, a set of equations describe the interface boundary conditions betweena ferromagnet and a normal metal. The charge and spin sectorsare described by the usual spin-dependent interface resistancesr b σ, namely Eqs. (8) and (9) of Ref. [ 26]. The heat sector is given by (neglecting interface thermoelectric effects) /summationdisplay αnαjq α=LRSh 4rb∗(1−γ2)(θN−θF), (12)where θNandθFare the temperatures on both sides of the ferro-normal interface and the interface resistances have beenparametrized according to the usual Valet-Fert notation r b ↑,↓= 2rb ∗(1±γ).nαare the components of the normal unit vector pointing towards the magnetic side of the interface. Last, theboundary conditions at the metallic electrodes are given byEqs. (12) and (13) of Ref. [ 26] for the spin and charge sector while the heat sector reads ( n αpoints towards the system) /summationdisplay αnαjq α+θ=kB/Delta1T, (13) where kB/Delta1T is the temperature difference applied to the reservoir with respect to the reference temperature T∗. III. APPLICATION TO THERMALLY INDUCED S T TI NAS P I NV A L V E Let us now turn to a spin valve made of the following stack: Cu20\|CoLCo\|Cu2\|PyLPy(ϕ)\|Cu10where the indices indicate the corresponding thicknesses in nm and ϕis the angle of the magnetization of the free (permalloy) layer with respect tothe ﬁxed cobalt layer. Following usual practice [ 11,23], the torque τexerted on the free layer is deﬁned as the difference of spin currents on both sides of the layer (spin relaxationonly provides extremely small corrections here; see [ 26]). We used standard material parameters for the mean free paths andspin-diffusion lengths of Cu, Co, and Py, as extracted fromgiant magnetoresistance measurements [ 26]) while we focus on the values given in Ref. [ 20] for the spin-resolved Seebeck coefﬁcients (see Supplemental Material). Figure 1shows the angular dependence of the spin torque for three different types of setups (see the right part of Fig. 2 for a cartoon). In the ﬁrst setup ( Vfor V oltage), we apply τ ϕ/πτ ϕ∗ FIG. 1. (Color online) Spin-transfer torque obtained when apply- ing a voltage ( τV, bottom curve), a temperature gradient ( τM,t o pf u l l curve), and a temperature gradient in the open-circuit conﬁguration (τP, top dashed curve), vs the magnetization angle ϕof the Py layer with respect to that of the Co layer. Symbols represent the simulations including spin-ﬂip scattering, while lines correspond to the analytical calculation Eq. ( 20). Here LCo=LPy=5 nm. (Inset) Sketch of the spin valve. 144430-2THERMALLY DRIVEN MAGNETIC PRECESSION IN SPIN . . . PHYSICAL REVIEW B 90, 144430 (2014) FIG. 2. (Color online) (Left) Waviness angle ϕ∗of the pure thermal torque τPas a function of LCoandLPy. The white cross indicates value LCo=LPy=5 nm corresponding to Fig. 1.T h e presence of a letter V , M or P in a given region means that the angular dependence of the corresponding torkance τV,τMorτPis wavy. NW indicates the region where none of them are wavy. (Right)Cartoon of our three measurement setups V , M, and P. In M and P a temperature difference is applied across the pillar. a voltage bias Vbacross the spin valve and calculate the torkance τV=dτ/dV b. We recover the usual feature of STT in metallic spin valve with a stronger torque in the antiparallelconﬁguration than in the parallel one (black curves). In the second setup (P for pure), we apply a temperature difference /Delta1T across the spin valve in an open circuit conﬁguration so that no current can ﬂow through the device. Because of that,only spin currents may ﬂow, leading to a torque τ P=dτ/d/Delta1T originating from the “pure” spin-dependent Seebeck effect.In the last closed circuit or “mixed” setup (M for mixed),a temperature difference is applied and a current can ﬂowthrough the spin valve (i.e., the two electrodes of the spinvalve are electrically—but not thermally—short circuited).In this last conﬁguration the Seebeck effect induces a ﬁnitecurrent density which in turn induces a STT very similar to thevoltage driven one. Hence, one ﬁnds that the mixed thermaltorkance τ M=dτ/d/Delta1T contains contributions of purelike and voltagelike torquance and is somehow intermediatebetween the two. The most remarkable feature of Fig. 1is the appearance in the pure case τ Pof a ﬁnite angle ϕ∗≈π/3 where τPvanishes. Depending on the sign of the thermal gradient, this new ﬁxed point will be stabilized or destabilized.When stabilized, it corresponds to a fast precessional statethat forms an STO. In the context of voltage-induced torques,these “wavy” structures, which do not require magnetic ﬁeldin contrast to more conventional STOs, have been discussedfor highly asymmetric spin valves [ 15–18]. Figure 2shows the “waviness phase diagram” of the spin valve as a function of thethicknesses L CoandLPyof the ﬁxed cobalt and free permalloy layers. This diagram illustrates the main point of this letter:thermally induced torque is wavy in a much broader rangeof parameters than the voltage-induced torque. We also ﬁndthat the various torques behave quite differently. A thickerCo layer is beneﬁcial for the waviness of τ V, whereas it is detrimental for that of τPandτM. In the limit of a very thin Co layer, the waviness angle for τPcomes close to π/2. As a comparison, the maximum waviness angle in this diagram forτ V(not represented) is ﬁve times lower.To proceed, we introduce a minimum model to estimate the critical value of the temperature gradient needed totrigger magnetic switching or STO behavior. In the macrospinapproximation in the presence of a purely uniaxial anisotropy,the critical torque (per unit angle and per unit surface of the spinvalve) needed to destabilize the initial (parallel or antiparallel)conﬁguration is given by [ 11,30]∂τ/∂ϕ =αM sLPyBu, where Buis the uniaxial anisotropy ﬁeld, αthe Gilbert damping coefﬁcient, and Msthe magnetization. Using τ=τP/Delta1T,w e obtain the critical value of the temperature gradient /Delta1TP needed to get magnetic switching (or STO) as /Delta1TP=αMsBuLPy ∂τP/∂ϕ≈LPy ∂τP/∂ϕ×1.67 kJ m−3rad−1.(14) The numerical value of the right-hand side of the previ- ous expression was obtained by simulating the spin valvePy 24\|Cu10\|Py6(ϕ)o f[ 31] for which a critical switching current Icrit=107Ac m−2has been reported. We calculate a corresponding critical torque of the order of 10−5Jm−2rad−1 which allows us to estimate globally the product αMsBu. Critical currents of the order of 107Ac m−2are rather standard values for current driven STT [ 31–33] and values up to two orders of magnitude smaller have been reported [ 34], so that the previous expression is a rather conservative estimate.Within the macrospin model, the criteria for observing theRF oscillations in the P (M) setup are that (i) the torkance iswavy and (ii) the applied temperature gradient is larger than/Delta1T P(M)for both the parallel and antiparallel conﬁgurations. Figures 3(a)–3(d) show the critical temperature difference in the mixed ( /Delta1TM, left column) and pure setup ( /Delta1TP, right column) for the parallel conﬁguration (upper raw) andantiparallel conﬁguration (middle row) as a function of thelayer thicknesses. The resulting phase diagram and shapeof the iso- /Delta1T M(/Delta1TP) lines is very generic and depend weakly on the material parameters. In stark contrast, theactual values of the temperature gradient needed to triggerthe RF oscillations are strongly dependent on the asymmetryof the Seebeck coefﬁcient of the free layer. For the smallvalue of /Delta1s Py=0.044 (which corresponds to the one obtained in Ref. [ 20]), one ﬁnds that unrealistically large thermal gradients are needed (particularly to destabilize the parallelconﬁguration, see the top panels). However, the lower panels ofFigs. 3(e) and3(f)show the critical temperatures as a function of/Delta1s Py(taken as a free parameter) for sPy=1.3. One ﬁnds that Seebeck coefﬁcients roughly ﬁve times bigger than theproposed value for permalloy would be sufﬁcient to bring thesecritical temperatures down to experimentally reachable valuesof a few degrees or smaller. The waviness angle (symbols)remains large on the whole range of parameters. IV . ANALYTICAL APPROACH: BUILDING EFFICIENT EFFECTIVE MATERIALS In the absence of spin-ﬂip scattering, ignoring the ﬁnite penetration of transverse spins and keeping only the ﬁrst-order terms from the Seebeck-Peltier effect, close analyticalexpressions can be obtained for our model. A ﬁrst result isthat many collinear materials (or interfaces) put in series canbe combined to obtain a unique effective material. After sucha procedure our spin valve can be reduced to two effective 144430-3DA VID LUC AND XA VIER W AINTAL PHYSICAL REVIEW B 90, 144430 (2014) 2 4 6 8 10 12 1424681012141400 K 1300 K1300 K 1400 K 1800K 3000 K 50K200 K1000 K KKKK (a)2 4 6 8 10 12 142468101214 300 K400 K600 K 1000K 1500 K 800K 250 K 80 K (b) 2 4 6 8 10 12 142468101214 10 K20 K30 K40 K50 K (c)2 4 6 8 10 12 142468101214 50K100 K200 K300 K400K (d) 0.360.380.40.42 ϕ/π 0.1 0.2 0.3 0.4 0.5 ΔsPy110100 ΔTM [K] (e)0.50.550.60.650.7 ϕ/π 0.1 0.2 0.3 0.4 0.5 ΔsPy110100 ΔTP [K] (f) FIG. 3. (Color online) Dependence of /Delta1TP(left column) and /Delta1TM(right column) in a Cu 40\|CoLCo\|Cu2\|PyLPy(ϕ)\|Cu10stack as a function of the layer thicknesses LCoandLPyin the parallel [(a) and (b)] and antiparallel [(c) and (d)] conﬁgurations. The blue cross indicates LCo=LPy=5n m( c f .F i g . 1). The background displays the waviness domains of Figs. 2(e) and2(f): dependence on /Delta1sB, the full (dashed) line corresponds to the parallel (antiparallel) conﬁguration for LCo=50 nm, LPy=2 nm, and sPy=1.3. Red symbols indicate the value of the waviness angle (right scale). layers A and B whose magnetizations make an angle ϕ.T h e effective parameters read r=/summationdisplay iri, (15) rβ=/summationdisplay iriβi, (16) r(1−β2)/L=/summationdisplay iri/parenleftbig 1−β2 i/parenrightbig/slashbig Li, (17)r(1−β2)s/L=/summationdisplay iri/parenleftbig 1−β2 i/parenrightbig si/Li, (18) r(1−β2)/Delta1s/L=/summationdisplay iri/parenleftbig 1−β2 i/parenrightbig /Delta1si/Li, (19) where the resistance riof a bulk layer is given by the ratio ri=2ρ∗Li/RShfor a bulk layer ( Liis the thickness of the layer) while the resistance of an interface is ri=2r∗ b/RSh.Li is the dimensionless Lorenz number of the material, introduced 144430-4THERMALLY DRIVEN MAGNETIC PRECESSION IN SPIN . . . PHYSICAL REVIEW B 90, 144430 (2014) before Eq. ( 9). Equations ( 15)–(19) express the effective parameters ( r,β,s,/Delta1s) in terms of the values ( ri,βi,si,/Delta1si) of the individual layers and interfaces. They form the basis forputting several material in series and can be used for optimizingthe effective parameters and increasing the torkance of thestack. By placing two of these effective materials A and Bin series, we obtain a general description of a spin valveF A\|N\|FB(ϕ). After some algebra, we obtain the expression of the torque on layer B, τ=−F 2sinϕ/braceleftbigg/bracketleftbigg βArB+1 rB−βBcosϕ/bracketrightbigg (GYeV b+SkB/Delta1T) +/bracketleftbigg/Delta1sA LA/parenleftbig 1−β2 A/parenrightbigrB+1 rB−/Delta1sB LB/parenleftbig 1−β2 B/parenrightbig cosϕ/bracketrightbigg ×KkB/Delta1T/bracerightbigg , (20) where F,G,Y,K, andSare expressions involving the various (effective) material parameters whose signs do not changewhenϕvaries (see Appendix Afor the explicit expressions and details on the derivation). We have checked Eq. ( 20)a g a i n s t our numerical simulations and found excellent agreement (seeFig.1). We ﬁnd that the current j c∝GYeV b+SkB/Delta1Tso that the open-circuit condition for τPis obtained using GYeV b+ SkB/Delta1T=0. While the expression of Eq. ( 20) is somewhat cumbersome, the analysis of its angular dependence allowsone to obtain simple criteria for the existence of a wavy regime.We ﬁnd cosϕ V ∗=βA βBrB+1 rB, (21) cosϕP ∗=1−β2 A 1−β2 BLB LA/Delta1sA /Delta1sBrB+1 rB, (22) where the above expressions provide ﬁrst a criterion for waviness ( \|cosϕ∗\|/lessorequalslant1) and second the value of ϕ∗for wavy structures. We ﬁnd that the criterion for waviness in the“pure” thermal case contains two conﬂicting contributions:In order to obtain a wavy structure one needs the polarizationof the resistivity of the free (B) layer to be small while thecorresponding Seebeck coefﬁcient is highly spin polarized.As both are not necessarily correlated (the former is relatedto the polarization of the density of state while the latter toits variation with respect to energy), this leaves much roomfor material optimization. The limit /Delta1s A≈0, which can be obtained by stacking two materials whose /Delta1shave opposite signs, is particularly interesting: The corresponding stack isalways wavy with a large angle ϕ ∗≈π/2. V . CONCLUSION We have developed a quantitative theory for spin-dependent Seebeck and Peltier effects in magnetic metallic devices.The theory relies entirely on measured(able) material pa-rameters so that its results do not depend on a—alwaysprecarious—detailed microscopic modeling. We ﬁnd that(i) the angular dependence of the thermally induced torqueis generally “wavy” in asymmetric devices and (ii) low (a fewdegrees) critical temperature gradient can be achieved by usingmaterials with a spin asymmetry of the Seebeck coefﬁcientof the order of 20 μVK −1. Many strategies can be used to improve it: doping with resonant impurities [ 31], resonant tunneling barriers or using materials with larger thermoelectricpower such as semi-conductors or topological insulators. Arecent work from the group of Kimura [ 35] indicates that CoFeAl alloy has a very high spin asymmetry of 35 μVK −1 which is very promising. Overall, our ﬁndings suggest that such temperature-induced torque can be a very effective toolto manipulate the magnetic states of spintronics devices. ACKNOWLEDGMENTS Funding was provided by the FP7 project STREP MACALO and the consolidator ERC grant MesoQMC. Wethank Professor T. Kimura for useful discussions about hisnew experimental data. APPENDIX A: DERIVATION OF THE SPIN TORQUE EXPRESSION Eq. ( 20) Equation ( 20) was derived in the case of a spin valve FA\|N\|FB(ϕ), under the following assumptions: (i) the spin valve has no variations along the yandzdirections, (ii) spin- ﬂip scattering is neglected, (iii) the transverse spin is absorbedat the normal-ferro interface ( /lscript ⊥very short), (iv) the Seebeck coefﬁcients sand/Delta1sare only considered at ﬁrst order, and (v) the various layers that make the two effective materials FA andFBhave a single orientation of their magnetization. The normal spacer and its interfaces with the ferromagnetic layersare taken to be perfectly transparent without loss of generalityas any ﬁnite resistance can be incorporated in the effectivematerial F AorFB. We note that in the numerics presented in the main text, conditions (ii) and (iv) are relaxed which onlylead to small corrections to the results. Within this set of approximations, Eqs. ( 4)–(11) become for each material, −/lscript ∗dμc dx=jc−βj/bardbl−s/primejq(A1) −/lscript∗dμ/bardbl dx=j/bardbl−βjc−/Delta1s/primejq(A2) −/lscript∗dθ dx=−s/primejc−/Delta1s/primej/bardbl+(1−β2) Ljq, (A3) and the conservation equations are djc dx=0, (A4) djq dx=0, (A5) dj/bardbl dx=0, (A6) withj/bardbl=j·m,μ/bardbl=μ·m, and s/prime=1−β2 Ls, (A7) /Delta1s/prime=1−β2 L/Delta1s. (A8) The conservation equations imply that jcandjqare constant, and the absence of spin ﬂip makes j/bardblpiecewise 144430-5DA VID LUC AND XA VIER W AINTAL PHYSICAL REVIEW B 90, 144430 (2014) constant. As a consequence, μc,θ, andμ/bardblare piecewise linear so that Eqs. ( A1)–(A3) can be easily integrated leading to the effective materials described in Eqs. ( 15)–(19). Once we have the effective Ohm law for the two materials— in the basis parallel to their respective magnetization—we needto combine the two materials together. This is done usingEq. (8) of Ref. [ 26] assuming vanishing mixing reﬂection (or equivalently a mixing conductance equal to the Sharvinconductance). We introduce μ=(μ x,0,μz) and j=(jx,0,jz) the spin-resolved potential and spin current in the normallayer, and μ A(respectively, μB) the value of the spin-resolved potential in the relevant magnetic layer, inﬁnitely close to theinterface. The equation relating the two magnets reads μ·m A=μA·mA, (A9) μ·mB=μB·mB, (A10) j·mA=jA·mA, (A11) j·mB=jB·mB, (A12) for the longitudinal part and −[j−(j·mA)mA]=[μ−(μ·mA)mA], (A13) [j−(j·mB)mB]=[μ−(μ·mB)mB], (A14) for the transverse part. Explicitly, they give μz=μA·mA, (A15) μxsinϕ+μzcosϕ=μB·mB, (A16) jz=jA·mA, (A17) jxsinϕ+jzcosϕ=jB·mB, (A18) jx=−μx, (A19) jxcosϕ−jzsinϕ=μxcosϕ−μzsinϕ, (A20) which translates eventually by eliminating jx,μx,jz, and μzto (μA−jA)·mA=cosϕ(μB−jB)·mB, (A21) (μB+jB)·mB=cosϕ(μA+jA)·mA. (A22) The last set of equations that we need are the boundary conditions at the reservoirs. They read jc+μL c=eVb, (A23) jc−μR c=0, (A24) j/bardblA+μL /bardbl=0, (A25) j/bardblB−μR /bardbl=0, (A26) jq+θL=kB/Delta1T, (A27) jq−θR=0, (A28)withμL/R c,μL/R /bardbl,θL/Rthe value of the potential, spin-resolved potential and temperature inﬁnitely close to the left (L) andright (R) reservoir. Finally, after some algebra, we can obtain the expressions of the currents and potentials given by Eqs. ( A29)–(A36): j c=FG[GYeV b+SkB/Delta1T], (A29) jq=FG[SeVb+KkB/Delta1T], (A30) j/bardblA=FγA[GYeV b+SkB/Delta1T]+FKδ AkB/Delta1T, (A31) j/bardblB=FγB[GYeV b+SkB/Delta1T]+FKδ BkB/Delta1T, (A32) μc=2rBF/parenleftbigg/bracketleftbigg/parenleftbigg 1+1 2rB/parenrightbigg G−βBγB/bracketrightbigg [GYeV b+SkB/Delta1T] −[s/prime BG+/Delta1s/prime BγB]KkB/Delta1T/parenrightbigg , (A33) μ/bardblA=F/bracketleftbigg βA/parenleftbigg 1+1 rB/parenrightbigg −βBcosϕ−γA/bracketrightbigg ×[GYeV b+SkB/Delta1T]+F/bracketleftbigg /Delta1s/prime A/parenleftbigg 1+1 rB/parenrightbigg −/Delta1s/prime Bcosϕ−δA/bracketrightbigg KkB/Delta1T, (A34) μ/bardblB=−F/bracketleftbigg βB/parenleftbigg 1+1 rA/parenrightbigg −βAcosϕ−γB/bracketrightbigg ×[GYeV b+SkB/Delta1T]−F/bracketleftbigg /Delta1s/prime B/parenleftbigg 1+1 rA/parenrightbigg −/Delta1s/prime Acosϕ−δB/bracketrightbigg KkB/Delta1T, (A35) θ=2rBF/bracketleftbigg/parenleftbigg1−β2 B LB+1 2rB/parenrightbigg (SeVb+KkB/Delta1T) −Y(s/prime BG+/Delta1s/prime BγB)eVb/bracketrightbigg , (A36) with the following notations: G=1 2/parenleftbigg sin2ϕ+1 rA+1 rB+1 rArB/parenrightbigg , (A37) Y=/parenleftbigg1−β2 B LB+1 2rB/parenrightbigg1 2rA+/parenleftbigg1−β2 A LA+1 2rA/parenrightbigg1 2rB (A38) γi=/parenleftbigg1 2sin2ϕ+1 2rj/parenrightbigg βi+1 2riβjcosϕ, (A39) δi=/parenleftbigg1 2sin2ϕ+1 2rj/parenrightbigg /Delta1s/prime i+1 2ri/Delta1s/prime jcosϕ, (A40) 144430-6THERMALLY DRIVEN MAGNETIC PRECESSION IN SPIN . . . PHYSICAL REVIEW B 90, 144430 (2014) TABLE I. Material parameters for the bulk materials. Bulk ρ∗ /lscriptsf material ( /Omega1nm) β (nm) s/Delta1 s L Cu 5 0 500 0.0185 0 π2/3 Co 75 0.46 60 −0.25 −0.02 π2/3 Py 291 0.76 5.5 −0.21 −0.044 π2/3 with (i,j)=(A,B )o r(B,A) K=1 2/parenleftbigg1 rA+1 rB+1 rArB/parenrightbigg G−rBβBγB+rAβAγA 2rArB, (A41) S=1 2rA(s/prime BG+/Delta1s/prime BγB)+1 2rB(s/prime AG+/Delta1s/prime AγA),(A42) F=1 2rA1 2rB1 KGY. (A43)TABLE II. Material parameters for the interfaces. Interface r∗ b material ( f/Omega1 m2) γδ T mx Rmx L Cu\|Co 0.51 0.77 0 0 0 π2/3 Cu\|Py 0.5 0.7 0 0 0 π2/3 The torque on layer B is deﬁned by τ=JN−JB=2/planckover2pi1 e2RSh(jxex+jzez−j/bardblB)=2/planckover2pi1 e2RShτe1, (A44) where e1is the in-plane normal vector orthogonal to the magnetization of FB. We obtain τ=−1 2sinϕ(μ/bardblA+j/bardblA)=−F 2sinϕ/braceleftbigg/bracketleftbigg βA/parenleftbigg 1+1 rB/parenrightbigg −βBcosϕ/bracketrightbigg (GYeV b+SkB/Delta1T) +/bracketleftbigg/Delta1sA LA/parenleftbig 1−β2 A/parenrightbig/parenleftbigg 1+1 rB/parenrightbigg −/Delta1sB LB/parenleftbig 1−β2 B/parenrightbig cosϕ/bracketrightbigg KkB/Delta1T/bracerightbigg . (A45) The expression of the waviness angle is, for any applied temperature gradient and/or voltage, cosϕ∗=βA/parenleftbig 1+1 rB/parenrightbig (GYeV b+SkB/Delta1T)+/Delta1sA LA/parenleftbig 1−β2 A/parenrightbig/parenleftbig 1+1 rB/parenrightbig KkB/Delta1T βB(GYeV b+SkB/Delta1T)+/Delta1sB LB/parenleftbig 1−β2 B/parenrightbig KkB/Delta1T. (A46) APPENDIX B: MATERIAL PARAMETERS The following Tables Iand IIcontain the material pa- rameters used in the numerical simulations. We used RSh= 2f/Omega1m2for the Sharvin resistance and the other parametersare given in terms of the Valet-Fert theory, as deﬁned in the text. 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PhysRevB.98.054429.pdf	PHYSICAL REVIEW B 98, 054429 (2018) Micromagnetic view on ultrafast magnon generation by femtosecond spin current pulses Henning Ulrichs* I. Physical Institute, Georg-August University of Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany Ilya Razdolski Physical Chemistry Department, Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, 14195 Berlin, Germany (Received 26 January 2018; revised manuscript received 2 July 2018; published 27 August 2018) In this paper, we discuss a micromagnetic modeling approach to describe the ultrafast spin-transfer torque excitation of coherent and incoherent magnons on the nanoscale. Implementing the action of a femtosecond spincurrent pulse entering an orthogonally magnetized thin ferromagnetic ﬁlm, we reproduce recent experimentalresults and reveal the factors responsible for the unequal excitation efﬁciency of various spin waves. Our ﬁndingsare in an excellent agreement with the results of an analytical description of spin-wave excitation based onclassical kinetic equations. Furthermore, we suggest an experimental design allowing for the excitation oflaterally propagating spin waves beyond the optical diffraction limit. Our ﬁndings demonstrate that the classicalmicromagnetic picture retains its predictive and interpretative power for modeling the reaction of the magnetizationon dynamical stimuli that reside on femtosecond temporal and nanometer spatial scales. DOI: 10.1103/PhysRevB.98.054429 I. INTRODUCTION Stimulated by the seminal experiment by Beaurepaire et al. [1] and the quest for ultrafast optomagnetic recording, an immense amount of knowledge on the incoherent laser-inducedspin dynamics (i.e., ultrafast demagnetization) in a large varietyof materials has been accumulated over the years [ 2]. Simul- taneously, femtosecond optical excitations of coherent spindynamics were discovered [ 3,4], encompassing an extremely broad range of timescales that are governed by the intrinsicproperties of magnetic systems. The temporal limitation for theexcited spin modes is often pertinent to the pulse duration ofthe light source, typically on the order of 10–100 fs. However,the spectrum of the accessible inhomogeneous (with nonzerowave vector k) spin-wave modes is governed by the spatial inhomogeneity scale of the excitation. In other words, on topof the temporal requirements, nonuniform spin-wave modescan only be generated if their wave vectors kare contained in the spectrum of the spatially inhomogeneous stimulus.For the excitation with visible (VIS) or near-infrared (NIR)light, the optical penetration depth δ≈10–15 nm serves as a good estimation for the excitation limit of perpendicular spinwaves in metallic media [ 5]. Optical excitation of the in-plane propagating spin waves is even more restrictive, as the allowedkvalues are governed by the diffraction-limited beam spot size (/greaterorsimilar10 3–104nm) [ 6,7]. Yet, in recent experiments, a strong interfacial localization of the spin-transfer torque exerted by spin polarized currentsenabled the excitation of spin waves with much larger wave vectors [ 8,9]. In particular, perpendicular standing spin waves (PSSW) with f=0.55 THz and k∼1n m −1have been detected [ 8]. The wavelengths of these excitations approach the exchange length lex(a few nanometers in Fe [ 10]), where *hulrich@gwdg.demacroscopic spin models are likely to break down [ 11]. En- abling the expansion of ultrafast photomagnonics [ 12,13] onto the nanometer scale, these ﬁndings simultaneously question the applicability of conventional modeling of spin dynamicsin these extreme conditions. For conventional magnonics, micromagnetic simulation is an indispensable tool for both prediction, and interpretation of static and dynamic magnetic properties [ 14–20]. In this paper, we show that the micromagnetic modeling approachis also suitable for ultrafast processes on nanometer scales[8,9,21,22]. In particular, we set up a micromagnetic model to ﬁrst reproduce the recent experimental ﬁndings [ 8]o f ultrafast optical magnon generation. In this study, Fe/Au/Fetrilayers were optically pumped from one side, generating a superdiffusive [ 22] spin current pulse that traverses the Au spacer and then interacts with the second Fe layer, resulting inthe excitation of high-frequency spin dynamics in the terahertz(THz) domain. Here we develop a micromagnetic modelfeaturing ultrafast spin-transfer torque perturbation and verifythat it can accurately reproduce the above mentioned spectralfeatures of experimentally observed spin dynamics. We further identify important factors governing the excitation efﬁciency and energy transfer into the PSSW modes in thin ferromagneticﬁlms. Complementing a recent theoretical work [ 23] on laser- generated superdiffusive spin transport in noncollinear spinvalve systems and resulting macrospin dynamics in ferromag-nets, our results open the door to understanding spin current-driven magnetism on the nanoscale. Later on, we include thermally activated, incoherent magnetic ﬂuctuations in the model, and show that ultrafast spin currents can effectivelycool or heat such thermal magnon ensembles, in agreementwith experimental observations on slower timescales [ 24]. In the outlook, we outline the topological possibilities for the spincurrent-mediated generation of in-plane propagating, large- k spin waves beyond the optical diffraction limit. 2469-9950/2018/98(5)/054429(11) 054429-1 ©2018 American Physical SocietyHENNING ULRICHS AND ILY A RAZDOLSKI PHYSICAL REVIEW B 98, 054429 (2018) FIG. 1. Model details. (a) Sketch of the micromagnetic model. Besides incorporating the spin current, the model optionally includes thermal ﬂuctuations. (b) Spatiotemporal dependence of the spincurrent entering the ﬁlm, assuming λ STT=2 nm. II. MODEL DETAILS Our model is visualized in Fig. 1(a). It consists of a single ferromagnetic layer of Fe, with a thickness of d=14.2n m . We take as material parameters a saturation magnetizationofμ 0M0=2.1 T, an exchange constant of Aex=19 pJ/m (corresponding to the exchange stiffness D=280 ˚A2meV in Fe from Ref. [ 25]), and a uniaxial anisotropy along xwith strength Ku=45956 J /m3, and assume a Gilbert damping factor of α=0.008. For the demonstration of coherent magnon generation, we simulate a cube of size 1 .16×1.16×14.2n m3subdivided intoNx×Ny×Nz=2×2×24 cells. Periodic boundary conditions in xandydirection enlarge this cube into an inﬁnitely extended ﬁlm. Micromagnetism as a classical ﬁeldtheory can describe spin-wave dynamics for wavelengths largerthan a few nanometers. The wavelength has to signiﬁcantlyexceed the atomic lattice constant, which is for iron about0.3 nm. Note that using mesh cells with comparable size isnot an attempt to claim the applicability of our micromagneticapproach on this atomic scale. Instead, this is done in order tosmoothly describe spin waves on the somewhat larger scale,that is, with wavelengths of several nanometers. As such, inthis work, we intentionally reach and explore the limits ofmicromagnetism, whereas going further beyond this limitrequires other models [ 26]. For the demonstration of incoherent magnon creation and annihilation, we simulate a larger cube of size 150 ×150× 14.2n m 3subdivided into Nx×Ny×Nz=256×256×24 cells. Note that in this case, an additional magnetic ﬁeldrepresenting thermal ﬂuctuations is switched on [ 27]. Further temperature dependencies of the material parameters, like, e.g.,the saturation magnetization M 0are neglected, because we stay far enough away from the Curie temperature of iron ( ∼1000 K). The spin current jswith polarization penters the system at z=0, and has the following empirical spatiotemporal form: js=¯h 2ej0e−z/λ STTe−t/τ2 1+e−(t−t0)/τ1. (1) Here the penetration depth of the spin current is λSTT=2 nm, as estimated in Ref. [ 8], and the temporal proﬁle of the spin current pulse [Fig. 1(b)] is approximated with an analytic function ( 1) with t0=50 fs, τ1=10 fs, and τ2=150 fs. The degree of spin polarization is set to 100 %, followingthe results of Ref. [ 28] and, most importantly, the spirit of the nonthermal mechanism of spin current generation at noblemetal/ferromagnet interfaces discussed in that work. Note thatin Eq. ( 1) we neglect the actual propagation of the spin current pulse inside the Fe ﬁlm with the Fermi velocity ∼1n m/fs, due to the fact that this speed is much larger than the phase andgroup velocities of the involved magnons. We use the softwarepackage MUMAX3(Ref. [ 27]) to model the effect of a spin cur- rent pulse on the local magnetization inside the Fe ﬁlm by aug-menting the Landau-Lifshitz-Gilbert equation with the spin-transfer torque (STT) term τ STTproposed by Slonczewski [ 29]: τSTT=γNz dμ 0M0jsM×M×p. (2) Following theoretical considerations [ 30,31], a ﬁeldlike torque term was not taken into account in Eq. ( 2) due to the condition d/greatermuchλSTT. Further experimental support can be obtained from Fig. 4(b) in Ref. [ 8], where the P-MOKE signal proportional to mzresponds to the STT stimulus with a signiﬁcant delay. On the contrary, the L-MOKE responseimmediately follows the spin current-driven accumulation ofmagnetic moment in Fe, corroborating the dominant role of thedamping-like torque term in the STT-induced spin dynamics.If a ﬁeldlike torque term would be active, the P-MOKE signalshould respond to the spin current excitation directly. The reported in-plane excursion of the magnetization of m y=My M0=0.023 directly after the spin current pulse arrival allows us to determine the respective current density to be usedin the simulations. For this purpose, we systematically variedthe current density and analyzed the temporal evolution ofthe in-plane component m y. The maximum excursion appears shortly after the spin current pulse maximum, which is inagreement with Ref. [ 8][ s e eF i g . 4(b) therein]. Figure 2 shows that the maximum depends linearly on the appliedcurrent density. The linear interpolation intersects with thehorizontal dashed line deﬁned by the experimental value of m y atj0=5.9×1012A/m2. This result in an excellent agreement with the one obtained from the spin-transfer density (7 μB/nm2) evaluated in Ref. [ 8](≈6×1012A/m2), thus reinforcing our micromagnetic model. Further, this j0value is realistic, as it corresponds to the ∼10% spin transport-induced demagnetiza- tion of a 10-nm-thick Fe ﬁlm within 200 fs, in agreement withthe latest results obtained within the superdiffusive transportmodel [ 23]. Moreover, it is close to the values reported in other works (10 12–1013A/m2,R e f .[ 32], and 1013A/m2,R e f .[ 33]). Similar numbers can be further obtained from the work of Choiet al. [34](∼10 12A/m2). 054429-2MICROMAGNETIC VIEW ON ULTRAFAST MAGNON … PHYSICAL REVIEW B 98, 054429 (2018) FIG. 2. Determination of the charge current density j0carrying the spin current [see prefactor in Eq. ( 1)] from the simulated dependence of the maximum of myonj0(closed rectangles). The continuous line is a linear ﬁt to these data. The horizontal dashed line marks the experimentally found maximum of my=0.023 [ 8], which corresponds to j0=5.9×1012A/m2(see vertical dashed line). III. RESULTS A. Coherent magnon generation In the ﬁrst numerical experiment, conducted at temperature T=0, we prepared a spin current pulse with polarization p/bardbly⊥m. Then, according to Eq. ( 2), the spin torque will be τSTT⊥m. The subsequent spatially resolved spin dynamics was computed for a total time of 1 ns after the spin current pulse peak. Figure 3(a) shows mdyn i, which is the dynamic part of the laterally averaged magnetization component /angbracketleftmi/angbracketrightx,y(z,t) (i=x, y, z ) in each layer of the Fe ﬁlm for the ﬁrst 8 ps. In particular, we deﬁne mdyn x=/angbracketleftmx/angbracketrightx,y−1,mdyn y=/angbracketleftmy/angbracketrightx,y, and mdyn z=/angbracketleftmz/angbracketrightx,y. See also movie 1 in Ref. [ 35] for an animated version of the data. One can see how the spin current inducesthe formation of a localized wave packet which then expands.Note that the quadratic dispersion of exchange-dominated spinwaves [ 36] f(k)=γμ 0 2π/radicalBigg/parenleftbigg Han+2Aex M0k2/parenrightbigg/parenleftbigg Han+2Aex M0k2+M0/parenrightbigg (3) is responsible for the quick spatial broadening of the spin- wave packet. Here,γ 2π≈28 GHz /T is the gyromagnetic ratio, Han=2Ku/M 0is the in-plane crystalline anisotropy ﬁeld, Aex the exchange constant, M0the saturation magnetization, and μ0=4π×10−7N/A2the magnetic permeability of vacuum. FIG. 3. Magnetization dynamics driven by the ultrafast spin-transfer torque. (a) Spatiotemporal plot of the dynamic part /angbracketleftmdyn i/angbracketrightof laterally averaged dynamic magnetization components. The left panel of (b) shows a spatially resolved, temporal Fourier transform of the /angbracketleftmz/angbracketrightdata shown in the right panel of (a), and the right panel of (b) shows the spin-wave dispersion f(k) from Eq. ( 3). The dashed lines relate the numerical response to the analytic theory. (c) Simulated ultrafast dynamics of myandmzprojections at the Au/Fe interface. The red shaded area represents the spin current pulse proﬁle according to Eq. ( 1). 054429-3HENNING ULRICHS AND ILY A RAZDOLSKI PHYSICAL REVIEW B 98, 054429 (2018) The front of the spin-wave packet in Fe (not to confuse with the wavefront of the spin current pulse formed by hotspin-polarized electrons which propagate in Au with the Fermivelocity) travels with a characteristic speed of about 7 nm /ps. This value corresponds to the group velocity of the magnonswith the largest wave vectors contained in the excitationspectrum. After about 2 ps, the wave packet has reached thesurface of the Fe ﬁlm. The spin dynamics for times t>2p si s formed by an interference pattern, which can be well analyzedby a Fourier transformation in time, as shown in Fig. 3(b). There, the local Fourier amplitude A FFT{/angbracketleftmz/angbracketrightx,y}(z,f)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN−1/summationdisplay j=0/angbracketleftmz/angbracketrightx,y(z,tj)e−i2πf tj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(4) is depicted, with frequencies f=0, 1 tmax,. . .,N tmax. The sub- script FFT indicates that we utilize a numerical fast Fourier transform algorithm ( FFTW 3), which we apply locally ( z- resolved) to the laterally averaged, out-of-plane magnetizationcomponent /angbracketleftm z/angbracketrightx,y. The temporal sampling goes from time t=0u pt o tN=tmax=1 ns, in equidistant steps of 10 fs. In agreement with the experiment [ 8], this representation reveals that the pattern from Fig. 3(a) can be understood as a superposition of standing spin waves. This is emphasizedby the right panel in Fig. 3(b), which shows the dispersion (3), plotted as a function of a dimensionless wave number κ=kd/π . Prominent spin dynamics can be found at integer κ n=knd/π=0,1,2, ... , and the corresponding eigenmode frequencies fn=f(κn). Having established the agreement between the results of micromagnetic simulations and the experimental data in thequasiequilibrium (nanosecond) regime, we now turn to muchfaster timescales. Taking advantage of the subnanometer spa-tial resolution along the out-of-plane direction in the Fe ﬁlm,in Fig. 3(c), we plot the evolution of the magnetic moment at the Au/Fe interface, i.e., where the spin current pulse isinjected. In contrast to the experimental MOKE data, whichare given by the convolution of a magnetization proﬁle andMOKE sensitivity functions [ 37], these data allow for a deeper insight into the spin-wave dynamics. First, we turn to thesubpicosecond timescale (the left-hand side) where the spin-wave propagation has not played a signiﬁcant role yet. It isseen that the in-plane projection of the magnetic momentm yquickly grows as a result of the spin injection, whereas the dynamics of the out-of-plane component mzis delayed. Similar behavior found in the experimental data (c f .F i g .4 ( b ) in Ref. [ 8])corroborates our understanding of the spin injection picture on the ultrafast timescale. Later on, the magnetization dynamics at the interface represents a compendium of oscillations corresponding to theindividual spin waves, see Fig. 3(c), the right-hand part. A close inspection reveals the excitation of up to nine spin-waveeigenmodes of the 14.2-nm-thick Fe ﬁlm, thus exceeding theexperimental reports. However, we emphasize that the afore-mentioned intrinsic spatial averaging of the MOKE signals pre-cludes the observation of extremely short-wavelength modes.In that regard, the data in Fig. 3illustrate that the qualitative agreement between the simulations and the experiment holdson femtosecond through nanosecond timescales. As such, itis reasonable to use the predictions of the micromagnetic approach to model the picosecond spin dynamics as well.We note, however, that aiming at the quantitative agreementwith the experimental data in the presence of nonequilib-rium magnons would require more sophisticated modeling ofthe measurement process and measured quantities, as wellas reﬁned description of the spin current pulse injection.Furthermore, such comparisons would strongly beneﬁt fromexperimental probing methods with better spatial resolution,as compared to MOKE. Besides reproducing experimental ﬁndings, the micromag- netic model allows for an identiﬁcation of the parameters thatgovern the mode-speciﬁc excitation efﬁciency. As such, wewill now discuss the energy transfer into the different spin-wave modes. For this purpose, we analyze in detail squaredFFT amplitude spectra at the interface of the Fe ﬁlm, z=0. This choice is motivated by the ﬁrst order boundary conditionsfor the spin-wave modes ensuring the largest amplitude of theoscillations of the spin projections at the interfaces of the Feﬁlm [ 36] .T h ei n t e g r a t i o no ft h e A 2 FFT(f) spectrum in the vicin- ity of the peaks corresponding to the excitation of the differentspin-wave modes yields the partial energies ε nassociated with each of the modes: εn=/integraldisplayfn+/Delta1f fn−/Delta1fA2 FFT(f)df. (5) This quantity is proportional to the energy transferred into this mode. In what follows, we shall focus on various factorsthat contribute to ε n, aiming at understanding their importance for a comparative analysis of the excitation efﬁciency of thePSSW modes in thin ferromagnetic ﬁlms. In Appendix, wedevelop an analytic formalism based on the Holstein-Primakofftransformations, which is capable of deriving the contributingfactors in detail [see Eqs. ( A1)t o( A15)]. Importantly, the energy supplied by a spin current pulse j s(z,t) is proportional to the product of spatial Fourier power, \|˜β\|2(kn)=β(kn)β∗(kn), (6) and the temporal Fourier power \|˜θ\|2(ωn)=˜θ(ωn)˜θ∗(ωn)( 7 ) of the spin current js(z,t) deﬁned by Eq. ( 1), evaluated at k=knandω=ωn. Note that in the appendix, we use analytic formulas. In particular there, we thus refer to β(kn)=γ¯hj0 2eμ0M02π/integraldisplayd 0dze−z/λ STT−inπz/d(8) and ˜θ(ωn)=1 2π/integraldisplay∞ −∞dte−t/τ2 1+e−(t−t0)/τ1e−iωnt, (9) where ωn=2πfn. In contrast, to numerically evaluate the two factors \|˜β\|2(kn) and\|˜θ\|2(ωn), and also to compute the partial powers εn, we use discrete fast Fourier transform algorithms. To avoid having to unify different prefactor conventions, wewill from now on disregard all irrelevant prefactors, and onlydiscuss proportionalities. The two factors \|˜β\| 2(kn) and \|˜θ\|2(ωn)a r es h o w na sa function of the mode index nin Fig. 4(a) forλSTT=2n m . 054429-4MICROMAGNETIC VIEW ON ULTRAFAST MAGNON … PHYSICAL REVIEW B 98, 054429 (2018) FIG. 4. PSSW excitation efﬁciency. (a) Squared spatial \|˜β\|2and temporal \|˜θ\|2Fourier amplitudes of the spin-current pulse ( 1) as a function of the PSSW mode index for d=14.2n ma n d λSTT=2 nm. The lines are shown as a guide to the eye. (b) Additional ξfactor as a function of the PSSW mode index for two different Fe ﬁlm thicknesses d(14.2 nm, open squares, and 70 nm, full squares). (c) Correlationlike double logarithmic plot of the partial energy εnvs the product of the various factors, which govern the energy transfer, \|˜β\|2\|˜θ\|2ξ. The different colors indicate various PSSW eigenmodes while within a single color, the data points are obtained for different values of λSTT. The solid line is a linear ﬁt to the data. Note that the temporal factor \|˜θ\|2is only important when the oscillation period Tn=1/fnapproaches the duration of the spin current stimulus ∼τ2. Thus, in our case for n< 5 and ∼250 fs spin current pulse duration, the spatial factor \|˜β\|2 plays a dominant role in determining the relative excitation efﬁciency of the PSSW modes. Further, we identify in Appendix the material parameter- dependent susceptibility of the different spin-wave modes,which, together with their ellipticity E k, gives rise to another factor ξ=/Gamma1k ω2 k1−Ek 2−Ek. Figure 4(b) shows the dependence of ξon the mode number nfor two different ﬁlm thicknesses. For thin ﬁlms exempliﬁed here as d=14.2n m ,t h e ξfactor peaks at n=1 and further decays for higher modes. As this dependence is similar to the behavior of βkn, the role of ξfor the relative excitation efﬁciency consists in emphasizing the mode withn=1, consistent with the experimental data [ 8]. However, for thicker ﬁlms, the maximum of ξis shifted towards higher modes ( n=5f o r d=70 nm). Note that because in those ﬁlms both ωandkonly slightly increase with n, for small n (when k nλSTT/lessmuch1,ωnτ2/lessmuch1) the other factors \|˜β\|2(kn) and \|˜θ\|2(ωn) are both almost constant. As such, up to much larger n, the dynamical response is dominated by ξand thus can be enhanced at higher ( n> 1) order spin-wave modes. Supported by the analytic theory, we expect in summary a linear relation εn∝\|˜β\|2(kn)\|˜θ\|2(ωn)ξ. (10) Indeed, Fig. 4(c) shows that a linear scaling law holds over four decades. The data shown here were obtained by varyingλ STTbetween 0.5 and 5 nm. The excellent agreement between the predictions of the analytic calculations and the results ofnumerical simulations emphasizes that all signiﬁcant factors are accounted for in Eq. ( 10). Having outlined the factors contributing to the excitation efﬁciency of the PSSW modes in thin ﬁlms, we can now builda bridge to the experiment. We note that in Ref. [ 8] an upper boundary for λ STThas been identiﬁed, based on the spin current pulse ability to excite the spin-wave mode with n=4. Here, the above mentioned formalism enables the determination ofalower λ STTboundary. The critical conditions for that rely on the fact that the excitation efﬁciency for n=5 mode was found insufﬁcient for its unambiguous detection in the experimentaldata. Clearly, for smaller λ STT,t h en=5 mode will be more strongly excited. As such, we can calculate the ratio r15=s15A1/A5 (11) for various λSTT, where Anis the Fourier amplitude of the nth PSSW mode at the interface. The correction factor s15takes into account that in the experiment the MOKE in-depth sensitivityfunction w(z) is responsible for the fact that different PSSW modes contribute unequally to the total MOKE signal. Wecalculate w(z) using an optical transfer matrix method [ 37], and determine s 15, in order to enable direct comparison with the experimental peak-to-noise ratio rexp=Aexp 1 AN, where ANis the experimental noise level. We arrive at the following conditionforλ STT: r15(λSTT)/greaterorequalslantAexp 1 AN. (12) In Fig. 5, we show the dependence of the peak ratios on λSTT. The intersection of the red (experimental) and black 054429-5HENNING ULRICHS AND ILY A RAZDOLSKI PHYSICAL REVIEW B 98, 054429 (2018) FIG. 5. Determination of a lower boundary for the spin-transfer torque characteristic depth λSTT. Simulated (black) peak ratio of the amplitudes of the ﬁrst and ﬁfth spin-wave eigenmodes r15[Eq. ( 11)] as a function of λSTT. The red horizontal line indicates the signal-to- noise level in the experiments [ 8], the red shaded region indicates the corresponding error bar. The vertical dashed line shows the estimatedlower boundary λ c. (simulated) curves indicates the lower boundary for λSTTof about 1.56 nm. B. Incoherent magnon creation and annihilation As a second numerical experiment, we prepared a spin current pulse whose polarization pis collinear to x, and thus to the Fe magnetization m. Then, according to Eq. ( 2)a t temperature T=0 the spin torque τSTT vanishes, and no coherent spin dynamics can be expected. At ﬁnite temperaturesthis expectation is, however, misleading due to the magneticﬂuctuations present in the ferromagnet. Practically, these ﬂuc-tuations manifest as a reduction of the average magnetization.Averaging in time gives the transverse components /angbracketleftm y,z/angbracketrightt= 0, and the longitudinal component /angbracketleftmx/angbracketrightt<1. The larger the temperature, the smaller is the longitudinal magnetizationcomponent. Switching on a spin current pulse with pcollinear toxacts only on the transverse components, which are momen- tarily nonzero. It is well known from conventional magnonics,that the resulting torque is damping- or antidamping-like,and that thermal ﬂuctuation will therefore be suppressed orenhanced [ 24]. This sort of magnon cooling or heating should also manifest on ultrafast time scales in either an increase, or afurther decrease of the longitudinal magnetization component. In the following, we will discuss simulation results obtained forλ STT=2n ma t T=100,200,and 300 K. Note that before the spin current pulse is injected, we determined a thermalaverage by ﬁrst increasing the temperature to the desiredvalue, and then letting the system evolve for 5 ns. This valuewas found sufﬁcient to ensure that the system reaches theequilibrium and does not evolve further. In Fig. 6(a),w ep l o t the temporal form of the spin current pulse according to Eq. ( 1). In Fig. 6(b), we show the temporal evolution of the spatially averaged longitudinal magnetization component /angbracketleftm x/angbracketrightxyzfor T=100 K. See also movie 2 in Ref. [ 35] for an animated version of the data. When the spin current pulse penetratesinto the ﬁlm, the ﬂuctuations increase (decrease), in caseofp/harpoondownleft/harpoonuprightx(p/bardblx). Therefore simultaneously the longitudinal magnetization component further decreases (increases). Note FIG. 6. Suppression and enhancement of incoherent magnon dy- namics by spin currents. (a) Temporal shape of the spin-current pulse js(t) of the two opposite polarities. (b) Simulated time dependence of the spatially averaged longitudinal magnetization component /angbracketleftmx/angbracketright atT=100 K. (c) Variations of spatially averaged longitudinal magnetization component /angbracketleftmx/angbracketright, normalized to that obtained in the case of no spin current injection. Multiple lines show the resultscalculated for three various temperatures, as indicated by dark arrows. The dashed line in (a) through (c) indicates the peak time of the spin current pulse. (d) Layer-resolved, only laterally averaged, normalizedchange of the longitudinal magnetization component /angbracketleftm x/angbracketrightobtained at the temperature of 200 K. that we only show single time series in Fig. 6(b). The determin- istic nature of the thermal noise in the numerical simulationenables us to apply a normalization procedure to the casewhen no spins are injected into the ferromagnet. Normalizing and shifting yields the quantity /angbracketleft/Delta1m x/angbracketrightxyz=/angbracketleftmx/angbracketrightxyz(js/negationslash=0) /angbracketleftmx/angbracketrightxyz(js=0)−1, which is shown in Fig. 6(c). This representation clearly shows that the maximum of the change in the magnetic momentis shifted by 360 fs with respect to the maximum of thespin current, marked by the vertical dashed line. It can beshown that the front of the transient normalized /angbracketleft/Delta1m x/angbracketrightxyzcan 054429-6MICROMAGNETIC VIEW ON ULTRAFAST MAGNON … PHYSICAL REVIEW B 98, 054429 (2018) be approximated with the time-integrated spin current pulse proﬁle, indicating the accumulative nature of the effect. Atlater times, both signals decay approximately exponentiallywith a time constant of 1.79(1) ps. Since in our modeling (bothnumerical and analytical, see Appendix) this decay originatesin Gilbert damping only, one can deduce that the spin-wavemodes with frequencies of a few THz dominate the dynamicresponse. We note that the change of the longitudinal moment is with only 10 −3rather small. As a matter of fact, it increases proportional to the temperature, as Fig. 6(c) shows. Recall that so far we were discussing spatial averages. By applyingthe normalization procedure to each layer of the simulatedﬁlm, we have obtained the spatially resolved data shown inFig.6(d). There one can see that at the injection side the change increases by one order of magnitude, compared to the averageddynamics. Note that Fig. 6(d) looks qualitatively similar to the left panel in Fig. 3(a). However, in contrast to the data shown there, here the spin injection does not cause spatiotemporalcoherence of the spin dynamics. Instead, the data plotted inFig.6(d) should be interpreted as a spatiotemporal modulation of the density of thermal magnons in the ﬁlm. IV . OUTLOOK: OPTICAL SPIN-WA VE EXCITATION BEYOND THE DIFFRACTION LIMIT So far, we have been discussing the spin-wave (magnon) eigenmodes which are quantized in the direction perpendicularto the ﬁlm. When considering optical excitation of laterallypropagating magnons, one has to acknowledge that this processis usually limited by diffraction. As already mentioned in theintroduction, this implies that optical excitation of magnonswith wavelengths below the light wavelength is rather inef-ﬁcient. We will now show that spin currents offer a uniquepossibility to overcome this limitation. Consider a multilayeras in Ref. [ 8], but with the top Fe layer patterned into stripes, as depicted in Fig. 7(a). There, the spin current generation, and subsequently the magnon generation in the bottom Felayer is governed by the geometry of the stripes instead of thelaser spot size. To model this situation, we again employed thespatiotemporal shape of the spin current pulse given by Eq. ( 1). In addition, a lateral mask deﬁned by six stripes (width w= 10 nm and spatial period s=20 nm) was imposed to model the structure shown in Fig. 7(a). As such, the lateral cross-section of the laser-generated spin current pulse reproduces the maskstripe pattern. We used 8192 ×1×10 cells of size 0 .5 3nm3, which were enlarged again into a ﬁlm by applying periodicboundary conditions in xandydirection. In Fig. 7(b), we show a map of the transient magnetization component /angbracketleftm z/angbracketrighta,yz(x,t), averaged across y,z, and across a width of a=2 nm around equidistantly probed locations at a distance xtowards the center of the stripes. The data clearly show the lateral emission of a spin-wave packet, moving witha velocity of about 2500 ±200 m /s. A Fourier transformation in time and space reveals the spectral properties of the spindynamics, as shown in Fig. 7(c). Here, one can see that the dispersion f(k) of dipole-exchange spin-waves in Fe [ 36][ s e e the dashed line in Fig. 7(c)] falls on top of the regions of strong response in the spectrum. Reﬂecting the periodicity ofthe stripe pattern, the strongest response can be seen around FIG. 7. Excitation of the in-plane propagating spin waves. (a) Conceptual device design. Femtosecond laser pulse excites the pe- riodically striped top Fe layer, resulting in generation of a spatiallymodulated spin current pulse. The latter reaches the underlying orthogonally magnetized Fe layer. The resulted spin-transfer torque excites propagating spin waves. Magnifying the active area, the insetshows the spin orientations in the Fe layers, and the spin current ﬂow in the nonmagnetic spacer. The stripe period is denoted with s, while L=500 nm is the maximum probing length. (b) Spatiotemporal plot of the averaged out-of-plane magnetization component /angbracketleftm z/angbracketrighta,yz(x,t). (c) Two-dimensional Fourier transform of the data shown in (b), indicating efﬁcient excitation of spin waves with wave numbersgoverned by the stripes periodicity s, i.e., at k=2π/s.T h ew h i t e dashed line shows the dispersion of dipole-exchange spin waves according to Ref. [ 36]. f=75 GHz and kx=2π/s. The group velocity at this wave number vg=∂ω ∂k(2π/s)≈2600 m /s is in a good agreement with the propagation speed of the spin-wave packet in Fig. 7(b). V . CONCLUSIONS In summary, we have shown that the action of ultrafast spin currents penetrating into a magnetic thin ﬁlm can bemodeled in great agreement with experimentally found 054429-7HENNING ULRICHS AND ILY A RAZDOLSKI PHYSICAL REVIEW B 98, 054429 (2018) ultrafast response and the spectral properties of the subsequent magnetization dynamics between 10 and 500 GHz by includinga Slonczewski-like spin-torque term in the micromagneticequation of motion. Note that, so far, experimental limitationsinhibit the detection of spin-wave modes with frequenciesabove ∼1T H z[ 9]. In principle, an optical experiment similar to Refs. [ 8,9] can be conceived, where smaller thicknesses of the collector Fe ﬁlm would effectively shift the frequenciesof the spin-wave modes up into the THz range. If, inaddition, probe light with much shorter wavelength, and thusshorter penetration depth would be used, this would enablethe observation of high-frequency spin-wave modes andcomparisons with the predictions of our model. In this work, we have shown that depending on the polarization of the spin current, the resulting torque caneither create or annihilate coherent or incoherent magnons.In particular, we reproduced the recent experimentaldemonstration of coherent magnon generation by spincurrents, and obtained further insights into the spatial scalesinvolved in this process. We have identiﬁed the factorscontributing to the relative excitation efﬁciency and shownthat the linear proportionality law holds over four ordersof magnitude. Lastly, employing numerical simulations, wecomplemented the experimentally estimated constraints onthe characteristic spin-transfer torque depth λ STTin Fe. Further, our analysis of the spin current excitation of incoherent magnons indicates that the simulated ultrafastheating and cooling should be detectable by magneto-opticalmethods. We note here that the heating effects which arenot explicitly accounted for in our model might introduceadditional complications. Both thermal and spin current-drivensignals will be overlaid in time, but can in principle be dis-tinguished by their symmetry properties. Considering thinnerferromagnetic ﬁlms and larger densities of the injected spincurrent, as in Ref. [ 38], our modeling suggests that the hot electrons-driven spin-transfer torque can be a relevant andviable ansatz to understand the observed spin dynamics. Wenevertheless acknowledge that under more extreme conditionsa transient change of the magnetization itself, and increasedthermal ﬂuctuations cannot be neglected in order to obtain acomplete picture. Then, more elaborate modeling techniquessuch as those described in Refs. [ 26,39] are needed. In the outlook section, we discussed a way to overcome the diffraction limit when exciting propagating in-plane spinwaves. The basic idea relies on the fact in Fe/Au/Fe trilayer spinvalve trilayers [ 8,28], nanostructuring of the top, laser-excited layer enables spatial tailoring of the spin current proﬁle. In thesimplest case, spin currents can only be excited where the topFe layer exists, thus introducing an in-plane inhomogeneityinto the STT stimulus. We have shown that geometrical pat-terning enables the excitation of propagating spin waves withwavelengths considerably smaller than the optical diffractionlimit of typically used VIS to NIR-VIS laser sources. Note thatthe proposed device design lifts the restriction pertinent to theuse of epitaxial Fe ﬁlms for setting up their magnetizations di-rections. Indeed, the latter can be achieved exploiting the shapeanisotropy of the stripes, and an external magnetic ﬁeld canbe used to ensure an orthogonal magnetization in the bottomFe layer, instead of relying on magneto-crystalline anisotropy.As such, spin current-driven excitation of high-frequency spinwaves in amorphous ferromagnets (e.g., low-loss CoFeB) or even insulating materials (such as yttrium iron garnet attractingincreased attention recently) remains an intriguing perspective. To conclude, we emphasize that this work shows that exper- imental observations from conventional magnonics and recentultrafast experiments can be explained on equal theoreticalfootings. We are convinced that our ﬁndings open a fruitful per-spective for the application of the predictive and interpretativepower of micromagnetic simulation in experimental ultrafastmagnetism. ACKNOWLEDGMENTS H.U. acknowledges ﬁnancial support by the Deutsche Forschungsgemeinschaft within project A06 of the SFB 1073“Atomic scale control of energy conversion.” The authorsthank A. Melnikov and C. Seick for valuable comments andM. Wolf for continuous support. APPENDIX: A CLASSICAL HAMILTONIAN VIEW The theoretical concepts for the following description were ﬁrst outlined by H. Suhl [ 40] and V .S. L’vov [ 41]. This theory can be regarded as analytic micromagnetic modeling. Notethat it has been successfully employed to describe spin-currentdriven magnetization dynamics in conventional magnonicstudies [ 42–45]. Consider a thin ferromagnetic ﬁlm of thick- nessd=14.2 nm, supporting magnons with amplitudes b k, frequencies ωk, and relaxation rates /Gamma1k. According to Suhl, the magnetization dynamics, as described by the Landau-Lifshitz(LL) equation ˙M=−γμ 0M×Heff (A1) can be analyzed into plane spin waves, and the dynamics of these modes can be described by simple kinetic equations. Theﬁrst step is to linearize the LL equation ( A1), and then apply the ﬁrst Holstein-Primakoff transformation (HPT) to calculate ˙m +=/summationdisplay k˙akeikr=˙my+i˙mz, (A2) where mi=Mi/M 0. The second HPT takes into account ellipticity of the precession in a tangentially magnetizedﬁlm. It ﬁnally maps a kto the amplitudes bk. In total, the HPTs diagonalize the Hamiltonian H, which generates the LL equation ( A1). Without dissipation and interactions, Hthen simply reads H=/summationdisplay k¯hωkbkb∗ k. (A3) Note that for the dispersion ωk=ω(k) we take the approx- imation Eq. ( 3) in the main text. The canonical equation of motion for the spin-wave amplitudes is then ˙bk+iωkbk=0. (A4) Adding a Gilbert-like dissipation term to the Eq. ( A1), one gets ˙bk+[iωk+/Gamma1k]bk=0, (A5) where the relaxation rate is given by /Gamma1k=αωH∂ω ∂ωH, (A6) 054429-8MICROMAGNETIC VIEW ON ULTRAFAST MAGNON … PHYSICAL REVIEW B 98, 054429 (2018) withωH=γμ 0H+γμ 02Aex M0k2. a. Coherent magnon generation For the case of our spatially inhomogeneous spin current, similar to Ref. [ 42], we ﬁrst introduce the quantity β(t,r)=γ¯hj0 2eμ0M0e−z/λ STTe−t/τ2 1+e−(t−t0)/τ1. (A7) It is then convenient to consider a Fourier representation of β, and separate out the time-dependence: β(t,r)=θ(t)/summationdisplay kβkeikr. (A8) Ifp⊥M, linearizing the STT term [Eq. ( 2)] results for the ﬁrst HPT in ˙m+=˙my+i˙mz−θ(t)/summationdisplay kβkeikr. (A9) The rate equation for a particular mode amplitude bkthen reads ˙bk+[iωk+/Gamma1k]bk−θ(t)βk=0. (A10) On short timescales t/lessmuch1 /Gamma1k,2π ωk, the second term in Eq. ( A10) can be neglected. Direct integration yields, if the initial dynamic amplitude is small: bk(t)=βk/integraldisplayt 0θ(t/prime)dt/prime. (A11) Summation over the all kmodes and Fourier transformation back into real space gives my(t,z)=β(z)/integraldisplayt 0θ(t/prime)dt/prime, (A12) whereas mz(t,z)=0. This result implies that one can obtain the temporal shape of the spin current pulse θ(t) by taking the time derivative of my(t) probed in the corresponding MOKE geometry (e.g., by L-MOKE in Fig. 4(b) from Ref. [ 8]). Furthermore, here it is seen why we call this process coherent.In a stroboscopic pump-probe experiment, one always inducesdeterministic growth of a transverse magnetization componentwith the same phase. For longer time scales, one needs to consider all terms in Eq. ( A10). Note that a constant spin current θ(t)=θ 0leads to a new equilibrium orientation of the magnetization, whereas atime-limited spin current pulse θ(t) pumps energy into the dif- ferent spin-wave modes. Here we assume that the perturbationis small so that the orientation of the magnetization remainsunchanged. Applying a time-domain Fourier transformation toEq. ( A10) and its complex conjugate, we get iω˜b k+[iωk+/Gamma1k]˜bk−˜θ(ω)βk=0, iω˜b∗ k+[−iωk+/Gamma1k]˜b∗ k−˜θ∗(ω)β∗ k=0. (A13) Thus the power spectrum of a given mode kreads pk(ω)=˜bk˜b∗ k=˜θ(ω)˜θ∗(ω)βkβ∗ k ω2 k−ω2+/Gamma12 k+2iω/Gamma1k. (A14)To obtain the absorbed partial energy εkintroduced in Eq. ( 5), we integrate pk(ω) over the frequency range: εk∼Im/bracketleftbigg/integraldisplay∞ 0pkdω/bracketrightbigg ≈˜θ(ωk)˜θ∗(ωk)βkβ∗ ktan−1/Gamma1k ωk ωk ≈˜θ(ωk)˜θ∗(ωk)βkβ∗ k/Gamma1k ω2 k. (A15) Here, we can already identify the factors \|˜θ\|2(ω)= ˜θ(ωk)˜θ∗(ωk), and \|˜β\|2(k)=βkβ∗ kfrom Eq. ( 10). Conﬁrm- ing intuitive expectations, both spectral Fourier powers ofthe spatial and temporal factors directly impact the partialenergy uptake of a particular spin current-driven spin-wavemode. Furthermore, we now explicitly derive the additionalfactor ξ, which appears crucial for the comparisons of the excitation efﬁciency between different modes. Recall that inthe numerical simulation, we actually analyze the dynamicsof the out-of-plane component m z. When passing back from the amplitudes bkto the magnetization, one has to take into account the ellipticity Ekof the precession. We thus have to acknowledge that (see, e.g., Chap. 1 in Ref. [ 46]) \|mk,z\|2 \|mk,y\|2=1−Ek, (A16) where Ek=/parenleftBigg 1+γμ 0H+γμ 02A M0k2 γμ 0M0/parenrightBigg−1 . (A17) When analyzing the partial energies of the modes found in the spectrum of mz, the mode speciﬁc correction yields for the third factor in Eq. ( 5): ξ=/Gamma1k ω2 k1−Ek 2−Ek. (A18) b. Incoherent magnon generation We will now consider spin injection into a thermally occupied magnon ensemble. If we assume a spatially homoge-neous spin current, whose polarization p/bardblM, the linearized contribution of the STT term to the LL equation ( A1) results in ˙m +=˙my+i˙mz+β(t)m+, (A19) with the temporal dependence of βgiven by Eq. ( 2). The rate equation for bkreads [ 42] ˙bk+[iωk+/Gamma1k+β(t)]bk=Fk, (A20) where β(t)=γjs(t) dμ0M0, and Frepresents a thermal noise source. Note that in Eq. ( A20) higher order terms in bk, which result from the second HPT, are not taken into account [ 42]. The product β(t)bkin Eq. ( A20) explains why the excitation process is incoherent. The random phase of a thermally drivenmagnon gets imprinted on the spin current term. Therefore,in a pump-probe experiment, the temporal evolution of theindividual mode’s phases will differ from shot to shot in arandom fashion. 054429-9HENNING ULRICHS AND ILY A RAZDOLSKI PHYSICAL REVIEW B 98, 054429 (2018) If the spin current is spatially inhomogeneous, one should again consider β=θ(t)/summationtext kβkeikr. Then, in principle, differ- entβkinduce a mixing between the magnon modes bk.T h e rate equation then reads ˙bk+[iωk+/Gamma1k]bk+θ(t)/summationdisplay k/prime,k/prime/primeδ(k/prime+k/prime/prime−k)βk/primebk/prime/prime=Fk. (A21) For the experimental situation of a laterally homogeneous, but vertically inhomogeneous spin current, the mixing couplesthe modes with different k zbut equal kxandky. To simulate the dynamics of a thermal magnon ensemble, we consider avolume of v=dL 2, and quantize the wave numbers according tokx=nπ L,ky=mπ L,kz=oπ d, with L=10d. Modes up to m,n=± 256 and o=24 are taken into account. For the implementation of Fk, we have chosen a Gaussian random number generator, which obeys /angbracketleftFkF∗ k/prime/angbracketright=2δ(k−k/prime)/Gamma1kkBT ¯hωk. (A22) In the absence of a spin current, this provides in the temporal average an equilibrium magnon number density of /vextendsingle/vextendsinglebeq k/vextendsingle/vextendsingle2=/angbracketleftbkb∗ k/angbracketright=kBT ¯hωk. (A23) To probe the dynamics, we determine the total number of magnons/summationtext kbkb∗ k. Since each magnon carries 2 μB,t h i s reduces the magnetization to mx=/radicalBigg 1−2μB vMs/summationdisplay kbkb∗ k. (A24) Similar to the analysis of the numerical simulations output, we then normalize the results by the equilibrium magnetizationmeq x, deﬁned by meq x=/radicalBigg 1−2μB vMs/summationdisplay kkBT ¯hωk, (A25) and shift by 1 to obtain /Delta1mx=mx meq x−1. (A26) Note that for comparing this analytic model with the micromagnetic model, we neglect the cross coupling terms FIG. 8. Comparison of the spin dynamics according to the classi- cal Hamiltonian model and to the numerical micromagnetic modeling. The red and blue lines are obtained for the opposite polarities ofthe spin current pulse. (a) Relative variations of averaged magnon number /angbracketleft/Delta1m x/angbracketrightfrom Eq. ( A26) as a function of time, obtained from the Hamiltonian model given by Eq. ( A21). Computation for different temperatures as indicated. (b) Corresponding micromagnetic simulation data, also shown in Fig. 6(c). The dashed line indicates the peak time of the injected spin current pulse. in Eq. ( A21) since those cancel out in the statistical average. Figure 8shows the results obtained for positive and negative spin current pulses of equal magnitude, at three differenttemperatures. The data look quite similar to those shown inFig. 6(c) in the main text. For instance, for positive spin injection, after a fast initial increase which saturates 350 fsafter the maximum of the injection density, the magnonnumbers decrease approximately exponentially with a decaytime of 1.48(1) ps. 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PhysRevB.99.054430.pdf	PHYSICAL REVIEW B 99, 054430 (2019) Editors’ Suggestion Characterizing breathing dynamics of magnetic skyrmions and antiskyrmions within the Hamiltonian formalism B. F. McKeever,1,2,D. R. Rodrigues,1,2D. Pinna,1Ar. Abanov,3Jairo Sinova,1,4and K. Everschor-Sitte1 1Institute of Physics, Johannes Gutenberg-Universität, 55128 Mainz, Germany 2Graduate School Materials Science in Mainz, Staudingerweg 9, 55128 Mainz, Germany 3Department of Physics & Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA 4Institute of Physics ASCR, v.v.i, Cukrovarnicka 10, 162 00 Prag 6, Czech Republic (Received 25 November 2018; published 26 February 2019) We derive an effective Hamiltonian system describing the low-energy dynamics of circular magnetic skyrmions and antiskyrmions. Using scaling and symmetry arguments, we model (anti)skyrmion dynamics through a ﬁnite set of coupled, canonically conjugated, collective coordinates. The resulting theoretical descrip-tion is independent of both micromagnetic details as well as any speciﬁcity in the ansatz of the skyrmion proﬁle.Based on the Hamiltonian structure, we derive a general description for breathing dynamics of (anti)skyrmionsin the limit of radius much larger than the domain wall width. The effective energy landscape reveals twoqualitatively different types of breathing behavior. For small energy perturbations, we reproduce the well-known small breathing mode excitations, where the magnetic moments of the skyrmion oscillate around theirequilibrium solution. At higher energies, we ﬁnd a breathing behavior where the skyrmion phase continuouslyprecesses, transforming Néel to Bloch skyrmions and vice versa. For a damped system, we observe the transitionfrom the continuously rotating and breathing skyrmion into the oscillatory one. We analyze the characteristicfrequencies of both breathing types, as well as their amplitudes and distinct energy dissipation rates. Forrotational (oscillatory) breathing modes, we predict on average a linear (exponential) decay in energy. We arguethat this stark difference in dissipative behavior should be observable in the frequency spectrum of excited(anti)skyrmions. DOI: 10.1103/PhysRevB.99.054430 I. INTRODUCTION Topological magnetic textures have attracted substantial attention in spintronics [ 1–4] in light of prospects to harness their favorable properties for magnetic memory technologies,[5–7] information processing [ 8–12], and novel approaches to computation [ 13–19]. An important building block in this direction is understanding their dynamical excitationsto test stability and to devise ways to efﬁciently manipu-late them. Among the magnetic textures studied are domainwalls [ 5,20–22] and skyrmions [ 4,6,23–28]. With respect to possible applications in memory devices, skyrmions presentseveral advantages over domain walls due to their smallersizes, lower threshold for current driven mobility [ 3,7,27,29], and their tendency to avoid obstacles and boundaries [ 30]. Recently, a growing collection of other exotic relatives arealso receiving attention. This includes magnetic solitons ofhigher topological order [ 31], chiral bobbers [ 32], nontopo- logical counterparts such as the skyrmionium particle [ 33], and antiskyrmions [ 34–37], where the winding number is opposite in sign compared to the skyrmion. Following the pioneering work by Schryer and Walker [38], effective descriptions for current-driven and ﬁeld-driven domain-wall motion have been considered widely [ 20,39,40], including recent models leveraging canonically conjugated bmckeever@uni-mainz.devariables derived from the spin Berry phase action [ 41–43]. For a magnetic skyrmion, it has been shown [ 33,44,45] that itsmotion can be described with two conjugated variables describing its position. Effective descriptions for the internaldynamics of magnetic skyrmions, however, are few in number.Previous works have often focused on eigenmode analysis ofmagnons to obtain the small amplitude internal excitationsof isolated magnetic skyrmions [ 46–51]. Skyrmion breathing modes, in which the core of the spin structure grows andshrinks periodically in time, were ﬁrst described theoreticallyby numerical simulations of skyrmion lattices [ 52] and later found experimentally [ 53] in the insulator Cu 2OSeO 3from microwave response experiments. In this paper, we derive a nonlinear effective model for rotationally symmetric (anti)skyrmion breathing modes interms of two collective coordinates whose validity extendsbeyond approaches based on an eigenmode analysis, similarto an earlier effective breathing model found in the contextof dynamically stabilized skyrmions [ 54]. This is obtained by the method of Hamiltonian collective coordinate dynamics,independently of microscopic details, and is applicable to thecircular internal modes of both skyrmions and antiskyrmions.We consider an experimentally relevant model for chiralthin ﬁlms to study the equilibrium and breathing propertiesof (anti)skyrmions in detail where all material details canbe collapsed into a single effective parameter. In model-ing the nonlinear excitations above equilibrium, we describetwo dynamical regimes of coherent magnetization behavior: 2469-9950/2019/99(5)/054430(15) 054430-1 ©2019 American Physical SocietyB. F. MCKEEVER et al. PHYSICAL REVIEW B 99, 054430 (2019) (i) oscillation around the local equilibrium magnetization direction and (ii) rotational breathing mode dynamics wherethe local magnetization continuously rotates [ 54,55]. By anal- ogy, these regimes may be thought of as a pendulum which,depending on its energy, either (i) swings about its equilibriumposition or (ii) fully rotates around its pivot point. This paper is organized as follows. In Sec. II, we describe how to pass from a micromagnetic dynamical description toa Hamiltonian mechanics in terms of collective coordinates.In Sec. IIIwe derive the Hamiltonian mechanics describing the dynamics of soft modes for (anti)skyrmions. We ﬁrstreview the Poisson bracket for the translational motion of rigidtextures in two dimensions and then derive its analog for cir-cular breathing modes. In Sec. IV, we introduce a micromag- netic model for chiral thin ﬁlms with perpendicular magneticanisotropy, construct its effective energy as a one parameterphysical model, and study its equilibrium properties. Lastly,in Sec. V, as an application of the effective Hamiltonian formalism, we study the breathing modes of circularly shaped(anti)skyrmions and make precise predictions regarding theirdissipative behavior. II. COLLECTIVE COORDINATES AND HAMILTONIAN FORMALISM The magnetization dynamics below the critical temperature in ferromagnetic materials is well described in a continuumapproximation by the Landau-Lifshitz-Gilbert (LLG) equa-tion [ 56], ˙m=1 Jm×δH[m] δm+αm×˙m,(m)2=1, (1) where the magnetization conﬁguration is represented by the unit vector m(r,t)=M(r,t)/Ms.H e r e J=Ms/γ0is the an- gular momentum density, γ0is the gyromagnetic constant, Ms is the constant saturation magnetization, the Hamiltonian of the system is H≡H[m],αis the Gilbert damping parameter, and the overdots indicate total time derivatives ˙m≡dm/dt. The LLG is a nonlinear differential equation with an in- ﬁnite number of degrees of freedom. This poses an obstaclefor the comprehensive study of the magnetization dynamics,prohibiting a full analytical solution and usually requiring ex-tensive micromagnetic simulations. However, the low-energydynamics of magnetic textures depend only on the system’ssoft modes , which can be described by conjugated variables in a Hamiltonian formalism [ 40,42,43,57]. This allows access to generic features of the magnetization textures independentlyfrom the microscopic characteristics of the material. The conservative, precessional part of the LLG equation may be derived from the ﬁrst variation of the action S=/integraldisplay dtL=S B−/integraldisplay dtH, (2) with the constraint of a constant magnetization amplitude. Here Lis the Lagrangian and SBis the spin Berry phase action [ 41,44,58]. The spin Berry phase couples the dynamical degrees of freedom of the local magnetization.In a spherical representation of the magnetization ﬁeld,m(θ(r,t),φ(r,t))=(sinθcosφ,sinθsinφ,cosθ) /intercal,u s i n g the “north-pole” parametrization [ 59], it is possible to writethe spin Berry phase as SB=J/integraldisplay dt/integraldisplay dV(1−cosθ)˙φ. (3) Two canonically conjugated ﬁelds, in this case given by φ(r,t) andJ[1−cos(θ(r,t))], are sufﬁcient to describe the magnetization dynamics. In addition to the energy-conservingpart, the phenomenological damping term may be intro-duced directly into the Euler-Lagrange equations by use of aRayleigh dissipative functional [ 56], such that the full Eq. ( 1) is derived from 0=d dtδL[M,˙M] δ˙M−δL[M,˙M] δM+δR[˙M] δ˙M, (4) where the Rayleigh functional is R[˙M]=αJ 2M2s/integraldisplay dV(˙M)2. (5) Since the magnetization is characterized by a ﬁeld, it has inﬁnitely many modes. It is possible to map the dynamicsto an inﬁnite number of time-dependent functions ξ(t)= {ξ 1(t),ξ2(t),...}, i.e., m[r,ξ(t)]. The unique equation of motion with inﬁnite degrees of freedom, Eq. ( 1), becomes an inﬁnite set of equations of motion for these dynamical param-eters in this approach. The utility of this mapping is that thesedifferent parameters may have different timescales. Therefore,the low-energy excitations can be described as a reducedset of collective coordinates, ξ(t)={ξ 1(t),ξ2(t),...,ξ 2N(t)} describing the soft modes which dominate the dynamics [ 40], and whose relaxation time is much longer than the rest ofthe inﬁnite set. Examples include the position and tilt angleof a domain wall when subjected to small driving currents,or the position ( X,Y) /intercalof a rigid homogenous domain in steady translational motion as described originally by Thiele[60]. Identifying collective coordinates therefore offers the possibility to work with a reduced number of degrees offreedom, i.e., a minimal number of equations of motion justfor the corresponding soft modes, rather than the full LLGﬁeld equation, which is usually analytically intractable. There are several methods to obtain the equations of mo- tion for the collective coordinates [ 40,42,60,61]. In this paper, we outline the Hamiltonian approach. Within this formalism,the equations of motion are obtained in an explicit and di-rect manner from an effective energy which is a functionof collective coordinates for the soft modes. We providethe comparison between this Hamiltonian approach and thegeneralized Thiele approach in Appendix B. Recasting Eq. ( 1) in a Hamiltonian language yields [ 44] ˙m={m,H} /Phi1,/Pi1+γm, (6) containing an energy-conserving part expressed using Poisson brackets for two canonically conjugated ﬁelds /Phi1(r,t) and /Pi1(r,t), and a damping contribution described by γm≡αm× ˙m. The conventions for Poisson brackets used throughout this paper and their properties are listed in Appendix A. The effective equations of motion for the collective coordi- nates in the Hamiltonian language are ˙ξi={ξi,H}q,p+γξi, (7) 054430-2CHARACTERIZING BREATHING DYNAMICS OF MAGNETIC … PHYSICAL REVIEW B 99, 054430 (2019) where the Poisson brackets {·,·}q,pare now deﬁned for pairs of canonical variables ( qi,pi) in terms of the independent collective coordinates ξi(t) and their time derivatives ˙ξi(t). In general, the canonical coordinates may always be taken as collective coordinates themselves, qi≡ξi; meanwhile, the momenta are gauge-dependent functionals pi=Ai[ξ], so the corresponding Lagrangian is L=/summationtext iAi˙ξi−H[62]. T h e canonical momenta arising from the vector potential corre-spond to a monopole ﬁeld in the spin Berry phase action,i.e.,∇×A=Jm[57,63], and are related to the gyrotropic tensor by [ 40,57]G ij=J−1(∂Ai/∂ξ j−∂Aj/∂ξ i). For the commonly used gauge choice giving Eq. ( 3) the functional is simply Ai[ξ]=/integraltext dV(1−cosθ)∂ξiφ. In terms of the magneti- zation, the gyrotropic tensor is Gij=/integraldisplay dVm·/parenleftbig ∂ξim×∂ξjm/parenrightbig . (8) In the following, we will only consider Hamiltonians H= H[ξ] that are not explicitly time dependent. In this case, the dissipative part γξiis derived by comparing the rate of energy dissipation that is given by the dissipative functional, ˙H= −2R=−αJ/summationtext ijDij˙ξi˙ξj, to its expansion ˙H=/summationtext i∂H ∂ξi˙ξi, where Dij=/integraldisplay dV/parenleftbig ∂ξim·∂ξjm/parenrightbig (9) is the viscosity tensor. Solving for γξiyields the result, γξi=αJ/summationdisplay j,k{ξi,ξj}ξ,pξDjk˙ξk. (10) We point out that Eqs. ( 7) and ( 10) taken together are not speciﬁc to magnetization dynamics; rather, they are genericfor conservative mechanical systems to which frictional forces(force terms linear in velocities) are included in the equationof motion by the use of a Rayleigh function. Using the identity{ξ i,H}ξ,pξ=/summationtext j{ξi,ξj}ξ,pξ∂H ∂ξj, we may write Eqs. ( 7) and ( 10) together for a reduced set of soft modes, ˙ξi=2N/summationdisplay j=1{ξi,ξj}ξ,pξ/parenleftBigg ∂H ∂ξj+αJ/summationdisplay kDjk˙ξk/parenrightBigg , (11) which is our ﬁrst main theoretical result. As an explicit matrix equation, the above result becomes ˙ξ=1 J(G−αD)−1∂H ∂ξ(12) by use of the relation between the Poisson brackets and the gyrocoupling tensor [ 63]J{ξi,ξj}ξ,pξ=(G−1)ij. III. EFFECTIVE HAMILTONIAN DESCRIPTIONS FOR MAGNETIC SKYRMIONS The LLG equation contains topologically nontrivial so- lutions. In 1D, they include domain walls, and in 2D theyinclude different types of solitons that are distinguished bytheir integer topological charge or winding number, Q=1 4π/integraldisplay dxdy m·(∂xm×∂ym). (13) An important example of magnetic solitons in 2D are skyrmions. In the following, we will apply the Hamiltoniandescription given by Eqs. ( 7) and ( 10) to the dynamics of skyrmions and antiskyrmions. First we review the steadytranslational motion for rigid topological structures using thisapproach, and second we apply the technique to study thebreathing mode. A. Translational modes of rigid topological textures The translational motion of a rigid structure can be de- scribed in terms of a position rs(t)=(X(t),Y(t))/intercal, where XandYare collective coordinates describing a soft mode [44,63,64]. The following discussion does not require a spe- ciﬁc deﬁnition of rs(t). To obtain the Poisson bracket structure from Eq. ( 3), we need to peturbatively expand the ﬁelds θ(r,t) and∂tφ(r,t) in terms of small deviations in X,Yand ˙X,˙Y, respectively. Considering a rigid texture ansatz for the magnetization, m(r,t)=m0[r−rs(t)], implies [ 60]∂Xm= −∂xmand∂Ym=−∂ym. Performing an expansion in the spin Berry Phase action Eq. ( 3) up to quadratic order and discarding terms that do not contribute to the dynamics leadsto (see Appendix B) S B≈/integraldisplay dtz 0JX˙Y/integraldisplay dxdy [(∂ycosθ)∂xφ−(x↔y)], (14) where z0is the thickness of the system and the spatial integral is proportional to the topological charge deﬁned in Eq. ( 13). Hence the effective action is Seff=/integraldisplay dt(4πQz0JX˙Y−H). (15) Noting that the canonical momentum to YispY= 4πQz0JX, we therefore read off the Poisson bracket for topologically nontrivial textures [ 44]: {Y,X}Y,pY=1 4πQz0J. (16) Inserting the Poisson bracket of Eq. ( 16) into Eq. ( 11)g i v e s the dynamical equations of motion for the translational mode, ˙X=−1 4πQz0/bracketleftbigg1 J∂H ∂Y+α(DYX˙X+DYY˙Y)/bracketrightbigg ,(17a) ˙Y=1 4πQz0/bracketleftbigg1 J∂H ∂X+α(DXX˙X+DXY˙Y)/bracketrightbigg , (17b) which are equivalent to the traditional Thiele equations for skyrmions [ 60,65], and in the case of circular states, like for a simple skyrmion, the off-diagonal elements DXY=DYX vanish. B. Internal dynamics of skyrmions and antiskyrmions For a study of the internal dynamics of magnetic solitons, such as skyrmion breathing modes [ 46–48], one must go be- yond the rigid texture (or traveling-wave) approximation usedin the last section. We will consider a thin ﬁlm system thatis translationally invariant along the zdirection and contains localized rotationally symmetric states [ 6,23,24], m(ρ)=[sinθ(ρ) cosφ(ψ),sinθ(ρ)s i nφ(ψ),cosθ(ρ)] /intercal, (18a) ρ=(ρcosψ,ρ sinψ)/intercal, (18b) 054430-3B. F. MCKEEVER et al. PHYSICAL REVIEW B 99, 054430 (2019) FIG. 1. Schematic ﬁgure of a (a) Néel skyrmion, (b) anti- skyrmion, and (c) the common mzcomponent for both structures. (d) The in-plane spins for prototypical (anti)skyrmions with phase η, where η=0 corresponds to the spin structures to the left in (a) and (b). where ρandψare the polar coordinates in the x–yplane. Since the magnetic angle θonly changes with distance ρfrom the skyrmion core, these ﬁeld conﬁgurations are referred toascircular in this paper. In view of describing the breathing modes of skyrmions as well as antiskyrmions, we parametrizethe azimuthal angle of the magnetization by φ(ψ)=mψ+η, (19) where m∈Zis the vorticity and ηis the relative azimuthal angle. For a simple skyrmion, one has m=1 and the angle η describes its helicity: for a Bloch-skyrmion η=(n+1/2)π and for a Néel-skyrmion η=nπ, where n∈Z, see Fig. 1. This corresponds to Q=−m/2[cosθ(∞)−cosθ(0)] with the boundary conditions θ(0)=πandθ(∞)=0. In other words, the ferromagnetic background points in the +ˆzdirec- tion. For circular (anti)skyrmions there are two characteristiclength scales: the radius Rand the width d for the twisted do- main over which m zvaries, see Fig. 1(c). We deﬁne the radius Rby the circle where mz=0. We assume in the following that during the breathing dynamics, θ(ρ) retains its smooth and monotonic variation, ensuring the deﬁnition of Rto be unique. This paper will consider only large (anti)skyrmions inthe regime where d/lessmuchR, where the skyrmion’s wall width can be considered constant even as its radius is allowed to vary. The relevant soft mode for the breathing dynamics of a circular (anti)skyrmion is described by the radius R(t) and the relative azimuthal angle η(t)[54]. Unlike in Sec. III A ,t h e magnetic texture is not rigid but soft in its overall shape. Forskyrmions, the phase ηis the global in-plane angle of the local magnetization pointing away from the radial direction (suchthatη=0,πcorrespond to Néel and η=±π/2 to Bloch conﬁgurations, respectively), while for antiskyrmions, chang-ingηcorresponds to a rigid rotation of the entire magnetic texture. We will now follow the general recipe of Sec. II.B y estimating the volume integral for the Berry phase term inEq. ( 3) we obtain the effective action (see Appendix C) S eff=/integraldisplay dt(aJR2˙η−H), (20)in terms of the collective variables Randη. The length scale ais given by a=2πz0C, where Cis a dimensionless constant (with 0 <C<5) arising from the integral over ρ. Deﬁning the canonical momentum conjugate to ηaspη=aJR2,w e read off the Poisson bracket as {η,R2}η,pη=1 Ja. (21) Next, via the identity {η,R2}η,pη=(∂RR2){η,R}η,pη= 2R{η,R}η,pη, we obtain {η,R}η,pη=(2aJR)−1. Exploiting Eq. ( 11), the dynamical equations for ηand Rare readily derived ˙η=1 2aJR∂H ∂R+αDRR 2a˙R R, (22a) ˙R=−1 2aJR∂H ∂η−αDηη 2a˙η R. (22b) Above we used that ∂Rm·∂ηm=0 by virtue of the rota- tionally symmetric ansatz Eqs. ( 18) and ( 19). Equation ( 22) describes the effective internal dynamics of a rotationallysymmetric magnetic texture subject to the ansatz Eqs. ( 18) and (19) for time-independent Hamiltonians H. IV . EFFECTIVE ENERGY FOR CIRCULAR SKYRMIONS Previous works have assumed an explicit domain wall ansatz for the skyrmion’s radial proﬁle [ 66–68]. By using scaling arguments, however, one does not need to assumea speciﬁc ansatz for the skyrmion. The energy can, in fact,be expanded in powers of the collective Rcoordinate for skyrmions satisfying d/lessmuchR. We illustrate this procedure for a micromagnetic model including exchange, anisotropy, andinterfacial Dzyaloshinskii-Moriya interaction (DMI), whosemagnetic free energy is given by H[m]=/integraldisplay dV/braceleftBigg A/summationdisplay i(∂im)2+K/parenleftbig 1−m2 z/parenrightbig +D(mz∂xmx−mx∂xmz) ±D(mz∂ymy−my∂ymz)/bracerightBigg , (23) where the +(−) sign stands for the isotropic (anisotropic) DMI which stabilizes circular skyrmions (antiskyrmions).Performing the expansion of Eq. ( 23)i nRfor a large radius skyrmion, we obtain the following dimensionless effectiveenergy in units of E DW=A3/2K−1/2, ˜E(˜r,η)=(c1−c3gcosη)˜r+c2 ˜r, (24) where ˜ r≡R//Delta1is the dimensionless (anti)skyrmion radius in units of the one-dimensional domain wall width /Delta1=√A/K. The single coupling constant gis the reduced DMI strength g=πD/(4√ AK), selecting either a ferromagnetic \|g\|<1o r helical \|g\|>1 mean-ﬁeld ground state [ 23,24,67]. All rescal- ing constants are summarized in Table I. The dimensionless values c1,c2, and c3are uniquely determined by the material parameters through the coupling constant g(see Appendix D)[49], since we have chosen to focus on the limit where 054430-4CHARACTERIZING BREATHING DYNAMICS OF MAGNETIC … PHYSICAL REVIEW B 99, 054430 (2019) TABLE I. Natural unit system for a study of skyrmion breathing modes in a system with DMI, exchange, and perpendicular magnetic anisotropy. Quantity Characteristic Deﬁnition SI unit /Delta1 length A1/2K−1/2m 1/ωFM time Msγ−1 0K−1s EDW energy A3/2K−1/2J g coupling constant πD(4√ AK)−11 the skyrmion wall width dis time independent. The effective energy is identical for both skyrmions and antiskyrmions asa consequence of Eq. ( 19), taken with the appropriate choice ofm=+ 1(−1) for skyrmions (antiskyrmions). As such, we will reduce our discussion to skyrmions only from now oneven though the results derived are valid for antiskyrmionsas well. In Appendix D, we discuss how to introduce other interactions, such as dipole-dipole and bulk DMI, into thisframework. Energy landscape analysis The energy landscape described by the effective model Eq. ( 24) has two extrema. The ﬁrst is a global energy mini- mum with corresponding equilibrium coordinates: ˜req=/radicalbiggc2 c1−c3\|g\|,η eq=/braceleftbigg 0i f 0 <g<1 π if−1<g<0. (25) The second represents an energy saddle point with coordi- nates: ˜rsad=/radicalbiggc2 c1+c3\|g\|,η sad=/braceleftbigg π if 0 <g<1 0i f −1<g<0, (26) where we note that ˜ req>˜rsadand that their respective effective energies are ˜Eeq(sad)=2c2/˜req(sad) . Upon rescaling the radii by the equilibrium radius ( r≡ ˜r/˜req) and energy by the equilibrium energy ( /epsilon1≡˜E/˜Eeq)i n Eq. ( 24), the effective energy reduces to the simpliﬁed form /epsilon1=1 2r/parenleftbigg1−Bcosη 1−\|B\|r2+1/parenrightbigg , (27) where we have deﬁned B=c3g/c1as the single parameter which encapsulates the entire contribution from the materialproperties on the physics of the system. In these units, thesaddle point radius is exactly the inverse of the correspondingsaddle point energy [ r sad=/epsilon1−1 sad=√(1−\|B\|)/(1+\|B\|)]. A cut along the /epsilon1=/epsilon1sadplane partitions the energy land- scape into three distinct regions. A schematic view of theenergy landscape is shown in Fig. 2, where individual energy sectors have been color coded to guide the reader. The ( r,η) coordinates in the bowl andhorn regions both correspond to high energy states ( /epsilon1>/epsilon1 sad) as opposed to the basin ’s low energy states ( /epsilon1</epsilon1 sad). In the inset of Fig. 2, we show a cut through η=0 and η=πto emphasize the structure of the extrema introduced above. The constant energy trajectoriesfollowed by skyrmions in their conﬁguration space in the FIG. 2. Effective energy landscape for circular skyrmion breath- ing modes, Eq. ( 27), with B=0.45, highlighting the distinct basin , bowl andhorn energy partitioned by the constant saddle energy curve (shown in white). Inset: a cut through η=0a n dη=π. absence of damping [see Fig. 3(a)] are obtained by solving Eq. ( 27) for the rescaled radius: r±(η)=(1−\|B\|)/epsilon1 1−Bcosη/bracketleftBigg 1±/radicalBigg 1−1−Bcosη (1−\|B\|)/epsilon12/bracketrightBigg . (28) While the two solutions in Eq. ( 28) represent distinct horn andbowl orbits in the /epsilon1>/epsilon1 sadregime, they represent the two branches of the same basin orbit in the /epsilon1</epsilon1 sadcase. In all scenarios, constant energy orbits describe skyrmion breathingmotions as the radius grows and shrinks as a function of η. The qualitative nature of orbits in the basin andbowl/horn regions are, however, very different from each other as thedynamical range of ηis limited in the basin orbits while it takes all values from 0 to 2 πin the horn/bowl orbits. This leads us to denominate the high and low energy breathingdynamics as rotating and oscillating modes, respectively. Thedegeneracy of the rotating modes disappears as their energyis lowered through the saddle energy and into the basin. Thehorn rotations are unphysical, however, as they would predictuncollapsable skyrmions in the limit of very small skyrmionsizes. Furthermore, since our theory is only applicable forskyrmions much larger than the proﬁle wall width d, it cannot reliably describe their behavior at such small radii. Overall,the dynamical spectrum of this model is reminiscent of that ofa simple pendulum which exhibits rotations and oscillations FIG. 3. Breathing mode dynamics from the energy landscape with B=0.45. (a) Constant energy orbits: two are degenerate on the bowl and horn at /epsilon1=2/epsilon1sad, while the third one lies in the basin at /epsilon1=3 4/epsilon1sad. (b) A damped trajectory starting on the bowl that spirals in towards the energy minimum. 054430-5B. F. MCKEEVER et al. PHYSICAL REVIEW B 99, 054430 (2019) FIG. 4. Time evolution of the dynamical system for the rescaled radius r(top row), azimuthal angle η(middle row), and rescaled energy (bottom row). Orange and blue curves correspond to undamped and damped trajectories, respectively. (a) Numerical results based on the effective model, Eq. ( 31) with B=0.45 for an initial energy starting in the /epsilon1>/epsilon1 sadbowl region (light blue background). The black dashed lines in (a) denote the maximum and minimum radii of the undamped trajectory [ r+ 1andr+ 2from Eq. ( 35), respectively]. The transition from rotating to oscillating phase (light pink background) is best seen in the behavior of the azimuthal angle where ηrotates a number of times before oscillating around its equilibrium value. Inset: Projection of the trajectory on the ( rcosη,rsinη) polar coordinate plane where the dashed magenta line represents saddle energy /epsilon1sadseparatrix. Subplots (b) and (c) show the breathing behavior of skyrmions obtained by micromagnetic simulations for an energy above and below the saddle point energy, respectively. The parameters of the simulations are Ms=1.0×106Cm−1s−1,α=0.02,D=2.8×10−3Jm−2,A=1.5×10−11Jm−1,K=1.1×106Jm−3, i.e., g≈0.54. The qualitatively different energy decay behavior is shown in the lower row to transition from linearlike to exponentially decreasing in the rotating and oscillatingphases, respectively. around its suspension point, depending on whether the kinetic energy is greater or less than the potential energy of its “upsidedown” unstable equilibrium. Lastly, the full range of the radial oscillations can be readily obtained from Eq. ( 28) by noting that all radial max- ima/minima in the orbits appear on the η=0,πline (as shown in Fig. 2). For B>0 the rotating modes have r max= r±(0) and rmin=r±(π) whereas in the oscillating phase one hasrmax=r+(0) and rmin=r−(0). Since the lower energy branch on this line is independent of the coupling B,t h e amplitude for oscillating breathing modes below the saddleenergy is insensitive to material properties (see discussionbelow). V . SKYRMION BREATHING MODES A. Equations of motion In terms of the variables randη, the Poisson bracket Eq. ( 21) becomes {η,r2}η,˜pη=c0/˜r2 eq, (29) where we assume a constant shape factor c0coming from the dimensionless integral c−1 0=2π˜z0/integraltext∞ 0dxx [1−cosθ(Rx)] in the spin Berry phase action. Moreover, the dimensionlessdissipation factors both scale linearly with the radius ˜ r, D rr=cr˜r=cr˜reqr,Dηη=cη˜r=cη˜reqr, (30)where crand cηare time-independent proportionality con- stants that may depend on the coupling strength B.U s i n g Eqs. ( 29) and ( 30) in the Hamiltonian formalism Eq. ( 12) gives the dynamical system for skyrmion breathing, dr dτ=−/parenleftbiggB 1−\|B\|sinη+αc0cη ˜reqr/epsilon1−1 r3/parenrightbigg , (31a) dη dτ=/parenleftbigg 2r/epsilon1−1 r3−αc0cr˜req 2B 1−\|B\|sinη/parenrightbigg ,(31b) with/epsilon1=/epsilon1(r,η) as deﬁned in Eq. ( 27) andτis the rescaled time: τ=pαt=/parenleftbigg2c0c1 4+(αc0)2cηcr1−\|B\| ˜req/parenrightbigg t. (32) Numerical solutions of Eqs. ( 31) with and without damping are shown in Fig. 4(a), where the initial energy is set above the saddle energy. These results show the characteristic transitionfrom rotating to oscillating phase in the r,η, and/epsilon1damped evolutions as compared to their undamped, constant energyanalogs. All three variables relax toward the equilibrium stateas expected. The rate of energy loss is qualitatively differentin the two phases, transitioning from a linearlike to an ex-ponential decay as further discussed below. Micromagneticsimulations conﬁrm the qualitative predictions of the effec-tive energy model as it pertains to the skyrmion breathingmodes. In particular, we observe the rotating [Fig. 4(b)] and 054430-6CHARACTERIZING BREATHING DYNAMICS OF MAGNETIC … PHYSICAL REVIEW B 99, 054430 (2019) oscillating [Fig. 4(c)] breathing regimes and their distinct energy loss behavior. B. Results In the following, we present analytical results pertaining to the expected dynamical periods, breathing amplitudes, and energy decay rates. 1. Dynamical periods The undamped periods of motion can be calculated from Eqs. ( 31a) and ( 31b)a s T(/epsilon1)=⎧ ⎪⎨ ⎪⎩sgn(B)(1−\|B\|)2/integraltextηmax ηmindη2/epsilon12(1−\|B\|)+Bcos(η)−1 (Bcos(η)−1)2√ (1−\|B\|)(/epsilon12(1−\|B\|)+Bcos(η)−1), with/epsilon1</epsilon1 sad (1−\|B\|)2/bracketleftbig/integraltext2π 0dη2/epsilon12(1−\|B\|)+Bcosη−1 2(Bcosη−1)2√ (1−\|B\|)(/epsilon12(1−\|B\|)+Bcosη−1)∓2π/epsilon1 (1−B2)3/2/bracketrightbig ,with/epsilon1>/epsilon1 sad.(33) where the limits of integration are ηmax=cos−1{[1−/epsilon12(1− \|B\|)]B−1}andηmin=−ηmax+θ(−B)2πwithθ(x)=1i f x>0 (or 0 otherwise) being the Heaviside function. For the case of /epsilon1>/epsilon1 sad, the positive and negative signs refer to bowllike and hornlike rotations, respectively. Direct numericalintegrations of these formulas show that the period scales lin-early in energy for both the rotating and oscillating breathingmodes (see Fig. 5). Our theory allows us to explore dynamics beyond the small amplitude limit in an ansatz independentmanner. It includes, however, the small amplitude breath-ing calculated previously by using a nonenergy-minimizingansatz. For small oscillations around equilibrium, the periodcan be computed up to O(/epsilon1−1) as T=π/radicalBigg 2/parenleftbigg1−\|B\| \|B\|/parenrightbigg . (34) Upon converting back to physical time using Eq. ( 32) and recalling from Eq. ( 25) how the physical equilibrium ra- dius scales with the material parameters ˜ r−1 eq∼√1−\|B\|, one recovers that the period of small oscillations around FIG. 5. Numerical calculations of the periods T(/epsilon1) with B= 0.82. In the regime /epsilon1>/epsilon1 sad, the rotational “bowl” breathing modes are shown. As expected from Eq. ( 33), the period scales linearly with the system’s energy T(/epsilon1)∝/epsilon1in both regimes.equilibrium scales as T∝˜r2 eqin agreement with previous literature [ 49,69]. In Fig. 6, we emphasize the predictive power of the effec- tive model by comparing theory to micromagnetic modeling FIG. 6. Comparison of micromagnetic simulations (black points) and effective model predictions (orange lines) in the undamped limit for (a) rotational breathing (light blue background) and (b)–(c) oscil- lating breathing (light pink background). We ﬁnd strong agreementfor the weak oscillatory breathing, including the expected simple harmonic motion for small oscillations in (c) about equilibrium, but our model shows a deviation from the micromagnetic simulationsin the rotational regime. The mismatch arises from the breakdown of the large skyrmion radius approximation [see inset in (a)] when the skyrmion contracts to its smallest size. The parameters in allthree simulations are M s=1.0×106Cm−1s−1,α=10−8,D= 3.2×10−3Jm−2,A=1.5×10−11Jm−1,a n d K=1.1×106Jm−3, i.e.,g≈0.62. 054430-7B. F. MCKEEVER et al. PHYSICAL REVIEW B 99, 054430 (2019) in the undamped dynamical limit. Radial and angular trajec- tories obtained by integrating Eq. ( 31) compare well with similar micromagnetic simulations in both rotating [Fig. 6(a)] and oscillating [Figs. 6(b)–6(c)] regimes. While the oscillating breathing modes match almost perfectly, a small deviationis seen between physical and predicted radial dynamics inthe rotating regime. This is due to a breakdown in the largeskyrmion approximation, underpinning the theory wheneverthe skyrmion contracts to sizes comparable to the proﬁle’swall width. To illustrate this, the inset of Fig. 6(a) shows the different proﬁles observed for maximum and minimumskyrmion radii throughout one rotational period. 2. Breathing amplitudes The maximum and minimum possible skyrmion radii de- rived from the model are1 r± 1=/epsilon1±/radicalbig /epsilon12−1, (35a) r± 2=1−B 1+B/epsilon1/parenleftBigg 1±/radicalBigg 1−1+B (1−B)/epsilon12/parenrightBigg , (35b) which may be obtained directly from Eq. ( 28) upon setting η=0o rη=π. At energies below the saddle energy ( /epsilon1< /epsilon1sad),r+ 1is the maximum radius of oscillations and r− 1is the minimum radius of oscillations; meanwhile, for ener-gies above the saddle energy ( /epsilon1>/epsilon1 sad),r+ 1is the maximum radius of rotations on the bowl, r+ 2is the minimum radius of rotations on the bowl, r− 2is the maximum radius of rotations on the horn, and r− 1is the minimum radius of rotations on the horn. We see from Eq. ( 35a) that the stationary points of the roscillations for breathing modes below the saddle energy are independent of the material properties. This maybe likened to a mass-on-a-spring system where the amplitudeis fully determined by the initial extension from equilibriumeven though the speciﬁc dynamics connecting the two extremaof motion do depend on the size of the spring constant and themass. In this case, this is not a trivial consequence of smallharmonic oscillations around equilibrium, however, becauseit is true for the entire 1 </epsilon1</epsilon1 sadrange. 3. Energy decay rates From the expression for the skyrmion’s energy Eq. ( 27) and the breathing equations of motion Eq. ( 31), one can quantify the energy dissipated by the system as d/epsilon1 dτ=/epsilon1r−1 r2dr dτ+B 2(1−B)rsinηdη dτ =−αc0cη ˜req/bracketleftBigg (/epsilon1r−1)2 r5+cr˜r2 eq cη/parenleftbiggB 2(1−B)/parenrightbigg2 rsin2η/bracketrightBigg , (36) which we will use to analytically explain the distinction between linear and exponential decay observed in the nu-merical calculations (see Fig. 4). Since Eq. ( 36)i sg l o b a l l y negative, except at the energy minimum ( r,η)=(1,0) where 1From here on, we select B>0 for deﬁniteness.it vanishes, it correctly describes a dissipative process that relaxes the skyrmion to its equilibrium state. If the skyrmion’s rotational and oscillatory mode precesses on timescales sufﬁciently small compared to those for energydissipation, the two dynamics can be effectively decoupledby averaging Eq. ( 36) over constant energy orbits to obtain a single ordinary differential equation describing the energylost by the system over time. One then has /angbracketleftbiggd/epsilon1 dτ/angbracketrightbigg =−α T(/epsilon1)c0cη ˜req/bracketleftbigg/contintegraldisplay dη(/epsilon1r(η)−1)2 r5(η)(37) +cr˜r2 eq cη/parenleftbiggB 2(1−B)/parenrightbigg2/contintegraldisplay dηr(η)s i n2η/bracketrightbigg , where the integrals are performed over one complete oscilla- tory/rotational orbit Eq. ( 28). In what follows we will leverage the fact that orbital periods Eq. ( 33) scale linearly with the orbit’s energy T(/epsilon1)∝/epsilon1(see Fig. 5). The above integrals are not exactly solvable but a series of upper and lower boundscan still be constructed (see Appendix E). For the rotational breathing mode, one can show that /angbracketleftbiggd/epsilon1 dτ/angbracketrightbigg \|rot/greaterorequalslant−α T(/epsilon1)c0cη ˜req/bracketleftBigg 2π/epsilon16 sad /epsilon1+2π/epsilon1cr˜r2 eq 4cη/parenleftbiggB 1−B/parenrightbigg2/bracketrightBigg /similarequal−α/bracketleftbigg C1+C2 /epsilon12/bracketrightbigg , (38) where we absorb all constants into C1,2>0. By construction, the solution of this new bounding equation is guaranteed todecay faster than the true solution of Eq. ( 36). Since /epsilon1/greatermuch1 for rotational modes, this upper bound guarantees at most alinear decay to the skyrmion’s energy. Following a similar reasoning for the oscillatory breathing modes by constructing a lower bound to the energy dissipationrate, one ﬁnds (see Appendix E) /angbracketleftbiggd/epsilon1 dτ/angbracketrightbigg \|osc/lessorequalslant−α T(/epsilon1)c0cη ˜req/bracketleftbigg2 /epsilon12cr˜r2 eq 4cη/parenleftbiggB 1−B/parenrightbigg5/2 ×[sinη+(/epsilon1) cos2η+(/epsilon1)−cosη+(/epsilon1)]/bracketrightbigg /similarequal−αC3/epsilon1√ /epsilon1−1, (39) where η+(/epsilon1)=cos−1{[1−/epsilon12(1−B)]B−1}is the maximum range of domain-wall tilt angle attained during a single con-stant energy oscillation. Since the solution of Eq. ( 39)i s guaranteed to decay slower than that of Eq. ( 36), the true energy loss in the oscillating regime must decay at leastexponentially. These arguments conﬁrm the sharp transitionobserved in the dissipation rate when the skyrmion breathingdynamics cross the saddle separatrix when transitioning be-tween /epsilon1>/epsilon1 sadand/epsilon1</epsilon1 sadstates, see Fig. 4. VI. CONCLUSIONS In this paper, we have derived the Hamiltonian system for the low-energy excitations of rotationally symmetric mag-netic (anti)skyrmions in an ansatz-independent manner. Bymeans of scaling and symmetry arguments, we modeled thebreathing mode of (anti)skyrmions in terms of collectivecoordinates, where the area of reversed spins in the skyrmion 054430-8CHARACTERIZING BREATHING DYNAMICS OF MAGNETIC … PHYSICAL REVIEW B 99, 054430 (2019) core and the skyrmion phase are conjugated variables in phase space. As seen from the form of the energy landscape, ourmodel exhibits a rich behavior which is conﬁrmed by micro-magnetic simulations of (anti)skyrmion structures in magneticsubstrates with translational invariance along the out-of-planeeasy anisotropy axis. The main results presented in this paperinclude the analytical and numerical investigation of the well-known oscillatory breathing mode where small amplitude os-cillations in the radius and skyrmion phase around equilibriumproceed in tandem, as well as the description of rotationalbreathing behavior, characterized by large-radius oscillationsand a continuous nonuniform precession of the phase. Further-more, we predict two distinct regimes of energy dissipationwhere the average power loss of large-amplitude rotationalbreathing modes decays linearly as opposed to exponentiallyfor the oscillating modes. We expect that these distinctiveenergy decays will allow us to detect the different modesexperimentally. It must be stated that the limit of our modellies in the implicit assumption of ﬁxed skyrmion wall proﬁles.The next order approximation would be to incorporate theskew of the wall proﬁle by introducing an additional pair ofcollective coordinates. Doing so would allow extension of thisanalysis to skyrmion radii much smaller than those allowedby this paper. We would like to emphasize that the resultsdescribed here hold for both skyrmions and antiskyrmions.Therefore, a perfect test system will be one where both ofthem occur simultaneously. This is, for example, naturallythe case when skyrmion and antiskyrmion pairs are created[36,70–72] or in systems with certain symmetries [ 37]. ACKNOWLEDGMENTS The groups at Mainz acknowledge funding from the Ger- man Research Foundation (DFG) including Project No. EV196/2 1, the Transregional Collaborative Research Center(SFB/TRR) 173 SPIN+X, and the Graduate School of Excel-lence Materials Science in Mainz (MAINZ, GSC 266). Fund-ing is also acknowledged from the Alexander von HumboldtFoundation. B. M., D. R., and D. P. contributed equally to this paper. APPENDIX A: POISSON BRACKETS In the main text, we applied techniques from Hamiltonian mechanics. Poisson brackets entered at the level of the col-lective coordinates, where we were concerned with just time-dependent functions, and also at the level of the LLG ﬁeld equation, where the magnetization ﬁeld depends on both spaceand time. Below we review equations for Poisson bracketsrelevant to this work. 1. Time-dependent functions The Poisson bracket convention for time-dependent func- tions A(t) and B(t)i s {A,B}q,p=/summationdisplay i/parenleftbigg∂A ∂qi∂B ∂pi−∂A ∂pi∂B ∂qi/parenrightbigg , (A1) corresponding to the simple action S=/integraltext dt(/summationtext ipi˙qi−H). For example, a basic result is the Poisson bracket between thecanonical coordinates and momenta, {qi,pj}q,p=δij. (A2) Using the Poisson bracket, the time derivative of any function f(q,p,t) can be calculated, ˙f=/summationdisplay i/parenleftbigg∂f ∂qi˙qi+∂f ∂pi˙pi/parenrightbigg +∂f ∂t ={f,H}q,p+∂f ∂t, (A3) where Hamilton’s equations ˙ pi=−∂H ∂qi,˙qi=∂H ∂piwere used in the second line. A further rule can be derived for the Poisson bracket between quantities that have a known dependenceon functions of the canonical variables, e.g., A(t)= A[f 1(q,p,t),f2(q,p,t),... ] and similar for B(t): {A,B}q,p=/summationdisplay i,j∂A ∂fi∂B ∂fj{fi,fj}q,p. (A4) 2. Fields There is an analogous description to Eqs. ( A1)–(A4)f o r ﬁelds. The Poisson bracket between scalar ﬁelds A(x) and B(x/prime)i s {A(x),B(x/prime)}/Phi1,/Pi1 =/summationdisplay i/integraldisplay dy/parenleftbiggδA(x) δ/Phi1i(y)δB(x/prime) δ/Pi1i(y)−δA(x) δ/Pi1i(y)δB(x/prime) δ/Phi1i(y)/parenrightbigg ,(A5) corresponding to an action of the form S=/integraltext dt/integraltext dV(/summationtext i/Pi1i∂t/Phi1i−H). So the Poisson bracket between the canonical ﬁelds is {/Phi1i(x),/Pi1 j(x/prime)}/Phi1,/Pi1=δijδ(x−x/prime). (A6) The time derivative for a ﬁeld F(x,t), similar to Eq. ( A3), is ˙F={F,H}/Phi1,/Pi1+∂F ∂t, (A7) by use of Hamilton’s equations ˙/Phi1i=δH δ/Pi1iand ˙/Pi1i=−δH δ/Phi1i. Finally, the useful identity in analogy to Eq. ( A4), for the Pois- son bracket between quantities that have known dependenceon a set of functions, say {g i(x)},i s {A(x),B(x/prime)}/Phi1,/Pi1 =/summationdisplay i,j/integraldisplay dy/integraldisplay dzδA(x) δgi(y)δB(x/prime) δgj(z){gi(y),gj(z)}/Phi1,/Pi1.(A8) 3. Example: Hamiltonian formulation of the LLG The LLG Eq. ( 1) may be written as a Hamiltonian Eq. ( 6) if one assumes that the local magnetization obeys the algebra {mi(x),mj(x/prime)}/Phi1,/Pi1=−1 J/summationdisplay k/epsilon1ijkmk(x)δ(x−x/prime),(A9) where J=Ms/γ0. This may be veriﬁed explicitly. For exam- ple, using the spherical parametrization of the magnetizationemployed in the main text, we may identify the two canonicalﬁelds as /Phi1(x)=φ(x) and/Pi1(x)=J(1−cosθ(x)) from the 054430-9B. F. MCKEEVER et al. PHYSICAL REVIEW B 99, 054430 (2019) spin Berry phase action Eq. ( 3). Then the three nonzero Poisson brackets between the magnetization components mx, my, and mzin Eq. ( A9) are straightforwardly veriﬁed using Eq. ( A5) with these canonical ﬁelds. Moreover, in this situa- tion Eq. ( A8) reduces to a cross product structure: {A(x),B(x/prime)}/Phi1,/Pi1=−1 J/integraldisplay dym(y)·/parenleftbiggδA(x) δm(y)×δB(x/prime) δm(y)/parenrightbigg . (A10) Hence, by using Eqs. ( A7)–(A9), evaluating the time evolu- tion of a component ml(x)g i v e s {ml(x),H}/Phi1,/Pi1 =/summationdisplay i,j/integraldisplay dy/integraldisplay dzδml(x) δmi(z)δH δmj(y){mi(z),mj(y)}/Phi1,/Pi1 =−1 J/summationdisplay jk/epsilon1lj kδH δmj(x)mk(x) =1 J/bracketleftbigg m(x)×δH δm(x)/bracketrightbigg l, (A11) which is the anticipated precessional term for undamped motion. APPENDIX B: COMPARISON BETWEEN GENERALIZED THIELE METHOD AND THE HAMILTONIAN FORMALISM The generalized Thiele method is based on the idea of describing the dynamics of certain magnetic conﬁgurationsjust in terms of the time evolution of a ﬁnite number ofcollective coordinates describing the soft modes [40,57], for w h i c hw eh a v et h a t ˙m= 2N/summationdisplay i=1/parenleftbigg∂m ∂ξi/parenrightbigg ˙ξi. (B1) This decomposition has been successfully applied in the de- scription of the low-energy excitations of topological mag-netic textures. For example, for the ﬁeld-driven or current-driven motion of domain walls, the physics is well describedup to a certain magnitude of the applied driving ﬁeld or current[21] by a soft mode described by the two collective coordi- nates: the domain-wall position and the tilt angle of the mag-netization inside the wall. Another example is the dynamics ofthe position ( X,Y) /intercalof a rigid homogenous domain in steady translational motion as described by Thiele [ 60]. Considering the expansion Eq. ( B1), performing the projection of the LLG Eq. ( 1) onto m×∂ξimand integrating over volume gives ageneralization of Thiele’s equations [ 40,60,61], /summationdisplay jGij˙ξj=1 J∂H ∂ξi+α/summationdisplay jDij˙ξj, (B2) where the matrix elements GijandDijwere deﬁned in Eqs. ( 8) and ( 9), respectively. By assuming that the matrix Gij[ξ]i s invertible, one may also write ˙ξk=/summationdisplay i,jG−1 kiGij˙ξj=1 J/summationdisplay iG−1 ki∂H ∂ξi+α/summationdisplay i,jG−1 kiDij˙ξj, (B3) where G−1 ki≡(G−1)kiare the elements of the inverse matrix. Comparing this equation with the Hamiltonian Eq. ( 7), the equivalence between the generalized Thiele approach andHamiltonian approach is embedded in the following identities {ξ i,ξj}ξ,pξ≡J−1G−1 ij, (B4a) γξk≡α/summationdisplay i,jG−1 kiDij˙ξj. (B4b) The result for the dissipative term γξjin terms of the Poisson brackets {ξi,ξj}ξ,pξis also derived in a more general manner (see the main text) and therefore this structure is generalfor including viscous damping into Hamilton’s equations,regardless of the system of study. To further illustrate the Hamiltonian approach, we present as an example the Poisson bracket for XandY, describing the position of a skyrmion, for the soft mode associated withtranslational motion. Example: Derivation of the X,YPoisson bracket for the translational modes from the spin Berry phase action By virtue of Thiele’s traveling wave ansatz, m(r,t)= m0[r−rs(t)], the unit magnetization has the properties ˙m= −(˙rs·∇)mand∂X,Ym=−∂x,ym, where rs=(X,Y)/intercal.T h e Poisson brackets between XandY, and hence access to the conservative dynamics, are derived by expanding the spinBerry phase action S B=/integraltext dtL Baround rs=0, while making use of these properties. The Lagrange function here is LB=/integraldisplay dVJ(1−cosθ)˙φ =/integraldisplay dVJ(1−cosθ)(˙X∂Xφ+˙Y∂Yφ). (B5) Performing the expansion, denoting θ=θ[r,rs] and φ= φ[r,rs], this becomes to lowest order in X,Y,˙X, and ˙Y: LB≈/integraldisplay dVJ[(1−cosθ[r,0])+X∂xcosθ[r,0]+Y∂ycosθ[r,0]][˙X(−∂xφ[r,0])+˙Y(−∂yφ[r,0])] =J/braceleftbigg −˙X/bracketleftbigg/integraldisplay dV(1−cosθ[r,0])∂xφ[r,0]/bracketrightbigg −˙Y/bracketleftbigg/integraldisplay dV(1−cosθ[r,0])∂yφ[r,0]/bracketrightbigg −X˙X/bracketleftbigg/integraldisplay dV∂xcosθ[r,0]∂xφ[r,0]/bracketrightbigg −Y˙Y/bracketleftbigg/integraldisplay dV∂ycosθ[r,0]∂yφ[r,0]/bracketrightbigg −X˙Y/parenleftbigg/integraldisplay dV∂xcosθ[r,0]∂yφ[r,0]/parenrightbigg −Y˙X/parenleftbigg/integraldisplay dV∂ycosθ[r,0]∂xφ[r,0]/parenrightbigg/bracerightbigg . (B6) 054430-10CHARACTERIZING BREATHING DYNAMICS OF MAGNETIC … PHYSICAL REVIEW B 99, 054430 (2019) The ﬁrst four terms in the expansion can be written as total derivatives of XorYor their squares and therefore do not enter into the equations of motion. Next, by integrating theﬁnal term by parts in the action with respect to time, theboundary contribution similarly vanishes and it follows thatthe spin Berry phase action reduces to lowest order to S B=/integraldisplay dtJ/braceleftbigg X˙Y/integraldisplay dV(∂ycosθ[r,0]∂xφ[r,0] −∂xcosθ[r,0]∂yφ[r,0])/bracerightbigg (B7) This is Eq. ( 14) of the main text. Finally, by integrating over z from 0 to z0and comparing Eq. ( B7) to the topological charge, Q=1 4π/integraldisplay dxdy m·(∂xm×∂ym) =1 4π/integraldisplay dxdy sinθ(∂yφ∂xθ−∂yθ∂xφ) =1 4π/integraldisplay dxdy (∂ycosθ∂xφ−∂yφ∂xcosθ), (B8) one ﬁnds the effective action Seff=/integraltext dt(4πQz0JX˙Y−H). This gives the Poisson bracket {Y,X}Y,pY=(4πQz0J)−1in t h em a i nt e x t[ s e eE q .( 16)]. APPENDIX C: DERIV ATION OF EQ. ( 21) For the effective breathing dynamics, the volume integral from the spin Berry phase action can be approximated inthree pieces. Let ρ=Rx, where Rdeﬁnes the dimension- full skyrmion radius located at m z=0 and where xis a dimensionless coordinate, LB=2πJz0˙η/integraldisplay∞ 0dρρ[1−cosθ(ρ)] =2πJz0R2(I1+I2+I3). (C1) Above, Iiare the dimensionless integrals, I1=/integraldisplayr0/R 0dxx[1−cosθ(Rx)], (C2) I2=/integraldisplay(r0+d)/R r0/Rdxx[1−cosθ(Rx)], (C3) I3=/integraldisplay∞ (r0+d)/Rdxx[1−cosθ(Rx)], (C4) where, for the bubblelike magnetic skyrmions, Fig. 7,w e assume an extended region of core spins pointing along −ˆz up to a radius r0, followed by a small skyrmion wall-width d,2 mz=cosθ(ρ)=⎧ ⎨ ⎩−1i f ρ< r0 0i f ρ=R 1i f ρ> r0+d, (C5) 2In the limit of very large bubblelike skyrmions, dequals the 1D domain wall width /Delta1=√A/K, although for smaller radii it is expected to pick up a dependence on DMI. FIG. 7. Length scales for a bubblelike skyrmion shape, i.e., d< R,w h e r e mz=− 1(+1) inside (outside) the innermost (outermost) circle. and meanwhile θ(ρ) is an a monotonically decreasing func- tion from ρ=r0toρ=r0+d. An auxiliary constant 0 < b<1 may accommodate for different rigid shapes for the θ proﬁle across the domain wall, Fig. 7, such that the skyrmion radius need not be in the center of the wall: r0=R−bd. (C6) Using Eqs. ( C5) and ( C6), we may calculate upper bounds on the integrals for all d<R, independent of a speciﬁc θ-proﬁle, 0<I1=2/integraldisplayr0/R 0dxx=/parenleftBigr0 R/parenrightBig2 <1, (C7) 0<I2<2d R/bracketleftbigg 1+d R(1−b)/bracketrightbigg <4d R<4, (C8) I3=0, (C9) where I2was bounded by a rectangle of height 2( r0+d)/R and width d/R. By focusing on situations d/lessmuchR(t)w i t ha constant wall width d, and with a large extended core, ( R− r0)/r0/lessmuch1, any small time variations in the integrals I1andI2 are considered negligible. Thus, for Eq. ( C1), we crudely ﬁnd LB=2πCJz0R2˙η, where Cis such that 0 <C<5. To ﬁnally ﬁnd the Poisson bracket written in the main text, we calculatethe canonical momentum, p η=∂L ∂˙η=2πCJz0R2, which, by using{η,pη}η,pη=1, gives Eq. ( 21): {η,R2}η,pη=1 2πCJz0. (C10) This was also calculated using a domain wall ansatz in Ref. [ 49]. APPENDIX D: GENERALIZATIONS OF THE EFFECTIVE MODEL The arguments used to expand the micromagnetic model Eq. ( 23)i np o w e r so f Rcan be used for other interactions, including dipole-dipole interactions and bulk DMI. In thisAppendix, we brieﬂy outline how to modify the effectiveenergy Eq. ( 24) to take into account these terms. The dipole-dipole interaction is known to produce in thin- ﬁlms with strong perpendicular magnetic anisotropy to mod-ify the strength of the anisotropy [ 73],K eff=K−1 2μ0M2 s. Moreover, for circular skyrmions, it also produces a couplingfor the ηangle. From symmetry arguments, we argue that this interaction is invariant under the transformation η→−η. And, from the scaling argument, we argue that it decays 054430-11B. F. MCKEEVER et al. PHYSICAL REVIEW B 99, 054430 (2019) with the inverse of the radius. Therefore, the contribution to theηcoupling may be written as −cdd(cos2η)/r, where cdd depends on the exact proﬁle of the skyrmion. Usually, cddis at least an order of magnitude smaller than the other c’s in the effective energy Eq. ( 24). This contribution was calculated using the domain wall ansatz in a previous paper [ 74]. Another example of possible modiﬁcations to the energy is the generalization of the DMI to include bulk and hybrid DMI.In the case of hybrid DMI, we take into account a combinationof bulk and interfacial DMI [ 75]. The general contribution to the energy density becomes ˜H DMI=DN(mz∂xmx−mx∂xmz) ±DN(mz∂ymy−my∂ymz) −DB(mz∂ymx−mx∂ymz) ±DB(mz∂xmy−my∂xmz), (D1) where BandNstand for bulk and interfacial DMI, respec- tively, and the different signs correspond to the energies thatstabilize skyrmions and antiskyrmions, as stated in the maintext. In this case, the effective contribution is given by ˜E DMI(˜r,η)=− ˜r(gNcosη+gBsinη), (D2) where gB,N=πDB,N/4√ AK. If neither gBorgNis zero, it produces an equilibrium ηthat is different from the usual Néel and Bloch skyrmion and can take any value between 0 and 2 π. The new equilibrium angle is given by the ratio of gNandgB, i.e.,η=arctan( gB/gN). For the case where we consider that the domain wall width dis also a function of the radius, we need to keep its explicit dependence in the effective energy. In this case, byanalyzing the scaling factors before the reparametrization bythe domain-wall width /Delta1, it is possible to obtain the effective energy for the model Eq. ( 23), ˜E(˜r,η)=/parenleftbiggc 12 d+c11d−c3gcosη/parenrightbigg ˜r+c21d ˜r, (D3) where c12andc21are contributions from the exchange inter- action, c11is due to the anisotropy interaction, and c3is due to DMI. The dependence of don the DMI strength may be obtained in two ways. One method is by analyzing the scalingbehavior for a rotationally symmetric solution of the LLGequation in two dimensions. An explicit method is to considerr,η, and das collective variables and minimizing the energy for these three parameters. Thereby we obtain d=c 3\|g\| 2c11. (D4) An important remark is that the effective description men- tioned in this paper is only valid for the case that ˜ r/greatermuchd.I f the radius of the skyrmion becomes comparable to the widthof the circular domain wall, it is necessary to consider thesolutions for small skyrmions studied in Refs. [ 23,24]. The effective energy Eq. ( 24) gives the static properties obtained in Ref. [ 67] with cvalues independent of DMI, while the model obtained in Eq. ( D3) with the width of the skyrmion given by Eq. ( D4) corresponds to the one obtained in Ref. [ 49]. For the circular domain wall ansatz, the values ofthe constants are given by: c 3=2,c1=2, and all other con- stants care equal to 1. The conversion between the constants in the two approaches [Eqs. ( 24) and ( D3)] is given by c1=2c11c12 c3\|g\|+c3 2\|g\|, c2=c21c3\|g\| 2c11. (D5) APPENDIX E: CONSTANT ENERGY ORBIT A VERAGES To estimate the average energy decay during the skyrmion breathing mode, one can construct upper and lower bounds forthe two integrals entering in Eq. ( 37): I 1(/epsilon1;B)≡/contintegraldisplay dη(/epsilon1r(η)−1)2 r5(η)(E1a) I2(/epsilon1;B)≡/contintegraldisplay dηr(η)s i n2η. (E1b) These integrals need to be performed over one full ro- tational/precessional constant energy orbit as deﬁned by theorbit trajectory: r rot(η;/epsilon1)=1−B 2(1−Bcosη)/bracketleftBigg 1+/radicalBigg 1−1−Bcosη (1−B)/epsilon12/bracketrightBigg ,(E2) r± osc(η;/epsilon1)=1−B 2(1−Bcosη)/bracketleftBigg 1±/radicalBigg 1−1−Bcosη (1−B)/epsilon12/bracketrightBigg ,(E3) where, as discussed in the main text, rrot(η) is deﬁned for η∈ [0,2π] whereas the two branches r± osc(η) are only deﬁned for η∈[−η+,η+] with cosη+=1−(1−B)/epsilon12 B. (E4) For rotational phase integrals we have /contintegraldisplay dηf(η)=2/integraldisplayπ 0dηf(η), (E5) whereas for oscillatory phase integrals we have /contintegraldisplay dηf±(η)=/integraldisplayη+ −η+dηf+(η)+/integraldisplay−η+ η+dηf−(η) =2/integraldisplayη+ 0dη[f+(η)−f−(η)], (E6) one can therefore write Eq. ( E1a) explicitly as Irot 1(/epsilon1;B)=/epsilon12Jrot 3−2/epsilon1Jrot 4+Jrot 5, (E7) Irot 2(/epsilon1;B) =2(1−B)/epsilon1/integraldisplayπ 0dηsin2η 1−Bcosη/parenleftBigg 1+/radicalBigg 1−1−Bcosη (1−B)/epsilon12/parenrightBigg , (E8) 054430-12CHARACTERIZING BREATHING DYNAMICS OF MAGNETIC … PHYSICAL REVIEW B 99, 054430 (2019) for integrals in the rotational regime, and Iosc 1(/epsilon1;B)=/epsilon12Josc 3−2/epsilon1Josc 4+Josc 5, (E9) Iosc 2(/epsilon1;B) =4/radicalbig B(1−B)/integraldisplayη+ 0dηsin2η 1−Bcosη√cosη−cosη+, (E10) for integrals in the oscillatory regime, where we have further deﬁned Jrot k≡2 (1−B)k/epsilon1k/integraldisplayπ 0dη(1−Bcosη)k ×/bracketleftBigg 1+/radicalBigg 1−1−Bcosη (1−B)/epsilon12/bracketrightBigg−k , (E11) and Josc k≡4 (1−B)k/epsilon1k/integraldisplayη+ 0dη(1−Bcosη)k ×⎧ ⎨ ⎩/bracketleftBigg 1+/radicalBigg 1−1−Bcosη (1−B)/epsilon12/bracketrightBigg−k −/bracketleftBigg 1−/radicalBigg 1−1−Bcosη (1−B)/epsilon12/bracketrightBigg−k⎫ ⎬ ⎭. (E12) The necessary upper bounds to the Irot 1andIrot 2are obtained by approximating 1 +√ 1−1−Bcosη (1−B)/epsilon12/lessorequalslant2 and writing /integraldisplayπ 0dηsin2η 1−Bcosη/lessorequalslantπ 2(1−B), (E13) /integraldisplayπ 0dη(1−Bcosη)k/lessorequalslantπ(1+B)k. (E14) These bounds are independent of the energy of the orbit thus justifying Eq. ( 38) of the main text. For the oscillatory phase integrals, one just focuses on Iosc 2 asJosc kis always bounded from below by 0. One ﬁnds Iosc 2(/epsilon1;B)/greaterorequalslant4 /epsilon12/radicalbigg B 1−B/integraldisplayη+ 0dηsin2η(cosη−cosη+) /greaterorequalslant2 /epsilon12/radicalbigg B 1−B(sinη+cos2η+−cosη+),(E15) where from the deﬁnition of η+(/epsilon1), the term sin η+cos2η+ can be expanded to dominant order in /epsilon1as sinη+cos2η+=/radicalBigg 1−/parenleftbigg1−(1−B)/epsilon12 B/parenrightbigg/parenleftbigg1−(1−B)/epsilon12 B/parenrightbigg2 ∼2/radicalbigg (1−B)5 B/epsilon14√ /epsilon1−1[1+O(/epsilon1−1)], (E16) allowing for the reconstruction of result Eq. ( 39)i nt h em a i n text.APPENDIX F: DETAILS ON MICROMAGNETIC SIMULATIONS All micromagnetic simulations reported in this paper were performed with an enhanced version of MicroMagnum [ 76]. 1. Simulation geometry details for Figs. 4–6 All simulations used a mesh of uniformly discretized cubic ﬁnite difference cells with periodic boundary conditions toapproximate an inﬁnite thin ﬁlm system. Figures 4(b) and 4(c) in the main text used 1024 ×1024×1 nodes with cell length 0 .25nm. Figure 6used for (a): 2048 ×2048×1 nodes with cell length 0 .5nm; for (b): 512 ×512×1 nodes with cell length; and for (c): 256 ×256×1 nodes with cell length 0.25nm. 2. Calculation of randη For the calculation of the skyrmion radius in the sim- ulations, we used a linear interpolation between the ﬁnitedifference cells where m zchanges sign. For this, we picked a ﬁxed cut through the diameter of the skyrmion since it hasno translational motion. FIG. 8. Transition from rotational breathing to oscillatory breath- ing from a micromagnetic simulation, showing one full rotation before traversing the saddle energy. The point where the energy drops below the energy saddle is estimated by closely inspecting theenergy signal to ﬁnd a sharp drop. The parameters in this simulation areα=0.1,D=2.8×10 −3Jm−2,A=1.5×10−11Jm−1,a n d K= 1.1×106Jm−3, i.e., g=0.54. 054430-13B. F. MCKEEVER et al. PHYSICAL REVIEW B 99, 054430 (2019) For the calculation of η, we used a marching-squares algorithm to approximate the magnetization along the mz=0 contour by using image processing tools [ 77] and then took a simple average of the ηvalues for the spins along these points, η=(1/N)/summationtextN i=1ηi, where ηi=arctan( my,i/mx,i)a r e theNwall angles taken along the interpolated contour. 3. The transition from rotations to oscillations Observing a transition from the rotatinglike breathing be- havior and small oscillatorylike breathing in the micromag-netic simulations is nontrivial. This can be understood fromthe results reported in the main text as follows. The amplitudeof the breathing motion is larger for higher energies, so thetotal size of the magnetic material has to be sufﬁciently large. 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PhysRevB.95.165106.pdf	PHYSICAL REVIEW B 95, 165106 (2017) Pumping of magnons in a Dzyaloshinskii-Moriya ferromagnet Alexey A. Kovalev, Vladimir A. Z yuzin, and Bo Li Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, Nebraska 68588, USA (Received 26 January 2017; revised manuscript received 17 March 2017; published 5 April 2017) We formulate a microscopic linear response theory of magnon pumping applicable to multiple-magnonic-band uniform ferromagnets with Dzyaloshinskii-Moriya interactions. From the linear response theory, we identify theextrinsic and intrinsic contributions where the latter is expressed via the Berry curvature of magnonic bands.We observe that in the presence of a time-dependent magnetization Dzyaloshinskii-Moriya interactions can actas ﬁctitious electric ﬁelds acting on magnons. We study various current responses to this ﬁctitious ﬁeld andanalyze the role of Berry curvature. In particular, we obtain an analog of the Hall-like response in systemswith nontrivial Berry curvature of magnon bands. After identifying the magnon-mediated contribution to theequilibrium Dzyaloshinskii-Moriya interaction, we also establish the Onsager reciprocity between the magnonmediated thermal torques and heat pumping. We apply our theory to the magnonic heat pumping and torqueresponses in honeycomb and kagome lattice ferromagnets. DOI: 10.1103/PhysRevB.95.165106 I. INTRODUCTION It is well known that an electric ﬁeld can drive a charge current, whereas in order to understand how to drive a spincurrent one needs to resort to the ﬁeld of spintronics [ 1]. Magnetization dynamics generates spin currents in adjacentnormal metal by a phenomenon known as spin pumping [ 2–4]. The discovery of spin pumping had a great deal of inﬂuenceon the development of the ﬁeld of spintronics as it led tonew insights into the spin Hall [ 5], spin torque [ 6,7], and spin Seebeck effects [ 8]. The phenomena related to the spin Seebeck effect are studied within the ﬁeld of spincaloritronics[9] in which the focus is on interplay between the spin degrees of freedom and heat currents. As heat and spin currents are also carried by magnons, one naturally arrives at a concept of magnon-mediated spin torqueswhich can lead to thermally induced motion of magneticdomain walls [ 10–12]. Such torques exist only in noncollinear magnetic structures or when the Dzyaloshinskii-Moriya inter-actions (DMI) are present. In the latter case, such spin torqueshave been termed as DMI torques [ 13]. Recently, both ﬁeldlike and antidampinglike contributions to DMI torques have beenstudied theoretically [ 14–18]. It has been noted [ 13] that DMI torques can be seen as magnon analogs of spin-orbit torques[19–24]. This suggests that the phenomenology developed for spin-orbit torques can be readily applied to DMI torques[25,26]. In particular, the intrinsic contribution to DMI torques has been identiﬁed [ 16]. Continuing this analogy, one can identify ﬁctitious electric ﬁelds acting on magnons due totime-dependent magnetization dynamics [ 11,27,28]. One can also identify the magnon-mediated equilibrium contribution toDMI. Due to such contribution the electron-mediated energycurrent calculated in response to magnetization dynamicsfrom the Kubo formalism contains an unphysical ground-state contribution [ 26] which needs to be subtracted. Similar unphysical contributions have been identiﬁed for anomalousresponses induced by statistical forces [ 29–31]. There is a considerable interest in magnets on lattices with nontrivial geometry as they allow observation of Berry phaserelated phenomena such as the thermal Hall effect of magnons[32–41]. Theoretically, the increased magnon damping [ 42], Dirac magnons [ 43], and the magnon-mediated spin Hall effect [16,44,45] have been predicted for kagome and honeycomb lattice ferromagnets. In addition, other manifestations of theBerry phase physics can arise in layered kagome [ 40] and honeycomb [ 46] ferromagnets as examined in this work. In this work we analyze magnon currents arising in response to magnetization dynamics (see Fig. 1). In the presence of a time-dependent magnetization, DMI can act as ﬁctitiouselectric ﬁelds acting on magnons. As has been noted earlier inthe Introduction, the energy current carried by such magnonscontains the ground state contribution associated with magnon-mediated equilibrium DMI. Note that such corrections areimportant only in systems with nontrivial Berry curvatureof magnon bands. Here we concentrate on systems withnontrivial Berry curvature by considering various currentresponses in honeycomb and kagome lattice ferromagnets.Our linear response calculation of heat currents agrees withthe calculation of magnon-mediated thermal torques [ 16], thus conﬁrming the Onsager reciprocity principle (see Fig. 1 ). We also study the feasibility of experimental observation of suchcurrent responses. The paper is organized as follows. In Sec. IIwe introduce the Hamiltonian describing magnons with multiple bandsand calculate the equilibrium DMI. Next, within the samesection, we describe pumping of magnons in response tomagnetization dynamics and thermal torques within the linearresponse theory. In the ﬁnal part of Sec. II, we formulate the Onsager relations. In Sec. IIIwe apply our theory to honeycomb and kagome lattice ferromagnets. We concludeour paper in Sec. IV. The Appendices A, B, C, and D contain very detailed derivations of our results. II. THEORY OF MAGNON PUMPING AND DMI TORQUES In this section we develop a microscopic linear response theory of magnon pumping and nonequilibrium magnonictorques applicable to multiple-magnonic-band uniform fer-romagnets with Dzyaloshinskii-Moriya interactions. We note 2469-9950/2017/95(16)/165106(14) 165106-1 ©2017 American Physical SocietyALEXEY A. KOV ALEV , VLADIMIR A. ZYUZIN, AND BO LI PHYSICAL REVIEW B 95, 165106 (2017) xyz FIG. 1. Two effects related by the Onsager reciprocity principle. Left: Magnetization dynamics pumps magnon current Jpand spin current Js=−¯hJp. This process also involves heat current Jq carried by magnons. Right: A temperature gradient leads to a thermal torque with two components TxandTyacting on the uniform magnetization. that in our theory magnons are treated as conserved particles. Gilbert damping αcould broaden magnonic bands and intro- duce magnon nonconserving processes. In realistic situationsαis typically small. In what follows, to simplify formulas, we take the system volume V=1 and recover it in the ﬁnal expressions ( 19), (20), and ( 28). A. Preliminaries We consider a noninteracting boson Hamiltonian describing the magnon ﬁelds, which could be, e.g., a result of the Holstein-Primakoff transformation: H=/integraldisplay dr/Psi1 †(r)H/Psi1(r), (1) where His a Hermitian matrix of the size N×Nand/Psi1†(r)= [a† 1(r),..., a† N(r)] describes Nbosonic ﬁelds corresponding to the number of modes within a unit cell (or the number ofspin-wave bands). The Fourier transformed Hamiltonian reads H=/summationdisplay ka† kH(k)ak, (2) where a† kis the Fourier transformed vector of creation operators. The Hamiltonian in Eq. ( 2) can be diagonalized by a unitary matrix Tk, i.e.,Ek=T† kH(k)TkandT† kTk=1N×N, where Ekis the diagonal matrix of band energies and 1 N×Nis theN×Nunit matrix. B. Magnon-mediated Dzyaloshinskii-Moriya interaction As magnons can exert a torque on magnetization even in equilibrium, we begin by considering an equilibrium state ofthe system. Such equilibrium DMI torques can be capturedby calculating the DMI tensor in the presence of magnons inequilibrium state. The torque operator is introduced as T=∂ mH×m, (3) where mis a unit vector in the direction of the spin density. We then interpret DMI in terms of the moments of the torque: Dαβ=1 2/angbracketleftbigg/integraldisplay dr/Psi1†(r)(Tαxβ+xβTα)/Psi1(r)/angbracketrightbigg eq, (4) where we assume a ﬁnite system. In order to represent an inﬁnite system, we will eliminate the position operator fromthe ﬁnal result. The average in Eq. ( 4) has been calculated inRef. [ 16] in a form of a tensor M βdeﬁned as Mβ=1 2Tr[(xβ∂mH+∂mHxβ)g(E)], (5) where g(E) is the Bose distribution function g(E)= 1/[exp(βE)−1]. In particular, it has been found that Mβ=/summationdisplay kn/braceleftbigg1 βln(1−e−βεnk)B(n) mβ(k)−g(Ekn)A(n) mβ(k)/bracerightbigg ,(6) where (for details of this calculation see Appendix A1) A(n) mβ(k)=/summationdisplay m/negationslash=nIm/bracketleftbigg [/tildewideηk]nm1 εnk−εmk[/tildewidevkβ]mn/bracketrightbigg (7) and B(n) mβ(k)=/summationdisplay m/negationslash=nIm/bracketleftbigg [/tildewideηk]nm2 (εnk−εmk)2[/tildewidevkβ]mn/bracketrightbigg , (8) with the velocity vk=∂kHk, the effective ﬁeld ηk=−∂mHk, and their eigenbasis representations /tildewidevk=T† kvkTkand/tildewideηk= T† kηkTk. Finally, the expression for the DMI tensor is given by Dαβ=[Mβ×m]α. (9) It is easy to notice that B(n) mβ(k)=−/Omega1(n) mβ(k), where now /Omega1(n) mβ(k)≡i[(∂mT† k)(∂βTk)]nn−(m↔β) is the mixed space Berry curvature of the nth band. The second term in Eq. ( 6) has a clear analogy to the orbital moment [ 47] which can be seen after a substitution ηk→vk[25]. C. Heat and spin pumping by magnetization dynamics In this subsection we derive the magnon-mediated current response to slow magnetization dynamics in a system withbroken inversion symmetry and spin-orbit interactions. TheKubo linear response energy current contains the groundstate energy contribution related to the magnon-mediated DMIwhich have been calculated in the previous subsection. Thus,we will use the results calculated earlier in order to identifyvarious transport contributions. We are interested in the heat, particle, and spin current density responses described by a tensor t aα: Jaα=−taα·∂tm, (10) where aisqfor the heat current, pfor the particle current, and sfor the spin current. Here the spin current is related to the magnon particle current density Jpby a relation Js=−¯hJp. In the presence of magnetization dynamics, Hamiltonian H acquires a perturbation of the form H/prime=/integraldisplay dr/Psi1†(r)H/prime/Psi1(r), (11) where H/prime=∂mH·δm(t) and we assume that δm(t) is small. We are interested in a linear response to the time derivativeofm(t), thus we write δm(t)=(1/iω)∂ tm. Note that this calculation is similar to the calculation of dc current responseto electric ﬁeld with the correspondence A(t)→δm(t), where the perturbation in Eq. ( 11) leads to an analog of equilibrium diamagnetic current correction. Using the linear response 165106-2PUMPING OF MAGNONS IN A DZYALOSHINSKII-MORIYA . . . PHYSICAL REVIEW B 95, 165106 (2017) Kubo theory we obtain for the heat and particle current density response: JK aα=/angbracketleftbig J[0] aα/angbracketrightbig ne+/angbracketleftbig J[1] aα/angbracketrightbig eq(12) or JK aα=lim ω→0/braceleftbig −/Pi1R α(ω)/iω/bracerightbig ∂tm+/angbracketleftbig J[1] aα/angbracketrightbig eq, (13) where /Pi1R α(ω)=/Pi1α(ω+i0) is the retarded correlation function related to the following correlator in Matsub- ara formalism: /Pi1α(iω)=−/integraltextβ 0dτeiωτ/angbracketleftTτJ[0] aαh/angbracketright, with h= −/integraltext dr/Psi1†(r)∂mH/Psi1(r) being the nonequilibrium ﬁeld, and j[0] q(r)=(1/2)/Psi1†(r)(vH+Hv)/Psi1(r) and J[0] q=/integraltext drj[0] q(r) being the heat current density and the macroscopic heatcurrent, respectively. For the particle current we have sim-ilar expressions j [0] p(r)=/Psi1†(r)v/Psi1(r) and J[0] p=/integraltext drj[0] p(r). Here the velocity operator is given by v=(1/i¯h)[r,H]. We also introduce a gradient correction to the heat andparticle currents due to perturbation, i.e., J [1] a=/integraltext drj[1] a(r), where j[1] q(r)=(1/2)/Psi1†(r)[(δm(t)·∂m)(vH+Hv)]/Psi1(r) and j[1] s(r)=/Psi1†(r)[δm(t)·∂mv]/Psi1(r). This analog of diamagnetic current cancels with the term /Pi1R α(0) resulting in the Kubo contribution of the form JK aα=lim ω→0/braceleftbig/bracketleftbig /Pi1R α(0)−/Pi1R α(ω)/bracketrightbig /iω/bracerightbig ∂tm. (14) The correlation function in Eq. (14) is calculated by consid- ering the simplest bubble diagram for /Pi1αand performing the analytic continuation, see, e.g., Ref. [ 16]. We express the result through a response tensor tK aαcontaining two contributions tK aα=tI aα+tII aα, which are given by tI aα=1 ¯h/integraldisplaydω 2πg(ω)d dωReTr/angbracketleftJaαGRηGA−JaαGRηGR/angbracketright, tII aα=1 ¯h/integraldisplaydω 2πg(ω)ReTr/angbracketleftbigg JaαGRηdGR dω−JaαdGR dωηGR/angbracketrightbigg , (15) where g(ω) is the Bose distribution function g(ω)=1/ [exp(¯hω/k BT)−1], GR=¯h(¯hω−H+i/Gamma1)−1,GA= ¯h(¯hω−H−i/Gamma1)−1,η=−∂mH,Jq=(vH+Hv)/2, and Jp=v. In our calculations we adopt a phenomenological treatment and relate the quasiparticle broadening to theGilbert damping, i.e., /Gamma1=α¯hω. Note that the Kubo response for the energy current density in Eq. ( 14) contains the bound energy current associated with DMI: J D q=ˆD·(m×∂tm), (16) where tensor ˆDis given in Eq. ( 9). This current needs to be subtracted from the Kubo current in Eq. ( 14) in order to obtain a transport heat current: Jqα=JK qα−JD q. (17) To express the response tensor tK aα, we use the Fourier transformed operators and the eigenbasis representation forthe velocity ¯ h/tildewidev k=∂kEk−iAkEk+iEkAk, and the effective ﬁeld−/tildewideηk=∂mEk−iAmEk+iEkAm, where Ak=iT† k∂kTk and Am=iT† k∂mTk. For the details of derivation of intrinsiccontribution to the heat current see Appendix A2. We obtain tK qα=/summationdisplay kn/braceleftbigg g(εnk)/bracketleftbig −εnkB(n) mα(k)+A(n) mα(k)/bracketrightbig −εnkg/prime(εnk) 2/Gamma1(n) k(∂mεnk)(∂kαεnk)/bracerightbigg , (18) which after combining with DMI energy current JD qleads to the response tensor describing the heat current (see Appendix A3): tex qα=−1 V/summationdisplay kN/summationdisplay n=11 2/Gamma1(n) k(∂mεnk)(∂kαεnk)εnkg/prime(εnk), (19) tin qα=1 V/summationdisplay kN/summationdisplay n=1c1(εnk)/Omega1(n) mkα(k), where εnk=[Ek]nn,/Gamma1(n) k=αεnk,g/prime(εnk)=(2kBT)−1{1− cosh(εnk/kBT)}−1, c1[εnk]=g(εnk)εnk−(1/β)l n ( 1− e−βεnk),Vis volume, and we separated the total tensor tqαinto the intrinsic and extrinsic contributions, i.e., tqα=tex qα+tin qα. For the particle current response only tK pαtensor needs to be considered, thus we obtain the following expression for thetotal tensor: t pα=tex pα+tin pα, divided into the intrinsic and extrinsic contributions (for details of calculations of intrinsiccontribution see Appendix B): t ex pα=−1 V/summationdisplay kN/summationdisplay n=11 2/Gamma1(n) k(∂mεnk)(∂kαεnk)g/prime(εnk), tin pα=1 V/summationdisplay kN/summationdisplay n=1g(εnk)/Omega1(n) mkα(k). (20) The last tensor also describes the spin current response, i.e., tsα=−¯htpα. D. Thermal torques In this subsection we derive the magnon-mediated magne- tization torque response to a temperature gradient in a systemwith broken inversion symmetry and spin-orbit interactions.For details of derivations see Appendix C. The thermal torque is deﬁned according to equation T=−β α∂αT, (21) where βαis the thermal torkance tensor and Tdescribes torque acting on magnetization and leading to modiﬁcation ofthe Landau-Lifshitz-Gilbert equation, i.e., s(1+αm×)˙m= m×H eff+T, where Heffis the effective magnetic ﬁeld and sis the spin density. We use the Luttinger linear response method [ 48] in which the temperature gradient is replicated by a perturbation to Hamiltonian Hof the form H/prime=1 2/integraldisplay dr/Psi1†(r)(Hχ+χH)/Psi1(r), (22) where we introduce the temperature gradient as ∂iχ= −∂iT/T . The torque response can be found by calculating the effective magnon-mediated ﬁeld: h=h[0]+h[1]=− /angbracketleft∂mH/angbracketrightne−/angbracketleft∂mH/prime/angbracketrighteq, (23) 165106-3ALEXEY A. KOV ALEV , VLADIMIR A. ZYUZIN, AND BO LI PHYSICAL REVIEW B 95, 165106 (2017) where for the second term the averaging is done over the equilibrium state and for the ﬁrst term over nonequilibriumstate induced by the temperature gradient. The magnon-mediated torque acting on the magnetization is given by T=m×h. (24) Within the linear response theory, the response h [0]to a temperature gradient can be calculated from expression h[0]=lim /Omega1→0/braceleftbig/bracketleftbig /Pi1R α(/Omega1)−/Pi1R α(0)/bracketrightbig /i/Omega1/bracerightbig ∂αχ, (25) where /Pi1R α(/Omega1)=/Pi1α(/Omega1+i0) is the retarded correlation function related to the following correlator in Matsubara formalism: /Pi1α(i/Omega1)=−/integraltextβ 0dτei/Omega1τ/angbracketleftTτhJ[0] qα/angbracketright. Note that this correlator differs from the one arising in Eq. ( 13) in the order of operators. In the correlator we reduce the perturbation H/primeto the energy current by employing the equality ˙H/prime=(i/¯h)[H,H/prime]= J[0] q∂χand integration by parts. Following the notations in Ref. [ 16], we introduce the linear response tensors Sαand Mαfor the ﬁelds h[0]andh[1]and the total response tensor Lα=Sα+Mαaccording to equation h[0]+h[1]=−Lα∂αχ, (26) where Mαis given by Eq. ( 5) as it follows from Eq. ( 23). For the tensors Sαwe obtain Sα=/summationdisplay kn/braceleftbigg g(εnk)/bracketleftbig −εnkB(n) mβ(k)+A(n) mβ(k)/bracketrightbig +εnkg/prime(εnk) 2/Gamma1(n) k(∂mεnk)(∂kβεnk)/bracerightbigg . (27) We can also separate the total response tensor into the intrinsic and extrinsic contributions: Lex α=1 V/summationdisplay kN/summationdisplay n=11 2/Gamma1(n) k(∂mεnk)(∂kαεnk)εnkg/prime(εnk), Lin α=1 V/summationdisplay kN/summationdisplay n=1c1(εnk)/Omega1(n) mkα(k). (28) For the thermal torkance tensor we obtain βα=Lα×m/T. (29) E. Onsager reciprocity relation We are now in the position to combine the results from pre- vious subsections into one expression that emphasized the On-sager reciprocity relation. In principle, the result of calculationof thermal torques in the last section can be extracted from theOnsager relations without performing the calculation. Writingthe response tensors in terms of the torkance tensors, we obtain ⎛ ⎝J pα Jqα T⎞ ⎠=⎛ ⎝ˆσ(m) ˆ/Pi1T(−m)αα(−m) ˆ/Pi1(m)Tˆλ(m)Tβα(−m) αα(m)Tβα(m)−ˆΛ(m)⎞ ⎠⎛ ⎝−∂αϕ ∂αχ m×∂tm⎞ ⎠, (30) where summation over repeated indices is implied, and we introduced the conductivity tensor ˆ σ(m), the magnonic heat conductivity tensor ˆλ(m), the tensor ˆ/Pi1(m) describing the1 23 a1 a2+ -d1 d2d3 xy FIG. 2. Schematics of the graphene layer parameters for the tight-binding model. Vectors connecting nearest neighbors are τ1= 1 2(1√ 3,1),τ2=1 2(1√ 3,−1), and τ3=1√ 3(−1,0) are used in deriving the Hamiltonian for magnons. Vectors a1=1 2(√ 3,1) and a2= 1 2(√ 3,−1) are used in deriving the second-nearest neighbor DMI. magnon Seebeck and Peltier effects, and the tensor ˆΛ(m) corresponding to LLG equation. The tensor αα(m)w a s introduced by analogy with the tensor βα(m) and it is given in Eq. ( 20), i.e., αα(−m)=tpα×m. For completeness we also added a response to an analog of electric ﬁeld formagnons −∂ αϕ[49]. Equation ( 30) immediately follows from Eqs. ( 19), (20), and ( 28) given that intrinsic contributions are odd and extrinsic contributions are even under magnetizationreversal. The Onsager reciprocity relation in Eq. ( 30)i s similar to expressions obtained for similar electron-mediatedeffects in Ref. [ 25]. Equation ( 30) can be modiﬁed to account for the possibility of magnon accumulation resulting from themagnon motive force [ 28] or temperature gradient [ 49]. III. RESULTS FOR HONEYCOMB AND KAGOME FERROMAGNETS In this section we apply our theory to single layer honey- comb and kagome ferromagnets with DMI. In our models, weintroduce two types of DMI. The Rashba DMI correspondto mirror asymmetry in the system (see Figs. 2and 4). The remaining DMI make the second quantized Hamiltonianof magnons to be asymmetric under time reversal. Suchasymmetries make our systems exhibit behavior analogousto electronic systems lacking the center of inversion andtime reversal symmetry [ 26]. To demonstrate explicitly how ﬁctitious electric ﬁelds result in magnon currents, we describethe honeycomb system analytically. Our results could also berelevant to three-dimensional layered structures with weaklycoupled layers. Note that the magnon pumping could in prin-ciple be modiﬁed by DMI induced anharmonic interactionsof magnons [ 42]. We do not expect this effect to be large when magnetization substantially deviates from the directionorthogonal to DMI vector. A. Application to honeycomb ferromagnet In this subsection we study a model of an insulating ferromagnet on a honeycomb lattice. This model containsphysics discussed above in a transparent and analytical way.For the details of further derivations see Appendix D.W e assume a Heisenberg exchange of ferromagnetic sign, in-plane 165106-4PUMPING OF MAGNONS IN A DZYALOSHINSKII-MORIYA . . . PHYSICAL REVIEW B 95, 165106 (2017) DMI of Rashba type, and second-nearest neighbor DMI. The Hamiltonian is H=−J/summationdisplay /angbracketleftij/angbracketrightSiSj+/summationdisplay /angbracketleftij/angbracketrightD[R][Si×Sj] +D[z]/summationdisplay /angbracketleft/angbracketleftij/angbracketright/angbracketrightνij[Si×Sj]z. (31) The vectors of the Rashba type DMI are shown in Fig. 2, where d1=1 2(√ 3,−1),d2=1 2(−√ 3,−1), and d3=(0,1), such as D[R]=D[R]d. Note that all vectors, such as τiandai, are measured in units of lattice spacing a 0which is recovered in the ﬁnal result. The vector of the second-nearest neighborDMI is in the zdirection, and the signs of ν ijare depicted in green in Fig. 2for the directions shown by dashed green arrows. For analytical results, we assume that all DMI aresmall, i.e., J/greatermuchD [R]andJ/greatermuchD[z]. In our model, initially, we assume that the order is in general ( mx,my,mz) direction, which can be realized by application of the magnetic ﬁeld.Our strategy would be to ﬁrst understand the role of theDMI in the behavior of magnons for a general direction ofthe ferromagnetic order. After that we will assume that themain order is in the zdirection, while the perturbations that deviate the order are in the x-yplane (see Fig. 1). To study the magnons, we perform the Holstein-Primakoff transformation.The unit cell of the honeycomb ferromagnet has two spinsS AandSB, hence the two sets of boson operators, a†(r),a(r) andb†(r),b(r) corresponding to the A and B sublattices are introduced. The Holstein-Primakoff transformation reads asusualS z A=S−a†aandS+ A=(Sx A+iSy A)=√ 2S−a†aa(S is the total spin), and the same for B spins. The Fourier imageof the Hamiltonian describing noninteracting magnons writtenin terms of the /Psi1=(a k,bk)Tspinor is H=JS/bracketleftbigg 3+/Delta1k−˜γk −˜γ∗ k 3−/Delta1k/bracketrightbigg , (32) where /Delta1k=2/Delta1[sin(ky)−2s i n(ky 2) cos (√ 3kx 2)], with /Delta1= mzD[z]/J. This type of DMI is a k-dependent mass of magnons. Deriving ˜ γkwe considered Rashba DMI in the lowest order in D[R]/J/lessmuch1 parameter. With this assumption ˜γk=2ei˜kx 2√ 3cos/parenleftbigg˜ky 2/parenrightbigg +e−i˜kx√ 3, (33) where ˜kx=kx−√ 3D[R] Jmyand ˜ky=ky+√ 3D[R] Jmx.W e observe that Rashba DMI plays an effective role of magnoncharge, while order direction ( m x,my,0) is an effective vector potential felt by magnons. The eigenvalues of the Hamiltonian are calculated to be /epsilon1k,±=JS/parenleftbig 3±/radicalBig /Delta12 k+\|˜γk\|2/parenrightbig , (34) with corresponding eigenfunctions vk,+=[cos( ˜ξk/2)ei˜χk,−sin(˜ξk/2)]T(35) and vk,−=[sin( ˜ξk/2),cos(˜ξk/2)e−i˜χk]T, (36) where sin( ˜ξk)=\|˜γk\|/√ /Delta12 k+\|˜γk\|2and ˜γk=\|˜γk\|ei˜χk, and the tilde symbol here means that corresponding kmomenta areshifted by the Rashba DMI. Unitary matrix that diagonalizes the Hamiltonian is readily constructed and it is given by Tk=⎡ ⎣cos/parenleftbig˜ξk 2/parenrightbig ei˜χk sin/parenleftbig˜ξk 2/parenrightbig −sin/parenleftbig˜ξk 2/parenrightbig cos/parenleftbig˜ξk 2/parenrightbig e−i˜χk⎤ ⎦. (37) We are now ready to derive spin and heat currents which are driven by magnetization dynamics. We set the dominantcomponent of the ferromagnetic order in the zdirection and assume that the magnetization dynamics is in the x-yplane. We only focus on the intrinsic contribution to the currents,i.e., due to nontrivial Berry curvatures of the magnon bandstructure. An expression deﬁning the Berry curvature is /Omega1 α,mβ=2Im[(∂αT† k)(∂mβTk)]=1 2sin(˜ξk) ×[(∂α˜χk)(∂mβ˜ξk)−(∂mβ˜χk)(∂α˜ξk)]/bracketleftbigg 10 0−1/bracketrightbigg .(38) In the following, we focus on the α=xandβ=xcase, and mention β=ycase at the end. Recall that /Delta1kdoes not depend on mβforβ=(x,y) components, hence ∂mβ/Delta1k=0. The derivative with respect to the direction of the order mβof the remaining functions that depend on ˜kis ∂ ∂mx=√ 3D[R] J∂ ∂˜ky≡√ 3D[R] J∂y, (39) ∂ ∂my=−√ 3D[R] J∂ ∂˜kx≡−√ 3D[R] J∂x. (40) This straightforward transformation makes the mixed Berry curvature a regular kspace one, except for the ∂mβ/Delta1k=0 condition. The Berry curvature has extrema at the K/prime=(0,4π 3) andK=(0,−4π 3) points, and can be approximated as /Omega1x,mx\|K(K/prime)≈−27 8D[R] J/Delta1 /parenleftbig 27/Delta12+3 4k2/parenrightbig3/2/bracketleftbigg 10 0−1/bracketrightbigg .(41) The curvature is the same for both K/primeandKpoints. The spectrum at these points is ﬁnite, /epsilon1k,±≈JS(3±3√ 3\|/Delta1\|), but the Berry curvature is of the monopole type. Hence atsmall temperatures, despite the exponential suppression ofthe magnon number at the K /primeandKpoints, there might be a contribution to the magnon currents due to this Berrycurvature. At the /Gamma1=(0,0) the spectrum of the lowest band is/epsilon1 k,−≈1 4SJk2, and it will be populated by the magnons the most at low temperatures. The Berry curvature is approximatedclose to this point as /Omega1 x,mx\|/Gamma1≈−D[R] J/Delta1 48k2 yk2 x/bracketleftbigg 10 0−1/bracketrightbigg . (42) According to Eqs. ( 10) and ( 20), the particle current density due to the Berry curvature at small temperatures SJ/greatermuchTreads Jpx=D[R] J√ 3 a0π/bracketleftbigg sinh/parenleftBigg 1 z3√ 3D[z] J/parenrightBigg e−3 z +D[z] J√ 3ζ(3) 36z3/bracketrightbigg (∂tm)x, (43) 165106-5ALEXEY A. KOV ALEV , VLADIMIR A. ZYUZIN, AND BO LI PHYSICAL REVIEW B 95, 165106 (2017) num anal 0.25 0.5kBT/SlashSJ0.00030.0006a0xyeven/SlashkB numanal 0.25 0.5kBT/SlashSJ0.0020.004a0xyeven/SlashkB FIG. 3. Left: The even component under magnetization reversal of the tensor αijas a function of temperature. Right: The even component under magnetization reversal of the torkance tensor βijas a function of temperature. In both cases the magnetization is along thezaxis. For the strength of DMI we use D[z]=D[R]=J/6. Red curves correspond to numerical results and blue curves correspond to analytical results in Eqs. ( 43)a n d( 44). where we introduced z=T/SJ for brevity, and set mz=1. Similarly, from Eq. ( 18), the heat current due to the Berry curvature at small temperatures SJ/greatermuchTreads Jqx=JSD[R] J3√ 3 a0π/bracketleftbigg sinh/parenleftBigg 1 z3√ 3D[z] J/parenrightBigg e−3 z +D[z] J√ 3I 216z4/bracketrightbigg (∂tm)x. (44) In both cases a term ∝e−3SJ Tis due to K/primeandKpoints, while the remaining one is due to /Gamma1point. We introduced a numerical constant I=/integraltext∞ 0dxx2[xex ex−1−ln(ex−1)]= 4π4/45≈8.65, and Riemann zeta-function ζ(3)≈1.2. It is straightforward to show that Berry curvature parts of the JpxandJqxcurrents driven by ( ∂tm)ymagnetization dynamics vanish. The JsyandJqycurrents driven by ( ∂tm)y magnetization dynamics will have the same expressions as in Eqs. ( 43) and ( 44). Thus, we calculated even under magnetiza- tion reversal components αeven xy=−αeven yxandβeven xy=−βeven yx as it follows from Eq. ( 30). As can be seen from Fig. 3,E q s .( 43) and ( 44) only qualitatively agree with the numerical results at higher temperatures as the Berry curvature from other parts ofthe Brillouin zone starts to contribute to the result. B. Application to kagome ferromagnet Here we apply our theory to the kagome lattice ferromagnet with the nearest neighbor DMI. The lattice of the system andits magnon spectrum are shown in Fig. 4. Note that all vectors, such as a 1anda2, are measured in units of lattice spacing a 0 which is recovered in the ﬁnal result. We consider a model considered in Ref. [ 16] with a Hamiltonian given by H=−J/summationdisplay /angbracketleftij/angbracketrightSiSj−B/summationdisplay iSz i+/summationdisplay /angbracketleftij/angbracketrightνijDij[Si×Sj],(45) where J> 0 corresponds to ferromagnetic nearest neighbor exchange, Bis the external magnetic ﬁeld, and νijdescribes a sign convention for the nearest neighbor DMI, i.e., νij=1f o r the clockwise sense of direction and νij=−1 otherwise (see Fig.4). Note that vectors Dij=D[z]ˆz+D[R] ijhave an in-plane Rashba-like component D[R] ijdirected orthogonally to bonds and outwards with respect to bond triangles (see Fig. 4). The Rashba-like DMI could result from mirror asymmetry with FIG. 4. Left: A two-dimensional kagome lattice with lattice vectors a1=1 2(√ 3,−1) and a2=1 2(√ 3,1) where atoms are placed in the corners of triangles. Rashba-like DMI vectors D[R] ijare shown by blue vectors perpendicular to the bonds. The clockwise ordering of bonds corresponding to ν=1 is shown by black arrows. Right: Magnon spectrum of a kagome ferromagnet with DMI D[z]=0.3J and magnetization pointing in the zdirection. The distribution of the Berry curvature over the Brillouin zone is plotted by the color codingon top of the spectrum for each subband. respect to the kagome planes. At sufﬁciently low temperatures the Hamiltonian in Eq. ( 45) can be analyzed by applying the Holstein-Primakoff transformation. The correspondingmagnon spectrum is shown in Fig. 4where the lower, middle, and upper bands have the Chern numbers −1, 0, and 1, respectively. We begin by analyzing an effect of magnon pumping by magnetization dynamics. This effect is characterized bytensor α αor equivalently by Eq. ( 10). It is also clear from Eq. ( 30) that the same tensor also describes a magnetization torque induced by an analog of electric ﬁeld for magnons.We assume a small-angle precession of magnetization aroundthezaxis. By symmetry consideration, it is sufﬁcient to consider only α even yx=−αeven xyandαodd xx=αodd yycomponents of the tensor where we separate tensor ααinto the parts that are odd and even under magnetization reversal, i.e.,α α=αodd α+αeven α. The results of our calculations for the two components are shown in Fig. 5. Note that we use a simple phenomenological treatment by relating the quasiparticlebroadening to the Gilbert damping as /Gamma1=α¯hω. Under a simple circular precession of the magnetization described byangleθwe have ∂ tm=θω[−sin(ωt),cos(ωt),0]Tand Jpx=θω/bracketleftbig αodd xxcos(ωt)−αeven yxsin(ωt)/bracketrightbig , Jpy=θω/bracketleftbig αodd xxsin(ωt)+αeven yxcos(ωt)/bracketrightbig . (46) D/Equal0.1JD/Equal0.2JD/Equal0.3J 0.5 1. 1.5kBT/SlashSJ0.30.6a0xxodd D/Equal0.1JD/Equal0.2JD/Equal0.3J 0.5 1. 1.5kBT/SlashSJ0.010.02a0yxeven FIG. 5. Left: The odd component of the tensor αijas a function of temperature. The plot is rescaled by multiplying it with the Gilber damping α. Right: The even component of the tensor αijas a function of temperature. In both cases the magnetization is along the zaxis. For the strength of the Rashba DMI we use D[R]=D[z]=D. 165106-6PUMPING OF MAGNONS IN A DZYALOSHINSKII-MORIYA . . . PHYSICAL REVIEW B 95, 165106 (2017) D/Equal0.1JD/Equal0.2JD/Equal0.3J 0.5 1. 1.5kBT/SlashSJ0.10.2a0xxodd/SlashkB D/Equal0.1JD/Equal0.2JD/Equal0.3J 0.5 1. 1.5kBT/SlashSJ0.020.04a0yxeven/SlashkB FIG. 6. Left: The odd component of the torkance tensor βijas a function of temperature. The plot is rescaled by multiplying it with the Gilber damping α. Right: The even component of the torkance tensor βijas a function of temperature. In both cases the magnetization is along the zaxis. For the strength of the Rashba DMI we use D[R]=D[z]=D. We can now estimate the amplitude of ac spin current asθ¯hω√ (αodd xx)2+(αeven yx)2. For a three-dimensional system containing weakly interacting kagome layers, we can writeα 3D ij=αs ij/c, where c∝a0is the interlayer distance which is comparable to the lattice constant a 0. For parameters D[z]= 0.1J,D[R]=0.1J,θ=0.1 degrees, ω=2π×10 GHz, kBT=0.5SJ, and the Gilbert damping α=0.1, we obtain the spin current of amplitude Js≈10−8J/m2. We suggest to detect such spin currents by the ac inverse spin Hall effect [ 50]. We also consider an effect of heat pumping by magneti- zation dynamics. This effect is characterized by tensor βα. Here we again assume a small-angle precession of magne-tization around the zaxis. Similar symmetry considerations result in relations β even yx=−βeven xyandβodd xx=βodd yybetween nonzero components of tensor βα=βodd α+βeven αseparated into the odd and even under magnetization reversal parts.The results of our calculations for the two components areshown in Fig. 6. The amplitude of ac heat current is given byθTω√ (βodd xx)2+(βeven yx)2which for the above parameters andT=50 K results in the heat current of amplitude Jq≈ 50 kW /m2. After invoking the Onsager relation ( 30) one can conﬁrm that estimates obtained in this subsection are comparable toestimates for thermal torques obtained in Ref. [ 16]. Note also that the phenomenology discussed in this paper is similar toRef. [ 26], however, the heat current is carried by magnons in contrast to electronic mechanisms considered before. IV . CONCLUSIONS In this work we explored ﬁctitious electric ﬁelds acting on magnons in response to time-dependent magnetizationdynamics in the presence of DMI. We ﬁnd that such ﬁctitiouselectric ﬁelds can drive sizable spin and energy currents. Wesuggest a detection scheme relying on the ac inverse spinHall effect [ 50]. Additionally, we obtain an analog of the Hall-like response in systems with nontrivial Berry curvatureof magnon bands. This leads to even under magnetizationreversal contributions to the response tensors. By the Onsagerreciprocity relation, this Hall-like response can be relatedto the antidamping thermal torque [ 16]. Finally, we identify the ground state energy current associated with the magnon-mediated equilibrium contribution to DMI. This contributionneeds to be subtracted from the Kubo linear response resultaccording to our analysis.ACKNOWLEDGMENTS We gratefully acknowledge useful discussions with K. Belashchenko. This work was supported by the U.S. Depart-ment of Energy, Ofﬁce of Science, Basic Energy Sciences,under Award No. DE-SC0014189. APPENDIX A: HEAT CURRENT AS A RESPONSE TO MAGNETIZATION DYNAMICS Measurable heat current consists of three parts. Free energy contribution, nonequilibrium heat current, and orbitalmagnetization heat current carried by magnons. 1. Free energy heat current Magnon mediated Dzyaloshinskii-Moriya interaction con- tribution to the free energy of the system is FDMI=D/bracketleftbigg m(r)×∂m(r) ∂r/bracketrightbigg , (A1) where DDMI is the Dzyaloshinskii-Moriya tensor we will calculate below. For instance, functionality on xmight be due to the boundary or it might be due to spatially dependentmagnetization proﬁle. Assuming a time dependence of themagnetization, via a r→r+ωt/k shift, one can derive the current due to time dependence of DMI part of free energy using continuity equation ∂FDMI ∂t+∇JDMI=0, where JDMI α=−1 VDαβ(∂tm)β, (A2) where Vis the volume of the system. The Dzyaloshinskii- Moriya interaction constant is Dαβ=1 2/angbracketleftbigg/integraldisplay dr/Psi1†(r)/parenleftbig rαTβ+Tβrα/parenrightbig /Psi1(r)/angbracketrightbigg eq, (A3) where Tβ=(∂mH×m)βis the torque operator. To calculate the DMI, we introduce Aαβ(η)=iTr/bracketleftbigg vαkdG+ dη¯vβkδ(η−Hk) −vαkδ(η−Hk)¯vβkdG− dη/bracketrightbigg , (A4) Bαβ(η)=iTr[vαkG+¯vβkδ(η−Hk)−vαkδ(η−Hk)¯vβkG−], (A5) where ¯ vβk=∂mβHk≡i[Hk,rmβ] is equivalent to the velocity operator deﬁnition, with rmβ≡i∂mβequivalent to the position operator. It was shown that Aαβ−1 2dBαβ dη=1 4πTr[rα(GA−GR)rmβ −rαrmβ(GA−GR)]−(α↔β) +1 2Tr/bracketleftbigg (rα¯vβk−vαkrmβ)d dηδ(η−Hk)/bracketrightbigg . (A6) 165106-7ALEXEY A. KOV ALEV , VLADIMIR A. ZYUZIN, AND BO LI PHYSICAL REVIEW B 95, 165106 (2017) Also, we derive the Berry curvature parts of AαβandBαβ: Aαβ(η)=−i/summationdisplay n/parenleftbig ∂αT† k∂mβTk/parenrightbig nnδ[η−(/epsilon1k)nn]−(α↔β), (A7) and a Berry curvature part of the Bαβas Bαβ(η)=i/summationdisplay n[∂αT† k(η−Hk)∂mβTk]nnδ[η−(/epsilon1k)nn]−(α↔β). (A8) Therefore, Dαβ=/summationdisplay k/integraldisplay∞ −∞d˜η/bracketleftbigg Aαβ(˜η)−1 2dBαβ(˜η) d˜η/bracketrightbigg/integraldisplay˜η 0dηg(η), (A9) and it can be shown that Dαβ=/summationdisplay n/integraldisplay+∞ −∞d˜η/bracketleftbigg Aαβ(˜η)−1 2dBαβ(˜η) d˜η/bracketrightbigg/integraldisplay˜η 0dηg(η) =/summationdisplay n/integraldisplay+∞ −∞d˜η/braceleftbigg −i(∂αT† k∂mβTk)nnδ[˜η−(/epsilon1k)nn]/integraldisplay˜η 0dηg(η)/bracerightbigg +i 2/summationdisplay n/integraldisplay+∞ −∞d˜η{[∂αT† k(˜η−Hk)∂mβTk]nng(˜η)δ[˜η−(/epsilon1k)nn]}−(α↔β). (A10) 2. Heat current due to magnons We assume that the magnetizaion is varying in time. Next, we assume that due to that there is a time-dependent term inthe Hamiltonian. For example, since the DMI depends on thedirection of the order, this DMI will be time dependent. TheHamiltonian of the spin waves is then H T=1 2/integraldisplay dr/Psi1†(r)[ˆH+ˆH/prime(t)]/Psi1(r). (A11) We deﬁne ˆHT=ˆH+ˆH/prime(t). Microscopic expression for the heat current current is derived via commutation relationship jQ(r)=1 2/Psi1†(r)(ˆHTV+VˆHT)/Psi1(r), (A12) hereV=i[ˆHT,r] is the full velocity. Velocity has two parts V=v+v/prime, where v=i[ˆH,r] and v/prime=i[ˆH/prime,r]. Assuming that the magnetic order is m(t)=m+δm(t), we write the perturbation as ˆH/prime(t)=(∂mˆH)δm(t). We will use analogy between magnetization dynamics and the electromagneticwaves. The direction of the local magnetization can be seenas a vector potential for effective electromagnetic ﬁeld electricand magnetic ﬁelds. Then, ∂m ∂tis analogous to the electric ﬁeld, while∇×mis analogous to the magnetic ﬁeld. We will then writeδm(t)=1 ω∂m(t) ∂t≡1 ω∂tm(in Matsubara frequency). The heat current is separated into two parts j[0] Q(r)=1 2/Psi1†(r)(ˆHv+vˆH)/Psi1(r), (A13) j[1] Q(r)=1 2/Psi1†(r)(ˆH/primev+vˆH/prime)/Psi1(r) +1 2/Psi1†(r)(ˆHv/prime+v/primeˆH)/Psi1(r) =1 2/Psi1†(r)[(δm(t)·∂m)(ˆHv+vˆH)]/Psi1(r).(A14) TheH/prime(t) will be treated as a perturbation. We will be working with global currents JQ≡1 V/integraltext drjQ(r). The heat current isconveniently written as /angbracketleftJQ/angbracketright=/angbracketleftBig J[0] Q/angbracketrightBig ne+/angbracketleftBig J[1] Q/angbracketrightBig eq, (A15) where the former one is estimated over nonequilibrium states and is given by Kubo formula, while the later one is due toanalog of diamagnetic current for magnons and is estimatedover equilibrium states. a. Nonequilibrium heat current, Kubo formula Kubo formula for an arbitrary operator A(ω), where ωis Matsubara frequency, is /angbracketleftA(ω)/angbracketrightne=/integraldisplayβ 0dτeiωτ/angbracketleftTτA(0)H/prime(−τ)/angbracketrighteq, (A16) where H/prime(τ)=/integraltext dr/Psi1†(τ,r)ˆH/prime/Psi1(τ,r) is the perturbing Hamiltonian /angbracketleftbig J[0] Qα/angbracketrightbig ne=/integraldisplay1/T 0dτeiωτ/angbracketleftbig TτJ[0] Qα(0)H/prime(−τ)/angbracketrightbig eq ≡1 VSαβ(ω)1 ω(∂tm)β. (A17) After all of the transforms, we get Sαβ=1 2/summationdisplay k(/epsilon1k˜vαk+˜vαk/epsilon1k)nm[T† k(∂βHk)Tk]mn ×g[(/epsilon1k)nn]−g[(/epsilon1k)mm] iω+(/epsilon1k)nn−(/epsilon1k)mm, (A18) 165106-8PUMPING OF MAGNONS IN A DZYALOSHINSKII-MORIYA . . . PHYSICAL REVIEW B 95, 165106 (2017) where ˜vαk=T† kvαkTk=∂α/epsilon1k+Aαk/epsilon1k−/epsilon1kAαk, (A19) ˜¯vβk=T† k¯vβkTk=∂mβ/epsilon1k+¯Aβk/epsilon1k−/epsilon1k¯Aβk, (A20) where Aαk=T† k∂αTkand ¯Aβk=T† k∂mβTk≡T† k∂βTk, and where a bar over ¯Aβksymbolizes information that the derivative is overβcomponent of the magnetization direction mβ. After the transformations we get Sαβ(ω)=1 2/summationdisplay kng[(/epsilon1k)nn]−g[(/epsilon1k)mm] iω+(/epsilon1k)nn−(/epsilon1k)mm(˜vαk)nm[(/epsilon1k)nn+(/epsilon1k)mm](∂mβ/epsilon1k+¯Aβk/epsilon1k+/epsilon1k¯Aβk)mn. (A21) Expand Sαβ(ω)i nωand get Sαβ(ω)=S[1] αβ(0)+S[2] αβ(0)+∂ ∂ωS[2] αβ(ω)\|ω=0ω, (A22) where n=mparts of Sαβare S[1] αβ(0)=/summationdisplay kn∂g(/epsilon1) ∂/epsilon1\|/epsilon1=(/epsilon1k)nn(/epsilon1k)nn(∂α/epsilon1k)nn(∂mβ/epsilon1k)nn =−1 2/summationdisplay kng[(/epsilon1k)nn]/parenleftbig ∂α∂mβ/epsilon12 k/parenrightbig nn, (A23) where we integrated by parts over k. Term with n/negationslash=melements reads S[2] αβ(ω)=−1 2/summationdisplay kng[(/epsilon1k)nn]−g[(/epsilon1k)mm] iω+(/epsilon1k)nn−(/epsilon1k)mm[(/epsilon1k)nn+(/epsilon1k)mm][(/epsilon1k)nn−(/epsilon1k)mm]2(Aαk)nm(¯Aβk)mn, (A24) which we expand in ω, and get S[2] αβ(0)=−1 2/summationdisplay kn{g[(/epsilon1k)nn]−g[(/epsilon1k)mm]}/bracketleftbig (/epsilon1k)2 nn−(/epsilon1k)2 mm/bracketrightbig (Aαk)nm(¯Aβk)mn (A25) and ∂ ∂ωS[2] αβ(ω)\|ω=0=i1 2/summationdisplay kn{g[(/epsilon1k)nn]−g[(/epsilon1k)mm]}[(/epsilon1k)nn+(/epsilon1k)mm](Aαk)nm(¯Aβk)mn. (A26) Overall the Kubo part of the current is presented as lim ω→0Sαβ(ω)1 ω=S[1] αβ(0)1 ω+S[2] αβ(0)1 ω+∂ ∂ωS[2] αβ(ω)\|ω=0. (A27) b. Analog of diamagnetic current for magnons In this section we calculate an expectation value of the perturbed current over the equilibrium ground state, /angbracketleftbig J[1] Qα/angbracketrightbig =1 V1 2Tr/summationdisplay kg[(/epsilon1k)]T† k{[δm(t)·∂m](Hkvαk+vαkHk)}Tk≡1 VMαβ1 ω(∂tm)β. (A28) Quantity of interest is T† k[∂β(Hkvαk+vαkHk)]Tk=∂mβ(/epsilon1k˜vαk+˜vαk/epsilon1k)+¯Aβk/epsilon1k˜vαk−˜vαk/epsilon1k¯Aβk+¯Aβk˜vαk/epsilon1k−/epsilon1k˜vαk¯Aβk. (A29) We then get Mαβ1 ω=1 2ω/summationdisplay kn(Aαk)nm(¯Aβk)mn/bracketleftbig (/epsilon1k)2 nn−(/epsilon1k)2 mm/bracketrightbig{g[(/epsilon1k)nn]−g[(/epsilon1k)mm]}+1 2ω/summationdisplay kn∂mβ∂α/parenleftbig /epsilon12 k/parenrightbig nng[(/epsilon1k)nn]. (A30) 165106-9ALEXEY A. KOV ALEV , VLADIMIR A. ZYUZIN, AND BO LI PHYSICAL REVIEW B 95, 165106 (2017) c. Overall Overall response is J[0] Qα+J[1] Qα=1 V/bracketleftbigg Sαβ1 ω+Mαβ1 ω/bracketrightbigg (∂tm)β=∂ ∂ωS[2] αβ(ω)\|ω=0(∂tm)β =1 V/braceleftBigg i 2/summationdisplay kn{g[(/epsilon1k)nn]−g[(/epsilon1k)mm]}[(/epsilon1k)nn+(/epsilon1k)mm](Aαk)nm(¯Aβk)mn/bracerightBigg (∂tm)β =1 V/braceleftBigg i 2/summationdisplay kng[(/epsilon1k)nn][(/epsilon1k)nn+(/epsilon1k)mm](Aαk)nm(¯Aβk)mn−(α↔β)/bracerightBigg (∂tm)β. (A31) 3. Overall heat current Summing up the Dzyaloshinskii-Moriya current and current carried by magnons, we get J/Sigma1 Qα=J[0] Qα+J[1] Qα+JDMI α=1 V/parenleftbigg Sαβ1 ω+Mαβ1 ω+Dαβ/parenrightbigg (∂tm)β =i1 V/braceleftBigg/summationdisplay kn(Aαk¯Aβk)nnc1[(/epsilon1k)nn]−(α↔β)/bracerightBigg (∂tm)β ≡1 V/summationdisplay kn[/Omega1αβ]nnc1[(/epsilon1k)nn](∂tm)β, (A32) where c1(x)=/integraltextx 0dηηdg dη, where /Omega1αβ=2Im(∂αT† k)(∂mβTk) is the mixed Berry curvature. APPENDIX B: SPIN CURRENT AS A RESPONSE TO MAGNETIZATION DYNAMICS Again, we study a ferromagnetic system with time- dependent magnetization direction. The Hamiltonian is HT=1 2/integraldisplay dr/Psi1†(r)/bracketleftbigˆH+ˆH/prime(t)/bracketrightbig /Psi1(r). (B1) We deﬁne ˆHT=ˆH+ˆH/prime(t). Microscopic expression for the spin density current current is derived via commutationrelationship j S(r)=/Psi1†(r)V/Psi1(r), (B2) hereV=i[ˆHT,r] is the full velocity. Velocity has two parts V=v+v/prime, where v=i[ˆH,r] and v/prime=i[ˆH/prime,r]. Assuming that the magnetic order is m(t)=m+δm(t), we write the perturbation as ˆH/prime(t)=(∂mˆH)δm(t). The spin current splits into two parts jS(r)=j[0] S(r)+j[1] S(r). (B3) We again consider macroscopic currents JS=1 V/integraltext drjS(r). We write J[0] Sα=1 VSαβ1 ω(∂tm)β, (B4) J[1] Sα=1 VMαβ1 ω(∂tm)β. (B5) The later term is due to analog of diamagnetic current for magnons, while the former current is given by Kubo formula Sαβ(ω)=/summationdisplay kn[˜vαk]nm[˜¯vβk]mng[(/epsilon1k)nn]−g[(/epsilon1k)mm] iω+(/epsilon1k)nn−(/epsilon1k)mm,(B6)where again ˜vαk=T† kvαkTk=∂α/epsilon1k+Aαk/epsilon1k−/epsilon1kAαk, (B7) ˜¯vβk=T† k¯vβkTk=∂mβ/epsilon1k+¯Aβk/epsilon1k−/epsilon1k¯Aβk.(B8) After straightforward transformations, expanding the expres- sion above in ω, and taking corresponding integral over kby parts, we obtain an expression Sαβ=−/summationdisplay kng[(/epsilon1k)nn]∂α∂mβ(/epsilon1k)nn −/summationdisplay kn(Aαk)nm(¯Aβk)mn/bracketleftbig (/epsilon1k)nn−(/epsilon1k)mm/bracketrightbig/braceleftbig g/bracketleftbig (/epsilon1k)nn/bracketrightbig −g[(/epsilon1k)mm]/bracerightbig +iω/summationdisplay kn(Aαk)nm/parenleftbig¯Aβk/parenrightbig mn{g[(/epsilon1k)nn]−g[(/epsilon1k)mm]}. (B9) The analog of diamagnetic current for magnons is given by Mαβ=/summationdisplay kn[T† k/parenleftbig ∂α∂mβHk/parenrightbig Tk]nng[(/epsilon1k)nn] =/summationdisplay kn[∂α∂mβ(/epsilon1k)nn]g[(/epsilon1k)nn] +/summationdisplay kn(Aαk)nm/parenleftbig¯Aβk/parenrightbig mn[(/epsilon1k)nn−(/epsilon1k)mm]{g[(/epsilon1k)nn] −g[(/epsilon1k)mm]}. (B10) 165106-10PUMPING OF MAGNONS IN A DZYALOSHINSKII-MORIYA . . . PHYSICAL REVIEW B 95, 165106 (2017) Hence we observe Sαβ+Mαβ=iω/summationdisplay kn(Aαk)nm(¯Aβk)mn{g[(/epsilon1k)nn]−g[(/epsilon1k)mm]}. (B11) The overall spin current is readily obtained: JSα=1 V(Sαβ+Mαβ)1 ω(∂tm)β =i1 V/summationdisplay kn(Aαk)nm(¯Aβk)mn{g[(/epsilon1k)nn]−g[(/epsilon1k)mm]}(∂tm)β ≡1 V/summationdisplay kn[/Omega1αβ]nng[(/epsilon1k)nn](∂tm)β, (B12) where /Omega1αβ=2Im(∂αT† k)(∂mβTk) is the mixed Berry curvature. APPENDIX C: TORQUE AS A RESPONSE TO TEMPERATURE GRADIENT We adopt the Luttinger formalism to study the response of the system to the temperature gradient. In this formalism theHamiltonian acquires extra terms, written compactly as H=/integraldisplay dr˜/Psi1 †(r)ˆH(r)˜/Psi1(r), (C1) where ˜/Psi1(r)=(1+r∇χ 2)/Psi1(r)≡ξ(r)/Psi1(r), with ∇χbeing the temperature gradient. We deﬁne the torque as T=/angbracketleft∂mH/angbracketright. For the response of the torque on the temperature gradient, weagain deﬁne two terms: /angbracketleft∂ mH/angbracketright≡/angbracketleft∂mH/angbracketrightne+1 2/angbracketleft∂m[rβH+Hrβ]/angbracketrighteq∇βχ. (C2) The ﬁrst term is again described by a Kubo formula /angbracketleft∂mH/angbracketrightne= 1 VSmβ∇βχ, the second term we again deﬁne as Mmβ= 1 2/angbracketleft∂m[rβH+Hrβ]/angbracketrighteq, we then formally rewrite the expression for torque /angbracketleft∂mH/angbracketright=1 V(Smβ+Mmβ)∇βχ=1 VLmβ∇βχ. (C3) Calculations for the torque are similar to the ones presented for the particle current in Appendix Bwith a deﬁnition of rm operator as in Appendix A1. As a result, we get /angbracketleft∂mH/angbracketright=1 V/braceleftbigg/summationdisplay kn/Omega1(n) mβ(k)c1[(/epsilon1k)nn] +/summationdisplay kn(∂mεnk)(∂βεnk)εnk1 2/Gamma1nkg/prime[(/epsilon1k)nn]/bracerightbigg ∇βχ, (C4) where now /Omega1(n) mβ(k)≡i[(∂mT† k)(∂βTk)]nn−(m↔β)i st h e mixed space Berry curvature of the nth band. APPENDIX D: A MODEL OF HONEYCOMB FERROMAGNET WITH DZYALOSHINSKII-MORIYA INTERACTION 1. Hamiltonian We study a model of a ferromagnet on a honeycomb lattice. We assume a Heisenberg exchange, in-plane Dzyaloshinskii-1 23 a1 a2+ -d1 d2d3 xy FIG. 7. Schematics of the graphene layer parameters for the tight-binding model. Vectors connecting nearest neighbors are τ1= 1 2(1√ 3,1),τ2=1 2(1√ 3,−1), and τ3=1√ 3(−1,0) are used in deriving the Hamiltonian for magnons. Vectors a1=1 2(√ 3,1) and a2= 1 2(√ 3,−1) are used in deriving the second-nearest neighbor DMI. Moriya interaction (DMI) of Rashba type, and second-nearest neighbor DMI. In our model we assume that the order isin general ( m x,my,mz) direction, which can be realized by application of the magnetic ﬁeld. The Hamiltonian is H=J/summationdisplay /angbracketleftij/angbracketrightSiSj+/summationdisplay /angbracketleftij/angbracketrightD[R][Si×Sj]+D[z]/summationdisplay /angbracketleft/angbracketleftij/angbracketright/angbracketright[Si×Sjx]z. (D1) Dzyaloshinskii-Moriya interaction originating from the Rashba type spin-orbit coupling for 1,2,3 links (see Fig. 7) is H[R] 1=D[R]/parenleftBigg −1 2[SA×SB]y+√ 3 2[SA×SB]x/parenrightBigg ,(D2) H[R] 2=D[R]/parenleftBigg −1 2[SA×SB]y−√ 3 2[SA×SB]x/parenrightBigg ,(D3) H[R] 3=D[R][SA×SB]y. (D4) In Holstein-Primakoff bosons, Rashba DMI reads [SA×SB]x=Sy ASz B−Sz ASy B=−iSmx(b†a−a†b),(D5) [SA×SB]y=−Sx ASz B+Sz ASx B=−iSmy(b†a−a†b).(D6) Together with Heisenberg exchange and second-nearest neigh- bor DMI written in Holstein-Primakoff bosons, we get H=JS/bracketleftbigg 3+/Delta1k−˜γk −˜γ∗ k 3−/Delta1k/bracketrightbigg , (D7) where /Delta1k=2/Delta1[sin(ky)−2s i n(ky 2) cos (√ 3kx 2)], where /Delta1= mzD[z]/J.D e r i v i n g ˜ γkwe considered Rashba DMI in the lowest order in D[R]/J/lessmuch1 parameter. With this assumption ˜γk=2ei˜kx 2√ 3cos/parenleftbigg˜ky 2/parenrightbigg +e−i˜kx√ 3, (D8) where ˜kx=kx−√ 3D[R] Jmyand ˜ky=ky+√ 3D[R] Jmx.W e observe that Rashba DMI plays an effective role of magnoncharge, while order direction is an effective vector potentialfelt by magnons. 165106-11ALEXEY A. KOV ALEV , VLADIMIR A. ZYUZIN, AND BO LI PHYSICAL REVIEW B 95, 165106 (2017) The eigenvalues of the Hamiltonian are calculated as /epsilon1k,±=JS/parenleftbig 3±/radicalBig /Delta12 k+\|˜γk\|2/parenrightbig , (D9) with corresponding eigenfunctions /Psi1k,+=/bracketleftBigg cos/parenleftbig˜ξk 2/parenrightbig ei˜χk −sin/parenleftbig˜ξk 2/parenrightbig/bracketrightBigg ,/Psi1 k,−=/bracketleftBigg sin/parenleftbig˜ξk 2/parenrightbig cos/parenleftbig˜ξk 2/parenrightbig e−i˜χk/bracketrightBigg , (D10) where sin ( ˜ξk)=\|˜γk\|√ /Delta12 k+\|˜γk\|2and ˜γk=\|˜γk\|ei˜χk, where the tilde symbol means that corresponding kmomenta are shifted by the Rashba DMI. Unitary matrix that diagonalizes the Hamiltonian is readily constructed: Tk=/bracketleftBigg cos/parenleftbig˜ξk 2/parenrightbig ei˜χk sin/parenleftbig˜ξk 2/parenrightbig −sin/parenleftbig˜ξk 2/parenrightbig cos/parenleftbig˜ξk 2/parenrightbig e−i˜χk/bracketrightBigg . (D11) An expression deﬁning the Berry curvature is /Omega1αβ(k)=2Im[(∂αT† k)(∂βTk)]=1 2sin(˜ξk)[(∂α˜χk)(∂β˜ξk)−(∂β˜χk)(∂α˜ξk)]/bracketleftbigg 10 0−1/bracketrightbigg ≡/bracketleftBigg /Omega1(+) αβ(k)0 0 /Omega1(−) αβ(k)/bracketrightBigg , (D12) where /Omega1(+) xmx(k)=−/Omega1(−) xmx(k) and (∂α˜χk)(∂β˜ξk)−(∂β˜χk)(∂α˜ξk)=(∂αIm ˜γk) \|˜γk\|2/parenleftbig /Delta12 k+\|˜γk\|2/parenrightbig[(∂β\|˜γk\|)(Re ˜γk)/Delta1k−(∂β/Delta1k)(Re ˜γk)\|˜γk\|] −(∂αRe ˜γk) \|˜γk\|2/parenleftbig /Delta12 k+\|˜γk\|2/parenrightbig[(∂β\|˜γk\|)(Im ˜γk)/Delta1k−(∂β/Delta1k)(Im ˜γk)\|˜γk\|]−(α↔β) =/Delta1k \|˜γk\|/parenleftbig /Delta12 k+\|˜γk\|2/parenrightbig[(∂αIm ˜γk)(∂βRe ˜γk)−(∂βIm ˜γk)(∂αRe ˜γk)] +∂α/Delta1k \|˜γk\|/parenleftbig /Delta12 k+\|˜γk\|2/parenrightbig[Re ˜γk(∂βIm ˜γk)−Im ˜γk(∂βRe ˜γk)]. (D13) Recall that βhere stands for the component of the ferromagnetic order, i.e., mβ. Recall that /Delta1kdoes not depend onmβ, hence ∂β/Delta1k=0. The derivitive with respect to the direction of the order mβof the remaining functions that depend on ˜kis ∂ ∂mx=√ 3D[R] J∂ ∂˜ky≡√ 3D[R] J∂y, (D14) ∂ ∂my=−√ 3D[R] J∂ ∂˜kx≡−√ 3D[R] J∂x, (D15) this straightforward transformation makes the mixed Berry curvature a regular kspace one. 2. Berry curvature at the K/primeand K points We ﬁrst show that the Berry curvature has peaks at the K/prime andKpoints. Let us study the spectrum close to K/prime=(0,4π 3), (/Delta1k)K/prime≈−3√ 3/Delta1, (D16) (˜γk)K/prime≈−√ 3 2(˜ky+i˜kx). (D17) AtK=(0,−4π 3) point we expand as (/Delta1k)K≈3√ 3/Delta1, (D18)(˜γk)K≈√ 3 2(˜ky−i˜kx). (D19) Hence, under the mentioned above approximations the mixed Berry curvature becomes a regular, kspace one. To the lowest order in Rashba DMI, we can disregard all tildes in ˜k.U s i n g these approximations, we get for the Berry curvature which isclose to the K /primepoint /Omega1(+) xmx(k)=1 2sin(ξk)/bracketleftbig (∂xχk)/parenleftbig ∂mxξk/parenrightbig −/parenleftbig ∂yχk/parenrightbig/parenleftbig ∂mxξk/parenrightbig/bracketrightbig ≈−√ 3D[R] J3√ 3/Delta1 2/parenleftbig 27/Delta12+3 4k2/parenrightbig3/2[(∂xReγk)(∂yImγk) −(∂yReγk)(∂xImγk)x] ≈−27 8D[R] J/Delta1 /parenleftbig 27/Delta12+3 4k2/parenrightbig3/2. (D20) Note that the Berry curvature is of the same sign for both KandK/primepoints ( /Delta1kand Re γkchange sign under the point interchange). 3. Berry curvature at the /Gamma1point We note that since the /Gamma1=(0,0) point is not gapped, it might contribute to currents at low temperatures. In the 165106-12PUMPING OF MAGNONS IN A DZYALOSHINSKII-MORIYA . . . PHYSICAL REVIEW B 95, 165106 (2017) following we estimate the Berry curvature at the point. For that we expand all functions entering the current close to /Gamma1 point in small kas /Delta1k≈1 4/Delta1ky/parenleftbig 3k2 x−k2 y/parenrightbig , (D21) Re ˜γk≈3−1 4˜k2, (D22) Im ˜γk≈1 24√ 3˜kx/parenleftbig˜k2 x−3˜k2 y/parenrightbig . (D23) We recall that ∂β/Delta1k=0f o rβ=x,y. a.α=xa n d β=mx /Omega1(+) xmx(k)≈1 2sin(˜χk)[(∂x˜χk)(∂mx˜ξk)−(∂mx˜χk)(∂x˜ξk)] ≈−D[R] J/Delta1 48k2 yk2 x. (D24) b.α=xa n d β=my /Omega1(+) xmy(k)≈1 2sin( ˜χk)[(∂x˜χk)(∂my˜ξk)−(∂my˜χk)(∂x˜ξk)] ≈−D[R] J/Delta1 192kykx/parenleftbig k2 x−k2 y/parenrightbig , (D25) which will vanish upon angle integration. Same for α=yand β=nxcombination. 4. Spin current The spin current is deﬁned as J[S] x=1 V/summationdisplay n=±/integraldisplay k/Omega1(n) xmx(k)g(/epsilon1k,n)(∂tm)x. (D26) We approximate the integrals at small temperatures SJ/greatermuchT. AtK/primeandKpoints we use the following approximations: g(/epsilon1k,+)−g(/epsilon1k,−)≈−2s i n h/bracketleftBigg SJ T3√ 3D[z] J/bracketrightBigg e−3SJ T,(D27) in which /epsilon1k±≈SJ(3±3√ 3\|/Delta1\|) was used: /integraldisplay∞ 0kdk3√ 3/Delta1 /parenleftbig 27/Delta12+3 4k2/parenrightbig3/2=4 3. (D28)At/Gamma1point only the /epsilon1k−≈1 4SJk2contributes to the current. We use the following integrations: /integraldisplay∞ 0k5dk1 e−1 4JSβk2−1=1 2/parenleftbigg1 4JS/parenrightbigg−3/integraldisplay∞ 0z2dz e−z−1 =/parenleftbigg1 4JS/parenrightbigg−3 ζ(3), (D29) where ζ(3) is the Riemann zeta function. Summing all the contributions, we get J[S] x=1 VD[R] J√ 3 π/bracketleftbigg sinh/parenleftBigg 1 z3√ 3D[z] J/parenrightBigg e−3 z +D[z] J√ 3ζ(3) 36z3/bracketrightbigg (∂tm)x, (D30) where z=T SJwas introduced for brevity. 5. Heat current J[Q] x=1 V/summationdisplay n=±/integraldisplay k/Omega1(n) xmx(k)c1(/epsilon1k,n)(∂tm)x. 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PhysRevB.103.214407.pdf	PHYSICAL REVIEW B 103, 214407 (2021) Geometric magnonics with chiral magnetic domain walls Jin Lan (/ZdZ1692/ZdZ18173),1,2Weichao Yu ( /ZdZ1157/ZdZ1099/ZdZ17073),3,2and Jiang Xiao ( /ZdZ14675/ZdZ8587)2,4,5,* 1Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, 92 Weijin Road, Tianjin 300072, China 2Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China 3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 4Institute for Nanoelectronics Devices and Quantum Computing, Fudan University, Shanghai 200433, China 5Shanghai Research Center for Quantum Sciences, Shanghai 201315, China (Received 25 February 2021; revised 24 May 2021; accepted 24 May 2021; published 2 June 2021) Spin wave, the collective excitation of magnetic order, is one of the fundamental angular momentum carriers in magnetic systems. Understanding the spin wave propagation in magnetic textures lies in the heart of developingpure magnetic information processing schemes. Here we show that the spin wave propagation across a chiraldomain wall follows simple geometric trajectories, similar to the geometric optics. And the geometric behaviorsare qualitatively different in normally magnetized ﬁlm and tangentially magnetized ﬁlm. We identify the lateralshift, refraction, and total reﬂection of spin wave across a ferromagnetic domain wall. Moreover, these geometricscattering phenomena become polarization dependent in antiferromagnets, indicating the emergence of spinwave birefringence inside antiferromagnetic domain walls. DOI: 10.1103/PhysRevB.103.214407 I. INTRODUCTION Spin wave, the propagating disturbance of ordered magne- tization, is one of the basic excitations in magnetic systems.As an alternative spin current carrier besides the spin-polarized electrons [ 1], spin wave manipulation is not only important for fundamental physics, but also attractive forindustrial applications [ 2,3]. Due to recent developments in experimental techniques, including excitation in short wave-length [ 4] and large amplitude [ 5], propagation in long distance [ 4,6], as well as detection with high sensibility [ 7,8], magnonics as a discipline devoted to manipulate spin wave isreceiving increasing interest [ 9,10]. Multiple approaches have been developed to control the spin wave, such as applying external magnetic ﬁeld [ 11–13], electric or spin current [ 14,15], heat [ 16–18], and coupling with microwave [ 19] and acoustic waves [ 20,21]. Restricted by the external sources introduced in these approaches, spinwave devices are typically difﬁcult to miniaturize. An alter-native approach is to utilize the magnetic textures, such asthe magnetic domain wall, magnetic vortex, and magneticskyrmion for the purpose of spin wave manipulation. Sinceboth magnetic texture and spin wave are of intrinsic magneticnature, they can coexist in single magnetic material, and in-timately interact with each other. Using magnetic texture tostore information, and spin wave to process information, puremagnetic computing schemes can be developed [ 5,22,23]. An up-to-date review on the topic of magnonics based on texturescan be found in Ref. [ 10]. The inﬂuence of magnetic texture on spin waves is mostly focused on the wave aspects of the spin wave, including its *Corresponding author: xiaojiang@fudan.edu.cnamplitude, phase, and polarization. The domain wall naturallyacts as the waveguide for spin wave [ 22,24–26], and magnetic vortex functions as spin wave emitter [ 27]. A Mach-Zehnder interferometer for spin wave can be constructed, by preparingdomain wall in one arm of a two-arm structure [ 28,29]. In the presence of the Dzyaloshinskii-Moriya interaction (DMI), anantiferromagnetic domain wall naturally serves as spin wavepolarizer and retarder [ 30]. However, the existing investiga- tions on spin wave trajectory, dictating the particle aspectof the spin wave, rely heavily on the wave-based equations[31,32] or effects [ 33–36], with straightforward and quantita- tive trajectory analysis missing. In this work, we investigate the spin wave scattering by magnetic domain walls in both normally magnetized ﬁlmand tangentially magnetized ﬁlm. Based on the semiclassicalanalysis and micromagnetic simulations, we identify variousgeometric relations between incident and out-going spin wavebeams, including lateral shift, refraction, and the total re-ﬂection, similar to the classical geometric optics. And thesegeometric magnonic phenomena become polarization depen-dent when extending to an antiferromagnetic environment.The geometric magnonics as demonstrated in this work, offersus simple yet intuitive paradigms in constructing magnonicdevices of different functionalities. This paper is organized as follows. In Sec. II,as e m i - classical scheme that describes the spin wave scattering bychiral domain wall is established. Based on the semiclas-sical trajectory analysis and the micromagnetic simulations,various geometric magnonic phenomena in normally andtangentially magnetized ﬁlms are then demonstrated, andfurther understanding by magnonic Snell’s law is pro-vided. Section IIIis devoted to an extension of the above geometric magnonic phenomena to an antiferromagnetic envi-ronment. The spin wave constriction by chiral domain wall is 2469-9950/2021/103(21)/214407(9) 214407-1 ©2021 American Physical SocietyJIN LAN, WEICHAO YU, AND JIANG XIAO PHYSICAL REVIEW B 103, 214407 (2021) FIG. 1. Schematics of spin wave scattering across a chiral domain wall. (a) is in bulk material, and (b),(c) are in normally/tangentially magnetized ﬁlms, which are slice cuts of (a) in the x-yandx-zplanes, respectively. (a) A magnetic domain wall along xdirection, and has translational invariance in the y-zplane. The black/green arrows denote domain wall magnetization m0and ﬁctitious magnetic ﬁeld b, respectively, and the gray-scale background is for the scalar potential φ. The blue/red slicing cut of the three-dimensional magnetic texture corresponds to domain walls in a normally magnetized and tangentially magnetized two-dimensional magnetic ﬁlm. The magenta arrow denotes the incident spin wave beam, and the blue/red arrows denote the out-going beams in the x-yandx-zplanes, respectively. In (b) and (c) the gray arrow depicts the electric ﬁeld, the green/orange colors encode the positive/negative magnetic ﬁeld, and the blue/red lines are the typical trajectories for normally incident spin wave on the domain wall. In the upper region, the magnetization distributions are depicted by arrows, with the in-plane magnetizations highlighted in blue/red color. presented in Sec. IV, and a short conclusion is drawn in Sec. V. II. BASIC MODEL A. Spin wave dynamics in chiral domain wall Consider a ferromagnetic system with its magnetization di- rection denoted by unit vector m, then its magnetic dynamics is governed by the Landau-Lifshitz-Gilbert (LLG) equation ˙m=−γm×h+αm×˙m, (1) where ˙m≡∂tm,γis the gyromagnetic ratio, and α is the Gilbert damping constant. The effective magneticﬁeld h=−(1/μ 0Ms)δu[m]/δm, where u[m]=(μ0Ms/2) {K[1−(m·ˆez)2]+A(∇m)2+Dm·(∇×m)}is the mag- netic energy density with Kthe easy-axis anisotropy along ˆz,Athe exchange coupling constant, Dthe DMI constant, μ0the vacuum permeability, and Msthe saturation magneti- zation. For exchange-type spin wave propagating in the thinﬁlm of interest in this work, the dipolar interaction simplyrenormalizes the easy-axis anisotropy Kfor out-of-plane mag- netization, and thus is neglected in following investigations. The total magnetization naturally divides into the static and dynamical parts: m(r,t)=m 0(r)+δm(r,t), where m0(r) represents the static magnetic texture, and δm(r,t)i st h e dynamical spin wave excitation. In spherical coordinate withe r≡m0(r) and the accompanying two transverse directions ˆeθ,φ, the spin wave is expressed as δm(r,t)=mθ(r,t)ˆeθ+ mφ(r,t)ˆeφ, or equivalently as a complex ﬁeld ψ(r,t)= mθ(r,t)−imφ(r,t). We deﬁne u0≡u[m0] as the energy den- sity due to the static background m0, andδu≡u[m]−u0as the energy density due to the spin wave excitation. For a homogeneous domain with its static background magnetization m0(r)=±ˆz,w eh a v e u0=0. A domain wall arises when two different domains meet, and hasﬁnite energy u0>0. Without loss of generality, we suppose that the domain wall magnetization varies along the xaxis, i.e.,m0(x) rotates continuously from −ˆzto+ˆzalong the x axis with m0(±∞)=±ˆz, and is translational invariant along they/zaxis. Due to the DMI, the magnetization inside the domain wall is enforced to rotate counterclockwisely alongthe advancing direction −ˆz→+ ˆy→+ ˆzalong the xaxis, as shown in Fig. 1(a). Upon this chiral domain wall, the spin wave dynamics is governed by a Schrödinger-like equation[22,37] i˙ψ=γ[A(−i∇+a) 2+K−φ]ψ, (2) where the vector potential [ 38,39]a=˜Dm0with ˜D=D/2A, and the scalar potential φ=2u0/(μ0Ms) is caused by the reduction of domain wall energy density u0due to spin wave excitation, which reduces the local magnetization m0→ m0√1−δm·δmwith the condition \|m\|=1. B. Semiclassical description To investigate the spin wave scattering behavior by a chiral domain wall, we construct a spin wave packet ψq[r(t)] with central position rand central momentum q. When spin wave wavelength is much shorter than the length scale of domainwall (short wavelength condition), the constructed spin wavepacket can be simultaneously localized in real and momentumspace, thus is treated as a pointlike object. Following the time-dependent variable principles [ 37,40,41] in the semiclassical approach proposed by Sundaram et al., the Lagrangian density corresponding to Eq. ( 2) reads L=k·˙r−a·˙r−ω, (3) where ω=γ(Ak 2+K−φ) is the local spin wave frequency with the canonical momentum k=q+a. Invoking the Euler- Lagrangian rule on Eq. ( 3), the dynamics of the spin wave 214407-2GEOMETRIC MAGNONICS WITH CHIRAL MAGNETIC … PHYSICAL REVIEW B 103, 214407 (2021) packet is then governed by mFM¨r=q0(−e−˙r×b), (4) where v≡˙r=∂kω=2γAkis the spin wave velocity, and mFM=¯hk/v=¯h/(2γA) is the effective mass for the spin wave in ferromagnets, and q0is the elementary charge. Here e=−(γ¯h/q0)∂xφˆxand b=(¯h/q0)∇×aare the ﬁctitious electromagnetic ﬁelds induced by inhomogeneous magnetictexture, i.e., the chiral domain wall here. The semiclassicalequation ( 4) is similar to the eikonal equations for the phase of propagation wave, which has been widely used in studyingthe trajectory of light and gravitational wave [ 42,43]. By denoting the domain wall proﬁle in spherical coordi- nate m 0(r)=[0,sinθ0(x),cosθ0(x)] with θ0the polar angle of magnetization m0with respect to ˆz, it is straightforward to ﬁnd that the ﬁctitious magnetic ﬁeld b=(¯h/q0)˜Dθ/prime 0(x)m0 always points opposite [because θ/prime 0<0; see Fig. 1(a)]t ot h e magnetization m0, and its strength is controlled by the mag- netization gradient θ/prime 0(x). Apparently, the projection of the magnetic ﬁeld on the x-z(y-z) plane is (anti-)symmetric about the domain wall center, i.e., byis negative in the whole region butbzis positive/negative in the left/right region, as depicted in Fig. 1(a). In the meantime, the scalar potential φinside the domain wall is a potential well, as illustrated by the gray-scalebackground centered at the domain wall in Fig. 1(a), which gives rise to an electric ﬁeld e xthat is antisymmetric about the domain wall. With the above knowledge that the domain wall manifests itself as ﬁctitious ﬁelds eandb, we may treat the spin wave scattering governed by Eq. ( 4) as a negatively charged particle deﬂected by these ﬁelds. The electrostatic (Lorentz) forceﬂips as electric (magnetic) ﬁeld reverses, thus the spin wavedeﬂection pattern depends on the symmetry of these ﬁcti-tious electromagnetic ﬁelds. Nevertheless, once the spin wavepacket moves away from the domain wall, it takes straighttrajectories, therefore simple geometric relations between in-cident and out-going spin wave beams are expected. For theoretical simplicity as well as experimental rele- vance, here we focus on two scenarios: (i) the normally magnetized thin ﬁlm case with the easy-axis anisotropy per- pendicular to the ﬁlm, corresponding to the ﬁlm plane beingx-yplane in Fig. 1(a); and (ii) the tangentially magnetized thin ﬁlm case with the easy-axis anisotropy lying in theﬁlm, corresponding to the ﬁlm plane being the y-zplane in Fig.1(a). More speciﬁcally, Figs. 1(b)and1(c)show the slice cut for these two scenarios. The perpendicular (to the ﬁlmplane) magnetic ﬁeld in normally/tangentially magnetizedﬁlms are b zand−by, respectively, which are antisymmetric and symmetric, while the electric ﬁeld exfor both cases are antisymmetric. For the special case of spin wave incidenting normally on the domain wall [Figs. 1(b) and1(c)], the spin wave experi- ences two qualitatively distinct fates across the domain wall:in the normally magnetized ﬁlm, the spin wave is shiftedlaterally [Fig. 1(b)], due to the opposite Lorentz forces in the left/right domain wall region caused by the antisymmetricmagnetic ﬁeld; while in the tangentially magnetized ﬁlm, thespin wave is bent upward [Fig. 1(c)], because of the symmetric magnetic ﬁeld.C. Numerical results To analyze the spin wave scattering problem more systematically, we turn to the numerical calculations.Here two types of numerical calculations are performedin parallel: the full scale micromagnetic simulation (seeAppendix A) based on the original LLG equation ( 1) and the trajectory simulation based on the semiclassicalequation ( 4). We assume that the domain wall takes the Walker proﬁle with m 0=[0,sech( x/W),tanh( x/W)] or θ0(x)= 2a r c t a n [ e x p ( −x/W)], where W=√A/Kis the characteris- tic domain wall width [ 22,33]. The effect of DMI is only to pin the domain wall as a Bloch type, and does not al-ter the proﬁle. Upon this magnetization proﬁle, the scalarpotential is φ(x)=(2K/μ 0Ms)sech2(x/W), which is a po- tential well since the magnetic energy density u0is larger inside the domain wall. This scalar potential φ(x) a spe- cial Pöschl-Teller type reﬂectionless potential [ 30,33,44,45], which always passes spin wave perfectly. The ﬁeld compo-nents that can inﬂuence the spin wave trajectories are themagnetic (electric) ﬁeld lying in the out-of-plane (in-plane) direction of the ﬁlm plane, which are calculated for the nor- mally magnetized and tangentially magnetized cases as in thefollowing: e NM=eTM=ex=−4γ¯hK q0Wsech2x Wtanhx W,(5a) bNM=bz=−¯hD 2q0AWsechx Wtanhx W,(5b) bTM=−by=¯hD 2q0AWsech2x W, (5c) where the correspondence between bNM/TM andby/zfollow the coordinate setting in Fig. 1. With the ﬁctitious electromagnetic ﬁelds eNM/TM and bNM/TM in Eq. ( 5), the spin wave trajectories calculated from Eq. ( 4) with different incident angles are overlaid with the micromagnetic simulation results, as shown in Fig. 2for the normally and tangentially magnetic ﬁlm cases. As expected,they agree well for all incident angles, and the out-going tra-jectory develops a lateral shift /Delta1rwith respect to the incident trajectory in normally magnetized ﬁlm, but forms an angledeﬂection /Delta1βin tangentially magnetized ﬁlm. In addition, in both normally magnetized and tangentially magnetized ﬁlms,when the incident angle exceeds a critical angle, the spin wavepacket is totally reﬂected by the domain wall. The reﬂectiononly occurs when spin wave incidents along the upward direc-tion, i.e., +ˆy(+ˆz) in normally/tangentially magnetized ﬁlm, highlighting the chiral nature of the underlying Lorentz force.These trajectory shifting or bending behaviors can be mostlyunderstood from the magnetic ﬁeld distributions. However,the effective electric ﬁeld also contributes in manipulating thespin wave trajectory, which is shown as the difference betweenthe solid and dashed trajectories for including and excludingthe effect of the electric ﬁeld in the main panels of Figs. 2(a) and2(b). For most incident angles, the electrostatic force is dominated by the Lorentz force due to the large spin wavevelocity, but its contribution becomes non-negligible aroundthe total reﬂection situation. Nevertheless, due to the attractive 214407-3JIN LAN, WEICHAO YU, AND JIANG XIAO PHYSICAL REVIEW B 103, 214407 (2021) FIG. 2. Numerical simulations of spin wave scattering by a chiral domain wall in (a) normally magnetized ﬁlm and (b) tangentially magnetized ﬁlm. In the left panel, each line depicts a spin wave trajectory calculated from semiclassical equation ( 4) starting from the same source point but with a speciﬁc incident angle, with solid/dashed lines denoting trajectories including/excluding the electric ﬁeld. The green/orange colors encode the positive/negative magnetic ﬁelds, and the arrows in the upper region denote the domain wall magnetizations. In the right panels, three typical spin wave trajectories extracted from micromagnetic simulations are plotted in orange, and the semiclassicaltrajectories are plotted in blue/red lines as in the left panel. The Gaussian spin wave beam is prepared in the gray antenna region with a discrepancy between beam direction and antenna direction in (b) (see Appendix A), and the spin wave trajectories are extracted based on spin wave ﬂux (see Appendix B). The insets in (a) and (b) plot the lateral shift /Delta1rand angle deﬂection /Delta1βas a function of incident angle β, respectively, and the gray area denotes the total reﬂection range. In (a) inset, the gray/black dots are for the lateral shift with/without electric ﬁeld; and in (b) inset, the solid line is the theoretical angle deﬂection /Delta1β=180−2βfor the total reﬂection case. The spin wave leaking in the upper right in (a) is due to the subwave spreading in different angles for spin wave generated in the antenna. For all numericalcalculations and micromagnetic simulations, the spin wave frequency is f=40 GHz, and the magnetic parameters are for yttrium iron garnet Y 3Fe5O12(YIG) [ 22,30,33,45]: the exchange coupling constant A=3.28×10−11A m, the anisotropy K=3.88×105A/m, the DMI constant D=3×10−3A, and the damping constant α=1×10−4. electrostatic force to the domain wall center, spin wave gains a higher longitudinal velocity and less traveling time withindomain wall, and thus experiences less deﬂection. The lateral shift /Delta1rin normally magnetized ﬁlm and the angle deﬂection /Delta1β in tangentially magnetized ﬁlm, as a function of incident angle β, are summarized in the inset of Figs. 2(a) and2(b), respectively. Typically, as spin wave deviates from the normal incident direction of the domainwall, the velocity v xdecreases and the passing time increases, thus both the lateral shift /Delta1rand angle deﬂection /Delta1βin- crease. However, these two geometric quantities /Delta1rand/Delta1β are both asymmetric with respect to the incident angle β, since the spin wave is subject to chiral Lorentz force inside thedomain wall. In Fig. 2(a) inset, the lateral shift /Delta1rinclud- ing/excluding electric ﬁeld shows a discrepancy, highlightingthe role of electrostatic force in developing the lateral shift.And in Fig. 2(b) inset, the angle deﬂection /Delta1βmaximizes for a certain positive incident angle and starts to decrease linearly,indicating the emergence of the total reﬂection of the spinwave beam.D. Magnonic Snell’s law For a straight domain wall under consideration in this work, because of the translational invariance along the y/z axes, the wave vector qy/zis conserved. Note that the canon- ical wave vector k=q+a, the angle βformed between the spin wave beam and the normal direction of the domain wallobeys the following generalized magnonic Snell’s law: normally magnetized: ksinβ−˜Dmy 0=const,(6a) tangentially magnetized: ksinβ−˜Dmz 0=const,(6b) where the in-plane magnetization component ( my/z 0for normally/tangentially magnetized ﬁlm) plays the role of gen-eralized refraction index characterizing the magnetic medium.The magnonic Snell’s law in Eq. ( 6) holds everywhere inside the continuum medium, thus is an extension of the previouslyproposed Snell’s laws that only concerns two sides of aninterface [ 33,34,46,47]. The magnonic Snell’s law formulated in Eq. ( 6)i s schematically illustrated by the matching of corresponding 214407-4GEOMETRIC MAGNONICS WITH CHIRAL MAGNETIC … PHYSICAL REVIEW B 103, 214407 (2021) (a) (b) FIG. 3. Schematics of magnonic Snell’s law across chiral domain wall in (a) normally magnetized ﬁlm and (b) tangentially magnetizedﬁlm. The isofrequency circles in the wave-vector space ( q x,qy/z)a r e plotted at the left/right domains and the domain wall center, respec- tively. The blue/red line plots the proﬁle of in-plane magnetizationm y/z 0, which acts as the generalized refraction index. The black arrow denotes the local momentum vector k, which forms angle βwith the xaxis, and the evolutions of angle βare connected by dashed lines. The magenta arcs describe the modes with/without corresponding propagation modes in other regions. The dotted arrow represents the momentum kof spin wave generated in the other side of the antenna. In upper region, the magnetization proﬁle is depicted by arrows, with the in-plane component highlighted by blue/red colors. isofrequency circles, as depicted in Fig. 3. Three represen- tative positions are the focus: the left/right domain withm 0=∓ˆzand the domain wall center m0=+ ˆy. For each isofrequency circle, the center is shifted to q=− a=− ˜Dm0, and the radius is k(x)=√(ω/γ)−K+φ(x). Speciﬁcally for the normally magnetized case in Fig. 3(a), the in-plane magnetization my 0maximizes at the domain wall center and vanishes in left/right domains, therefore the domain wallmimics a three-layer system with low/high/low refraction in-dices. Consequently, the spin wave experiences a lateral shift,similar to the lateral shift of light ray as passing throughan air/glass/air structure. As for the tangentially magnetizedcases in Fig. 3(b), the in-plane magnetization m z 0monoton- ically decreases along the xdirection, therefore the domain wall mimics a two-layer structure with low/high refractionindices, giving rise to the spin wave refraction, similar to thecase of light refraction in an air/water interface. And sincethere is an interface of effectively low/high refraction indicesfor both normally and tangentially magnetized cases, the spinwave total reﬂection arises due to the lack of a corresponding propagation mode in the other regions. III. SPIN WA VE DEFLECTIONS BY ANTIFERROMAGNETIC DOMAIN WALL The spin wave deﬂections in ferromagnetic environ- ment discussed above naturally extend to antiferromagnets,and their features are enriched by the additional polar-ization degree of freedom. In antiferromagnets, due totwo sublattices with opposite magnetizations, there existsboth left/right circular polarization modes for spin wave[8,30,48–50]. Since these two circular modes precess in op- posite fashion, they experience opposite ﬁctitious magneticﬁelds induced by DMI, and thus are deﬂected in oppositedirections. Here we denote the magnetization in two sublattices of antiferromagnets as m 1/2; then the staggered magnetization is n=(m1−m2)/\|m1−m2\|, and the net magnetization is m= m1+m2. Under the approximation n·m=0, the magnetic dynamics in antiferromagnets is governed by an LLG-likeequation [ 45,51–54] 1 γJn×¨n=−γn×h+αn×˙n, (7) where h=A∇2n+Knzˆz−D∇×nis the effective ﬁeld tak- ing similar form as in Eq. ( 1), and Jis the intersublattice exchange coupling constant. And similarly, the total magneti-zation ndivides into the static background n 0and the dynam- ical antiferromagnetic spin wave excitation δn:n=n0+δn, withδn=nθˆeθ+nφˆeφ. The domain wall has the same mag- netization proﬁle as in the ferromagnetic case [ 45,54], and for spin wave dynamics, Eq. ( 7) is reduced to a Klein-Gordon-like equation [ 39,45] −¨ψs=γ2J[A(−i∇+sa)2+K−φ]ψs, (8) where ψs=nθ−isnφdenotes the left/right circularly polar- ized spin wave with s=∓1 the chirality, and potentials φ andafollow deﬁnitions in Eq. ( 2). Following similar proce- dures as in Eq. ( 4), the spin wave dynamics is recast from Eq. ( 8)t o mAFM¨r=q0(−e−˙r×sb), (9) where the right/left circular spin waves take analogy to charged particles traveling in the same electric ﬁeld e= −(γ¯h/q0)(γJ/2ω)∇φand opposite magnetic ﬁelds ∓b with b=(¯h/q0)∇×a. The second scaling factor in the electric ﬁeld eis because the real potential in the Klein- Gordon equation corresponds to a dispersion shift. Weshould note that only the effective magnetic ﬁeld changessign for the two polarizations; the effective electric ﬁeld isthe same, therefore the two circular polarizations do not cor-respond to the positive/negative charges. Here the spin wave dispersion in homogeneous domains is ω=γ/radicalbig J(K+Ak2), the group velocity is v=∂kω=γ2JAk/ω, and the effective mass is mAFM=¯hk/v=¯hω/(γ2JA). The polarization-dependent trajectories calculated from the semiclassical equation (9) and simulated based on micromag-netics are depicted in Fig. 4, and they agree well with each other as expected. In the normally magnetized case [Fig. 4(a)], 214407-5JIN LAN, WEICHAO YU, AND JIANG XIAO PHYSICAL REVIEW B 103, 214407 (2021) FIG. 4. Numerical simulations of spin wave scattering by an antiferromagnetic domain wall in (a) normally magnetized ﬁlm and (b) tangentially magnetized ﬁlm. In each main panel, the green/orange/purple color plots the trajectory of left/right circular and linear spin wave extracted from micromagnetic simulations, and the blue/red/black lines are corresponding trajectories calculated from semiclassical equation(9). The lower two panels plot the trajectories of left/right circular spin wave separately. A linearly polarized oscillating magnetic ﬁeld is exerted in the antenna region (gray rectangle) at the domain wall center to generate spin wave. The spin wave trajectory with polarization information is based on the extraction of spin wave ﬂux (see Appendix B). In all numerical simulations, the spin wave frequency is f=50 GHz, and the magnetic parameters are as follows: the exchange coupling constant A=3.28×10 −11A m, the anisotropy K=3.88×105A/m, the DMI constant D=2×10−3A, the interlayer coupling constant J=1×106A/m, and the damping constant α=1×10−4. The DMI constant D here is slightly lower than in the FM case for better stability. a linearly polarized spin wave beam is injected by the antenna. For a large incident angle, the left/right circular modes beamsexperience opposite lateral shift and split into two parallelbeams, causing a double refraction. For small incident angles,one of the polarizations would bend so much that it is totallyreﬂected by the domain wall, while the other polarization stillexperience a lateral shift and penetrates into the other domain.In the tangentially magnetized case [Fig. 4(a)], the left/right circular polarizations of the same frequency do not have thesame wave-vector direction, thus they split as soon as theyleave the antenna. As they hit the domain wall, they are bentin opposite directions due to the opposite effective magneticﬁelds in the domain wall region, and total reﬂection can alsohappen for one of the polarizations if the incident angle issmaller than a critical angle. All these polarization-dependentscattering patterns shown in Fig. 4can be straightforwardly understood by the effective electromagnetic ﬁelds as in theferromagnetic case in Fig. 2, or by extending magnonic Snell’s law to antiferromagnetic environment, by using ∓n y/z 0as the the generalized refraction indices for left/right circular modes. The spin wave birefringence phenomenon observed in Fig. 4refers to the polarization-dependent trajectories in two-dimensional magnetic ﬁlm, which is different fromthe polarization-dependent phase demonstrated in one-dimensional magnetic wire in previous reports [ 30,48,55]. Recently, the bireﬂection of spin wave induced by the hy-bridization with elastic wave is also reported, where the ﬁlmboundary rather than a domain wall serves as the scatteringinterface [ 56].IV . SPIN WA VE CONSTRICTION BY DOMAIN WALL We have seen that the normally magnetized case can be considered as an analogy of air/glass/air for light (see Fig. 3), where the domain wall serves as the middle high refractionindex “glass” layer. It is known that light can be conﬁned inthe glass and travel along the glass layer without leaking intothe air because of the total internal reﬂection, as widely usedin optical ﬁber. In the normally magnetized ﬁlm, a domainwall can also be used to guide spin waves just as glass guidinglight. Figure 5shows exactly this phenomena in normally magnetized ferromagnetic and antiferromagnetic ﬁlms. In theferromagnetic case, when the spin wave is excited within thedomain wall with a shallow incident angle, the downward-going spin wave is constricted within the domain wall with asnakelike trajectory, while the upward-going spin wave leaksinto the bulk domains. Therefore, this spin wave constrictionis unidirectional. This constriction is due to the opposite effectLorentz force due to the opposite effective magnetic ﬁeldsat the two sides of the domain wall [see the main ﬁgure inFig.2(a)]. The AFM case is quite similar, but the constriction depends on the spin wave polarization. This unidirectionalconstriction can be also understood using the isofrequencycircle mismatching in Fig. 3(a). This unidirectional constricted spin wave mode is different from the spin wave bound state found within the domain wall[22,24–27]. There are two major differences: (i) the spin wave bound state previously discussed has frequency lower thanthe bulk spin wave gap in the domains, while the constrictedspin wave mode discussed above has frequency above the 214407-6GEOMETRIC MAGNONICS WITH CHIRAL MAGNETIC … PHYSICAL REVIEW B 103, 214407 (2021) FIG. 5. Constriction of spin wave by a chiral domain wall in nor- mally magnetized ﬁlm in (a) ferromagnets and (b) antiferromagnets. The green/orange color plots the trajectory of left/right circular and linear spin wave extracted from micromagnetic simulations, and theblue/red lines are corresponding trajectories calculated from semi- classical equation ( 9), and all other settings follow Fig. 4.T h es p i n wave leaking in the upper right in (a) is due to the subwave spreadingin different angles for spin wave generated in the antenna, and similar leaking also occurs in (b). spin wave gap; (ii) the spin wave bound state exists because the scalar potential φis a potential well (or the asymmetric effective electric ﬁeld) and regardless of the vector potentialor the effective magnetic ﬁeld, while the constricted spin wavemode here exists only because of the Lorentz force caused bythe asymmetric effective magnetic ﬁeld. In this work, the spin wave ﬁber is based on a single do- main wall in normally magnetized ﬁlm, thus is also differentfrom the previously reported spin wave ﬁber relying on twoparallel magnetic domain walls by present authors [ 33]. Xing et al. [57] also reported the ﬁberlike spin wave propagation behavior within a chiral domain wall, but their results arebased on single mode spin wave without demonstration of thetotal internal reﬂection. V . CONCLUSIONS In conclusion, we demonstrate that the spin wave scatter- ing by a chiral domain wall can be simpliﬁed to geometricrelations between incident and out-going beams. Underlyingthese geometric scattering behaviors is the deﬂection of spinwave by a ﬁctitious electromagnetic ﬁeld induced by domainwall, where the deﬂection chirality is a collaboration of thechirality of DMI, domain wall, and spin wave. The geometricmagnonics demonstrated in this work offers us new designingprinciples in controlling spin wave propagation and separatingspin waves of opposite chiralities. ACKNOWLEDGMENTS J.L. is grateful to X. Feng for discussions about the eikonal equations. This work is supported by National Natural ScienceFoundation of China (Grant No. 11904260), Natural ScienceFoundation of Tianjin (Grant No. 20JCQNJC02020), and theStartup Fund of Tianjin University. W.C. is supported bythe JSPS Kakenhi (Grant No. 20K14369). J.X. is supportedby the Science and Technology Commission of ShanghaiMunicipality (Grant No. 20JC1415900) and Shanghai Mu- nicipal Science and Technology Major Project (Grant No.2019SHZDZX01). APPENDIX A: MICROMAGNETIC SIMULATIONS The micromagnetic simulations are performed in COMSOL MULTIPHYSICS , where the LLG equation is transformed into a weak form and then solved by the generalized-alpha method[58]. In each simulation, a domain wall is placed at the mag- netic ﬁlm center, and a spin wave beam is prepared in therectangle antenna region and incident on the domain wall.Near the ﬁlm boundaries, the damping constant αis gradually increased from 1 ×10 −4to 2×10−1in 50 nm to absorb undesired spin wave. To generate spin wave beam, the excitation magnetic ﬁeld is set to take the Gaussian form [ 33,35,59] hex=h0cos(2πft)e x p/parenleftbigg(x/prime−x/prime c)2 2λ2/parenrightbigg /Theta1/parenleftbiggwa 2−\|x/prime−x/prime c\|/parenrightbigg /Theta1/parenleftbiggha 2−\|y/prime−y/prime c\|/parenrightbigg , (A1) where /Theta1(x) is the Heaviside step function, h0and fare the strength and frequency of the excitation magnetic ﬁeld, andλis the Gaussian distribution width. Here x /primeandy/primeare the positions along the width/height direction of the antenna, x/prime c andy/prime care the central positions of the antenna, and waand haare the width/height of the antenna. For micromagnetic simulations in this work, the antenna size is set to wa= 250 nm, ha=15 nm, and the Gaussian distribution width is λ=60 nm. We denote the velocity angle βas the angle between the propagation direction of the spin wave beam and the xaxis, and antenna angle β/primeas the angle between the normal direc- tion of antenna and the xaxis. The velocity angle βis then related to velocity v(or canonical momentum k), and the antenna angle β/primeis related to the wave vector q, with β=arccosvx x,β/prime=arccosqx q. (A2) Due to the vector potential aexperienced by the spin wave packet, or the relation k=q+a, the velocity angle βand antenna angle β/primeare not necessarily the same. More explicitly, the vector potential avanishes (maintains) in the uniformed domains in normally/tangentially magnetized ﬁlms, thus thevelocity angle βequals to (deviates from) the antenna angle β /prime, i.e.,β=β/primein the normally magnetized case while β/negationslash=β/prime in the tangentially magnetized case. The antiferromagnetic simulations are performed in a syn- thetic antiferromagnetic ﬁlm consisting of two ferromagneticlayers that are coupled antiferromagnetically [ 45]. Denoting m 1/2as the magnetization in the upper/lower magnetic layer, the magnetic dynamics is governed by coupled LLG equations ˙mi=−γmi×hi+αmi×˙mi, (A3) where hi=A∇2mi+Kmz i−Jm¯i/2 is the effective ﬁelds acting on miwith ¯1=2,¯2=1. Deﬁning staggered mag- netization n=(m1−m2)/\|m1−m2\|, net magnetization 214407-7JIN LAN, WEICHAO YU, AND JIANG XIAO PHYSICAL REVIEW B 103, 214407 (2021) m=(m1+m2), and using the approximation n·m=0, Eq. ( A3) is then recast to Eq. ( 7). APPENDIX B: TRAJECTORY TRACKING OF SPIN WA VE BEAM To visualize the trajectory with polarization information of the spin wave beam, we deﬁne the local spin wave ﬂux j(r,t)=m0(r)·[˙m(r,t)×m(r,t)], (B1) where m0is the static magnetic background at t=0, and m is the total magnetization at the time tunder consideration. The spin wave ﬂux jis only nonzero when local magnetiza- tion precesses, and its sign is directly determined by chiralitydenoting the precession direction. For right circular spin wavein ferromagnets, the corresponding ﬂux jis always negative, as shown in Figs. 2and5(a). In antiferromagnets, the polarized spin waves generally have both left/right circular polarization components, andtheir mixture complicated the trajectory analysis. 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PhysRevB.73.212501.pdf	Voltage dependence of Landau-Lifshitz-Gilbert damping of spin in a current-driven tunnel junction Hosho Katsura,1,Alexander V . Balatsky,2,†Zohar Nussinov,3,‡and Naoto Nagaosa1,4,5 1Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 3Department of Physics, Washington University, St. Louis, Missouri 63160, USA 4CERC, AIST Tsukuba Central 4, Tsukuba 305-8562, Japan 5CREST, Japan Science and Technology Agency (JST), Saitama 332-0012, Japan /H20849Received 25 January 2006; revised manuscript received 4 May 2006; published 5 June 2006 /H20850 We present a theory of Landau-Lifshitz-Gilbert damping /H9251for a localized spin Sin the junction coupled to the conduction electrons in both leads under an applied voltage V. We ﬁnd the voltage dependence of the damping term reﬂecting the energy dependence of the density of states. This effect is linear in the voltage andcontrolled by particle-hole asymmetry of the leads. DOI: 10.1103/PhysRevB.73.212501 PACS number /H20849s/H20850: 75.80. /H11001q, 71.70.Ej, 77.80. /H11002e Spintronics is an emerging subﬁeld that holds the poten- tial to replace conventional electronic devices with spin ana-logues where the manipulation, control, and readout of spinswill enable functionality with no or little electronic chargedynamics. 1In order to realize this promise, the spin dynam- ics of the small scale devices needs to be well controlled.One of the most pressing questions concerns a set up whichwould preserve coherence and allow a manipulation of spins.In most systems, the relevant spin degrees of freedom arecoupled to some bath, such as a fermionic bath of electrons.The detailed dynamics of single spins when in contact withsuch a bath plays a pivotal role in addressing decoherence inpotential spintronic systems. The conventional way to treat this problem is via a Caldeira-Leggett approach where the external bath is mod-eled by collective excitations which are capable of destroy-ing coherent spin dynamics. Often, spin dynamics is de-scribed by a Landau-Lifshitz-Gilbert equation 2,3 dS/H20849t/H20850 dt=−S/H20849t/H20850/H11003h−/H9251S/H20849t/H20850/H11003dS/H20849t/H20850 dt, /H208491/H20850 where his, up to constant prefactors, the external magnetic ﬁeld and the coefﬁcient /H9251captures the damping due to the external bath. A schematic view of the solution to thisequation 4is shown in Fig. 1. There are standard methods to calculate /H9251in an equilibrium situation when, say, one con- siders a spin in a Fermi liquid.5,6 In this paper, we address a related question concerning the effect of an applied voltage bias on the Gilbert coefﬁcient /H9251. Our work complements the recent results of Ref. 7 whereinthe effects of the “retarded” electronic contributions in theequations of motion for a system of spins were studied. Bothsuch retarded correlations, 8as well as additional “Keldysh” correlations, generally manifest themselves in the single spinequations of motion see, e.g., Ref. 9 for general spin equa-tions of motion entailing the effects of both correlations. Inthe current work, we examine the voltage dependence ofGilbert damping. For the sake of clarity, we depart from theKeldysh contour formalism of Refs. 7–9, and use a Caldeira-Leggett approach.In what follows, we consider the case of a junction be- tween two electrodes that contains one spin S, see Fig. 2. This spin Smay be the spin of a single magnetic impurity or it may portray the spin of a cluster at low temperature whenthe spins in the cluster are locked. Upon applying a ﬁnitebias between the electrodes of Fig. 2, a current ﬂow is gen-erated. Thereafter, at long times, the system is at a steady butnonequilibrium state so long as the voltage bias Vis applied. We will focus on the voltage dependence of the dampingterm /H9251/H20849V/H20850in Eq. /H208491/H20850. We ﬁnd that the change in the density of states associated with the chemical potential gradient across the junction triggers a modiﬁcation to the damping /H9251/H20849V/H20850that is linear in voltage and is proportional to the particle-hole asymmetry of the density of states. The scale of the correction is set by the Fermi energy of the metal in theleads E Fand by particle-hole asymmetry in the density of states /H9251/H20849V/H20850=/H92510+/H92511/H20849V/H20850=/H92510/H208511+O/H208491/H20850eV/EF/H20852. /H208492/H20850 This result vividly illustrates the presence of voltage induced damping in such junctions. Spin unpolarized electrons tun-neling across the junction interact via exchange interactionwith the spin Sand produce random magnetic ﬁelds that disorder the local spin. This noise augments that already FIG. 1. Sketch of the dissipative spin dynamics. /H20849a/H20850depicts the Larmor precession of the spin about the direction of an appliedmagnetic ﬁeld /H20849B/H20850.I n /H20849b/H20850, the spin dynamics in the presence of Landau-Lifshitz-Gilbert damping is shown.PHYSICAL REVIEW B 73, 212501 /H208492006 /H20850 1098-0121/2006/73 /H2084921/H20850/212501 /H208494/H20850 ©2006 The American Physical Society 212501-1present equilibrium magnetic noise in a Fermi liquid bath. Such a behavior of /H9251/H20849V/H20850with the external voltage is in line with the works of Ref. 10. An analysis of a related single spin problem in a Josephson junction /H20849instead of the normal junction studied here /H20850was advanced in Refs. 8, 9, and 11. We will shortly derive the effective single spin action from which the principle equation of motion of Eq. /H208491/H20850fol- lows. The physical system under consideration is illustratedin Fig. 2. It consists of two /H20851left/H20849L/H20850and right /H20849R/H20850/H20852electrodes across which a voltage bias is applied; a magnetic impurity/H20849S/H20850is situated in between /H20849or lies on one of /H20850the electrodes. An external magnetic ﬁeld Bis present. In the absence of effects stemming from conduction electrons in the tunnelingbarrier, the single spin would precess at the Larmor preces-sion frequency about the applied ﬁeld direction /H20851see, e.g., Fig. 1 /H20849a/H20850/H20852. With the external circuit elements present, the spin motion becomes dissipative /H20851as schematically shown in Fig. 1/H20849b/H20850/H20852. With the spin embedded in the tunneling barrier, the work function is modiﬁed and the conduction electron tunnelingmatrix element is supplanted by a Kondo like exchange termJ/H20849S· /H9268c/H20850, with/H9268cdenoting the conduction electron spin.12In what follows, we will dispense with the csubscript. The Hamiltonian governing this system is given by H=He+Hs+HT, /H208493/H20850 He=/H20858 /H9251,k,/H9268/H9264/H9251kc/H9251k/H9268†c/H9251k/H9268, Hs=−h·S/H20849t/H20850, HT=1 /H9024/H20858 /H9251,k,/H9268/H20858 /H9252,p,/H9268/H11032c/H9251k/H9268†/H20849T/H9251/H9252/H20850/H9268/H9268/H11032c/H9252p/H9268/H11032, /H208494/H20850 where c/H9251k/H9268†/H20849c/H9251k/H9268/H20850creates /H20849annihilates /H20850an electron with mo- mentum kand spin /H9268/H33528/H20853↑,↓/H20854in the lead /H9251/H33528/H20853L,R /H20854. The abbreviation /H9264/H9251k=/H9280/H9251k−/H9262/H9251, where /H9280/H9251kis the energy of elec- tron with momentum kin/H20849the lead /H20850/H9251and/H9262/H9251is the chemical potential in /H20849the lead /H20850/H9251. The second term in Eq. /H208493/H20850,Hs,i s the Zeeman energy of the spin in an external magnetic ﬁeldB. Here, h/H11013g /H9262BBwith gis the gyromagnetic ratio and /H9262Bis the Bohr magneton. The last term in Eq. /H208493/H20850,HT, represents both the Kondo coupling and direct tunneling process, wherethe amplitudes /H20853T /H9251/H9252/H20854=/H20853TLL,TLR,TRL,TRR/H20854are the tunneling matrix elements and their explicit forms areTLL=JLL„/H9268·S/H20849t/H20850…, TRR=JRR„/H9268·S/H20849t/H20850…, TLR=TRL=/H20851T0/H9254/H9268/H9268/H11032+JLR„/H9268·S/H20849t/H20850…/H20852. /H208495/H20850 Here, T0is the direct tunneling matrix element and J/H9251/H9252is the Kondo coupling, while /H9024denotes the volume of each lead /H20849assumed, for simplicity, to be the same /H20850. Typically, from the expansion of the work function for tunneling, JLR/T0/H11011J/U, where Uis the height of a spin-independent tunneling barrier andJthe magnitude of the spin exchange interaction.13Typi- cal values of the ratio between the spin dependent and spinindependent tunneling amplitudes in Eq. /H208495/H20850,J /H9251/H9252/T0, are O/H2084910−1/H20850, with a typical Fermi energy EFRof the order of sev- eral electron volts. From the Hermiticity of the Hamiltonian, we can ﬁnd that the matrix element /H20849T/H9251/H9252/H20850/H9268/H9268/H11032satisﬁes /H20851/H20849T/H9251/H9252/H20850/H9268/H9268/H11032/H20852=/H20849T/H9252/H9251/H20850/H9268/H11032/H9268. In the following, we derive the effective action for the single impurity spin via an imaginary time path integral for-malism. The full action is given by S=/H20885 0/H9252 d/H9270/H20858 /H9251k/H9268c/H9251k/H9268†/H11509/H9270c/H9251k/H9268+iS/H9275„S/H20849/H9270/H20850…+/H20885 0/H9252 d/H9270H/H20849/H9270/H20850,/H208496/H20850 where the second, Wess-Zumino-Novikov-Witten, term in Eq. /H208496/H20850depicts the Berry phase accumulated by the spin.14In our action, we have the following quadratic form of fermi-ons: /H20885 0/H9252 d/H92701 /H9024/H20858 /H9251k/H9268/H20858 /H9252p/H9268/H11032c/H9251k/H9268†/H20851/H9254/H9251/H9252/H9254/H9268/H9268/H11032/H20849/H11509/H9270+/H9264/H9251k/H20850+/H20849T/H9251/H9252/H20850/H9268/H9268/H11032/H20852c/H9252p/H9268/H11032 /H11013/H20885 0/H9252 d/H92701 /H9024/H20858 /H9251k/H9268/H20858 /H9252p/H9268/H11032c/H9251k/H9268†/H20851„M/H9251/H9252/H20849/H9270/H20850…/H9268/H9268/H11032/H20852kpc/H9252p/H9268/H11032. /H208497/H20850 We may integrate over the lead electrons to obtain the effec- tive action for the spin Seff„S/H20849/H9270/H20850…/H11011iS/H9275„S/H20849/H9270/H20850…+/H20885 0/H9252 d/H9270h·S/H20849/H9270/H20850− ln det M./H208498/H20850 From the third term in Eq. /H208498/H20850, we can extract a quadratic form of spins with the aid of the well-known identityln det M=Trln M. In order to tabulate the expansion of Tr ln Mperturbatively, we deﬁne matrices M 0andM1. MKP=/H20849M0/H20850KP+/H20849M1/H20850KP, /H20849M0/H20850KP/H11013/H20851 /H20849−i/H9275+/H9264/H9251k/H20850/H9254/H9251/H9252/H9254/H9268/H9268/H11032/H20852/H9254KP, /H20849M1/H20850KP/H11013/H208491//H20881/H9252/H20850/H20851T/H9251/H9252/H20849/H9275−/H9275/H11032/H20850/H20852/H9268/H9268/H11032, where K/H11013/H20849k,/H9275/H20850andP/H11013/H20849p,/H9275/H11032/H20850with fermionic Matsubara frequencies /H9275and/H9275/H11032. Employing the expansion ln /H208491+x/H20850 =−/H20858n=1/H11009/H20849−x/H20850n/n, we can write the effective action as FIG. 2. Magnetic impurity coupled to two electrodes. /H9262Land/H9262R denotes the chemical potentials of the left and right leads, respec- tively. The voltage drop across the two electrodes is eV=/H9262L−/H9262R.BRIEF REPORTS PHYSICAL REVIEW B 73, 212501 /H208492006 /H20850 212501-2Seff„S/H20849/H9270/H20850…/H11011S0−T rl n M0−T r /H20849M0−1M1/H20850+1 2Tr/H20849M0−1M1/H208502 +¯, /H208499/H20850 where S0is the sum of the ﬁrst and the second term in Eq. /H208498/H20850. The third term in Eq. /H208498/H20850/H20851and consequently the last term shown in Eq. /H208499/H20850/H20852is the ﬁrst nontrivial contribution to the spin equation of motion. Its evaluation is straightforward, Tr/H20849M0−1M1/H208502=/H20858/H20849M0−1/H20850K1K1/H20849M1/H20850K1K2/H20849M0−1/H20850K2K2/H20849M1/H20850K2K1 =1 2/H9252/H20858/H20849−i/H9275+/H9264/H9251k/H20850−1/H20851T/H9251/H9252/H20849/H9275−/H9275/H11032/H20850/H20852/H9268/H9268/H11032 /H11003/H20849−i/H9275/H11032+/H9264/H9252p/H20850−1/H20851T/H9252/H9251/H20849/H9275/H11032−/H9275/H20850/H20852/H9268/H11032/H9268, where repeated indices are implicitly summed over. Then, we ﬁnd /H9004S/H20849S/H20849/H9270/H20850/H20850=1 2Tr/H20849M0−1M1/H208502 =−2/H20885d/H9270/H20885d/H9270/H11032S/H20849/H9270/H20850·S/H20849/H9270/H11032/H20850K/H20849/H9270−/H9270/H11032/H20850,/H2084910/H20850 where K/H20849/H9270/H20850=/H20858 /H9251,/H9252/H33528/H20853L,R /H20854K/H9251/H9252/H20849/H9270/H20850, K/H9251/H9252=/H20858 k/H33528/H9251/H20858 p/H33528/H9252J/H9251/H9252J/H9252/H9251 21 /H9252/H20858 /H9275mf/H20849/H9264k/H20850−f/H20849/H9264p/H20850 i/H9275m+/H9264k−/H9264pe−i/H9275m/H9270, /H2084911/H20850 with f/H20849/H9264/H20850denotes the Fermi distribution function. We should note here that we focus on the semiclassical dynamics of the single spin, i.e., large spin Slimit, and hence we neglect the higher order terms in Eq. /H208499/H20850. The effective action /H9004Sof Eq. /H208498/H20850can be decomposed into two /H20849trivial and nontrivial /H20850com- ponents as /H9004S=/H9004Sloc+/H9004Sdis, with /H9004Sloc=−2 K/H20849/H9275=0/H20850/H20885d/H9270„S/H20849/H9270/H20850…2, /H9004Sdis=/H20885d/H9270/H20885d/H9270/H11032„S/H20849/H9270/H20850−S/H20849/H9270/H20850…2K/H20849/H9270−/H9270/H11032/H20850. /H2084912/H20850 Here, K/H20849/H9275=0/H20850is the zero-frequency Fourier component of K/H20849/H9270/H20850. The ﬁrst term /H20849/H9004Sloc/H20850is a trivial constant as S/H20849/H9270/H208502=S2. The nonlocal part /H20849/H9004Sdis/H20850represents the dissipative effect due to the coupling between S/H20849/H9270/H20850and electrons bath. Carrying out the Matsubara sum in Eq. /H2084911/H20850as a contour integral,17the integral kernel K/H20849/H9270/H20850is calculated as K/H20849/H9270/H20850=/H20858 /H9251/H9252J/H9251/H9252J/H9252/H9251 2/H20858 k/H33528/H9251/H20858 p/H33528/H9252/H20886dz 2/H9266i/H20873e−z/H9270 e/H9252z−1/H9258/H20849−/H9270/H20850 +e−z/H9270 1−e−/H9252z/H9258/H20849/H9270/H20850/H20874f/H20849/H9264k/H20850−f/H20849/H9264p/H20850 z+/H9264k−/H9264p =P/H20885 0/H11009 d/H9275/H20873−J/H9251/H9252J/H9252/H9251 2/H11003/H20858 k/H33528/H9251/H20858 p/H33528/H9252/H20851f/H20849/H9264k/H20850−f/H20849/H9264p/H20850/H20852/H9254/H20849/H9275+/H9264k−/H9264p/H20850/H20874 /H11003cosh /H20849/H9252/2 − /H20841/H9270/H20841/H20850/H9275 sinh /H20849/H9252/H9275/2/H20850 /H11013/H20885 0/H11009 d/H9275J/H20849/H9275/H20850cosh /H20849/H9252/2 − /H20841/H9270/H20841/H20850/H9275 sinh /H20849/H9252/H9275/2/H20850, /H2084913/H20850 where J/H20849/H9275/H20850denotes the spectral density in the Caldeira- Leggett theory.15,16The symbol Pin Eq. /H2084913/H20850denotes the principal part of the integral. The spectral density of Eq./H2084913/H20850,J/H20849 /H9275/H20850, is estimated as J/H20849/H9275/H20850=/H20858 /H9251/H9252J/H9251/H9252J/H9252/H9251 2/H20885 −EF/H9251/H11009 d/H9264/H9251N/H20849/H9264/H9251/H20850/H20885 −EF/H9252/H11009 d/H9264/H9252N/H20849/H9264/H9252/H20850 /H11003/H20851f/H20849/H9264/H9251/H20850−f/H20849/H9264/H9252/H20850/H20852/H9254/H20849/H9275+/H9264/H9251−/H9264/H9252/H20850 /H11011/H20858 /H9251/H9252J/H9251/H9252J/H9252/H9251 2N/H20849/H9264/H9251=0/H20850N/H20849/H9264/H9252=0/H20850/H9275 =/H20858 /H9251/H9252J/H9251/H9252J/H9252/H9251 2D/H9251/H20849EF/H9251/H20850D/H9252/H20849EF/H9252/H20850/H9275, /H2084914/H20850 where D/H9251/H20849EF/H9251/H20850denotes the density of states at the Fermi en- ergy level EF/H9251of the lead /H9251. It is obvious that J/H20849/H9275/H20850in Eq. /H2084914/H20850 is proportional to /H9275, i.e., J/H20849/H9275/H20850is Ohmic. If the spectral den- sity is expressed as J/H20849/H9275/H20850=/H9257/H9275/2/H9266, from Eq. /H2084913/H20850, the nonlo- cal in time kernel of the action /H20851Eq. /H2084910/H20850/H20852isK/H20849/H9270/H20850/H11011/H9257 2/H92661 /H92702.W e thus obtain from Eq. /H2084912/H20850, /H9004Sdis=/H9257 2/H9266/H20885d/H9270/H20885d/H9270/H11032„S/H20849/H9270/H20850−S/H20849/H9270/H11032/H20850…2 /H20849/H9270−/H9270/H11032/H208502. /H2084915/H20850 The functional derivative of /H9004Sdiswith respect to S/H20849/H9270/H20850is /H9254/H9004Sdis /H9254S/H20849/H9270/H20850=/H9257 /H9266/H20885d/H9270/H11032„S/H20849/H9270/H20850−S/H20849/H9270/H11032/H20850… /H20849/H9270−/H9270/H11032/H208502=i/H9257d d/H9270S/H20849/H9270/H20850. /H2084916/H20850 From the free portion of the action /H20851the ﬁrst two terms of Eq. /H208498/H20850/H20852, we have /H9254S0 /H9254S/H20849/H9270/H20850=i1 S2dS/H20849/H9270/H20850 d/H9270/H11003S/H20849/H9270/H20850+h. /H2084917/H20850 Adding Eqs. /H2084916/H20850and /H2084917/H20850, equating the sum to zero, cross multiplying with S/H20849/H9270/H20850, and changing /H9270→it, we obtain Eq. /H208491/H20850 with/H9251=/H9257=2/H9266 /H9275J/H20849/H9275/H20850. In other words, the voltage dependence of/H9251in Eq. /H208491/H20850is identical to J/H20849/H9275/H20850. Next we examine the voltage dependence of J/H20849/H9275/H20850. We apply a voltage leading to a chemical drop of EFL −EFR=/H20849EF+/H9004/H9262L/H20850−/H20849EF+/H9004/H9262R/H20850=/H9004/H9262L−/H9004/H9262R=eV. Assuming, for example, that the net charge on both the right and left leads is unchanged, we also have DL/H20849EF/H20850/H9004/H9262L+DR/H20849EF/H20850/H9004/H9262R =0. With these constraints we get /H9004/H9262L=DR/H20849EF/H20850 DL/H20849EF/H20850+DR/H20849EF/H20850eV,BRIEF REPORTS PHYSICAL REVIEW B 73, 212501 /H208492006 /H20850 212501-3/H9004/H9262R=−DL/H20849EF/H20850 DL/H20849EF/H20850+DR/H20849EF/H20850eV, /H2084918/H20850 the Gilbert coefﬁcient /H9251may be approximated as /H9251/H20849V/H20850/H110112/H9266/H20873JLL2 2DL/H20849EF+/H9004/H9262L/H208502+JRR2 2DR/H20849EF+/H9004/H9262R/H208502 +JLR2DL/H20849EF+/H9004/H9262L/H20850DR/H20849EF+/H9004/H9262R/H20850/H20874 /H110112/H9266/H20875/H20851JLLDL/H20849EF/H20850/H208522 2+/H20851JRRDR/H20849EF/H20850/H208522 2 +JLR2DL/H20849EF/H20850DR/H20849EF/H20850 +eV DL/H20849EF/H20850+DR/H20849EF/H20850/H20873JLL2DL/H20849EF/H20850DR/H20849EF/H20850/H11509DL/H20849EF/H20850 /H11509EF −JRR2DL/H20849EF/H20850DR/H20849EF/H20850/H11509DR/H20849EF/H20850 /H11509EF +JLR2DR/H20849EF/H20850DR/H20849EF/H20850/H11509DL/H20849EF/H20850 /H11509EF −JLR2DL/H20849EF/H20850DL/H20849EF/H20850/H11509DR/H20849EF/H20850 /H11509EF/H20874/H20876. /H2084919/H20850 The change in the density of states associated with the chemical potential gradient across the junction triggers amodiﬁcation of the damping /H9251that is linear in voltage. For typical Fermi energy EFL/Rof the order of several electronvolts, the voltage dependence of /H9251may become very notable. This voltage driven effect may be expressed in terms of /H92510 and/H92511/H20849V/H20850with/H9251/H20849V/H20850=/H92510+/H92511/H20851Eq. /H208492/H20850/H20852. Here /H92510=/H9266/H20853JLL2/H20851D/H20851/H20849EFL/H20850/H208522+2JLR2D/H20849EFL/H20850D/H20849EFR/H20850+JRR2/H20851D/H20849EFR/H20850/H208522/H20854, /H2084920/H20850 /H92511=1 4e/H20841T0/H208412/H20851D/H20849EFL/H20850+D/H20849EFR/H20850/H20852/H20873IoL /H20851D/H20849EFL/H20850/H208522+IoR /H20851D/H20849EFR/H20850/H208522/H20874 /H11003/H20875D/H20849EFL/H20850D/H20849EFR/H20850/H20873JLL2/H11509DL/H20849EF/H20850 /H11509EF−JRR2/H11509DR/H20849EF/H20850 /H11509EF/H20874 +JLR2/H20873/H20851D/H20849EFR/H20850/H208522/H11509DL/H20849EF/H20850 /H11509EF−/H20851D/H20849EFL/H20850/H208522/H11509DR/H20849EF/H20850 /H11509EF/H20874/H20876 /H2084921/H20850 with Io/H9251=4/H9266e2/H20851D/H20849EF/H9251/H20850/H208522/H20841T0/H208412V. In conclusion, we present a theoretical study of Landau- Lifshitz-Gilbert damping /H20851Eq. /H208491/H20850/H20852for a localized spin Sin a junction. The exchange interactions between the localizedspin and tunneling electrons leads to additional dissipation ofthe spin motion, see Fig. 1. In the presence of an appliedvoltage bias V, the damping coefﬁcient, i.e., Gilbert damp- ing, is modiﬁed in linear order in Vfor the leads with particle-hole asymmetry in the density of states. Work at LANL was supported by the U.S. DOE under LDRD X1WX. *Electronic address: katsura@appi.t.u-tokyo.ac.jp †Electronic address: avb@lanl.gov ‡Electronic address: zohar@wuphys.wustl.edu 1D. D. Awschalom, M. E. Flatte, and N. Samarth, Spintronics /H20849Sci- entiﬁc American, 2002 /H20850, pp. 67–73. 2T. L. Gilbert, Phys. Rev. 100, 1243 /H208491955 /H20850. 3L. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153 /H208491935 /H20850. 4S. Chikazumi, Physics of Ferromagnetism /H20849Oxford University Press, Oxford, 1997 /H20850. 5B. Heinrich, D. Fraitová, and V . Kamberský, Phys. Status Solidi 23, 501 /H208491967 /H20850. 6Y . Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl. Phys. Lett. 84, 5234 /H208492004 /H20850. 7M. Onoda and N. Nagaosa, Phys. Rev. Lett. 96, 066603 /H208492006 /H20850. 8J.-X. Zhu, Z. Nussinov, A. Shnirman, and A. V . Balatsky, Phys. Rev. Lett. 92, 107001 /H208492004 /H20850.9Z. Nussinov, A. Shnirman, D. P. Arovas, A. V . Balatsky, and J.-X. Zhu, Phys. Rev. B 71, 214520 /H208492005 /H20850. 10O. Parcollet and C. Hooley, Phys. Rev. B 66, 085315 /H208492002 /H20850;L . N. Bulaevskii, M. Hruska, and G. Ortiz, ibid. 68, 125415 /H208492003 /H20850; A. Shnirman, D. Mozyrsky, and I. Martin, Europhys. Lett. 67, 840 /H208492004 /H20850. 11L. Bulaevskii, M. Hruska, A. Shnirman, D. Smith, and Yu. Makh- lin, Phys. Rev. Lett. 92, 177001 /H208492004 /H20850. 12Using the Anderson Hamiltonian, spin decoherence by ac voltage is studied in A. Kaminski, Y . V . Nazarov, and L. I. Glazman,Phys. Rev. B 62, 8154 /H208492000 /H20850. 13J.-X. Zhu and A. V . Balatsky, Phys. Rev. B 67, 174505 /H208492003 /H20850. 14E. Fradkin, Field Theories of Condensed Matter Systems /H20849Addison-Wesley, Redwood City, 1991 /H20850. 15A. O. Caldeira and A. J. Leggett, Ann. Phys. 149, 374 /H208491983 /H20850. 16A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46,2 1 1 /H208491981 /H20850. 17G. D. Mahan, Many-Particle Physics /H20849Kluwer, New York, 2000 /H20850.BRIEF REPORTS PHYSICAL REVIEW B 73, 212501 /H208492006 /H20850 212501-4
PhysRevB.100.144414.pdf	PHYSICAL REVIEW B 100, 144414 (2019) Modulation and phase-locking in nanocontact vortex oscillators Jérémy Létang ,1Sébastien Petit-Watelot ,2Myoung-Woo Yoo,1Thibaut Devolder,1 Karim Bouzehouane,3Vincent Cros,3and Joo-V on Kim1 1Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Sud, Université Paris-Saclay, 91120 Palaiseau, France 2Institut Jean Lamour, CNRS, Université de Lorraine, 54011 Nancy, France 3Unité Mixte de Physique, CNRS, Thales, Université Paris-Sud, Université Paris-Saclay, 91767 Palaiseau, France (Received 21 June 2019; revised manuscript received 18 September 2019; published 8 October 2019) We have conducted experiments to probe how the dynamics of nanocontact vortex oscillators can be modulated by an external signal. We explore the phase-locking properties in both the commensurate and chaotic regimes,where chaos appears to impede phase-locking while a more standard behavior is seen in the commensurate phase.These different regimes correspond to how the periodicity of the vortex core reversal relates to the frequency ofcore gyration around the nanocontact; a commensurate phase appears when the reversal rate is an integer fractionof the gyration frequency, while a chaotic state appears when this ratio is irrational. External modulation wherethe power spectral density exhibits rich features appears due to the modulation between the external sourcefrequency, gyration frequency, and core-reversal frequency. We explain these features with ﬁrst- or second-ordermodulation between the three frequencies. Phase-locking is also visible between the external source frequencyand internal vortex modes (gyration and core-reversal modes). DOI: 10.1103/PhysRevB.100.144414 I. INTRODUCTION Spin-torque nano-oscillators [ 1–5] (STNO) have strong potential for applications such as rf communications, mi- crowave generation [ 6], ﬁeld sensing [ 7], and neuro-inspired computing [ 8–10]. An important aspect involves phase- locking [ 11–14] and modulation [ 15–19] with external sig- nals, which have been studied extensively in vortex-basedsystems [ 20,21]. However, the role of vortex core reversal [22] in this context has remained largely unexplored. Indeed, in nanocontact-based systems, core reversal can give rise to more complex states such as rich modulation patterns but alsoa chaotic dynamic [ 23–28]. Because of the sensitivity to initial conditions, chaos is potentially useful for information pro-cessing as a large number of patterns can be generated rapidly[29], and therefore be used as a random number generator in symbolic dynamics or even neuromorphic computing. The difference between nanopillar [ 30] and nanocontact [31] systems in terms of modulation and phase-locking orig- inates mainly in the geometry. For vortex-based oscillators,the geometry is important because different spin-torque com-ponents are at play. In the nanopillar geometry, the primarycontribution to dynamics arises from spin-transfer torques ofthe Slonczewski form due to the current ﬂowing perpendicularto the ﬁlm plane (CPP) [ 1,2], where the out-of-plane compo- nent of the spin torque determines which sense of gyration ispossible relative to the core polarity, p[32]. In other words, self-gyration of the vortex core is only possible for certaincombinations of the current density J, the polarity p, and the spin polarization direction p z. The condition for oscillations is Jpp z>0. This is in stark contrast to the nanocontact system, where the primary driving torques in a steady state are of theZhang-Li form [ 33] due to currents ﬂowing in the ﬁlm plane (CIP). As such, there is no condition on the core polarity forself-sustained gyration, which allows for phenomena such as periodic core reversal to occur [ 23]. In this paper, we present an experimental and theoretical study in which we investigated how modulation and phase-locking due to the injection of an external current affectthe vortex dynamics in a nanocontact oscillator. Particularfocus is given to the core-reversal regime, where periodiccore reversal occurs in addition to the usual vortex gyrationaround the nanocontact [ 23]. A notable feature is the existence of both commensurate states, where the ratio between thecore-reversal and gyration frequencies is an integer fraction,and incommensurate or chaotic states, where this ratio isirrational. We ﬁnd that external modulation affects these twoperiodic processes differently, which offers insight into howchaos may be induced and controlled in such oscillators. This paper is organized as follows. In Sec. II, we de- scribe the materials and sample fabrication, the experimentalsetup for the electrical measurements, and the simulationmethods employed. In Sec. III, an overview is given of the three oscillator regimes studied. In Sec. IV, the response of the nanocontact vortex oscillator to alternating currents inthe different regimes is presented. In Sec. V, we describe the effects of current modulation on the periodic core reversal. Adiscussion and concluding remarks are given in Sec. VI. II. EXPERIMENTAL SETUP AND SIMULATION METHODS A. Materials and sample fabrication The oscillator comprises a metallic nanocontact adjacent to a pseudo spin valve, as illustrated in Fig. 1.T h em u l - tilayer is deposited using sputtering and has the compo-sition Ta(5) /Cu(40) /Co(20) /Cu(10) /NiFe(20) /Au(5), where the ﬁgures in parentheses denote the ﬁlm thickness in nm. 2469-9950/2019/100(14)/144414(11) 144414-1 ©2019 American Physical SocietyJÉRÉMY LÉTANG et al. PHYSICAL REVIEW B 100, 144414 (2019) ~rf generatorIdcBias T spectrumamplifier analyzerf (MHz) Idc (mA) PSD (nV2/Hz)1000 800 600 400 200 13.5 14.0 14.5 15.0 15.5 16.00.10.20.51.0Power (dBm)(a) (b)f0fext 2f0 -80-70-40 -50 -60 200 400 1000 800 600 f (MHz)dc ac+dc ac Idc = 14.2 mAAu Resist NiFe Co TaCu FIG. 1. Experimental setup and device geometry. The current ﬂows from the top electrode into the multilayer stack, until the other electrode. (a) Spectrum as read on the spectrum analyzer for a DC current of 14.2 mA. After treatment, spectra are aggregated to give(b) a PSD map. An insulating resist layer is then deposited on top of the Au cap layer, through which a nanocontact is formed using ananoindentation technique involving the conductive tip of anatomic force microscope [ 34]. The nanocontact has the shape of a truncated pyramid, with a lateral size of approximately20 nm in contact with the spin valve, 40 nm thick. Electricalmeasurements are made possible via gold electrodes to thenanocontact. Further details on the fabrication technique canbe found in previous work [ 23,35]. The NiFe layer is the free layer in which the vortex dynam- ics takes place, while the Co layer is the reference magneticlayer allowing the giant magnetoresistance signal. The mag-netic properties of the ﬁlms before patterning were determinedwith vector network analyzer ferromagnetic resonance beforepatterning. The NiFe layer is found to have a coercivityof 1 mT, a saturation magnetization of 1 .053±0.003 T, a spectroscopic splitting factor ( g-factor) of 2 .111±0.003, and a Gilbert damping constant of (7 ±1).10 −3. The Co layer isalso relatively soft with a coercivity of 2 mT, a saturation magnetization of 1 .768±0.011 T, a g-factor of 2 .133± 0.009, and a Gilbert constant of (10 ±1).10−3without any inhomogeneous broadening of the ferromagnetic resonanceline. These parameters correspond to a polycrystalline cobaltﬁlm with an fcc structure. B. Electrical measurements The electrical measurements of the nanocontact device have been performed at 77 K. The main contribution to thesignal arises from the gyrotropic dynamics of the magneticvortex that is induced by the applied DC currents. The gyra-tion leads to a resistance variation that translates into voltageoscillations in the sub-GHz range. These oscillations weregenerally measured with a spectrum analyzer in the range of100 to 1000 MHz. The spectrum analyzer was mainly usedwith a resolution bandwidth of 50 kHz, a video bandwidth of5 kHz. The signal is ampliﬁed with a 50-dB broadband am-pliﬁer before being fed into the spectrum analyzer. In additionto the DC current, we also apply a radio-frequency currentwith a synthesizer as an additional modulation. The circuitis illustrated in Fig. 1. For the majority of the measurements here, we ﬁx either the DC current or the frequency of themodulation signal, with the other parameter being varied. An example is given in Fig. 1(b), where the power spectral density (PSD) is shown as a color map as a function ofthe DC current I DCat a ﬁxed value of external modulation frequency [white line in Fig. 1(b)]. The PSD at each current is represented using a color code, which allows features in thepower spectrum to be followed as I DCis varied. For the sake of brevity, such plots are referred to as maps in this paper. C. Electrical and micromagnetics simulations It has previously been shown that an accurate description of the electrical current and associated Ørsted-Ampère ﬁeldproﬁles in the nanocontact geometry is necessary to provide agood quantitative agreement with experimental observations[36–38]. To this end, we have employed the ﬁnite-element code C OMSOL to compute the current and Ørsted-Ampère ﬁeld proﬁles in the nanocontact devices studied using the methoddescribed in Ref. [ 37]. By assuming cylindrical symmetry, we model the multilayer cross section as a 2 μm×100 nm rectangle with the nanocontact at one end. The full multilayerstack is simulated with the bulk values of the conductivityused for each material. The nanocontact itself is taken tobe a right trapezium whose 13.5-nm-smaller parallel sideis in contact with the multilayer stack [ 38]. Temperature and electrical wave propagation effects have been neglectedin this calculation. The simulations give the dependence ofthe perpendicular-to-plane and in-plane current densities asa function of radial distance from the nanocontact and ﬁlmthickness in the ferromagnetic free layer. Since we neglectthe thickness dependence when considering the magnetiza-tion dynamics, the current and Ørsted-Ampère ﬁeld proﬁlesare averaged over the ﬁlm thickness of the free layer. WithC OMSOL , we have calculated that the Ørsted-Ampère ﬁeld is around 800 A /m for a DC current of 1 mA, which corresponds to an increase of 1 mT for every 1 mA. This ﬁeld is in-planebut arises from both CIP and CPP components [ 38]. 144414-2MODULATION AND PHASE-LOCKING IN NANOCONTACT … PHYSICAL REVIEW B 100, 144414 (2019) The magnetization dynamics is studied with the micro- magnetics code M UMAX 3[39], which performs a numerical time integration of the Landau-Lifshitz-Gilbert equation withspin-transfer torques [ 40–42], dm dt=−γ0m×Heff+αm×dm dt+/Gamma1ST, (1) γ0is the gyromagnetic ratio, m(r,t) is a unit vector repre- senting the magnetization ﬁeld, Heffis the effective magnetic ﬁeld,αis the Gilbert damping constant, and /Gamma1STrepresents nonconservative spin-transfer torques. The effective ﬁeld isgiven by the variational derivative of the total magnetic en-ergy density Uwith respect to the magnetization unit vector, H eff=−(1/μ0Ms)δU/δm, and comprises contributions from the exchange, dipole-dipole, and the Zeeman interactions,where the latter includes contributions from the static externalapplied ﬁeld and the Ørsted-Ampère ﬁeld generated by thecurrent ﬂow through the nanocontact. The simulation geometry comprises a 1280 ×1280×20 nm system that is discretized using 512 ×512×1 ﬁnite differ- ence cells. We use micromagnetic parameters suitable forPermalloy; we take the saturation magnetization to be M s= 800 kA /m, the exchange stiffness Aex=10 pJ/m, and the Gilbert damping constant α=0.013 (a standard value for NiFe). For the spin-transfer torques, the dominant contribu-tion comes from the current in-plane terms, so we use /Gamma1 ST=−[u(r)·∇]m, (2) where u=J(r)PμB/(eMs) represents an effective spin-drift velocity, where Jis the in-plane current density, μBis the Bohr magneton, eis the electron charge, with the spin polarization taken to be P=0.5. We have veriﬁed that the nonadiabatic and Slonczewski terms are negligible in nanocontact vortexdynamics, so no further considerations to these terms willbe given here. The spatial proﬁles for J(r) and the Ørsted- Ampère ﬁeld it generates, H Oe(r), determined using the C OM- SOLsimulations described above, are used as inputs into the micromagnetics simulations. The initial magnetic state in the free layer is obtained by mimicking the experimental procedure, which is described inRef. [ 23] and later in Sec. III. In this procedure, once the transient dynamics has died out, we obtain a self-gyratingvortex around the nanocontact, with a remnant antivortexstructure pinned to one edge of the simulation box [ 23]. This serves as the initial condition for subsequent simulations. Forcalculations that involve sweeping the applied current, we usethe ﬁnal state of the simulation at a given current value as theinitial state for the subsequent value. The current dependence of the PSD of vortex oscillations is computed as follows. For each value of the applied current,we conduct the simulation over an interval of 100 ns, fromwhich we extract the spatially averaged m xcomponent, which is representative of the giant magnetoresistance signal, andthe core polarity, which is a measure of the core polarity, p. Since an adaptative time step is used in the numerical timeintegration, this data is resampled using cubic interpolationto recreate a time series with equal time steps. Fast Fouriertransforms are then applied to this time series data, fromwhich we compute the PSD. In what follows, the PSD undermodulation is studied either as a function of DC current,where current steps of 0.1 mA are used, or as a function of the modulation frequency, where frequency steps of 15 MHzare used. III. OVERVIEW OF OSCILLATOR REGIMES Experimentally, the ground state of the magnetic free layer is the uniformly magnetized state. As such, a nucleation pro-cedure is required before measurements to generate the vortexstate for self-oscillations. To achieve this, a 10 mT in-planemagnetic ﬁeld is applied to saturate the magnetization alongone direction. A large DC current is then applied (around 16 or17 mA), which generates a strong Ørsted-Ampère ﬁeld aroundthe nanocontact with a circulating proﬁle that favors onevortex chirality. The applied ﬁeld is then swept quasistaticallyto−10 mT, during which the magnetization in the free layer reverses through domain wall nucleation and propagation. Asthe domain wall sweeps through the nanocontact region, avortex is nucleated [ 43], which results in the well-deﬁned features in the power spectrum [Fig. 1(b)]. After nucleation, we change the applied ﬁeld to tune the oscillator regime. This nucleation procedure strongly depends on the ini- tial conditions, so the ease with which nucleation occurscan ﬂuctuate between experiments. To preserve the overalltopology of the magnetization state, it is conjectured that thevortex nucleation is always accompanied by the nucleationof an antivortex [ 43,44]; while the former is attracted to the Zeeman potential associated with the Ørsted-Ampère ﬁeld,the latter is repelled by this potential [ 45]. The presence of an antivortex is supported by the observation of a largenumber of harmonics [ 23] in the experimental power spec- trum at low currents and by simulations. The need for suchnucleation processes means that the measured power spectracan exhibit small quantitative differences between successivenucleation events [ 46], however, the power spectra remains unchanged between measurements after a given nucleationevent. The measurements are typically done by decreasing theDC current from nucleation current, down to a critical value,where the vortex can be annihilated. If the DC current is keptabove this critical value (around 10 mA), vortex annihilationis unlikely and measurements can be performed for bothincreasing and decreasing current sweeps. V ortex annihilationis certain under 4 mA. An example of the experimental power spectra is shown in Fig. 2. The current dependence of the PSD is presented in Fig. 2(d). Peaks in the sub-GHz range appear above a threshold around 7 mA, where the power is concentrated inthe fundamental frequency which is indicated by the blueline. This ﬁrst oscillation mode corresponds to vortex gy-ration around the nanocontact. The trajectory of the vortexis conjectured to be noncircular, since a number of higherorder harmonics are clearly visible in the power spectrum.An example of the PSD in this regime is shown in Fig. 2(a), where these harmonics can be clearly seen. This steady-stategyration regime extends from 7 to 11 mA, though this intervalmay vary between nucleation events (e.g., from 5 to 13 mA ina different experiment not shown here). As the current is increased, a second threshold is reached above which periodic core reversal takes place [ 23]. This corresponds to the appearance of additional sidebands in the 144414-3JÉRÉMY LÉTANG et al. PHYSICAL REVIEW B 100, 144414 (2019) Idc (mA)f (MHz) PSD (nV2/Hz) PSD (nV2/Hz) PSD (nV2/Hz) (a) (c)(b)f02f0 f0 2f0 (e)(d) 1/61/2 1/3 1/4 Idc (mA)2fcrfcr/f0(a) (c) (b)f0 fcr 200 6 81 0 1 2 1 4 1 64006008001000 0 PSD ( nV2/Hz) 6 8 10 12 14 160.00.10.20.30.40.5200 100 400 600 800 10000510152025 f (MHz)200 100 400 600 800 1000051015202535 30200 100 400 600 800 1000051015202535 30 10 2.557.5f (MHz)f (MHz) f0 f0-fcrpure gyration commensurate incommensurate (c) (b) FIG. 2. Spectra of PSD (in nV2/Hz) vs frequency (in MHz), at 8.5 mA (steady gyration regime) (a), at 12 mA (modulatedregime) (b), and at 16 mA (chaotic regime) (c). Aggregated spectra measurements with DC current varying give the PSD map (d). Red triangles give the above spectra correspondence. Upper (blue) dotscorrespond to gyration frequency. Lower (green) dots correspond to f 0−fcr. (e) gives ratio between fcrand f0versus DC current. PSD, which can be observed in Fig. 2(b) for an applied current of 12 mA. In this ﬁgure, the fundamental frequencyis labeled by f 0and the core-reversal frequency is labeled byfcr(note that fcris below the measurement range for a few current values). This example represents a commensurate state because the ratio between the core-reversal and gyrationfrequencies is a rational fraction, as shown in Fig. 2(e). These ratios vary as the current is increased and the presence ofplateaus is indicative of a self-phase-locked state, whereby aninteger multiple of the core-reversal frequency is locked to thegyration frequency. Physically, this means that core reversaloccurs after integer revolutions around the nanocontact. In between the plateaus, we can observe instances in which the ratio f cr/f0is irrational. A clear example can be seen between the 1 /3 and 1 /2 plateaus in Fig. 2(e), where this ratio appears to vary linearly with current. An example of the PSDi ss h o w ni nF i g . 2(c) at a current of 16 mA. In contrast to the commensurate state, the PSD in this regime is characterizedby broad peaks with no obvious relationship between f crand f0. This regime is termed the incommensurate state and cor- responds to temporal chaos; while core reversal occurs afteran integer number of revolutions around the nanocontact, thisnumber itself is characterized by a chaotic sequence [ 23,46]. In other words, the dynamics in this regime is characterizedby vortex gyration that is modulated by chaotic vortex corereversal. However, this behavior contrasts with other results[47–50] where two modes coexist without chaos. This is due to two weakly coupled parameters, such as two layers [ 47,48] or two nanocontacts [ 49,50]. The main features of the experimental spectra are re- produced in the micromagnetics simulations. The simulatedcurrent dependence of the PSD is presented in Fig. 3(a). We observe a ﬁnite lower threshold current for oscillationsat 7.9 mA. Below this threshold, the vortex core is immobileand localized at a distance of around 160 nm from the centerof the nanocontact, as shown in Fig. 3(a). This position results from a competition between the attractive central potential ofthe Zeeman interaction associated with the Ørsted-Ampèreﬁeld, and the attractive interaction between the vortex andthe antivortex, where the latter is pinned at the edge of thesimulation box [ 23]. The potential asymmetry is also due to a small contribution from CIP currents as discussed inRef. [ 38]. Once this lower threshold is overcome, we observe a steady-state gyration of the vortex around the nanocontact[Fig. 3(c)], where the trajectory represents a limit cycle with an egglike form that results from the balance between theasymmetric potential landscape, as discussed above, withthe radial symmetry of the spin torques due to the in-planecurrents ﬂowing from the nanocontact [ 38]. The absence of a radial symmetry of this trajectory gives rises to the richharmonic content of the power spectra [Fig. 2(a)]. Core reversal does not occur at arbitrary points along the trajectory but takes place close to the nanocontact, wherethe vortex velocity is higher, as shown in Figs. 3(d)–3(f). The core-reversal process involves the strong deformationof the vortex core, where a “dip” in the m zcomponent is generated in the direction opposite to the core polarity[51,52]. Once a critical deformation is reached, the dip trans- forms into the nucleation of a vortex-antivortex pair withan opposite polarity, and the original vortex annihilates withthe antivortex [ 22], leading to a burst of spin waves [ 53]. Because the core reversal process is actually mediated bythe annihilation and nucleation of a vortex with an oppositepolarity, a discontinuity appears in the core position and isrepresented by the sharp near-vertical lines in Figs. 3(d)–3(f) above the nanocontact. The periodic core reversal is analogous 144414-4MODULATION AND PHASE-LOCKING IN NANOCONTACT … PHYSICAL REVIEW B 100, 144414 (2019) PSD (nV2/Hz) Idc (mA)f (MHz)1000 800 600 400200 01.0 0.8 0.6 0.4 0.2 0(a) (e) 15.0 mA (f) 16.5 mA(b) 7.0 mA (c) 10.0 mA (d) 13.0 mA17 15 13 11 9 7 100 nm FIG. 3. Simulated power spectra of vortex gyration, with (a) cur- rent dependence of the power spectral density and (b)–(f) show different trajectories: (b) Static vortex core, below the threshold current; (c) Steady-state gyration; (d) Commensurate 1 /4r e g i m e ; (e) Chaotic regime; (f) Commensurate 1 /2 regime. In (b)–(f), an antivortex is situated on same side than the static vortex, much further. to relaxation oscillations; after a reversal, the core spirals outward from the nanocontact center, gaining in energy, untilthe critical deformation is reached and energy is released at thesubsequent reversal [ 23]. We also observe that the trajectories shrink as the current is increased, which results in f 0increas- ing faster than a linear function in the current, as observedexperimentally [Fig. 2(d)] and in simulation [Fig. 3(a)]. We note also that reversals in the core polarity result in a changein the sense of the gyration around the nanocontact (i.e.,clockwise or counterclockwise). This leads to the modulationsidebands seen in the power spectra. For the commensurate states, the trajectories have a clear overlap where the core-reversal events occur at near-identicalpositions, as can be seen for the 1 /4 and 1 /2 states in Figs. 3(d) and3(f), respectively. In the chaotic regime, on the other hand, the point at which the core reverses canvary greatly between revolutions around the nanocontact[Fig. 3(e)]. This results in a set of trajectories that cover a greater area around the nanocontact, which translates intothe broad spectral peaks as seen in Fig. 2(c). Because of the large qualitative differences in the trajectories betweenthe steady-state, commensurate, and chaotic regimes, we cananticipate that external forcing with AC currents will havedifferent effects on the core dynamics.(a) NC vortex(b) FIG. 4. Schematic representation of circular vortex gyration around the nanocontact. (a) V ortex gyration in a counterclockwisedirection with velocity ˙X 0.uindicates the direction of the effect spin drift velocity. (b) Force diagram for the four terms in the Thiele equation [Eq. ( 3)]. The current-dependent terms are outlined by a box. IV . MODULATION DUE TO ALTERNATING CURRENTS In this section, we describe the effects of external forcing due to AC currents on the different oscillatory regimes ofthe nanocontact vortex oscillator. AC currents lead to peri-odic modulations in the Zeeman potential, associated withthe Ørsted-Ampère ﬁeld, and to periodic modulations in thespin torques exerted on the vortex core. To see how theseterms might inﬂuence the core dynamics, consider the Thieleequation, which provides a good description of the gyrationfar below the threshold for core reversal [ 54–56]: G×(˙X 0−u(X0,I))+D·˙X0=−∂U(I) ∂X0. (3) X0is the position of the vortex core in the ﬁlm plane, ˙X0≡ ∂tX0,Gis the gyrovector, Dis the Gilbert dissipation tensor, u is the effective spin-drift velocity that measures the strength ofthe spin torques, Uis the total energy of the vortex system, and Iis the applied current. This equation of motion can be derived from the Landau-Lifshitz equation [Eq. ( 1)] by assuming a rigid core for the vortex. As such, it captures the gyrotropicdynamics but it cannot account for vortex core reversal. There are two current-dependent terms, the spin current u and the potential energy density U; modulations in the current, I=I DC+iAC, will therefore result in modulations in these two terms. To see how these enter the dynamics, consider thecircular motion around the nanocontact as a result of a purecentral potential, U(/bardblX 0/bardbl), i.e., we neglect contributions from exchange interactions with the antivortex. We will also as-sume a counterclockwise gyration (when viewed from above,+z), which corresponds to a gyrovector G=2πM sdp/γ where dis the ﬁlm thickness and pis the core polarization. A schematic of this motion is given in Fig. 4(a). A pictorial representation of the four force terms in Eq. ( 3)i sg i v e ni n Fig. 4(b). This ﬁgure gives a clear interpretation of how the four forces counterbalance each other. The restoring force dueto the Zeeman potential is directed radially inward, whichfavors the vortex core centered on the nanocontact, while thegyrotropic term is directed radially outward. The equilibriumorbit is therefore determined by a balance not only betweenthese two forces, but from all forces as two of them share acommon term, ˙X 0. Modulations in the strength of the Zeeman potential, due to the AC current, amounts to a modulation of 144414-5JÉRÉMY LÉTANG et al. PHYSICAL REVIEW B 100, 144414 (2019) FIG. 5. Map of the power spectral density as a function of modulation frequency. Parts (a)–(c) correspond to experimental measurements, while (d)–(f) correspond to results of micromagnetics simulations. Left [(a), (d)], central [(b), (e)], and right [(c), (f)] columns correspond tothe pure gyration regime, core modulated regime, and chaotic regime, respectively. The inset in (b) is a zoom on the region between 350 and 450 MHz on both scales; 1:1 phase-locking and modulation sidebands are visible. The applied current I DC, in-plane ﬁeld H, and perpendicular- to-plane ﬁeld H⊥for each part are as follows: (a) IDC=12.8m A , μ0H=294μT; (b) IDC=15 mA, μ0H=68μT; (c) IDC=16.7m A , μ0H⊥=90.4m T ;( d ) IDC=10 mA, μ0H=2m T ;( e ) IDC=13 mA, μ0H=2 mT; (f) IDC=11.5m A , μ0H=2m T . the radial force and therefore acts to modulate the gyrovector, therefore the radius of the vortex gyration. Moreover, sincethe potential also determines the gyration frequency [ 32,56], this modulation is akin to a parametric excitation. Let us nowdiscuss the two other forces: damping and spin torque. Bothact tangentially to the circular orbit, where the damping actslike friction in the direction opposite to the motion, whilethe adiabatic spin-torque term acts in the direction of themotion as a velocity “boost.” Compensation between thesetwo is required for the vortex to maintain steady-state gyrationaround the nanocontact. Modulations in the current lead to amodulation in the adiabatic torque, which acts to modulate the“boost” of the vortex core along its trajectory. This is akin toa phase modulation of the vortex oscillator. In the following,we will discuss how these two contributions affect the vortexdynamics in the three regimes. A. Forcing in the steady gyration regime We ﬁrst examine the effects of current modulation in the low current regime in which no core reversal is present. Theexperimental power spectra are presented in Fig. 5(a), which correspond to the following operating conditions: a DC cur-rent of 12.8 mA in Fig. 5(a) resulting in a gyration frequency of 200 MHz. An AC modulation of i AC=0.3 mA (peak to peak) is applied whose frequency fextis swept between 180 and 620 MHz, which is clearly present in the experimentalspectra as a narrow line with unity slope, f=f ext. We note that harmonics in the external forcing can also be seen (i.e.,fainter lines with f=2f extand f=3fext). We attribute this to nonlinearities in the gain of the ampliﬁers we used, whichmeans the sample does not receive those harmonics.Phase-locking to the external signal can be seen in Fig. 5(a) asf extcrosses fgyrat around 200 MHz, which is evidenced by a vacant horizontal segment in the power spectrum at whichthe oscillator frequency is entrained by the external signal.Because of the elliptical trajectory of the vortex core aroundthe nanocontact, this entrainment also manifests itself in thehigher harmonics, notably at 2 f gyr. We also note that fractional synchronization is seen in Fig. 5(a), whereby phase-locking occurs at integer multiples or integer fractions of the gyrationfrequency. Phase-locking is perceptible at f ext=fgyr,fext= 2fgyrand fext=3fgyr. Overall, this behavior is similar to the phase-locking phenomenon observed in nanopillar vortexoscillators [ 20,21], though it seems we do not have fractional phase-locking in this regime as in nanopillar oscillators [ 57]. To understand the modulation process in this pure gyration case, we can use a simple model of r(t) the resistance of the system and i(t) the current ﬂowing through the system, r(t)=R 0+/Delta1Rcos(ωgyrt), i(t)=IDC+iACcos(ωextt),(4) where R0is the mean resistance of the device, /Delta1Ris the resistance variation due to gyration, and ωgyr=2πfgyrand ωext=2πfext. The spectrum analyzer measures the power given by P(t)=p0+p1cos(ωextt)+p2cos(2ωextt)+p3cos(ωgyrt) +p4cos[(ωgyr+ωext)t]+p4cos[(ωgyr−ωext)t]) +p5cos[(ωgyr+2ωext)t]+p5cos[(ωgyr−2ωext)t], (5) 144414-6MODULATION AND PHASE-LOCKING IN NANOCONTACT … PHYSICAL REVIEW B 100, 144414 (2019) where p0=R0(I2 DC+i2 ac/2),p1=2R0IDCiAC,p2=R0i2 AC/2, p3=/Delta1R(I2 DC+i2 AC/2),p4=/Delta1RIDCiAC, and p5=/Delta1Ri2 AC/4. A number of these frequencies are visible experimentally,namely f ext,2fext,fgyr, and fgyr+fext. Some of the other frequencies are not clearly visible experimentally, due to theirlow intensity. However, they are much more visible in thesimulation data, which can be seen in Fig. 5(d). For the micromagnetics simulations presented in Fig. 5(d), we considered a DC current of I DC=10 mA which leads to a gyration frequency of 200 MHz. To study higher ACcurrents than those attainable experimentally, and to bettervisualize the modulation sidebands, we considered an ACcurrent i AC=1 mA that was swept from 100 to 900 MHz. For fext<fgyr, we can clearly observe modulation effects in the power spectra, where the f=fextsignal is accompanied by sidebands not only at f=fgyr+fextand f=fgyr−fext, but also at f=2fgyr−fextand to a lesser extent at f= 3fgyr−fext. We can also slightly see 2 fext−fgyrorfgyr− 2fextsidebands. As fextenters the locking window, frequency entrainment can clearly be observed over a range of 25 MHzin which the gyration frequency is controlled by the frequencyofi AC. This entrainment is also visible in the ﬁrst harmonic, where a segment with f=2fextis visible in the locking window. Phase-locking is also observable at fext=3fgyrand fext=4fgyr, where at each harmonic the entrainment of the gyration frequency varies like fgyr=fext/(n+1) with ndenoting the nth harmonic. This is accompanied by clear modulation signals at f=(n+1)fgyr−fext, which are most visible in the frequency range below the gyration frequency f<fgyr. While most of these frequencies correspond to those predicted by the simple model, we see additional contributionsin the experimental spectra. These can be understood as higherorder modulation effects. The simulation results are similar tothe phase-locking phenomena observed in nanopillar vortexoscillators. B. Forcing in the commensurate regime We now examine the effects of current modulation in the commensurate regime, where periodic core reversal oc-curs at a rate that is an integer fraction of the gyra-tion frequency. The experimental spectra are presented inFig.5(b). The operating conditions consist of a DC current of I DC=15 mA, which in one experiment leads to a gyration fre- quency of fgyr=410 MHz, as shown in Fig. 5(b).I nF i g . 5(b), we can see that the phase-locking and modulation patterns aresimilar to the previous case in which the dynamics comprisespure gyration, though these phenomena are more visible andmodulation occurs on a larger range in the commensurateregime. The important difference here is that the externalsignal now modulates two distinct processes, the gyration andthe periodic core reversal, where the frequency for the latter isdenoted by f cr. Besides phase-locking at fext=fgyr, we also ﬁnd evidence of entrainment when the external signal crosses one of themodulation sidebands due to vortex core reversal, namely,atf ext=fgyr±fcr, as shown in Fig. 5(b). Phase-locking of modulation sidebands and fractional synchronization are phe-nomena that have already been reported in previous studies onSTNOs [ 58], though it is in a feedback loop, and not related to core reversal.The spectral features with constant frequencies in Fig. 5(b), i.e., which are independent of f ext, can be expressed as linear combinations of the gyration frequency fgyrand the core reversal frequency fcr. These are the natural frequencies of the vortex dynamics. A similar spectrum with natural frequen-cies is given in Fig. 2(b), without any external forcing. For instance, when f gyr=4fcr, one of the natural frequencies can be the sum of the ﬁrst sideband of the gyration frequency (i.e., fgyr+fcr) and of the third sideband of the second harmonic of the gyration frequency (i.e., 2 fgyr−3fcr). Since the dynamics are in the commensurate regime, we can write fgyr=afcr with integer a, the commensurate ratio between these two frequencies [ a=4i nF i g s . 5(b) and5(e)], which allows us to express the kth natural frequency fksimply as fk=kfcr, (6) with ibeing an integer constant. Therefore, we can write f1=fcr,f2=2fcr,f3=3fcr=fgyr−fcr,f4=4fcr=fgyr, etc. We observe then that the external signal modulates allthe natural frequencies f kto a certain degree. This simpler description of the system enables us to reuse the simple model,previously discussed for the pure gyration regime, for thecommensurate regime, r(t)=R 0+/summationdisplay k/Delta1Rkcos(ωkt), i(t)=IDC+iACcos(ωextt),(7) where R0,/Delta1R,ωext, and ωkare deﬁned as in the previous section. Here, ω1=2πfcr,ω2=2π2fcr,ω3=2π3fcr,ω4= fgyr[due to the 1 /4 ratio between fcrand fgyrin Fig. 5(b)]. The measured power is therefore P(t)=p0+p1cos(ωextt)+p2cos(2ωextt) +/summationdisplay kp3,kcos(ωkt)+/summationdisplay kp4,kcos[(ωk+ωext)t] +/summationdisplay kp4,kcos[(ωk−ωext)t] +/summationdisplay kp5,kcos[(ωk+2ωext)t] +/summationdisplay kp5,kcos[(ωk−2ωext)t], (8) where p0=R0(I2 DC+i2 AC/2), p1=2R0IDCiAC,p2= R0i2 AC/2,p3,k=/Delta1Rk(I2 DC+i2 AC/2),p4,k=/Delta1RkIDCiAC, and p5,k=/Delta1Rki2 AC/4. We can now compare this simple model to the frequencies exhibited in Fig. 5(b). Ascending branches correspond to the f=fk+fextfrequencies, while descending branches correspond to the f=fk−fextsidebands. Only the ﬁrst order of modulation is visible, such that no sidebands f=fk±nfext, with ndenoting the nth order, appear. No signals collinear to the harmonics of the external signalappear. Phase-locking, like in the pure gyration regime, occurs when f extis equal to a multiple of fgyr. But in this com- mensurate regime, it can occur also for any natural frequencyof the vortex. When such a phase-locking occurs, it is alsovisible on the other natural frequencies of the vortex, at ahigher or smaller frequency. Here, fractional synchronization 144414-7JÉRÉMY LÉTANG et al. PHYSICAL REVIEW B 100, 144414 (2019) 10 mA 11.5 mA 13 mA 15 mA + 160 MHz modulation+ 250 MHz modulation+ 300 MHz modulation+ 500 MHz modulation 100 nm(a) (d) (c) (b) FIG. 6. Role of current modulation on vortex core reversal. Simulated trajectories; top ones correspond to the motion under DC currents alone while bottom ones correspond to the addition ofcurrent modulation (at the frequencies indicated). (a)–(d) correspond to different operating conditions, with a given DC current. is possible. However, fextcan cross some natural frequency fkwithout inducing phase-locking. The reason why some nat- ural frequencies are more likely to be phase-locked remainsunknown, though we can at least say that frequencies like f gyr, fcrand their harmonics are more likely to respond to external excitation than any other fk. It should be noticed that core reversal corresponds to a square signal that contains only oddharmonics. Therefore, we see phase-locking at f cr,3fcr,5fcr but not at 2 fcr, for instance. Simulation in Fig. 5(e) exhibits a similar behavior than experimental curves, showing nonetheless a regime changewhich we did not observe in our measurements, but whichhave been shown on a different device [ 59]. This effect is slightly visible in Fig. 5(d), but is wider in Fig. 5(e):T h e system changes from a commensurate to an incommensurateregime, mainly around the phase-locking region. Indeed, be-tween 260 and 310 MHz, and between 350 and 460 MHz,there are no phase-locking or commensurate regimes: Wedon’t see any sharp peaks but rather a diffuse signal, which ischaracteristic of the incommensurate regime. Such a regimemodiﬁcation will be described with more detail in Sec. Vand Fig.6. Over a frequency range of approximatively 700 MHz, the f extsignal appears stronger in Fig. 5(e) [indeed, it is also the case for Figs. 5(d) and5(f)]. This seems to indicate that for fext=fgyr, the inﬂuence of fextover the vortex decreases, and therefore less energy is transferred to it, leading to a moreintense f ext. C. Incommensurate states When the vortex exhibits a chaotic behavior, the appear- ance of the PSD map changes in terms of phase-locking andmodulation. Indeed, in Fig. 5(c), we apply a DC current of IDC=16.7 mA, leading to a gyration frequency of fgyr= 520 MHz. We still apply an AC current of iAC=0.3 mA. We can see that there is no phase-locking and modulation whenthe external and vortex frequencies are incommensurate. Thisindicates that a chaotic behavior prevents phase-locking andmodulation of such oscillators. Such a result is a consequenceof the Kolmogorov-Arnold-Moser theory on dynamical sys-tems [ 60–62]: An incommensurate regime is more robust to small perturbations. Indeed, a chaotic oscillator emits widerband frequencies, and therefore cannot be synchronized toa single frequency. Incommensurate states, which appear inchaotic regimes in vortex nanocontact oscillators [ 23], are less subject to phase-locking. Core reversal is aperiodic ina chaotic incommensurate regime. Therefore, a periodic ex-ternal forcing barely induces locking of vortex frequencies.However, it might be possible that increasing the couplingstrength between the chaotic oscillator and the external signal,in other terms, a higher AC current, induces phase-lockingand modulation, making chaotic regime oscillators have asimilar behavior to steady oscillation or core-reversal regimeoscillators. Indeed, in Fig. 5(f), where a higher AC current sent into the device is simulated, we can see a 30-MHz phase-lockingrange between f extand fgyr. Modulation sidebands are also visible at low frequency. V . IMPACT OF CURRENT MODULATION ON CORE REVERSAL To better understand the simulated spectra presented in Figs. 5(d)–5(f), we examine how the trajectories of the vortex core change when the current modulation is present. Thetrajectories can be classiﬁed into four categories: a ﬁxed point(no gyration), a limit cycle representing steady state gyration,a limit cycle with core reversal, and a chaotic attractor [ 46]. In Fig. 6, we illustrate examples of trajectories in the steady state gyration regime [Fig. 6(a)], the commensurate state [Figs. 6(b) and 6(c)], and the incommensurate or chaotic regime [Fig. 6(d)]. Current modulation can change the oscillation regime. Fig- ure6(a) shows transitions between the steady-state gyration toward a core-reversal state ( I DC=10 mA, fext=160 MHz), and the opposite transition from a commensurate state towardsteady-state gyration ( I DC=11.5 mA, fext=250 MHz) can be seen in Fig. 6(b). We also observe modulation-induced transitions back and forth between the commensurate andincommensurate states, which are shown in Fig. 6(c)forI DC= 13 mA, fext=300 MHz and in Fig. 6(d) forIDC=15 mA, fext=500 MHz. This indicates that the conditions for core reversal can be suppressed or delayed as a result of the currentmodulation. Let us now discuss these modulation-induced transitions in more detail. Figure 7illustrates the simulated power spectra of the magnetoresistance signal and the vortex core polarity, p,a s a function of the DC current I DCand the modulation frequency fext. In the absence of forcing, we can observe two clear thresholds for oscillations as the current is increased, one forthe onset of steady-state gyration at around 8 mA [Fig. 7(a)] and the other for the onset of periodic core reversal at around 144414-8MODULATION AND PHASE-LOCKING IN NANOCONTACT … PHYSICAL REVIEW B 100, 144414 (2019) (a) fext (MHz)200 400 600 8002004006008001000 0f (MHz) 01 . 0 0.2 0.4 0.6 0.8PSD (nV2/Hz) (b)(c) (d)7 9 11 13 15 17 Idc (mA)2004006008001000 07 9 11 13 15 17f (MHz)2004006008001000 0 2004006008001000 0200 400 600 800no external signal with external signal mx frequencies polarity frequencies FIG. 7. PSD maps of magnetoresistance signal [(a), (c)] and polarity frequencies [(b), (d)] while a DC current sweep without external signal [(a), (b)] or while an external frequency sweep with 13 mA DC current-(c), (d)]. 10.5 mA [Fig. 7(b)]. These ﬁgures clearly demonstrate that core reversal is at the origin of the intrinsic modulation thatappears above 10.5 mA. Moreover, we can see that in themodulation regime, the spectra for the magnetoresistance andcore polarization oscillations differ: only odd harmonics of f cr are visible in the PSD of the core polarization, whereas all har- monics of fcrare visible in the PSD of the magnetoresistance variations. Because the core polarity signal closely resemblesa square wave, its Fourier series only contains odd harmonics.If there is jitter in the reversal events, even harmonics alsomight appear. On the other hand, the power spectrum ofthe magnetoresistance comprises the vortex gyration, whichprovides the dominant term in the PSD, along with the corereversal and the different orders of modulation between thesetwo frequencies. This indicates that spectral lines such as f= 2f cratIDC≈15 mA originates from second-order modulation processes between the gyration and core-reversal frequenciesbecause the core reversal signal on its own cannot produce aneven f crharmonic. We now present a similar analysis for a ﬁxed current of 13 mA, where the modulation frequency is swept from 100to 850 MHz [Figs. 7(c) and7(d)]. Similar modulation effectsas those described previously are observed, except around the phase-locking region where the core-reversal frequency f crdecreases [Fig. 7(d)]. This is a consequence of the core reversal being impeded by the modulation, which leads tolonger intervals between reversal events. As such, we canobserve that the modulation frequencies disappear around the1:1 locking in the power spectrum of the magnetoresistancevariations [Fig. 7(c)]. VI. DISCUSSION AND CONCLUDING REMARKS We have performed a detailed experimental and numerical study of the role of current modulation on the vortex dynamicsin magnetic nanocontact oscillators. These oscillators can pos-sess two intrinsic modes, which can coexist: steady-state gyra-tion around the nanocontact and periodic core reversal, whichare characterized by the frequencies f gyrand fcr, respec- tively. We have shown how modulation in the applied current,which affects both the Zeeman potential and the spin-transfertorques, inﬂuence these regimes. In particular, the modulationof both f gyrand fcrby the external frequency fextcan lead to rich intermodulation spectra. External modulation can also 144414-9JÉRÉMY LÉTANG et al. PHYSICAL REVIEW B 100, 144414 (2019) lead to transitions between the natural oscillation regimes, namely simple gyration, commensurate, and chaotic states. Beyond phase-locking applications such as spectrum anal- ysis [ 63], we suggest that the nanocontact vortex oscillator might also be suitable for neuro-inspired applications [ 9,10] since it offers a rich variety of oscillatory modes that can beharnessed with external modulation. In terms of chaos-basedinformation processing, this study sheds light on how unstableperiodic orbits might be targets using external modulation. Itis indeed a ﬁrst step in chaos control in nanocontact vortexoscillators. Another step would be to prove experimentallywhat has been seen in simulation, namely, the regime chargeand phase-locking of chaos at higher forcing strength.ACKNOWLEDGMENTS The authors thank S. Girod and C. Deranlot for assistance in sample fabrication, and J.-P. Adam and D. Rontani forfruitful discussions. This work is supported by the AgenceNationale de la Recherche (France) as part of the Investisse-ments d’Avenir program (LabEx NanoSaclay, Reference No.ANR-10-LABX-0035) and under Contract No. ANR-17-CE24-0008 (CHIPMuNCS). J.L. acknowledges ﬁnancial sup-port from the EOBE doctoral school of Université Paris-Saclay. 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PhysRevB.89.024409.pdf	PHYSICAL REVIEW B 89, 024409 (2014) Propagation of thermally induced magnonic spin currents Ulrike Ritzmann, Denise Hinzke, and Ulrich Nowak Fachbereich Physik, Universit ¨at Konstanz, D-78457 Konstanz, Germany (Received 29 August 2013; revised manuscript received 19 December 2013; published 13 January 2014) The propagation of magnons in temperature gradients is investigated within the framework of an atomistic spin model with the stochastic Landau-Lifshitz-Gilbert equation as underlying equation of motion. We analyze themagnon accumulation, the magnon temperature proﬁle, as well as the propagation length of the excited magnons.The frequency distribution of the generated magnons is investigated in order to derive an expression for theinﬂuence of the anisotropy and the damping parameter on the magnon propagation length. For soft ferromagneticinsulators with low damping a propagation length in the range of some micrometers can be expected for exchangedriven magnons. DOI: 10.1103/PhysRevB.89.024409 PACS number(s): 75 .30.Ds,75.30.Sg,75.76.+j I. INTRODUCTION Spin caloritronics is a new, emerging ﬁeld in magnetism describing the interplay between heat, charge, and spin transport [ 1,2]. A stimulation for this ﬁeld was the discovery of the spin Seebeck effect in Permalloy by Uchida et al. [3]. Analog to the Seebeck effect, where in an electric conductoran electrical voltage is created by applying a temperaturegradient, in a ferromagnet a temperature gradient excites aspin current leading to a spin accumulation. The generatedspin accumulation was detected by measuring the spin current locally injected into a platinum contact using the inverse spin Hall effect [ 3,4]. A ﬁrst explanation of these effects was based on a spin-dependent Seebeck effect, where the conductionelectrons propagate in two different channels and, due to aspin-dependent mobility, create a spin accumulation in thesystem [ 5]. Interestingly, it was shown later on that this effect also appears in ferromagnetic insulators [ 6]. This shows that in addition to conduction-electron spin currents, chargeless spin currents exist as well, where the angular momentum is transported by the magnetic excitations of the system, so-calledmagnons. A ﬁrst theoretical description of such a magnonicspin Seebeck effect was developed by Xiao et al. [7]. With a two temperature model including the local magnon (m) andphonon (p) temperatures the measured spin Seebeck voltageis calculated to be linearly dependent on the local differencebetween magnon and phonon temperature, /Delta1T mp=Tm−Tp. This temperature difference decays with the characteristiclength scale λ. For the ferromagnetic material yttrium iron garnet (YIG) they estimate the length scale in the range ofseveral millimeters. The contribution of exchange dominated magnons to the spin Seebeck effect was investigated in recent experimentsby Agrawal et al. [8]. Using Brillouin light scattering the difference between the magnon and the phonon temperature ina system with a linear temperature gradient was determined.They found no detectable temperature difference and estimatea maximal characteristic length scale of the temperaturedifference of 470 μm. One possible conclusion from this result might be that instead of exchange magnons, magnetostaticmodes mainly contribute to the spin Seebeck effect and areresponsible for the long-range character of this effect. Alterna-tively, phonons might contribute to the magnon accumulationas well via spin-phonon drag [ 9,10]. A complete understanding of these different contributions to the spin Seebeck effect isstill missing. In this paper thermally-excited magnonic spin currents and their length scale of propagation are investigated. Usingan atomistic spin model simulation which describes thethermodynamics of the magnetic system in the classical limitincluding the whole frequency spectra of excited magnons,we describe spin currents by exchange magnons in thevicinity of a temperature step. After introducing our model,methods, and basic deﬁnitions in Sec. IIwe determine the magnon accumulation as well as the corresponding magnontemperature and investigate the characteristic length scale ofthe decay of the magnon accumulation in Sec. III. In Sec. IV we introduce an analytical description which is supported byour simulations shown in Sec. Vand gives insight into the material properties dependence of magnon propagation. II. MAGNETIZATION PROFILE AND MAGNON TEMPERATURE For the investigation of magnonic spin currents in temper- ature gradients we use an atomistic spin model with localizedspins S i=μi/μsrepresenting the normalized magnetic mo- mentμsof a unit cell. The magnitude of the magnetic moment is assumed to be temperature independent. We simulate athree-dimensional system with simple cubic lattice structureand lattice constant a. The dynamics of the spin system is described in the classical limit by solving the stochasticLandau-Lifshitz-Gilbert (LLG) equation, ∂S i ∂t=−γ μs(1+α2)Si×[Hi+α(Si×Hi)],( 1 ) numerically with the Heun method [ 11] with γbeing the gyromagnetic ratio. This equation describes a precession ofeach spin iaround its effective ﬁeld H iand the coupling with the lattice by a phenomenological damping term with dampingconstant α. The effective ﬁeld H iconsists of the derivative of the Hamiltonian and an additional white-noise term ζi(t), Hi=−∂H ∂Si+ζi(t). (2) The Hamiltonian Hin our simulation includes exchange interaction of nearest neighbors with isotropic exchange 1098-0121/2014/89(2)/024409(7) 024409-1 ©2014 American Physical SocietyULRIKE RITZMANN, DENISE HINZKE, AND ULRICH NOW AK PHYSICAL REVIEW B 89, 024409 (2014) m0α=1α=0.1α=0.06Tp space coordinate z/a phonon temperature kBTp/Jmagnetization m0.1 0 40 30 20 10 0 -10 -20 -30 -401 0.995 0.99 0.985 0.98 0.975 FIG. 1. (Color online) Steady state magnetization mand equilib- rium magnetization m0over space coordinate zfor a given phonon temperature proﬁle and for different damping parameters αin a small section around the temperature step. constant Jand a uniaxial anisotropy with an easy axis in z direction and anisotropy constant dz, H=−J 2/summationdisplay /angbracketlefti,j/angbracketrightSiSj−dz/summationdisplay iS2 i,z.( 3 ) The additional noise term ζi(t) of the effective ﬁeld Hi includes the inﬂuence of the temperature and has the following properties: /angbracketleftζ(t)/angbracketright=0, (4) /angbracketleftbig ζi η(0)ζj θ(t)/angbracketrightbig =2kBTpαμ s γδijδηθδ(t). (5) Herei,jdenote lattice sites and ηandθCartesian components of the spin. We simulate a model with a given phonon temperature Tp which is space dependent and includes a temperature step inzdirection in the middle of the system at z=0f r o ma temperature T1 pin the hotter area to T2 p=0 K (see Fig. 1). We assume that this temperature proﬁle stays constant during thesimulation and that the magnetic excitations have no inﬂuenceon the phonon temperature. The system size is 8 ×8×512, large enough to minimize ﬁnite-size effects. All spins are initialized parallel to the easy axis in z direction. Due to the temperature step a nonequilibrium inthe magnonic density of states is created. Magnons propagatein every direction of the system, but more magnons exist in thehotter rather than in the colder part of the system. This leadsto a constant net magnon current from the hotter towards thecolder area of the system. Due to the damping of the magnonsthe net current appears around the temperature step with aﬁnite length scale. After an initial relaxation time the systemreaches a steady state. In this steady state the averaged spincurrent from the hotter towards the colder region is constantand so the local magnetization is time independent. We cannow calculate the local magnetization m(z) depending on the space coordinate zas the time average over all spins in the plane perpendicular to the zdirection. We use the phonon temperature T 1 p=0.1J/k Bin the heated area, the anisotropy constant dz=0.1J, and vary the damping parameter α. The resulting magnetization versus the spaceﬁtequilibrium data magnon temperature kBTm/Jmagnetization m0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.101 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 FIG. 2. (Color online) Equilibrium magnetization m0over the magnon temperature Tm. Red points show the simulated equilibrium magnetization and the black line shows a ﬁt of the data. coordinate zfor different damping parameters in a section around the temperature step is shown in Fig. 1. For comparison the particular equilibrium magnetization m0of the two regions is also calculated and shown in the ﬁgure. Far away from the temperature step on both sides the amplitudes of the local magnetization m(z) converge to the equilibrium values, only in the vicinity of the temperaturestep deviations appear. These deviations describe the magnonaccumulation, induced by a surplus of magnons from thehotter region propagating towards the colder one. This leadsto a less thermal excitation in the hotter area and the valueof the local magnetization increases. In the colder area thesurplus of incoming magnons decrease the value of the localmagnetization. For smaller values of αthe magnons can propagate over larger distances before they are ﬁnally damped.This leads to a damping-dependent magnon accumulationwhich increases with decreasing damping constant α. For a further analysis in the context of the spin-Seebeck effect we deﬁne a local magnon temperature T m(z)v i a the magnetization proﬁle m(z). For that the equilibrium magnetization m0(T) is calculated for the same model but homogeneous phonon temperature Tp. In equilibrium magnon temperature Tmand the phonon temperature Tpare the same and we can determine the (magnon) temperature dependenceof the equilibrium magnetization m 0(Tm) of the system. The magnetization of the equilibrium system decreases for increas-ing magnon temperature as shown in Fig. 2and the behavior can be described phenomenologically with a function [ 12] m 0(T)=(1−Tm/Tc)β, where Tcis the Curie temperature. Fitting our data we ﬁnd Tc=(1.3326±0.000 15) J/k Band for the exponent we get β=0.329 84 ±0.000 65. This ﬁt of the data is also shown in Fig. 2and it is a good approximation over the whole temperature range. The inverse function is usedin the following to determine the magnon temperature for agiven local magnetization and with that a magnon temperatureproﬁle T m(z). The resulting magnon temperature proﬁles are shown in Fig. 3. Far away from the temperature step the magnon temperature Tm(z) coincides with the given phonon temper- atureTp, and deviations—dependent on the damping constant α—appear only around the temperature step. These deviations 024409-2PROPAGATION OF THERMALLY INDUCED MAGNONIC . . . PHYSICAL REVIEW B 89, 024409 (2014) T0 mα=1α=0.1α=0.06Tp space coordinate z/a phonon temperature kBTp/Jmagnon temperature kBTm/J 0.1 0 40 30 20 10 0 -10 -20 -30 -400.1 0.08 0.06 0.04 0.02 0 FIG. 3. (Color online) Magnon temperature Tmover the space coordinate zfor different damping parameters αcorresponding to the results in Fig. 1. correspond to those of the local magnetization discussed in connection with Fig. 1. III. MAGNON PROPAGATION LENGTH To describe the characteristic length scale of the magnon propagation around the temperature step we deﬁne the magnonaccumulation /Delta1m(z) as the difference between the relative equilibrium magnetization m 0(z) at the given phonon temper- atureTp(z) and the calculated local magnetization m(z): /Delta1m(z)=m0(z)−m(z). (6) We investigate the magnon propagation in the colder part of the system, where Tp(z)=0. For a small magnon temperature, the temperature dependence of the magnetization can beapproximated as m(T m)≈1−β TcTm.( 7 ) This linear equation is in agreement with an analytical solution for low temperatures presented by Watson et al. [12]. For low phonon temperatures one obtains for the difference betweenphonon and magnon temperature /Delta1T=T m−Tp=β Tc/Delta1m.( 8 ) Note, that the proportionality between magnon accumulation and temperature difference holds for higher temperaturesas well, as long as the magnon and phonon temperaturesare sufﬁciently close so that a linear approximation applies,though the proportionality factor increases. Note also, that thisproportionality was determined in theoretical descriptions of amagnonic spin Seebeck effect [ 7]. Our results for the magnon accumulation should hence be relevant for the understandingof the magnonic spin Seebeck effect where the temperaturedifference between the magnons in the ferromagnet and theelectrons in the nonmagnet plays a key role. We further investigate our model as before with a tempera- ture in the heated area of T 1 p=0.1J/k B, anisotropy constant dz=0.1J, and different damping parameters. The magnon accumulation /Delta1m versus the space coordinate zin the colder region of the system at Tp=0Ki ss h o w ni nF i g . 4. Apartα=1.00α=0.50α=0.10α=0.08α=0.06 space coordinate z/amagnon accumulation Δm 250 200 150 100 50 01 10−2 10−4 10−6 10−8 10−10 FIG. 4. (Color online) Magnon accumulation /Delta1m over space coordinate zin the colder region of the system at Tp=0Kf o r different damping constants αshows exponential decay with magnon propagation length ξ. The points show the data from our simulation and the lines the results from an exponential ﬁt. from a sudden decay close to the temperature step the magnon accumulation /Delta1m(z) then decays exponentially on a length scale that depends on the damping constant α. To describe this decay we ﬁt the data with the function /Delta1m(z)=/Delta1m(0)e−z/ξ.( 9 ) We deﬁne the ﬁtting parameter ξas the propagation length of the magnons. Here, the deviations from the exponential decayat the beginning of the system are neglected. The ﬁts for thedata are shown in Fig. 4as continuous lines. The propagation length dependence on the damping pa- rameter αi ss h o w ni nF i g . 5. The values of the propagation length from our simulations, shown as points, are inverselyproportional to the damping constant αand, furthermore, show also a strong dependence on the anisotropy constantd z. This behavior will be discussed in the next two sections with an analytical analysis of the magnon propagation and aninvestigation of the frequencies of the propagating magnons.A simple approximation for the propagation length leads toEq. ( 19) which is also shown as solid lines in Fig. 5. dz=0.01Jdz=0.05Jdz=0.10Jdz=0.50J damping constant αpropagation length ξ/a 1 0.1100 10 1 FIG. 5. (Color online) Propagation length ξover the damping constant αfor different anisotropy constant dz. Numerical data is shown as points and the solid lines are from Eq. ( 19). 024409-3ULRIKE RITZMANN, DENISE HINZKE, AND ULRICH NOW AK PHYSICAL REVIEW B 89, 024409 (2014) IV . ANALYTICAL DESCRIPTION WITH LINEAR SPIN-WA VE THEORY For the theoretical description of the magnon accumulation, excited by a temperature step in the system, we solve the LLGequation [Eq. ( 1)], analytically in the area with T p=0K .W e consider a cubical system with lattice constant awhere all spins are magnetized in zdirection parallel to the easy axis of the system. Assuming only small ﬂuctuations in the xandy directions we have Sz i≈1 andSx i,Sy i/lessmuch1. In that case we can linearize the LLG equation and the solution of the resultingequation consists of a sum over spin waves with wave vectorsqand the related frequency ω qwhich decay exponentially in time dependent on their frequency and the damping constantαof the system, S ± i(t)=1√ N/summationdisplay qS± q(0)e∓iqri±iωqte−αωqt. (10) The frequency ωqof the magnons is described by the usual dispersion relation /planckover2pi1ωq=1 (1+α2)/parenleftbigg 2dz+2J/summationdisplay θ[1−cos(qθaθ)]/parenrightbigg . (11) The dispersion relation includes a frequency gap due to the anisotropy constant and a second wave-vector-dependent termwith a sum over the Cartesian components [ 13]. Considering now the temperature step, magnons from the hotter area propagate towards the colder one. We investigatethe damping process during that propagation in order todescribe the propagating frequencies as well as to calculate thepropagation length ξof the magnons for comparison with the results from Sec. III. The magnon accumulation will depend on the distance to the temperature step and—for small ﬂuctuationsof the S xandSycomponents—can be expressed as /Delta1m(z)=1−/angbracketleftSz(z)/angbracketright≈1 2/angbracketleftSx(z)2+Sy(z)2/angbracketright, (12) where the brackets denote a time average. We assume that the local ﬂuctuations of the SxandSycomponents can be described with a sum over spin waves with different frequencies anddamped amplitudes a q(z), Sx(z)=/summationdisplay qaq(z) cos(ωqt−qr), (13) Sy(z)=/summationdisplay qaq(z)s i n (ωqt−qr). (14) In that case for the transverse component of the magnetization one obtains /angbracketleftSx(z)2+Sy(z)2/angbracketright=/angbracketleftBigg/summationdisplay qaq(z)2/angbracketrightBigg , (15) where mixed terms vanish upon time averaging. The magnon accumulation can be written as /Delta1m(z)=1 2/angbracketleftBigg/summationdisplay qaq(z)2/angbracketrightBigg . (16) The amplitude aq(z) of a magnon decays exponentially as seen in Eq. ( 10) dependent on the damping constant and thefrequency of the magnons. In the next step we describe the damping process during the propagation of the magnons. In theone-dimensional limit magnons only propagate in zdirection with velocity v q=∂ωq ∂q. Then the propagation time can be rewritten as t=z/vqand we can describe the decay of the amplitude with aq(z)=aq(0)f(z) with a damping function f(z)=exp/parenleftBigg −αωqz ∂ωq ∂qz/parenrightBigg . (17) The amplitudes are damped exponentially during the propagation which deﬁnes a frequency-dependent propagationlength ξ ωq=2a/radicalBig J2−/parenleftbig1 2(1+α2)(/planckover2pi1ωq−2dz)−J/parenrightbig2 α(1+α2)/planckover2pi1ωq,(18) where we used γ=μs//planckover2pi1. In the low anisotropy limit this reduces to ξωq=λ/πα , where λ=2π/q is the wavelength of the magnons. The total propagation length is then the weighted average over all the excited frequencies. The minimal frequency isdeﬁned by the dispersion relation with a frequency gap ofω min q=2dz/[/planckover2pi1(1+α2)]. For small frequencies above that minimum the velocity is small, so the magnons are dampedwithin short distances. Due to the fact that the damping processis also frequency dependent higher frequencies will also bedamped quickly. In the long wavelength limit the minimaldamping is at the frequency ω max q≈4dz/[/planckover2pi1(1+α2)] which can be determined by minimizing Eq. ( 17). In a three-dimensional system, besides the zcomponent of the wave vector, also transverse components of the wave vectorhave to be included. The damping of magnons with transversecomponents of the wave vector is higher than described inthe one-dimensional case, because the additional transversepropagation increases the propagation time. In our simulationsthe cross section is very small, so that transverse componentsof the wave vectors are very high and get damped quickly. Thisfact and the high damping for high frequencies described inEq. ( 17) can explain the very strong damping at the beginning of the propagation shown in Fig. 4. V . FREQUENCIES AND DAMPING OF PROPAGATING MAGNONS In this section we investigate the frequency distribution of the magnonic spin current while propagating away fromthe temperature step. First we determine the frequencies ofthe propagating magnons in our simulations with Fouriertransformation in time to verify our assumptions from thelast section. As before a system of 8 ×8×512 spins with a temperature step in the center of the system is simulated withan anisotropy of d z=0.1J. The temperature of the heated area isT1 p=0.1J/k Band the damping constant is α=0.1. After an initial relaxation to a steady state the frequency distributionof the propagating magnons in the colder area is determinedby Fourier transformation in time of S ±(i)=Sx(i)±iSy(i). The frequency spectra are averaged over four points in the x-y 024409-4PROPAGATION OF THERMALLY INDUCED MAGNONIC . . . PHYSICAL REVIEW B 89, 024409 (2014) 10 8 6 4 2 04× × × × × × × ×10−3 31 0−3 21 0−3 11 0−3 0z=2 0az=1 0az=1a(a) frequency ¯hωq/Jamplitude \|S+(ωq)\| 1.4 1.2 1 0.8 0.6 0.4 0.2 081 0−5 61 0−5 41 0−5 21 0−5 0ωminz=1 0 0 az=9 0az=8 0a(b) frequency ¯hωq/Jamplitude \|S+(ωq)\| FIG. 6. (Color online) Amplitude \|S+(ωq)\|versus the frequency ωqfor a system with 8 ×8×512 spins. (a) After propagation over short distances from 1 to 20 lattice constants. (b) After propagation over longer distances from 80 to 100 lattice constants. plane and analyzed depending on the distance zof the plane to the temperature step. The results for small values of zare shown in Fig. 6(a) and for higher values of z, far away from the temperature step, for the regime of the exponential decay, in Fig. 6(b). For small values of z, near the temperature step, the frequency range of the propagating magnons is very broad. The minimumfrequency is given by ω min q=2dz/[/planckover2pi1(1+α2)] and far away from the temperature step the maximum peak is aroundω max q=4dz/[/planckover2pi1(1+α2)]. These characteristic frequencies are in agreement with our ﬁndings in Sec. IV. Furthermore, a stronger damping for higher frequencies can be observed. This effect corresponds to the strong damping ofmagnons with wave-vector components transverse to the z direction and it explains the higher initial damping, which wasseen in the magnon accumulation in Fig. 4. A much narrower distribution propagates over longer distances and reaches thearea shown in Fig. 6(b). In that area the damping can be described by one-dimensional propagation of the magnonsinzdirection with a narrow frequency distribution around the frequency with the lowest damping ω max q=4dz/[/planckover2pi1(1+α2)]. The wavelength and the belonging group velocity of themagnons depending on their frequency in the one-dimensionalanalytical model are shown in Fig. 7(a). In the simulated system magnons with the longest propagation length havea wavelength of λ=14a. Depending on the ratio d z/Jthe wavelength increases for systems with lower anisotropy. Asdiscussed in the last section, magnons with smaller frequencies2 1.5 1 0.5 0 4 3.5 3 2.5 2 1.5 1 0.5 0100 80 60 40 20 0ωmin qvqλ(a) frequency ¯hωq/J velocity vq¯h/(Ja)wave length λ/a damping functionΔ=1 0 aΔ=2 0 aΔ=5 0 a(b) frequency ω[J/¯h]damping ratio 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.21 0.98 0.96 0.94 0.92 0.9 FIG. 7. (Color online) (a) Wavelength λand group velocity vqof the magnons in a one-dimensional model dependent on the frequency ωq. (b) Damping ratio as explained in the text versus the frequency ωqfor different distances /Delta1and compared to the damping function [Eq. ( 17)]. are less damped in the time domain, but due to their smaller velocity the magnons very close to the minimum frequencyalso have a smaller propagation length. To investigate the frequency-dependent damping process during the propagation of the magnons we calculate the ratioof the amplitude of the magnons \|S +(ωq,z)\|forz=80a andz=80a+/Delta1with/Delta1=10a,20a,50aand normalize it to a damping per propagation of one spin. The resultingratios [ \|S +(ωq,z)\|]/[\|S+(ωq,z−/Delta1)\|]1//Delta1are shown in Fig. 7 in comparison with the frequency-dependent damping func-tion [Eq. ( 17)]. The ﬁgure shows good agreement between simulation and our analytical calculations. These results explain the dependence of the magnon propagation length on the model parameters. The frequencywith the maximal amplitude is determined by the anisotropyconstant. Under the assumption that the frequency with thelowest damping is dominant and the contribution of otherfrequencies can be neglected the propagation length can becalculated as ξ=a 2α/radicalBigg J 2dz, (19) where the square-root term is the domain wall width of the model. This formula is also plotted in Fig. 5. The comparison with our simulations shows good agree- ment though the equation above gives only the propagationof those magnons with the smallest damping during the 024409-5ULRIKE RITZMANN, DENISE HINZKE, AND ULRICH NOW AK PHYSICAL REVIEW B 89, 024409 (2014) propagation. In the considered system with α=0.1 and dz=0.1Jwe get a propagation length of about ξ=11aat a wavelength of the magnons λ=14a. For smaller values of the anisotropy and smaller damping parameters the frequencydistribution of the thermal magnons is broader and Eq. ( 19) is an overestimation of the real propagation length since themagnon accumulation is no longer exponentially decaying dueto the broader spectrum of propagating frequencies. Howeverwe would expect for soft ferromagnetic insulators with a smalldamping constant of 10 −4–10−3and an anisotropy constant in the range of 10−3J–10−2Ja propagation length of 103a–105a which would be in the micrometer range. VI. SUMMARY AND DISCUSSION Using the framework of an atomistic spin model we describe thermally-induced magnon propagation in a model containing a temperature step. The results give an impressionof the relevant length scale of the propagation of thermally-induced exchange magnons and its dependence on systemparameters as the anisotropy, the exchange, and the dampingconstant. In the heated area magnons with a broad frequencydistribution are generated and because of the very strongdamping for magnons with high frequency, especially thosewith wave-vector components transverse to the propagationdirection in zdirection, most of the induced magnons are damped on shorter length scales. Behind this region ofstrong damping near the temperature step, the propagation ofmagnons is unidirectional and the magnon accumulation de-cays exponentially with the characteristic propagation lengthξ. This propagation length depends on the damping parameter but also on system properties as the anisotropy of the system,because of the dependence on the induced frequencies.In contrast to long-range magnetostatic spin waves, which can propagate over distances of some millimeters [ 14,15], we ﬁnd that for exchange magnons the propagation lengthis considerably shorter and expect from our ﬁndings forsoft ferromagnetic insulators with a low damping constanta propagation length in the range of some micrometers forthose magnons close to the frequency gap and the lowestdamping. These ﬁndings will contribute to the understandingof length-scale-dependent investigations of the spin Seebeckeffect [ 8,16–18]. Recent experiments investigate the longitudinal spin See- beck effect, where the generated spin current longitudinal tothe applied temperature gradient is measured [ 19–22]. In this conﬁguration Kehlberger et al. show that the measured spin current is dependent on the thickness of the YIG layer andthey observe a saturation of the spin current on a length scaleof 100 nm [ 16]. This saturation can be explained by the length scale of the propagation of the thermally excited magnons.Only those magnons reaching the YIG/Pt interface of thesample contribute to the measured spin current and—as shownhere—exchange magnons thermally excited at larger distancesare damped before they can reach the interface. In this paper,we focus on the propagation length of those magnons withthe lowest damping, however, the length scale of the magnonaccumulation at the end of a temperature gradient is dominatedby a broad range of magnons with higher frequencies whichare therefore damped on shorter length scales. ACKNOWLEDGMENTS The authors would like to thank the Deutsche Forschungs- gemeinschaft (DFG) for ﬁnancial support via SPP 1538 “SpinCaloric Transport” and the SFB 767 “Controlled Nanosystem:Interaction and Interfacing to the Macroscale.” [1] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391(2012 ). [2] G. E. W. Bauer, A. H. MacDonald, and S. Maekawa, Solid State Commun. 150,459(2010 ). [3] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature (London) 455,778 (2008 ). [4] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett.88,182509 (2006 ). [5] K. Uchida, S. Takahashi, J. Ieda, K. Harii, K. Ikeda, W. Koshibae, S. Maekawa, and E. Saitoh, J. Appl. Phys. 105,07C908 (2009 ). [6] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y . Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer,S. Maekawa, and E. Saitoh, Nat. Mater. 9,894(2010 ). [7] J. Xiao, G. E. W. Bauer, K.-c. Uchida, E. Saitoh, and S. 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PhysRevB.93.064418.pdf	PHYSICAL REVIEW B 93, 064418 (2016) Thermally driven spin torques in layered magnetic insulators Scott A. Bender and Yaroslav Tserkovnyak Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA (Received 12 November 2015; revised manuscript received 19 December 2015; published 17 February 2016) Thermally driven spin-transfer torques have recently been reported in electrically insulating ferromagnet \|normal-metal heterostructures. In this paper, we propose two physically distinct mechanisms for such torques. The ﬁrst is a local effect: out-of-equilibrium, thermally activated magnons in the ferromagnet,driven by a spin Seebeck effect, exert a torque on the magnetization via magnon-magnon scattering with coherentdynamics. The second is a nonlocal effect which requires an additional magnetic layer to provide the symmetrybreaking necessary to realize a thermal torque. The simplest structure in which to induce a nonlocal thermaltorque is a spin valve composed of two insulating magnets separated by a normal metal spacer; there, a thermalﬂux generates a pure spin current through the spin valve, which results in a torque when the magnetizations ofthe layers are misaligned. DOI: 10.1103/PhysRevB.93.064418 I. INTRODUCTION The growing ﬁeld of spin caloritronics [ 1] complements the electrical control of spin current with a new experimental bias:temperature gradient. In contrast to electrical biasing, whichcouples to the electron charge, transport by the application ofa thermal ﬂux is possible for neutral carriers. If, for example,a temperature gradient is applied to a magnetic insulator, anet ﬂow of angular momentum, carried by thermally activatedspin-wave excitations, results [ 2]. When integrated into larger structures, magnonically-active elements open the possibil-ity of new effects and devices based on thermally driventransport [ 3]. One such effect is that of a thermal spin-transfer torque at a normal-metal \|insulating ferromagnet interface, which has been recently observed [ 4] via the modulation of ferromagnetic resonance linewidth. Thermally driven magnetic dynamicswere predicted [ 5–7] and reported [ 8]f o rconducting ferromag- netic layers, where the spin-transfer torque can be provided byspin-polarized electric current injected into the magnetic layerby an interfacial spin-dependent Seebeck effect [ 9]. In contrast, for an insulating ferromagnetic layer, spin-transfer torque canarise only from a thermally driven pure spin current mediatedby ferromagnetic magnons. A general framework, describingthe interplay between magnon transport and the ferromagneticorder-parameter dynamics, however, has been lacking. In this paper, we provide an account of the physics of thermal magnon-mediated spin-transfer torques arising innormal-metal (N) \|insulating-ferromagnet (F) heterostructures, building on the formalism developed in our previous works[10,11]. In Sec. II, we construct a local mechanism, which utilizes SU(2) symmetry breaking of an anisotropic F tocouple thermally activated magnons to the spin-density orderparameter. In conjunction with the spin Seebeck effect, thisengenders a thermally driven torque in F. Reference [ 12] similarly addresses the issue of a thermally induced torque atan N\|F interface; while the mechanism therein relies on phonon drag at the interface, ours is routed in the interfacial exchangeinteraction. In Sec. III, we investigate a nonlocal mechanism, where the SU(2) symmetry is structurally broken. The simplestexample of this is an F \|N\|F trilayer, in contact with normal- metal leads that serve as reservoirs of angular momentum.A thermomagnonic ﬂux passing through the ferromagnetic components results in a spin accumulation in the normal-metalspacer, which exerts a torque on the ferromagnetic layers,in close analogy with a traditional electronic spin valve. Forboth mechanisms, we obtain the change in magnetic dampingin linear response to a temperature gradient and considermagnetic dynamics induced beyond linear response. II. LOCAL MECHANISM For illustrative purposes, we discuss the local mechanism for thermal spin-transfer torque in the simplest possiblestructure: an N \|F bilayer. A spin current entering F (assumed to form a single domain) through the N \|F interface is comprised of two orthogonal, physically distinct components. The ﬁrst isthe spin current collinear with the spin density order parameterunit vector n, with ntaken to be spatially uniform in the thin ﬁlm limit; physically, this current arises from thermalﬂuctuations and on the F side is transported by magnons. Thesecond current, which is orthogonal to nand linear in n×μ /prime (where μ/primeis the spin accumulation in N along the interface) and ˙n, gives rise to the spin-transfer torque on n[13]. In the presence of a temperature gradient across the N \|F interface, a spin current of the ﬁrst kind ﬂows, which results in thebuildup of a thermally induced spin accumulation in N alongthe interface. Crucially, this spin accumulation is collinearwith the order parameter nand therefore cannot produce a spin-transfer torque on n. A temperature gradient maintained across an N \|F interface cannot exert a spin torque on F in the absence of SU(2) symmetry breaking. A thermal spin torque on nin an N \|F structure (Fig. 1) therefore requires an SU(2) symmetry breaking anisotropy bythe F layer itself, which is not explicitly provided by Ref. [ 4]. In the simplest case, which we consider here, this is providedby local uniaxial anisotropy. The ferromagnetic Hamiltonianis ˆH=/integraldisplay d 3x/parenleftbigg −A 2sˆs·∇2ˆs+Hˆsz+K 2sˆs2 z/parenrightbigg . (1) Here, Ais the exchange stiffness, sis the saturation spin density (in units of /planckover2pi1), and Kis the anisotropy constant (in units of energy), which is easy plane when K> 0 and 2469-9950/2016/93(6)/064418(6) 064418-1 ©2016 American Physical SocietySCOTT A. BENDER AND Y AROSLA V TSERKOVNY AK PHYSICAL REVIEW B 93, 064418 (2016) k1 k2k3damping k2k3 k1antidamping(a) τl τl 00 TTxyz nφθ αμ=μz nxδT(b) N F FIG. 1. (a) Schematic for the N \|F bilayer. An interfacial temper- ature drop δTdrives spin current into F, which is absorbed by the magnons. (b) Magnon scattering processes, opened when the spin density order parameter nis misaligned with the F broken-symmetry axisz. The annihilation of one ﬁnite k(thermal) magnon and the corresponding creation of two robs nof/planckover2pi1of angular momentum in thezdirection, resulting in a damping torque; the inverse process generates an antidamping torque. Wavy lines represent ˆ ϕmagnons, while straight represent /Phi1magnons. easy axis when K< 0. The spin density operator ˆsconsists of a coherent piece /angbracketleftˆs/angbracketright= ˜snaround which the spin density ﬂuctuates incoherently: δˆs=ˆs−/angbracketleftˆs/angbracketright. These ﬂuctuations are composed of magnons, which reduce the effective spin densityto˜s=s(1−n x/s), where nxis the thermal magnon density. The anisotropy couples the thermal cloud and n, allowing for an exchange of angular momentum between the two viathe scattering of thermal magnons. The local mechanismfor the thermally driven torque works as follows: Whena temperature gradient is applied across the N \|F interface, angular momentum is driven into the normal cloud by thespin Seebeck effect. The out-of-equilibrium cloud relaxes theexcess angular momentum to the order parameter via thiscoupling, thereby exerting a torque on n. 1. Scattering Using the Holstein-Primakoff transformation, the spin density may be mapped to boson ﬁeld operators ˆ/Phi1(x) and ˆ/Phi1†(x)[14]: ˆsz=ˆ/Phi1†ˆ/Phi1−s, ˆs−(x)=/radicalbig 2s−ˆ/Phi1†ˆ/Phi1ˆ/Phi1, (2) where ˆs±=ˆsx±iˆsyand [ ˆ/Phi1(x),ˆ/Phi1†(x/prime)]=δ(x−x/prime). It is convenient to decompose ˆ/Phi1into/Phi1=/angbracketleftˆ/Phi1/angbracketright, corresponding to a coherent condensate magnon, and ˆ ϕ, which describes ﬂuctuations around /Phi1: ˆ/Phi1(x)=/Phi1+ˆϕ(x). (3) The quanta of ˆ ϕare incoherent magnons, each of which carry angular momentum −/planckover2pi1n≈/planckover2pi1z; For our purposes, these magnons are thermally activated, so that the thermal magnoncloud density is given by n x=/angbracketleftˆϕ†ˆϕ/angbracketright. Writing /Phi1≡√nce−iφ, which plays the role of the condensate wave function, the totalaverage angular momentum in the zdirection becomes /planckover2pi1/angbracketleftˆs z/angbracketright= /planckover2pi1(nx+nc)−/planckover2pi1s.U s i n gE q .( 2) to compute the remaining components of /angbracketleftˆs/angbracketright= ˜sn, one identiﬁes φas the azimuthal angle between nand the xaxis; we assume that nc+nx/lessmuchs, so that the condensate density nc, which parametrizes themisalignment of nwith−z, can be written as nc≈(˜s/2)θ2, withθas the polar angle between them (see Fig. 1). Expanding Eq. ( 2)i nˆ/Phi1†ˆ/Phi1/s and inserting Eq. ( 3) generates terms of various powers of ˆ ϕand/Phi1, which divide into two classes: ˆ ϕmagnon number conserving and noncon- serving. In the absence of driving, the former terms relax the thermal magnon distribution fk≡/angbracketleftˆϕ† kˆϕk/angbracketright(with ˆ ϕk=/integraltext d3xeik·xˆϕ/√ VandVas the volume of F) towards a Bose-Einstein proﬁle: fBE(/epsilon1k)=1/[eβ(/epsilon1k−μ)+1] with a well- deﬁned magnon temperature T=1/β(in units of energy) and chemical potential μ. The magnon spectrum /epsilon1k=Ak2+Uis shifted by the Hartree-Fock mean-ﬁeld potential U=/planckover2pi1/Omega1+ 2Knc/s, where /planckover2pi1/Omega1=H−K(1−2nx/s). The relaxation time associated with these processes depends on both theexchange and anisotropy terms in Eq. ( 1). Focusing on high temperatures ( T/greatermuch/planckover2pi1/Omega1), the exchange mechanism dominates, and the relaxation time is fast [ 10]; we shall therefore suppose that the thermal magnon cloud is parametrized byTandμ, even in the presence of driving. In addition, in equilibrium Gilbert damping establishes μ=0, while inelastic spin-preserving magnon-phonon scattering ﬁxes the magnontemperature to that of the phonons. Because the former typeof magnon-lattice interaction relies on spin-orbit coupling, itis generally weaker than the latter; in this spirit, we shallsuppose that the magnon temperature always remains pinnedto that of the phonon temperature, while μm a yb ed r i v e nf r o m its equilibrium value. The spin torque on narises from terms in ˆHthat break ˆϕmagnon number conservation. The exchange interaction, which is independent of nby SU(2) symmetry, does not contribute. However, when nis misaligned with the zaxis, anisotropy generates a contribution to ˆH, ˆH cx(n)=(K/s )/Phi1∗/integraldisplay d3xˆϕ†(x)ˆϕ(x)ˆϕ(x)+H.c.,(4) opening a magnon scattering channel that redistributes z angular momentum between the thermal cloud and orderparameter: two ˆ ϕmagnons are annihilated (created), creating (annihilating) one ˆφmagnon and one /Phi1magnon. Because the total z-angular momentum carried by the spin density is conserved by rotational symmetry, the corresponding loss(gain) of /planckover2pi1zangular momentum by the thermal magnon cloud is compensated by the absorption (emission) of angularmomentum by /Phi1, which is translated as an antidamping (damping) spin torque on n. The resulting scattering rate at which angular momentum is transferred between the thermalcloud and the order parameter is obtained by Fermi’s goldenrule and given by [ 15]: /Gamma1=2η(/planckover2pi1ω−μ)n c=/planckover2pi1˙nx\|cx=−/planckover2pi1˙nc\|cx. (5) Here, fi=fBE(/epsilon1ki),/planckover2pi1ω≡/planckover2pi1/Omega1+Knc/sis the precessional frequency of naround −z, andηis given by: η=(K/s )2 T(2π)5/integraldisplay d3k1/integraldisplay d3k2/integraldisplay d3k3δ(k1−k2−k3) ×δ(/planckover2pi1ω+/epsilon1k1−/epsilon1k2−/epsilon1k3)(1+f1)f2f3, (6) which can be written as: η=¯ηI, where ¯ η≡(T/T c)3(K/T )2, Tc=As2/3approximates the Curie temperature, and Iis a dimensionless integral that depends on ( μ−U)/T and 064418-2THERMALLY DRIVEN SPIN TORQUES IN LAYERED . . . PHYSICAL REVIEW B 93, 064418 (2016) /planckover2pi1ω/T . According to Eq. ( 5), when μ=/planckover2pi1ω, the thermal cloud and order parameter are in equilibrium, corresponding to anentropic maximum of the closed magnetic subsystem of F. 2. Driven magnetic dynamics At zero temperature, the thermal cloud is absent, and the dynamics of the order parameter of F are described by theLandau-Lifshitz-Gilbert phenomenology: (1+αn×)/planckover2pi1˙n+n×H=(α /prime i+α/prime rn×)(μ/prime×n−/planckover2pi1˙n),(7) where H=(H+Kz·n)zis comprised of the applied ﬁeld Hand the anisotropy ﬁeld Kz·n, andαis the bulk Gilbert damping of F, describing the ﬂow of angular momentumfrom nto the lattice. The quantities α /prime randα/prime idescribe angular momentum transfer with the normal metal N andare the real and imaginary parts of g ↑↓/4πsdF, where g↑↓ is the spin-mixing conductance at the interface and dFis the thickness of F. In general, the spin accumulation μ/primemust be self-consistently determined by complementing magneticdynamics with a treatment of spin transport in N; we shallcircumvent this inessential complication by taking N to be agood spin sink, so that the spin accumulation is electricallytunable (by, e.g., the spin Hall effect) independently of thetemperature gradient [ 16] and chosen to be along the zaxis: μ /prime=μ/primez. Provided H−K> 0, the equilibrium ( μ/prime=0) solution to Eq. ( 7)i sn=−z. At ﬁnite temperatures, the scattering by the thermal cloud of magnons modiﬁes the order parameter dynamics, andthe coefﬁcients α /prime randα/prime iacquire temperature dependent corrections ∼nx/s. The temperature dependent magnon gap /planckover2pi1/Omega1must be positive (which we will assume through the remainder of Sec. II)f o r n=−z(nc=0) to be a stable equilibrium in the absence of driving. Here, it is convenient torecast Eq. ( 7) as a rate equation for n c, to which the scattering rate/Gamma1is phenomenologically added; denoting α/prime r=α/prime, and neglecting higher order terms in the α’s, one has, for small angle dynamics ( θ/lessmuch1): /planckover2pi1˙nc=2α/primeμ/primenc−2(α+α/prime)/planckover2pi1ωnc−/Gamma1, (8) where the ﬁrst two terms on the right-hand side follow from Eq. ( 7), while /Gamma1is given by Eq. ( 5). Equation ( 8) can be recast back as a ﬁnite-temperature Landau-Lifshitz-Gilbert equation(valid for small angle dynamics): (1+αn×)/planckover2pi1˙n+n×˜H=α /primen×(μ/prime×n−/planckover2pi1˙n)+τl,(9) which is one of the central results of this paper. Here, τl= ηn×(μ×n−/planckover2pi1˙n) is the local spin torque, with μ=μz, and ˜H=[/planckover2pi1/Omega1+K(1+z·n)]z. Complementing the dynamics of the order parameter, Eq. ( 8), is that of the thermal cloud. In response to a spin accumulation μ/primeand/or a temperature gradient, angular momentum in the −ndirection is driven into (out of) F and absorbed (emitted) by the thermal cloud, creating anout-of-equilibrium chemical potential μ> 0(<0). For ﬁxed magnon temperature, the rate of change of the chemicalpotential resulting from these biases may be obtained fromthe total rate equation for the thermal-cloud density: /planckover2pi1˙n x=˙μ∂μnx=j/bardbl/dF−Gdμ/dF+/Gamma1. (10)In the ﬁrst term on the right-hand side, j/bardblis the current injected from N across the interface, which in linear response is givenby j /bardbl=G(μ/prime−μ)+SδT . (11) The quantities [ 11,17]G∼g↑↓(T/T c)3/2andS∼Gare the temperature-dependent interfacial magnon conductance andspin Seebeck coefﬁcients, which are both proportional tog ↑↓;δT=T/prime−Tis the difference between the electron temperature T/primeand magnon temperature T. The second term on the right-hand side of Eq. ( 10), parametrized by Gd∼ α, describes the Gilbert damping of thermal-cloud angular momentum into the F lattice, which, in the absence of drivingby N, relaxes μto zero. Together, Eqs. ( 8) and ( 10) form a closed set of coupled equations for the condensate density n cand the thermal cloud chemical potential μ. Separating the “fast” dynamics of the thermal magnons from the “slow” dynamics of the orderparameter, we solve Eq. ( 10) for the magnon steady-state condition ˙n x=0 to obtain a chemical potential μ=(SδT+ Gμ/prime+2ηnc/planckover2pi1ωdF)/(Gd+G+2dFαnc). 3. Ferromagnetic resonance linewidth Focusing on behavior near equilibrium ( μ/prime=δT=μ= nc=0), Eq. ( 8) may be written as /planckover2pi1˙nc=2αtot/planckover2pi1ωnc, where αtot=α+α/prime+η−(α/primeμ/prime+ημ)//planckover2pi1ωis the total damping ofn, and ηis evaluated in equilibrium ( μ=nc=0). To lowest order in nc,μ=(SδT+Gμ/prime)/(Gd+G), which when inserted into Eq. ( 8) yields αtot=α+α/prime+η+/Delta1αμ/prime+ /Delta1αT, governing the relaxation of nc. The contribution /Delta1αμ/prime=− [α/prime+η/(1+Gd/G)](μ/prime//planckover2pi1/Omega1) consists of the zero- temperature term ∝α/primeand the thermal enhancement ∝η.T h e change in damping resulting from a temperature gradient, /Delta1αT=−ηS Gd+GδT /planckover2pi1/Omega1, (12) is due to the thermal magnons in its entirety and is one of the central results of this paper. The sign of Eq. ( 12) can be understood from the fact that when δTis positive, magnons carrying spin in the +zdirection are injected into F, reducing the damping of n(/Delta1αT<0). Both /Delta1αμ/primeand/Delta1αTcan be deduced from ferromagnetic resonance measurements. 4. DC-pumped magnon condensates Finite-angle dynamics of nmay be excited upon the application of a sufﬁciently large spin accumulation and/ortemperature gradient. This was the subject of Ref. [ 10], in the limit in which the condensate and cloud are stronglycoupled ( η→∞ ). There, when F is in normal phase ( n c=0), μ</planckover2pi1/Omega1is determined from the steady state condition ˙nx=0, as in Sec. II 3; when F is in condensate phase ( nc>0), the Bose-Einstein gas of thermal magnons becomes saturated(μ=/planckover2pi1/Omega1), andn cis determined from the steady state condition ˙nc=0. Together, Eqs. ( 8) and ( 10) represent a generalization of Ref. [ 10] to ﬁnite cloud-condensate coupling, with the structure of the phase diagram determined by the steady-statesolutions for μandn cto the joint condition ˙nc=˙nx=0. In the strong condensate-cloud coupling regime ( η/greatermuchα,α/prime), the phase diagram of Ref. [ 10] is reproduced. More generally, 064418-3SCOTT A. BENDER AND Y AROSLA V TSERKOVNY AK PHYSICAL REVIEW B 93, 064418 (2016) BEC μ=2Ω μ=Ω μ=0 0−1 −212 3 1 20 μ=−Ω−δT/Ω μ/Ωμ=μc FIG. 2. Phase diagram for F \|N bilayer corresponding to the stable solutions to the coupled Eqs. ( 8)a n d( 10), demonstrating normal phase, Bose-Einstein condensation (BEC), and swasing ( μ/prime> /planckover2pi1/Omega1[1+(α+η)/α/prime]). Here we have taken η=α/2=α/prime/2,K=/planckover2pi1/Omega1, T=102/planckover2pi1/Omega1,a n ds(A//planckover2pi1/Omega1)3/2=104.W h e n μ>/planckover2pi1/Omega1(below the phase transition), the thermal cloud is oversaturated. (See Ref. [ 10].) the cloud chemical potential must overcome a threshold μc= [1+(α+α/prime)/η]/planckover2pi1/Omega1−α/primeμ/prime/ηin order to realize a steady-state condensate; thus, the cloud may become oversaturated, withμ>/planckover2pi1/Omega1[18], but damping by the lattice and N relaxes angular momentum of the condensate more quickly than itis replenished by cloud-condensate scattering when μ<μ c, so that F remains in normal phase. The corresponding phasediagram is shown in Fig. 2, with the terminology borrowed from Ref. [ 10]. III. NONLOCAL MECHANISM The second mechanism for thermal spin-transfer torque relies on the presence of an additional ferromagnetic layer toprovide the SU(2) symmetry breaking required to realize atorque on n. Let us now consider the simplest such structure: a spin valve, composed of two ferromagnet layers (one freeand one ﬁxed) separated by a normal metal spacer, as depictedin Fig. 3. In a conducting spin valve, electrical or thermal biasing generates a two channel spin current, [ 7,19] carried by electrons parallel and antiparallel to the order parameter in themagnetic layers, which exerts a torque on the free layer. In our electrically insulating structure, thermal biasing (applied perpendicularly to the plane) generates a pure spincurrent that is single channel, carried through the ferromag-netic layers by magnons; this current results in a nonlocaltorque on the free magnetic layer order parameter nas a consequence of the misalignment of the free and ﬁxed layers.Slonczewski [ 20] has proposed a similar scheme. There, a heat current is converted into a spin current via a ferrite layer, whichis coupled to a paramagnetic monolayer by superexchange; free layer spacer xyz nz φθ nxδTlead δT δT δTfixed layer lead αl αs αl αsμ FIG. 3. Thermally biased spin valve. A heat ﬂux drives spin accumulation μ/prime(in the plane deﬁned by nandz) into the normal metal spacer. When free layer spin density is misaligned with thezaxis,μ /primeis no longer collinear with n, and the component of μ/prime perpendicular to nprovides a torque. spin current is subsequently transferred to the conduction electrons of a spacer and ultimately to a free magnet. Incontrast, our proposal relates the thermal spin ﬂux directly tothe spin-mixing conductance, a readily measurable quantity,circumventing the need for a paramagnetic monolayer. The spin valve we consider is a ﬁve layer structure (N- lead\|free-F\|N-spacer \|ﬁxed-F \|N-lead), the mirror symmetry of which is broken by the fact that the pinned layer is ﬁxed(e.g., by exchange biasing) with spin density oriented in the−zdirection. For simplicity, we assume all of the transport coefﬁcients for the free and ﬁxed layers to be identical.In equilibrium, the free layer is oriented either parallel orantiparallel to the ﬁxed layer. In order to maximize theefﬁciency of spin transport across the structure, let us assumethat the thickness d Fof each of the (monodomain) ferromagnet layers is much shorter than the thermal magnon diffusionlength. Likewise, we take the normal-metal spacer thicknessd sto be much shorter than the electronic spin diffusion length therein, which may be accomplished by using a poor spin sink(e.g., Cu). In contrast, let us for simplicity assume that thenormal-metal leads attached to the ferromagnets are excellentspin sinks, such as Pt, so that no spin accumulates inside them.Spin transport between the magnetic layers and the spacer (s) is parametrized by the spin-mixing conductance g ↑↓ sand between F and the leads ( l)b yg↑↓ l. In response to a temperature gradient applied across the structure, a spin current of magnons ﬂows through the mag-netic layers, resulting in a spin accumulation μ /primein the normal metal spacer. By symmetry, when the magnetic moments ofthe two ferromagnetic layers are parallel, μ /primevanishes; for all other orientations, μ/primebuilds up in the plane deﬁned by n(the free layer spin density order parameter) and z(the ﬁxed layer). When the two magnetic layers are misaligned, thespin accumulation exerts a dampinglike nonlocal torque τ nl= −α/prime sn×n×μ/primeon the free layer (with α/prime s=/Rfracturg↑↓ s/4πsdFas the effective Gilbert damping coefﬁcient due to contact withthe spacer). Following the approach of Sec. IIof separating order parameter and magnonic timescales and focusing on the latter,we have the magnon spin current density j iinto ferromagnetic layer F i, for a ﬁxed orientation of n, is (with i=1 as the free layer and i=2 the ﬁxed layer): /planckover2pi1˙nidF=ji=jl→i+js→i+˜ji, (13) 064418-4THERMALLY DRIVEN SPIN TORQUES IN LAYERED . . . PHYSICAL REVIEW B 93, 064418 (2016) where niis the thermal cloud magnon density in layer i, jl→i=−Glμi−(−1)iSlδTis the current entering F ifrom the lead, js→i=−Gs(μi+ni·μ)+(−1)iSsδTis the spin current entering F ifrom the spacer, ˜ji=−Gdμiis the spin current lost to Gilbert damping of thermal magnons, μiis the thermal magnon chemical potential in each ferromagnet,n 1=n, and n2=−z. The quantities GlandGsare the magnon conductances of the interfaces of F iwith the leads (l) and spacer ( s), respectively, and SlandSsare the spin Seebeck coefﬁcients of the interfaces of F with the leads andspacer. While a thorough treatment of spin transport involvesa detailed account of how magnon, phonon, and electrontemperature proﬁles are established throughout the structure,we have assumed, for our proof-of-principle calculation, thatthe electron/magnon temperature difference δTis the same across all interfaces (see Fig. 3). The rate of change of the spin density ρ=D Fμ(/planckover2pi1/2) (with DFas the Fermi surface density of states) inside the spacer is: ˙ρds=(−n)j1→s+zj2→s, (14) withji→s=−js→i. In a steady state of magnon ﬂux, we require ˙n1,˙n2, and ˙ρto vanish, which, employing Eqs. ( 13) and (14), yields a closed set of ﬁve equations for μ1,μ2, andμ/prime. Solving for the latter and inserting into τnl=−α/prime sn×n×μ/prime yields the thermally driven torque on n. In order to characterize linear response and magnetic dynamics, it sufﬁces to ﬁnd τnlnear the parallel conﬁguration of the free and ﬁxed layers ( n=−z) and the antiparallel conﬁguration ( n=z). Near the parallel conﬁguration, we obtain τnl≈σpδTn×n×z, (15) where σp=α/prime s(Gd+Gl)Ss+GsSl 2Gs(Gd+Gs+Gl); (16) near the antiparallel conﬁguration, τnl≈σapδTn×n×z (17) where σap=α/prime s 2/parenleftbiggSs Gs+Sl Gd+Gl/parenrightbigg . (18) The dynamics of the free layer are captured by the LLG equation: (1+αn×)/planckover2pi1˙n+n×H=−/planckover2pi1(α/prime l+α/prime s)n×˙n+τnl,(19) where we’ve assumed τl/lessmuchτnlso that the local torque is neglected. In order to characterize the small angle dynamicsof the free layer near the poles n=±z, we parametrize n by spherical coordinates as above and expand Eq. ( 19)i nθ, neglecting for simplicity the local torque τ l. Near the parallel conﬁguration ( θ=0), we obtain an equation of motion /planckover2pi1˙θ≈ −/epsilon1pθ, where /epsilon1p=αp/planckover2pi1/Omega1, with αp=α/prime l+α/prime s+α+/Delta1αpas the total damping and /Delta1αp=σpδT//planckover2pi1/Omega1, (20) as the change in damping near the parallel conﬁguration resulting from the δT. Near the antiparallel conﬁguration (θ=π),/planckover2pi1˙θ≈/epsilon1ap(π−θ), where /epsilon1ap=αap/planckover2pi1˜/Omega1, with /planckover2pi1˜/Omega1= δT δTH K K<0 H>0 STO P AP easy axis easy plane ap>0 p>0BS ll(easy axis) p>0 ap>0H −H K−K FIG. 4. Phase diagrams for the free layer in the spin valve for constant K(<0) and constant H(>0), showing parallel (P), antiparallel (AP), bistable (BS), and spin-torque oscillator (STO) phases. H+K(1−2nx/s) as the magnon gap there, αap=α/prime l+α/prime s+ α+/Delta1αapas the total damping, and /Delta1αap=−σapδT//planckover2pi1˜/Omega1 (21) as the change in damping near the antiparallel conﬁguration. In order to understand the signs of Eqs. ( 20) and ( 21), and hence of induced torques Eqs. ( 15) and ( 17), suppose that δTis negative so that a heat ﬂux ﬂows from right to left in Fig. 3. Via magnons in the ﬁxed layer, this heat current is accompanied by a spincurrent carrying angular momentum in the +zdirection, which when transferred to the free layer order parameter decreases thedamping in the parallel conﬁguration ( /Delta1α p<0) or increases the damping in the antiparallel conﬁguration ( /Delta1αap>0). Beyond linear response, the spin valve can be driven into different phases, the boundaries of which are deﬁned by theplanes /epsilon1 p=0 and/epsilon1ap=0i nH−K−δTspace. When both /epsilon1pand/epsilon1apare positive, the free layer is bistable. When /epsilon1pis positive (negative) and /epsilon1apis negative (positive), the free layer is stabilized in the −z(+zdirection). Last, when both /epsilon1pand /epsilon1apare negative, the dynamics stabilize to a limit cycle with 0<θ<π , i.e., the magnet is a spin-torque oscillator. The resulting phase diagram is show in Fig. 4. IV . CONCLUSION We have shown how both local and nonlocal thermally driven spin torques τlandτnlmay arise in magnet/metal het- erostructures, which modify the magnetic dynamics [Eqs. ( 9) and ( 19)]. In linear response, these torques manifest in both the bilayer and spin valve as changes in the damping [Eqs. ( 12), (20), and ( 21)]; at a threshold bias for each structure, the net effective damping reverses its sign, resulting in ﬁnite-angledynamics. For the case of the spin valve, the change in thedamping of the free layer, near either orientation, goes as/Delta1α∼α /prime sδT//planckover2pi1/Omega1. When F is sufﬁciently thin, α/prime sbecomes comparable to α, so that a temperature difference δT∼/planckover2pi1/Omega1 results in a change in damping /Delta1α that can overcome the intrinsic Gilbert damping; taking α∼10−4and using yttrium- iron-garnet (YIG) magnetic layers with platinum leads, thisis the case when the thickness of F is less than ∼100 nm. For an F \|N bilayer, which relies on the local mechanism, assuming again that α /prime>α, the change in damping goes as∼ηδT/ /planckover2pi1/Omega1. Supposing that K≈4πM2 s/scorresponds to shape anisotropy, and using YIG parameters (with Ms≈ 064418-5SCOTT A. BENDER AND Y AROSLA V TSERKOVNY AK PHYSICAL REVIEW B 93, 064418 (2016) 150 emu /cm3as the saturation magnetization), one arrives at ¯η≈10−6, which may be further enhanced by the factor I. Our model makes several assumptions. First, we rely on strong magnon-magnon scattering to thermalize the cloudof incoherent magnons to a Bose-Einstein proﬁle. At lowtemperatures and high driving, the magnons may no longer beparametrizable by a local chemical potential and temperature.Additionally, implicit in our treatment is the assumption ofstrong inelastic spin-preserving magnon-phonon coupling thatﬁxes the magnon temperature to that of the phonons, which istaken to increase in a steplike fashion across the structure.If the magnon-phonon coupling is not sufﬁciently strong,the magnon temperature proﬁle must be determined from anappropriate theory for the magnon heat transport. Likewise,in order to make quantitative experimental predictions, amore detailed account of how heating in the normal metalsestablishes a phononic temperature proﬁle across the structureis necessary. We have exploited the separation of magnonic and order-parameter dynamical time scales, which may need to beexamined more carefully in practice. Last, the scattering ratecoefﬁcient ηdepends sensitively on the low-energy magnon spectrum and inelastic scattering that may not be sufﬁcientlystrong to achieve full thermalization; further work is warrantedto address the problem of the magnon equilibration at thebottom of the magnon band, for a given material underconsideration. Future efforts may also apply and extend ourapproach to the problem of thermal spin torques to newstructures, e.g., more complex multilayers, superlattices, andantiferromagnets. ACKNOWLEDGMENT This work was supported by the US DOE-BES under AwardNo. DE-SC0012190. [1] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012 ). [2] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y . Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer,S. Maekawa, and E. Saitoh, Nat. Mater. 9,894(2010 ); H. Adachi, K.-i. Uchida, E. Saitoh, and S. Maekawa, Rep. Prog. Phys. 76, 036501 (2013 ). [3] A. V . Chumak, V . I. Vasyuchka, A. A. Serga, and B. 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Kimura, NPG Asia Mat.6,e127 (2014 ).[10] S. A. Bender, R. A. Duine, A. Brataas, and Y . Tserkovnyak, Phys. Rev. B 90,094409 (2014 ). [11] S. A. Bender and Y . Tserkovnyak, P h y s .R e v .B 91,140402 (2015 ). [12] Y . Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa, Phys. Rev. B92,224404 (2015 ). [13] J. C. Slonczewski, J. Magn. Magn. Mater. 159,L1(1996 ). [14] T. Holstein and H. Primakoff, Phys. Rev. 58,1098 (1940 ). [15] E. Zaremba, T. Nikuni, and A. Grifﬁn, J. Low Temp. Phys. 116, 277 (1999 ). [16] In practice, the generation of a spin accumulation by electrical means introduces also Joule heating in N, resulting in atemperature gradient which is quadratic in the bias. [17] Apart from the inclusion of Gilbert damping, Eq. ( 10)i st h e linear response version of Eq. ( 21)i nR e f .[ 10] online, with the coefﬁcients g /primeandS/primeobtained by expanding Eqs. ( 18)a n d( 19) in Eq. ( 10). [18] Strictly speaking, μcannot exceed U(=/planckover2pi1/Omega1in normal phase) for the Bose-Einstein function. We, however, focuson scattering and transport at high energies ∼T,w h e r e the deviations from a Bose-Einstein distribution should beinsigniﬁcant. [19] M. Johnson and R. H. Silsbee, P h y s .R e v .L e t t . 55,1790 (1985 ); K. Uchida, T. Ota, K. Harii, S. Takahashi, S. Maekawa, Y . Fujikawa, and E. Saitoh, Solid State Commun. 150,524 (2010 ). [20] J. C. Slonczewski, P h y s .R e v .B 82,054403 (2010 ). 064418-6
PhysRevB.97.094424.pdf	PHYSICAL REVIEW B 97, 094424 (2018) Spin-wave beam propagation in ferromagnetic thin ﬁlms with graded refractive index: Mirage effect and prospective applications Pawel Gruszecki and Maciej Krawczyk Faculty of Physics, Adam Mickiewicz University in Poznan, Umultowska 85, Pozna ń61-614, Poland (Received 8 December 2017; revised manuscript received 14 February 2018; published 23 March 2018) Using analysis of isofrequency contours of the spin-wave dispersion relation, supported by micromagnetic simulations, we study the propagation of spin-wave (SW) beams in thin ferromagnetic ﬁlms through the areas ofthe inhomogeneous refractive index. We compare the transmission and reﬂection of SWs in areas with gradualand step variation of the SW refractive index. In particular, we show the mirage effect for SWs with narrowingSW beam width and an application of the gradual modulation of the SWs refractive index as a diverging lens.Furthermore, we study the propagation of SWs in ferromagnetic stripe with modulated refractive index. Wedemonstrate that the system can be considered as the graded-index waveguide, which preserves the width of theSW beam for a long distance—the property essential for prospective applications of magnonics. DOI: 10.1103/PhysRevB.97.094424 I. INTRODUCTION Spin waves (SWs) are promising information carriers con- sidered for efﬁcient and low energy consuming informationprocessing devices—magnonic units, being able to supplementor even replace standard CMOS circuits [ 1–4]. However, before practical utilization of SWs, methods for efﬁcient excitation, transduction, and control of propagating SWs innanoscale planar structures need to be developed. AlthoughSWs are characterized by complex dispersion relation, manyphenomena and practical solutions known from photonics can be exploited and transferred to magnonics [ 5,6]. In photonics, control of light propagation with the design of spatially variedrefractive index has found broad spectra of applications, rangedfrom ﬁbers to metamaterials [ 7–9]. Especially proﬁtable are graded-index (GI) materials, i.e., materials with a gradualchange of the refractive index [ 10]. Design of the refractive index topography enables manipulation of the direction, veloc- ity, and phase of the propagating waves. A naturally occurringoptical phenomenon related to the gradual decrease of therefractive index is the mirage [10]. This effect takes place when light bends near a warmed-up region (e.g., a ground or a road),where, due to a gradient of the air temperature, the gradual decrease of the refractive index occurs. A well-known example of the mirage is a Fata Morgana . In ﬁber communication, additional dielectric cladding to the core is used to improvetransmission properties. It protects the transmitted signal fromleaking energy by reducing the inﬂuence of any roughness and irregularities of the outer surfaces of the ﬁber. The refractive index between cladding and core region can be changed eithersteplike or continuously, providing step-index and GI ﬁbers,respectively. Usage of GI ﬁbers reduces modal dispersion andsigniﬁcantly improves the efﬁciency of signal transmission,especially in multimode ﬁbers [ 10,11]. In magnonics, there are many ways to modulate SW RI. That can be done by modiﬁcation of materials properties,such as the saturation magnetization, the exchange stiffness,or the magnetic anisotropy, but also by structural design(geometrical pattern), a change of the magnetic ﬁeld magnitude or the magnetic conﬁguration. All these properties and relatedrefractive index values can be varied in a continuous way. Verygood example of such nonuniformity is the demagnetizing ﬁeldnaturally existing at the edges of ferromagnetic ﬁlms [ 12–14] or a noncollinear magnetization [ 15–17]. The gradual change of the refractive index can be also introduced during devicefabrication by nanostructuralization, ion implantation, or volt-age [ 18]. Furthermore, it is possible to modulate the refractive index dynamically by a change of the external magnetic ﬁeld,e.g., using a magnetic ﬁeld generated by DC current [ 19–22], the voltage across the ﬁlm [ 18], or temperature [ 23,24]. The inﬂuence of nonuniformity of the static external mag- netic ﬁeld on propagating SWs has already been studied.However, the normal incidence of magnetostatic SWs onto aregion with a perturbed proﬁle of the static external magneticﬁeld with collinearly [ 20,21,25,26] or noncollinearly [ 27] magnetized thin ﬁlms has been considered. Recently, we havereported an investigation of SW beam reﬂection from thevicinity of the interface with gradual refractive index due to thedemagnetizing ﬁeld [ 12,13]. Also, the SW propagation in non- collinear magnetization has been exploited to demonstrate GImagnonics as a promising ﬁeld of research for utilization [ 15]. A prospective application of magnetic media with the gradual change of the refractive index in magnonics is theguiding of SWs. In the recent theoretical papers guiding alongthe domain walls was considered [ 16,28], also the conﬁnement in the region between domain walls with chirality appearingdue to the presence of the Dzyaloshinskii-Moriya interactionwas studied [ 17,29]. Nonetheless, an oblique incidence of SWs onto a region with a gradual change of magnetic properties inferromagnetic ﬁlms and GI magnonic waveguides have not yetbeen extensively explored [ 30– 32], and we contribute to this ﬁeld in this paper. In the paper, using isofrequency dispersion contours analy- sis [33] in order to develop ray optics approximation for SWs, supported by micromagnetic simulations, we study the SWbeam propagation in thin ferromagnetic ﬁlms and waveguides, 2469-9950/2018/97(9)/094424(9) 094424-1 ©2018 American Physical SocietyPAWEL GRUSZECKI AND MACIEJ KRAWCZYK PHYSICAL REVIEW B 97, 094424 (2018) which are made from thin yttrium iron garnet (YIG) ﬁlm. YIG is a dielectric magnetic material highly suitable for magnonicapplications due to its low damping [ 34]. Recently, fabrication of very thin YIG ﬁlms with thicknesses down to tens ofnanometers, preserving low damping [ 27,35–37], which can be patterned in nanoscale [ 38–40] has been demonstrated. For the sake of simplicity, our attention is concentrated onthe investigation of thin YIG ﬁlms, out-of-plane (OOP) mag-netized by the external magnetic ﬁeld. The change of therefractive index is obtained by variation of the magnitude ofthe static effective magnetic ﬁeld. Nonetheless, the model canbe extended for an in-plane magnetized ﬁlm, after taking intoaccount proper dispersion relation. The analytical predictionsare validated by micromagnetic simulations. In particular, weshow that with a decrease (increase) of the internal magneticﬁeld value H, the SW refractive index increases (decreases). We deﬁne conditions for total internal reﬂection and show howthat phenomenon depends on grad H. Interestingly, for a slow increase of Hin space, we observe a mirage effect for SWs. For a rapid change of Hvalue, we get the signiﬁcant lateral shift of the SW beam along the interface. Moreover, comparison of theresults obtained for gradual and step changes of the refractiveindex in the ferromagnetic stripe suggests that GI waveguidescan offer improved transmission of the SW beam. The paper is organized as follows. In Sec. IIwe present the analytical model and the micromagnetic simulations. Theobtained results for the extended thin YIG ﬁlm and stripe waveguides are discussed in Sec. III. Conclusions are provided in Sec. IV. II. MODEL AND METHODS A. Spin-wave dynamics We consider a thin ferromagnetic ﬁlm with the thickness (Lz) much smaller than the lateral dimensions of the ﬁlm (Lz/lessmuchLx,Ly). The ﬁlm is saturated with the static external magnetic ﬁeld H. Magnetization dynamics is described by the Landau-Lifshitz-Gilbert (LLG) equation of motion for themagnetization vector M[41]: dM dt=−\|γ\|μ0 1+α2M×Heff−α\|γ\|μ0 MS(1+α2)M×(M×Heff), (1) where αis the damping parameter, γis the gyromagnetic ratio, Heffis the effective magnetic ﬁeld, and MSis magnetization saturation. The ﬁrst term in the LLG equation describes theprecessional motion of the magnetization around the effectivemagnetic ﬁeld, and the second term enriches that precession bydamping. The effective magnetic ﬁeld, in general, can consistof many terms. In this paper we consider the contributions ofthe external magnetic ﬁeld, the exchange ﬁeld H ex, and the dipolar ﬁeld Hd:Heff=H+Hex+Hd. In the case of OOP uniformly magnetized thin ﬁlm the SW dispersion relation is given by [ 42,43]: ω2=/parenleftbig ωH+l2 exωMk2/parenrightbig/parenleftbig ωH+l2 exωMk2+ωMF(kLz)/parenrightbig ,(2) where ω=2πfis the angular frequency of SWs, fis the frequency, μ0is the permeability of vacuum, ωH= \|γ\|μ0(H−MS),ωM=γμ0MS, and the exchange length lex=/radicalBig 2A/(μ0M2 S),Ais the exchange constant, kis the wave FIG. 1. Schematic representation of the gradual bending of obliquely incident SW onto the region with increased [(a),(c) anddecreased [(b),(d)] value of the external magnetic ﬁeld H.T h e alternate isofrequency contours in (c) and (d) correspond to the alternate regions of decreasing and increasing values of the internalmagnetic ﬁeld H iin (a) and (b), respectively. The wave vectors ki, vectors of the group velocities vi, and their angles with respect to theyaxis,θiandϕi, respectively, are indicated for each magnetic ﬁeld segment i. The vertical dashed lines in (c) and (d) mark the conservation of the tangential component of the wave vector k/bardblat successive refractions and reﬂection. The insets on the right side of(a) and (b) correspond to Hvalues in accordingly marked regions. number, and F(x)=1−(1−e−x) x. (3) IfkLz/greatermuch1, the dispersion relation can be simpliﬁed to the following equation: ω2=/parenleftbig ωH+l2 exωMk2/parenrightbig/parenleftbig ωH+l2 exωMk2+ωM/parenrightbig . (4) It is clear from Eq. ( 2) that the SW dispersion relation is isotropic for any frequency in the OOP magnetized ﬁlm.This makes the analysis simpler, and we will study onlyOOP magnetized thin ﬁlms and stripes in this paper. Inthe case of in-plane magnetized ﬁlms at low frequencies,where dipolar contribution dominates, the dispersion relationis anisotropic. Nevertheless, with increasing frequency, theisofrequency contours smoothly transform through ellipticalto almost circular at high frequencies, where SW dynamic isdetermined by the exchange interactions [ 31,33]. Thus, the implementation of the analytical model developed below forOOP conﬁguration to the in-plane magnetized thin ﬁlms canbe done by taking the analytical dispersion relation for SW inthe in-plane magnetized ﬁlms from Ref. [ 42] instead of Eq. ( 2). B. Ray optic approximation Let us analyze SWs propagation in a medium with a slow but stepwise change of the internal magnetic ﬁeld value along the y axis from H0, as shown schematically in the insets in Figs. 1(a) and1(b). We assume that the dispersion relation obtained for 094424-2SPIN-WA VE BEAM PROPAGATION IN FERROMAGNETIC … PHYSICAL REVIEW B 97, 094424 (2018) a uniform thin ﬁlm magnetized by the homogeneous in space external magnetic ﬁeld [Eq. ( 2)] can be used in each segment i of the constant magnetic ﬁeld Hi. SW ray propagating through an area with a gradual change of the internal magnetic ﬁeldwill be gradually bent due to the change of the magneticﬁeld resulting in the variation of the SWs refractive index[see Figs. 1(c)and1(d)]. The bending of SW ray can be estimated from the conser- vation of the tangential to the interface component of the wavevector k /bardbl,ki,/bardbl=ki+1,/bardbl[6]. As the interface, we refer to the xzplanes being perpendicular to the gradient of the magnetic ﬁeld, which separates two successive segments with differentmagnetic ﬁelds, H i, andHi+1. Schematically it is shown in Figs. 1(c)and1(d). To model SW beam propagation we will consider SW ray which can be treated as a curve on which arelaying centroids of the SW beam. Rays follow the changes ofthe SWs group velocity direction ( v i) as it is shown in Figs. 1(a) and1(b) where Hincreases and decreases along the yaxis, respectively. If we mark the angle of SW beam propagation byϕ(the angle of ( v i) with respect to the yaxis) we can write dy=cotϕ(y)dx, (5) and the function deﬁning SW ray path y(x) can be expressed in the recursive form: yN=yN−1+cotϕ(yN−1)dx, (6) where yN=y(xN),xN=x0+Ndx and SW starts propaga- tion from the point ( x0,y0) with the angle of incidence ϕ0≡ ϕ(y0). The direction of propagation ϕis related to the direction of the energy transfer, i.e., the direction of the group velocityvector [ v g=∇ kω(k), where ω(k)=ω(kx,ky) is a dispersion relation]. The group velocity is normal to the isofrequencycontours. In the case of isotropic dispersion (considered in thepaper), the isofrequency contours are circular and the directionof the group and phase velocities are equal ϕ=θwhere θ=cot −1ky/kx. The angle of incidence can be expressed as cotϕ(kx,ky)=vg,y vg,x=∂ ∂kyω(kx,ky) ∂ ∂kxω(kx,ky), (7) where vg,nis the nth component of the group velocity. The initial conditions for the incident SW are known, hence a valueofk xand angular frequency ωat the starting point of SW are known. The normal component of the wave vector ky(ω,H) can be calculated from Eq. ( 2) numerically or analytically. Exemplary isofrequency contours lines for different values ofthe internal magnetic ﬁeld H iwith marked directions of group velocities corresponding to the ﬁxed value k/bardblare shown in Figs. 1(c)and1(d). Apart from refraction at the interface, there is possible also reﬂection. If the variation of the magnetic ﬁeld betweensuccessive segments is small, most of the SW energy istransmitted (in the analytical model transmission is consideredonly), unless total reﬂection condition is fulﬁlled. In totalreﬂection, there are no available solutions corresponding tothe given k /bardblof the incident wave at some value of the internal magnetic ﬁeld, e.g., in Fig. 1(d)\|k3\|<k 2,/bardbl. In such a case SW is reﬂected from the interface, see Figs. 1(b) and1(d), where such a situation is presented at the interface betweenH 2andH3. According to the law of reﬂection θinc=θrefandky,ref=−ky,inc, where superscripts “inc” and “ref” refer to the incident and reﬂected waves, respectively. Taking into accountboth the bending and reﬂection of SWs, we can predict the SWsray path for the conserved tangential component of the wavevector to the interface (aligned along the xaxis,k x=const.) using the following procedure: dy=/braceleftbigg cotϕ(y)dx ifk(ω,H(y))/greaterorequalslantkx −cotϕ(y−dy)dxifk(ω,H(y))<kx.(8) A similar model of the SWs propagation in a medium with a gradual change of the Hcaused by the demagnetizing ﬁeld induced in the vicinity of the ﬁlm’s edge was presented inRef. [ 13]. However, that model was valid only for isotropic SWs dispersion. This limitation is removed here due to theanalysis of the group velocity direction instead of the wavevector. C. Micromagnetic simulations Micromagnetic simulations have been proven to be an efﬁcient tool for the calculation of SW dynamics in ferro-magnetic materials. Presented results were obtained usingthe MuMax3 [ 44] which solves the time-dependent LLG equation ( 1) with included Landau damping term with the ﬁnite difference method. In simulations, we consider an obliqueSW beam propagation in YIG thin ﬁlm saturated by an OOP magnetic ﬁeld. We assume typical magnetic parameters of YIG at 0 K; it is A=0.4×10 −11J/m,MS=0.194×106A/m, γ=176 rad GHz /T, and the value of damping α=0.0005. The system of size Lx×Ly×Lzwas discretized with cuboid elements of dimensions lx×ly×Lz. Lateral dimensions of the single cell lx×lyand ﬁlm thickness Lz=10 nm are less than the exchange length of YIG, 13 nm. The simulations havebeen performed for two geometries: (i) 6 μm×4μm×10 nm discretized with the cell of lateral dimensions 2 ×2n m 2for high-frequency exchange SWs and (ii) 32 μm×4μm×10 nm discretized with the cell of size 8 ×8n m2for SWs of lower frequency (15 GHz). Every simulation comprises two parts. First, we get the equilibrium static magnetic conﬁguration, which in our study isalways OOP magnetization. Then, the results of the ﬁrst stageare used in the dynamic part of simulations, which are aimedat obtaining the steady state. SWs are continuously generatedin the form of a Gaussian beam which propagates throughthe ﬁlm. At the edges of the ﬁlm x=0 andx=L x, absorbing boundary conditions are applied [ 45]. SW beams are excited by means of the spatially nonuniform dynamic external magneticﬁeld at a given frequency and the spatial proﬁle designedto excite the SW beam of appropriate width. The proﬁle ofthe dynamic magnetic ﬁeld used to excite the SW beam issimilar to the proﬁle generated by a coplanar waveguide withmodulated width [ 46] multiplied by the Gaussian function changing its value along the axis of the coplanar waveguide.The exact description of the SWs’ beam excitation can befound in Ref. [ 47]. After a sufﬁciently long time of continuous excitation, when the beam is clearly visible and doesn’t changequalitatively in time, a steady state is achieved. The datanecessary for further analysis are stored. From the storedmicromagnetic simulations results rays corresponding to the 094424-3PAWEL GRUSZECKI AND MACIEJ KRAWCZYK PHYSICAL REVIEW B 97, 094424 (2018) excited SW beam are extracted. Firstly, the time average SW intensity color maps are obtained for simulation according to the equation: I(x,y)=f 4/integraltext4/f 0[mx(x,y,t )]2dt, where mx is the normalized component of the magnetization vector. Then, the Gaussian ﬁtting is applied to get ray line coordinates(details can be found in Ref. [ 12]). Those ray lines are directly compared with the results of the analytical model. III. RESULTS A. Analytical model In OOP magnetized thin ﬁlm the SW ray angle with respect to the yaxis is given by cotϕ(y)=cotθ(y)=ky kx=/radicalbig k2(ω,H(y))−k2x kx,(9) where the value of k(ω,H(y)) can be calculated numerically from the dispersion relation Eq. ( 2). The results of the an- alytical analysis of the SW rays are shown in Figs. 1(a) and1(b) for the increased and decreased internal magnetic ﬁeld, respectively. The knowledge about the group velocity of SWs during their propagation through the ﬁlm is important from an applicationpoint of view. Exemplary results showing how v gdepends on the SWs frequency for different values of the external magneticﬁeld are presented in Fig. 2(a). It is shown that v gfor OOP conﬁguration monotonously increases (apart from the region of dominating magnetostatic interactions at low frequencies,invisible in the ﬁgure) [ 43] with an increase of the frequency and decrease of the external magnetic ﬁeld value. Therefore,SWs falling at a region with decreased H, and thus the in- creased refractive index, accelerates. That situation is oppositeto optics, where an increase of the refractive index is relatedto a decrease of the group velocity of electromagnetic waves. The effect of total internal reﬂection is important for wave applications, especially in designing of the waveguides.We will analyze the total internal reﬂection of SWs in theOOP magnetized ﬁlm with spatially modulated magnitude ofthe static external magnetic ﬁeld H(y)=H 0+H/prime(y), where H/prime(y) is its modulation, which in the case under investigation, depends only on the ycoordinate. Therefore, we will analyze the critical ﬁeld Hcrat which the total internal reﬂection takes place in dependence on frequency, H0, and the angle of incidence. The results for μ0H0=0.5–1.5 T with an interval 0.2 T and ϕ=60◦, and additionally, for angles of incidence 50° and 70° at μ0H0=0.5 T are shown in Fig. 2(b). It is visible that with the increase of ϕthe smaller value of the ﬁeld Hcr is required to obtain total internal reﬂection. Furthermore, the greater ϕthe smaller slope of Hcr(f), and interestingly, these dependencies are linear. The dependencies of Hcron the angle of incidence for different values of H0andfare in details presented in Figs. 2(c)–2(f).I nF i g s . 2(c)and2(d)are dependencies of Hcr on the angle of incidence for SWs at frequencies 100 GHz and 15 GHz, respectively, and for different values of H0. It is visible that while the angle of incidence increases, the value of Hcr decreases. Moreover, the higher value of H0the smaller Hcris needed to obtain total internal reﬂection. It is, because, whileHincreases the FMR (ferromagnetic resonance) frequencyFIG. 2. (a) The group velocity of SWs in the OOP magnetized YIG ﬁlm of thickness 10 nm in dependence on frequency, plottedfor several values of the homogeneous, static external magnetic ﬁeld, H 0. (b) Critical external magnetic ﬁeld Hcrin dependence on SWs frequency for the angle of incidence 60° and several values of theexternal magnetic ﬁeld (solid lines), and also for μ 0H0=0.5Ta n d the angle of incidence 50° (black dash-dotted line) and 70° (black dashed line). Hcrin dependence on the angle of incidence and several values of H0for SWs of frequency, in (c) for 100 GHz and in (d) for 15 GHz. Hcrin dependence on the angle of incidence and several values of SWs frequency for (e) μ0H0=1.5Ta n d (f)μ0H0=0.5T . increases as well. Therefore, the greater length of wave vector (frequency is much higher than the FMR frequency) the highervalue of H cris required to obtain total internal reﬂection. In Figs. 2(e)and2(f)are presented dependencies of Hcron the angle of incidence for μ0H0=1.5 and 0.5 T, respectively, for different values of f. It is shown that the higher the frequency, the larger modulation of H/prime(y) is needed to obtain total internal reﬂection, which is consistent with previous analysis. B. Micromagnetic simulations The predictions of the analytical model we validated by means of micromagnetic simulations. Those simulations wereperformed for (i) high-frequency f=100 GHz exchange dominated SWs and (ii) lower frequency SWs, f=15 GHz. In Fig. 3we show the SW intensity maps (the square of the dynamic component of the magnetization averaged over time:/angbracketleftm 2 x(x,y,t )/angbracketright1/f). The results are presented for the exchange dominated SW beams of the beam width c.a. 700 nm andwavelength 33.6 nm (at 100 GHz) incident under the angle60° at the region with varied magnetic ﬁeld H(y)=H 0+ H/prime(y), where μ0H0=1.5 T and H/prime(y) is a Gaussian function. Additionally, we show SW rays obtained from the analyticalmodel (black-dashed lines) from Eq. ( 8) and extracted from 094424-4SPIN-WA VE BEAM PROPAGATION IN FERROMAGNETIC … PHYSICAL REVIEW B 97, 094424 (2018) FIG. 3. The micromagnetic simulations results presenting SWs intensity maps obtained for SW beams of frequency f=100 GHz incident under the angle θ=60° in the 10 nm thick YIG ﬁlm. Solid green lines and dashed black lines correspond to the results interpolated from simulations and calculated using the analytical model, respectively. The static magnetic ﬁeld (directed OOP) depends only on the ycoordinate. Its proﬁle is presented in the insets on the right side of each ﬁgure. In (g)–(i), the horizontal overlapping lines correspond to the plane of reﬂection and their positions obtained from the analytical model and extracted from micromagnetic simulations perfectly agree. The insets inright top corners of (h) and (i) present zoomed in regions of the simulated area marked via dashed green rectangle. (a) SW beam propagation in the medium with uniform H; incline SW beam propagation through the region with the (b) gradual and (c) step decrease of the magnetic ﬁeld. (d) and (e) gradually increased Hv a l u eu pt o2 . 0Ta t y=0 with a different gradient of Hchanges; (f) step-index change of Hup to 2.0 T in the central region of the ﬁlm. (g) and (h) gradually increased Hvalue up to 2.25 T for y=0 with different proﬁles of H; (i) step-index change ofHup to 2.25 T in the central region of the ﬁlm. In (d)–(e) the SW beam is spread due to small ﬁeld increase, this Hmodulation effectively works as an diverging lens; in (e) the small part of SW beam energy is also reﬂected; in (f) a part of the energy is reﬂected at both ﬁrst and second interface. In (g)–(i) the total internal reﬂection of SW beam is visible. (g) The total internal reﬂections at the region with the gradually changing refractive index is referred to as a mirage. In (h) is visible a large shift between the incident and reﬂected beam spots, at some range,SW beam propagates parallel to the interface. the MS results (green-solid line) for different proﬁles of H/prime(y) in (a)–(i). Overall, the very good agreement between micromagnetic simulations and the analytical model is found. Accordingly with the model predictions in Fig. 1(a) the decrease of Hresults in the gradual decrease of the angle of refraction, see Figs. 3(b) and3(c). It causes the shift of the transmitted beam along the xaxis with respect to the case of unbent SW beam [Fig. 3(a)]. In the case of gradually decreasing H(y) the SW beam reﬂections from the nonuniform region are not visible [Fig. 3(b)], whereas in the case of step-index change ofH(y) the reﬂected SW beams are apparent [Fig. 3(c)]. In the case of step-index change of H(y), the interference pattern of the incident and reﬂected waves is visible as horizontal stripeswith higher and lower intensities near the interface. The increase of H(y) causes the increase of the angle of refraction [Figs. 3(d)–3(i)], which is also according to the model estimation shown in Fig. 1(a). When the increase ofthe magnetic ﬁeld is slow and the maximal value of H(y)i s smaller than H 0+Hcr, the refracted SW beam is spread, see Fig.3(d). Hence, that region with increased Hcan be treated as a diverging lens for SWs in analogy to optics [ 10]. For more abrupt changes of H(y), clearly visible reﬂected SW beams appear [Fig. 3(e)]. In the limit of the step-index change of H(y) the incident SW beam is split into transmitted and pronouncedreﬂected SW beams, see Fig. 3(f). For higher maximal values of the external magnetic ﬁeld, exceeding the condition for total internal reﬂectionmax (H(y))>H 0+Hcr, the incident beam is totally reﬂected [Figs. 3(g)–3(i)]. It is noteworthy that the magnetic ﬁeld at which the total internal reﬂection takes place is almost identicalin the case of micromagnetic simulations and analytical model,see two overlapping horizontal dashed lines in Figs. 3(g)–3(i). These horizontal lines correspond to the value of the externalmagnetic ﬁeld H=H 0+Hcrand can be referred to as the 094424-5PAWEL GRUSZECKI AND MACIEJ KRAWCZYK PHYSICAL REVIEW B 97, 094424 (2018) FIG. 4. Spin-wave beam propagating under the angle of 60° at frequency f=15 GHz in 10 nm thick and 4 μm wide YIG stripe in the presence of gradual (a)–(c) and step (d)–(f) changes of the external magnetic ﬁeld across the stripe’s widths. The respective proﬁles of the magnetic ﬁeld across the stripe are shown in (c) and (f). In (a) and (d) the SWs intensity obtained from micromagnetic simulations is shown; in(b) and (e) the amplitude of SWs from the fragment of the structures presented in (a) and (d) (marked by the green rectangle with dashed sides), respectively, are plotted. The dashed red lines show the result of the ray model which match well with the ray extracted from micromagnetic simulations, marked with the solid yellow lines. interface at which total internal reﬂection takes place. Inter- estingly, for the slow increase of H(y), the equivalent of the mirage effect for SWs is apparent [see Fig. 3(g)]. It means that the wavefronts of incident SW beam are gradually bent withoutreﬂection, and the interference pattern near the interface is notvisible. For the rapid change of the H(y) magnitude, the interfer- ence pattern near the area of the varied magnetic ﬁeld is present,pointing at the reﬂection of waves. In this scenario, the SW raynear the interface becomes almost parallel to its line for somedistance, see Fig. 3(h). It means that between the incident and reﬂected SW beam spots appears lateral shift along theinterface of value almost 1 μm. One may call this phenomenon as Goos-Hanchen effect for SWs [ 12,48] of surprisingly high value. However, we fully recreated this effect using ray opticsapproximation only (see the dashed black line), whereas theliterature Goos-Hanchen effect is a wave phenomenon, deﬁnedas a lateral shift between the incident and reﬂected beam spotsdue to phase change occurring at the interface [ 49]. It means the Goos-Hanchen effect cannot be explained by the use ofray optics approximation, therefore, the observed lateral shiftshouldn’t be referred to as the Goos-Hanchen shift, althoughthe observed phenomenon looks equivalently in the far ﬁeld.For the step change of H(y) [Fig. 3(i)], neither lateral shift nor bending are visible. The interference pattern is clearly apparentand there is no transmission to the upper part of the sample. In the last part of our study, we verify analytical predictions for lower frequency SWs and the ferromagnetic stripe ofthe ﬁnite width. We analyze the SW beam at the frequencyf=15 GHz (wavelength 115 nm) with the beam width 680 nm propagating in the 10 nm thick and 4 μm wide YIG stripe OOP magnetized by the external magnetic ﬁeld. The value of H(y) in the middle part is set as μ 0H0=0.5 T and its magnitude gradually increases when moving to the stripe edges up toμ 0H(w)=0.58 T near the sides of the stripe, at a distance w=1.6μm from the center of the stripe [see Fig. 4(c)]. The quadratic change of the ﬁeld H(y)=H0+H/prime my2/w2, where μ0H/prime m=0.08 T, is assumed. In Fig. 4(a)the SW beam propagates under the angle 60° with respect to the yaxis counted in the middle part of the stripe and is multiple times reﬂected in the consideredpart of the waveguide. Under the assumption of the realistic value of the damping in YIG, the SW beam propagates for adistance up to 30 μm with reasonable intensity. The bending of the wavefronts in the region with the gradually increasedmagnetic ﬁeld is demonstrated in Fig. 4(b), which is similar to the observation made in Fig. 3(g) for high-frequency SWs. The ray of the propagating SW beam obtained from theanalytical model, Eq. ( 8), is shown with the red dashed line. The satisfactory agreement between analytical and simulationresults is found at the beginning part of the waveguide, butthis negligible discrepancy increases with a distance, pointingout that wave effects, which are not taken into account in theray model, exist in the propagation through GI media andaccumulate with a distance. For a comparison, micromagnetic simulations have been performed for the step-index change of H(y)a ty=±w,s e e the ﬁeld proﬁle in Fig. 4(f). The SW beam in such a system [see Figs. 4(d) and4(e)] can propagate in the form of the narrow beam for a shorter distance than in the GI waveguide. Theadditional simulation has been performed also for the SW beamof the same proﬁle in the beam waist propagating along thexaxis (ϕ=90 ◦) in the waveguide saturated by the uniform external magnetic ﬁeld of value μ0H0=0.5 T. The results of simulations are presented in Fig. 5(a), where proﬁles of intensities of SW beams propagating in above described threewaveguides (with GI, step-index, and constant proﬁle of themagnetic ﬁeld) at eight different distances along the xaxis are shown. It is clear that the SW beam propagating in theGI waveguide preserves its width for a much longer distancethan the other two waveguides. For instance, at x=26μm the SW beam width is comparable with that at x=2μm, whereas for step-index waveguide SW beam is spread acrossthe whole width of the waveguide. Interestingly, the SW beamin GI waveguide is much narrower than the input beam in theareas where the total internal reﬂection occurs (see proﬁle forx=6μm). Furthermore, we analyze the change of the averaged inten- sity of SW beams along the waveguide’s width with increasingxfor different H(y) proﬁles, see Fig. 5(b). For all proﬁles of H(y) the dependencies are similar because it is due to the damping constant αin Eq. ( 1). One can observe that decay 094424-6SPIN-WA VE BEAM PROPAGATION IN FERROMAGNETIC … PHYSICAL REVIEW B 97, 094424 (2018) graded-index (/g661=60°) step-index (/g661=60°) uniform (/g661=90°) Intensity [a.u.](a) (b) (c)2 μm -2-10 00000000 0 . 0 5 0.03 0.15 0.06 0.43 0.23 0.99 0.29126 μm 10 μm 14 μm 18 μm 22 μm 26 μm 30 μmy [μm] 00.51averaged intensity [a.u.] 00.51Amplitude [a.u.] 6 1 01 41 82 22 63 0 x [μm]6 1 01 41 82 22 63 0 x [μm] (d) 100700 500 300FWHMy [nm] 6 1 01 41 82 22 63 0 x [μm] FIG. 5. (a) Spin-wave beam proﬁle across the width of stripe presented in Fig. 4for several values of x. Solid violet and dashed yellow lines have been obtained for the SW beam proﬁles excited withϕ=60◦and propagating in the GI and the step index stripes, respectively. The thin black dashed line corresponds to the SW beamof the same width but propagating along the xaxis (ϕ=90 ◦)i n the uniform magnetic ﬁeld. The SWs intensity is normalized to the maximal intensity in the whole waveguide for all simulations. (b) Theaveraged intensity of SWs across the stripe’s width as the function of xfor two different proﬁles of the external magnetic ﬁeld: step-index (dash-dotted magenta line) and GI (dashed yellow line). These resultsare compared with a SW beam propagating along the xaxis in the uniformly magnetized stripe. (c) Maximal amplitude of the SW beam in GI stripe in dependence on x. (d) SW beam’s width (full width at half maximum) evolution in the GI stripe in dependence on the x coordinate. Golden dots in (c) and (d) correspond to the proﬁles of SW beam in GI stripe presented in (a). of the SW beam amplitude excited with ϕ=90◦is slightly slower than for beams excited with ϕ=60◦in the GI and step-index waveguides. However, the difference results fromdifferent optical paths of the beams in the analyzed examples,i.e., the shortest path is obtained for the SW beam excitedwithϕ=90 ◦in a waveguide magnetized by the uniform ﬁeldH(y)=H0, and the longest for the SW beam excited withϕ=60◦propagating in the step-index ﬁber. SW beam propagating in the GI waveguide, due to its bending, possessesthe moderate effective length of the beam ray.Finally, we analyze in details SW beam behavior in the GI waveguide. In Figs. 5(c)and5(d)the SW beam amplitude and the full width at half maximum (FWHM y) taken along theyaxis, respectively, are plotted in dependence on x. The local maxima in Fig. 5(c) correspond to the SWs beam narrowing while it propagates near the waveguide edges. Inthe region near the critical value of HSWs intensity is around three times larger than in the central part of the waveguide.Therefore, the more pronounced maxima, the more collimatedthe SW beam is. This is conﬁrmed by the FWHM y(x), which oscillates between its maximal width (located near the centerof the waveguide, FWHM y=780 nm) and the minimal width (located near the critical value of the H, FWHM y=160 nm). Any systematic beam’s spreading along the xis visible and the ratio between maximal and minimal widths is kept beingequal to 4. Overall, it means that the stripe with GI modulationofH(y) preserves the best transmission properties from all analyzed cases. The signiﬁcant difference in SW beam propagation between the stripe with the gradual and step change of the staticmagnetic ﬁeld proves that the stripes with graded refractiveindex are promising candidates for transmitting signals carriedby SWs. The system under investigation can be referred toas a wide, multimode GI waveguide for SWs. We believethe proposed system can be a good playground to study SWbeam guiding and can be directly used for an experimentaldemonstration of the investigated effects, e.g., using recently experimentally validated in Ref. [ 50] method of SW beams excitation. IV . SUMMARY We have studied SW beam propagation in the ferromagnetic ﬁlm and waveguide with a gradual change of the magneticﬁeld. These structures can be referred to as magnonic GImedia, for which nonuniformity of the refractive index isintroduced by the variation of the external magnetic ﬁeldmagnitude. For the sake of simplicity, we performed allinvestigations for the out-of-plane magnetized thin YIG ﬁlms.Nevertheless, the study can be extended into other materialsand other magnetic conﬁgurations. We have proposed the rayoptic approximation to describe SW beam propagation throughan area of the ferromagnetic ﬁlm with the gradual variationof the magnetic ﬁeld value. We have successfully veriﬁedthe analytical model by micromagnetic simulations for highand relatively low-frequency SWs in an exchange regime,for which the magnetostatic effects are negligible. We havedemonstrated bending of the SW beam propagating obliquelythrough regions with GI, the total internal reﬂection, and themirage effect with narrowing of the SW beam. We have alsodemonstrated that a region with the inhomogeneous magneticﬁeld value can be used to obtain a diverging SW lens. A thinferromagnetic stripe with an inhomogeneous magnetic ﬁeldacross its width has been used to study GI materials for SWguiding. The obtained results have been compared with resultsfor the stripe with a step-index change of the external magneticﬁeld. We show that the stripe with the gradual modulation of theSW refractive index possesses better transmission propertiesthan the step-index stripe. Interestingly, SW beam propagatingin the GI stripe is periodically narrowed in the region of the 094424-7PAWEL GRUSZECKI AND MACIEJ KRAWCZYK PHYSICAL REVIEW B 97, 094424 (2018) reﬂection at areas with the internal ﬁeld fulﬁlling the condition for total internal reﬂection and then again spreads. Ultimately,any systematic beam spreading has not been observed for stripewith GI, as opposed to the step-index stripe, or SW beam prop-agating along the stripe’s axis. It points out that this approachcan be used to transmit the SW beam without beam wideningfor very long distances, limited only by damping. We believethat the use of SWs waveguides with additional GI claddingshall signiﬁcantly reduce the inﬂuence of defects at the edgesof the ferromagnetic stripes, reduce spreading of the SW beamwidth, and enhance the transmission. Further investigations arenecessary for the optimization of the multi- and single-mode SW waveguides and for other magnetization conﬁgurations. 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PhysRevB.78.104410.pdf	Spin transfer and critical current for magnetization reversal in ferromagnet-ferromagnet- ferromagnet double-barrier tunnel junctions Xi Chen, Qing-Rong Zheng, and Gang Su * College of Physical Sciences, Graduate University of Chinese Academy of Sciences, P .O. Box 4588, Beijing 100049, People’ s Republic of China /H20849Received 2 May 2008; revised manuscript received 6 August 2008; published 16 September 2008 /H20850 In terms of the Keldysh nonequilibrium Green’s-function method and invoking the generalized Landau- Liﬁshiz-Gilbert equation in presence of the spin-transfer torque /H20849STT /H20850, we have systematically investigated the spin-transfer effect as well as the critical current for magnetization reversal in the ferromagnet /H20849FM /H20850- ferromagnet-ferromagnet double-barrier magnetic tunnel junctions. It has been found that the tunnel magne-toresistance /H20849TMR /H20850increases dramatically with the increase in the molecular ﬁeld of the middle FM, and the larger the molecular ﬁeld, the greater the TMR. The STT is found to oscillate with the bias voltage for ﬁnitethickness of the middle FM, while the electrical current as a function of the bias is almost linear with slightoscillations. It has been shown that the molecular ﬁeld and the polarization dependences of the critical voltageand critical electrical current show steplike behaviors for ﬁnite thicknesses of the middle FM. The order ofmagnitude of the critical current is estimated to be about 10 5–106A/cm2. The present results are expected to be instructive for manufacturing the relevant spintronic devices. DOI: 10.1103/PhysRevB.78.104410 PACS number /H20849s/H20850: 73.40.Gk, 75.60.Jk, 75.70.Cn I. INTRODUCTION In 1988, two different groups, independently, discovered an unusual large magnetoresistance in magneticmultilayers. 1,2This surprising discovery was then named as giant magnetoresistance /H20849GMR /H20850and thought to be a totally new phenomenon that is caused by spin-dependent scatter-ings of conduction electrons. Since then, a great number ofresearch works focus on various circumstances of GMR ef-fect in different systems, substantiating it to be a source ofnew scientiﬁc and technological applications. On the otherhand, the development of GMR effect triggers the rediscov-ery of tunneling magnetoresistance /H20849TMR /H20850in magnetic tun- nel junctions /H20849MTJs /H20850. A single-barrier MTJ consists of two metallic ferromagnet /H20849FM /H20850electrodes separated by an insu- lating barrier. In 1975, Jullière 3ﬁrst studied the magnetore- sistance in the Fe-Ge-Co MTJ and found that the TMR ratiocan be as high as 14%. A breakthrough has been made in1995 that a large TMR has been reproducibly observed atroom temperature in CoFe /Al 2O3/Co /H20849NiFe /H20850and Fe /Al2O3/Fe MTJs.4Recently, the use of a single-crystal MgO barrier in a MTJ has already generated a rather highTMR ratio, reaching 500%. 5In addition, the current- perpendicular-to-plane geometry of the MTJ makes it easy tobe integrated into a nanoelectronic device, and in fact, theTMR-based read heads have been commercialized. 6These advances enable the TMR effect to possess even more essen-tial industrial applications than the GMR effect in informa-tion storage and spin-based electronic devices. A signiﬁcantprogress has been made both experimentally and theoreti-cally in the last decade. 7–15 An opposite phenomenon of GMR and TMR effects, coined as the spin-transfer effect /H20849STE /H20850, was predicted inde- pendently by Berger16and Slonczewski17in 1996. STE states that when the spin-polarized electrons ﬂowing from one FMlayer into another FM layer with the magnetization alignedby a relative angle may transfer the transverse angular mo-menta to the local spins of the second FM layer, therebyexerting a torque on the magnetic moments. By means of the STE, it is possible to switch the magnetic state of the freeFM layer of a MTJ or spin valve by applying an electricalcurrent instead of a magnetic ﬁeld. This proposal was soonconﬁrmed experimentally /H20849e.g., Ref. 18/H20850. It is now well es- tablished that the STE could be applied to develop writingheads for magnetic random access memory /H20849MRAM /H20850or hard disk drivers. A nonvolatile STE memory has been demon-strated recently. 19In view of potentially wide use of the STE, a plenty of investigations concerning spin-transfer torque/H20849STT /H20850have been presented for different systems /H20849e.g., Refs. 20–32/H20850. On the other hand, the double-barrier magnetic tunnel junction /H20849DBMTJ /H20850, in which the formation of quantum well states and the resonant tunneling phenomenon are antici-pated, has attracted much attention in recent years. In orderto observe the coherent tunneling through the DBMTJ,people have attempted to improve the junction quality toeliminate the inﬂuences from the interface roughness andimpurity scattering, and remarkable advances have beenachieved on this aspect /H20849e.g., Refs. 33–36/H20850. One of the inter- esting DBMTJs is the FM-FM-FM DBMTJ, which is com-prised of three metallic FM layers separated by double insu-lating barriers. It has been shown that the FM-FM-FMDBMTJs can be used as the basic elements for MRAM orDBMTJ-based spin transistors and have successfully beenfabricated recently, where some peculiar properties havebeen observed. 33,35On account of the importance of such a nanostructure, it would be interesting to explore systemati-cally the STE as well as the critical current for magnetizationreversal in the presence of the STT in FM-FM-FM DBMTJs,as they are directly related to the practical realizations inpossible spintronic devices. The other parts of this paper are arranged as follows. In Sec. II, a model is proposed, and the relevant Green’s func-tions are obtained in terms of the nonequilibrium Green’s-function /H20849NEGF /H20850method. In Sec. III, the properties of STT in the system under consideration are numerically investi-PHYSICAL REVIEW B 78, 104410 /H208492008 /H20850 1098-0121/2008/78 /H2084910/H20850/104410 /H208499/H20850 ©2008 The American Physical Society 104410-1gated, and some discussions are presented. The critical cur- rent for magnetization reversal of the middle FM layer inpresence of the STT will be explored in Sec. IV . Finally, abrief summary will be given in Sec. V . II. MODEL AND GREEN’S FUNCTIONS The system under interest, as illustrated in Fig. 1, is com- posed of three metallic FMs separated by two thin insulators.The left /H20849L/H20850and right /H20849R/H20850ferromagnets, whose molecular ﬁelds are assumed to align along the zaxis, are stretched to inﬁnite, and the middle FM, whose orientation of magneti-zation is along the z /H11032axis, deviating an angle /H9258from the z axis, is supposed to be several nanometers thick. The Hamil-tonian of the system reads H=H L+HR+HC+HLC+HCR, /H208491/H20850 with H/H9251=/H20858 k/H9251/H9268/H9255k/H9251/H9268ak/H9251/H9268+ak/H9251/H9268/H20849/H9251=L,R/H20850, /H208492/H20850 HC=/H20858 k/H9268/H9255kck/H9268+ck/H9268−/H20858 k/H20849ck↑+,ck↓+/H20850/H9268ˆ·M/H6023/H20873ck↑ ck↓/H20874 =/H20858 k/H9268/H20851/H20849/H9255k−/H9268Mcos/H9258/H20850ck/H9268+ck/H9268−Msin/H9258ck/H9268+ck/H9268¯/H20852, /H208493/H20850 HLC=/H20858 kLk/H9268/H9268/H11032/H20849TkLk/H9268/H9268/H11032akL/H9268+ck/H9268/H11032+ H.c. /H20850, /H208494/H20850 HCR=/H20858 kRk/H9268/H9268/H11032/H20849TkRk/H9268/H9268/H11032akR/H9268+ck/H9268/H11032+ H.c. /H20850, /H208495/H20850 where ak/H9251/H9268and ck/H9268are annihilation operators of electrons with momentum kand spin /H9268in the /H9251electrode and in the middle FM layer, respectively, /H9255k/H9251/H9268=/H9255k/H9251−/H9268M/H9251−eV/H9251is the single-electron energy for the wave vector k/H9251with the mo- lecular ﬁeld M/H9251in the /H9251electrode, /H9255kis the single-electron energy, Mis the molecular ﬁeld that is proportional to the exchange constant in the middle FM layer, /H9268¯=−/H9268, and Tk/H9251k/H9268/H9268/H11032 are the tunneling matrix elements of electrons between the /H9251 electrode with spin /H9268and the middle FM layer with spin /H9268/H11032. In this paper, we assume that the middle FM is made of a soft magnetic material so that the magnetization is easier toreverse and can be regarded as the free layer, while the side FMs can be treated as pinned layers. The spins in the middleFM can be written as 11,20 S/H9258=/H6036 2/H20858 k/H9262/H9271ck/H9262+ckv/H20849R−1/H9273/H9262/H20850+/H9268ˆ/H20849R−1/H9273/H9271/H20850, /H208496/H20850 where R=/H20898cos/H9258 2− sin/H9258 2 sin/H9258 2cos/H9258 2/H20899, /H9268ˆis the Pauli matrix, and /H9273/H9262/H20849/H9271/H20850denotes spin states. Note that Eq. /H208496/H20850is written in the xyzcoordinate frame, while the spins S/H9258are quantized in the x/H11032y/H11032z/H11032coordinate frame. We can fur- ther write S/H9258=/H6036 2/H20858k/H20849cos/H9258ck/H9268+ck/H9268¯−/H9268sin/H9258ck/H9268+ck/H9268/H20850. Therefore, the spin torque, namely, the time evolution rate of the total spin of the middle FM, can be obtained by /H11509S/H9258//H11509t =i /H6036/H20855/H20851H,S/H9258/H20852/H20856. According to Refs. 20,37, and 38, the middle FM gains two types of torques: one is the equilibrium torquecaused by the spin-dependent potential and the other is fromthe tunneling of electrons that is just what we are interestedin. After cautiously separating the current-induced torquesfrom the equilibrium one, the STT can be obtained, /H9270/H9258=i /H6036/H20855/H20851HT,S/H9258/H20852/H20856=Re/H20875/H20858 kLkTr/H9268/H20849/H9268ˆ1cos/H9258 −/H9268ˆ3sin/H9258/H20850TˆkLkGˆ kkL/H11021/H20849t,t/H20850/H20876, /H208497/H20850 where /H9268ˆ1=/H2087301 10/H20874,/H9268ˆ3=/H20873100− 1/H20874 are Pauli matrices, TˆkLk=/H20873TkLk↑↑TkLk↓↑ TkLk↑↓TkLk↓↓/H20874, with the entities TkLk/H9268/H9268/H11032being the tunneling matrix elements, Gˆ kkL/H11021=/H20873GkkL/H11021↑↑GkkL/H11021↓↑ GkkL/H11021↑↓GkkL/H11021↓↓/H20874 are the lesser Green’s functions in spin space, with the enti- ties deﬁned as Gkk/H9251/H11021/H9268/H9268/H11032/H20849t,t/H11032/H20850=i/H20855ak/H9251/H9268/H11032+/H20849t/H11032/H20850ck/H9268/H20849t/H20850/H20856, and Tr /H9268stands for the trace of the matrix taking over the spin space. From Eq. /H208497/H20850, one may see that the current-induced STT can be obtained as long as we get the lesser Green’s func- tions Gkk/H9251/H11021/H9268/H9268/H11032. In the following, we will use Keldysh’s NEGF technique to determine all lesser Green’s functions.39They are closely related to the retarded Green’s functions that aredeﬁned by G k/H9268k/H9251/H9268/H11032r/H20849t,t/H11032/H20850=−i/H9258/H20849t−t/H11032/H20850/H20855/H20853ck/H9268/H20849t/H20850,ak/H9251/H9268/H11032+/H20849t/H11032/H20850/H20854/H20856,FIG. 1. /H20849Color online /H20850A schematic illustration of spin-transfer torque in a FM-I-FM-I-FM tunnel junction. Note that the electronsﬂow along the xaxis.CHEN, ZHENG, AND SU PHYSICAL REVIEW B 78, 104410 /H208492008 /H20850 104410-2Gk/H9268k/H9268/H11032r/H20849t,t/H11032/H20850=−i/H9258/H20849t−t/H11032/H20850/H20855/H20853ck/H9268/H20849t/H20850,ck/H9268/H11032+/H20849t/H11032/H20850/H20854/H20856. By using the equation of motion, we have /H20849/H9255−/H9255k/H9251↑/H20850Gk↑k/H9251↑r/H20849/H9255/H20850=/H20858 k/H11032Tk/H9251k/H11032/H11569↑↑Gk↑k/H11032↑r/H20849/H9255/H20850+/H20858 k/H11032Tk/H9251k/H11032/H11569↑↓Gk↑k/H11032↓r/H20849/H9255/H20850, /H208498/H20850 /H20849/H9255−/H9255k/H9251↑/H20850Gk↓k/H9251↑r/H20849/H9255/H20850=/H20858 k/H11032Tk/H9251k/H11032/H11569↑↑Gk↓k/H11032↑r/H20849/H9255/H20850+/H20858 k/H11032Tk/H9251k/H11032/H11569↑↓Gk↓k/H11032↓r/H20849/H9255/H20850, /H208499/H20850/H20849/H9255−/H9255k/H9251↓/H20850Gk↑k/H9251↓r/H20849/H9255/H20850=/H20858 k/H11032Tk/H9251k/H11032/H11569↓↓Gk↑k/H11032↓r/H20849/H9255/H20850+/H20858 k/H11032Tk/H9251k/H11032/H11569↓↑Gk↑k/H11032↑r/H20849/H9255/H20850, /H2084910/H20850 /H20849/H9255−/H9255k/H9251↓/H20850Gk↓k/H9251↓r/H20849/H9255/H20850=/H20858 k/H11032Tk/H9251k/H11032/H11569↓↓Gk↓k/H11032↓r/H20849/H9255/H20850+/H20858 k/H11032Tk/H9251k/H11032/H11569↓↑Gk↓k/H11032↑/H11021/H20849/H9255/H20850. /H2084911/H20850 Obviously, to obtain the solution of Gk/H9268k/H9251/H9268/H11032r, we have to get Gk/H9268k/H11032/H9268/H11032r. For this purpose, we may apply the same procedure as that of obtaining Eqs. /H208498/H20850–/H2084911/H20850and have /H20849/H9255−/H9255k/H11032↑+Mcos/H9258/H20850Gk↑k/H11032↑r/H20849/H9255/H20850=−Msin/H9258Gk↑k/H11032↓r/H20849/H9255/H20850+/H20858 /H9251/H20858 k/H9251Tk/H9251k/H11032↑↑Gk↑k/H9251↑r/H20849/H9255/H20850+/H20858 /H9251/H20858 k/H9251Tk/H9251k/H11032↓↑Gk↑k/H9251↓r/H20849/H9255/H20850+/H9254kk/H11032, /H2084912/H20850 /H20849/H9255−/H9255k/H11032↑+Mcos/H9258/H20850Gk↓k/H11032↑r/H20849/H9255/H20850=−Msin/H9258Gk↓k/H11032↓r/H20849/H9255/H20850+/H20858 /H9251/H20858 k/H9251Tk/H9251k/H11032↑↑Gk↓k/H9251↑r/H20849/H9255/H20850+/H20858 /H9251/H20858 k/H9251Tk/H9251k/H11032↓↑Gk↓k/H9251↓r/H20849/H9255/H20850, /H2084913/H20850 /H20849/H9255−/H9255k/H11032↓−Mcos/H9258/H20850Gk↑k/H11032↓r/H20849/H9255/H20850=−Msin/H9258Gk↑k/H11032↑r/H20849/H9255/H20850+/H20858 /H9251/H20858 k/H9251Tk/H9251k/H11032↓↓Gk↑k/H9251↓r/H20849/H9255/H20850+/H20858 /H9251/H20858 k/H9251Tk/H9251k/H11032↑↓Gk↑k/H9251↑r/H20849/H9255/H20850, /H2084914/H20850 /H20849/H9255−/H9255k/H11032↓−Mcos/H9258/H20850Gk↓k/H11032↓r/H20849/H9255/H20850=−Msin/H9258Gk↓k/H11032↑r/H20849/H9255/H20850+/H20858 /H9251/H20858 k/H9251Tk/H9251k/H11032↓↓Gk↓k/H9251↓r/H20849/H9255/H20850+/H20858 /H9251/H20858 k/H9251Tk/H9251k/H11032↑↓Gk↓k/H9251↑r/H20849/H9255/H20850+/H9254kk/H11032. /H2084915/H20850 By combining Eqs. /H208498/H20850–/H2084915/H20850, we get a set of coupled equations /H20849/H9255−/H9255k/H11032↑+Mcos/H9258/H20850Gk↑k/H11032↑r/H20849/H9255/H20850=−Msin/H9258Gk↑k/H11032↓r/H20849/H9255/H20850+/H20858 k/H11033AGk↑k/H11033↑r/H20849/H9255/H20850+/H20858 k/H11033CGk↑k/H11033↓r/H20849/H9255/H20850+/H9254kk/H11032, /H2084916/H20850 /H20849/H9255−/H9255k/H11032↑+Mcos/H9258/H20850Gk↓k/H11032↑r/H20849/H9255/H20850=−Msin/H9258Gk↓k/H11032↓r/H20849/H9255/H20850+/H20858 k/H11033AGk↓k/H11033↑r/H20849/H9255/H20850+/H20858 k/H11033CGk↓k/H11033↓r/H20849/H9255/H20850, /H2084917/H20850 /H20849/H9255−/H9255k/H11032↓−Mcos/H9258/H20850Gk↑k/H11032↓r/H20849/H9255/H20850=−Msin/H9258Gk↑k/H11032↑r/H20849/H9255/H20850+/H20858 k/H11033BGk↑k/H11033↓r/H20849/H9255/H20850+/H20858 k/H11033CGk↑k/H11033↑r/H20849/H9255/H20850, /H2084918/H20850 /H20849/H9255−/H9255k/H11032↓−Mcos/H9258/H20850Gk↓k/H11032↓r/H20849/H9255/H20850=−Msin/H9258Gk↓k/H11032↑r/H20849/H9255/H20850+/H20858 k/H11033BGk↓k/H11033↓r/H20849/H9255/H20850+/H20858 k/H11033CGk↓k/H11033↑r/H20849/H9255/H20850+/H9254kk/H11032, /H2084919/H20850 with A=−i/H9003L↑/H20849/H9255/H20850 2−i/H9003R↑/H20849/H9255/H20850 2−i/H925322/H9003L↓/H20849/H9255/H20850 2−i/H925342/H9003R↓/H20849/H9255/H20850 2, B=−i/H9003L↓/H20849/H9255/H20850 2−i/H9003R↓/H20849/H9255/H20850 2−i/H925312/H9003L↑/H20849/H9255/H20850 2−i/H925332/H9003R↑/H20849/H9255/H20850 2, C=−i/H92531/H9003L↑/H20849/H9255/H20850 2−i/H92533/H9003R↑/H20849/H9255/H20850 2−i/H92532/H9003L↓/H20849/H9255/H20850 2−i/H92534/H9003R↓/H20849/H9255/H20850 2,where /H9003/H9251/H9268/H20849/H9255/H20850is the linewidth function deﬁned by /H9003/H9251/H9268/H20849/H9255/H20850 =2/H9266/H20858k/H9251/H9267/H20849k/H9251/H20850/H20841Tk/H9251/H9268k/H9268/H208412with/H9267/H20849k/H9251/H20850as the density of states of electrons with momentum k/H9251and spin /H9268in the /H9251FM elec- trode. For convenience, we introduce four parameters, /H92531 =TkLk↑↓/TkLk↑↑,/H92532=TkLk↓↑/TkLk↓↓,/H92533=TkRk↑↓/TkRk↑↑, and /H92534=TkRk↓↑/TkRk↓↓, which characterize the spin-ﬂip scattering effect. By consid- ering the symmetry, we may further assume /H92531=/H92533and/H92532 =/H92534for simplicity. The retarded Green’s functions can be obtained in terms of Eqs. /H2084916/H20850–/H2084919/H20850. On the other hand, the lesser self-energy can be approximated by Ng’s ansatz:40 /H9018/H11021=/H90180/H11021/H20849/H90180r−/H90180a/H20850−1/H20849/H9018r−/H9018a/H20850, where /H9018r−/H9018a=Gr−1−Ga−1./H90180r and/H90180/H11021are given by the following equations:SPIN TRANSFER AND CRITICAL CURRENT FOR … PHYSICAL REVIEW B 78, 104410 /H208492008 /H20850 104410-3/H20873/H90180↑↑r/H90180↓↑r /H90180↑↓r/H90180↓↓r/H20874=/H20898−i/H9003L↑/H20849/H9255/H20850 2−i/H9003R↑/H20849/H9255/H20850 20 0 −i/H9003L↓/H20849/H9255/H20850 2−i/H9003R↓/H20849/H9255/H20850 2/H20899, /H20873/H90180↑↑/H11021/H90180↓↑/H11021 /H90180↑↓/H11021/H90180↓↓/H11021/H20874=/H20898i/H9003L↑/H20873/H9255+eV 2/H20874fL/H20849/H9255/H20850+i/H9003R↑/H20873/H9255−eV 2/H20874fR/H20849/H9255/H20850 0 0 i/H9003L↓/H20873/H9255+eV 2/H20874fL/H20849/H9255/H20850+i/H9003R↓/H20873/H9255−eV 2/H20874fR/H20849/H9255/H20850/H20899. After some algebras, we can arrive at /H9270/H9258=−1 4/H9266/H20885d/H9255Tr/H9268/H20877/H20849/H92681cos/H9258−/H92683sin/H9258/H20850/H9003L/H20873/H9255+eV 2/H20874/H20851fR/H20849/H9255/H20850−fL/H20849/H9255/H20850/H20852Gr/H9003R/H20873/H9255−eV 2/H20874BGa/H20878, /H2084920/H20850 with /H9003/H9251=/H20873/H9003/H9251↑ /H9003/H9251↓/H20874, B=/H20849/H90180r−/H90180a/H20850−1/H20849/H9018r−/H9018a/H20850, and f/H9251/H20849/H9255/H20850as the Fermi distribution function of the electrons in the /H9251region. It is evident that these equations should be solved numerically in a self-consistent manner. To proceed further the numerical calculations, we need to make some assumptions. Suppose that the two side FM elec-trodes are made of the same materials, i.e., M L=MR,PL =PR=P, where PL/H20849R/H20850=/H20851/H9003L/H20849R/H20850↑−/H9003L/H20849R/H20850↓/H20852//H20851/H9003L/H20849R/H20850↑+/H9003L/H20849R/H20850↓/H20852is the polarization of the left /H20849right /H20850FM layer. Then, the linewidth function can be written as /H9003L↑,↓=/H9003R↑,↓=/H90030/H208491/H11006P/H20850, where /H90030=/H9003L/H20849R/H20850↑/H20849P=0/H20850=/H9003L/H20849R/H20850↓/H20849P=0/H20850will be taken as an energy scale. I0=e/H90030 /H6036andG0=e2 /H6036will be taken as scales for the tunnel current and the differential conductance, respectively. Thefollowing material parameters are used for the middle FM:the uniaxial anisotropy ﬁeld H k=500 Oe, the molecular ﬁeld M=10/H90030/H110111200 Oe, where /H90030is taken as 20 meV , the Fermi level 130 /H90030,kBT=0.02 /H90030, the damping coefﬁcient /H9251 =0.1, and the junction area of 80 /H1100380 nm2. The thickness of the middle FM is 5.6 nm throughout the paper unless it isspeciﬁed otherwise. III. SPIN-TRANSFER TORQUE Let us ﬁrst look at the case in absence of the spin-ﬂip scatterings. The /H9258dependence of TMR, STT /H20849/H9270/H9258/H20850, as well as the ratio between the STT and electrical current /H20849/H9270/H9258/I/H20850for different molecular ﬁelds /H20849M/H20850of the middle FM are pre- sented in Figs. 2/H20849a/H20850–2/H20849c/H20850. It is observed that the TMR in- creases strikingly with increase in the molecular ﬁeld Mat a given /H9258. The stronger the molecular ﬁeld, the larger the TMR. The results also show that the STT as a function of /H9258 behaves as a proﬁle similar to a sine curve and vanisheswhen the relative alignment of magnetizations of the sideFMs is parallel /H20849 /H9258=0/H20850or antiparallel /H20849/H9258=/H9266/H20850with respect to the middle FM. It is easily understood because the spin-polarized electrons along the zor − zaxis cannot feel the spin-transfer effect owing to the property of sˆ1,2/H11003/H20849sˆ1/H11003sˆ2/H20850. However, the maximum values of the STT and their posi-tions vary for different M. When /H9258is much less /H20849larger /H20850than /H9266/2, the STT increases /H20849decreases /H20850with increasing the mo- lecular ﬁeld. From Fig. 2/H20849c/H20850, one may ﬁnd that /H9270/H9258/Iis also a nonmonotonic function of /H9258, and the maximum appears near /H9258=/H9266/2. It is interesting to note that /H9270/H9258/Ifor different Mhas a crossing point at /H9258=/H9266/2. The/H9258dependences of TMR, STT /H9270/H9258, and /H9270/H9258/Ifor differ- ent polarizations Pof the side FMs are shown in Figs. 2/H20849d/H20850–2/H20849f/H20850. It is no doubt that the larger the polarization P, the larger the TMR and the STT. This result suggests that theferromagnetic materials with large polarization should be FIG. 2. /H20849Color online /H20850The angular dependence of /H20849a/H20850TMR, /H20849b/H20850 spin-transfer torque /H9270/H9258//H90030, and /H20849c/H20850/H9270/H9258/Ifor different M, where P =0.8. The /H9258dependence of /H20849d/H20850TMR, /H20849e/H20850spin-transfer torque /H9270/H9258//H90030, and /H20849f/H20850/H9270/H9258/Ifor different P, where M=30/H90030,eV=100/H90030.CHEN, ZHENG, AND SU PHYSICAL REVIEW B 78, 104410 /H208492008 /H20850 104410-4chosen if the STE as a mechanism is used to design a spin- tronic device, being consistent with the intuition. At ﬁrst glance, the property of the STT in the FM-FM-FM DBMTJ system is similar to the previous spin-valve systems.However, we will ﬁnd the differences if we concentrate on the maximum of the STT /H9270max/H9258and the corresponding angle /H9258max. In previous works, /H9258maxdecreases with increasing the polarization Pwhile /H9270max/H9258increases with increasing P.17 However, both maxima of the STT /H9270max/H9258and/H9258maxin the present system increase monotonically with P, as shown in Figs. 3/H20849a/H20850and3/H20849b/H20850. The Mdependences of /H9258maxand/H9270max/H9258are also presented in Figs. 3/H20849c/H20850and3/H20849d/H20850. It is found that neither /H9258maxnor/H9270max/H9258is a monotonic function of M. Now let us discuss the /H9258dependence of the tunnel current and the STT /H9270/H9258in presence of the spin-ﬂip scattering effect, as shown in Fig. 4. The angular dependence of the electrical current exhibits a cosinelike behavior, while the STT shows asinelike behavior, a reminiscent of the spin-valve effect. Itcan be seen that the spin-ﬂip effect leads to not only animperfect spin-valve effect as the maximum of the tunnelcurrent does not appear at /H9258=0 but also a nonvanishing spin torque at /H9258=0 or /H9266. It appears that the spin-ﬂip scatterings cause an additional spin torque in the parallel or antiparallelalignments of magnetizations between the middle and sideFMs. This observation is similar to those uncovered insingle-barrier MTJs. 11,20In addition, when /H92531=/H92532=/H9253, the angle shift is proportional to /H9253; when /H92531/HS11005/H92532, the effects of /H92531and/H92531onI/I0and/H9270/H9258//H90030are various. For instance, the behaviors for /H92531=0.1, /H92532=0.2 and /H92531=0.2, /H92532=0.1 look different, where the angle shift in the latter case is moreobvious.The bias dependence of the current as well as the STT for different thicknesses L mof the middle FM are shown in Fig. 5when /H9258is nearly equal to zero /H20849for instance, /H9258=0.005 /H9266/H20850. We assume that the system can be viewed as a quantum well,and the lowest 20 energy levels of the middle region areincluded in the calculations. In Fig. 5, it is seen that the STT oscillates obviously with the bias for different thicknesses ofthe central FM. The oscillation varies for different L m, while the electrical current is almost linear /H20849but with slight oscilla- tions /H20850with increase in the bias. The oscillations origin from the quantum resonant tunneling of electrons, as the middleFM region is taken as a quantum well. When we change thethickness L mof the middle FM, it gives rise to the shift of energy levels, leaving the resonant oscillations slightly dif-ferent from various thicknesses but the qualitative behaviorslook similar. As will be demonstrated later, it is this oscillat-ing behavior of the STT with the bias that leads to unusualcharacteristics of the critical current for magnetization rever-sal.FIG. 3. The polarization dependence of /H20849a/H20850/H9258maxand /H20849b/H20850/H9270max/H9258and the molecular-ﬁeld dependence of /H20849c/H20850/H9258maxand /H20849d/H20850/H9270max/H9258, where eV =100/H90030.FIG. 4. /H20849Color online /H20850The angular dependence of /H20849a/H20850the tunnel current Iand /H20849b/H20850the spin-transfer torque /H9270/H9258for different /H92531and/H92532, where P=0.8, M=30/H90030, and eV=100/H90030. FIG. 5. /H20849Color online /H20850The bias dependence of the spin-transfer torque /H9270/H9258and electrical current I/H20849inset /H20850, where M=15/H90030and P =0.7.SPIN TRANSFER AND CRITICAL CURRENT FOR … PHYSICAL REVIEW B 78, 104410 /H208492008 /H20850 104410-5IV. CRITICAL CURRENT FOR MAGNETIZATION REVERSAL In terms of the STE, one can apply directly the electrical current to switch the magnetic state of a FM in absence of amagnetic ﬁeld. It is this property of STE that makes it pos-sible to fabricate the current-controlled spintronic devices,which is much expected in information industry, because us-ing a current to manipulate a nanomagnetic device may beeasier to realize in fabrication than using a magnetic ﬁeld.Therefore, to enable to reverse the magnetization by a cur-rent through STE, there should exist a critical current atwhich the alignment of magnetic moments in a FM is re-versed, which is more concerned in the device engineering.For this purpose, as the FM-FM-FM DBMTJ can be a basicelement for MRAM and spin transistors, we shall pay atten-tion to the critical current for magnetization reversal in theFM-FM-FM DBMTJ by invoking the Landau-Liﬁshiz-Gilbert /H20849LLG /H20850equation with inclusion of the STT. The gen- eralized LLG equation can be written as 41 /H208491//H9253/H20850dnˆm dt=nˆm/H11003/H20851H/H6023eff−/H9251nˆm/H11003/H20849H/H6023eff+snˆs/H20850/H20852, where /H9253is the gyromagnetic ratio g/H9262B//H6036,/H9251is the LLG damping coefﬁcient, nˆsis a unit vector whose direction is that of the initial spin-polarized current, nˆm=M/H6023//H20841M/H20841,H/H6023effis the effective magnetic ﬁeld including the external ﬁeld, theanisotropy ﬁeld, the exchange ﬁeld, the demagnetizationﬁeld, and the random ﬁeld, etc. The ﬁrst two terms have beeninvestigated extensively in the past decades, and the lastterm, the STT, is induced by the spin-polarized current whichis under our interest. According to Ref. 42, the critical ﬁeld h s, which is deﬁned through /H9270/H6023=snˆm/H11003/H20849nˆs/H11003nˆm/H20850=2lmKhsnˆm/H11003/H20849nˆs/H11003nˆm/H20850and de- duced from the stability condition of the magnetization at /H9258=0, reads hs=−/H9251/H208731+h+1 2hp/H20874, /H2084921/H20850 where K=1 2MH k,Hkis the Stoner-Wohlfarth switching ﬁeld, h=H//H208492K/M/H20850,hp=Kp/K,His an applied ﬁeld, and Kpis the easy-plane anisotropy energy that is assumed to be 2 /H9266Ms2for a thin ﬁlm. In the previous work,42the quantity sis assumed to be proportional to the electronic current: s=/H20849/H6036/2e/H20850/H9257I, with /H9257 =/H20849I↑−I↓/H20850//H20849I↑+I↓/H20850as the spin-polarization factor of the inci- dent current I, which can be treated as a constant. The critical current is given by Ic=1 /H9257/H208732e /H6036/H20874/H9251/H20849a2lmHkMs/H20850/H208731+2/H9266Ms Hk+H Hk/H20874, /H2084922/H20850 where Msis the saturation magnetization and the magnetic ﬁeld His applied along the zaxis. For a thin-ﬁlm device with current-perpendicular geometry, we may assume that thereare ways to neutralize the easy-plane anisotropy ﬁeld, lead- ing to the critical current I c=1 /H9257/H208492e /H6036/H20850/H9251/H20849a2lmHkMs/H20850, which indi- cates that the current is proportional to Ms.However, our preceding calculations show that the rela- tionship between STT and electrical current is more compli-cated than a simple proportionality. Therefore, it is not suit-able to use Eq. /H2084922/H20850directly in our system, and a self- consistent way that incorporates the NEGF method and LLGequation should be adopted. For a given M, the critical ﬁeld h sand thus the critical spin torque /H9270/H9258can be determined by Eq. /H2084921/H20850that was derived from the LLG equation. As /H9270/H9258is a function of voltage and M, a proper voltage Vshould be chosen to get the critical /H9270/H9258by the NEGF method. Once a critical voltage Vcis determined, the critical current is then obtained.43 The relationship between the critical voltage, the critical current, and the molecular ﬁeld Mof the middle FM for different thicknesses Lmof the middle FM is presented in Fig. 6. It can be seen that it is much complicated than a linearity. The steps appear at different molecular ﬁelds fordifferent L m. The steplike behaviors of the critical voltage and current are not very unexpected. To switch the magneticstate of the middle FM, the larger the magnetization M, the larger the critical STT /H20849 /H9270/H9258/H20850is needed. However, the STT dose not increase monotonously with the bias, which implies thatin order to get the same amount of the STT, different biasesare needed, and this causes steps in M−eV c/H20849Ic/H20850curves. Since the oscillations are various for different thicknesses ofmiddle FM, the positions of steps appear at different places. Because of I 0=e/H90030 /H6036, the order of the critical current may be estimated from our calculations to be about 105–106A/cm2, which is comparable to the results calculated for othersystems. 42 Figure 7shows the molecular-ﬁeld dependence of the critical voltage and critical current for different polarizations P. We can see that with the increase in M, both VcandIc increase with steplike features. Not only the thickness of the middle FM but also the polarization Pof the side FMs can inﬂuence the steplike characters of the critical voltage andcurrent. The reason of the polarization Peffecting on the steplike characters of the critical voltage and current is thatthe tunneling electrons with spin up and spin down haveFIG. 6. /H20849Color online /H20850The molecular-ﬁeld dependence of /H20849a/H20850 critical voltage Vcand /H20849b/H20850critical current Icfor different Lm, where P=0.7.CHEN, ZHENG, AND SU PHYSICAL REVIEW B 78, 104410 /H208492008 /H20850 104410-6opposite spin-transfer effects on the middle FM. Because the electrons with different spins have different energy levels inquantum wells of the middle FM, the bias dependence of theSTT induced by spin-up and spin-down electrons is different.When the polarization Pis changed, the tunneling rate of spin-up and spin-down electrons is changed and so is thebias dependence of the total STT. As discussed above, thelarger the polarization P, the larger the maximum of STT, which suggests that for larger P, even a small amount of tunneling electrons could generate the sufﬁcient STT thatenables to reverse the magnetization, leading to the observa-tion that the larger P, the lesser the critical voltage /H20849current /H20850 eV c/H20849Ic/H20850is needed. By using the data of Fig. 7, we can depict the critical current versus the critical voltage, as shown inFig. 8. It is observed that the relationship between I candVc looks nearly linear in trend, and for different P, all curves of Icagainst Vcalmost fall into the same curve, in particular, at small Vcregions, which also reveals that the critical differ- ential resistance remains almost constant with polarization P. The relationships between the critical voltage, the critical current, and the polarization Pof the side FMs for different Mare also considered. The results are shown in Fig. 9.I ti s easy to see that the larger the polarization is, the smaller thecritical voltage /H20849current /H20850is needed. However, the relationship between the critical voltage /H20849current /H20850and the polarization isnot a simple linearity either; the steps are observed for dif- ferent Mthat result from the resonant tunneling through quantum wells owing to the ﬁnite thickness of the middleFM. It is nontrivial to note that when the magnetization ex-ceeds a certain value, some curves are ended. This is becausethe larger M, the larger critical STT is needed. However, for a smaller P, a smaller STT is obtained. So in some regions, the critical voltage and current cannot be procured by ourself-consistent calculations. V. SUMMARY By means of the Keldysh nonequilibrium Green’s- function method and invoking the generalized LLG equationwith inclusion of the STT, we have systematically investi-gated the spin-transfer effect as well as the critical current formagnetization reversal in the FM-FM-FM DBMTJ system.The angular dependence of the TMR, the spin-transfertorques /H9270/H9258and/H9270/H9258/Ifor different molecular ﬁelds Mof the middle FM, and the polarization Pof the side FMs have been calculated. We have found that, in absence of spin-ﬂip scat-terings, the TMR increases dramatically with the increase inthe molecular ﬁeld of the middle FM, and the larger themolecular ﬁeld, the greater the TMR. The angular depen-dence of the STT shows a shape similar to a sine curve. It isalso observed that the larger the polarization of the side FMs,the greater the TMR and STT. In presence of the spin-ﬂipscatterings, the angular dependence of the electrical currentshows a cosinelike character, while the STT reveals a sine-like behavior. The spin-ﬂip scatterings lead to an imperfectspin-valve effect. The STT is found to oscillate with the biasvoltage for ﬁnite thickness of the middle FM, while the elec-trical current as a function of the bias is almost linear withslight oscillations. These oscillations may origin from theresonant tunneling from the quantum well states in themiddle FM. By applying the generalized LLG equation in presence of the STT, we have studied the critical current for magnetiza-tion reversal in the FM-FM-FM DBMTJ. It is uncovered thatFIG. 7. /H20849Color online /H20850The molecular-ﬁeld dependence of /H20849a/H20850 critical voltage Vcand /H20849b/H20850critical current Icfor different P. FIG. 8. /H20849Color online /H20850The critical current Icversus critical volt- ageVcfor different P.FIG. 9. /H20849Color online /H20850The polarization dependence of /H20849a/H20850criti- cal voltage Vcand /H20849b/H20850the critical current Icfor different M.SPIN TRANSFER AND CRITICAL CURRENT FOR … PHYSICAL REVIEW B 78, 104410 /H208492008 /H20850 104410-7the molecular-ﬁeld dependence of the critical voltage and electrical current shows steplike behaviors for ﬁnite thick-nesses of the middle FM. With increasing the molecular ﬁeldof the middle FM, both the critical voltage and electricalcurrent increase with steps. The order of magnitude of thecritical current is estimated to be about 10 5−106A/cm2in the system under interest, which is comparable with the pre-vious results. The molecular-ﬁeld dependence of the criticalvoltage and current shows a steplike increasing behaviorwith increasing Mand, for a given molecular ﬁeld of the middle FM, the larger the polarization Pof the side FMs, the smaller the critical voltage and current. It has been unveiled that the relationship between V cand Icexhibits approxi- mately a linear behavior, which is almost independent ofpolarization Pof the side FMs. The polarization dependence of the critical current and voltage also shows a steplike de-creasing behavior at a given M. These steplike behaviors of V candIcare closely related to the fact that with increasing the bias voltage, the STT increases with oscillations, whichmay be originated from the resonant tunneling between the quantum well states in the middle FM. Finally, we would liketo remark that since the FM-FM-FM DBMTJ can be an im-portant element for MRAM as well as spin transistors, ourabove-obtained results could provide useful information andguidance for choosing appropriate magnetic materials to fab-ricate the relevant spintronic devices. ACKNOWLEDGMENTS We are grateful to S. S. Gong, X. F. Han, W. Li, X. L. Sheng, Z. C. Wang, Z. Xu, Q. B. Yan, L. Z. Zhang, and G. Q.Zhong for helpful discussions. This work is supported in partby the National Science Fund for Distinguished YoungScholars of China /H20849Grant No. 10625419 /H20850, the National Sci- ence Foundation of China /H20849Grants No. 90403036 and No. 20490210 /H20850, the MOST of China /H20849Grant No. 2006CB601102 /H20850, and the Chinese Academy of Sciences. *Author to whom correspondence should be addressed; gsu@gucas.ac.cn 1M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Eitenne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. 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PhysRevB.98.184428.pdf	PHYSICAL REVIEW B 98, 184428 (2018) Resonance modes and microwave-driven translational motion of a skyrmion crystal under an inclined magnetic ﬁeld Masahito Ikka,1Akihito Takeuchi,1and Masahito Mochizuki2,3 1Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa 229-8558, Japan 2Department of Applied Physics, Waseda University, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan 3PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan (Received 22 June 2018; revised manuscript received 24 August 2018; published 27 November 2018) We theoretically investigate the microwave-active resonance modes of a skyrmion crystal on a thin-plate specimen under application of an external magnetic ﬁeld that is inclined from the perpendicular direction tothe skyrmion plane. In addition to the well-known breathing mode and two rotation modes, we ﬁnd novelresonance modes that can be regarded as combinations of the breathing and rotation modes. Motivated by theprevious theoretical work of [Wang et al. ,P h y s .R e v .B 92,020403(R) (2015 )], which demonstrated skyrmion propagation driven by breathing-mode excitation under an inclined magnetic ﬁeld, we investigate the propagationof a skyrmion crystal driven by these resonance modes using micromagnetic simulations. We ﬁnd that thedirection and velocity of the propagation vary depending on the excited mode. In addition, it is found that amode with a dominant counterclockwise-rotation component drives much faster propagation of the skyrmioncrystal than the previously studied breathing mode. Our ﬁndings enable us to perform efﬁcient manipulation ofskyrmions in nanometer-scale devices or in magnetic materials with strong uniaxial magnetic anisotropy such asGaV 4S4and GaV 4Se4, using microwave irradiation. DOI: 10.1103/PhysRevB.98.184428 I. INTRODUCTION Noncollinear spin structures in magnets such as spirals, vortices, and chiral solitons with ﬁnite helicity and/or chiralityshow nontrivial collective excitations and thus offer intriguingspintronics and magnonics functions. One of the most impor-tant examples of such spin structures is magnetic skyrmionsrealized in magnets with broken spatial inversion symmetry[1–7], in which keen competition between the Dzyaloshinskii- Moriya (DM) interaction and the ferromagnetic-exchange in-teraction takes place. The skyrmion structures usually appearin a plane that lies perpendicular to an external magneticﬁeldH ex, where the magnetizations at its periphery (center) are oriented parallel (antiparallel) to Hex. The skyrmions are classiﬁed into three types, i.e., the Bloch type, the Néeltype, and the antivortex type, according to the way of their magnetization rotation [Fig. 1(a)][2,4]. In bulk specimens, layered skyrmion structures are stacked to form a tubular structure along the H exdirection. The skyrmions often appear in a hexagonally packed form knownas a skyrmion crystal, as shown in Fig. 1(b) [8–12]. These skyrmions show speciﬁc microwave-active collective modes[13–17]. When an external ﬁeld H exis applied normal to the plane of the skyrmion, the skyrmion crystal has one (two)resonance mode(s) activated by a microwave magnetic ﬁeld H ωperpendicular (parallel) to the thin-plate plane. It shows a breathing mode under the perpendicular Hωﬁeld, in which the crystallized skyrmions expand and shrink uniformly inan oscillatory manner. In contrast, the two modes that occurunder the in-plane H ωﬁeld are rotation modes, for which the rotational sense is counterclockwise (clockwise) for thelower-frequency (higher-frequency) mode.Recent theoretical studies and experiments have revealed that these resonance modes of skyrmions host interesting microwave and spintronics functions [ 6,18], including a gi- gantic microwave directional dichroism [ 19–22], induction of spin voltages [ 23,24], generation of spin currents [ 25], spin-torque oscillator functions [ 26,27], microwave sensing functions [ 28], and magnonic crystal functions [ 29–31]. These phenomena have all been investigated for the three collectivemodes mentioned above. Situations in which the skyrmion plane and the skyrmion tube are inclined from the H exdirec- tion rarely occur in bulk specimens because they can easilyfollow the H exdirection. However, when the skyrmions are conﬁned within a quasi-two-dimensional thin-plate specimen[see Fig. 1(c)][10,32], a situation can be realized in which distributions of the magnetizations and scalar spin chirali-ties have disproportionate weight and are slanted from the skyrmion center, as shown in Figs. 1(d) and1(e) [33]. Such a situation can also occur in magnets with strong uniaxial mag-netic anisotropies, in which the orientations of the skyrmionplane and the skyrmion tubes are ﬁxed, irrespective of the H ex direction. We expect the emergence of characteristic resonance modes for skyrmion crystal under the application of an in-clined H exﬁeld. However, such modes have not been studied systematically to date, although novel modes are expectedto host previously unrecognized functions and phenomena.Indeed, a recent theoretical study proposed that the transla-tional motion of skyrmions can be driven by application ofa microwave magnetic ﬁeld H ωto a skyrmion-hosting two- dimensional system under an inclined Hexﬁeld [ 34]. In addi- tion, it was found that a skyrmion crystal appears always on 2469-9950/2018/98(18)/184428(9) 184428-1 ©2018 American Physical SocietyIKKA, TAKEUCHI, AND MOCHIZUKI PHYSICAL REVIEW B 98, 184428 (2018) Hex xyz Hz/J (θ=0˃)Ferromag. Skyrmion Crystal Helical xy111 sites θ=30˃θ=0˃96sitesBloch-type Neel-type Antivortex θ=0˃ θ=0˃ θ=0˃ θ=30˃ θ=30˃ θ=30˃z -1 1 miz -0.0045 0.0045Ci(a) (d)(b) (c) (e) (f) 0.0168 0.0567xy Hzθ Hx FIG. 1. (a) Three types of magnetic skyrmion structures. Distri- butions of the magnetizations (top panels) and those of the scalar spin chiralities Ci=(mi+ˆx×mi+ˆx+ˆy)·mi(bottom panels) are shown. (b) Top view of the skyrmion crystal under a perpendicular magnetic ﬁeld. (c) Thin-plate specimen hosting a skyrmion crystal under a magnetic ﬁeld Hex=(Hx,0,Hz) that is inclined with a ﬁnite in-plane component Hx=Hztanθ. (d) Skyrmion structures under the inclined magnetic ﬁeld. (e) Bloch-type skyrmion crystal under the inclined magnetic ﬁeld. (f) Theoretical phase diagram of thespin model given by Eq. ( 1)a tT=0 as a function of H zfor a perpendicular magnetic ﬁeld Hex=(0,0,Hz) with θ=0◦. the (111) plane, irrespective of the Hexdirection in insulating vanadates GaV 4S8and GaV 4Se8with lacunar spinel structure because of their strong uniaxial magnetic anisotropy [ 35–37]. In insulating skyrmion-hosting materials of this type, speciﬁcresonance modes can be sources of interesting microwavemagnetoelectric phenomena owing to their multiferroic naturewith magnetically induced electric polarizations. Under thesecircumstances, clariﬁcation of the microwave-active modesand the microwave-related phenomena of skyrmions underapplication of an inclined H exﬁeld becomes an issue of major importance. In this paper, we theoretically investigate the microwave- active resonance modes of a skyrmion crystal in a two-dimensional system under an inclined H exﬁeld. By numeri- cally solving the Landau-Lifshitz-Gilbert (LLG) equation, wetrace the dynamics of the magnetizations that constitute theskyrmion crystal to calculate microwave absorption spectraand obtain real-space snapshots for each eigenmode. In ad-dition to the well-known breathing and two types of rotationmodes, we ﬁnd that characteristic modes appear, which canbe regarded as combinations of the breathing and rotationmodes. Using micromagnetic simulations, we demonstratethat continuous excitation of these resonance modes via mi-crowave application results in propagation of the skyrmioncrystal where its direction and velocity sensitively dependon the excited mode (or the microwave frequency) and themicrowave intensity. Furthermore, we ﬁnd that a mode with a dominant counterclockwise-rotation component drives muchfaster propagation of the skyrmion crystal than the previouslyexamined breathing mode. The knowledge of these resonancemodes and the microwave-driven motion of skyrmion crystalsunder an inclined magnetic ﬁeld lead to techniques to ma-nipulate skyrmions using microwaves and provide a means torealize unique skyrmion-based devices. II. SPIN MODEL AND METHOD We employ a classical Heisenberg model on a square lattice to describe the magnetism in a thin-plate specimen ofa skyrmion-hosting magnet, which contains the ferromagneticexchange interaction, the DM interaction among the normal-ized magnetization vectors m i, and the Zeeman coupling to the external magnetic ﬁeld Hex[38,39]. The Hamiltonian is given by H0=−J/summationdisplay /angbracketlefti,j/angbracketrightmi·mj+/summationdisplay i,ˆγDγ·(mi×mi+ˆγ) −Hex·/summationdisplay imi. (1) Types of skyrmions are determined by a structure of the Moriya vectors Dγ(γ=x,y); that is, Dx=(D,0) and Dy=(0,D) produce the Bloch-type skyrmion, Dx=(0,D) andDy=(−D,0) produce the Néel-type skyrmion, and Dx=(0,D) and Dy=(D,0) produce the antivortex-type skyrmion. We adopt J=1 for the energy units and set D/J= 0.27. The external magnetic ﬁeld Hexis inclined from the perpendicular direction ( /bardblz) towards the xdirection as Hex= (Hx,0,Hz), with Hx=Hztanθ, where θis the inclination angle [see Fig. 1(c)]. Figure 1(f) shows a theoretical phase diagram of this spin model at T=0 as a function of Hzwhen Hexis applied normal to the two-dimensional plane ( θ=0). This phase diagram exhibits the skyrmion-crystal phase in aregion of moderate ﬁeld strength sandwiched by the helicalphase and the ﬁeld-polarized ferromagnetic phase. The unitconversions when J=1 meV are summarized in Table I. The LLG equation is given by dm i dt=−γmi×Heff i+αG mmi×dmi dt. (2) HereαG(=0.02) is the Gilbert-damping coefﬁcient. The effective magnetic ﬁeld Heff iacting on the local magnetization mion the ith site is calculated from the Hamiltonian H= TABLE I. Unit conversion table when J=1m e V . Dimensionless Corresponding value quantity with units Exchange coupling J=11 m e V Time t=1 0.66 ps Frequency f=ω/2πω =1 241 GHz (ω=0.01 2.41 GHz) Magnetic ﬁeld H=1 8.64 T 184428-2RESONANCE MODES AND MICROW A VE-DRIVEN … PHYSICAL REVIEW B 98, 184428 (2018) H0+H/prime(t) in the form Heff i=−1 γ¯h∂H ∂mi. (3) The ﬁrst term, H0, is the model Hamiltonian given by Eq. ( 1). The second term, H/prime(t), represents the coupling between the magnetizations and a time-dependent magnetic ﬁeld H(t)i n the form H/prime(t)=−H(t)·/summationdisplay imi. (4) The calculations are performed using a system of N=96× 111 sites where the periodic boundary condition is imposed. III. RESULTS FOR THE RESONANCE MODES To identify the resonance modes of a skyrmion crystal un- der the application of an inclined Hexﬁeld, we ﬁrst calculate dynamical magnetic susceptibilities, χα(ω)=/Delta1Mα(ω) Hα(ω)(α=x,y,z ). (5) HereHα(ω) and/Delta1Mα(ω) are Fourier transforms of the time-dependent magnetic ﬁeld H(t) and the simulated time proﬁle of the total magnetization /Delta1M(t)=M(t)−M(0), withM(t)=1 N/summationtextN i=1mi(t). For these calculations, we use a short rectangular pulse for the time-dependent ﬁeld H(t) whose components are given by Hα(t)=/braceleftbiggHpulse 0/lessorequalslantt/lessorequalslant1, 0 otherwise ,(6) where t=(J/¯h)τis the dimensionless time, with τbeing the real time. An advantage of using the short pulse is that fora sufﬁciently short duration /Delta1twithω/Delta1t/lessmuch1, the Fourier component H α(ω) becomes constant, being independent of ωup to the ﬁrst order of ω/Delta1t. The Fourier component is calculated as Hα(ω)=/integraldisplay/Delta1t 0Hpulseeiωtdt=Hpulse iω(eiω/Delta1t−1)∼Hpulse/Delta1t. (7) As a result, we obtain the relationship χα(ω)∝/Delta1Mα(ω). In Fig. 2(a), we present calculated microwave absorption spectra, i.e., imaginary parts of the dynamical magnetic sus-ceptibilities Im χ xand Im χyfor the in-plane microwave ﬁelds Hω/bardblxandHω/bardbly, respectively, as functions of microwave frequency ω(=2πf) for several values of inclination angle θ.H e r ew eﬁ x Hz=0.036. Note that while these spectra are calculated for the Bloch-type skyrmion crystal, we conﬁrmedthat the Néel type and the antivortex type show perfectlyequivalent spectra. We ﬁnd that while only two rotation modeswith counterclockwise- and clockwise-rotation senses (re-ferred to as modes 1 and 3, respectively) are active under theperpendicular H exﬁeld with θ=0◦, novel modes (modes 2 and 4) appear when the Hexﬁeld is inclined with θ/negationslash=0. Mode 2 appears between the two original rotation modes(modes 1 and 3) in frequency, whereas mode 4 has a higherfrequency than mode 3. The intensities of these novel modes grow as the inclination angle θincreases. As will be discussed below, modes 2 and4 can be regarded as combinations of the breathing and clockwise-rotation modes, where the former (latter) compo-nent is dominant for mode 2 (mode 4). On the other hand,the intensity of mode 3 is either enhanced or suppressed withincreasing θ, depending on the polarization of the microwave ﬁeldH ω. When the microwave ﬁeld Hωis oriented parallel (perpendicular) to the direction toward which the Hexﬁeld is inclined, i.e., Hω/bardblx(Hω/bardbly), the intensity becomes suppressed (enhanced) as θincreases. Figure 2(b) shows imaginary parts of the dynamical magnetic susceptibilities Im χzcalculated for the out-of-plane microwave ﬁeld Hω/bardblzfor several values of θ.A g a i n ,w e ﬁnd that novel modes (modes 1, 3, and 4) appear whentheH exﬁeld is inclined, whereas the single breathing mode (mode 2) alone exists under the perpendicular Hex ﬁeld. The intensities of these novel modes increase as θ increases, while the intensity of the original mode, mode 2,decreases. As will be discussed below, mode 1 (modes 3 and4) can be regarded as combined oscillations of the dominantcounterclockwise- (clockwise-) rotation component and thesubsequent breathing component. Comparison of the calculated dynamical magnetic suscep- tibilities in Figs. 2(a) and2(b) shows that four types of collec- tive modes activated by H ω/bardblx,yhave identical resonance frequencies with four corresponding collective modes acti-vated by H ω/bardblz, indicating that both the in-plane microwave ﬁelds Hω/bardblx,yand the out-of-plane microwave ﬁeld Hω/bardblz activate equivalent modes under application of the inclined Hexﬁeld. In Fig. 3, snapshots of the magnetization distributions for each mode focusing on a skyrmion constituting the Bloch-type skyrmion crystal are shown for H ω/bardbly(left panels) and Hω/bardblz(right panels) when Hz=0.036 and θ=30◦.T h e y are simulated via application of a microwave magnetic ﬁeldH α(t)=Hω αsinωt(α=x,y,z ) with a corresponding reso- nance frequency as the time-dependent magnetic ﬁeld H(t) in Eq. ( 4). In the simulations of these eigenmode dynamics, we have monitored time proﬁles of net magnetization andtheir Fourier transforms to conﬁrm that a pure eigenmode witha single-frequency component is excited while other modesare absent. Noticeably, mode 2 under H ω/bardblyand mode 2 under Hω/bardblzatω=0.0666 are identical and are regarded as a breathing mode. It is also found that mode 1 (mode 3)under H ω/bardblyand mode 1 (mode 3) under Hω/bardblzatω= 0.0494 ( ω=0.0872) are again identical and can be regarded as combined oscillations of the dominant counterclockwise(clockwise) rotation and the subsequent breathing component.While the absorption intensity is too weak to enable snapshotsof mode 4 to be obtained under H ω/bardblz, we expect that they would be equivalent to those of mode 4 under Hω/bardbly. It is known that a single skyrmion in a constricted geometry exhibits quantized higher-harmonic radial and azimuthal spin-wave modes when the system is activated by a sinusoidal acﬁeld, especially in the presence of relatively strong dampingeffects [ 40,41]. In the present calculations, we used an inﬁnite and uniform system with periodic boundary conditions, andthus, such higher harmonics should be absent. However, itmight be important to check their absence. For this purpose,we calculate the dynamical magnetic susceptibilities by ap-plying a sinusoidal ac magnetic ﬁeld. Figure 3(b) [Fig. 3(c)] 184428-3IKKA, TAKEUCHI, AND MOCHIZUKI PHYSICAL REVIEW B 98, 184428 (2018) FIG. 2. (a) and (b) Imaginary parts of the calculated dynamical magnetic susceptibilities of skyrmion crystal conﬁned in a two-dimensional system under perpendicular ( θ=0◦) and inclined ( θ/negationslash=0◦) magnetic ﬁelds as functions of the angular frequency ω. (a) Those for the in- plane microwave polarization with Hω/bardblx,y. (b) Those for the out-of-plane microwave polarization with Hω/bardblz. Note that these spectra are calculated for the Bloch-type skyrmion crystal, but it was conﬁrmed that the Néel-type and the antivortex-type skyrmion crystals have perfectly equivalent spectra. Here the inclined magnetic ﬁeld is given by Hex=(Hztanθ,0,Hz), with Hz=0.036. A dominant component of the oscillation is indicated below the name of each mode, where CW and CCW indicate the clockwise and the counterclockwise rotations,respectively. The spectra for mode 4 are magniﬁed in the insets. (c) and (d) Spectral intensities as functions of θfor modes 1–4 for (c) H ω/bardblx, yand (d) Hω/bardblz. (e) Resonant frequencies ωRas functions of θfor modes 1–4. shows the calculated spectrum for Hω/bardbly(Hω/bardblz)f o rθ= 30◦, which has peaks at the same frequencies as that in Fig. 2, indicating the absence of higher-harmonic modes and thevalidity of the identiﬁed eigenmodes. Note that calculationsof the spatial distributions of power and phase may provideuseful information to identify the eigenmodes [ 42]. In addition to the resonance modes of the Bloch-type skyrmion crystal, we also examined those of the Néel-typeand the antivortex-type skyrmion crystals. We found that themicrowave absorption spectra for these three different typesof skyrmion crystals overlap perfectly, indicating that thethree skyrmion crystals have resonance modes with identicalfrequencies and identical intensities. However, differencesappear in terms of the rotation senses of modes 1, 3, and 4.The rotation sense of the antivortex skyrmion crystal is alwaysopposite that of the corresponding modes of the Bloch-type and Néel-type skyrmion crystals. Figure 4shows snapshots of the calculated magnetization distributions of mode 1 for thethree types of skyrmion crystals. The rotation sense is coun-terclockwise for the Bloch type and the Néel type, whereas itis clockwise for the antivortex type [ 43]. To close this section, it is worth mentioning that the present study is based on a pure two-dimensional model, although realspecimens have ﬁnite thickness. In a real specimen, magneticstructures at the surfaces might be different from those insidethe specimen because magnetizations at the surfaces haveneighbors only on one side [ 44,45]. Such surface magnetic structures might affect the properties of magnetic resonancesquantitatively. However, we expect that our results will not bechanged qualitatively or semiquantitatively even if we adopt 184428-4RESONANCE MODES AND MICROW A VE-DRIVEN … PHYSICAL REVIEW B 98, 184428 (2018) FIG. 3. (a) Snapshots of the four resonance modes (modes 1–4) of skyrmion crystal activated by the in-plane microwave ﬁeld Hω/bardbly(left panels) and those of the three resonance modes (modes 1–3) activated by the out-of-plane microwave ﬁeld Hω/bardblz(right panels) under an inclined magnetic ﬁeld Hex=(Hztanθ,0,Hz), with Hz=0.036 and θ=30◦. One skyrmion in the skyrmion crystal is focused on because all the skyrmions in the skyrmion crystal show identical motions. We show here the results for the Bloch-type skyrmion crystal, but the other two skyrmion types show equivalent behaviors, with the exception of the rotation sense of the antivortex-type (see text and Fig. 4). (b) and (c) Imaginary parts of the dynamical magnetic susceptibilities calculated by tracing the time proﬁles of net magnetization through applying a sinusoidal ac magnetic ﬁeld H(t)f o r( b ) Hω/bardblyand (c) Hω/bardblz. Both spectra exhibit peaks at frequencies equivalent to those of the spectra forθ=30◦s h o w ni nF i g . 2, which were calculated by applying a short rectangular pulse. This indicates that both the short pulse and the ac ﬁeld excite identical modes and validates our method to study the modes. FIG. 4. Snapshots of mode 1 for Bloch-type, Néel-type, and antivortex-type skyrmion crystals under an inclined magnetic ﬁeld Hex=(Hztanθ,0,Hz), with Hz=0.036 and θ=30◦activated by the in-plane microwave ﬁeld Hω/bardbly. Note that the rotation sense of the antivortex type is opposite those of the Bloch type and the Néel type.a three-dimensional model. In addition, the demagnetization effects due to the magnetic dipole interactions are not incor-porated in the present study. A recent study on the microwave-active skyrmion resonances revealed that frequencies andamplitudes of the skyrmion modes vary depending on theshape and thickness of the specimens due to the demagne-tization effects [ 16]. However, the results of Ref. [ 16]a l s o indicated that the demagnetization effects do not change theproperties of resonant modes qualitatively. IV . RESULTS OF TRANSLATIONAL MOTION A recent theoretical study by Wang et al. discovered that translational motion of skyrmions can be driven by an out-of-plane microwave ﬁeld H ω/bardblzunder an inclined magnetic ﬁeldHexthrough activation of their breathing oscillations [34]. Motivated by this study, we investigate the motions of a two-dimensional skyrmion crystal driven by several differentresonance modes under the H exﬁeld inclined towards the xdirection. In the numerical simulations, we ﬁnd that the translational motion can be driven not only by the previouslyexamined out-of-plane microwave ﬁeld H ω/bardblzbut also by 184428-5IKKA, TAKEUCHI, AND MOCHIZUKI PHYSICAL REVIEW B 98, 184428 (2018) FIG. 5. Translational motion of Bloch-type skyrmion crystal in a thin-plate specimen driven by microwave irradiation through ac- tivation of the resonance modes under an inclined magnetic ﬁeld Hex=(Hztanθ,0,Hz), with Hz=0.036 and θ=30◦. Here, the microwave ﬁeld is given by Hω αsinωt(α=x,y), with Hω α= 0.0006. The skyrmion crystal moves approximately towards the positive (negative) ydirection when a microwave ﬁeld Hω/bardblx (Hω/bardblz)w i t h ω=0.0494 ( ω=0.0666) activates mode 1 (mode 2) with a dominant counterclockwise-rotation (breathing) component. Displacement vectors connecting the original position (dashed cir-cles) and the position after microwave irradiation for 400 ns (solid circles) are indicated by the thick arrows in the right panels. The motion driven by mode 1 is much faster than that driven by mode 2. in-plane microwave ﬁelds Hω/bardblx,yvia activation of the rotational oscillations of the skyrmions. Figure 5shows snapshots of the skyrmion crystal driven byHω/bardblx(top right panel) and the same skyrmion crystal driven by Hω/bardblz(bottom right panel) at t=400 ns after the microwave irradiation commences. Figure 5also shows the initial conﬁguration of the skyrmion crystal at t=0 (left panel) under application of an inclined magnetic ﬁeld Hex=(Hztanθ,0,Hz), with Hz=0.036 and θ=30◦.H e r e the microwave ﬁeld is given by Hω αsinωt(α=x,y), with Hω α=0.0006. An area composed of 96 ×96 sites is magni- ﬁed, although the simulations are performed using a systemof 96×111 sites. The displacement vectors connecting the original position and the position at t=400 ns are indicated by the thick arrows shown in the right panels. When themicrowave ﬁeld H ω/bardblxwithω=0.0494 activates mode 1 with a dominant counterclockwise-rotation component, theskyrmion crystal propagates in a direction close to the positiveydirection, whereas the same skyrmion crystal propagates in a direction close to the negative ydirection when H ω/bardblzwith ω=0.0666 activates mode 2 with breathing oscillations. We also ﬁnd that the travel distance in the former case is muchlonger than that in the latter case, which indicates that thein-plane microwave ﬁeld H ω/bardblxdrives much faster motions of the skyrmion crystal than the out-of-plane microwave ﬁeld Hω/bardblz. Next, we investigate the microwave frequency dependence of the velocity v=(vx,vy) of the driven skyrmion crystal un- der an inclined magnetic ﬁeld Hex=(Hztanθ,0,Hz), with FIG. 6. Calculated ωdependence of the velocity v=(vx,vy) for translational motion of the Bloch-type skyrmion crystal induced by a microwave ﬁeld Hωunder an inclined magnetic ﬁeld Hex= (Hztanθ,0,Hz), with Hz=0.036 and θ=30◦. Here, the mi- crowave ﬁeld is given by Hω αsinωt(α=x,y), with Hω α=0.0006, andω=2πfis its angular frequency. The velocities show peaks at the resonant frequencies of the modes, while their signs vary depending on the mode. Hz=0.036 and θ=30◦.I nF i g s . 6(a) and6(b),w ep l o tt h e simulated ωdependence of vxandvy, respectively, for differ- ent microwave polarizations where the microwave amplitudeis set to be H ω α=0.0006. We ﬁnd that the velocities are enhanced to have peaks at frequencies that correspond to theresonant modes, whereas their signs vary depending on themode. We also ﬁnd that the velocity is highest when the in-plane microwave ﬁeld H ω/bardblx(ω=0.0494) activates mode 1 with the dominant counterclockwise-rotation component. Inthis case, the value of v xbecomes ∼0.04 m/s. In contrast, the velocity when mode 2 is activated at ω=0.666 is highest in the case of the out-of-plane microwave ﬁeld Hω/bardblz, where it reaches −0.01 m/s. Note that the in-plane microwave ﬁeld Hω/bardblxdrives the skyrmion crystal approximately four times faster than the out-of-plane microwave ﬁeld Hω/bardblz. We then investigate the θdependence of the velocity of the skyrmion crystal when driven by mode 1 and mode 2,where θis the inclination angle of the external magnetic ﬁeldH ex=(Hztanθ,0,Hz), with Hz=0.036. In Fig. 7(a), the calculated absolute values of velocity v=√ v2 x+v2 yare plotted for mode 1. The velocity increases noticeably as θ increases and seems to become saturated. Figure 7(b) shows the direction of propagation for different microwave polariza-tions. We ﬁnd that the skyrmion crystal under the in-planemicrowave ﬁelds H ω/bardblx,ymoves approximately in the positive ydirection, irrespective of the value of θ. In contrast, the skyrmion crystal under the out-of-plane microwave ﬁeld Hω/bardblzmoves approximately in the positive xdirection when θis small, whereas the propagation direction becomes slanted towards the positive ydirection as θincreases. 184428-6RESONANCE MODES AND MICROW A VE-DRIVEN … PHYSICAL REVIEW B 98, 184428 (2018) FIG. 7. Calculated θdependence of the velocity v=√ v2 x+v2 y of the skyrmion crystal driven by a microwave ﬁeld Hωthrough activation of (a) mode 1 with the dominant counterclockwise-rotation component and (c) mode 2 with breathing oscillations for different microwave polarizations. Here, θis the inclination angle of the external magnetic ﬁeld Hex=(Hztanθ,0,Hz), with Hz=0.036, and the microwave ﬁeld is given by Hω αsinωt(α=x,y), with Hω α= 0.0006. All velocity data are measured at the resonant frequencies of the modes. Calculated θdependence of the propagation direction of the skyrmion crystal driven by (b) mode 1 and (d) mode 2 for different microwave polarizations. Figure 7(c) shows calculated speeds of v=√ v2 x+v2 yfor mode 2 activated under different microwave polarizations.ForH ω/bardblz, the velocity initially increases as θincreases to reach a maximum at θ∼15◦and, subsequently, decreases gradually with increasing θ. In contrast, this type of peak- maximum behavior is not clear in the Hω/bardblx,ycase, but the velocity shows saturation behavior or a slight decreaseafter the initial increase in the small θregion. Figure 7(d) indicates that the direction of propagation under the in-planemicrowave ﬁelds H ω/bardblx,yis approximately in the negative xdirection. In contrast, the direction of propagation under the out-of-plane microwave ﬁeld Hω/bardblzis approximately in the negative ydirection and is slanted slightly towards the positive xdirection. In both cases, the changes in the propagation direction upon variation of θare small. Note that each of the velocity data is measured at the resonant frequency of the mode, which varies depending onthe inclination angle θ.T h eθdependence of the resonant fre- quency ω Rfor each mode is summarized in Fig. 2(e). It should also be noted that the data shown in Fig. 7are calculated for the Bloch-type skyrmion crystal, but we have examined theother two types of skyrmion crystals as well and have foundthat the absolute speed data plotted in Figs. 7(a) and7(c) do not change. In contrast, the directions of propagation differamong the three types of skyrmion but they are related toeach other. The plots in Figs. 7(b) and7(d) also hold for the Néel-type and antivortex-type skyrmion crystals if we replacethe deﬁnitions of φin the insets. These deﬁnitions should be replaced with those in Fig. 8(a) for the Néel-type skyrmionFIG. 8. Deﬁnitions of the angle φthat describes the direction of propagation of the (a) Néel-type and (b) antivortex-type skyrmion crystal when driven by modes 1 and 2. The θdependence of the direction of propagation for the Bloch-type skyrmion crystal in Fig. 7(b) also holds for the Néel-type and antivortex-type skyrmion crystals if we replace the deﬁnitions of φwith them. Note that they are related to each other through 90◦rotation around the zaxis and mirror operation with respect to the zxplane. crystal and those in Fig. 8(b) for the antivortex-type skyrmion crystal. Finally, we examine the Hzdependence of the ve- locity. Figure 9(a) shows the calculated absolute velocity FIG. 9. Calculated Hzdependence of the velocity v=√ v2 x+v2 y of skyrmion crystal when driven by a microwave ﬁeld Hωthrough activation of (a) mode 1 with the dominant counterclockwise-rotation component and (b) mode 2 with breathing oscillations for different microwave polarizations. Here, Hzis the perpendicular component of the external magnetic ﬁeld Hex=(Hztanθ,0,Hz), with θ=30◦, while the microwave ﬁeld is given by Hω αsinωt(α=x,y), with Hω α=0.0006. 184428-7IKKA, TAKEUCHI, AND MOCHIZUKI PHYSICAL REVIEW B 98, 184428 (2018) v=√ v2 x+v2 yof a skyrmion crystal when driven by a mi- crowave ﬁeld Hωthrough activation of mode 1. Here Hz is a perpendicular component of the inclined magnetic ﬁeld Hex=(Hztanθ,0,Hz), with θ=30◦. The microwave am- plitude is again ﬁxed at Hω α=0.0006. The plots for Hω/bardblx, yshow a maximum at Hz∼0.35, which is deep inside the skyrmion crystal phase, whereas the plot for Hω/bardblzmono- tonically increases with increasing Hz. In contrast, the plots for mode 2 in Fig. 9(b) show more complex behavior that is dependent on the microwave polarization. The microscopicmechanism of these characteristic behaviors is an issue ofimportance, which should be clariﬁed in a future. Note that the present study is based on a pure two- dimensional model. The absolute velocities may slightly varydepending on the thickness of real thin-plate specimens be-cause the relative weight of the inﬂuences from the surfacemagnetic structures and pinning effects due to impurities,grains, and defects must differ depending on the sample thick-ness. A quantitative investigation of the skyrmion velocities inreal three-dimensional specimens is left for future studies. V . SUMMARY In summary, we have theoretically studied the microwave- active resonance modes of skyrmion crystal on a thin-platespecimen under a perpendicular or inclined external magneticﬁeldH ex. We have found that while only two rotation modes or a single breathing mode are active under the perpendicularﬁeld, novel microwave-active modes emerge when the Hex ﬁeld is inclined. The modes that exist originally under the perpendicular ﬁeld are enhanced or suppressed depending onthe polarization of the microwave ﬁeld H ω. Recent studies re- vealed that the collective modes of skyrmions host rich phys-ical phenomena and may provide potentially useful devicefunctions, which are attracting a great deal of research interest.As an example of these phenomena, we have investigated themicrowave-driven translational motion of a skyrmion crystalunder an inclined H exﬁeld [ 34]. Our numerical simulations have demonstrated that the propagation velocity of skyrmioncrystal is enhanced at resonant frequencies of the modes,while the velocity and direction of the skyrmion propagationare sensitively dependent on the activated modes. Importantly,the in-plane microwave ﬁeld that activates the dominant coun-terclockwise rotation drives the skyrmion crystal much morerapidly than the out-of-plane microwave ﬁeld that activatesthe breathing mode studied in Ref. [ 34]. The knowledge obtained in this study will help to open novel research intothe spintronics and magnonics functions of skyrmions basedon microwave irradiation. 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PhysRevB.85.014416.pdf	PHYSICAL REVIEW B 85, 014416 (2012) Composition dependence of magnetic properties in perpendicularly magnetized epitaxial thin ﬁlms of Mn-Ga alloys S. Mizukami, T. Kubota, F. Wu, X. Zhang, and T. Miyazaki WPI–Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan H. Naganuma, M. Oogane, A. Sakuma, and Y . Ando Department of Applied Physics, Tohoku University, Sendai 980-8579, Japan (Received 5 August 2011; revised manuscript received 17 December 2011; published 18 January 2012) Mn-Ga binary alloys show strong magnetism and large uniaxial magnetic anisotropy even though these alloys do not contain any noble, rare-earth metals or magnetic elements. We investigate the composition dependenceof saturation magnetization M Sand uniaxial magnetic anisotropy Kuin epitaxial ﬁlms of Mn xGa1−xalloys (x∼0.5–0.75) grown by magnetron sputtering. The MSvalues decrease linearly from approximately 600 to 200 emu /cm3with increasing x, whereas the Kuvalues decrease slightly from approximately 15 to 10 Merg /cm3 with increasing x. These trends are distinct from those for known tetragonal hard magnets obtained in a limited composition range in Mn-Al and Fe-Pt binary alloys. These data are analyzed using a localized magnetic momentmodel. DOI: 10.1103/PhysRevB.85.014416 PACS number(s): 75 .20.Hr, 73 .61.At, 75 .30.Gw, 75 .70.Ak I. INTRODUCTION The demand for perpendicular magnetization ﬁlms has increased considerably since these ﬁlms are crucial in ad- vanced spintronics applications, such as high-density per- pendicular magnetic recording and Gbit-class spin-transfer- torque magnetic random access memory (STT-MRAM).1–3 As sizes of magnets are reduced to several tens of nm to increase storage densities in these applications, thermal ﬂuctuations in the magnetization direction become substantial, thereby causing memory loss in such storage devices. A high magnetic anisotropy of over 10 Merg /cm3is required in practice to maintain the magnetization direction in such nanoscale magnets, and therefore, a variety of perpendicular magnetization ﬁlms has been investigated thus far. Mn-Ga binary alloys are similar to Mn-Al binary alloys in the sense that they are hard magnets. Both these alloy types show strongmagnetism even though they consist of nonmagnetic and light elements. Despite Mn-Ga alloys exhibiting strong magnetism over a much wider compositional range when compared with Mn-Al alloys, the former have been studied less extensively. Alloys of Mn-Ga with L1 0structure are thermodynamically stable for x∼0.5–0.65 [Fig. 1(a)].4Although magnetization decreases as the Mn composition xincreases, these alloys have a large magnetic anisotropy with magnetically easy axis parallel to the caxis.4In the past decade, several groups have investigated the structural, magnetic, and transportproperties for off-stoichiometric L1 0Mn-Ga ﬁlms with a view to their magneto-optical and spintronics applications.5–8 Forx∼0.65–0.75, the D022structure appears to exhibit strong ferrimagnetism [Fig. 1(b)], as conﬁrmed by neutron scattering.9A high Curie temperature of up to ∼800 K, large magnetic anisotropy, and composition-sensitive magnetiza- tion were also reported in D022-phase polycrystalline bulk samples.10 Recently, renewed interest in D022Mn 3Ga has been expressed in the context of Heusler alloys because the D022 structure can be depicted as a tetragonally distorted D03structure that is similar to the L21structure in Heusler alloys. In the case of D03Mn 3Ga that is not stable in reality, the band structure calculated from ﬁrst principles exhibits a spinpolarization close to unity, namely that of a half metal. 11,12 Tetragonally distorted D022Mn 3Ga is not a half metal; however, it still exhibits a spin polarization as large as 88%,as predicted by the calculation from ﬁrst principles. 13The structural and magnetic properties of bulk D022Mn-Ga alloys with off-stoichiometric compositions have been investigated,and the results suggest that vacancies produced by Mndeﬁciencies increase the magnetic moment signiﬁcantly in theD0 22phase.14 We have recently reported on c-axis-oriented, off- stoichiometric D022Mn 2.5Ga epitaxial ﬁlms, prepared by the sputtering technique, that exhibited a uniaxial perpendicularmagnetic anisotropy constant K uof over 10 Merg /cm3.15–17 Moreover, the Gilbert damping constant, which determines the critical current density for STT switching, was smallerfor Mn-Ga alloy ﬁlms with D0 22orL10phases than for the perpendicularly magnetized ﬁlms previously reported.18In addition, we also predicted the presence of large tunnel magne-toresistance (TMR) effect of over 600% in MgO-based D0 22 Mn 3Ga magnetic tunnel junctions (MTJs).19The experimental TMR ratio in a D022-Mn 2.4Ga/MgO/CoFe MTJ was 22.1% at 10 K,19,20even though the experimental spin polarization estimated by using a point contact Andreev reﬂection wasas high as 58%. 21Large perpendicular magnetic anisotropy, small damping constant, and large TMR ratio are importantrequirements for STT-MRAM; 2,3consequently, Mn-Ga alloys may be one of the key materials for this type of application. Several groups have recently reported the structural and magnetic properties of Mn-Ga alloy ﬁlms with D022- ordered22,23orL10-ordered24–27structures. These reports focused on the limited composition range belonging to D022 orL10phases. Further, it is noteworthy that all previous studies on compositional dependence have been performedwith polycrystalline bulk samples, and consequently, it hasthus far been difﬁcult to estimate exact values for M sand 014416-1 1098-0121/2012/85(1)/014416(6) ©2012 American Physical SocietyS. MIZUKAMI et al. PHYSICAL REVIEW B 85, 014416 (2012) particularly Kuagainst varying Mn composition because of the exceedingly large saturation ﬁelds involved.4,10,14Recent studies have reported the composition dependence of the TMRratio in MgO-based Mn-Ga MTJ. 28In this study, we report the wide-range composition dependence of MsandKuin thin epitaxial ﬁlms of Mn-Ga alloys, and we discuss thesedependencies using a localized magnetic moment model. II. EXPERIMENTAL AND COMPUTATIONAL PROCEDURES In our experiment, 30-nm-thick Mn xGa1−xﬁlms were deposited on Cr-buffered single-crystal (100) MgO substratesunder an Ar pressure of 0.1 Pa using magnetron sputtering witha base pressure of less than 10 −6Pa. We prepared alloys ﬁlms withx=0.54, 0.62, and 0.72 using Mn-Ga alloy targets of different compositions, and the ﬁlms with x=0.65 and 0.75 were prepared using the co-sputtering technique. Films withx=0.75 were grown at a substrate temperature of 400 ◦C because the stoichiometric Mn 3Ga ﬁlms were not available on a Cr buffer after a postannealing procedure. Films withother compositions were grown at room temperature andsubsequently annealed at 400–500 ◦Cin situ . After cooling to room temperature, all the ﬁlms were capped with a Ta layerto prevent oxidation. The composition of the Mn-Ga ﬁlms was carefully an- alyzed several times by inductively coupled plasma massspectrometry. Structural analysis was performed by using anout-of-plane and in-plane x-ray diffractometer (XRD) with a9-kW rotating anode, and the Cu K αline was used for the analysis. Magnetization measurements were carried out usinga vibrating sample magnetometer as well as a superconductingquantum interference device magnetometer. The electronic band structures of L1 0-ordered MnGa and D022-ordered Mn 3Ga were calculated with linear mufﬁn-tin orbitals in the atomic sphere approximation (LMTO-ASA)based on the density functional formalism. To calculate theK uvalues, we used LMTO-ASA including the spin-orbit interaction and the force theorem.18,29,30 III. EXPERIMENTAL RESULTS Figure 2shows examples of the XRD patterns obtained for 30-nm-thick Mn xGa1−xalloy ﬁlms deposited on a Cr buffer layer. The XRD patterns showed only the sharp (002) and (004)diffraction peaks for these ﬁlms; the other diffraction peakscorrespond to the substrate. The (002) and (004) diffractionpeaks are respectively attributed to the superlattice peak for theL1 0-type ordering and the fundamental peak of the L10and D022Mn-Ga alloys for the unit cells shown in Figs. 1(a) and 1(b). From the XRD results, the Mn xGa1−xalloy ﬁlms appear to exhibit a single phase solid solution in the range of 0 .5< x/lessorequalslant0.75, and this is consistent with the results reported for the bulk samples.10However, the (002) and (004) diffraction peaks for the alloy ﬁlms became broader at x=0.75, thereby indicating that the ﬁlm structure is degraded when comparedwith those of alloy ﬁlms with intermediate compositions. Figure 3(a) shows the Mn concentration dependence of the lattice constants estimated from the XRD patterns forthe alloy ﬁlms. The in-plane lattice constant ais roughly (b) (a)MnIIGa MnI (b) (a) MnIIGa or Mn (c) FIG. 1. (Color online) Schematic of a unit cell of crystal structure for (a) L10-ordered MnGa, (b) D022-ordered Mn 3Ga, and (c) MnxGa1−x(0.5<x/lessorequalslant0.75) alloys ordered maximally in the L10 structure with no vacancies. The unit cell in (a) is doubled along thecaxis for comparison with the D022unit cell. In (b), Mn atoms that occupy Wyckoff positions 2 band 4dare denoted by Mn Iand Mn II, respectively; Ga atoms are located at the Wyckoff position 2 a (Ref. 14). The magnetic moment of Mn Iis antiparallel to that of Mn II. The Mn Iatoms in the D022structure are replaced by Ga atoms in the L10structure. In (c), Ga and Mn atoms are randomly located at both 2aand 2bpositions. (002) (004)MgO MgO(a) 30 40 50 60020406080tensity (kcps) (002) (004)MgO MgO(b)30 40 50 60Diffraction in t 30 40 50 60020406080 2θ (o) FIG. 2. Example of x-ray diffraction patterns for 30-nm-thick MnxGa1−xalloy ﬁlms deposited on a Cr buffer layer. (a) x=0.54 and (b) x=0.62. 014416-2COMPOSITION DEPENDENCE OF MAGNETIC PROPERTIES ... PHYSICAL REVIEW B 85, 014416 (2012) 0.5 0.6 0.7 0.83.03.54.0a (Å)7.07.58.0c (Å) 09 51.00a(a) (b) 0.5 0.6 0.7 0.80.850.900.95c / 2a x(b) FIG. 3. (Color online) Composition dependence of (a) in-plane (out-of-plane) lattice constant a(c) and (b) tetragonal distortion ratio c/2afor Mn xGa1−xalloy ﬁlms with thicknesses of 30 nm ( •) and 100 nm (/triangle) deposited on a Cr buffer layer (Ref. 15), and with thickness of 100 nm deposited on a MgO substrate ( /diamondmath)( R e f . 18). Values used in the ﬁrst-principles calculations in Ref. 30and in this report are represented by /squareand⋆, respectively. The cvalues in L10ﬁlms are doubled, for comparison with those in D022ﬁlms. independent of x, whereas the out-of-plane lattice constant c decreases slightly with increasing x. Figure 3(b) shows the plot of the tetragonal distortion ratio c/2aas a function of x. Most of the alloy ﬁlms show a c/2aratio of 0.90–0.92. The magnitude of the quantities and the curve trends in Figs. 3(a) and3(b) are comparable with those reported for bulk samples, thereby indicating good ﬁlm quality.10The values for cand c/2aatx=0.75 are slightly different from those at other x values, probably owing to structural strains that may be relatedto an instability in the D0 22structure in the stoichiometric Mn 3Ga.10 Figure 4shows examples of the hysteresis curves for 30-nm- thick Mn xGa1−xalloy ﬁlms deposited on a Cr buffer layer. The magnetization curves for all the ﬁlms show rectangular shapeswith a squareness close to unity if a magnetic ﬁeld is appliedperpendicular to the ﬁlm plane; this enables us to evaluate thesaturation magnetization M S. Figure 5(a) shows the xdependence of MSfor the alloy ﬁlms. It is noteworthy that the value for MSdecreases very linearly and systematically from about 600 to 200 emu /cm3 with increasing x.M o s to ft h e MSvalues for our ﬁlms are larger for any xvalues than those in previous reports; however, the MSvalue for the Mn 2Ga ﬁlms reported in Ref. 22is slightly larger than that in our ﬁlms, as shown in Fig. 5(a). The values for Kuagainst xin the alloys ﬁlms are also shown in Fig. 5(b).T h eKuvalues are determined using the relation Ku=MSHeff k/2+2πM2 S, where Heff kdenotes the effective magnetic anisotropy ﬁeld estimated from the mag-netization curves for measurements taken when the appliedmagnetic ﬁeld is in the ﬁlm plane. Although the values forK utend to be larger for ﬁlms with lower Mn compositions and with large thicknesses, all the ﬁlms show large Kuvalues01000emu/cm3) H // film(a) -20 -10 0 10 20-100001000M (emu/cm3) H (kOe)-20 -10 0 10 20-1000M (e Hfilm filmH // film(b) H FIG. 4. (Color online) Example of hysteresis curves for 30-nm- thick Mn xGa1−xalloy ﬁlms deposited on a Cr buffer layer. (a) x= 0.54 and (b) x=0.62. ranging from 10 to 15 Merg /cm3.T h eKuvalues for Mn 3Ga and Mn 2Ga alloys reported in Ref. 22are larger than those in our ﬁlms by a factor of 1.5–2.0, as shown in Fig. 5(b). A signiﬁcant change in the coercivity Hcis also observed along with increasing xvalues, as shown in Fig. 5(c). Thinner ﬁlms show larger Hcvalues when compared with those of thicker ﬁlms; this phenomenon is commonly observed inperpendicularly magnetized ﬁlms. The large change in H ccan be ascribed to a large change in the values for Heff kagainst x (not shown here). IV . DISCUSSION In order to understand the composition dependence of MS, we evaluated the magnetic moment mper unit cell for the alloy ﬁlms, as shown in Fig. 6(a). The measured mvalues for the bulk Mn-Ga alloys14and for the epitaxial ﬁlms of D022Mn 2Ga and Mn 3Ga22are also shown in Fig. 6(a) for the purpose of comparison. The values of min the previous reports are almost identical to those obtained by us for xvalues at around 0.70. Ourmvalues are slightly larger than those estimated in the bulk alloy samples at xvalues of around 0.66 and smaller than the previous values obtained for Mn 2Ga ﬁlms. In the following section, we discuss the origin of the linear reductionin magnetization against Mn composition. As mentioned earlier, vacancies in the crystal structure of the alloy can be introduced when the Mn content is reduced inD0 22Mn 3Ga. A Mn atom at the Wyckoff 2 bposition (Mn I) has a local magnetic moment directed opposite to the net magneticmoment and the local magnetic moment of the Mn IIatom located at the Wyckoff 4 dposition [Fig. 1(b)]; consequently, the Mn deﬁciencies at the 2 bsites tend to increase the value ofm. The previous theoretical studies suggest that the values ofmdepend nonlinearly on the number of vacancies.14The 014416-3S. MIZUKAMI et al. PHYSICAL REVIEW B 85, 014416 (2012) (a) 0.5 0.6 0.7 0.802004006008001000Ms (emu/cm3) 2030 erg/cm3) (b) (c)0.5 0.6 0.7 0.8010Ku (M 0.5 0.6 0.7 0.8051015Hc (kOe) x FIG. 5. (Color online) (a) Saturation magnetization MS,( b ) uniaxial magnetic anisotropy constant Ku, and (c) coercivity Hcas a function of composition xfor Mn xGa1−xalloy ﬁlms with thicknesses of 30 nm ( •) and 100 nm ( /triangle) deposited on a Cr buffer layer (Ref. 15) and with thickness of 100 nm deposited on MgO substrate ( /diamondmath) (Ref. 18). The values for Hcare obtained from hysteresis loops measured with the applied ﬁeld aligned with the ﬁlm normal. The measured values in Ref. 22(/triangleinv) and the calculated values in Ref. 30 (/square) and this report ( ⋆) are plotted in (a) and (b). theoretical mvalues obtained in Ref. 14are also plotted in Fig. 6(a) with different distributions of vacancies. Here, model I corresponds to the calculation taking into account the Mndeﬁciencies only at the 2 bsites, and model II corresponds to the calculation with Mn deﬁciencies at both the 2 band 4d sites in a D0 22Mn 3Ga unit cell [Fig. 1(b)].14Both models predict an increase in mwith decreasing x, and the variation inmfor model I is more pronounced than that in model II. The experimental mversus xcurves shows a linear dependence, and it is different from the predicted values for both models,thereby indicating these models are not applicable to our ﬁlms. We can also consider an alternative model that does not introduce vacancies. As pointed out for L1 0Mn-Ga alloys,4a Mn atom can occupy the 4 dsite preferentially to the 2 aand 2bsites, in a manner similar to Mn atom behavior in τMnAl alloy. For x> 0.5, the extra Mn atom can replace the Ga atom at the 2 aor 2bsites [Fig. 1(c)]. The local magnetic moment forCalc. Calc.(a) 51015m (μB / unit cell) (b) Calc.0.5 0.6 0.7 0.80 0.5 0.6 0.7 0.80.40.60.81.0S x FIG. 6. (Color online) Composition dependence of (a) magnetic moment mper unit cell and (b) L10long-range ordering parameter Sfor Mn xGa1−xalloy ﬁlms with thickness of 30 nm ( •) and 100 nm (/triangle) deposited on a Cr buffer layer (Ref. 15) and with thickness of 100 nm deposited on a MgO substrate ( /diamondmath)( R e f . 18). The measured m values in Ref. 14(×)a n dR e f . 22(/triangleinv) and the calculated mvalues in Ref. 30(/square) and in this report ( ⋆) are plotted in (a). The theoretical x dependencies of mare also shown for model I ( ◦) and model II ( +) (Ref. 14). The curves provide only a visual guide. The dashed and solid lines indicate the calculated mandSvalues with an occupation probability of Mn atoms at the Wyckoff 4 dposition for pII=1.0a n d 0.93, respectively. this extra Mn atom may couple antiferromagnetically to the magnetic moments of the Mn atoms at the 4 dsites, and this coupling could reduce the net magnetic moment for the Mn-Gaalloy. Here we assume that the local magnetic moment of theMn atom is independent of both xand the atom’s location at 2 a or 2b, and the magnetic moment depends only on the location of the Mn atom at the sites 4 dor 2a(2b). The following simple relation expresses the theoretical magnetic moment per unitcellm: m=4(m IIpII−\|mI\|pI). (1) Here,mI(mII) andpI(pII) denote the local magnetic moment and the occupation probability of a Mn atom at 2 aor 2b (4d) sites, respectively. The occupation probability obeys the following conservation law: pI+pII=2x. (2) From Eqs. ( 1) and ( 2),mis rewritten as m=4[(\|mI\|+mII)pII−2\|mI\|x]. (3) The values of mare calculated as a function of xusing Eq. ( 3), and the resulting curve is shown in Fig. 6(a) for different pII 014416-4COMPOSITION DEPENDENCE OF MAGNETIC PROPERTIES ... PHYSICAL REVIEW B 85, 014416 (2012) values. The experimental mvalue curve is reasonably ﬁtted to the calculated values with pII=0.93 (solid line) if mI= −3.2μBandmII=2.5μBare used. These mIandmIIvalues are comparable with those evaluated in D022Mn 3Ga13,14and L10MnGa,30and the values mentioned below. Interestingly, the values of mreported in the epitaxial Mn 2Ga ﬁlm22lie on the line calculated with pII=0, thereby possibly indicating that there may be negligible disorders in the ﬁlm. In order to verify the above possibility, the L10long- range ordering parameter Sis estimated from the integrated intensity ratio of the (002) and (004) diffraction peaks in theXRD patterns for the alloy ﬁlms using a standard method[Fig. 6(b)]. 31For this estimation, the Debye-Waller factor σ is set to 0.0156 (0.0104) nm for the L10(D022) ﬁlms; the value of this factor is obtained from the ratio of the (004) and(008) peak intensities for the 100-nm-thick ﬁlms deposited ona MgO substrate. Theoretically, Sis expressed as 31 S=2(pII−x). (4) The estimated value for Sis plotted as a function of x,a ss h o w n by the dashed (solid) line for pII=1.0 (0.93) in Fig. 6(b). The maximum value for Sis unity at x=0.5; subsequently, Sdecreases with increasing x(dashed line). The Svalue also reduces with decreasing pIIvalues, i.e., due to the the swapping o faM na t o ma tt h e4 dsite with a Ga atom at the 2 aor 2b sites. The experimental Svalue decreases with increasing x, and the ranges between the values for pII=1.0 and for 0.93 at 0.5<x< 0.7 are roughly consistent with the value of pII used for the calculation of min Fig. 6(a). The experimental S values are above the theoretical maxima at x> 0.7, thereby implying that the experimental Svalues are not reliable in this range. This might be due to an underestimation of theσvalue for x> 0.7; thus, microscopic characterization for atomic ordering is needed to conﬁrm this model of localizedmagnetic moment. In order to further discuss the validity of this picture, we calculated the electronic band structures of L1 0MnGa and D022Mn 3Ga using the lattice constants shown in Fig. 3(a). The resulting total and partial densities of states (DOSs) forL1 0MnGa and D022Mn 3Ga are shown in Figs. 7(a) and7(b), respectively. The largest peak of the density of states in theminority (majority) spin band is located at an energy levelgreater (lesser) than the Fermi energy level in L1 0MnGa, and this peak is chieﬂy attributed to the dorbitals of the Mn atoms at the Mn IIsites. These peaks also appear for D022 Mn 3Ga even though ﬁner structures appear owing to the D022 superlattice [Fig. 7(b)], thereby implying that the magnetic moment of the Mn atoms at Mn IIsites in D022Mn 3Ga is not largely different from that for L10MnGa. In fact, the magnetic moment of Mn at the Mn IIsite in D022Mn 3Ga is estimated as 2.5μB, and this value is roughly identical to that for L10MnGa (2.6μB). InD022Mn 3Ga, there are a few fairly narrow peaks attributed to the dorbitals of the Mn atoms at the Mn Isites, thereby indicating that these dorbitals are almost localized, and the resulting magnetic moment is estimated to be −3.1μB. [The derived MSvalues were also plotted in Fig. 5(a) for comparison.] This localized nature of dorbitals at Mn Isites has also been observed in the context of Heusler alloys.32 These physical insights gained from the electronic structures(a) majorityminority -6 -4 -2 0 2-10010es (state/eV unit cell) (b)minority majority Density of stat e -6 -4 -2 0 2-10010 E-E F (eV) FIG. 7. (Color online) Spin-dependent total and partial densities of states (DOSs) in (a) L10MnGa and (b) D022Mn 3Ga. Total DOS is denoted by solid curves, and partial DOSs for Mn I,M n II, and Ga are shown with thin solid, dot-dashed, and broken curves, respectively. are in accord with the localized and composition-insensitive magnetic moment model suggested above. The respective values of Kucalculated from ﬁrst principles are also plotted in Fig. 5(b). The order of magnitude for the theoretical Kuvalues is in good agreement with those obtained in the experiment; however, the values for the D022phases are larger than those obtained in our experiments by a factor of1.5–2.0. It is difﬁcult to simultaneously interpret the data of K u vsxon the basis of the speculated localized magnetic moment model. The orbital magnetic moments for the d(p) orbitals of the Mn (Ga) atom, evaluated from the ﬁrst-principlescalculation including the spin-orbit interaction, are so smallthat we cannot explain the large K uand its composition depen- dence. Further studies are required to obtain a uniﬁed physicalpicture that explains this composition dependence of M Sand Kuin Mn-Ga alloys. In addition, the studies need to utilize calculations from ﬁrst principles to account for disorders. V . SUMMARY In our study, we investigated the structural and magnetic properties of the Mn xGa1−xalloy ﬁlms with different compo- sition ratios. The MSvalue was approximately 600 emu /cm3 014416-5S. MIZUKAMI et al. PHYSICAL REVIEW B 85, 014416 (2012) atx=0.54, and it reduced to approximately 200 emu /cm3at x=0.75 while maintaining squared hysteresis curves and Ku values at around 10–15 Merg /cm3. These data were analyzed using the localized magnetic moment model. The Mn-Ga alloycan be a promising material not only for STT-MRAM but alsofor other applications requiring materials with a high K uvalue and widely tunable MSvalues.ACKNOWLEDGMENTS This work was partially supported by a Grant-in-Aid for Scientiﬁc Research from JSPS, the Strategic InternationalCooperative Program ASPIMATT from JST, and a Grant forIndustrial Technology Research from NEDO, World PremierInternational Research Center Initiative (WPI) from MEXT,and the Casio foundation. 1D. Weller, A. Moser, L. Folks, M. E. Best, W. Lee, M. F. Toney, M. Schwickert, J.-U. Thiele, and M. F. Doerner, IEEE Trans. Magn. 36, 10 (2000). 2H. Yoda, T. Kishi, T. Nagase, M. 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PhysRevB.101.195202.pdf	PHYSICAL REVIEW B 101, 195202 (2020) Spin pump induced inverse spin Hall effect observed in Bi-doped n-type Si A. A. Ezhevskii ,1,D. V . Guseinov ,1A. V . Soukhorukov ,1A. V . Novikov ,2D. V . Yurasov ,2and N. S. Gusev2 1Lobachevsky State University of Nizhny Novgorod, 23 Pr. Gagarina (Gagarin Avenue), 603950 Nizhny Novgorod, Russia 2Institute for Physics of Microstructures Russian Academy of Sciences, GSP-105, Afonino, Nizhny Novgorod 603950, Russia (Received 21 February 2020; revised manuscript received 6 April 2020; accepted 30 April 2020; published 20 May 2020) An inverse spin Hall effect (ISHE) in n-type silicon was observed experimentally when conduction electrons were scattered on the spin-orbit potential of bismuth. The spin current in the silicon layer was generatedby excitation of the magnetization precession during ferromagnetic resonance in a thin Permalloy (Py) layerdeposited on a Si layer doped by phosphor and bismuth. From the angular dependences of the dc voltage fordifferent Py /n-Si:Bi structures aligned along the [011] or [100] crystal axes, we were able to distinguish the planar Hall effect (PHE) and ISHE contributions. The ISHE dc voltage signal was proportional to the sin θsin 2θ product for the structure aligned to the [011] crystal axis and to sin θcos 2θfor the [100] direction. In addition, the PHE dc voltage was observed for the angles corresponding to the sin 2 θdependence. This means that for silicon as a many-valley semiconductor, the scattering of spins due to the spin-orbit potential induced by shallowdonor in n-type material is dependent on the orientation of the valley axes relative to the direction of the magnetic ﬁeld. DOI: 10.1103/PhysRevB.101.195202 I. INTRODUCTION There has been a growing theoretical and experimental interest focused on the spin Hall effect (SHE), which refersto the generation of spin currents from charge currents viathe spin-orbit interaction [ 1–6]. The spin-orbit interaction responsible for the SHE may also cause the inverse spin Halleffect (ISHE), which is the process that converts a spin currentinto an electric voltage [ 7–11]. The ISHE was observed and investigated in simple metallic ferromagnetic/paramagneticNi 81Fe19/Pt bilayer systems using a spin-pumping method operated by ferromagnetic resonance (FMR) [ 7–11]. The spin pumping driven by ferromagnetic resonance injects a spincurrent into the paramagnetic layer, which gives rise to anelectromotive force transverse to the spin current using theISHE in the paramagnetic layer. In a Ni 81Fe19/Pt ﬁlm, an electromotive force was found perpendicular to the appliedmagnetic ﬁeld at the ferromagnetic resonance condition. Theelectromotive force was observed also in a Pt /Y 3Fe4GaO 12 ﬁlm in which the metallic ferromagnetic layer was replaced by an insulating Y 3Fe4GaO 12layer, supporting that the spin- pumping-induced ISHE is responsible for the observed elec-tromotive force [ 10]. In [ 11] the ISHE was studied in p-type silicon by holes scattering on the lattice spin-orbit potential(SOP). A speciﬁc feature of n-type silicon which is considered as a material for spintronic future applications [ 12–23]i s a weak spin-orbit interaction for electrons, which on theone hand leads to weak spin accumulation effects due tothe small angles of the spin Hall effect, and on the other ezhevski@phys.unn.ruhand to signiﬁcantly lower spin-relaxation rates and longspin-diffusion length. Strong spin-dependent scattering on theSOP can be induced by doping of Si by a heavy donor ofgroup V . The latter could make it possible to control thespin-orbit contribution to the scattering, which may lead tothe generation and detection of spin currents under certainconditions. Earlier studies [ 24] of spin resonance spectra of conduc- tion electrons in n-type Si showed that spin scattering of conduction electrons on the spin-orbit potential of Bi makesa signiﬁcant contribution to their linewidths and gfactors. The suggestion to observe the ISHE experimentally in sili-con doped with heavy donors as a small contribution to theordinary Hall effect was made in [ 25]. The extrinsic spin Hall effect due to skew scattering at heavy substitutional impuritiesin silicon using ﬁrst-principles calculations was consideredin theoretical work [ 26]. In the case of Si(Pt) and Si(Bi), a spin Hall angle is calculated comparable to those found inmetals. In Ref. [ 27] the effect of enhancement of the spin- orbit coupling in silicon thin ﬁlms by doping with bismuthwas studied experimentally at low temperature by quantumcorrections to conductance measuring. Despite several studiesthat showed enhanced spin-orbit interaction in n-type Si by doping with the heavy elements, the ISHE effect itself has notyet been observed experimentally in n-Si. In this paper we report the observation of ISHE in n-type silicon. In this case conduction electrons which were spin-polarized by ferromagnetic resonance in the Permalloy layerdeposited on silicon are scattered on the spin-orbit potentialof bismuth. Due to the small value of ISHE in n-type silicon, some additional effects such as the anomalous Hall effect(AHE) and anisotropic magnetoresistance (AMR) have to beconsidered and compared with ISHE. 2469-9950/2020/101(19)/195202(7) 195202-1 ©2020 American Physical SocietyA. A. EZHEVSKII et al. PHYSICAL REVIEW B 101, 195202 (2020) FIG. 1. (a) Schematic view of the Py /n-Si:Bi structure for the ISHE study. Precession of the magnetization direction M(t)o ft h e ferromagnetic Py layer pumps spins into the adjacent Si:Bi:P layer by inducing a spin current. (b) Schematic illustration for the pure-spin-current injection J sby spin pumping (H – magnetic ﬁeld, σ– the spin-polarization vector). II. EXPERIMENTAL To study the effects associated with the excitation and detection of spin current in n-type silicon, the phenomenon of spin pumping [ 1] was exploited. The investigated structure was formed by the following procedure. At ﬁrst a 250-nm-thick Si layer was deposited by molecular beam epitaxy on thecommercial “silicon on isolator” (SOI) wafer (Soitec) with a250-nm-thick device Si layer and a 3- μm buried oxide (BOX) layer. As a result, the total thickness of the Si layer above theBOX was increased up to 500 nm, which corresponded to themaximum thickness of the Si layer which can be doped bybismuth using our ion implantation technique. Initially, thedoping level in this layer was of the order of 10 15cm−3(p type). The Si layer in the investigated structure was doped byP and Bi via ion implantation (Bi: /Phi1=0,005μC/cm 2,E= 24 keV; P: /Phi1=600μC/cm2,E=100 keV) and subsequent annealing (1000◦C, 30 min). The molecular beam epitaxy (MBE) used for growth of the additional Si layer above the BOX was implemented becausewe did not have the SOI wafer with a 500-nm-thick Si devicelayer. The increased Si layer thickness after the MBE growthas compared to the initial Si device layer thickness of SOIwafer allowed us to increase the magnitude of studied effectsand made them easier to measure. The Au/Ti (1 .0×0.5m m 2) Ohmic contacts and a Py layer (1 .0×1.8m m2) were formed on the Si layer (1 .0× 3.0m m2) using lift-off lithography and magnetron sputtering. The Py /n-Si:Bi structures with two different orientations (the longer side of the structure was aligned to the [101] or [100]crystal axis) were formed. Use of SOI substrates allowedeliminating the impact of substrate conductivity on the ob-tained results. The structure layout is shown in Fig. 1(a), and a schematic illustration of spin pumping is shown in Fig. 1(b). In order to proof the bismuth involvement in the ISHE spin scattering, the Py/Si:P structure without Bi doping was madeas a reference sample. Phosphorus doping of the silicon layerwas carried out under the same conditions as for the structureswith Bi doping. FMR spectra in the Py layer were measured at room temperature with the Bruker_EMX-plus-10 /12 electron spin resonance spectrometer operating at 9.4 GHz microwave(MW) frequency. The Py/Si:Bi/SOI structure was placed near the center of a high- QTE 011cylindrical microwave cavity at which the magnetic component of the microwave ﬁeld ( h)i s maximized while the electric-ﬁeld component is minimized.The voltage on the Au/Ti contacts was measured using alock-in ampliﬁer at a modulation frequency of the magneticﬁeld of 100 kHz and recorded using an oscillograph. III. PRECESSION-INDUCED SPIN PUMPING IN Py/n-Si:Bi FILM To describe the injection of spins from a thin layer of ferromagnetic material into the n-type silicon using the spin- pumping method, one can use the theoretical model proposedin [28,29] for a ferromagnetic-normal metal (F-N) structure. As is well known, an F-N interface leads to a dynamicalcoupling between the ferromagnetic magnetization and thespins of the conduction-band electrons in the normal metal[30,31]. When the magnetization direction precesses under the inﬂuence of an applied magnetic ﬁeld, a spin current I s is pumped out of the ferromagnet [ 29] and can be written as Ipump s=¯h 4πg↑↓ r/bracketleftbigg /vectorm(t)×d/vectorm(t) dt/bracketrightbigg . (1) Here, ¯ his the reduced Planck constant, m(t) is a unit vector of the time-dependent magnetization direction of the ferro-magnet, which at time trotates around the vector of the magnetic ﬁeld H eff[m(t)⊥Heff], and g↑↓ ris the real part of the mixing conductance, which will be determined below.This follows the conservation of energy /Delta1E F=−/Delta1ENand angular momentum /Delta1LF=−/Delta1LN, according to which the number of spins pumped from the ferromagnet into the normalmetal (being equal to N s) transfers the energy /Delta1EN=Nsμs/2 and the angular momentum /Delta1LN=Ns¯h/2. Here μsis the spin accumulation or nonequilibrium chemical potential im-balance, /Delta1E F=g/Delta1LFHeffthe magnetic energy, and gis the gyromagnetic ratio of the ferromagnetic material. The spincurrent from ( 1) leads to a damping of the ferromagnetic precession, resulting in an alignment of the magnetizationwith the applied magnetic ﬁeld. 195202-2SPIN PUMP INDUCED INVERSE SPIN HALL EFFECT … PHYSICAL REVIEW B 101, 195202 (2020) From a microscopic point of view, the spins sof the con- duction electrons in the normal conductor are coupled to thelocalized spins Sof 3dmagnetic electrons in the ferromagnet by the s-dexchange interaction in the vicinity of the F-N interface, and this can be described by the term 2 J sds·S. Following the concept described above in FMR-induced spin-pumping experiments used for the Py /n-Si:Bi system, an external magnetic ﬁeld Hand a microwave ﬁeld hare applied, and a precession motion of the magnetization Mis excited by absorbing the angular momentum from the microwaveat the resonance condition (see Fig. 1). The magnetization precession damping proportional to M×dM/dtproduces a pure spin current. The time average of M×dM/dtgener- ates a dc pure spin current J scarrying the spin-polarization vector σ. Here, the steady magnetization precession is maintained by balancing the absorption and the emission of the angular mo-mentum of the magnetization. This emission is proportionalto the M(t)× dM(t) dtterm in the following Landau-Lifshitz- Gilbert (LLG) equation [ 18]: dM(t) dt=−γM(t)×Heff+α MsM(t)×dM(t) dt,(2) where, γ,α, and Msare the gyromagnetic ratio, the Gilbert damping constant, and the saturation magnetization,respectively. When our semiconductor layer is connected to the fer- romagnetic Py ﬁlm [see Fig. 1(b)], the spin polarization propagates into the semiconductor, which gives rise to a purespin current with a spatial direction j salong the yaxis and a spin-polarization direction σ\|\|H. According to [ 1,4,5] [see expressions ( 1) and ( 2)] the precession of the magnetization direction Mis caused by the torque ∝M×Heff, and the direct-current component of a generated spin current at the Py /n-Si:Bi interface ( y=0) is expressed as j0 s=ω 2π/integraldisplay2π/ω 0¯h 4πg↑↓ r1 M2s/bracketleftbigg M(t)×dM(t) dt/bracketrightbigg zdt =g↑↓ rγ2h2¯h[4πMSγ+/radicalbig (4πMS)2γ2+4ω2] 8πα2[(4πMS)2γ2+4ω2],(3) where the real part of the mixing conductance g↑↓ ris given in [5]a s g↑↓ r=2√ 3πMSγdF gμBω/parenleftbig /Delta1HF/N pp−/Delta1HF pp/parenrightbig . (4) Here dFis the thickness of the Py ﬁlm; ωandγare the angular frequency of magnetization precession and the gyromagneticratio, respectively; and /Delta1H F/N pp/Delta1HF ppare the FMR spectral width for the Py/Si:Bi:P layer and the Py ﬁlm, respectively,which are proportional to the Gilbert damping constant α: /Delta1H pp=(2ω/√ 3γ)α. (5) In the Py /n-Si:Bi structure the spin current injected into the Si layer decays along the ydirection (see Fig. 1) due to spin relaxation. According to [ 25] it can be expressed as js(y)=sinh[( dN−y)/λN] sinh( dN/λN)j0 s. (6) FIG. 2. Conﬁgurations of the magnetic ﬁelds related to the plane of the Py /n-Si:Bi layer. The static magnetic ﬁeld Hrotates in the plane of the structure, and the microwave magnetic ﬁeld his orthogonal to this plane. Here dNis the thickness, and λN(dN>λ N)i st h es p i n - diffusion length in the uniformly doped n-Si:Bi:P layer. IV . SPIN CURRENT CONVERSION TO A CHARGE CURRENT BY ISHE IN Py /n-Si:Bi FILM Scattering of polarized spins of the spin current on the SOP of Bi atoms in the n-Si layer causes the inverse spin Hall effect that converts a spin current into a charge current [ 8]: /vectorjC=eθSH ¯h[/vectorJS×/vectorσ]. (7) Here/vectorσdenotes the spin-polarization vector of the spin current /vectorJS(see Fig. 1), and θSHis the spin Hall angle. Using Eqs. ( 6) and ( 7), one can obtain the averaged charge current density: /angbracketleftjc/angbracketright=1 dN/integraldisplaydN 0jc(y)dy, (8) /angbracketleftjc/angbracketright=θSHE/parenleftbigg2e ¯h/parenrightbiggλN dNtanh/parenleftbiggλN 2dN/parenrightbigg j0 S, (9) and for dc ISHE voltage one can write [ 11] VISHE=weθSHλNtanh( dN/2λN) dNσN+dFσF/parenleftbigg2e ¯h/parenrightbigg j0 S. (10) Herewis the width of the Py layer, dNandλNthe thickness and the spin-diffusion length of the nonmagnetic layer, and σF andσNare the electrical conductivity of the Py and n-Si:Bi layers, respectively. In all our experiments, the static magnetic ﬁeld lied in the plane of the structure and the microwave magnetic ﬁeld wasorthogonal to this plane, as shown in Fig. 2. Here the x /primeandz/primeaxes of the Cartesian coordinate system (x/prime,y,z/prime) are directed along the sample length and width, while the yaxis is the normal to the sample surface. The coordinate system ( x,y,z) is chosen so that the zaxis is parallel to the applied magnetic ﬁeld H. Detection of the ISHE was carried out by measuring the dc voltage signal appearing between the Au/Ti contacts (seeFig. 1). For such conﬁgurations of magnetic ﬁelds and mag- netization, one may not take into account the contribution of 195202-3A. A. EZHEVSKII et al. PHYSICAL REVIEW B 101, 195202 (2020) FIG. 3. (a) The Py /n-Si:Bi structure with the structure aligned to the [101] crystal axis. (b) Allowed (Y →X/prime/primevalley) and not allowed (Y→Z/prime/primevalley) transitions for spin scattering at the SHE (see text). the anomalous Hall effect [ 32], but in addition to ISHE it is necessary to consider the transverse part of the anisotropicmagnetoresistance (AMR) in Permalloy, which is known asthe planar Hall effect (PHE) [ 32] for the in-plane conﬁgura- tion of the static magnetic ﬁeld. In this case, the microwavecurrent, which is coupled to the microwave resistance, canproduce a dc electromotive force and voltage: /vectorE PHE=/Delta1ρ M2(/vectorJ·/vectorM)/vectorM, (11) VPHE(θ)=/angbracketleftRe{I(t)}·Re{H(t)}/angbracketright2π/ω =1 2I0h0∇R(H0) cos/Phi1=/Delta1RPHEI0h0sin 2θcos/Phi1. (12) Here/Phi1is a phase shift associated with the losses in the system, and the resistance is expanded in a Taylor seriesaround the dc ﬁeld H 0to ﬁrst order in the microwave ﬁeld h(t),R(H(t))=R(H0)+h(t)∇R(H0)[32]. The angular dependences of AMR are summarized in [ 32] for each measurement conﬁguration. For our case one cansee that V PHE∝sin 2θAs suggested in [ 32], the only way to distinguish between ISHE and PHE signals is to study theangular dependences of the dc voltage signal on the directionof the magnetic ﬁeld. Silicon is a many-valley semiconductor. After scattering, the conduction electron transferred to a valley on a differentcrystal axis in kspace. Since the valleys have a magnetic anisotropy ( g /bardbl/negationslash=g⊥), scattering of spins due to the spin-orbit potential induced by the shallow donor in n-type material may be dependent on the orientation of the valley axes relative tothe direction of the magnetic ﬁeld. If we consider the tran-sitions between different valleys, taking into account the spinconservation ( M S=const) in SHE scattering, then one could see (Fig. 3) that not all transitions are allowed due to the anisotropy of the valleys ( g/bardbl/negationslash=g⊥). This can be understood by considering the spin-orbit inter- action, which can be written as /slurabove HSO=λ(/slurabove LZ/slurabove SZ+/slurabove LX/slurabove SX+/slurabove LY/slurabove SY) =λ/bracketleftbig/slurabove LZ/slurabove SZ+1 2(/slurabove L+/slurabove S−+/slurabove L−/slurabove S+)/bracketrightbig . (13) We now consider the matrix elements of transitions be- tween different band (valley) states, which consists of sixdegenerate basis states ( A1,E, and T2). The only sufﬁcient nonvanishing matrix elements of the orbital momentum be-tween these states are /angbracketleftT Y\|/slurabove LX\|TZ/angbracketright=/angbracketleft TZ\|/slurabove LY\|TX/angbracketright=/angbracketleft Tx\|/slurabove LZ\|TY/angbracketright=− i. (14) Accordingly, for SHE processes the effective spin-orbit inter- action involves only the /angbracketleftTx\|/slurabove LZ\|TY/angbracketrightterm, because for other terms the spin operators/slurabove SX,/slurabove SYare responsible for spin-ﬂip processes and are not allowed for SHE scattering. As shownin Fig. 3, the main axes of the Z /prime/primeand X/prime/primevalleys are directed at an angle of 45° relative to the Z/prime\|\|[10-1] axis in the (010) plane of the layer. In this case, the transitions involved in theSHE occur only when the magnetization is orthogonal to themain axes of the Y and X /prime/primevalleys (H /bardblZ/prime/prime), and the ISHE signal is zero when H/bardblZ/prime. Therefore, the transitions between different valley states give the additional sin 2 θangular dependence to the general dependence of sin θ[11], and so, for the ISHE dc voltage one can write that VISHE∝sinθsin 2θdependence. V . EXPERIMENTAL RESULTS AND DISCUSSION According to [ 24], bismuth in silicon can lead to dra- matic increase of spin-relaxation rate, which may suppressthe ISHE. For that in the spin-pumping experiment we wereforced to use rather a low bismuth concentration (10 16cm−3 or lower). In order to enhance the spin-pumping effect and spin current we implemented the heavy ( >1019cm−3)P doping. Figure 4(a) shows the angular dependence of the dc voltage signal for the rotation of the magnetic ﬁeld in the plane of thelayer. As can be seen, the signal has a maximum when theangle between the magnetic ﬁeld and the Z /prime(see Fig. 3) axis is θ=45°, and no signal was found when the direction of H was parallel to the Z/primeaxis. For observation of ISHE it was unusual, because for Py/Pt and Py/p-type Si layers (see Refs. [ 1,2]) the ISHE dc voltage signal has a maximum at the direction ofthe magnetic ﬁeld H /bardblZ /prime(θ=90◦in Fig. 3) and is described by sin θdependence for the in-plane rotation of the static magnetic ﬁeld [ 6]. On the other hand, this dependence does not completely coincide with the angular dependence for thePHE voltage proportional to sin 2 θ. 195202-4SPIN PUMP INDUCED INVERSE SPIN HALL EFFECT … PHYSICAL REVIEW B 101, 195202 (2020) FIG. 4. The angular dependences of the dc voltage containing PHE and ISHE contributions for the Py /n-Si:Bi structure in which the structure is aligned to the [101] axis. (a) Experimental data (diamonds) and modeling (solid line) which take into account both contributions ( V=VISHE+VPHE), and (b) theoretical approximation of the ISHE (blue line) and PHE (red line) ( VISHE=asinθsin 2θand VPHE=bsin 2θ,a/b=0.4). We suggested that the observed angular dependences of the dc voltage signal contains both PHE and ISHE contribu-tions: V=V ISHE+VPHE=asinθsin 2θ+bsin 2θ. The an- gular dependences of the dc voltage signal, containing PHEand ISHE contributions for the Py /n-Si:Bi structure in which the structure is aligned to the [101] crystal axis are shown inFigs. 4(a) and4(b). To verify our hypothesis about the valley contribution to the SHE in n-type silicon, the structure aligned to the [001] crystal axis was studied (Fig. 5). The angular dependences of FIG. 5. The Py /n-Si:Bi structure aligned to the [001] crystal axis. FIG. 6. The angular dependences of the dc voltage containing PHE and ISHE contributions for the Py /n-Si:Bi structure in which the structure is aligned to the [001] crystal axis. (a) Experimental data (diamonds) and modeling (solid line), which takes into account both contributions ( V=VISHE+VPHE), and (b) theoretical approximation of the ISHE (blue line) and PHE (red line) ( VISHE=a/primesinθcos 2θ andVPHE=b/primesin 2θ,a/prime/b/prime=1). the dc voltage containing the PHE and ISHE contributions for such Py /n-Si:Bi structures are shown in Fig. 6. In this case the ISHE dc voltage signal is proportional to sin θcos 2θ, which results in the fact that the ISHE signal could be observed atθ=90 and 270 ◦, in addition to the PHE dc voltage at the angles corresponding to the sin 2 θdependence. This followed the valley involvement to the SHE scattering when magneti-zation is orthogonal to the main axes of the Y and X /primevalleys (Fig. 5). In this case the additional cos 2 θmultiplier is added to the sin θdependence. Using the experimental parameters and the obtained re- sults, we evaluated the angle θSHE of the spin Hall effect. Taking σf=1.49×106(/Omega1m)−1,σn=5.0×104(/Omega1m)−1, andVISHE=200 nV [this value corresponded to the max- imum in the dependence for ISHE in Fig. 5(b)] for 200- mW microwave power, 4 πMs=0.997 08 T, WPy/Si:Bi-WPy= 0.6m T , α=0.008 93, g↑↓=4.450×1017m−2, and js= 6.2×10−11J/m2, we have determined this angle to be θSHE≈ 0.0001. This value coincides with the calculations made in [26] and is of the same order that was found in p-type silicon [11] and in gallium arsenide [ 26]. Since the spin-diffusion length for n-type Si is much longer than for p-type silicon, it is still of great interest to obtain the Si:Bi:P layers with higher thickness to obtain a stronger ISHEdc voltage signal. Nevertheless, in the process of Bi dopingof silicon, we faced the problem of obtaining the uniformlydoped layers with a thickness higher than half a micron, since 195202-5A. A. EZHEVSKII et al. PHYSICAL REVIEW B 101, 195202 (2020) FIG. 7. The angular dependences of the dc voltage, contain- ing PHE contributions for the Py /n-Si structure doped only with phosphorus ( NP>1019cm−3): experimental data (diamonds) and modeling (solid line) taking into account the PHE contribution (VPHE∝sin 2θ). The structure is aligned to the [001] crystal axis. we used the method of ion implantation for Bi and P doping, which limited the thickness of the doped layer. We estimated the value of the PHE contribution to the dc voltage, taking into account the known AMR ratio of/Delta1R/R(0)∼0.4% for the Permalloy layer [ 32]. Using the parameters of our structure and microwave cavity, we couldestimate that the dc voltage on the edges of the Py layermay be of the order of 20 nV and a slightly smaller valuebetween the Au/Ti contacts (here we also not included somepower loss in the microwave cavity). This value is almost oneorder of magnitude smaller than we have measured in ourexperiment. The difference can be understood on the basis ofthe spin-orbit interaction contribution to the value of /Delta1R[33]. In the case of dynamic magnetization in the bilayer systemssuch as Py /n-Si:Bi, both layers should be taken into account. In order to make measurements of the PHE without the ISHE involvement to the dc voltage, the structure of Py /n-Si was made without bismuth doping. Phosphorus doping of thesilicon layer was carried out by ion implantation and anneal-ing at the same conditions as for structures with bismuth. The angular dependences of the dc voltage, containing PHE contributions for the Py /n-Si structure, are shown in Fig. 7. As can be seen, only the PHE dc voltage signal is observed, with angular dependence proportional to sin 2 θ. Again, the magnitude of the PHE is much higher than wasestimated from parameters of Py ﬁlm but smaller than thatobserved in the Bi-doped structures. For Si doped with phos-phorus, the spin scattering due to the spin-orbit potentialinduced by phosphorus is much smaller than for silicon dopedwith bismuth. However, owing to the three to four orders ofmagnitude higher concentration of phosphorus as comparedto bismuth, it can lead to a high spin-relaxation rate, whichmay contribute to the damping coefﬁcients in the Eq. ( 2)f o r FMR. In this case, one may suggest that the contribution ofspin-orbit interaction may come from the n-Si layer and the interface layer between the Py and n-Si layers. It is consistent with FMR linewidth behavior for different structures such asPy/n-Si:Bi, Py /n-Si:P, and Py on high-resistivity Si (Fig. 8). It is seen that FMR linewidths for Py /n-Si:Bi and Py /n- Si:P are almost the same, and they are remarkably higher thanthat for Py deposited on high-resistivity Si. Such behavior ofFMR linewidths and PHE for Py /n-Si bilayer systems was not FIG. 8. The FMR spectra of Py for different structures: (1) Py /n- Si:Bi, (2) Py /n-Si:P, and (3) Py on Si of high resistivity (5000 /Omega1cm). considered in literature earlier and requires some additional experimental studies and theoretical analysis. VI. SUMMARY In summary, we have studied the inverse spin Hall effect that was induced in n-type silicon due to scattering of the spin-polarized conduction electrons on the spin-orbit potentialof bismuth using the spin-pumping effect. The spin current inthe silicon layer was generated by excitation of the magneti-zation precession at ferromagnetic resonance in the thin Pylayer deposited on the n-Si layer doped by bismuth. From the angular dependences of the dc voltage for Py /n-Si:Bi structures with different orientation relative to crystal axes,the planar Hall effect (PHE) and ISHE contributions wereevaluated. It was obtained that the ISHE dc voltage signalwas proportional to sin θsin 2θfor the structure which was aligned to the [011] crystal axis and to sin θcos 2θfor the structure aligned toward the [100] direction. In addition, thesin 2θdependence of the PHE dc voltage was observed. This leads to the contribution of the intervalley scattering to theSHE, and this gives an additional sin 2 θor cos 2 θfactor to the usual sin θdependence. This means that for silicon as a many-valley semiconductor, the scattering of spins due tothe spin-orbit potential induced by a shallow donor in n-type material is dependent on the orientation of the valley axesrelative to the direction of the magnetic ﬁeld. Using the obtained results, the angle of the spin Hall effect was estimated from the magnitude of the ISHE voltage in thePy/n-Si:Bi structures to be θ SHE∼0.0001. The value of the PHE contribution to the dc voltage is almost one order ofmagnitude smaller than we measured in our experiment. Thedifference can be understood on the basis of the spin-orbitinteraction contribution to the value of /Delta1R, which in the case of dynamic magnetization in bilayer systems such asPy/n-Si:Bi can be determined by both layers. Our results can be helpful for understanding some aspects of the spin-orbitinteraction in semiconductors and for the engineering of Si-based spintronic devices. ACKNOWLEDGMENTS We are grateful to Professor A. A. Fraerman and Dr. E. A. Karashtin (Institute for Physics of Microstructures of theRussian Academy of Sciences) for interest in this work and 195202-6SPIN PUMP INDUCED INVERSE SPIN HALL EFFECT … PHYSICAL REVIEW B 101, 195202 (2020) stimulating discussions. Sample fabrication was supported by the Russian Science Foundation (RSF) (RU) 16-12-10340.Ferromagnetic resonance and spin-pumping experiments were supported by RFBR (RU), 18-03-00235-a. [1] M. I. Dyakonov and V . I. Perel, Phys. Lett. A 35,459(1971 ). [2] J. E. Hirsch, Phys. Rev. Lett. 83,1834 (1999 ). 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PhysRevB.90.174415.pdf	PHYSICAL REVIEW B 90, 174415 (2014) Atomistic modeling of magnetization reversal modes in L10FePt nanodots with magnetically soft edges Jung-Wei Liao,1,2Unai Atxitia,2,3,Richard F. L. Evans,2,†Roy W. Chantrell,2and Chih-Huang Lai1,‡ 1Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan 2Department of Physics, The University of York, York YO10 5DD, United Kingdom 3Departamento de Fisica de Materiales, Universidad del Pais Vasco, UPV/EHU, ES-20018 San Sebastian, Spain (Received 30 June 2014; revised manuscript received 31 October 2014; published 13 November 2014) Nanopatterned FePt nanodots often exhibit low coercivity and a broad switching ﬁeld distribution, which could arise due to edge damage during the patterning process causing a reduction in the L10ordering required for a high magnetocrystalline anisotropy. Using an atomistic spin model, we study the magnetization reversal behavior ofL1 0FePt nanodots with soft magnetic edges. We show that reversal is initiated by nucleation for the whole range of edge widths studied. For narrow soft edges the individual nucleation events dominate reversal; for wider edges,multiple nucleation at the edge creates a circular domain wall at the interface which precedes complete reversal.Our simulations compare well with available analytical theories. The increased edge width further reduces andsaturates the required nucleation ﬁeld. The nucleation ﬁeld and the activation volume manipulate the thermallyinduced switching ﬁeld distribution. By control of the properties of dot edges using proper patterning methods,it should be possible to realize exchange spring bit-patterned media without additional soft layers. DOI: 10.1103/PhysRevB.90.174415 PACS number(s): 75 .60.Jk,75.75.Jn,75.10.−b,75.50.Ss I. INTRODUCTION Continuing requirements for greater digital data storage capacity has led to continued growth in data storage density inmagnetic recording media. Future improvements are limitedby the magnetic recording trilemma, caused by competingrequirements of reduced signal to noise ratio, thermal stabilityof written information, and writability [ 1]. Two solutions for the magnetic recording trilemma have been proposed: ﬁrstheat assisted magnetic recording (HAMR) [ 2,3], where laser heating during recording is used to lower the anisotropysufﬁciently to achieve writing; the second solution is bit-patterned media (BPM), where each bit is deﬁned by a singledot in a lithographically deﬁned array [ 4] and the larger magnetic volume reduces the requirement for high anisotropyrequired for long-term thermal stability. Bit-patterned media can be made using a variety of methods including patterning [ 5–11] and self-assembly of magnetic nanoparticles [ 12]. Controlling the microstructural properties of magnetic nanoparticles is quite challenging, however,lithographic patterning techniques allow a continuous L1 0 FePt ﬁlm to be patterned into an array of isolated magnetic islands or “dots” [ 5–11]. However, during lithography ions near the dot edge can reduce the L10ordering, resulting in magnetically soft edges [ 6,7,9–11]. The presence of damaged edges in the dots could reduce both the coercivity [ 6,10,11] and the thermal stability [ 9]. In addition to decreasing the coercivity, a broad switching ﬁeld distribution (SFD) can also lead to write errors inneighboring bits during the writing process. The SFD (thevariation of switching ﬁelds between dots) includes bothextrinsic and intrinsic components [ 13–17]. The extrinsic SFD Present address: Department of Physics, University of Konstanz, Konstanz D-78457, Germany. †richard.evans@york.ac.uk ‡chlai@mx.nthu.edu.twmay be caused by dipolar interaction between dots, and the intrinsic SFD arises from variations of intrinsic magneticproperties of dots, including anisotropy K, volume V, and the easy axis alignment [ 14]. Furthermore, thermal ﬂuctuations also broaden the intrinsic SFD, known as the thermal SFD[15–17]. Within the simple Stoner-Wohlfarth approximation (monodomain), the thermal SFD is mainly related to theanisotropy energy barrier KV, and the measurement time scale. This makes the thermal SFD pronounced at high ﬁeldsweep rates associated with the recording process [ 16,17]. Reversal behavior in relatively large dots with magnetically soft edges of ﬁxed width, associated with ion damage fromthe etching process, has been studied previously [ 18], where the presence of soft edges was shown to change the reversalmode. In addition, small-sized dots with ring-shaped soft edgesof varied width have been investigated by macrospin analyticmodels without including the thermal ﬂuctuations [ 19]. The increased width of edge is found to reduce the coercivityof dots [ 19], suggesting a strong relationship between the edge width and the reversal mode. Further control of themagnetic properties of edges with ﬁxed width in large sizeddots can also be done via soft He +irradiation [ 20]. The experimental observations can only be explained by the modelincluding the thermal ﬂuctuations [ 20]. All the reported works indicate that either the edge width [ 19] or the thermal ﬂuctuations [ 20] affect the reversal mode and could result in different switching ﬁeld distributions of patterned dots.However, the edge-width dependence of the reversal modeincluding thermal ﬂuctuations is still not understood. Here we develop a computational model to study magneti- zation reversal modes in L1 0FePt dots with magnetically soft edges. We employ an atomistic spin model formalism, whichprovides detailed information on reversal modes unreachableby standard micromagnetic simulations [ 21,22]. In particular, soft edges of only a few nanometers are tractable, and we canfurther study the effect of the reduced exchange coupling atthe interface, possibly resulting from the core/edge interfaceroughness. Moreover, thermal effects are consistently taken 1098-0121/2014/90(17)/174415(8) 174415-1 ©2014 American Physical SocietyLIAO, ATXITIA, EV ANS, CHANTRELL, AND LAI PHYSICAL REVIEW B 90, 174415 (2014) into account within our model using the Langevin dynamics formalism that allows us to study the relationship betweenthe coercivity, the thermal SFD, and reversal modes. Wefurther compare the atomistic spin modeled results withavailable analytic approaches, which were originally presentedfor hard/soft nanowires [ 23–28], to examine the validity of these approaches for the core/shell nanostructure. Thesesimpler approaches are capable of highlighting the key physics.Additionally, since the atomistic resolution in the simulationmakes this method computationally intensive restricting thesize of the calculated system to nanometer length scales, thesevalidated analytic approaches could be potentially utilized toinvestigate properties of large sized systems, e.g., a dot array. II. ATOMISTIC SPIN MODEL The studied nanodots are composed of a magnetically hard core and a magnetically soft edge, as illustrated inFig. 1. In the case of patterned dots, we hypothesize that the edge region loses its L1 0atomic order due to the patterning process, making it magnetically soft. We focus exclusively onthe width of the edge W edge, with the ﬁxed core size, rcore, on the magnetization reversal. We therefore ﬁx the diameter ofthe core 2 r core=25 nm and the dot thickness, td=4 nm, while the edge width is varied systematically from Wedge=0–18 nm. We note that in this approach the different Wedgevaries the total volume of dot and therefore affects the corresponding thermalstability, which is beyond the scope of the present work. Thesystem is constructed from a single face-centered cubic crystaland cut into the shape of a nanodot with the desired geometry. The nanodots are modeled using an atomistic spin model approach [ 29] with the VA M P I R E software package [ 30]. The energetics of the system are described by the spin Hamiltonianwith the Heisenberg exchange, given by H=H core+Hedge, (1) Hcore=−/summationdisplay i,jJcoreSi·Sj−/summationdisplay i,νJceSi·Sν −kcore/summationdisplay i/parenleftbig Sz i/parenrightbig2−μcore/summationdisplay iHapp·Si, (2) Hedge=−/summationdisplay ν,δJedgeSν·Sδ−/summationdisplay ν,jJceSν·Sj −kedge/summationdisplay ν/parenleftbig Sz ν/parenrightbig2−μedge/summationdisplay νHapp·Sν, (3) where S=μ/μare spin unit vectors, i,jlabel core sites with moment μcore, and ν,δlabel edge sites with moment μedge. Here we assume the same moment for both core and edge such that μcore=μedge=1.5μB, which compares well to the saturation magnetization of L10FePt as obtained in experiment [ 31].JcoreandJedgeare the exchange interactions between moments of the same type in the core and the edge,respectively. We consider only nearest neighbor interactionsbetween the moments. We select values of the exchangeenergy to give a Curie temperature around 700 K comparablewith experiment, namely J core=Jedge=3×10−21J/link.Jce represents the interfacial exchange interaction between the core and the edge and is varied as a parameter between 0 Wedge2rcore 0 - 18 nmWedge 0 - 18 nm td 4 nm25 nm FIG. 1. Schematic diagram of the atomistic modeled dot. Dark and white gray regions represent the core and the edge atoms, respectively. andJcore.kcore=4.9×10−23J/atom is the uniaxial anisotropy constant of the core spins (with easy axis perpendicular tothe ﬁlm plane) and k edge=1×10−24J/atom is the uniaxial anisotropy of the edge spins. Happis the external applied ﬁeld. The hysteresis loops are calculated dynamically using the stochastic Landau-Lifshitz-Gilbert (LLG) equation at theatomic level, given by ∂S i ∂t=−γ (1+λ2)Si×[Hi,eff+λ(Si×Hi,eff)], (4) where λis the intrinsic damping parameter, γ=1.76×1011 T−1s−1is the absolute value of the gyromagnetic ratio, and Hi,effis the effective magnetic ﬁeld in each spin. The ﬁeld is derived from the spin Hamiltonian and is given by Hi,eff=−1 μi∂H ∂Si+Hdemag ,i+Hi,th, (5) where Hdemag ,iand Hi,thare the demagnetization and the thermal ﬁelds, respectively. Since the calculation of thedemagnetization ﬁeld at the atomic level is computationally ex-pensive, we have instead calculated the demagnetization ﬁeldby applying the approach developed by Boerner et al. [32]. Within this approach, the dot is divided into regular macrocellswith the volume V k=(1.77)3nm3which contains 250 atomic spins. The value of spin’s moments within each macrocellare then summed to obtain the macrocell magnetic moment,μ k=/summationtext δ∈/trianglekμδSδ, where klabels macrocell sites, and δ labels spin sites in each macrocell, /trianglek. We then calculate the demagnetization ﬁeld of each macrocell, Hdemag ,k,b y using the corresponding magnetic moment and treat it as thedemagnetization ﬁeld of each spin in the macrocell, H demag ,i. Hdemag ,kis calculated by direct pairwise summation including the macrocell self-demagnetization [ 29], Hdemag ,k=μ0 4π/summationdisplay k/negationslash=l3(μl·ˆrkl)ˆrkl−μl \|rkl\|3−μ0 3μk Vk,(6) where μ0=4π×10−7T2J−1m3is the vacuum permeability, rklis the vector between kandlmacrocell sites, and ˆrkl= rkl/\|rkl\|is the corresponding unit vector. This is a computa- tionally efﬁcient approach since the number of macrocells isrelatively small and moreover, since the magnetostatic ﬁeld 174415-2ATOMISTIC MODELING OF MAGNETIZATION REVERSAL . . . PHYSICAL REVIEW B 90, 174415 (2014) varies rather slowly with time it needs updating only on a time scale of around 1000 time steps [ 29]. The thermal ﬂuctuations are represented using Langevin dynamics [ 33,34], where the thermal ﬁeld Hi,this given by Hi,th=/Gamma1(t)/radicalBigg 2λkBT γμi/Delta1t, (7) where kBis the Boltzmann constant, Tis the heat bath tempera- ture,λis the Gilbert damping parameter, γis the absolute value of the gyromagnetic ratio, and /Delta1tis the integration time step. The thermal ﬂuctuations are represented by a vector Gaussiandistribution in space /Gamma1(t) with a mean of zero and generated from a pseudorandom number generator. The simulations inthis work are carried out at a heat-bath temperature of T= 300 K. We set the damping parameter λ=1.0 to reduce the computational time required for reaching an equilibrium state. The LLG equation is integrated using the Heun integrationscheme [ 34] with an integration time step /Delta1t=1f s . III. RESULTS In order to study reversal modes we simulate hysteresis loops as a function of the width of edges, Wedge. To calculate the hysteresis loops we apply an external ﬁeld in a range from−5t o+5 T, which lies above the anisotropy ﬁeld in the core, at intervals of 5 mT. The ﬁeld sweep rate is 5 T/ns. Initially weconsider that the interfacial core and edge spins are stronglycoupled by setting J ce=Jcore=Jedge=3.0×10−21J/link. Figure 2(a) shows representative out-of-plane hysteresis loops for a range of edge widths. One can observe thatby increasing edge width, both the nucleation and coercive -101 -4 -3 -2 -1 0M/M S Happl (T)0n m 1n m 2n m 3n m 7n m(a) (b)0n m 1n m 2n m 3n m 7n m FIG. 2. (Color online) (a) Simulated out-of-plane hysteresis loops for dots with different edge widths. Magnetization is normalizedto the saturation magnetization at 0 K. Snapshots of domain conﬁgurations during reversal, observed along the dot plane normal direction, are shown in (b). Symbols in (a) and on the left in (b)indicate the position of snapshots during the reversal process. The color scale (blue to red) represents the magnetization component along the easy axis direction. Black dotted circles denote the positionof the core/edge interface.-1-0.500.51 -1 -0.5 0 0.5 1M/M S Happl (T)Core EdgeCore+Edge FIG. 3. (Color online) Whole hysteresis loop of the dot with Wedge=12 nm and two individual loops of its core and its edge, decomposing the whole loop. ﬁelds decrease. Furthermore, the squarelike hysteresis loop for narrow edges turns into a two-step reversal as the edgewidth increases, indicating a change in the reversal process.Figure 2(b) illustrates the corresponding snapshots of spin conﬁgurations during reversal for various W edge. The reversal mode strongly depends on the edge width, which will bediscussed in more detail in the following sections. To obtaindetailed information on the observed reversal behavior, we alsocalculate hysteresis loops for each edge width for 30 differentrealizations of the random number generator. Therefore, weaverage over 60 statistically independent values to obtainthe mean coercivity and standard deviation. Since we areconsidering dots with the same magnetic properties in oursimulations, the deviation from the mean arises completelyfrom the thermal ﬂuctuations. Thus the standard deviation isa manifestation of the intrinsic SFD resulting from thermalﬂuctuations [ 17]. This is an important parameter since it increases with increasing ﬁeld sweep rate and is signiﬁcantat time scales associated with data transfer in informationstorage. Additionally, we separately calculate the coercivity ﬁelds of both the core, H core c, and the edge, Hedge c,s h o w n in Fig. 3. To do so, we calculate the individual reduced magnetization of the core and edge as follows, μcore=\|μcore\| Ncore/summationdisplay i∈coreSi,μedge=\|μedge\| Nedge/summationdisplay i∈edgeSi, (8) where Ncore(edge) denotes the number of atoms in the core (edge). Figure 4(a) shows the variation of the mean coercivity Hcore(edge) c as a function of Wedge, and the corresponding standard deviation, σcore(edge) , is shown in Fig. 4(b).T h e coercivities and the standard deviation are strongly dependenton the edge width, which will be discussed in the followingsections. Furthermore, to understand the reversal mode, wewill compare coercive ﬁelds obtained from atomistic spinmodel simulations with those obtained from different theo-retical approaches [ 23–28], given by the lines in Fig. 4.T h e additional models, to be discussed later, are all based on aconventional micromagnetic approach. In such models the 174415-3LIAO, ATXITIA, EV ANS, CHANTRELL, AND LAI PHYSICAL REVIEW B 90, 174415 (2014) 1234Hc(T)(a) Heﬀ KHn Hp 0.020.040.060.080.10.120.14 0 2 4 6 8 1 01 21 41 61 8σ(T) Wedge (nm)(b) σK σn σpledge DWCore Edge FIG. 4. (Color online) (a) Mean coercivity of both the core and the edge as a function of the width of edges. The gray dashed linerepresents the effective anisotropy ﬁeld calculated by the linear chain model. Purple solid and black dashed lines indicate the nucleation and pinning ﬁelds, respectively. (b) Standard deviation of coercivity ofboth the core and the edge as a function of the width of edges giving an estimate of the thermal switching ﬁeld distribution. The gray dashed line represents the deviation approached by the effective anisotropyﬁeld. Purple solid and black dashed lines are the deviation calculated by the nucleation and the pinning ﬁelds. The vertical dashed line denotes the domain-wall length in the edge. temperature dependence of magnetic properties is not intrinsic to the formalism, as it is in the atomistic approach, and mustbe introduced explicitly. Consequently, in the micromagnetic-based models we will introduce the effect of temperature(T=300 K) by normalizing the micromagnetic parameters in the theoretical calculations. For the anisotropy constants,K core(edge) , we use the Callen-Callen law [ 35], Kcore(edge) (T=300 K) ≈Kcore(edge) (T=0K )m3 e, (9) where me=Ms(T=300 K) /M s(T=0K )=0.82 is ob- tained directly from our computational atomistic spin cal-culations. M sis the saturation magnetization of the core (edge). We note that experimentally the exponent value for thedecrease in anisotropy constant of L1 0FePt is close to 2.1 [ 31], which can be reproduced using the multiscale atomistic spinmodel simulations [ 36]. However, this multiscale simulation is computationally expensive for the calculation of dots withthe diameter of 25 nm (Fig. 1). Therefore, we have used a simpliﬁed atomistic spin model, where K core(edge) (T) follows the Callen-Callen law. In fact at room temperature the 2.1scaling law and the Callen-Callen law give similar lengthscales, e.g., the exchange length or the domain-wall length, which can be estimated by the ratio m 2.1/2 e/m3/2 e∼0.93. We also note that both surface and interface effects can slightlyvary the Callen-Callen law for K edge(T)[37]. The exchange stiffness of the core (edge), Acore(edge) (T), has been shown to scale with meas [38,39] Acore(edge) (T=300 K) ≈Acore(edge) (T=0K )m1.745 e.(10)A. Narrow soft edge: individual nucleation The magnetization reversal in the absence of soft edges, as shown in the spin conﬁguration snapshot for Wedge=0n m in Fig. 2(b), starts by the nucleation of a small region (red area in the snapshot) in the boundary and proceeds with thesubsequent expansion to the entire dot. At this point it is worth-while considering the physical origin of the nucleated reversal.The origin of the incoherent nucleated reversal process lies inthe combination of high magnetocrystalline anisotropy andthermal ﬂuctuations. At applied ﬁelds in the vicinity of thecoercivity thermal ﬂuctuations break the symmetry of the dotand cause a nucleation event. The narrow domain wall width,arising from the high magnetocrystalline anisotropy in thecore, stabilizes the nucleated domain. Following the nucleationthe lowest energy barrier for switching is then propagation ofthe domain wall, leading to an incoherent reversal mechanism.The combination of short time scales, high anisotropy, andsystem size greater than the domain wall width gives thefundamental physical origin of the thermal switching ﬁelddistribution. For longer time scales more nucleation attemptsare made reducing the effective thermal SFD since the materialswitches at the same ﬁeld, while for lower anisotropy materialsthe nucleated domain is unstable and so the effect of thermalﬂuctuations is also lower. For dots with a narrow soft edge, W edge=1 or 2 nm, the reversal mechanism is the same as for dots with no soft edges,although due to the low coercivity of the edge the nucleationﬁeld is reduced signiﬁcantly. The thermal SFD also reducesrapidly with narrow soft edges due to the reduced stability ofthe nucleated domain owing to the lower effective anisotropy.In an attempt to quantify the reduction in the coercivity as afunction of the edge width we have developed an atomisticone-dimensional (1D) linear chain model, details of whichare given in Appendix. By estimating the coercive ﬁeld as aneffective anisotropy ﬁeld of the nucleated area, H eff K, the linear chain model predicts a linear decrease in the coercivity givenby H core(edge) c =Heff K=Hcore K(1−bW edge). (11) In Fig. 4(a) we can see that for Wedge/lessorequalslant2 nm, both the coercivity of the core and the edge are equal and linearlydecrease as a function of the edge width. It can be seen that Eq. ( 11) gives reasonable agreement with the numerical results. B. Wide soft edge: an incomplete to a complete circular domain wall With a further increase in the edge width we observe the re- versed region with a negative curvature, shown by Fig. 2(b) for Wedge=3 nm with nucleated areas denoted by red regions. The negative curvature could suggest that more than one reversedregion nucleates during the reversal. The deviation betweenH core(edge) andHeff K[Eq. ( 11)] reﬂects the reversal dominated by multireversed regions. These multiple nucleation events also mark an increasing difference between Hedge c andHcore c values with further increases in the edge width [Fig. 4(a)]. From the spin conﬁguration snapshots in Fig. 2(b) we observe this behavior corresponds to an incomplete circular domainwall formed at the core/edge interface. In this region we cannotapproach H core(edge) using Eq. ( 11) because this is only valid 174415-4ATOMISTIC MODELING OF MAGNETIZATION REVERSAL . . . PHYSICAL REVIEW B 90, 174415 (2014) for the reversal dominated by a single reversed region. Instead, we ﬁnd that Hedge capproaches the domain wall nucleation ﬁeld Hn, obtained from the analytical expression derived for the limit of strong hard/soft coupling with a soft layer thickerthan the domain-wall width ( ≈5 nm in this study) in the hard layer [ 23–25] given by H n=Hedge K+/parenleftbiggπ 2/parenrightbigg2/parenleftbiggledge EX Wedge/parenrightbigg2 Medge, (12) where Hedge K=2Kedge/M edge is the anisotropy ﬁeld of the edge, and Medgeis the saturation magnetization of the edge andledge EX=/radicalbigAedge/K edgeis the exchange length in the edge. For applied ﬁelds larger than Hnbut less than the domain- wall pinning ﬁeld at the edge/core interface Hp, the increased ﬁeld compresses the domain wall in the edge and therefore reduces the corresponding domain-wall width ledge DW.A te v e n wider edge widths [for example, see Fig. 2(b) atWedge= 7 nm], the nucleation occurs in the entire edge, but the domainwall is then pinned at the core/edge interface, showing acircular domain wall. As the reversal continues, the domainwall propagates inwards until collapse and full magnetizationreversal. In addition, the propagated domain wall shows anoncircular symmetry [Fig. 2(b) atW edge=7 nm], in contrast to the circular symmetry of the domain wall pinned at thecore/edge interface. The suggests the depinning of part of the circular domain wall during the reversal of spins in the core. However, in contrast to the analytical model of thesingle nucleation region proposed in Ref. [ 20], the reversed region in the core shows a negative curvature [Fig. 2(b) at W edge=7 nm], indicating that the reversal could be dominated by the multinucleation events. On the other hand, Hcore c saturates at HpwhenWedge/greaterorequalslantledge DW, which reads [ 26–28] ledge DW=π/radicalBigg 2Aedge Kcore+Kedge. (13) Hpis given by [ 26–28] Hp=1 42[Kcore−Kedge] Medge. (14) Figure 4(a) shows that our simulation results ﬁt perfectly to theHp(black dashed line). Thus it conﬁrms that for soft edges wider than ledge DW atHp, the reversal mechanism is through depinning of part of a circular domain wall at the edge/coreinterface driven by the multiple nucleation events in the corewithH core c=Hp. C. Thermally induced switching ﬁeld distribution The calculated thermal switching ﬁeld distribution σcore(edge) from the simulations for different edge widths is shown in Fig. 4(b). Similarly to the coercive ﬁelds, our simulations show that for Wedge/lessorequalslant2n m , σcore/similarequalσedge, the thermal SFD displays a linear decrease with the increasing Wedge.I n order to quantify the thermal ﬂuctuations in the coerciveﬁeld within a micromagnetic framework it is necessary toassociate the magnetic moment μin Eq. ( 7) with a volume characteristic of magnetization reversal. For this we use theactivation volume V act, which is an equilibrium quantity anddeﬁned as the volume associated with the magnetization change between positions of minimum and maximum staticenergy [ 40]. Furthermore, we average the thermal ﬂuctuation ﬁeld over a speciﬁc time equal to the inverse of an “attemptfrequency” used in phenomenological models of thermalactivation processes. The attempt frequency is generally takenas the natural frequency of oscillation in the local minimum,i.e.,f 0=γH KwithHKthe anisotropy ﬁeld. This leads to a variance in the ﬁeld components, which we take as the standarddeviation of coercivity, σ core(edge) , given by σcore(edge) =/radicalBigg 2λkBTH K MsVact. (15) ForWedge/lessorequalslant2 nm, the single nucleated region dominates the reversal. However, the observed nucleation is a nonequilibrimquantity [ 41]. For V actone should estimate the volume of the equilibrium domain change during reversal. Considering thedot size is smaller than the domain size ( ∼26 nm in this study), we can treat the dot as a single domain particle and thereforeapproach V actto the total volume of the dot, Vact∼π(rcore+ Wedge)2td. Taking HK=Heff K[Eq. ( 11)], we arrive at σcore(edge) =σK=/radicalBigg 2λkBTHeff K Ms[π(rcore+Wedge)2td], (16) where σKisσcore(edge) in this region. It can be seen that Eq. ( 16) [indicated by the gray dashed line in Fig. 4(b)] gives results reasonably close to the numerical results. ForWedge/greaterorequalslant3 nm, the common behavior of spins in the core starts to deviate from that in the edge, as we show in Fig. 4(a). Similarly we ﬁnd that σcoredeviates from σedge [Fig. 4(b)]. In this region, the different reversal mode of core spins withthat of edge spins suggests that V actin the edge approaches to the edge volume, Vact∼π[(rcore+Wedge)2−(rcore)2]td.U s i n g Eq. ( 15) withHK∼Hedge c=Hn[Eq. ( 12)]σedgein this region, σn, is [purple solid line in Fig. 4(b)] σn=/radicalBigg 2λkBTH n Msπ[(rcore+Wedge)2−(rcore)2]td. (17) ForWedge/greaterorequalslantledge DW,Hcore c saturates at Hp[Eq. ( 14)]. The different reversal behavior of core spins to that of edge spinsbrings us to the estimation of the activation volume in the coreas the core volume, V act∼π[((rcore)2]td.U s i n gE q .( 15) with HK∼Hcore c=Hp[Eq. ( 14)] we arrive at [black dashed line in Fig. 4(b)] σp=/radicalBigg 2λkBTH p Msπr2coretd, (18) where σpisσcorein this region. Equations ( 17) and ( 18)g i v e values of σcore(edge) roughly a factor of 2 different from the numerical results [Fig. 4(b)]. Given the assumptions involved the difference is reasonable agreement. D. Effect of interfacial exchange coupling on the reversal modes Finally we investigate the effect of core/edge exchange coupling strength Jceon the reversal modes in the nanodot. To do so we vary the normalized interfacial exchange coupling 174415-5LIAO, ATXITIA, EV ANS, CHANTRELL, AND LAI PHYSICAL REVIEW B 90, 174415 (2014) 01234Hcore c (T)(a) 01234 0 0.2 0.4 0.6 0.8 1Hedge c (T) Jce/Jcore(b)1n m 3n m 7n m FIG. 5. (Color online) (a) Coercivity of the core as a function of the normalized core/edge exchange coupling strength at varied widthof edges. The core/edge exchange coupling strength is normalized to the exchange interaction between spins in the core (edge). Dashed lines are guided by the eye. (b) Coercivity of the edge as a function ofthe normalized core/edge exchange coupling strength. Dashed curves represent a ﬁtting to the Langevin function [Eq. ( 19)]. strength ˜Jce=Jce/Jcore(edge) from 0 (no coupling) to 1 which corresponds to the strong coupling studied in detail inthe previous section. Here, we also perform hysteresis-loopcalculations for varied edge width W edge. Figure 5(a) shows the coercivity of the core Hcore c as a function of ˜JceforWedge=1,3, and 7 nm to cover the three well-separated regimes of reversal modes observed in oursystem. We observe that for W edge=1 nm the core coercive ﬁeldHcore cpresents a minimum at a relatively weak coupling strength, similar to results observed in hard/soft structures,where the observed minimum is related to the two-spinbehavior [ 42]. Considering that the local minimum of H core c happens when Jedge/greatermuchJce(˜Jce/lessorequalslant1/20) in narrowed edges (Wedge/lessorequalslant2 nm), all spins in the edge might behave as a single macrospin. During the reversal, the single-spin behavior in theedge could provide a torque to spins in the core and yield thelocal minimum of H core c, which has been previously observed in the two-spin model [ 42] as well as in the experiment [ 43]. As the coupling increases, the coercive ﬁeld saturates to some value which has been already discussed in previoussections of the present work. For edge widths larger thanor equal to 3 nm, the minimum of the core coercive ﬁelddisappears and a monotonous decrease in H core cto a saturation value is observed. For Wedge/greaterorequalslant7 nm [equal to ledge DWgiven by Eq. ( 13)], this saturation value corresponds to the domain-wall depinning ﬁeld. Therefore, the interface coupling dependenceof the core coercive ﬁeld for W edge/greaterorequalslant7 nm is similar. On the other hand, the edge coercive ﬁeld Hedge cconsistently increases with increasing interfacial exchange coupling to asaturation value [see Fig. 5(b)] following a Langevin law representing the effective bias ﬁeld created in the edge bythe coupling to the core, in direct analogy to a paramagnetmagnetization in the presence of an external ﬁeld and thermalﬂuctuations [ 44]. This effective bias ﬁeld is comparable to theexternal ﬁeld applied in the calculation of hysteresis loops and can be estimated by H edge c(˜Jce)=Hedge c,1L(βμ edgeHex), (19) where Hexestimates the average effective bias ﬁeld in the edge induced by the interfacial coupling, and μedge=MedgeVedge is the saturation magnetization of the edge. The Langevin function is L(x)=coth(x)−1/x.Hedge c,1is a ﬁtting constant and coincides with Hedge c at˜Jce=1.0 (calculated in the previous sections). We can assume that μedgeHex=Vedged˜Jce where dis a parameter that measures the energy transferred from the core to the edge via the interfacial coupling. Thisparameter is expected to depend on the volume of the edge,V edge∼W2 edgetd, so that as the thickness is ﬁxed for all Wedge, we expect that d∼1/W2 edgesimilar to that in a soft/hard bilayer structure, Hex∝1/t2 soft[23]. In Fig. 5(b) we show that indeed this relation ﬁts very well to simulations. IV . DISCUSSION AND CONCLUSIONS To summarize, using atomistic spin model simulations, we have investigated reversal modes in patterned L10FePt dots with damaged edges in the presence of thermal ﬂuctuations.Speciﬁcally, the calculated dot is composed of a hard magnetic core, which represents the undamaged part of the dot, and the damaged edge with soft magnetic properties. We haveinvestigated the effects of the extent of damage on the edgeby varying its width. We observe that the nucleation initiatesreversal for all width of edges. The increased edge widthlinearly decreases and then saturates the required ﬁeld fornucleation, with the curvature of the initially nucleated regionreducing from positive to negative. Furthermore, the increasededge width reduces the thermally induced switching ﬁeld dis-tribution, which is found related to both the nucleation ﬁeld andthe activation volume. We have further studied reversal modesin dots with varied core/edge interfacial coupling strength,which could possibly result from the core/edge interfacialroughness. For dots with narrow edges, the reversal behavesin a similar way with that obtained in the two-spin model,suggesting that we can treat all spins in the edge as a singleeffective macrospin. In addition, we describe the coercivityof the edge using the Langevin function, representing thecompetition between the effective ﬁeld generated from thecore/edge coupling strength and the thermal ﬂuctuations. While the numerical simulation by the atomistic spin model is sufﬁcient to explain the magnetization reversal, it isinsightful to digest these results by simpler analytic methods sothat the key physics can be highlighted. In some cases studiedhere, the reversal dynamics of the minority spins is mainlyone dimensional. We are thus motivated to employ the linearspin chain model to capture the one-dimensional dynamics.Speciﬁcally at narrow edges the linear chain model is able toestimate the required ﬁeld for the nucleation. As the edge widthincreases the nucleation ﬁeld of core spins ﬁts to the domain-wall pinning ﬁeld at the core/edge interface. Considering thecomputationally intensive nature of the atomistic spin modelsimulation, these analytic theories can provide a global sketchfor different parameters at minimal costs. 174415-6ATOMISTIC MODELING OF MAGNETIZATION REVERSAL . . . PHYSICAL REVIEW B 90, 174415 (2014) Comparing to previous studies focused on the reversal modes along the layer-growth direction in the typical exchangespring media, here we present detailed two-dimensionalreversal behaviors on the patterned dot plane as well as thecorresponding thermally induced switching ﬁeld distribution,both of which in fact dominate properties of typical patterneddots and cannot be investigated by standard micromagneticcalculations. We also note that different magnetic propertiesof the edge, which have been assumed constant values inthis study and have not been experimentally probed, onlyvary the characteristic length of different reversal modesand the corresponding coercivity ﬁelds without affectingthe validity of theories. According to our study here, thepresence of damaged edges with uniform magnetic propertiesreduces the thermally induced switching ﬁeld distribution,and the width of the damaged edge signiﬁcantly changes thecoercivity in patterned dots. Therefore, the experimentallyobserved broadening of the switching ﬁeld distribution inpatterned L1 0FePt dots with damaged edges [ 11] should be attributed to extrinsic properties of the nanodots createdby patterning processes, for example, the variation in eitherthe width or the magnetic properties of the damaged edges.As long as we can precisely control properties of damagededges by applying a proper patterning technique, for example,ion implantation [ 20], we could realize exchange spring bit-patterned media without additional soft layers. ACKNOWLEDGMENTS The authors would like to thank H.-H. Lin for insightful suggestions. Fruitful discussions with O. Hovorka, J. Wu,W. Fan, P. Chureemart, and S. Ruta are also acknowledged.J.-W. also highly appreciates the assistance from J. Barkerand T. Ostler on solving computational issues. This work hasbeen supported by the Ministry of Science and Technologyunder Grant No. MOST 101-2917-I-007-016. U.A. gratefullyacknowledges support from Basque Country Government un-der “Programa Posdoctoral de perfeccionamiento de doctoresdel DEUI del Gobierno Vasco.” The ﬁnancial support of theAdvanced Storage Technology Consortium (ASTC) and EUSeventh Framework Programme under Grant Agreement No.281043 FEMTOSPIN is gratefully acknowledged. APPENDIX: LINEAR CHAIN MODEL In order to quantify the variation of the coercivity for narrow edge thicknesses we have developed a 1D atomistic linearchain model, simplifying the whole dot into a one-dimensionalregion started from the center of the core to the edge. Eachspin in the chain model represents the average spin within agiven atomic plane, and we can write down the following spinHamiltonian: H i/prime=−/summationdisplay i/prime,j/primeJi/primeSi/prime·Sj/prime−ki/prime/parenleftbig Sz i/prime/parenrightbig2−μi/primeHapp·Si/prime,(A1) where i/prime,j/primelabel different spins with the identical moment μ.Jis the intralayer exchange coupling, Sis the unit vector representing the spin direction, kis the anisotropy constant, and Happis the external applied ﬁeld. We set μ=μcore=μedge=1.5μB,k=kcore=4.9×10−23J/link 0.40.60.81 012Knorm eﬀ (T) Wedge (nm)M/M S Atomic plane1n m 2n m Linear chain model Linear ﬁt0.51 40 45 50 55 FIG. 6. (Color online) Effective value of anisotropy, calculated by the linear chain model, as a function of the width of the edge. The gray dashed line is the linear ﬁtted function. Inset shows the calculated layer-resolved magnetization prior to magnetization reversal in thecore in the linear chain model. The atomic plane is counted from the center of the core toward the edge, and the vertical dashed line denotes the core/edge interface. The blue area indicates the nucleatedregion for the edge width of 2 nm. for spins in the core and k=kedge=1×10−24J/link for those in the edge. We allow reduced exchange coupling at thecore/edge interface by writing the exchange energy betweeninterface spins as H int=JintSi/prime·Sν/prime, (A2) where Jintis the interface exchange coupling, and ν/primelabels spins in separate regions (core or edge) from those labeledbyi /prime. The equilibrium state of the spin system is determined by solving the Landau-Lifshitz equation, with no precession term, ∂Si/prime ∂t=−γ (1+λ2)Si/prime×λ(Si/prime×Hi/prime,eff), (A3) where λis the intrinsic damping parameter, γis the absolute value of the gyromagnetic ratio, and Hi/prime,effis the effective magnetic ﬁeld in each atomic plane, given by Hi/prime,eff=−1 μi/prime∂(Hi/prime+Hint) ∂Si/prime. (A4) In the inset of Fig. 6, we show the calculated layer-resolved magnetization within the spin chain model with variousW edge, after positively saturating all spins and then applying a corresponding negative ﬁeld prior to magnetization reversal inthe core. We number the atomic plane from the center of thecore to the edge, and the vertical dashed line in the inset ofFig. 6denotes the core/edge interface. Increasing W edgegives rise to increasing penetration of the domain wall into the core.From the energy contributed to the reversal, we estimate anormalized effective value of the anisotropy constant K norm eff by integrating the anisotropy energy over the domain-wall width from the edge to the core in the nucleated region (seethe blue region in the inset of Fig. 6forW edge=2 nm) and then normalizing to Kcore. This quantiﬁes the reduction in the energy barrier due to the exchange spring. Figure 6illustrates 174415-7LIAO, ATXITIA, EV ANS, CHANTRELL, AND LAI PHYSICAL REVIEW B 90, 174415 (2014) the variation of Knorm eff withWedge. We observe a linear decrease inKnorm eff with the increase in Wedge, and we further describe the linear decrease as (gray dashed line in Fig. 6) Knorm eff=1−bW edge, (A5) where b=0.324 (nm−1) obtained from ﬁtting. Since a single nucleated area dominates the reversal in the region of narrowsoft edges, we then estimate the coercive ﬁeld as an effective anisotropy ﬁeld of the nucleated area Heff K, indicated by the gray dashed line in Fig. 4(a), Hcore(edge) c =Heff K=Hcore K(1−bW edge). (A6) [1] D. Weller and A. Moser, IEEE Trans. Magn. 35,4423 (1999 ). [2] R. Rottmayer, S. Batra, D. Buechel, W. Challener, J. Hohlfeld, Y . Kubota, L. Li, B. Lu, C. Mihalcea, K. Mountﬁeld, K. Pelhos,C. Peng, T. Rausch, M. A. Seigler, D. Weller, and X. Yang,IEEE Trans. Magn. 42,2417 (2006 ). [3] T. W. McDaniel, J. Phys.: Condens. Matter 17,R315 (2005 ). 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PhysRevB.82.184423.pdf	Spin and charge transport induced by magnetization dynamics in diffusive ferromagnetic metals Steven S.-L. Zhang and Shufeng Zhang Department of Physics, University of Arizona, Tucson, Arizona 85721, USA /H20849Received 27 May 2010; revised manuscript received 23 August 2010; published 19 November 2010 /H20850 In ferromagnetic metals, the temporal and spatial dependence of the magnetization vector generates spin-dependent electric and magnetic ﬁelds. We study spin and charge transport by using the linearalizedBoltzmann equation in response to these ﬁelds. A generalized spin-diffusion equation is derived and thenexactly solved for a nanowire where the spatial dependence of the magnetization is one dimensional. It is foundthat the relative length scales between the domain-wall width and the spin-diffusion length are important indetermining the local electromotive force and the enhanced damping induced by magnetization dynamics. DOI: 10.1103/PhysRevB.82.184423 PACS number /H20849s/H20850: 75.78.Fg, 75.76. /H11001j I. INTRODUCTION Recently, there have been intensive studies on domain- wall motion driven by electric current. The essential mecha-nism is the transfer of spin angular momentum carriedby spin-polarized conduction electrons to the localmagnetization. 1,2Conversely, domain-wall motion can trans- fer the spin angular momentum of magnetization to conduc-tion electrons through the ﬂow of an induced spin current. 3–7 Among other consequences, these induced spin currents lead to/H208491/H20850an electric voltage across a moving domain wall, known as the spin electromotive force /H20849emf /H20850,8–11and /H208492/H20850the electronic dissipation /H20849e.g., Joule heating /H20850of the spin current which provides a mechanism for the loss of the angular mo-menta of a processing ferromagnet, i.e., damping. 5,12–14 A simple description of the above phenomena can be made as follows. In a ferromagnet, the conductionelectron spin is strongly coupled with the magnetization,H /H11032=−J/H9268·m, where mdenotes the direction of the magneti- zation and /H9268is the Pauli matrix representing the spin of the conduction electron. If the magnet is uniformly magnetized,one can choose the quantization axis of the electron spin parallel to the magnetization and thus H /H11032simply gives rise to a constant energy − Jfor the spin-up electron and Jfor the spin-down electron. In a domain wall where the magnetiza-tion varies in space, H /H11032is no more spatially independent and thus H/H11032does not commute with the kinetic energy Hk, i.e., /H20851H/H11032,p2/2m/H20852/HS110050. To approximately solve the Hamiltonian H/H11032+Hk, one may make a canonical transformation such that the quantization axis is always parallel to the direction of thelocal magnetization. Along the local quantization axis, H /H11032is diagonalized in spin space, i.e., H/H11032=−J/H9268·m/H20849r,t/H20850→−J/H9268z. However, such transformation makes the kinetic energy morecomplicated due to the noncommutativity of H /H11032andHk. Ad- ditional terms are generated by the transformation and theyhave already been identiﬁed as the spin-dependent electricand magnetic ﬁelds 3–5,15,16given below, Eis=sEi=s/H6036 2em·/H20849/H11509tm/H11003/H11509im/H20850, /H208491/H20850 Bis=−s/H6036 4e/H9280ijk/H20849/H11509jm/H11003/H11509km/H20850·m, /H208492/H20850 where s=+1 for the spin-up and s=−1 for spin-down elec- trons, and the space and time partial derivatives of the mag-netization are notated as /H11509iand/H11509t. The spin transport of a moving domain wall is thus identical to a system of uniformmagnetization subject to the temporally and spatially depen-dent electric and magnetic ﬁelds given by Eqs. /H208491/H20850and /H208492/H20850. Up until now, theoretical considerations on these ﬁelds havebeen carried out either by performing averaging in space orby utilizing simple Ohm’s law to obtain spin-dependentcurrents. 17Since it has been well established that the spin transport properties depend on spin diffusion in magneticinhomogeneous structure 18where the local Ohm’s law for each spin channel is usually not valid, a natural question is:how does spin diffusion alter the spin emf and the enhanceddamping? Recently, Tserkovnyak et al. 4,13have considered a generalization of the spin-diffusion equation for these spin-dependent ﬁelds; we will brieﬂy discuss the difference be-tween our model and theirs in the next section. In this paper, we derive a generalized spin-diffusion equa- tion by using the Boltzmann equation. By explicitly solvingthe spin-diffusion equation with appropriate boundary condi-tions, we are able to reexamine various spin transport prop-erties such as spin emf and damping torques. The rest ofthe paper is organized as follows: in Sec. II, we derive the diffusion equation. In Sec. III, we analytically solve for the spin-diffusion equation in a magnetic nanowire wherethe magnetization depends on only one coordinate, i.e.,m=m/H20849x,t/H20850/H20849note that the direction of mremains three di- mensional /H20850. In Sec. IV, we discuss the dependence of the spin emf and a nonlocal damping torque on the relative val-ues of diffusion length and domain wall width. Finally, wesummarize our results in Sec. V. II. GENERALIZATION OF SPIN-DIFFUSION EQUATION We start with the semiclassical Boltzmann equation for each spin channel within the relaxation time approximation, /H11509fs /H11509t+v·/H11612rfs+eEs·v/H11509f0 /H11509/H9255=−fs−fs /H9270s−fs−f−s /H9270sf, /H208493/H20850 where s=/H110061 labels spin channel /H20849up or down /H20850, fs=fs/H20849p,r,t/H20850and f0=f0/H20849p/H20850are the nonequilibrium and equilibrium distribution functions, respectively, /H9270sand/H9270sf are the spin-conserving and spin-ﬂip relaxation times, Es is the effective ﬁeld given by Eq. /H208491/H20850,p=mvandPHYSICAL REVIEW B 82, 184423 /H208492010 /H20850 1098-0121/2010/82 /H2084918/H20850/184423 /H208496/H20850 ©2010 The American Physical Society 184423-1fs=/H20848d3pfs/H20849p,r,t/H20850//H20848d3p. The explicit time dependence of the nonequilibrium distribution function fs/H20849p,r,t/H20850is completely due to the time-dependent magnetization vector m/H20849r,t/H20850, i.e., fs/H20849p,r,t/H20850=fs/H20851p,r,m/H20849t/H20850/H20852. It is noted that although the gauge magnetic ﬁeld, Eq. /H208492/H20850, exerts a nonzero force /H20849ev/H11003Bs/H20850on the electron, it does not enter the above Boltzmannequation if isotropic band structure is assumed, i.e.,/H11612 vfs·/H20849v/H11003Bs/H20850/H11008v·/H20849v/H11003Bs/H20850=0. Before we proceed to solve the above Boltzmann equa- tion, we should bring up an interesting issue. In the earlierpapers by Duine 3and by Tserkovnyak and Mecklenburg,4the dissipative force, /H20849/H9252/H6036/2e/H20850/H11509tm·/H11509im, whose origin is the mis- tracking of conduction electron spins with local magnetiza-tion, has been introduced. One may consider this dissipativeforce as a reaction force of the nonadiabatic spin torque inthe current-driven domain-wall motion. 19As it has been well known, the nonadiabatic spin torque plays an important rolein domain-wall motion when the applied current densityreaches 10 7–108A/cm2.20Conversely, the domain-wall mo- tion can generate the spin current but its magnitude of theinduced spin current is only about 10 4–105A/cm2for a very fast wall velocity. Such current density is too small tohave a remarkable backreaction effect on the domain-wallmotion. Furthermore, as we shall show in Sec. IV, the spin emf signal owing to the gauge electric ﬁeld, Eq. /H208491/H20850,i so nt h e order of 0.1–1 /H9262V, whereas the additional voltage produced by the dissipative force would be in the range of 1–10 nVwhich is too small to be experimentally interested. Based onthe above considerations, we conclude that the dissipativeforce is present but has a negligible contribution to the spinemf. Therefore we can limit ourselves to the two-componentBoltzmann equation /H20849otherwise one has to resort to the spinor Boltzmann equation to deal with the transverseelectron-spin distribution function /H20850. Next we write f sas the sum of the equilibrium part /H20849i.e., Fermi-Dirac distribution /H20850f0/H20849p/H20850and small perturbations in- duced by Es, fs/H20849p,r,t/H20850=f0/H20849p/H20850+/H20873−/H11509f0 /H11509/H9255/H20874/H20851gs/H20849p,r,t/H20850+e/H9262s/H20849r,t/H20850/H20852, /H208494/H20850 where gs/H20849p,r,t/H20850is the anisotropic part satisfying /H20848d3pgs/H20849p,r,t/H20850=0 and e/H9262s/H20849r,t/H20850is the isotropic part. For a spherical band structure, one can conjecture that gs=p·g1 where g1is a vector parallel to the ﬁeld Es. By inserting Eq. /H208494/H20850into Eq. /H208493/H20850and separately writing the equation for the anisotropic and isotropic parts,21we have /H11509gs/H20849p,r,t/H20850 /H11509t−eE/H11032s·v=−gs/H20849p,r,t/H20850 /H9270s/H208495/H20850 and e/H11509/H9262s/H20849r,t/H20850 /H11509t+/H20855v·/H11612rgs/H20849p,r,t/H20850/H20856=−e/H9262s/H20849r,t/H20850−/H9262−s/H20849r,t/H20850 /H9270sf,/H208496/H20850 where E/H11032s/H11013Es−/H11612r/H9262sand /H20855/H20856denotes the average over the momentum. It is noted that the charge continuity equation, /H11509n/H20849t/H20850//H11509t+/H11612r·je=0, can be readily derived from Eq. /H208496/H20850by simply summing over the two spin channels. In solving thenonequilibrium distribution function explicitly, we may dis-card the time derivative terms in both Eqs. /H208495/H20850and /H208496/H20850. This is because both g s/H20849p,r,t/H20850and/H9262s/H20849r,t/H20850depend on time only through the time dependence of m/H20849r,t/H20850and thus they vary on the time scale of nanoseconds; whereas the spin-conservingand spin-ﬂip relaxation times are on the order of /H9270s/H1101510−14s and /H9270sf/H1101510−12s for metallic ferromagnets. Therefore the ﬁrst terms on the left sides of Eqs. /H208495/H20850and /H208496/H20850 are much smaller than those on the right sides of the equa-tions. We note that while the above approximation simpliﬁesthe solution, an apparent drawback would appear to be aviolation of charge conservation, i.e., that neglecting /H11509n/H20849t/H20850//H11509t results in a divergenceless spin current. We will clarify theabove concern and further discuss this approximation whenconsider the solution in one-dimensional /H208491D/H20850case in Sec. III. By dropping the time derivative terms in Eqs. /H208495/H20850and /H208496/H20850, we have g s/H20849p,r,t/H20850=e/H9270sE/H11032s·v /H208497/H20850 and e/H9262s/H20849r,t/H20850−e/H9262−s/H20849r,t/H20850=−/H9270sf/H20855v·/H11612rgs/H20849p,r,t/H20850/H20856. /H208498/H20850 By placing Eq. /H208497/H20850into Eq. /H208498/H20850and carrying out the integra- tion over a spherical Fermi surface, we have /H116122/H9262s/H20849r,t/H20850=/H9262s/H20849r,t/H20850−/H9262−s/H20849r,t/H20850 /H20849/H9011s/H208502+/H11612·Es, /H208499/H20850 where /H9011s=vF/H20881/H9270s/H9270sf/3. By replacing sto −sin Eq. /H208499/H20850, and by adding and subtracting the resultant equation with Eq. /H208499/H20850, we ﬁnd /H116122/H9262a/H20849r,t/H20850=/H9262a/H20849r,t/H20850 /H20849/H9011/H208502+/H11612·E /H2084910/H20850 and /H116122/H9262¯/rho1/H20849r,t/H20850=p/H9011/H11612·E, /H2084911/H20850 where we have used E−s=−Es/H20851note that Es=sE, see Eq. /H208491/H20850/H20852. Several parameters have been introduced: thediffusion length /H9011is deﬁned via /H20849/H9011/H20850 −2=/H20849/H9011s/H20850−2+/H20849/H9011−s/H20850−2, /H9262a/H11013/H20849/H9262↑−/H9262↓/H20850/2 is the spin accumulation, /H9262¯/rho1/H11013/H20849/H9011↑2/H9262↑+/H9011↓2/H9262↓/H20850//H20849/H9011↑2+/H9011↓2/H20850is the weighted average of the chemical potential and p/H9011=/H20849/H9011↑2−/H9011↓2/H20850//H20849/H9011↑2+/H9011↓2/H20850is the spin-diffusion length polarization. With speciﬁc boundaryconditions, one can solve Eqs. /H2084910/H20850and /H2084911/H20850for /H9262s/H20849r,t/H20850. Once the chemical potential is determined, gsis found through Eq. /H208497/H20850and the distribution fsof Eq. /H208494/H20850is obtained. The current density for spin sis js/H20849r,t/H20850=e h3/H20885d3pvsfs/H20849ps,r,t/H20850=/H9268s/H20849Es−/H11612r/H9262s/H20850, /H2084912/H20850 where we have used − /H20849/H11509f0//H11509/H9255s/H20850=/H9254/H20849/H9255s−/H9255F/H20850and we have de- ﬁned the Drude conductivity /H9268s/H11013nse2/H9270s/mfor spin swith ns=kF3/6/H92662being the corresponding electronic density. It is interesting to compare our spin-diffusion equation, Eq. /H2084910/H20850, and the generalized Poisson equation, Eq. /H2084911/H20850, with the conventional spin-diffusion equation associated witha static external electric ﬁeld. 18When current is driven by an external electric ﬁeld, the divergence of the electric ﬁeld isSTEVEN S.-L. ZHANG AND SHUFENG ZHANG PHYSICAL REVIEW B 82, 184423 /H208492010 /H20850 184423-2zero and thus there are no such source terms as in either Eq. /H2084910/H20850or Eq. /H2084911/H20850. However, the effective ﬁeld induced by magnetization dynamics, Eq. /H208491/H20850, is fundamentally different from a static electric ﬁeld in that both the curl and diver-gence of E sis not zero in general. This source term in the diffusion equation, Eq. /H2084910/H20850, is the origin of the spin- transport phenomena to be discussed below. To make con-crete statements on the role of the spin diffusion, we shallexplicitly solve Eqs. /H2084910/H20850and /H2084911/H20850with a proper boundary condition. We will take a simple case where the magnetiza-tion vector depends on one coordinate only, i.e., m=m/H20849x,t/H20850 and thus the diffusion equation becomes one dimensionaland can be readily solved. III. SOLUTIONS FOR ONE-DIMENSIONAL NANOWIRES In the present case, the electric ﬁeld in Eq. /H208491/H20850has only x component, i.e., E/H20849x,t/H20850=E/H20849x,t/H20850ex. Equations /H2084910/H20850and /H2084911/H20850 become d2/H9262a/H20849x/H20850 dx2=/H9262a/H20849x/H20850 /H20849/H9011/H208502+dE dx/H2084913/H20850 and d2/H9262¯/rho1/H20849x/H20850 dx2=p/H9011dE dx. /H2084914/H20850 We have suppressed the time variable for the moment in view of the fact that the chemical potentials depend on timeonly through the magnetization m/H20849x,t/H20850whose time variation is much slower than spin relaxation. To specify the boundaryconditions, we consider an inﬁnite long magnetic nanowirewhere these is only one domain wall at any time. The wallcenter X c/H20849t/H20850varies with time and the induced electric ﬁeld, Eq. /H208491/H20850, is nonzero when − W/2/H11021x−Xc/H20849t/H20850/H11021W/2, where Wis the width of the moving domain wall. The boundary conditions are thus E/H20849/H11006/H11009 /H20850=0, ord/H9262a/H20849x/H20850 dx/H20841x=/H11006/H11009=0 and d/H9262¯/rho1/H20849x/H20850 dx/H20841x=/H11006/H11009=0. We obtain the solutions of Eqs. /H2084913/H20850and /H2084914/H20850, /H9262a/H20849x,t/H20850=1 2/H20885 −/H11009/H11009 /H20851/H9258/H20849x−x/H11032/H20850−/H9258/H20849x/H11032−x/H20850/H20852e−/H20849x−x/H11032/H20850//H9011E/H20849x/H11032,t/H20850dx/H11032 /H2084915/H20850 and /H9262¯/rho1/H20849x,t/H20850=1 2p/H9011/H20885 −/H11009/H11009 /H20851/H9258/H20849x−x/H11032/H20850−/H9258/H20849x/H11032−x/H20850/H20852E/H20849x/H11032,t/H20850dx/H11032+A, /H2084916/H20850 where Ais an arbitrary constant for deﬁning zero chemical potential at any given position and /H9258is the step function, i.e., /H9258/H20849x/H20850is 0 for x/H110210 and 1 for x/H110220. The physical meaning of the two step functions in Eqs. /H2084915/H20850and /H2084916/H20850is the electron diffusion to the left x/H11032/H11021xand to the right x/H11032/H11022x. Since /H9262a and/H9262¯/rho1are linearly related to /H9262↑and/H9262↓/H20851see their deﬁnitions after Eq. /H2084911/H20850/H20852, the above two equations determine the chemi- cal potentials /H9262↑,↓. Subsequently, we can obtain the current density by placing these /H9262sinto Eq. /H2084912/H20850,jxs/H20849x,t/H20850=s/H9268s/H20849/H9011−s/H208502 /H9011/H20851/H20849/H9011s/H208502+/H20849/H9011−s/H208502/H20852/H20875/H20885 −/H11009+/H11009 e−/H20841x−x/H11032/H20841//H9011E/H20849x/H11032,t/H20850dx/H11032/H20876. /H2084917/H20850 Thus the spin current density is jxsp/H20849x,t/H20850/H11013j↑−j↓=Csp /H9011/H20875/H20885 −/H11009+/H11009 e−/H20841x−x/H11032/H20841//H9011E/H20849x/H11032,t/H20850dx/H11032/H20876, /H2084918/H20850 where we have deﬁned Csp=/H20849/H9268↑/H9011↓2+/H9268↓/H9011↑2/H20850//H20849/H9011↑2+/H9011↓2/H20850and the charge current is jxch/H20849x,t/H20850/H11013j↑+j↓=Cch /H9011/H20875/H20885 −/H11009+/H11009 e−/H20841x−x/H11032/H20841//H9011E/H20849x/H11032,t/H20850dx/H11032/H20876, /H2084919/H20850 where we have deﬁned Cch=/H20849/H9268↑/H9011↓2−/H9268↓/H9011↑2/H20850//H20849/H9011↑2+/H9011↓2/H20850. By inspecting the coefﬁcients CspandCch, we realize that the spin current is always larger than the charge current; thisis because the spin-up and spin-down electrons travel in op-posite directions. Furthermore, C spis always nonzero but Cch could be zero when /H9268↑/H9011↓2=/H9268↓/H9011↑2; in fact, this is the case when we use the deﬁnition of /H9011/H9268after Eq. /H208499/H20850for a spin- independent Fermi velocity. Therefore, the effective electricﬁeld generates spin current but not charge current. Ingeneral, however, charge current also exists when /H9268↑/H9011↓2/HS11005/H9268↓/H9011↑2. We point out that the position dependence of the charge current density in the 1D case does not violate the chargeconservation. It is noted that the effective electric ﬁeld gen-erated by magnetization dynamics is time dependent. There-fore, both charge densities and currents are time dependent.In fact, the charge conservation, /H11509ne//H11509t+/H11612rje=0, is automati- cally satisﬁed by the Boltzmann equation, Eq. /H208493/H20850.I no u r calculation of the current density, we have neglected /H11509ne//H11509t for the reasons given after Eq. /H208496/H20850. In principle, one could keep this time-dependent charge density in Eq. /H208496/H20850with the resulting diffusion equation becoming a partial differentialequation in space and in time. It is interesting to examine the limiting cases. When the spin-diffusion length is much smaller than thedomain-wall width, the variation in the electric ﬁeldover the diffusion length is small so that we can replace/H208491//H9011/H20850exp /H20849−/H20841x−x /H11032/H20841//H9011/H20850by 2/H9254/H20849x−x/H11032/H20850in Eqs. /H2084918/H20850and /H2084919/H20850,w e ﬁnd jxsp/H20849x,t/H20850=2CspE/H20849x,t/H20850,jxch=2CchE/H20849x,t/H20850. These local relations between the current and the electric ﬁeld are Ohm’s laws we used previously5for/H9011↑=/H9011↓; they are valid only in this limiting case /H20849/H9011/H11270W/H20850. In the opposite case where the spin-diffusion length is much largerthan the domain-wall width /H20849/H9011/H11271W/H20850, by approximating /H20841x−x /H11032/H20841//H9011=/H20841x−Xc+Xc−x/H11032/H20841//H9011/H11015/H20841x−Xc/H20841//H9011in Eqs. /H2084918/H20850and /H2084919/H20850, we haveSPIN AND CHARGE TRANSPORT INDUCED BY … PHYSICAL REVIEW B 82, 184423 /H208492010 /H20850 184423-3jxsp/H20849x,t/H20850=CspE¯/H9011exp /H20849−/H20841x−Xc/H20841//H9011/H20850, jxch/H20849x,t/H20850=CchE¯/H9011exp /H20849−/H20841x−Xc/H20841//H9011/H20850, /H2084920/H20850 where E¯/H9011=/H20849/H9011/H20850−1/H20848−/H11009/H11009E/H20849x/H11032,t/H20850dx/H11032can be viewed as an average electric ﬁeld over the spin diffusion length. Although theinduced electric ﬁeld is conﬁned to the region of the domain-wall width, the spin and charge currents can be found onediffusion length away from the domain wall. Clearly, thislatter case represents a nonlocal relation between the currentand the electric ﬁeld; we will discuss further in the nextsection. IV. SPIN EMF VOLTAGE AND ENHANCED DAMPING One of the most important consequences of the induced electric ﬁeld, Eq. /H208491/H20850, is the generation of a measurable volt- age across a moving domain wall or emf. To measure a localpotential, one needs to specify the voltmeter probe at a givenlocation. Let us assume the voltmeter probe is made of anonmagnetic material which does not interfere with the mag-netization of the wire. The measured chemical potential isthen /H9262/H20849x/H20850=/H20849/H9262↑+/H9262↓/H20850/2. If the probe is far away from the domain wall, /H9262↑=/H9262↓. Thus the voltage drop between x=/H11009 andx=−/H11009can be readily obtained from Eq. /H2084916/H20850, Vemf/H11013/H9262/H20849/H11009/H20850−/H9262/H20849−/H11009/H20850=p/H9011/H6036 2e/H20885 −/H11009+/H11009 m·/H20849/H11509tm/H11003/H11509xm/H20850dx, /H2084921/H20850 where we have used the effective ﬁeld deﬁned in Eq. /H208491/H20850. Interestingly, the emf is independent of the diffusion length.We point out that the emf signal is ﬁnite even if the chargecurrent is zero in the special case we have discussed after Eq./H2084919/H20850; this is because the measured voltage is directly related to the charge accumulation rather than the electric current.For a ﬁxed spin polarization p /H9011and for a given domain-wall motion, the emf is indeed universal, i.e., the emf is indepen-dent of the transport parameters such as the conductivity andspin-diffusion length. However, one should bear in mind thatdomain-wall motion could display a very complicated pat-tern. For example, a large applied magnetic ﬁeld induceswall motion via wall transformation, distortion, and oscilla-tion. Thus the emf signal is far from a simple universal con-stant. It is interesting to have a close look at the emf signal when a domain wall is injected into a nanowire in real time.Consider a domain wall is initially outside the voltmeter asshown schematically in Fig. 1/H20849a/H20850. When a magnetic ﬁeld is applied, the domain wall moves toward the voltmeter andlater leaves the voltmeter. As translational domain-wall mo-tion does not generate the electric ﬁeld, see Eq. /H208491/H20850,i ti s necessary to apply a magnetic ﬁeld larger than the Walker’sbreakdown ﬁeld 22so that the wall motion contains both translational and rotational components shown in Fig. 1/H20849b/H20850. In the process, a voltage begins to show up when the domainwall approaches one of the probe tips within a spin-diffusion length. Quantitatively, the measured electric potential by anonmagnetic probe tip at any location in the nanowire can beobtained from Eqs. /H2084915/H20850and /H2084916/H20850, V/H20849x,t/H20850= /H9262↑+/H9262↓ 2 =p/H9011/H6036 4e/H20885 −/H11009/H11009 /H20851/H9258/H20849x−x/H11032/H20850−/H9258/H20849x/H11032−x/H20850/H20852/H208491−e−/H20849x−x/H11032/H20850//H9011/H20850m ·/H20873/H11509m /H11509t/H11003/H11509m /H11509x/H11032/H20874dx/H11032. /H2084922/H20850 The example shown in Figs. 1/H20849b/H20850–1/H20849d/H20850is obtained via 1D micromagnetic simulation with following para-meters: the width, thickness, and length of the nano- wire are 128 nm, 8 nm and 5 /H9262m, respectively, the saturation magnetization Ms=800 emu /cc, the exchange constant A=1.3/H1100310−6erg /cm, the uniaxial anisotropy K=5.0/H11003103erg /cm3, the spin polarization p=0.4, and the Gilbert damping constant /H9251=0.01. Although the electron spin diffusion will give rise to an additional damping /H20849to be discussed next /H20850, we will ignore it in the determination of the induced voltage. This approximation is justiﬁed due to thelarge domain-wall width for a transverse wall in the presentcase, which results in an insigniﬁcant contribution of theadditional damping to the wall motion. 5,13,17 Once the magnetization is computed, the measured volt- age as a function of time is numerically obtained through Eq./H2084922/H20850. The oscillation of the voltage shown in Fig. 1/H20849c/H20850reﬂects the oscillatory motion of the domain wall. The smooth /H20849red/H20850 curve is obtained by ﬁltering high-frequency components,i.e., one only keeps those Fourier components with frequen-cies lower than the domain-wall oscillation frequency. In Fig.1/H20849d/H20850, we compare the ﬁltered voltage signals for three differ- ent spin-diffusion lengths. Note that t AandtBare the times when the domain-wall center reaches one of the probes. Thevoltage displays obvious time-dependent behavior: /H208491/H20850the rising time is given by the spin-diffusion length divided bythe domain-wall velocity during which the domain wall en-ters the voltmeter probe, /H208492/H20850when the domain wall is com- pletely inside the voltmeter probes, the average emf stayssaturated, and /H208493/H20850the domain wall leaves the probe with the decaying time same as the rising time. Next, we utilize our spin-diffusion equation to derive an enhanced damping due to the induced electric ﬁelds. Whilethere are many sources of damping, here we consider anadditional damping due to the transfer of the angular mo-mentum of the moving domain wall to the spin current. Thespin-transfer torque is deﬁned as the divergence of spincurrent. 23Before we explicitly relate the spin torque with the magnetization, we note that spin current is a tensor whichconsists of two vectors: the direction of electron ﬂow andspin-polarization vector of the transport electrons. Since theelectric ﬁeld is in xdirection, the spin current also ﬂows in x direction. Thus j xsp=mjxsp, where jxspis given by Eq. /H2084918/H20850. The spin-transfer torque isSTEVEN S.-L. ZHANG AND SHUFENG ZHANG PHYSICAL REVIEW B 82, 184423 /H208492010 /H20850 184423-4/H9270st/H11013−/H11509jxsp /H11509x =−g/H9262B/H6036 4e2Csp /H9011/H20875/H20885 −/H11009+/H11009 e−/H20841x−x/H11032/H20841//H9011m·/H20873/H11509m /H11509t/H11003/H11509m /H11509x/H11032/H20874dx/H11032/H20876/H11509m /H11509x. /H2084923/H20850 We show below that the above torque is a damping torque, i.e., the torque decreases the magnetic energy.The rate of magnetic energy density change isdE /dt/H11013−H eff·/H11509tM=−Heff·/H9270stwhere we only consider the energy change due to the spin torque. By using Eq. /H2084923/H20850,w e have dE dt=−g/H9262B/H6036 4e2Csp /H9011/H92530/H20885 −/H11009+/H11009 dx /H11003/H20875/H20885 −/H11009+/H11009 dx/H11032e−/H20841x−x/H11032/H20841//H9011Heff·/H11509m /H11509x/H11032/H20876Heff·/H11509m /H11509x,/H2084924/H20850 where we have made a ﬁrst-order approximation by replac- ing/H11509tmin Eq. /H2084923/H20850by −/H92530m/H11003Heffwith/H92530being the gyro- magnetic ratio. Let us ﬁrst consider two limiting cases. Whenthe spin-diffusion length is much smaller than the domain- wall width we have /H20879dE dt/H20879 /H9011/H11270W=−g/H9262B/H6036 2e2Csp/H92530/H20885 −/H11009+/H11009 dx/H20879Heff·/H11509m /H11509x/H208792 and the opposite limiting case gives /H20879dE dt/H20879 /H9011/H11271W=−g/H9262B/H6036 4e2Csp /H9011/H92530/H20875/H20885 −/H11009+/H11009 dxHeff·/H11509m /H11509x/H208762 . In both cases, the magnetic energy decreases and thus the spin current of conduction electrons takes away the angularmomentum and energy from the magnetization dynamics.Subsequently, both angular momentum and energy carried bythe spin current dissipate into lattice via spin-ﬂip scatteringand Joule heating. In Fig. 2, we show the total magnetic energy rate of change as a function of the spin-diffusionlength. Clearly, when spin-diffusion length is small, the dis-sipation of angular momentum and magnetic energy is moreeffective. Finally, we emphasize that the enhanced damping torque, Eq. /H2084923/H20850, is nonlocal since the damping at one location de- pends on the magnetization at other locations within thespin-diffusion length. Such nonlocal damping has already FIG. 1. /H20849Color online /H20850/H20849a/H20850Schematics of the voltage measurement as a domain wall, subject to an applied magnetic ﬁeld, passes through the voltage probes. The separation of the two probe tips is 800 nm /H20849much larger than the domain-wall width /H20850in the simulation. /H20849b/H20850 Domain-wall displacement as a function of time for an applied magnetic ﬁeld of 70 Oe. The inset shows the time dependence of thedomain-wall width. /H20849c/H20850The time-dependent voltage signal when the domain wall travels through the nanowire for a spin-diffusion length of 40 nm. The black curve is the real-time signal and the smooth red curve is obtained by ﬁltering the frequencies higher than the domain-walloscillation frequency /H208490.4 GHz /H20850./H20849d/H20850The ﬁltered voltage signals for three different spin-diffusion lengths of 5, 40, and 80 nm.SPIN AND CHARGE TRANSPORT INDUCED BY … PHYSICAL REVIEW B 82, 184423 /H208492010 /H20850 184423-5been pointed out by Foros et al.12where they have used the dissipation-ﬂuctuation relation to approximately estimate anenhanced damping due to spin current. Although the physicsof the damping described here is similar to theirs, the detailsdiffer signiﬁcantly. The advantage of our theory is a broader usage of the analytic damping torque which is valid for awide range of the spin-diffusion length. V. SUMMARY We have generalized the spin-diffusion equation in the presence of the spin-dependent electric ﬁeld induced bymagnetization dynamics. Compared with the conventionalspin diffusion, the main difference is that our generalizeddiffusion equation contains a divergence of the electric ﬁeldas a source term. By solving the equation in 1D case, wefound that the previously assumed local spin and chargecurrent 5is only valid in the limiting case that the spin- diffusion length is much smaller than the domain-wall width.Beyond this limit, the current, the spin emf, and the dampingtorque manifest nonlocal features depending on the relativelength scales between the spin-diffusion length and thedomain-wall width. ACKNOWLEDGMENTS This work is partially supported by the U.S. DOE /H20849Grant No. DE-FG02-06ER46307 /H20850and the NSF /H20849Grant No. DMR- 0704182 /H20850. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 3R. A. Duine, Phys. Rev. B 77, 014409 /H208492008 /H20850. 4Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77, 134407 /H208492008 /H20850. 5S. Zhang and S. S.-L. Zhang, Phys. Rev. Lett. 102, 086601 /H208492009 /H20850. 6J. Shibata and H. Kohno, Phys. Rev. Lett. 102, 086603 /H208492009 /H20850. 7Y. Ban and G. Tatara, Phys. Rev. B 80, 184406 /H208492009 /H20850. 8S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601 /H208492007 /H20850. 9S. A. Yang, G. S. D. Beach, C. Knutson, D. Xiao, Q. Niu, M. Tsoi, and J. L. Erskine, Phys. Rev. Lett. 102, 067201 /H208492009 /H20850. 10W. M. Saslow, Phys. Rev. B 76, 184434 /H208492007 /H20850. 11R. A. Duine, Phys. Rev. B 79, 014407 /H208492009 /H20850. 12J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. B 78, 140402 /H20849R/H20850/H208492008 /H20850.13C. H. Wong and Y. Tserkovnyak, Phys. Rev. B 80, 184411 /H208492009 /H20850. 14C. H. Wong and Y. Tserkovnyak, Phys. Rev. B 81, 060404 /H208492010 /H20850. 15G. E. Volovik, J. Phys. C 20, L83 /H208491987 /H20850. 16A. Takeuchi, K. Hosono, and G. Tatara, Phys. Rev. B 81, 144405 /H208492010 /H20850. 17S. S.-L. Zhang and S. Zhang, IEEE Trans. Magn. 46, 2297 /H208492010 /H20850. 18T. Valet and A. Fert, Phys. Rev. B 48, 7099 /H208491993 /H20850. 19S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 /H208492004 /H20850. 20M. Klaui, P.-O. Jubert, R. Allenspach, A. Bischof, J. A. C. Bland, G. Faini, U. Rüdiger, C. A. F. Vaz, L. Vila, and C. Vouille, Phys. Rev. Lett. 95, 026601 /H208492005 /H20850. 21Y. Qi and S. Zhang, Phys. Rev. B 67, 052407 /H208492003 /H20850. 22N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 /H208491974 /H20850. 23M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407 /H208492002 /H20850. FIG. 2. The time-averaged rate of change in the total magnetic energy as a function of spin-diffusion length, where the solid line issimple data ﬁtting. Inset shows the time evolution of the instanta-neous total magnetic energy rate of change for diffusion length of40 nm. We use C sp=0.08 /H20849/H9262/H9024cm/H20850−1for the nanowire.STEVEN S.-L. ZHANG AND SHUFENG ZHANG PHYSICAL REVIEW B 82, 184423 /H208492010 /H20850 184423-6
PhysRevB.83.094410.pdf	PHYSICAL REVIEW B 83, 094410 (2011) Linear-response theory of spin Seebeck effect in ferromagnetic insulators Hiroto Adachi,1,2,Jun-ichiro Ohe,1,2Saburo Takahashi,1,2,3and Sadamichi Maekawa1,2 1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan 2CREST, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan 3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan (Received 12 October 2010; published 11 March 2011) We formulate a linear response theory of the spin Seebeck effect, i.e., a spin voltage generation from heat current ﬂowing in a ferromagnet. Our approach focuses on the collective magnetic excitation of spins, i.e.,magnons. We show that the linear-response formulation provides us with a qualitative as well as quantitativeunderstanding of the spin Seebeck effect observed in a prototypical magnet, yttrium iron garnet. DOI: 10.1103/PhysRevB.83.094410 PACS number(s): 85 .75.−d, 72.15.Jf, 72.25.−b I. INTRODUCTION The generation of spin voltage, i.e., the potential for an electron’s spin to drive spin currents, by a temperature gradientin a ferromagnet is referred to as the spin Seebeck effect (SSE).Since the ﬁrst observation of the SSE in a ferromagnetic metal,Ni 81Fe19,1this phenomenon has attracted much attention as a new method of generating spin currents from heat energyand opened a new possibility of spintronics devices. 2The SSE triggered the emergence of the new ﬁeld dubbed “spincaloritronics” 3,4in the rapidly growing spintronics community. Moreover, as the induced spin voltage can be converted intoelectric voltage through the inverse spin Hall effect 5at the attached nonmagnetic metal, this phenomenon put a newtwist on the long and well-studied history of thermoelectricresearch. 6 One of the canonical frameworks to describe nonequilib- rium transport phenomena is linear-response theory.7Having been applied to a number of transport phenomena, linear-response theory has been so successful because it is intimatelyrelated to the universal ﬂuctuation-dissipation theorem. Up tonow, however, the linear-response formulation of the SSE hasnot been known mainly because, unlike the charge current, thespin current is not a conserved quantity. Therefore, it is of greatimportance to formulate the SSE in terms of linear-responsetheory. Concerning the SSE, a big mystery is now being estab- lished, which is, how can conduction electrons sustain thespin voltage over such a long range of several millimeters 1 in spite of the conduction electrons’ short spin-ﬂip diffusionlength, which is typically of several tens of nanometers? A keyto resolve this puzzle was reported by a recent experiment onelectric signal transmission through a ferromagnetic insulator 8 which demonstrates that the spin current can be carried bythe low-lying magnetic excitation of localized spins, i.e., the magnon excitations, and that it can transmit the spin current asfar as several millimeters. Subsequently, the SSE was reportedto be observed in the magnetic insulator LaY 2Fe5O12despite the absence of conduction electrons.9These experiments suggest that contrary to the conventional wisdom over thelast two decades that the spin current is carried by conduction electrons, 10the magnon is a promising candidate as a carrier for the SSE. The purpose of this paper is twofold. First, we analyze the SSE observed in LaY 2Fe5O129(hereafter referred to as YIG)in terms of magnon spin current, i.e., a spin current carried by magnon excitations. Second, we develop a frameworkfor analyzing the SSE by means of the standard linear-response formalism 7which is amenable to the language of the magnetism community.11This allows us to describe the spin transport phenomena systematically, and it can be easilygeneralized to a situation including other degrees of freedom,e.g., conduction electrons and phonons, to describe a morecomplicated process in the case of metallic systems. 1 The plan of this paper is as follows. In Sec. II, we present a linear-response approach to the “local” spin injection bythermal magnons, in which the spin injection is driven bythe temperature difference between the ferromagnet and theattached nonmagnetic metal. Next, in Sec. IIIwe develop a linear-response theory of the “nonlocal” spin injection bythermal magnons, in which the spin injection is driven by thetemperature gradient inside the ferromagnet. As one can seebelow, this process can explain the SSE observed in YIG. 9 Finally, in Sec. IVwe summarize and discuss our results. II. “LOCAL” SPIN INJECTION BY THERMAL MAGNONS We start by brieﬂy reviewing the SSE experiment for YIG.9Figure 1shows the experimental setup where several Pt terminals are attached on top of a YIG ﬁlm in a static magneticﬁeldH 0ˆz(/greatermuchanisotropy ﬁeld) which aligns the localized magnetic moment along ˆ z. A temperature gradient ∇Tis applied along the zaxis, and it induces a spin voltage across the YIG/Pt interface. This spin voltage then injects a spin currentI sinto the Pt terminal (or ejects it from the Pt terminal). A part of the injected/ejected spin current Isis converted into a charge voltage through the so-called inverse-spin Hall effect:5 VISHE=/Theta1H(\|e\|Is)(ρ/w ), (1) where \|e\|,/Theta1H,ρ, andware the absolute value of electron charge, spin Hall angle, resistivity, and width of the Pt terminal(see Fig. 1), respectively. Hence, the observed charge voltage V ISHE is a measure of the injected/ejected spin current Is. To investigate the SSE observed in YIG, we consider a model shown in Fig. 2(a). While YIG is a ferrimagnet, we model it as a ferromagnet since we are interested inthe low-energy properties. The key point in our model isthat the temperature gradient is applied over the insulatingferromagnet, but there is locally no temperature difference 094410-1 1098-0121/2011/83(9)/094410(6) ©2011 American Physical SocietyADACHI, OHE, TAKAHASHI, AND MAEKAW A PHYSICAL REVIEW B 83, 094410 (2011) VV TΔV H0VISHE xzy~ 6 mmz w Pt YIG FIG. 1. Experimental setup for observing the SSE.9Inset: Schematics of the spatial proﬁle of the observed voltage. between the ferromagnet and the attached nonmagnetic metals, i.e.,TN1=TF1=T1,TN2=TF2=T2, and TN3=TF3=T3. We assume that each domain is initially in thermal equilibriumwithout interactions with the neighboring domains, and thencalculate the nonequilibrium dynamics after we switch on theinteractions. Note that this procedure is essentially equivalentto that used by Luttinger 12to realize the initial condition mentioned above. Let us consider ﬁrst the low-energy excitations in the ferromagnet. In the following, we focus on the spin-waveregion where the magnetization M(r) ﬂuctuates only weakly around the ground state value M sˆzwith the saturation magnetization Ms, and we set M/Ms=(1−m2/2)ˆz+m to separate the small ﬂuctuation part m(⊥ˆz)f r o mt h e ground-state value. Then, the low-energy excitations of M are described by boson (magnon) operators a† qandaqthrough the relations13m+ q=√1/S0a† −qandm− q=√1/S0aqwhere m±≡(mx±imy)/√ 2,S0is the size of localized spins, and m(r,t)=NF−1/2/summationtext qmq(t)eiq·rwithNFbeing the number of localized spins in the ferromagnet. Consistent with thisboson mapping, the magnetization dynamics is described by Jex Jsd TN1Jex Jsd 1z2z3zT2T3 T1 z(b)(a) T(z)N3 N1 N2 2F1F3F = =TT 1 F1=F2 2 N2=T TT = =TN3 F3TT 3 FIG. 2. (a) System composed of ferromagnet ( F) and nonmag- netic metals ( N) divided into the three temperature domains of F1/N 1, F2/N 2,a n dF3/N 3with their local temperatures of T1,T2,a n dT3.( b ) Temperature proﬁle.the following action:14,15 SF=/integraldisplay Cdt/summationdisplay qm+ −q(t)[Xq(i∂t)]−1m− q(t), (2) where the integration is performed along the Keldysh contour C,16and the bare magnon propagator is given by ˇXq(ω)=/parenleftBigg XR q(ω), 0,XK q(ω) XAq(ω)/parenrightBigg (3) with the following equilibrium condition: XA q(ω)=/bracketleftbig XR q(ω)/bracketrightbig∗,XK q(ω)=2iImXR q(ω) coth/parenleftbigg¯hω 2kBT/parenrightbigg .(4) The retarded component of ˇXq(ω)i sg i v e nb y XR q(ω)= S−1 0(ω−/tildewideωq+iαω)−1where αis the Gilbert damping con- stant, and /tildewideωq=γH 0+ωqis the magnon frequency. Here, γis the gyromagnetic ratio and ωq=Dexq2, where Dex= 2S0Jexa2 Sis the spin-wave stiffness constant with Jexanda3 S being the exchange energy and the effective block spin volume. In the nonmagnetic metal, the dynamics of the spin density scan be described by the action17 SN=/integraldisplay Cdt/summationdisplay ks+ −k(t)[χk(i∂t)]−1s− k(t), (5) where s± k=(sx k±isy k)/2 is deﬁned by sk= NN−1/2/summationtext pc† p+kσcpwith σ,c† p=(c† p,↑,c† p,↓), and NN being the Pauli matrices, the electron creation operator for spin projection ↑and↓, and the number of atoms in the nonmagnetic metal. The equilibrium spin-density propagatoris given by ˇχ k(ω)=/parenleftbiggχR k(ω), 0,χK k(ω) χA k(ω)/parenrightbigg (6) with the following equilibrium condition: χA k(ω)=/bracketleftbig χR k(ω)/bracketrightbig∗,χK k(ω)=2iImχR k(ω) coth/parenleftbigg¯hω 2kBT/parenrightbigg .(7) The retarded part of ˇ χis given by18χR k(ω)=χN(1+λ2 Nk2− iωτ sf)−1withχN,λN, andτsfbeing the paramagnetic suscepti- bility, spin diffusion length, and spin relaxation time, the formof which is consistent with the corresponding diffusive Blochequation [see Eq. ( 10) below]. Finally, the interaction between magnons and spin density at the interface is given by S F−N=/integraldisplay Cdt/summationdisplay k,qS0Jk−q sd√NFNNm−q(t)·sk(t), (8) where Jk−q sd is the Fourier transform of Jsd(r)=Jsdξ0(r) with Jsdbeing the s-dexchange interaction between conduction-electron spins and localized spins, and ξ0(r)=/summationtext r0∈N−Ninterface a3 Sδ(r−r0). It is instructive to point out that in the spin-wave region and in the classical limit with negligible quantum ﬂuctuations, asystem described by Eqs. ( 2), (5), and ( 8) is equivalent 15,19to a system described by the stochastic Landau-Lifshitz-Gilbertequation, ∂ tM=/bracketleftbigg γ(Heff+h)−Jsd ¯hs/bracketrightbigg ×M+α MsM×∂tM,(9) 094410-2LINEAR-RESPONSE THEORY OF SPIN SEEBECK EFFECT ... PHYSICAL REVIEW B 83, 094410 (2011) coupled with the Bloch equation,20 ∂ts=/parenleftbig DN∇2−τ−1 sf/parenrightbig δs+Jsd ¯hMsM×s+l, (10) where Heff=H0ˆz+(Dex/γ)∇2(M/Ms),DN=λ2 N/τsfis the diffusion constant, and δs(r)=s(r)−s0ξ0(r)M(r)/Msis the spin accumulation with the local equilibrium spin density s0= χNS0Jsd/¯h. The noise ﬁeld hrepresents thermal ﬂuctuations in Fwith/angbracketlefthi(r,t)/angbracketright=0 and/angbracketlefthi(r,t)hj(r/prime,t/prime)/angbracketright=2kBT(r)α γMsδijδ(r− r/prime)δ(t−t/prime),21while the noise source linNsatis- ﬁes/angbracketleftli(r,t)/angbracketright=0 and /angbracketleftli(r,t)lj(r/prime,t/prime)/angbracketright=2kBT(r)χNa3 τsfδijδ(r− r/prime)δ(t−t/prime)22with the lattice constant a, both of which are postulated by the ﬂuctuation-dissipation theorem. In this section we focus on the “local” spin injection from F1intoN1. The spin current induced in N1can be calculated from the linear response expression of the magnon-mediatedspin injection given in the Appendix [Eq. ( A4)]. Consider the process P 1shown in Fig. 3(a) where magnons travel around the ferromagnet F1without feeling the temperature difference between F1andF2. Using the standard rules of constructing the Feynman diagram in Keldysh space,16the corresponding interface Green’s function ˇCk,q(ω) for the correlation between the magnons in F1and the spin density in N1[Eq. ( A4)] can be written in the form ˇCk,q(ω)=Jk−q sdS0√NNNFˇχk(ω)ˇXq(ω), (11) where NNandNFare the number of lattice sites in N1 andF1. Substituting Eq. ( 11) into Eq. ( A4) and employing the equilibrium conditions [Eqs. ( 4) and ( 7)], we obtain the expression for the injected spin current: IN1 s=−4NintJ2 sdS2 0√ 2¯h2NNNF/summationdisplay q,k/integraldisplay ωImχR k(ω)ImXR q(ω) ×/bracketleftbigg coth/parenleftbigg¯hω 2kBTN1/parenrightbigg −coth/parenleftbigg¯hω 2kBTF1/parenrightbigg/bracketrightbigg ,(12) where we have introduced the shorthand notation/integraltext ω=/integraltext∞ −∞dω 2π, andNintis the number of localized spins at the N1-F1 interface playing a role of the number of channels. The ω IsP3 P1P′2 P′1 3z2z1z 1NIsIsN2IsN3(a) (b)P2 TN1 zN3 N1 N2 2F1F3F = =TT 1 =F2 2 N2=T T = =TN3 F3 F1 TT T 3 FIG. 3. (a) Feynman diagrams expressing the spin current in- jected from the ferromagnet ( F) to the nonmagnetic metals ( N). The thin solid lines with arrows (bold lines without arrows) represent electron propagators (magnon propagators). (b) Spatial proﬁle of the calculated spin current.integration can be performed by picking up only magnon poles under the condition α¯h/tildewideωq/lessmuchkBTN1,kBTF1(always satisﬁed for YIG), giving/integraltext ωImχk(ω)ImXq(ω)[coth(¯hω 2kBT)]≈ −1 2Imχk(/tildewideωq)[coth(¯h/tildewideωq 2kBT)]. By making the classical approxi- mation coth(¯h/tildewideωq 2kBT)≈2kBT ¯h/tildewideωq, we obtain IN1 s=NintJ2 sdS0χNτsf 2√ 2π4¯h3(λN/a)3ϒ1kB(TN1−TF1), (13) where ϒ1=/integraltext1 0dx/integraltext1 0dyx2√y [(1+x2)2+y2(2JexS0τsf/¯h)2]with the dimen- sionless variables x=kλNandy=¯hωq/(2JexS0), and we used the relation N−1 F/summationtext q=(2π)−2/integraltext√ydy. III. MAGNON-MEDIATED SPIN SEEBECK EFFECT Equation ( 13) means that, through the “local” process P1shown in Fig. 3(a), the spin current is notinjected into the nonmagnetic metal N1when F1andN1have the same temperature. That is, the “local” process cannot explain theexperiment 9where no temperature difference exists between the YIG ﬁlm and the attached Pt ﬁlm. A way to account for theexperiment within the “local” picture is to invoke a differencebetween the phonon temperature and magnon temperature. 23 In this paper, on the other hand, we take a different route andconsider the effect of temperature gradient within the YIG ﬁlm on the spin injection into the Pt terminal. The basic idea of our approach is as follows. The above result [Eq. ( 13)] that the injected spin current vanishes when T F1=TN1originates from the equilibrium condition of the magnon propagator [Eq. ( 4)]. When magnons deviate from local thermal equilibrium by allowing the magnons to feelthe temperature gradient inside the ferromagnet, the magnonpropagator cannot be written in the equilibrium form, andit generates a nontrivial contribution to the thermal spininjection. The relevant “nonlocal” process P /prime 1is shown in Fig. 3(a) in which magnons feel the temperature difference between F1andF2. The interaction between F1andF2is described by the action SF−F=/integraldisplay Cdt/summationdisplay q,q/prime2Jq−q/prime exS2 0 NFmq(t)·m−q/prime(t), (14) where Jq−q/prime ex is the Fourier transform of Jex(r)=Jexξ1(r) with ξ1(r)=/summationtext r0∈F−Finterface a3 Sδ(r−r0). We now regard the whole of the magnon lines appearing in the process P/prime 1as a single magnon propagator δˇXq(ω), namely, δˇXq(ω)=1 N2 F/summationdisplay q/prime/vextendsingle/vextendsingleJq−q/prime ex/vextendsingle/vextendsingle2ˇXq(ω)ˇXq/prime(ω)ˇXq(ω).(15) Then the propagator is decomposed into the local-equilibrium part and nonequilibrium part as24 δˇXq(ω)=δˇXl−eq q(ω)+δˇXn−eq q(ω), (16) where δˇXl−eq q=/parenleftBigg δXl−eq,R q, 0,δXl−eq,K q δXl−eq,A q/parenrightBigg (17) 094410-3ADACHI, OHE, TAKAHASHI, AND MAEKAW A PHYSICAL REVIEW B 83, 094410 (2011) is the local-equilibrium propagator satisfying the local- equilibrium condition, i.e., δXl−eq,A q=[δXl−eq,R q ]∗and δXl−eq,K q =[δXl−eq,R q−δXl−eq,A q ] coth(¯hω 2kBT) with δXl−eq,R q (ω)=1 N2 F/summationdisplay q/prime/vextendsingle/vextendsingleJq−q/prime ex/vextendsingle/vextendsingle2/bracketleftbig XR q(ω)/bracketrightbig2XR q/prime(ω), (18) while δˇXn−eq q=/parenleftBigg 0, 0,δXn−eq,K q 0/parenrightBigg (19) is the nonequilibrium propagator with δXn−eq,K q (ω) given by δXn−eq,K q (ω)=/summationdisplay q/prime/vextendsingle/vextendsingle2Jq−q/prime exS0/vextendsingle/vextendsingle2 N2 F/bracketleftbig XR q/prime(ω)−XA q/prime(ω)/bracketrightbig/vextendsingle/vextendsingleXR q(ω)/vextendsingle/vextendsingle2 ×/bracketleftbigg coth/parenleftbigg¯hω 2kBTF2/parenrightbigg −coth/parenleftbigg¯hω 2kBTF1/parenrightbigg/bracketrightbigg . (20) Note that the local equilibrium propagator [Eq. ( 17)] does not contribute to the “nonlocal” spin injection. When we substitute Eq. ( 16) into Eq. ( A4) and use Eq. ( 11) with ˇXq(ω) being replaced by δˇXq(ω), we obtain the following expression for the magnon-mediated thermal spin injection: IN1 s=−4J2 sdS0(2JexS0)2NintN/prime int√ 2¯h2N3 FNN/summationdisplay q,q/prime,k/integraldisplay ωImχR k(ω)/vextendsingle/vextendsingleXR q(ω)/vextendsingle/vextendsingle2 ×ImXR q/prime(ω)/bracketleftbigg coth/parenleftbigg¯hω 2kBT1/parenrightbigg −coth/parenleftbigg¯hω 2kBT2/parenrightbigg/bracketrightbigg , (21) where N/prime intis the number of localized spins at the F1-F2interface, and we used TNi=TFi=Ti(i=1,2). Theωintegration can be performed as before, giving/integraltext ωImχR k(ω)\|XR q(ω)\|2ImXR q/prime(ω)[coth(¯hω 2kBT1)−coth(¯hω 2kBT2)]≈ −π 2α/tildewideωqδ(ωq−ωq/prime)ImχR k(/tildewideωq)[coth(¯h/tildewideωq 2kBT1)−coth(¯h/tildewideωq 2kBT2)], which suggests that the magnon modes with different q’s do not interfere with each other. With the classical approximation coth(¯h/tildewideωq 2kBT)≈2kBT ¯h/tildewideωq, we obtain IN1 s=Nint/parenleftbig J2 sdS0/parenrightbig χNτsf(a/λN)3 8√ 2π5¯h3α(/Lambda1/a S)ϒ2kBδT, (22) where δT=T1−T2,/Lambda1is the size of F1along the temperature gradient, and ϒ2=/integraltext1 0dx/integraltext1 0dyy2 [(1+x2)2+y2(2S0Jexτsf/¯h)2]which is approximated as ϒ2≈0.1426 ( ϒ2≈0.337¯h/2S0Jexτsf)f o r 2S0Jexτsf/¯h/lessorsimilar1 (for 2 S0Jexτsf/¯h/greatermuch1). The spin current IN3sinjected into the right terminal N3can be calculated in the same manner by considering the processP 3, which gives IN3s=−IN1sfrom the relation T1−T2= −(T3−T2). The spin current IN2sinjected into the middle terminal N2vanishes because the two relevant processes ( P2 andP/prime 2) cancel out. Therefore, we obtain the spatial proﬁle of the injected spin current as shown in Fig. 3(b). Note that the effect of the spatial dependence of magnetization M[T(r)] through the local temperature T(r) is already taken into account in our treatment because the temperature dependenceofMin the magnon region is automatically described by the number of thermal magnons discussed in this paper. For an order of magnitude estimation, we compare Eq. ( 22) with the experiment.9By using /Theta1H≈0.0037,25,26 ρ=15.6×10−8/Omega1m,w=0.1m m , λN≈7n m , τsf≈1p s , a=2˚A,aS=12.3˚A,S0=16,α≈5×10−5,8χN=1× 10−6cm3/g,27andNint=0.1×4m m2/a2 S,t h es-dexchange coupling extracted from the previous ferromagnetic reso-nance experiment 8(Jsd≈10 meV) can account for the spin Seebeck voltage VISHE/δT≈0.1μV/K observed at room temperature. Finally, we comment on the issue of length scales associated with the SSE. In the original SSE experiment for a metallic ferromagnet,1the signal maintained over several millimeters was a big surprise because the spin diffusion length for thatsystem is much shorter than a millimeter. Concerning themagnon-mediated SSE in an insulating magnet 9which we have discussed, it is of crucial importance to recognize thatthe length scale relevant to the SSE is related to magnondensity ﬂuctuations and is given by longitudinal ﬂuctuations of magnons, while the magnon mean free path is related tomagnon dephasing and is given by transverse ﬂuctuations of magnons. 28It was shown by Mori and Kawasaki30that these two length scales do not coincide with each other since theyobey quite different dynamics, and it was demonstrated that ina certain situation the length scale of magnon density ﬂuctua-tions (which is relevant to the SSE as well) is much longer thanthe magnon mean free path [see Eq. (6.33) in Ref. 30where the length scale of long-wavelength magnon density ﬂuctuationsis inﬁnitely long]. 31The notion of these two different length scales is the key to understanding the length scales observedin the SSE experiment in an insulating magnet. 9 IV . CONCLUSION We have developed a theory of the magnon-mediated spin Seebeck effect in terms of the canonical frameworkof describing transport phenomena, i.e., the linear-responsetheory, and shown that it provides us with a qualitativeas well as quantitative understanding of the spin Seebeckeffect observed in a prototypical magnet, yttrium iron garnet. 9 Because the carriers of spin current in this scenario aremagnons, we can obtain a bigger signal for a magnetic materialwith a lower magnon damping [see Eq. ( 22) where the injected spin current is inversely proportional to the Gilbert dampingconstant α]. An advantage of our linear-response formulation is that it can be easily generalized to a situation includingdegrees of freedom other than magnons, e.g., phonons andconduction electrons, to describe a more complicated processin the case of metallic 1and semiconducting systems,33and a calculation taking account of the effect of nonequilibriumphonons will be reported in a future publication. 34A numerical approach to the SSE is also developed in Ref. 35. We believe that the present approach stimulates further research on thespin Seebeck effect. ACKNOWLEDGMENTS We are grateful to E. Saitoh, K. Uchida, G. E. W. Bauer, and J. Ieda for helpful discussions. This work was ﬁnancially 094410-4LINEAR-RESPONSE THEORY OF SPIN SEEBECK EFFECT ... PHYSICAL REVIEW B 83, 094410 (2011) supported by a Grant-in-Aid for Scientiﬁc Research on Priority Areas (No. 19048009) and a Grant-in-Aid for Young Scientists(No. 22740210) from MEXT, Japan. APPENDIX: LINEAR-RESPONSE EXPRESSION OF MAGNON-INDUCED SPIN INJECTION The Gaussian action for conduction electrons in the nonmagnetic metal Ni(i=1,2,3) is given by SN=/integraldisplay Cdt/summationdisplay p,p/primec† p(t){i∂t−(/epsilon1pδp,p/prime +Up−p/prime[1+iηsoσ·(p×p/prime)])}cp/prime(t),(A1) where c† p=(c† p,↑,c† p,↓) is the electron creation operator for spin projection ↑and↓,Up−p/primeis the Fourier transform of the impurity potential Uimp/summationtext r0∈impurities δ(r−r0), and ηso measures the strength of the spin-orbit interaction.36 At the ferromagnet/nonmagnetic-metal interface, the mag- netic interaction between conduction-electron spin densityand localized spin is described by the s-dinteraction [Eq. ( 8)]. The spin current induced in the nonmagnetic metal N 1can be calculated as the rate of change of the spin accumulation in N1, i.e., IN1s(t)≡/summationtext r∈N1/angbracketleft∂tsz(r,t)/angbracketright= /angbracketleft∂t/tildewidesz k0(t)/angbracketrightk0→0, where /angbracketleft ···/angbracketright means the statistical average at a given time t, and/tildewidesk=√NNskwith sbeing deﬁned below Eq. ( 5).The Heisenberg equation of motion for /tildewideszk0gives ∂t/tildewidesz k0=/summationdisplay q,kiJk−q sdS0√2NFNN¯h/parenleftbig m+ −q/bracketleftbig s− k,sz k0/bracketrightbig +m− −q[s+ k,sz k0/bracketrightbig/parenrightbig =i/summationdisplay q,k2Jk−q sdS0√2NFNN¯h/parenleftbig m+ −qs− k+k0−m− −qs+ k+k0/parenrightbig ,(A2) w h e r ew eh a v eu s e dt h er e l a t i o n[ /tildewidesz k,/tildewides± k/prime]=± 2/tildewides± k+k/prime, and neglected a small correction term arising from the spin-orbitinteraction assuming that the spin-orbit interaction is weakenough at the neighborhoods of the interface. Then, thestatistical average of the above quantity gives the followingspin current: I N1 s(t)=/summationdisplay q,k−4Jk−q sdS0√2NFNN¯hReC< k,q(t,t), (A3) where C< k,q(t,t/prime)=−i/angbracketleftm+ −q(t/prime)s− k(t)/angbracketrightis the interface Green’s function. In the steady state, the Green’s function C< k,q(t,t/prime) depends only on the time difference t−t/primeasC< k,q(t−t/prime)=/integraltext∞ −∞dω 2πC< q,k(ω)e−iω(t−t/prime). Adopting the representation37ˇC= (CR,CK 0,CA) and using C<=1 2[CK−CR+CA], we ﬁnally obtain IN1 s=/summationdisplay q,k−2Jk−q sdS0√2NFNN¯h/integraldisplay∞ −∞dω 2πReCK k,q(ω)( A 4 ) for the spin current IN1sin a steady state. As in the case of tunneling charge current driven by a voltage difference,38the spin current IN1scan be calculated systematically. hiroto.adachi@gmail.com 1K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature (London) 455, 778 (2008). 2I.˘Zuti´c, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004). 3Spin Caloritronics ,e d i t e db yG .E .W .B a u e r ,A .H .M a c D o n a l d , and S. Maekawa, special issue of Solid State Commun. 150, 459 (2010). 4J. C. Slonczewski, Phys. Rev. B 82, 054403 (2010). 5E. Saitoh, M. Ueda, H. Miyajima, and G. 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PhysRevB.73.054413.pdf	Magnetic reversal in nanoscopic ferromagnetic rings Kirsten Martens and D. L. Stein† Department of Physics, University of Arizona, Tucson, Arizona 85721, USA A. D. Kent Department of Physics, New York University, New York, New York 10003, USA /H20849Received 29 September 2005; revised manuscript received 6 December 2005; published 10 February 2006 /H20850 We present a theory of magnetization reversal due to thermal ﬂuctuations in thin submicron-scale rings composed of soft magnetic materials. The magnetization in such geometries is more stable against reversalthan that in thin needles and other geometries, where sharp ends or edges can initiate nucleation of a reversedstate. The two-dimensional ring geometry also allows us to evaluate the effects of nonlocal magnetostaticforces. We ﬁnd a “phase transition,” which should be experimentally observable, between an Arrhenius and anon-Arrhenius activation regime as magnetic ﬁeld is varied in a ring of ﬁxed size. DOI: 10.1103/PhysRevB.73.054413 PACS number /H20849s/H20850: 05.40. /H11002a, 02.50.Ey, 75.60.Jk I. INTRODUCTION The dynamics of magnetization reversal in submicron- sized, single-domain particles and thin ﬁlms has attractedmuch attention given its importance in information storageand other magnetoelectronic applications. The problem canbe approached by stochastic methods: in the classical regime/H20849typically at temperatures above /H110111K /H20850the magnetization dynamics is governed by the Landau-Lifschitz-Gilbert equation 1perturbed by weak thermal noise. The classical Néel-Brown theory2,3of thermally induced reversal assumed a spatially uniform magnetization and uniaxial anisotropy.Experimental conﬁrmation of this theory has been providedfor certain simple single-domain systems /H2084915–30-nm Ni, Co, and Dy nanoparticles /H20850. 4 There, nevertheless, remain fundamental open questions, especially when there is spatial variation of the magnetiza-tion density. 5–8While in small particles that are spherical or nearly so, as in Ref. 4, the Néel-Brown theory appears towork reasonably well, it appears to break down for elongatedparticles, thin ﬁlms, and other geometries, which exhibit farlower coercivities than predicted. 9 Braun5made an initial step by studying the effects of spatial variation of magnetization density on magnetic rever-sal in an inﬁnitely long cylindrical magnet. However,Aharoni 10pointed out that the energy functional employed neglected important nonlocal magnetostatic energy contribu-tions, invalidating the result. Further, for submicron-scalemagnets with large aspect ratio, ﬁnite system effects arelikely to play an important role; for example, simulations 6,11 indicate that magnetization reversal in cylindrical-shapedparticles proceeds via propagation and coalescence of mag-netic “end caps,” nucleated at the cylinder ends. Both ofthese issues are addressed in Ref. 12. Here we consider a geometry that avoids these difﬁcul- ties: an effectively two-dimensional annulus. Such systemshave recently received increasing attention. 13,14They are typically constructed of soft magnetic materials /H20851quality fac- torQ/H11011O/H2084910−2/H20850/H20852, such as Fe, fcc Co, or permalloy, have radii of order /H20849102–103/H20850nm and thicknesses of order 10 nm or less.Our interest in these systems is twofold. The ﬁrst is tech- nological: because the magnetic bending length is muchsmaller than the typical system size, there are two oppositelypolarized stable states, each with magnetization vectorspointing everywhere along the circumferential direction;they are degenerate in the absence of an external magnetic ﬁeld. But a current running along the zˆdirection through the center, with zˆthe direction normal to the annulus plane, gen- erates a circumferential magnetic ﬁeld breaking the degen-eracy. By switching the direction of the current, the relativestability of the two states is switched. /H20849A slightly different method, but with similar wire dimensions and current mag-nitudes, was used in Ref. 13 /H20850. The utility of such a system as an information storage device depends on the magnetizationstate being relatively long-lived against thermal ﬂuctuations,even at relatively high temperatures. Unlike the cylindricalparticle, the micromagnetic ring has no ends where nucle-ation is easily initated, making its magnetic state more stableagainst thermally induced reversal. The second is physical: by developing a theory for ther- mally induced reversal that can be solved analytically, we are able to extract a number of interesting qualitative featuresthat would be more difﬁcult to uncover numerically andwhich should apply also to more complicated situations.While several important quantitative features require a nu-merical treatment, we show below that our most importantqualitative ﬁndings are robust /H20849see in particular the Discus- sion section /H20850. One of our central predictions is that a type of phase transition occurs in the thermally induced reversal rate, andmore importantly, that it can be realized experimentally. Thepossibility of such a transition in classical stochastic ﬁeldtheories /H20849which as we show below includes the physical problem of interest here /H20850was noted in Ref. 15, as system size was varied in a symmetric Ginzburg-Landau double-well /H92784 potential. It was further shown in Ref. 16 to apply moregenerally to asymmetric systems as well. In the present case,the transition depends on two parameters: the system size and the strength of the applied magnetic ﬁeld. Although theformer cannot be continuously varied, the latter can, facili-tating experimental tests of the predicted transition. In par-PHYSICAL REVIEW B 73, 054413 /H208492006 /H20850 1098-0121/2006/73 /H208495/H20850/054413 /H2084910/H20850/$23.00 ©2006 The American Physical Society 054413-1ticular, we will show that as magnetic ﬁeld varies for a ring of ﬁxed size, there should be a transition from a regimewhere activation is Arrhenius to one where it is non-Arrhenius. A preliminary account of this work has appearedin Ref. 17. II. MODEL We consider an annulus of thickness t, inner radius R1, and outer radius R2. We conﬁne our attention to rings satis- fying t/H11270/H9004R/H11270R, where /H9004R=R2−R1andR=/H20849R1+R2/H20850/2. A current run through the center leads to an applied ﬁeld Heat Rin the circumferential direction /H9258ˆ; the small variation /H11011O/H20849/H9004R/R/H20850of ﬁeld strength with radius can be ignored. As will be discussed below, magnetostatic forces produce strong anisotropies, forcing the magnetization vector to lie in theplane and preferentially oriented parallel to the inner andouter circumferences. We may therefore consider magnetiza- tion conﬁgurations that vary only along the /H9258ˆdirection. Suppose now that the system is initially in its metastable state; i.e., with magnetization vector M=−M0/H9258ˆ. We are in- terested in determining the mean rate for thermal ﬂuctuationsto reverse the magnetization to its stable direction. We con-sider temperatures above 1 K, where classical thermal acti-vation can be expected to apply. The magnetization dynam-ics are then governed by the Landau-Lifshitz-Gilbertequation 1 /H11509tM=−/H9253/H20851M/H11003Heff/H20852+/H20849/H9251/M0/H20850/H20851M/H11003/H11509tM/H20852, /H208491/H20850 where M0is the /H20849ﬁxed /H20850magnitude of M,/H9251is the damping constant, and /H9253/H110220 is the gyromagnetic ratio. The effective ﬁeld Heff=−/H9254E//H9254Mis the variational derivative of the total energy E, which /H20849with free space permeability /H92620=1/H20850is11,18 E/H20851M/H20849x/H20850/H20852=/H92612/H20885 /H9024d3x/H20841/H11612M/H208412+1 2/H20885 R3d3x/H20841/H11612U/H208412−/H20885 /H9024d3xHe·M, /H208492/H20850 where /H9024is the region occupied by the ferromagnet, /H9261is the exchange length, /H20841/H11612M/H208412=/H20849/H11612Mx/H208502+/H20849/H11612My/H208502+/H20849/H11612Mz/H208502, and U /H20849deﬁned over all space /H20850satisﬁes /H11612·/H20849/H11612U+M/H20850=0. The ﬁrst term on the right-hand side of Eq. /H208492/H20850is the bending energy, the second is the magnetostatic energy, and the last is theZeeman energy. Crystalline anisotropy terms are neglected,given their negligibly small contribution; they can be easilyincluded but will at most result in a small modiﬁcation of themuch larger shape anisotropies, to be discussed below. III. ENERGY SCALING AND THE MAGNETOSTATIC TERM The presence of the nonlocal magnetostatic term compli- cates analysis. However, the quasi-two-dimensional nature ofthe problem allows a signiﬁcant simpliﬁcation, as shown byKohn and Slastikov 19in an asymptotic scaling analysis that applies when the aspect ratio k=t/Rand the normalized ex- change length l=/H9261/Rare both small, and l2/H11011k/H20841lnk/H20841. These constraints restrict the range of ring geometries to which ouranalysis applies. Before discussing the Kohn-Slastikov /H20849KS /H20850 result, we recast the energy in dimensionless form, lettingX=x/Rand similarly for all other lengths, and let h =H e//H208492M0l2/H20850. Then, integrating along the direction normal to the plane, we have for the bending plus Zeeman energy contribution Eb+Ez M02R3=kl2/H20885 /H9275d2X/H20851/H20849/H11612Xm/H208502−2h·m/H20852, /H208493/H20850 where m=M/M0is the normalized magnetization and /H9275rep- resents the two-dimensional /H208492D /H20850surface with boundary /H11509/H9275. Before analyzing these terms further, we examine the magnetostatic energy contribution. The analysis of Ref. 19showed that this asymptotically separates into local bulk andsurface terms: E mag M02R3=1 2k/H20885 /H9275d2Xmz2+/H208491/4/H9266/H20850k2/H20841lnk/H20841/H20885 /H11509/H9275dsX/H20849m·rˆ/H208502 +1 2k2/H20885 /H9275d2X/H20841/H11612·m/H20841H−1/22, /H208494/H20850 where sXis dimensionless arc length along the boundary, and the ﬁnal integral is the squared H−1/2Sobolev norm of /H11612·m.20With current technology, the orders of magnitude k /H1101110−2andl2/H11011k/H20841lnk/H20841/H1101110−2–10−1are just attainable. Then the ﬁrst term of Eq. /H208494/H20850is larger than the others by roughly two orders of magnitude, forcing mz=0, and therefore in this topology we can ignore ﬂuctuations of mout of the plane /H20849we will discuss this further in Sec. III A /H20850. The second term, like the ﬁrst, is a /H20849local /H20850magnetostatic surface /H20849or shape anisotropy /H20850term. The third term represents a nonlocal magnetostatic bulk energy. When nonzero, thisterm will be roughly an order of magnitude smaller than theothers /H20849see Sec. V B /H20850, and so to a ﬁrst approximation 21it can be dropped. This will result, for some values of ring size andexternal ﬁeld, in an error of up to 10% in the computation ofthe action, so we can hope at best for reasonably good quan-titative predictions of the logarithm of the escape rate. Asnoted in the Discussion section, however, the importantqualitative features uncovered by our analysis should remainunaffected. We use a locally varying coordinate system where the angle /H9278/H20849s/H11032/H20850measures the deviation of the local magnetiza- tion vector from the local applied ﬁeld direction; i.e., /H9278=0 indicates that the local magnetization is parallel to the localﬁeld, /H9278=/H9266indicates that it is antiparallel, and so on. The parameter s/H11032=R/H9258is the arc length along the circumference. The geometry and variables used are displayed in Fig. 1. The normalized magnetization vector can therefore be written, in cylindrical coordinates, as m=/H20849mr,m/H9258,mz/H20850 =/H20849sin/H9278,cos/H9278,0/H20850. After integrating out the radial coordinate the bending plus Zeeman energy becomesKIRSTEN MARTENS, D. L. STEIN, AND A. D. KENT PHYSICAL REVIEW B 73, 054413 /H208492006 /H20850 054413-2Eb+Ez M02R3=kl2/H20877/H20873lnR2 R1/H20874/H20885 02/H9266 d/H9258/H208751+/H20873/H11509mr /H11509/H9258/H208742 +/H20873/H11509m/H9258 /H11509/H9258/H208742 +2/H20873mr/H11509m/H9258 /H11509/H9258−m/H9258/H11509mr /H11509/H9258/H20874/H20876−/H9004R R/H20885 02/H9266 d/H92582h·m/H20878. /H208495/H20850 Themr/H20849/H11509m/H9258//H11509/H9258/H20850−m/H9258/H20849/H11509mr//H11509/H9258/H20850term is a “winding number” of /H9278with respect to the local direction; it gives zero in all conﬁgurations considered here, but would give a nonzero contribution for uniform magnetizations, e.g., m=xˆ. For ﬁxed M0, it does not contribute to the magnetization equa- tions of motion given by Eq. /H208491/H20850. Finally, subtracting out constant terms and the ﬁrst deriva- tive term /H20849which gives zero contribution /H20850, noting that the boundary integral occurs over both inner and outer radii, andrescaling lengths again gives E=/H20885 0/H5129/2 ds/H20875/H20873/H11509/H9278 /H11509s/H208742 + sin2/H9278−2hcos/H9278/H20876, /H208496/H20850 where E=E/E0=E//H208512M02R2/H9004R/H20881ckl2/H20852,s=/H9258/H20881c,/H5129=2/H9266/H20881c,h =/H20841He/H20841//H208492M0l2c/H20850, and c=/H208491/2/H9266/H20850/H20849k/H20841lnk/H20841/l2/H20850/H20849R//H9004R/H20850.I nd e - riving Eq. /H208496/H20850we used the fact that ln /H20849R2/R1/H20850=/H9004R/R +O/H20851/H20849/H9004R/R/H208502/H20852. The error is negligible for the geometries con- sidered here; for example, with the ring parameters used in Fig. 7, /H9004R/R=1/5 and ln /H20849R2/R1/H20850=0.200 67. The parameter c/H208490/H11021c/H11021/H11009 /H20850depends on the ring size and material proper- ties; it represents the ratio of the anisotropy energy scale to the bending energy scale, and determines the width of aBloch wall. A. Energetics and topology The scaling results of the previous section are useful in- sofar as they provide results on how different contributionsto the energy scale in the thin-ﬁlm limit. Their effective ap- plication in a physical situation must also take into accountthe geometry, and in our case, the topology of the ferromag-netic particle under study. Consideration of both of theseaspects provides a guide for considering what types of mag-netization conﬁgurations might be relevant in different thin-ﬁlm geometries. As one example, we consider the ﬂat disk topology stud- ied by Shinjo et al. 22/H20849corresponding to our geometry with R1=0/H20850. They studied magnetization conﬁgurations in permal- loy disks of thickness 50 nm and diameters ranging from 0.3to 1 /H9262m. Interestingly, they observed vortex structures, par- ticularly in the larger diameter samples. The surface termenergy in Eq. /H208494/H20850/H20849which leads to the shape anistotropy /H20850is minimized by requiring the magnetization vector to remain tangential to the surface /H20849i.e., in the ± /H9258ˆdirection /H20850. But given the topology of the samples used here, this forces the interiormagnetization to do one of two things: either the magnetiza-tion magnitude goes to zero at the center, or the magnitude stays mostly constant but then the out-of-plane magnetiza-tion component m z/HS110050 in some interior region. Either choice costs energy, but /H20849when considered over the same region /H20850the ﬁrst costs more than the second. Given that there must be an out-of-plane magnetization component in the disk topology studied by Shinjo et al. ,22the KS analysis can determine the approximate length scale overwhich this occurs. Comparing the ﬁrst term in Eq. /H208494/H20850with the bending term /H208495/H20850, we estimate that their respective ener- gies are of the same order when the region where m z/HS110050i s roughly of the order of an exchange length, i.e., 10–20 nm.This appears to be exactly what is observed. /H20849Shinjo et al. 22 do not provide an estimate for the width of this region, not- ing only that they observe a contrast “spot” at the center ofeach disk that corresponds to out-of-plane magnetization; seeFig. 2 of their paper. /H20850Note that if the magnetization were to be out of plane in a region much larger than this, Eq. /H208494/H20850 predicts a prohibitively large energy cost. /H20849Simulation results consistent with these conclusions appear in Ref. 23. /H20850 In our ring topology, however, an “outer vortex” conﬁgu- ration /H20849i.e., magnetization circumferential at the outer bound- ary /H20850does not require an out-of-plane magnetization any- where. We can therefore ignore conﬁgurations with m z/HS110050, any of which are likely to have energies larger than the con-ﬁgurations considered here. IV. TRANSITION IN ACTIVATION BEHAVIOR The reversal rate /H9003due to thermal ﬂuctuations at temperature Tis given by the Kramers formula /H9003 /H11011/H90030exp /H20849−/H9004W/kBT/H20850, where the activation barrier /H9004W/H11271kBT is simply the energy difference between the /H20849meta /H20850stable and “saddle” states. The latter is the state of lowest energy with asingle negative eigenvalue /H20849and corresponding unstable di- rection /H20850of the linearized zero-noise dynamics. Equivalently, it is the conﬁguration of highest energy along the system’soptimal escape path. 24The rate prefactor /H90030is determined by ﬂuctuations about this optimal path, and its evaluation willbe presented in Sec. V C. FIG. 1. Ferromagnetic annulus viewed from above, showing coordinates used in text.MAGNETIC REVERSAL IN NANOSCOPIC … PHYSICAL REVIEW B 73, 054413 /H208492006 /H20850 054413-3Stable, unstable, and saddle states are all time- independent solutions of the Landau-Lifshitz-Gilbert /H20849LLG /H20850 equations. For ﬁxed M0, Eq. /H208491/H20850and the variational equation Heff=−/H9254E//H9254Myield a nonlinear differential equation that must be satisﬁed by any such time-independent solution: d2/H9278/ds2= sin/H9278cos/H9278+hsin/H9278. /H208497/H20850 There are three “constant” solutions /H20849i.e.,/H9278is independent of /H9258; these remain nonuniform conﬁgurations because mvaries with position /H20850for 0/H33355h/H110211: the stable state /H9278=0 /H20849m=/H9258ˆ/H20850; the metastable state /H9278=/H9266/H20849m=−/H9258ˆ/H20850, and a pair of degenerate un- stable states /H9278=cos−1/H20849−h/H20850, which constitute the saddle for a range of /H20849/H5129,h/H20850. The /H9278=0,/H9266solutions are degenerate when h=0, and the /H9278=/H9266solution becomes unstable at h=1. W e therefore conﬁne ourselves to ﬁelds in the range 0 /H33355h/H110211. We have also found a nonconstant /H20849“instanton” /H20850solution25 of Eq. /H208497/H20850, which we will see is the saddle for the remaining range of /H20849/H5129,h/H20850.I ti s /H9278/H20849s,s0,m/H20850= 2 cot−1/H20875dn/H20873s−s0 /H9254/H20841m/H20874sn/H20849R/H20841m/H20850 cn/H20849R/H20841m/H20850/H20876, /H208498/H20850 where dn /H20849·/H20841m/H20850,s n /H20849·/H20841m/H20850, and cn /H20849·/H20841m/H20850are the Jacobi elliptic functions with parameter m,0/H33355m/H333551;26s0is an arbitrary constant arising from the rotational symmetry of the prob-lem; and Rand /H9254are given by sn2/H20849R/H20841m/H20850=1 /m−h/2 − /H208491/2m/H20850/H20881m2h2+4 /H208491−m/H20850, /H208499/H20850 /H92542=m2 2− /H20851m+/H20881m2h2+4 /H208491−m/H20850/H20852. /H2084910/H20850 The period of the dn function equals 2 K/H20849m/H20850, the complete elliptic integral of the ﬁrst kind.26Accordingly, imposition of the periodic boundary condition yields a relation between /H5129 andm: /H5129=2K/H20849m/H20850/H9254. /H2084911/H20850 In the limit m→1, corresponding to /H5129→/H11009at ﬁxed h, Eqs. /H208498/H20850–/H2084911/H20850reduce to Braun’s solution.5In the limit m→0, dn/H20849x/H208410/H20850=1, and the instanton solution reduces to the constant state/H9278=cos−1/H20849−h/H20850.A tm=0, the critical length and ﬁeld are related by /H5129c=/H9266/H9254c=2/H9266//H208811−hc2. /H2084912/H20850 The solution /H208498/H20850–/H2084911/H20850corresponds to a pair of domain walls of width O/H20849/H9254/H20850. At ﬁxed h, the constant conﬁguration is the saddle for /H5129/H11021/H5129cand the instanton is the saddle for /H5129 /H11022/H5129c. This can be understood as follows: at ﬁxed ﬁeld, the bending energy becomes sufﬁciently large at small /H5129so that the constant state becomes energetically preferred. /H20849There is a second transition at even smaller /H5129, where the bending energy becomes so large that the magnetization lies along asingle Euclidean direction everywhere; we do not considersuch small length scales here. /H20850Conversely, at ﬁxed /H5129the constant conﬁguration is the saddle for h/H11022h c, and the non- constant for h/H11021hc./H20849However, when /H5129/H333552/H9266, the constant conﬁguration is the saddle for all h./H20850Here the Zeeman termdominates at sufﬁciently large ﬁeld, preferring a constant conﬁguration, while at smaller ﬁeld the shape anisotropy en-ergy dominates, preferring the instanton conﬁguration. The“phase boundary” /H20851Eq. /H2084912/H20850/H20852is the m=0 line in the /H20849/H5129,h/H20850 plane, and is shown in Fig. 2. We now compute the reversal rate in both regimes, and examine how it is affected by thetransition in the saddle state. V. REVERSAL RATE We turn now to a computation of the magnetization rever- sal rate /H9003due to thermal ﬂuctuations at temperature T.I n equilibrium, it is given, as noted in Sec. IV , by the Kramersformula /H9003/H11011/H9003 0exp /H20849−/H9004W/kBT/H20850.24We ﬁrst compute the acti- vation barrier /H9004Wfor each saddle conﬁguration. A. Activation energy As noted earlier, the exponential dependence of the mag- netization reversal rate on temperature is given by /H9004W, the energy difference between the saddle /H20849/H9278u/H20850and metastable /H20849/H9278s/H20850states /H20849the notation arises from the properties that the saddle is unstable along the longitudinal escape direction, while the metastable state is locally stable in all directions /H20850. With the latter given by /H9278s=/H9266, this is /H9004W/E0=E/H20851/H9278u/H20852−E/H20851/H9278s=/H9266/H20852 =/H20885 0/H5129/2 ds/H20875/H20873/H11509/H9278u /H11509s/H208742 + sin2/H20849/H9278u/H20850−4hcos2/H20849/H9278u/2/H20850/H20876. /H2084913/H20850 When the constant state /H9278=cos−1/H20849−h/H20850is the saddle con- ﬁguration, it easily seen that /H9004W=/H208491−h/H208502/H5129/2. When the nonconstant, or instanton, state is the saddle, the integral /H2084913/H20850 must be computed numerically. However, it can be analyti- cally computed in the m→0/H20851/H5129→/H5129c+/H20849h/H20850/H20852limit, where one ﬁnds/H9004W/H20849m→0/H20850→/H208491−h/H208502/H5129/2. So the energy /H20849and its ﬁrst derivative, which can also be computed /H20850is continuous at /H5129c/H20849h/H20850. Of course, the second derivative is discontinous there. FIG. 2. The phase boundary between the two activation regimes in the /H20849/H5129,h/H20850plane. In the shaded region the instanton state is the saddle conﬁguration; in the unshaded region, the constant state.KIRSTEN MARTENS, D. L. STEIN, AND A. D. KENT PHYSICAL REVIEW B 73, 054413 /H208492006 /H20850 054413-4Figure 3 shows the activation energy as a function of ring circumference for ﬁxed ﬁeld. The activation energy grows linearly with /H5129when the transition state is constant; it becomes almost ﬂat above /H5129cat ﬁxed h, because the width of the domain walls remains es- sentially constant /H20849cf. Fig. 4 of Ref. 16 /H20850. In Fig. 4 we show the activation energy dependence on hat ﬁxed /H5129, which is more relevant to experiment. B. Bulk magnetostatic energy contribution We can now go back and check whether the contribution of the bulk magnetostatic term is small compared to thebending energy. This requires an evaluation, or at least anestimate, of the H −1/2Sobolev norm of the divergence of the reduced /H20849i.e., in-plane /H20850magnetization. To do this, we need to introduce some additional notation. The L2/H20849/H9275/H20850norm of a quantity /H20849say, the gradient of the reduced magnetization /H20850is /H20648/H11612m/H20648L2/H20849/H9275/H20850=/H20875/H20885 /H9275d2x/H20849/H11612m/H208502/H208761/2 /H2084914/H20850 so that the dimensionless bending energy is simply kl2/H20648/H11612m/H20648L2/H20849/H9275/H208502. From here on we shall abbreviate L2/H20849/H9275/H20850toL2 for ease of notation.Formally, the H−1/2Sobolev norm of the magnetization divergence is given by20 /H20648/H11612·m/H20648H−1/2=/H20648/H20849/H11612/H20850−1/2/H20849/H11612·m/H20850/H20648L2. /H2084915/H20850 Its meaning becomes clearer through the use of Fourier transforms. Deﬁne the Fourier transform fˆ/H20849/H9264/H20850off/H20849x/H20850in the usual way: f/H20849x/H20850=/H20848d2/H9264fˆ/H20849/H9264/H20850ei/H9264·x. Then /H20648/H11612·m/H20648H−1/22=/H20885d2/H9264/H20873/H92641mˆ1+/H92642mˆ2 /H20841/H9264/H208411/2/H208742 =/H20885d2/H9264/H20849/H92641mˆ1+/H92642mˆ2/H208502 /H20841/H9264/H20841. /H2084916/H20850 It now follows in a straightforward fashion that /H20648/H11612·m/H20648H−1/2/H33355/H20648m/H20648H1/2/H33355/H20648m/H20648L21/2/H20648/H11612m/H20648L21/2/H33355/H20648/H11612m/H20648L21/2, /H2084917/H20850 where the last inequality follows because /H20648m/H20648L2=1. There- fore the bulk magnetostatic term is dominated by the bending energy. As noted earlier, the relevant scaling regime for the ap- proach presented here corresponds to /H51292/H11011k/H20841lnk/H20841 /H1101110−2–10−1. For the constant saddle conﬁguration the maxi- mal bulk magnetostatic energy arises when /H9278u=/H9266/2; this is also the maximum value of /H9278for the saddle, corresponding toh=0. For this conﬁguration, and with ring parameters used in Fig. 7, an upper bound for the magnetostatic bulk energy,computed using the inequalities /H2084917/H20850, is roughly an order of magnitude smaller than the bending energy. As hincreases from 0, and correspondingly /H9278u→/H9266, the magnetostatic bulk term decreases to zero. For the nonconstant, or instanton, saddle, the minimum value of /H5129is 2/H9266. At this length scale, an upper bound for the ratio of magnetostatic bulk energy to bending energy variesroughly from 0.05 to 0.1 as hvaries; smaller numbers are found as length scale increases, justifying the neglect of thisterm. Qualitatively, the instanton conﬁguration has nonzero di- vergence only over a region of O/H20849 /H9254/H20850, which remains smaller than O/H208491/H20850except close to m=1 /H20849/H5129→/H11009/H20850andh=1. The in- stanton conﬁguration contributes to the bending energy, how- ever, over the entire ring. It is therefore not surprising that, inthe appropriate scaling region, the instanton’s magnetostaticbulk energy is relatively small compared to its bending en-ergy. This is in contrast to instanton conﬁgurations in thecylinder; 5,10there, while the region contributing to a bulk divergence is O/H208491/H20850, the same region supplies the entire con- tribution to the bending energy as well, and so the magneto- static contribution cannot be neglected there. C. Rate prefactor The leading-order rate asymptotics are determined by the activation barrier /H9004W; the subdominant asymptotics appear as the rate prefactor /H90030. Because the magnitude of /H90030is controlled by the extent of ﬂuctuations about the optimalescape path, the prefactor is considerably more difﬁcult tocalculate than /H9004W. Although the reversal rate is only linearly dependent on the prefactor, as opposed to its exponential FIG. 3. Activation energy /H9004Wfor ﬁxed h=0.3 as /H5129varies. The dot indicates the transition from constant to instanton saddleconﬁguration. FIG. 4. Activation energy /H9004Wfor ﬁxed /H5129=7 as hvaries. The dot indicates the transition from instanton to constant saddleconﬁguration.MAGNETIC REVERSAL IN NANOSCOPIC … PHYSICAL REVIEW B 73, 054413 /H208492006 /H20850 054413-5dependence on the activation barrier, the rate can still be signiﬁcantly affected by /H90030, especially in the vicinity of a transition in saddle conﬁgurations /H20849cf. Fig. 5 /H20850. Moreover, an understanding of the prefactor is needed to study other quan-tities of physical interest, such as exit locationdistributions. 27 The prefactor computation procedure is summarized in Refs. 24 and 28 /H20849see also Refs. 29 and 30 /H20850. Consider a small perturbation /H9257about the metastable state, so that sufﬁciently close to it /H9278=/H9278s+/H9257. Then to leading order the time depen- dence of ﬂuctuations about the metastable state is given by /H9257˙=−/H9011s/H9257, where /H9011sis the linearized zero-noise dynamics at /H9278s. Similarly /H9011uis the linearized zero-noise dynamics around /H9278u. Then24,28 /H90030=1 2/H9266/H20881/H20879det/H9011s det/H9011u/H20879/H20841/H9261u,0/H20841, /H2084918/H20850 where /H9261u,0is the single negative eigenvalue of /H9011u. Its corre- sponding eigenvector is the direction along which the opti-mal escape trajectory approaches the transition state. In gen-eral, the determinants in the numerator and denominator ofEq. /H2084918/H20850can separately diverge: they are typically products of an inﬁnite number of eigenvalues with magnitude greaterthan 1. However, their ratio , which can be interpreted as the limit of a product of individual eigenvalue quotients, is ﬁnite. 1. Constant saddle When /H5129/H11021/H5129c/H20849h/H20850, or equivalently h/H11022hc/H20849/H5129/H20850, the saddle is the pair of constant conﬁgurations /H9278=cos−1/H20849−h/H20850, the prefac- tor can be determined by direct computation of eigenvalues of the stable and unstable states.16Linearizing around the stable state gives /H9257˙=−/H9011s/H9257=− /H20849−d2/dx2+1− h/H20850/H9257, /H2084919/H20850 and similarly /H9257˙=−/H9011u/H9257=− /H20849−d2/dx2+h2−1 /H20850/H9257 /H2084920/H20850 about the transition state. The spectrum of eigenvalues cor- responding to /H9011sis/H9261ns=4/H92662n2 /H51292+1− h,n=0 ,±1 ,±2 ... /H2084921/H20850 and the eigenvalues corresponding to /H9011uare /H9261nu=4/H92662n2 /H51292+h2−1 , n=0 ,±1 ,±2.... /H2084922/H20850 This simple linear stability analysis justiﬁes the claims that/H9278sis a stable state and /H9278ua saddle. Over the interval /H208510,/H5129c/H20850all eigenvalues of /H9011sare positive, while all but one of /H9011uare. Its single negative eigenvalue /H92610u=− /H208491−h2/H20850depends onhbut is independent of /H5129. Putting everything together, we ﬁnd /H90030−=/H92700−1/H208491−h2/H20850 /H9266sinh /H20849/H208811−h/H5129/2/H20850 sin/H20849/H208811−h2/H5129/2/H20850, /H2084923/H20850 where /H92700−1=/H9251/H9253E0/M0V/H208491+/H92512/H20850, with Vthe ring volume. The rate includes a factor of 2 because the system can escape over either of the saddles, which are rotationally equivalentwith respect to /H9278s=/H9266. The prefactor /H90030−diverges at /H5129c/H20849h/H20850, or conversely hc/H20849/H5129/H20850,a s expected /H20849cf. Fig. 5 /H20850; in this limit, /H90030−/H11011const/H11003/H20849/H5129c−/H5129/H20850−1as /H5129→/H5129c−at ﬁxed h,o ra s /H20849h−hc/H20850−1ash→hc+at ﬁxed /H5129. The prefactor in this region for ﬁxed /H5129ashvaries is plotted in Fig. 5. The divergence arises from the vanishing of the ei-genvalue of a pair of degenerate eigenfunctions at the criticalpoint. This indicates the appearance of a pair of soft modes,resulting in a transverse instability of the optimal escape tra-jectory as it approaches the saddle. The meaning and inter-pretation of the divergence is discussed in detail in Ref. 16.Near /H20849but not at /H20850the critical point the prefactor formulas hold, but in a vanishing range of Tas/H5129 cis approached. Exactly at /H5129cthe prefactor is ﬁnite but non-Arrhenius /H20851with a different exponent than that in Eq. /H2084930/H20850/H20852. Inclusion of higher- order ﬂuctuations31about the saddle can be used to compute the prefactor at criticality, and will be addressed elsewhere.We return to the prefactor divergence in Sec. VI. The independence of /H9003 0with respect to temperature leads to the well-known exponential temperature dependence ofthe overall reversal rate. By analogy with chemical kinetics,this exponential falloff of the rate is often called “Arrheniusbehavior.” 2. Nonconstant (instanton) saddle Computation of the determinant quotient in Eq. /H2084918/H20850is less straightforward when the transition state is nonconstant,i.e., when /H5129/H11022/H5129 c/H20849h/H20850or equivalently h/H11021hc/H20849/H5129/H20850. An additional complication follows from the translational degeneracy /H20849en- ergy invariance with respect to choice of s0/H20850of the noncon- stant state. This implies a soft collective mode in the linear-ized dynamical operator /H9011 uof Eq. /H2084918/H20850, resulting in a zero eigenvalue for all h/H11021hc/H20849/H5129/H20850/H20851not to be confused with the van- ishing of the lowest stable eigenvalue of the saddles exactly athc/H20849/H5129/H20850/H20852. To proceed, we use the McKane-Tarlie regularization procedure,32which allows the evaluation of det /H11032/H9011u, the functional determinant of the operator /H9011uwith the zero ei- FIG. 5. The prefactor /H90030/H20849in units of /H92700−1/H20850vshwhen /H5129=7 on the “constant saddle” side of the transition. The prefactor on the “in-stanton saddle” side of the transition acquires an additional tem-perature dependence, as discussed in the text.KIRSTEN MARTENS, D. L. STEIN, AND A. D. KENT PHYSICAL REVIEW B 73, 054413 /H208492006 /H20850 054413-6genvalue removed. We refer the reader to Ref. 32 for details, but sketch the main features here. With periodic boundary conditions, it is formally the case that det/H11032/H9011u /H20855/H92571/H20841/H92571/H20856=/H92572/H20849s+/H5129,s0;m/H20850−/H92572/H20849s,s0;m/H20850 /H92571/H20849s,s0;m/H20850detH/H20849s,s0;m/H20850, /H2084924/H20850 where /H92571/H20849s,s0;m/H20850and/H92572/H20849s,s0;m/H20850are two linearly independent solutions of /H9011u/H9257i=0, i=1,2, /H20855/H92571/H20841/H92571/H20856 =/H20848−/H5129/2/H5129/2dsy12/H20849s,0;m/H20850is the square of the norm of the zero mode, and det H/H20849s,s0;m/H20850=/H9257˙2/H20849s,s0;m/H20850/H92571/H20849s,s0;m/H20850 −/H9257˙1/H20849s,s0;m/H20850/H92572/H20849s,s0;m/H20850is the Wronskian; here a dot denotes a derivative with respect to s. The expression /H2084924/H20850is mean- ingful only as part of a determinant quotient, as noted above.The functions /H92571and/H92572can be found by differentiating the instanton solution /H208498/H20850with respect to s0andm, respec- tively; i.e., /H92571/H20849s,s0;m/H20850=/H11509/H9278/H20849s,s0;m/H20850//H11509s0and/H92572/H20849s,s0;m/H20850 =/H11509/H9278/H20849s,s0;m/H20850//H11509m. This yields /H92571/H20849s,s0;m/H20850=−2m /H9254sn/H20849R/H20841m/H20850cn/H20849R/H20841m/H20850 /H11003sn/H20873s−s0 /H9254/H20841m/H20874cn/H20873s−s0 /H9254/H20841m/H20874 cn2/H20849R/H20841m/H20850+s n2/H20849R/H20841m/H20850dn2/H20873s−s0 /H9254/H20841m/H20874 /H2084925/H20850 and /H92572/H20849s,s0;m/H20850=−2 cn2/H20849R/H20841m/H20850+s n2/H20849R/H20841m/H20850dn2/H20873s−s0 /H9254/H20841m/H20874 /H11003/H20877m/H20849s−s0/H20850 /H92542d/H9254 dmsn/H20849R/H20841m/H20850cn/H20849R/H20841m/H20850sn/H20873s−s0 /H9254/H20841m/H20874cn/H20873s−s0 /H9254/H20841m/H20874 +sn/H20849R/H20841m/H20850cn/H20849R/H20841m/H20850 2/H208491−m/H20850/H20875sn/H20873s−s0 /H9254/H20841m/H20874cn/H20873s−s0 /H9254/H20841m/H20874E/H20873s−s0 /H9254/H20841m/H20874−s n/H20873s−s0 /H9254/H20841m/H20874cn/H20873s−s0 /H9254/H20841m/H20874/H208491−m/H20850/H20873s−s0 /H9254/H20874 −s n2/H20873s−s0 /H9254/H20841m/H20874dn/H20873s−s0 /H9254/H20841m/H20874/H20876+d n /H20849R/H20841m/H20850dR dmdn/H20873s−s0 /H9254/H20841m/H20874/H20878, /H2084926/H20850 where E/H20849·/H20841m/H20850is the incomplete elliptic integral of the second kind.26 Inserting these solutions into Eq. /H2084924/H20850yields /H20879det/H11032/H9011u /H20855/H92571/H20841/H92571/H20856/H20879 =/H92543/H20851/H208492m//H9254/H20850/H20849d/H9254/dm /H20850K/H20849m/H20850−K/H20849m/H20850+E/H20849m/H20850//H208491−m/H20850/H20852 4m2sn/H20849R/H20841m/H20850cn/H20849R/H20841m/H20850dn/H20849R/H20841m/H20850dR/dm. /H2084927/H20850 Using a similar procedure, we ﬁnd the corresponding nu- merator for the determinant ratio in Eq. /H2084918/H20850to be det/H9011s= 4 sinh2/H20851/H9254/H208811−hK/H20849m/H20850/H20852, /H2084928/H20850 consistent with the numerator of Eq. /H2084923/H20850, obtained through direct computation of the eigenvalue spectrum. /H20849Recall, though, that it is only the ratio of the determinants that is sensible. /H20850This becomes clearer by noting that the expres- sions in Eqs. /H2084927/H20850and /H2084928/H20850are well behaved for all ﬁnite /H5129 /H11022/H5129c/H20849m/H110220/H20850. While both expressions separately diverge as m→1, it is easily checked that the divergences cancel.As already noted, the rotational symmetry of the instanton state /H20849corresponding to the arbitrariness of the constant s0/H20850 corresponds to a “soft mode,” resulting in appearance of azero eigenvalue /H9261 u,1=0 of the operator /H9011u. The correspond- ing eigenfunction is clearly /H92571given by Eq. /H2084925/H20850. The ap- pearance of a zero mode corresponds to the zero rotationalenergy of the instanton solution: the center of the domainwall pair can appear anywhere in the ring. This is in contrast FIG. 6. Lowest eigenvalue /H9261u,0as a function of hfor/H5129=7.MAGNETIC REVERSAL IN NANOSCOPIC … PHYSICAL REVIEW B 73, 054413 /H208492006 /H20850 054413-7to the situation in the ﬁnite cylinder,25where the instanton is “pinned.” The general procedure for including the correctionrequired due to removal of the zero eigenvalue is describedby Schulman. 30The correction in our problem is an addi- tional factor of 2 /H5129/H20881/H20855/H92571/H20841/H92571/H20856//H9266kBT, which vanishes asm→0/H20849and thereby removes the divergence of the prefactor asm→0+/H20850. Finally, we need to compute the eigenvalue /H9261u,0corre- sponding to the unstable direction. With the substitution z =/H20849s−s0/H20850//H9254, the eigenvalue equation /H9011u/H9257=/H9261/H9257becomes 0=d2/H9257 dz2+/H92542/H9257/H11003/H20877/H20849/H9261+1− h/H20850cn4/H20849R/H20841m/H20850+2 /H20849/H9261+3 /H20850sn2/H20849R/H20841m/H20850cn2/H20849R/H20841m/H20850dn2/H20849z/H20841m/H20850+/H20849/H9261−1− h/H20850sn4/H20849R/H20841m/H20850dn4/H20849z/H20841m/H20850 /H20851cn2/H20849R/H20841m/H20850+s n2/H20849R/H20841m/H20850dn2/H20849z/H20841m/H20850/H208522 /H20878. /H2084929/H20850 The lowest eigenvalue corresponds to a nodeless solution for /H9257. By continuity it must tend towards −1+ h2asm→0+.W e have solved Eq. /H2084929/H20850numerically for /H5129=7; the result appears in Fig. 6. The weak dependence on h/H20849and also /H5129/H20850is typical. Finally, we put all of the above results together to ﬁnd the formula for the prefactor per unit length : /H92700/H90030+//H5129=/H20841/H92610/H20849/H5129,h/H20850/H20841m/H20849kBT/H20850−1/2sinh /H20851/H9254/H208811−hK/H20849m/H20850/H20852 /H11003/H208732mK/H20849m/H20850dln/H9254 dm+1 1−m /H11003/H20851E/H20849m/H20850−/H208491−m/H20850K/H20849m/H20850/H20852/H20874−1/2 , /H2084930/H20850 where E/H20849m/H20850is the complete elliptic function of the second kind.26As noted above, /H92610/H20849/H5129,h/H20850is weakly dependent on h and/H5129, and is O/H208491/H20850everywhere. The most important qualitative feature to be noted from Eq. /H2084930/H20850is that the zero eigenvalue arising from the uniform translation mode leads to non-Arrhenius behavior—i.e., aT-dependent prefactor—everywhere on the low-ﬁeld side of the transition. Finally, we note that the eigenfunction /H92571given by Eq. /H2084925/H20850has a single pair of nodes. Because nodes arise in pairs, there must then be only a single /H20849nodeless /H20850solution of lower eigenvalue than /H92571. But/H92571has zero eigenvalue, proving that the solution /H208498/H20850has a single unstable eigenmode, and is therefore a proper saddle.The above results allow one to ﬁnd the overall reversal rate in any part of the /H20849/H5129,h/H20850phase plane. Results for a per- malloy ring with given dimensions are shown in Fig. 7. Among commonly used soft ferromagnetic materials, per- malloy has the largest magnetic exchange length. The discus-sion of scaling in Sec. III suggests that the effects of nonlocalmagnetostatic terms are minimized with larger exchangelengths. Where else might one ﬁnd magnetic materials withlarge exchange lengths? Such materials would require bothlow magnetization density and large exchange constants.This combination occurs naturally in certain ferrimagnets.One example is MgOFe 2O3, which has an exchange length a factor of 5 larger than that of permalloy. There are manyexamples of such materials that have been prepared as poly-crystalline thin ﬁlms, and thus are soft magnets /H20849i.e., have no or very weak magnetocrystalline anisotropies /H20850. Such materi- als might prove useful for experimental studies of the phe-nomena described in this paper. VI. DISCUSSION A theory of magnetization reversal in thin micromagnetic rings has been presented. Such systems are distinguished bytheir lack of edges or corners where nucleation is easily ini-tiated, leading to greater stability of magnetization conﬁgu-rations and facilitating comparison of theory to experiment. By utilizing a scaling analysis 19that uncovers a separation of energy scales in the thin-ﬁlm limit, we are able to retain FIG. 7. Total switching rate /H20849in units of s−1/H20850 vs/H9252=1/kBT/H20849in units of K−1/H20850, at ﬁelds of /H20849a/H20850 60 mT /H20849instanton saddle /H20850and /H20849b/H2085072 mT /H20849con- stant saddle /H20850. Parameters used are k=0.01, l =0.1, R=200 nm, R1=180 nm, R2=220 nm, M0 =8/H11003105A/m /H20849permalloy /H20850,/H9251=0.01, and /H9253=1.7 /H110031011T−1s−1. Deviation of low-ﬁeld switching rate in /H20849a/H20850from dashed line signals non-Arrhenius behavior.KIRSTEN MARTENS, D. L. STEIN, AND A. D. KENT PHYSICAL REVIEW B 73, 054413 /H208492006 /H20850 054413-8leading-order terms that allow for an analytic solution of the relevant magnetization conﬁgurations in the thermally in-duced reversal problem. The discarded terms, in particularthose corresponding to nonlocal magnetostatic energy contri-butions, are shown to contribute no more than O/H2084910% /H20850to the energy over most values in the /H20849/H5129,h/H20850phase plane. A complete solution that takes into account all terms must be numerical, and is planned for future work. Nevertheless, an analytic solution is highly useful and can uncover information that may be difﬁcult to extract from anumerical one. In particular, we predict an unusual transitionfrom Arrhenius to non-Arrhenius activation behavior /H20849cf. Fig. 7 /H20850. Our analysis suggests that such a transition should now be observable experimentally, by varying the externally applied magnetic ﬁeld for rings of ﬁxed size. A clear signa-ture of such a transition would be the observation of a cross-over from Arrhenius to non-Arrhenius behavior as ﬁeld var-ies, as seen in Fig. 7. Because this requires measurement ofthe prefactor, such an observation would require numerousruns where reversal occurs. Arrhenius behavior of magnetic reversal has already been found in several systems and geometries. In Ref. 4, measure-ments of switching ﬁeld and waiting times on nearly spheri-cal Ni, Co, and Dy nanoparticles found an activation volumeclose to the particle volume, indicating a uniform magneti-zation reversal /H20849analogous to the constant saddle case here /H20850 and conﬁrming the Néel-Brown theory for these systems. Incontrast, measurements on Ni wires with diameters40–100 nm revealed an activation volume considerablysmaller than the particle volume, indicating a nonuniformtransition state /H20849analogous to our instanton saddle /H20850. Here, too, Arrhenius switching behavior was found. But would notthe arguments given above imply that one should see non-Arrhenius behavior for these wires? No, because here the/H20849roughly /H20850cylindrical geometry with end caps /H20849and conse- quently the relevant boundary conditions on the magnetiza-tion /H20850, lead to the absence of a uniform translation or rota- tional symmetry present, and therefore no zero mode leadingto a temperature-dependent prefactor. In fact, these observa-tions support the robustness of our conclusions for a widervariety of cases than considered in this paper. We will dis-cuss this further below. Before doing that, however, we wish to suggest a second experimental test that may be easier to conduct: this is tomeasure the dependence of the activation energy on meanradius Rfor rings of identical composition. On the Arrhenius side of the transition, where magnetization reversal proceedsvia a uniform rotation of the magnetization, the activationbarrier scales linearly with the ring size. However, on thenon-Arrhenius side, where the instanton state governs thereversal, the activation barrier is almost independent of ringsize /H20849see Fig. 3 /H20850. In this set of measurements, one may needto alter the applied ﬁeld as ring size varies to keep the system on one or the other side of the transition, given that thecritical ﬁeld depends on R/H20851cf. Eq. /H2084912/H20850/H20852. How robust are our predictions of a transition in activa- tion behavior, and in particular, can the neglected energycontributions wash out or obscure the transition? It is indeedpossible, perhaps likely, that the details of the transition closeto the critical ﬁeld /H20849or circumference if ﬁeld is ﬁxed /H20850are sensitive to these terms. In particular, the second-order na-ture of the transition, and the corresponding divergence ofthe prefactor /H20849cf. Fig. 5 /H20850, could disappear. Inclusion of the magnetostatic terms could even in principle change the tran-sition from second- to ﬁrst-order, with a jump replacing thedivergence in the prefactor. Such ﬁrst-order transitions havebeen predicted to occur in thermally induced conductancejumps in monovalent metallic nanowires. 33 However, our central prediction, a transition from Arrhen- ius to non-Arrhenius activation behavior, should be robustbecause it is due to something much more fundamental: arotationally invariant transition state /H20851our “constant” state /H9278=cos−1/H20849−h/H20850/H20852at high ﬁelds and a rotationally noninvariant state /H20851our instanton state /H208498/H20850/H20852at low ﬁelds, with the crossover determined primarily through a competition between theshape anisotropy arising from magnetostatic forces and theZeeman energy arising from the external ﬁeld. In fact, thediscussion in Sec. V C 2 leads to the conclusion that theappearance at lower ﬁelds of anyrotationally noninvariant state should give non-Arrhenius switching behavior in thering geometry. Experimentally, what is then required is asymmetric enough ring so that the “domain wall” part of thetransition state /H20849centered at s 0in our instanton solution /H20850has more or less equal probability of nucleating anyplace alongthe ring. This “Goldstone mode,” arising from the rotation-ally invariant geometry, is ultimately where the non-Arrhenius factor comes from. Although the size constraintson the ring parameters leading to the speciﬁc instanton solu-tion /H208498/H20850are difﬁcult to realize at the present time, the gener- ality of the basic physical features determining the transitionshould lead to the predicted crossover from Arrhenius tonon-Arrhenius behavior in at least some ring geometries thatare outside of the scaling regime considered here. ACKNOWLEDGMENTS This research was partially supported by NSF Grant Nos. PHY-0351964 /H20849D.L.S. /H20850, FRG-DMS-0101439 and DMR- 0405620 /H20849A.D.K. /H20850, and the Akademisches Auslands Amt and the Evangelisches Studienwerk e.V . Villigst /H20849K.M. /H20850. K.M. thanks the Physics Department of the University of Arizonafor their hospitality and support during her stay. D.L.S.thanks Bob Kohn and Valeriy Slastikov for numerous valu-able discussions, and for providing several references. Present address: Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 19, 69120 Heidelberg, Germany. †Present address: Department of Physics and Courant Institute of Mathematical Sciences, New York, NY 10003.1F. H. deLeeuw, R. van den Doel, and U. Enz, Rep. Prog. Phys. 43, 689 /H208491980 /H20850. 2L. Néel, Ann. Geophys. /H20849C.N.R.S. /H20850/H20849C.N.R.S. /H208505,9 9 /H208491949 /H20850. 3W. F. Brown, Jr., Phys. 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Math. 55, 1408 /H208492002 /H20850. 21The magnetostatic energy also contains a mixing term between bulk and boundary contributions, which scales the same as thebulk term; it is therefore omitted here. Its contribution is the same order as that of the bulk term, which is shown in the text tobe at least an order of magnitude smaller than all of the otherterms. 22T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, Science 289, 930 /H208492000 /H20850. 23N. A. Usov and S. E. Pechany, J. Magn. Magn. Mater. 118, L290 /H208491993 /H20850. 24P. Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 /H208491990 /H20850. 25R. S. Maier, in Noise in Complex Systems and Stochastic Dynam- ics II , edited by Z. Gingl, J. M. Sancho, L. Schimansky-Geier, and J. Kertesz /H20849SPIE, Bellingham, 2004 /H20850, pp. 48–57, indepen- dently found a spatially varying solution for ﬁnite cylindricalgeometries. 26M. Abramowitz and I. A. 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PhysRevB.83.174447.pdf	PHYSICAL REVIEW B 83, 174447 (2011) Spin-density waves and domain wall interactions in nanowires N. Sedlmayr,1,2,V . K. Dugaev,3,4and J. Berakdar5 1Department of Physics, University of Kaiserslautern, DE-67663 Kaiserslautern, Germany 2Research Center OPTIMAS, University of Kaiserslautern, D-67663 Kaiserslautern, Germany 3Department of Physics, Rzesz ´ow University of Technology, al. Powsta ´nc´ow Warszawy 6, PL-35-959 Rzesz ´ow, Poland 4Department of Physics and CFIF , Instituto Superior T ´ecnico, TU Lisbon, av. Rovisco Pais, PT-1049-001, Lisbon, Portugal 5Department of Physics, Martin-Luther-Universit ¨at Halle-Wittenberg, Heinrich-Damerow-Str. 4, DE-06120, Halle, Germany (Received 18 February 2011; revised manuscript received 7 April 2011; published 31 May 2011) We investigated how the dynamics of a domain wall are affected by the presence of spin-density waves in a ferromagnetic wire. Domain walls and other scattering centers can cause coherent spin-density waves to propagatethrough a wire when a current is applied. In some cases, the spin torque due to these scattered electrons can beenhanced such that it is on a par with the exchange and anisotropy energies controlling the shape and dynamicsof the domain wall. In such a case, we ﬁnd that the spin-density waves enhance the current-induced domain-wallmotion, allowing for domain-wall motion with smaller current pulses. Here, we consider a system involving twodomain walls and focus on how the motion of the second domain wall is modiﬁed by the spin-density wavescaused by the presence of the ﬁrst domain wall. DOI: 10.1103/PhysRevB.83.174447 PACS number(s): 75 .60.Ch, 75 .60.Jk, 75.75.−c, 75.30.Fv I. INTRODUCTION The dynamics of a domain wall (DW) in a ferromagnetic wire has received much interest due to its fundamentalphysical importance and possible future applications. 1–4Cou- pling between the carriers of spin-polarized currents and themagnetic moments, forming DWs, alters not only the transportproperties 5–7of the wire but also the magnetization itself. The current traveling through the wire couples to the domain wallcausing it to move through the wire. Thus differently orientedcollinear regions of the wire may serve as discrete bits that canbe steered by the current. In order to increase the efﬁciencyof these devices, 3,8it is necessary to increase the density of the DWs. Therefore, even a weak interaction between DWscan become important. This coupling can also distort the DWduring its propagation through the wire. Some examples fromthe now voluminous literature can be found in Refs. 4,9, and10. So far most of the work has concentrated on how the spin-polarized currents will affect the DWs. However, there arealso feedback processes that can lead to interactions betweenthe DWs and spin-density ﬂuctuations. The relevant effectsare enhanced considerably when considering low-dimensionalstructures. 11An effective Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two domain walls in a wire canbe mediated by the current electrons. 12–14The energy proﬁle of this interaction then favors particular alignments and positionsof the domain walls. Furthermore, any change in the spindensity caused by the presence of one domain wall can havean effect on a wall further down the wire in the direction ofelectron ﬂow. For applications, it is most interesting to consider a high density of DWs in a wire. In this case, it is crucial toinspect how the DWs affect each other when a spin-polarizedcurrent is sent down the wire. In a previous article, 12we looked at how the current-mediated RKKY-like interactionchanges the magnetization dynamics with a focus on themotion of relatively sharp walls. These walls were treatedas localized moments. Here, we extend this idea to consider how a long (adiabatic) domain wall is distorted in the presenceof a spin-density wave caused by scattering from a previousdomain wall. The spin-density wave acts as an effective appliedmagnetic ﬁeld. However, because the spin torque of the currentelectrons is no longer uniform, it is not a homogeneous ﬁeld.This will tend to distort the shape of the domain walls. Furthermore, this nonuniform spin density allows the domain wall to be set into motion using a smaller current pulsethan would otherwise be needed. This is closely related to itseffect as an effective magnetic ﬁeld. This offers an alternativepossible solution 15to the problem of overheating associated with sending too large of a current through the nanowires.Though we note that in order to obtain a spin density of the appropriate order of magnitude, one must consider very narrow wires of a few atoms across. It is now possible to fabricate ferromagnetic wires com- posed of single atomic sites on a lattice. 16–19Surprisingly, these wires show both ferromagnetic order and contain regionsof noncollinearity, 20in other words, domain walls. In the low-dimensional limit of these metallic wires, the effect of the spin-density corrections on the domain wall may become of the order of that of the exchange and magnetic anisotropy.This can cause severe distortion of the domain-wall proﬁleand alter the way it travels through the wire. We note thatthe spin-density corrections must not necessarily originatefrom a DW; any spin-coherent scattering center will lead tothe behavior described in this paper, for example, a magnetic impurity or a region of noncollinearity in an otherwise single- domain ferromagnet. Although here, we concentrate on thecase of a sharp domain wall as the scatterer, the generalizationto other scenarios is straightforward. The spin density for aspeciﬁc material can be varied by two parameters: the overalland relative magnitudes of the two spin channels. By varyingthese parameters we can consider a variety of scatterers. Our set up thus consists of a ferromagnetic wire with two domain walls present. A current is then applied to the wire. We 174447-1 1098-0121/2011/83(17)/174447(6) ©2011 American Physical SocietyN. SEDLMAYR, V . K. DUGAEV , AND J. BERAKDAR PHYSICAL REVIEW B 83, 174447 (2011) focus on how the motion of the second domain wall (deﬁned with respect to the current direction) is affected by the spin-density waves caused by the presence of the ﬁrst domain wallin the wire. II. THEORETICAL FORMULATION We start with a zero-temperature general Hamiltonian for a nanowire describing noninteracting conduction electrons ofspinα,ˆa † α(r), coupled with a strength, J, to some nonuniform bulk magnetization, /vectorM(r),21,22 ˆH/prime=/integraldisplay dr/summationdisplay αˆa† α(r)[ˆξδαβ−J/vectorσαβ./vectorM(r)]ˆaβ(r).(1) Here, ˆξis the kinetic-energy operator. The inhomogene- ity can be partially dealt with by a local vector-gaugetransformation. 23–25After this transformation, we will have a uniform Zeeman-splitting term and a spin-dependent spatiallyvarying potential, U αβ(r), that describes the scattering from any noncollinear conﬁgurations of the magnetization. Thistransformation describes a local rotation in spin space to alignthe magnetization direction throughout the wire. It is possible for any form of /vectorM(r) that has a constant magnitude. The gauge transformation is/parenleftBigg ˆa old 1(r) ˆaold 2(r)/parenrightBigg =T(r)/parenleftbiggˆanew 1(r) ˆanew 2(r)/parenrightbigg , (2) deﬁned such that T†(r)/vectorσ·/vectorn(r)T(r)=σz, where /vectornis the unit vector along /vectorM.26Our new Hamiltonian is then ˆH=/integraldisplay dr/summationdisplay αβˆa† α(r)[ˆξδαβ−JMσz αβ+Uαβ(r)]ˆaβ(r)( 3 ) with the scattering potential given by U(r)=−1 2m[2/vectorA(r).∂r+∂r./vectorA+/vectorA2(r)], (4) where /vectorA(r)=T†(r)∇rT(r) is a gauge potential. For the case of two domain walls in a nanowire, with the ﬁrst DW, of width L/prime, located at z=0 and the second DW, of width L, located at z=z0,w eh a v e /vectorM(z)=M⎛ ⎜⎝cos[θ(z)] sin[ϕ(z)] sin[θ(z)] sin[ϕ(z)] cos[ϕ(z)]⎞ ⎟⎠, (5) where ϕ(z)=π−cos−1[tanh(z/L/prime)]/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright =ϕ1(z) +π−cos−1[tanh[( z−z0)/L]]/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright =ϕ2(z). (6) The angle, θ(z), is used to give the walls a different orientation. Around the ﬁrst wall we set it arbitrarily to zero and aroundthe second to an angle, θ 0, which therefore deﬁnes the relative orientation of the two walls. The cross terms between the wallsare very small and can be neglected provided that z 0/greatermuchL,L/prime, allowing us to approximate the scattering potential, U(z)≈U1(z)+U2(z), (7)-5 -2.5 0 2.5 5 (z-z0)/L-505Si(z) 107m-1Sx Sy Sz FIG. 1. (Color online) The spin density of the carrier electrons around the second domain wall centered at z0and measured in carrier- spin per unit length. One can clearly see both the spin density waves and the overall proﬁle of the second domain wall. The solid (black)curve is S x(z), the dashed (red) is Sy(z), and the dotted (blue) is Sz(z). as independent contributions from each DW. To see some interaction effects between the DWs, we must nonetheless,of course, have the distance between the DWs less than thespin-coherence length scale in the system. Now U 1(z)=I[ϕ/prime 1(z)]2 8m+iσy/bracketleftbiggϕ/prime/prime 1(z) 4m+ϕ/prime 1(z)∂z 2m/bracketrightbigg and (8) U2(z)=I[ϕ/prime 2(z)]2 8m+iσy/bracketleftbiggϕ/prime/prime 2(z) 4m+ϕ/prime 2(z)∂z 2m/bracketrightbigg cos(θ0) −iσx/bracketleftbiggϕ/prime/prime 2(z) 4m+ϕ/prime 2(z)∂z 2m/bracketrightbigg sin(θ0). (9) IfL>λ F, then we can also neglect the ϕ/prime/prime 2(z) and [ ϕ/prime 2(z)]2 terms, which are of the order of ( λF/L)2. The correction to the spin density of a single scattered wave of spin δis /Delta1/vectorSδ=δψ† εδ/vectorσδψ εδ, (10) where δψεδis the scattered wave function from an incoming particle of spin δat an energy ε. It can be calculated within the Born approximation provided that L/prime/greaterorsimilarλF.12This tells us the nature of the spin waves in our system. These spin-densitycorrections result in an inter-DW spin torque and an RKKY-like interaction, details of which can be found in Sedlmayret al. 12–14The result for the spin density around the second domain wall at an arbitrarily chosen conﬁguration, θ0=π/4, is shown in Fig. 1. Plotted for λF=0.367 nm, L/prime=λF, L=10λF, andJM=2.24 eV . The adiabatic change in the spin density across the width of the second DW is clearlyvisible and it is modiﬁed by the presence of the spin-densitywaves. These oscillations in the spin density will tend toinhomogeneously distort the domain-wall proﬁle away fromits equilibrium proﬁle. 174447-2SPIN-DENSITY WA VES AND DOMAIN WALL ... PHYSICAL REVIEW B 83, 174447 (2011) III. MOTION OF THE SECOND DOMAIN WALL We wish to ﬁnd the effect of the scattered electrons on the second domain wall. We take the spin density of the scatteredelectrons and calculate its effect on the second DW usingthe now standard Landau-Lifschitz-Gilbert (LLG) equationapproach. 21,27,28In our case, this gives us ∂t/vectorM(z,t)=−1 1+α2/bracketleftbigg γ/vectorM(z,t)×H +αγ M/vectorM(z,t)×[/vectorM(z,t)×H] +bj(1+αξ) M2/vectorM(z,t)×[/vectorM(z,t)×(/vectorj.∇)/vectorM(z,t)] +bj(ξ−α) M/vectorM(z,t)×(/vectorj.∇)/vectorM(z,t)/bracketrightbigg . (11) γis the gyromagnetic ratio, γ=gμB/¯h, where gis the Land ´e factor, and αis a phenomenological constant characterizing the magnetic damping of the system. The constants for the nonadiabatic terms are the following: /vectorjis the current density, ξ=τex/τsfis the ratio between the exchange and spin-ﬂip time scales, and bj=PμB/[eM(1+ξ2)], where Pis the polarization. The effective ﬁeld is H(z)=J/vectorS(z)+αex∂2/vectorM(z) ∂z2+KM x(z)/vectorx. (12) αexis the exchange coupling. The coefﬁcient K, characterizes the magnetic anisotropy in the wire. Here, we have chosenxas the anisotropy axis. We use 21,22αex=J/Ma andL2= π√K/α ex, where ais the lattice spacing. Note, however, that the domain-wall width can be changed by geometric effects as well as by changing the anisotropy or the exchange energy.The effective ﬁeld, H, which the DW experiences, is composed of the spin-density ﬂuctuations of the carriers, the exchangeinteraction of the DW itself, and the magnetic anisotropy of thesystem. The exchange interaction tends to oppose distortionson the wire and will impose an upper limit on how much theDW can be distorted by a given spin density.We calculate the dynamics resulting from Eq. ( 11)u s i n gt h e boundary conditions: /vectorM(0,z)=/vectorM 0(z) and /vectorM(t,z0±zb)= /vectorM0(z±zb), where /vectorM0(z) is the magnetization in Eq. ( 5). The value of zbmust be large enough to insert no artifacts in the results. Note that from Eq. ( 12) it is clear that for a sizable effect we require aL2>\|S\|, where Sis the spin density. This can be achieved by considering the experimentally realizable limit ofa chain of atoms with a cross section of approximately σ cs= λ2 F. The lattice spacing is a=1 nm and we use here the Fermi wavelength of iron, λF=0.367 nm. Also we take M=1.72× 106Am−1,JM=2.24 eV , je=−2.33×108Am−2, and ξ=0.011.28The second domain wall has a width of L=10λF and the width of the ﬁrst wall is of the order of the Fermi wavelength, L/prime=λF. The motion of considered DWs is therefore relatively narrow. This gives us the advantage ofmaking our results clearer. However, it could also limit theapplicability of the model. We note though, a DW would benarrower in a wire with a small cross section than in the bulk orthin ﬁlm case, i.e., upon patterning of the wire the DW widthcan be decreased. Nonetheless, the results here would holdfor DW widths of the order of 10 nm. The case of DWs withlarger widths is discussed below. Finally, for α=0.01 and the exchange and anisotropy as deﬁned above ( xis, therefore, a hard magnetic axis), the second DW’s conﬁguration is initiatedat an arbitrary angle, θ 0=π/4. Altering the initial angle of the second DW does not signiﬁcantly affect the current-inducedmotion of the second domain wall. The speed of the DWmotion is independent of θ 0. Qualitatively, the same motion is seen, except for some small distortions in the DW proﬁle,which depend on θ 0. Firstly, let us consider the effects of the spin-density wave without considering the current-induced motion in Eq. ( 11), plotted in Figs. 2and3. In the right-hand side (RHS) of Figs. 2 and3, the distortion over a short period of time of the wall into a low-energy position is shown, as in the case withoutthe spin-density corrections [see the left-hand side (LHS) ofFigs. 2and3], but as the spin density ﬂuctuates on a scale much shorter than the second wall we see no coherent evolution ofthe domain wall over a longer period of time. The motion seen Mx M z, z 0z0+2L z0−2L t 0 tf0 tf1 −1 FIG. 2. (Color online) A contour plot of the xcomponent of the magnetization dynamics, Mx(t,z), around the second wall, without an applied current, centered at z0. The left-hand ﬁgure is the component without the spin-density corrections, the right-hand ﬁgure includes the spin-density corrections. The left-hand ﬁgure shows motion caused by the anisotropy in the wire, on the right-hand side the spin-density wavescause additional distortions. Here, t f=2.69×10−13s. 174447-3N. SEDLMAYR, V . K. DUGAEV , AND J. BERAKDAR PHYSICAL REVIEW B 83, 174447 (2011) Mz M z, z 0z0+5L z0−5L t 0 tf0 tf1 −1 FIG. 3. (Color online) A contour plot of the zcomponent of the magnetization dynamics, Mz(t,z), around the second wall, without an applied current, centered at z0. The left-hand ﬁgure is the component without the spin-density corrections, the right hand ﬁgure includes the spin-density corrections. The left-hand ﬁgure shows motion caused by the anisotropy in the wire, on the right-hand side the spin-density wavescause additional distortions. Note that the presence of the spin density readily distorts the DW proﬁle. Here, t f=2.69×10−13s. when there is no spin-density correction present, as in the LHS of Figs. 2and3, is because we have not started the walls from their equilibrium positions with respect to the anisotropy inthe system. In this case, the domain wall attempts to ﬁnd itslowest energy conﬁguration. What is most clear from theseﬁgures is that the spin density is able to completely distortthe proﬁle of the domain wall. Rather than being able to shiftto its equilibrium conﬁguration the domain wall is severelydistorted by the spin-density torque. Its low-energy positionis now a compromise between not just the anisotropy andexchange energies but also the spin-density torque exertedon the wall. The low-energy proﬁle with respect to the spin-density contributions would have an oscillating shape but thisis opposed by the exchange-energy cost it brings. Now let us consider what happens when we include the current terms in Eq. ( 11). The case without any correction to the spin density from scattering from a ﬁrst domain walli ss h o w ni nt h eL H So fF i g . 4, here, we focus on the xcomponent of the magnetization. One simply sees the usual current-induced motion of the domain wall. The RHS ofFig.4includes the spin-density corrections. After some time, one also can notice a distortion of the domain-wall proﬁlecaused by the spin-density corrections. The new domain-wallproﬁle then undergoes coherent motion with this new shape.The presence of the spin-density wave also has a small butnoticeable transient effect at the beginning of the domain-wallmotion; the wall is set into motion more quickly than in thecase without the spin-density wave. This effect is short livedand the long-term motion is, in this case, similar to the onewithout the spin-density oscillations. After a while, some smalldistortion of the DW proﬁle does appear. However, distortionssimilar to those found when the current terms do not contributeare not seen. If we lower the applied current density we ﬁnd the effects more pronounced. In fact, the presence of the spin density ap-preciably speeds up the motion of the DW when a small current Mx M z, z 0z0+5L z0−5L t 0 tf0 tf1 −1 FIG. 4. (Color online) A contour plot of the xcomponent of the magnetization dynamics, Mx(t,z), around the second wall centered at z0. This includes the current terms for the magnetization dynamics for je=−2.33×108Am−2. The left-hand ﬁgure is the component without the spin-density corrections, the right-hand ﬁgure is the full result showing the accelerated start of the DW motion due to the spin-densityterms. Here, t f=6.73×10−14s. 174447-4SPIN-DENSITY WA VES AND DOMAIN WALL ... PHYSICAL REVIEW B 83, 174447 (2011) Mx M z, z 0z0+3L z0−3L t 0 tf0 tf1 −1 FIG. 5. (Color online) A contour plot of the xcomponent of the magnetization dynamics, Mx(t,z), around the second wall centered at z0. The left-hand ﬁgure is the component without the spin-density corrections, the right-hand ﬁgure is the full result including spin-density corrections. In this case, we apply a smaller current, je=−5.82×107Am−2. The faster motion of the domain wall due to the spin-density correction is now clearly visible in the right-hand ﬁgure as compared to the left-hand ﬁgure. Here, tf=6.73×10−14s. is applied, see Fig. 5. This opens up the possibility that in such systems the presence of closely-packed multiple domain wallsassists in the current-induced motion of the subsequent DWsallowing them to be shifted with a smaller applied currentpulse. The spin-density wave acts as a local magnetic ﬁeldand, in principle, gives us more parameters with which tocontrol the current-induced-DW motion. Namely, the overalland relative amplitudes of the spin-density correction channels,see Eq. ( 10). We can also consider what happens for larger widths of DWs. For a DW with a width L=10 nm, the main result is still valid. The spin-density wave aids the current-induced-DWmotion. However, if we increase the DW width to L=40 nm, these effects are already negligible for longer timescales. The Mx M z, z 0z0+3L z0−3L t 0 tf1 −1 FIG. 6. (Color online) A contour plot of the xcomponent of the magnetization dynamics, Mx(t,z), around the second wall centered atz0. Here, we consider a longer DW with a width L=40 nm. The initial current-induced motion of the DW is still enhanced by the spindensity; for longer times the change in DW velocity is negligible. Note a narrowing of the DW width as it moves caused by the spin- density wave. The applied current is j e=−2.33×108Am−2. Here, tf=6.73×10−14s.reason for this is clear: for a longer DW the spin-density wave is oscillating too much over its extension to have a coherenteffect. Nonetheless, for short timescales, one still sees a fasterinitial motion of the DW. Furthermore, one can see a gradualnarrowing of the DW caused by the spin-density wave, seeFig.6. The edges of the DW are gradually squeezed as they attempt to precess in the effective magnetic ﬁeld of the rapidlyoscillating spin-density wave. This process is limited by theincrease in the exchange energy for a narrower DW. The DWends with a width of approximately 10–20 nm undergoingcurrent-induced motion. This effect is, of course, not presentin the case of the absence of the spin-density wave. IV . DISCUSSION AND CONCLUSIONS We have investigated how scattering of spin-density waves (which are caused by spin-polarized currents) from domainwalls affects the domain-wall magnetization dynamics. In theone-dimensional limit of atomic chains the spin torque exertedby the spin-density waves can be sizable, even comparablewith the exchange energy of the bulk ferromagnetic system.For pinned domain walls, the inhomogeneous spin distortsthe domain-wall proﬁle into a new shape, which minimizesthe relevant energies of the domain wall. However, when thedomain wall is free to move through the system, the walls aredistorted only slightly. In this case the main effect of the spinwaves is to propel the domain wall into motion faster than intheir absence. The relatively strong effect of the current-induced spin- density wave on the motion of the domain wall is similarto the effect of an external magnetic ﬁeld. This is becausethe magnetization related to the spin wave appears in theequation of motion as an external ﬁeld and, on the other hand,it is known that by using a magnetic ﬁeld one can easilyput the domain wall into motion. However, in the case of aspin-density wave, the effective ﬁeld acting on the domainwall is not homogeneous. On the contrary, it is stronglyoscillating. Correspondingly, it can affect the short-rangeinteraction of magnetic moments, making unstable the usual 174447-5N. SEDLMAYR, V . K. DUGAEV , AND J. BERAKDAR PHYSICAL REVIEW B 83, 174447 (2011) shape of the domain wall. Being strongly inhomogeneous, the spin wave induces some displacements and certain disorderof the wall, which is important for the DW motion becausethe ﬁrst displacement allows the depinning of the wall to putit into motion. In this connection, we propose to combinethe spin-wave-induced dynamics with the short pulses ofcurrent. It can be realized with a combination of a dccurrent generating the spin density wave, and strong current pulses. ACKNOWLEDGMENTS This work is partly supported by FCT Grant No. PTDC/FIS/70843/2006 in Portugal, by the DFG contract BE2161/5-1, and by the Graduate School of MAINZ (MATCOR). sedlmayr@physik.uni-kl.de 1M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno, Nature (London) 428, 539 (2004). 2A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, P h y s .R e v .L e t t . 92, 077205 (2004). 3S. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 4C. H. Marrows, Adv. Phys. 54, 585 (2005). 5U. Ebels, A. Radulescu, Y . Henry, L. Piraux, and K. Ounadjela, P h y s .R e v .L e t t . 84, 983 (2000). 6H. D. Chopra and S. Z. Hua, Phys. Rev. B 66, 020403 (2002). 7C. R ¨uster, T. Borzenko, C. Gould, G. Schmidt, L. W. Molenkamp, X. Liu, T. J. Wojtowicz, J. K. Furdyna, Z. G. Yu, and M. E. Flatt ´e, P h y s .R e v .L e t t . 91, 216602 (2003). 8L. Thomas, R. Moriya, C. Rettner, and S. Parkin, Science 330, 1810 (2010). 9G. Bertotti, C. Serpico, I. Mayergoyz, R. Bonin, and M. d’Aquino,J. 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Gradmann, P h y s .R e v .B 57, R677 (1998). 20R. Wiesendanger, Rev. Mod. Phys. 81, 1495 (2009). 21E. M. Lifschitz and L. P. Pitaevskii, Statistical Physics Part 2: Theory of the Condensed State (Butterworth-Heinemann, Oxford, 2002). 22S. Blundell, Magnetism in Condensed Matter (Oxford University Press, New York, 2009). 23V . Korenman, J. L. Murray, and R. E. Prange, Phys. Rev. B 16, 4032 (1977). 24G. Tatara and H. Fukuyama, Phys. Rev. Lett. 78, 3773 (1997). 25V . K. Dugaev, J. Barna ´s, A. Łusakowski, and L. A. Turski, Phys. Rev. B 65, 224419 (2002). 26Note: in this work we will consider the transverse dynamics of the domain wall by setting /vectorM=M/vectorn,w h e r e /vectornis a unit vector ﬁeld and it is the dynamical variable here. The longitudinal dynamics in /vectorM occurs at higher energies and it is not considered here. The localcoupling constant, JM, is material and carrier-type dependent and varies in a considerable range, e.g., we estimate it to be 1 .7e Vf o r Co and 2 .24 eV for Fe. On the other hand, if a localized magnetic structure is considered, a scattering center instead of DW, say asingle Mn 12molecular magnet [H. B. Heersche, Z. de Groot, J. A. Folk, H. S. J. van der Zant, C. Romeike, M. R. Wegewijs, L. Zobbi,D. Barreca, E. Tondello, and A. Cornia, Phys. Rev. Lett. 96, 206801 (2006); M. H. Jo, J. E. Grose, K. Baheti, M. M. Deshmukh, J. J. Sokol, E. M. Rumberger, D. N. Hendrickson, J. R. Long, H. Park,a n dD .C .R a l p h , Nano Lett. 6, 2014 (2006)] we estimate JM= 1m e V ,s e eR . - Q .W a n g ,L .S h e n g ,R .S h e n ,B .W a n g ,a n dD .Y .X i n g ,Phys. Rev. Lett. 105, 057202 (2010) for the exchange coupling. 27T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 28S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). 174447-6
PhysRevB.96.024425.pdf	PHYSICAL REVIEW B 96, 024425 (2017) Relativistic theory of magnetic inertia in ultrafast spin dynamics Ritwik Mondal,Marco Berritta, Ashis K. Nandy, and Peter M. Oppeneer Department of Physics and Astronomy, Uppsala University, P .O. Box 516, SE-75120 Uppsala, Sweden (Received 7 March 2017; revised manuscript received 15 June 2017; published 18 July 2017) The inﬂuence of possible magnetic inertia effects has recently drawn attention in ultrafast magnetization dynamics and switching. Here we derive rigorously a description of inertia in the Landau-Lifshitz-Gilbertequation on the basis of the Dirac-Kohn-Sham framework. Using the Foldy-Wouthuysen transformation up to theorder of 1 /c 4gives the intrinsic inertia of a pure system through the second order time derivative of magnetization in the dynamical equation of motion. Thus, the inertial damping Iis a higher order spin-orbit coupling effect, ∼1/c4, as compared to the Gilbert damping /Gamma1that is of order 1 /c2. Inertia is therefore expected to play a role only on ultrashort timescales (subpicoseconds). We also show that the Gilbert damping and inertial damping arerelated to one another through the imaginary and real parts of the magnetic susceptibility tensor, respectively. DOI: 10.1103/PhysRevB.96.024425 I. INTRODUCTION The foundation of contemporary magnetization dynamics is the Landau-Lifshitz-Gilbert (LLG) equation which describesthe precession of spin moment and a transverse damping ofit, while keeping the modulus of magnetization vector ﬁxed[1–3]. The LLG equation of motion was originally derived phenomenologically and the damping of spin motion hasbeen attributed to relativistic effects such as the spin-orbitinteraction [ 1,4–6]. In recent years there has been a ﬂood of proposals for the fundamental microscopic mechanismbehind the Gilbert damping: the breathing Fermi surface modelof Kamberský, where the damping is due to magnetizationprecession and the effect of spin-orbit interaction at the Fermisurface [ 4], the extension of the breathing Fermi surface model to the torque-torque correlation model [ 5,7], scattering theory description [ 8], effective ﬁeld theories [ 9], linear response formalism within relativistic electronic structure theory [ 10], and the Dirac Hamiltonian theory formulation [ 11]. For practical reasons it was needed to extend the original LLG equation to include several other mechanisms [ 12,13]. To describe, e.g., current induced spin-transfer torques, theeffects of spin currents have been taken into account [ 14–16], as well as spin-orbit torques [ 17] and the effect of spin diffusion [ 18]. A different kind of spin relaxation due to the exchange ﬁeld has been introduced by Bar’yakhtar et al. [19]. In the Landau-Lifshitz-Bar’yakhtar equation nonlocal spin dissipations originate from the spatial dispersion ofexchange effects through the second order space derivativeof the effective ﬁeld [ 20,21]. A further recent work predicts the existence of extension terms that contain spatial as well astemporal derivatives of the local magnetization [ 22]. Another term, not discussed in the above investigations, is the magnetic inertial damping that has recently drawnattention [ 23–26]. Originally, magnetic inertia was discussed following the discovery of earth’s magnetism [ 27]. Within the LLG framework, inertia is introduced as an additional term[9,24,28,29] leading to a modiﬁed LLG equation, ∂M ∂t=−γM×Heff+M×/parenleftbigg /Gamma1∂M ∂t+I∂2M ∂2t/parenrightbigg ,(1) ritwik.mondal@physics.uu.sewhere /Gamma1is the Gilbert damping constant [ 1–3],γthe gyromag- netic ratio, Heffthe effective magnetic ﬁeld, and Iis the inertia of the magnetization dynamics, similar to the mass in Newton’sequation. This type of motion has the same classical analogas the nutation of a spinning symmetric top. The potentialimportance of inertia is illustrated in Fig. 1. While Gilbert damping slowly aligns the precessing magnetization to theeffective magnetic ﬁeld, inertial dynamics causes a tremblingor nutation of the magnetization vector [ 24,30,31]. Nutation could consequently pull the magnetization toward the equatorand cause its switching to the antiparallel direction [ 32,33], while depending crucially on the strength of the magneticinertia. The parameter Ithat characterizes the nutation motion is in the most general case a tensor and has been associatedwith the magnetic susceptibility [ 29,31,33]. Along a different line of reasoning, Fähnle et al. extended the breathing Fermi surface model to include the effect of magnetic inertia [ 9,34]. The technological importance of nutation dynamics is thusits potential to steer magnetization switching in memorydevices [ 23–25,32] and also in skyrmionic spin textures [35]. Magnetization dynamics involving inertial dynamics has been investigated recently, and it was suggested that itsdynamics belongs to the faster time scales [ 24,26], i.e., the femtosecond regime. However, the origin of inertial dampingfrom a fundamental framework is still missing, and, moreover,although it is possible to vary the size of the inertia inspin-dynamics simulations, it is unknown what the typicalsize of the inertial damping is. Naturally the question arises whether it is possible to derive the extended LLG equation including inertia whilestarting from the fully relativistic Dirac equation. Hickeyand Moodera [ 36] started from a Dirac Hamiltonian and obtained an intrinsic Gilbert damping term which originatedfrom spin-orbit coupling. However they started from onlya part of the spin-orbit coupling Hamiltonian which was anti-Hermitian [ 37,38]. A recent derivation based on Dirac Hamiltonian theory formulation [ 11] showed that the Gilbert damping depends strongly on both interband and intrabandtransitions (consistent with Ref. [ 39]) as well as the magnetic susceptibility response function, χ m. This derivation used the relativistic expansion to the lowest order 1 /c2of the Hermitian Dirac-Kohn-Sham (DKS) Hamiltonian including the effect ofexchange ﬁeld [ 40]. 2469-9950/2017/96(2)/024425(9) 024425-1 ©2017 American Physical SocietyMONDAL, BERRITTA, NANDY , AND OPPENEER PHYSICAL REVIEW B 96, 024425 (2017) MHeﬀ Precession Nutation FIG. 1. Schematic illustration of magnetization dynamics. The precessional motion of Maround Heffis depicted by the blue solid- dashed curve, and the nutation is shown by the red curve. In this paper we follow an approach similar to that of Ref. [ 11], but we consider higher order expansion terms of the DKS Hamiltonian up to the order of 1 /c4.T h i si ss h o w nt o lead to the intrinsic inertia term in the modiﬁed LLG equationand demonstrates that it stems from a higher-order spin-orbitcoupling term. A relativistic origin of the spin nutation angle,caused by Rashba-like spin-orbit coupling, was previouslyconcluded, too, in the context of semiconductor nanostructures[41,42]. In the following, we derive in Sec. IIthe relativistic correction terms to the extended Pauli Hamiltonian up tothe order of 1 /c 4, which includes the spin-orbit interaction and an additional term. Then the corresponding magnetizationdynamics is computed from the obtained spin Hamiltonian inSec. III, which is shown to contain the Gilbert damping and the magnetic inertial damping. Finally, we discuss the size ofthe magnetic inertia in relation to other earlier studies. II. RELATIVISTIC HAMILTONIAN FORMULATION We start our derivation with a fully relativistic particle, a Dirac particle [ 43] inside a material and in the presence of an external ﬁeld, for which we write the DKS Hamiltonian: H=cα·(p−eA)+(β−1)mc2+V1 =O+(β−1)mc2+E, (2) where Vis the effective crystal potential created by the ion-ion, ion-electron, and electron-electron interactions. In general,there can be an additional potential term for the DKS exchangeﬁeld, however, we suppress to write it explicitly here (seeRef. [ 11] for details). A(r,t) is the magnetic vector potential from the external ﬁeld, cis the speed of light, mis the particle’s mass, and 1 is the 4 ×4 unit matrix. αandβare the Dirac matrices that have the form α=/parenleftbigg0σ σ 0/parenrightbigg ,β =/parenleftbigg1 0 0−1/parenrightbigg , where σis the Pauli spin matrix vector and 1is 2×2 unit ma- trix. The Dirac equation is then written as i¯h∂ψ(r,t) ∂t=Hψ(r,t) for a Dirac bispinor ψ. The quantity O=cα·(p−eA) deﬁnes the off-diagonal or odd terms in the matrix formalism,andE=V1 are the diagonal, i.e., even terms. The latter haveto be multiplied by a 2 ×2 block diagonal unit matrix in order to bring them in a matrix form. To obtain the nonrelativisticHamiltonian and the relativistic corrections one can write downthe Dirac bispinor in double two component form as ψ(r,t)=/parenleftbiggφ(r,t) η(r,t)/parenrightbigg , and substitute those into the Dirac equation. The upper two components represent the particle, and the lower twocomponents represent the antiparticle. However the question of separating the particle’s and antiparticle’s wave functions is not clear for any given momentum. As the part α ·pis off-diagonal in the matrix formalism, it retains the odd components andthus links the particle-antiparticle wave functions. One wayto eliminate the antiparticle’s wave function is by an exacttransformation [ 44] which gives terms that require a further expansion in powers of 1 /c 2. Another way is to search for a representation where the odd terms become smaller andsmaller, and one can ignore those with respect to the eventerms and retain only the latter [ 45]. The Foldy-Wouthuysen (FW) transformation [ 46,47] was the very successful attempt to ﬁnd such a representation. It is a unitary transformation obtained by suitably choosing the FW operator, U FW=−i 2mc2βO. (3) The minus sign in front of the operator is because βand Oanticommute with each other. The transformation of the wave function adopts the form ψ/prime(r,t)=eiUFWψ(r,t) such that the probability density remains the same, \|ψ\|2=\|ψ/prime\|2. The time-dependent FW transformation can be expressed as[45,48] H FW=eiUFW/parenleftbigg H−i¯h∂ ∂t/parenrightbigg e−iUFW+i¯h∂ ∂t. (4) The ﬁrst term can be expanded in a series as eiUFWHe−iUFW=H+i[UFW,H]+i2 2![UFW,[UFW,H]]+.... +in n![UFW,[UFW,...[UFW,H]...]]+... . (5) The time dependency enters through the second term of Eq. ( 4) and for a time-independent transformation one works with ∂UFW ∂t=0. It is instructive to note that the aim of the whole procedure is to make the odd terms smaller and one cannotice that as it goes higher and higher in the expansion, thecorresponding coefﬁcients decrease of the order 1 /c 2due to the choice of the unitary operator. After a ﬁrst transformation,the new Hamiltonian will contain new even terms, E /prime,a sw e l l as new odd terms, O/primeof 1/c2or higher. The latter terms can be used to perform a next transformation having the new unitaryoperator as U /prime FW=−i 2mc2βO/prime. After a second transformation the new Hamiltonian H/prime FWis achieved that has the odd terms of the order 1 /c4or higher. The transformation is a repetitive process, and it continues until the separation of positive andnegative energy states is guaranteed. After a fourth transformation we derive the new trans- formed Hamiltonian with all the even terms that are correct up 024425-2RELATIVISTIC THEORY OF MAGNETIC INERTIA IN . . . PHYSICAL REVIEW B 96, 024425 (2017) to the order of1 m3c6as [48–50] H/prime/prime/prime FW=(β−1)mc2+β/parenleftbiggO2 2mc2−O4 8m3c6/parenrightbigg +E −1 8m2c4[O,[O,E]+i¯h∂tO] +β 16m3c6{O,[[O,E],E]}+β 8m3c6{O,[i¯h∂tO,E]} +β 16m3c6{O,(i¯h)2∂ttO}. (6) Here∂t≡∂/∂t stands for the ﬁrst-order time derivative. Note that [A,B] deﬁnes the commutator, while {A,B}represents the anticommutator for any two operators AandB.As i m - ilar Foldy-Wouthuysen transformation Hamiltonian up to anorder of 1 /m 3c6was derived by Hinschberger and Hervieux in their recent work [ 51], however there are some differences, for example, the ﬁrst and second terms in the third line of Eq. ( 6) were not given. Once we have the transformed Hamiltonian asa function of odd and even terms, the ﬁnal form is achieved bysubstituting the correct form of odd terms Oand calculating term by term. Evaluating all the terms separately, we derive the Hamiltonian for only the positive energy solutions, i.e., theupper components of the Dirac bispinor as a 2 ×2m a t r i x formalism [ 40,51,52]: H /prime/prime/prime FW=(p−eA)2 2m+V−e¯h 2mσ·B−(p−eA)4 8m3c2 −e¯h2 8m2c2∇·Etot+e¯h 8m3c2{(p−eA)2,σ·B} −e¯h 8m2c2σ·[Etot×(p−eA)−(p−eA)×Etot] −e¯h2 16m3c4{(p−eA),∂tEtot}−ie¯h2 16m3c4σ ·[∂tEtot×(p−eA)+(p−eA)×∂tEtot]. (7) The higher order terms (1 /c6or more) will involve similar formulations and more and more time derivatives of themagnetic and electric ﬁelds will appear that stem fromthe time derivative of the odd operator O[48,51]. The ﬁelds in the last Hamiltonian ( 7) are deﬁned as B=∇×A,t h e external magnetic ﬁeld, E tot=Eint+Eextare the electric ﬁelds where Eint=−1 e∇Vis the internal ﬁeld that exists even without any perturbation and Eext=−∂A ∂tis the external ﬁeld (only the temporal part is retained here because of theCoulomb gauge). A. The spin Hamiltonian The aim of this work is to formulate the magnetization dynamics on the basis of this Hamiltonian. Thus, we split theHamiltonian into spin-independent and spin-dependent partsand consider from now on electrons. The spin Hamiltonian isstraightforwardly given as H S(t)=−e mS·B+e 4m3c2{(p−eA)2,S·B} −e 4m2c2S·[Etot×(p−eA)−(p−eA)×Etot]−ie¯h 8m3c4S·[∂tEtot×(p−eA) +(p−eA)×∂tEtot], (8) where the spin operator S=(¯h/2)σhas been used. Let us brieﬂy explain the physical meaning behind each term thatappears in H S(t). The ﬁrst term deﬁnes the Zeeman coupling of the electron’s spin with the externally applied magneticﬁeld. The second term deﬁnes an indirect coupling of light tothe Zeeman interaction of spin and the optical Bﬁeld, which can be shown to have the form of a relativistic Zeeman-liketerm. The third term implies a general form of the spin-orbitcoupling that is gauge invariant [ 53], and it includes the effect of the electric ﬁeld from an internal as well as an external ﬁeld.The last term is the new term of relevance here that has onlybeen considered once in the literature by Hinschberger et al. [51]. Note that, although the last term in Eq. ( 8) contains the total electric ﬁeld, only the time derivative of the external ﬁeldplays a role here, because the time derivative of internal ﬁeldis zero as the ionic potential is time independent. In generalif one assumes a plane-wave solution of the electric ﬁeld inMaxwell’s equation as E=E 0eiωt, the last term can be written ase¯hω 8m3c4S·(E×p) and thus adopts the form of a higher-order spin-orbit coupling for a general Eﬁeld. The spin-dependent part can be easily rewritten in a shorter format using the identities: A×(p−eA)−(p−eA)×A=2A×(p−eA) +i¯h∇×A (9) A×(p−eA)+(p−eA)×A=−i¯h∇×A (10) for any operator A. This allows us to write the spin Hamilto- nian as HS=−e mS·B+e 2m3c2S·B/bracketleftbigg p2−2eA·p+3e2 2A2/bracketrightbigg −e 2m2c2S·[Etot×(p−eA)]+ie¯h 4m2c2S·∂tB +e¯h2 8m3c4S·∂ttB. (11) Here, the Maxwell’s equations have been used to derive the ﬁnal form that the spatial derivative of the electric ﬁeld willgenerate a time derivative of a magnetic ﬁeld such that ∇× E ext=−∂B ∂t, while the curl of an internal ﬁeld results in zero as the curl of a gradient function is always zero. The ﬁnal spinHamiltonian ( 11) bears much importance for the strong laser ﬁeld-matter interaction as it takes into account all the ﬁeld-spin coupling terms. It is thus the appropriate fundamentalHamiltonian to understand the effects of those interactions onthe magnetization dynamics described in Sec. III. B. Single Dirac particle spin dynamics Although the dynamics of a macroscopic magnetization is important for many technological applications, the dynamicsof a single spin- 1 2Dirac particle is of fundamental value in its own right. Assuming that the electron’s spin does not explicitlydepend on time, the single spin dynamics in the Heisenberg 024425-3MONDAL, BERRITTA, NANDY , AND OPPENEER PHYSICAL REVIEW B 96, 024425 (2017) picture reduces to ∂S ∂t=1 i¯h[S,HS(t)]. (12) The derivation of the ensuing spin dynamics is then straight- forward, substituting the Hamiltonian terms given in ( 11) in the equation of motion and carrying out the differentcommutators. We use the commutator algebra of two spins[S j,Sk]=i¯h/epsilon1jklSl. The spin dynamics of a single Dirac particle becomes ∂S ∂t=e mS×B−e 2m3c2S×B/bracketleftbigg p2−2eA·p+3e2 2A2/bracketrightbigg +e 2m2c2S×[Etot×(p−eA)] −ie¯h 4m2c2S×∂tB−e¯h2 8m3c4S×∂ttB. (13) This is an insightful result describing the relativistic (up to the order of 1 /c4) spin dynamics of a single electron. Even though this is given as spin angular momentum operator dynamics, itcan be recognized, ﬁrst, that the dynamics contains a spinprecession (ﬁrst two terms), stemming from the commonnonrelativistic precession (ﬁrst term) and a relativistic cor-rection to it (second term). Note that there is no exchangeﬁeld for a single spin- 1 2particle. The third term describes the dynamics due to the conventional spin-orbit interaction andthe spin-orbit torque due to the applied electromagnetic ﬁeld.The fourth term containing the ﬁrst-order time derivative ofthe magnetic ﬁeld causes the transversal relaxation of spins(reminiscent of the Gilbert damping). The last term, whichcontains the second-order time derivative of Bleads to nutation (see below) and thus establishes the existence of nutation evenfor a single spin. The obtained single spin dynamics thuscontains precession, damping, electromagnetic ﬁeld torque,and nutation and should be valid for any Dirac spin- 1 2particle under the inﬂuence of an external electromagnetic ﬁeld (cf.Refs. [ 54,55]). III. MAGNETIZATION DYNAMICS In general, magnetization is given by the magnetic moment per volume element in a magnetic solid. Our next goal isto derive the dynamics of such a magnetization element. Wework here in the framework of the DKS Hamiltonian thatcan be seen as an effective collective Hamiltonian describingall the electrons in a system. The general equation for theexpectation value of an observable Ois/angbracketleftO/angbracketright=Tr[ρO], where ρis the density matrix. Considering that we are working in the Heisenberg picture, the density matrix does not evolve withtime, so we can assume it to be the (diagonal) density matrix,which in the energy eigenstate representation adopts the form: ρ nk;nk(r)=f(Enk)ψ∗ nk(r)ψnk(r), (14) where f(Enk) is the Fermi-Dirac distribution, ψnk(r)a r e eigenstates of the DKS Hamiltonian in coordinate represen-tation, and E nkare the corresponding nth band electronic energies with momentum k. Here we are focusing on the magnetization of some small volume, which can be writtenas an expectation of the collective spin of the electrons, as M(t)=gμB /Omega1/integraldisplay /Omega1drTrσ[Sρ(r)], (15) where /Omega1is a suitably chosen volume, e.g., the unit cell volume. At this point we can make a partition of the unit volume of theconsidered material, for instance taking volume elements /Omega1 j enclosing the individual atoms of the unit cell. In this way we can split the integral in Eq. ( 15) in different volume elements and obtain information on the magnetization localized on eachindividual atom of the unit cell. We can thus write: M(t)=/summationdisplay jgμB /Omega1/integraldisplay /Omega1jdrTrσ[Sρ(r)]=/summationdisplay jgμB /Omega1/angbracketleftSj/angbracketright. (16) Next, we introduce a coarse graining for the macroscopic material, where the spacial coordinate is associated with theposition of one of the unit cells or atomic volumes at position R j, i.e.,M(r,t)=M(t)\|r=Rj. To derive the dynamics, we take the time derivative on both sides of Eq. ( 16) and, employing the Heisenberg equation of motion, we arrive at the equation for the magnetizationdynamics as [ 36,56,57] ∂M ∂t=/summationdisplay jgμB /Omega11 i¯h/angbracketleft[Sj,HS(t)]/angbracketright. (17) Now the task looks simple; one needs to substitute the spin Hamiltonian ( 11) and calculate the commutators in order to ﬁnd the equation of motion. Note that the dynamics onlyconsiders the local dynamics as we have not taken into accountthe time derivative of the particle density operator (for details,see Ref. [ 11]). Incorporating the latter would give rise to local as well as nonlocal processes (i.e., spin currents) within thesame footing. The ﬁrst term in the spin Hamiltonian produces the dynamics as ∂M (1) ∂t=−γM×B, (18) where γ=g\|e\|/2mdeﬁnes the gyromagnetic ratio and the Landé gfactor g≈2 for spins, the electronic charge e<0. Using the linear relationship of magnetization with the magnetic ﬁeld B=μ0(H+M), the latter is replaced in Eq. ( 18) to get the usual form in the Landau-Lifshitz equation, −γ0M×H, where γ0=μ0γis the effective gyromagnetic ratio. This gives the Larmor precession of magnetizationaround an effective ﬁeld H. The effective ﬁeld will always have a contribution from the exchange ﬁeld (and the relativisticcorrections to it), which has not been explicitly written outin this paper, as they are not in the focus here. For detailedcalculations yet including the exchange ﬁeld, see Ref. [ 11]. The second and third terms in the Hamiltonian ( 11) contain products of spin and orbital degrees of freedom. At this pointit is important to notice that neither the spin nor the orbitaldegrees of freedom commute with the Hamiltonian due to thespin-orbit coupling and the 1 /c 4corrections, which implies that the equilibrium density matrix ρcannot be expressed exactly as ρ=ρS⊗ρOwhere ρSis the reduced density matrix for the spin degrees of freedom and ρOis the reduced density matrix of the orbital degrees of freedom. Considering 024425-4RELATIVISTIC THEORY OF MAGNETIC INERTIA IN . . . PHYSICAL REVIEW B 96, 024425 (2017) an observable Oacting on the orbital degrees of freedom in Hilbert space (for instance the momentum or the orbitalangular momentum or some function depending on them) and Sthe spin, due to the impossibility to separate orbital and spin parts of the density matrix we are not, in principle, allowed towrite Tr[ρSO]=Tr[ρ SS]T r [ρOO]=M/angbracketleftO/angbracketright. (19) It is important to realize that the nonseparability (entangle- ment) of the orbital and spin parts of the density matrixis due to the spin-orbit coupling and its corrections (sinceit prevents both quantities to be conserved). However, inferromagnetic materials the energy separation of the spinstates is mostly due to the exchange magnetic ﬁeld whichis orders of magnitude larger than the spin-orbit couplingand its corrections. As a consequence, the separation of thedensity matrix as a direct product of spin and orbital partsis a good approximation, therefore we can employ Eq. ( 19). Moreover, due to the large splitting of spin bands and thecontinuous smooth behavior of the energy levels as a functionof momentum, the out-of-equilibrium dynamics on the latterdegrees of freedom is faster than the dynamics of the spindegrees of freedom. Using now the approximation ( 19), the second term in the spin Hamiltonian ( 11) will result in a relativistic correction to the magnetization precession. Within a uniform ﬁeldapproximation ( A=B×r/2), the corresponding dynamics will take the form ∂M (2) ∂t=γ 2m2c2M×B/angbracketleftbigg p2−eB·L+3e2 8(B×r)2/angbracketrightbigg , (20) withL=r×pthe orbital angular momentum. The presence ofγ/2m2c2implies that the contribution of this dynamics to the precession is relatively small, while the leading precessiondynamics is given by Eq. ( 18). For sake of completeness we note that a relativistic correction to the precession term ofsimilar order 1 /m 2c2was obtained previously for the exchange ﬁeld [ 11]. The next term in the Hamiltonian is a bit tricky to handle as the third term in Eq. ( 11) is not Hermitian, not even the fourth term which is anti-Hermitian. However together they form a Hermitian Hamiltonian [ 11,37,38]. Therefore one has to work together with those terms and cannot only perform thedynamics with an individual term. In an earlier work [ 11]w e have shown that taking a uniform magnetic ﬁeld along with thegauge A= B×r 2will preserve the Hermiticity. This uniform- ﬁeld condition is usually fulﬁlled for thin-ﬁlm samples, wherethe skin depth of the electromagnetic ﬁeld is longer than theﬁlm thickness. For thicker samples a ﬁeld that is uniform overa part of the sample could alternatively be introduced. Thedynamical equation of spin motion with the second and thirdterms can thus be written in a compact form for harmonicapplied ﬁelds as [ 11] ∂M (3,4) ∂t=M×/parenleftbigg A·∂M ∂t/parenrightbigg , (21)with the intrinsic Gilbert damping parameter Athat is a tensor deﬁned by Aij=γμ 0 4mc2/summationdisplay n,k[/angbracketleftripk+pkri/angbracketright−/angbracketleftrnpn+pnrn/angbracketrightδik] ×/parenleftbig 1+χ−1 m/parenrightbig kj. (22) Here χmis the magnetic susceptibility tensor of rank 2 (a 3×3 matrix), 1is the 3 ×3 unit matrix, and /angbracketleft ···/angbracketright stands for the expectation value with respect to the DKSelectronic states ψ nk. Note that for diagonal terms, i.e., i=kthe contributions from the expectation values of rkpi cancel each other. The damping tensor can be decomposed and shown to have contributions from isotropic Heisenberg-like, anisotropic Ising-like, and Dzyaloshinskii-Moriya-liketensors, as detailed in the following. First, the damping tensorA ijis decomposed into a symmetric and an antisymmetric part deﬁned as Asym ij=1 2(Aij+Aji) and Aanti ij=1 2(Aij−Aji). The symmetric tensor can further be written as Asym ij= Iij+αδij, where αdeﬁnes the isotropic diagonal components, i.e., the Heisenberg-like contribution, and Iijare the Ising- like contributions. Note that if the Heisenberg contributionsare such that α= 1 3Tr{Asym ij}, the trace of the Ising-like contributions becomes zero, Tr {Iij}=0. The antisymmetric matrix can be decomposed into a vector multiplied by theantisymmetric Levi-Civita tensor, A anti ij=/epsilon1ijkDk, which gives the Dzyaloshinskii-Moriya-like contribution. The completedamping dynamics can then be written as [ 11] ∂M (3,4) ∂t=αM×∂M ∂t+M×/bracketleftbigg I·∂M ∂t/bracketrightbigg +M×/bracketleftbigg D×∂M ∂t/bracketrightbigg . (23) The antisymmetric Dzyaloshinskii-Moriya term has been shown to lead to a chiral damping [ 11]; experimental observa- tions of such damping have been reported recently [ 58]. The other cross term having the form E×Ain Eq. ( 11) is related to the angular momentum of the electromagnetic ﬁeld andthus provides a torque on the spin that has been at the heartof angular magnetoelectric coupling [ 53]. This relativistic Hamiltonian providing spin-photon coupling has been shownrecently [ 59] to explain the coherent ultrafast magnetism observed in pump-probe experiments [ 60]. A possible effect in spin dynamics including the light’s angular momentum hasbeen investigated in the strong ﬁeld regime, and it has beenshown that one has to include this cross term in the dynamicsin order to explain the qualitative and quantitative strong ﬁelddynamics [ 55]. For the last term in the spin Hamiltonian ( 11) it is rather easy to formulate the spin dynamics because it is evidentlyHermitian. Working out the commutator with the spins givesa contribution to the dynamics as ∂M (5) ∂t=δM×∂2B ∂t2, (24) with the constant δ=γ¯h2 8m2c4. Let us work explicitly with the second-order time derivative of the magnetic induction by the relation B=μ0(H+M), 024425-5MONDAL, BERRITTA, NANDY , AND OPPENEER PHYSICAL REVIEW B 96, 024425 (2017) using a chain rule for the derivative: ∂2B ∂t2=∂ ∂t/parenleftbigg∂B ∂t/parenrightbigg =μ0∂ ∂t/parenleftbigg∂H ∂t+∂M ∂t/parenrightbigg =μ0/parenleftbigg∂2H ∂t2+∂2M ∂t2/parenrightbigg . (25) This is a generalized equation for the time derivative of the magnetic induction which can be used even for nonharmonicﬁelds. The magnetization dynamics is then given by ∂M (5) ∂t=μ0δM×/parenleftbigg∂2H ∂t2+∂2M ∂t2/parenrightbigg . (26) Thus the extended LLG equation of motion will have these two additional terms: (1) a ﬁeld-derivative torque and (2)magnetization-derivative torque, and they appear with theirsecond order time derivative. It deserves to be noted that, ina previous theory we also obtained a similar term—a ﬁeld-derivative torque in ﬁrst order-time derivative appearing in thegeneralized Gilbert damping. Speciﬁcally, the extended LLGequation for a general time-dependent ﬁeld H(t) becomes ∂M ∂t=−γ0M×H+M×/bracketleftbigg ¯A·/parenleftbigg∂H ∂t+∂M ∂t/parenrightbigg/bracketrightbigg +μ0δM×/parenleftbigg∂2H ∂t2+∂2M ∂t2/parenrightbigg , (27) where ¯Ais a modiﬁed Gilbert damping tensor (for details, see Ref. [ 11]). However for harmonic ﬁelds, the response of the fer- romagnetic materials is measured through the differentialsusceptibility, χ m=∂M/∂H, because there exists a net magnetization even in the absence of any applied ﬁeld. With this, the time derivative of the harmonic magnetic ﬁeld can befurther written as: ∂ 2H ∂t2=∂ ∂t/parenleftbigg∂H ∂M∂M ∂t/parenrightbigg =∂ ∂t/parenleftbigg χ−1 m·∂M ∂t/parenrightbigg =∂χ−1 m ∂t·∂M ∂t+χ−1 m·∂2M ∂t2. (28) In general the magnetic susceptibility is a spin-spin response function that, in reciprocal space, is wave-vector and frequencydependent, χ m=χm(q,ω). Substituting this expression in Eq. ( 26), the dynamics assumes the form with the ﬁrst and second order time derivatives as ∂M(5) ∂t=M×/parenleftbigg K·∂M ∂t+I·∂2M ∂t2/parenrightbigg , (29) where the parameters Iij=μ0δ(1+χ−1 m)ijandKij= μ0δ∂t(χ−1 m)ijare tensors. The dynamics of the second term is that of the magnetic inertia that operates on shorter timescales [ 25]. Having all the required dynamical terms, ﬁnally the full magnetization dynamics can be written by joining togetherall the individual parts. Thus the full magnetization dynamicsbecomes, for harmonic ﬁelds, ∂M ∂t=M×/parenleftbigg −γ0H+/Gamma1·∂M ∂t+I·∂2M ∂t2/parenrightbigg .(30)Note that the Gilbert damping parameter /Gamma1has two con- tributions, one from the susceptibility itself, Aij, which is of order 1 /c2and an other from the time derivative of it,Kijof order 1 /c4. Thus, /Gamma1ij=Aij+Kij. However we will focus on the ﬁrst one only as it will obviouslybe the dominant contribution, i.e., /Gamma1 ij≈Aij. Even though we consider only the Gilbert damping term of order 1 /c2in the discussions, we shall explicitly analyze the other term ofthe order 1 /c 4. For an ac susceptibility, i.e., χ−1 m∝eiωtwe ﬁnd that Kij∝μ0δ∂t(χ−1 m)ij∝iμ0ωδ χ−1 m, which suggests again that the Gilbert damping parameter of the order 1 /c4 will be given by the imaginary part of the susceptibility, Kij∝−μ0ωδIm(χ−1 m). The last equation ( 30) is the central result of this work, as it establishes a rigorous expression for the intrinsic magneticinertia. Magnetization dynamics including inertia has beendiscussed in a few earlier articles [ 24,30,31,61] .T h ev e r yl a s t term in Eq. ( 30) has been associated previously with the inertial magnetization dynamics [ 32,62,63]. As mentioned, it implies a magnetization nutation, i.e., a changing of the precessionangle as time progresses. Without the inertia term we obtainthe well-known LLG equation of motion that has already beenused extensively in magnetization dynamics simulations (see,e.g., Refs. [ 64–68]). IV . DISCUSSIONS Magnetic inertia was discussed ﬁrst in relation to the earth’s magnetism [ 27]. From a dimensional analysis, the magnetic inertia of a uniformly magnetized sphere undergoing uniformacceleration was estimated to be of the order of 1 /c 2[27], which is consistent with the here-obtained relativistic natureof magnetic inertia. The spin dynamics derived for a single Dirac particle [Eq. ( 13)] is a general and fundamental result, which estab- lishes the existence of nutation even for any Dirac particle.To describe the magnetization dynamics of a small volumeelement, we introduce a collective macroscopic variable M, stemming from the spin degrees of freedom, where the otherdegrees of freedom (e.g., electronic orbitals, environments) areaveraged out. The derived magnetization dynamics, based on the fun- damental Dirac-Kohn-Sham Hamiltonian, provides explicitexpressions for both the Gilbert and inertial dampings. Thus,a comparison can be made between the Gilbert dampingparameter and the magnetic inertia parameter of a pure system.As noticed above, both the parameters depend on the magneticsusceptibility tensor, however it should be noted that thequantity /angbracketleftr αpβ/angbracketrightis imaginary itself, because [ 11] /angbracketleftrαpβ/angbracketright=−i¯h 2m/summationdisplay n,n/prime,kf(Enk)−f(En/primek) Enk−En/primekpα nn/primepβ n/primen.(31) Here the momentum matrix elements pα nn/primeare taken with respect to the states ψnkthat follow from the DKS Hamiltonian (2) or (approximately) from Hamiltonian ( 6). The Gilbert damping parameter should consequently be given by theimaginary part of the susceptibility tensor [ 36,69]. On the other hand the magnetic inertia tensor must be given by the real partof the susceptibility [ 31]. This is in agreement with a recent 024425-6RELATIVISTIC THEORY OF MAGNETIC INERTIA IN . . . PHYSICAL REVIEW B 96, 024425 (2017) article where the authors also found the same dependence of real and imaginary parts of susceptibility to the nutationand Gilbert damping, respectively [ 33]. In our calculation, the Gilbert damping and inertia parameters adopt the followingforms, respectively, /Gamma1 ij=iγμ 0 4mc2/summationdisplay n,k[/angbracketleftripk+pkri/angbracketright−/angbracketleftrnpn+pnrn/angbracketrightδik] ×Im/parenleftbig χ−1 m/parenrightbig kj =−μ0γ¯h 4mc2/summationdisplay n,k/bracketleftbigg/angbracketleftripk+pkri/angbracketright−/angbracketleftrnpn+pnrn/angbracketrightδik i¯h/bracketrightbigg ×Im/parenleftbig χ−1 m/parenrightbig kj =−ζ/summationdisplay n,k/bracketleftbigg/angbracketleftripk+pkri/angbracketright−/angbracketleftrnpn+pnrn/angbracketrightδik i¯h/bracketrightbigg ×Im/parenleftbig χ−1 m/parenrightbig kj, (32) Iij=μ0γ¯h2 8m2c4/bracketleftbig 1+Re/parenleftbig χ−1 m/parenrightbig ij/bracketrightbig =ζ¯h 2mc2/bracketleftbig 1+Re/parenleftbig χ−1 m/parenrightbig ij/bracketrightbig , (33) withζ≡μ0γ¯h 4mc2. Note that the change of sign from damping tensor to the inertia tensor is also consistent with Ref. [ 33] and also that a factor of two is present in inertia. However, mostimportantly, the inertia tensor is ¯ h/mc 2times smaller than the damping tensor as is revealed in our calculations. Consideringatomic units we can evaluate ζ∼μ 0 4c2∼0.00066 4×1372∼8.8×10−9, ζ¯h 2mc2∼ζ 2c2∼8.8×10−9 2×1372∼2.34×10−13. This implies that the intrinsic inertial damping is typically 4×104times smaller than the Gilbert damping, and it is not an independently variable parameter. Also, becauseof its smallness magnetic inertial dynamics will be moresigniﬁcant on shorter timescales [ 24,26]. As recently outlined by Wegrowe and Olive the quite different time scales of Gilbertand inertial dampings can be exploited to study the effect ofthe fast inertial motion on the slower precessional motion [ 26]. A further analysis of the Gilbert and inertial parameters can be made. One can use the Kramers-Kronig transformationto relate the real and imaginary parts of a susceptibility tensorwith one another. This suggests a relation between the twoparameters that has been found previously by Fähnle et al. [34], namely I=−/Gamma1τ, where τis a relaxation time. We obtain here a similar relation, I∝−/Gamma1¯τ, where ¯ τ=¯h/mc 2 has time dimension. Even though the Gilbert damping is c2times larger than the inertial damping, the relative strength of the twoparameters also depends on the real and imaginary parts ofthe susceptibility tensor. In special cases, when the real partof the susceptibility is much higher than the imaginary part,their strength could be comparable to each other. We note inthis context that there exist materials where the real part of thesusceptibility is 10 2–103times larger than the imaginary part.Finally, we emphasize that our derivation provides the intrinsic inertial damping of a pure, isolated system. For the Gilbert damping it is already well known that environmentaleffects, such as interfaces or grain boundaries, impurities, ﬁlmthickness, and even interactions of the spins with quasiparti-cles, for example, phonons, can modify the extrinsic damping(see, e.g., Refs. [ 70–72]). Similarly, it can be expected that the inertial damping will become modiﬁed through environmentalinﬂuences. An example of environmental effects that canlead to magnetic inertia have been considered previously, forthe case of a local spin moment surrounded by conductionelectrons, whose spins couple to the local spin moment andaffect its dynamics [ 31,32]. V . CONCLUSIONS In conclusion, we have rigorously derived the magneti- zation dynamics from the fundamental Dirac Hamiltonianand have provided a solid theoretical framework for, andestablished the origin of, magnetic inertia in pure systems.For a single spin- 1 2Dirac particle under the inﬂuence of an electromagnetic ﬁeld we have derived the relativistic spindynamics and showed that it contains an inertia term. For thedynamics of a macroscopic magnetic volume element, we havederived expressions for the Gilbert damping and the magneticinertial damping on the same footing and have shown that bothof them have a relativistic origin. The Gilbert damping stemsfrom a generalized spin-orbit interaction involving externalﬁelds, while the inertial damping is due to higher-order (in1/c 2) spin-orbit contributions in the external ﬁelds. Both have been shown to be tensorial quantities. For general timedependent external ﬁelds, a ﬁeld-derivative torque with a ﬁrstorder time derivative appears in the Gilbert-type damping,and a second order time-derivative ﬁeld torque appears in theinertial damping. In the case of harmonic external ﬁelds, the expressions of the magnetic inertia and the Gilbert damping scale withthe real part and the imaginary part, respectively, of themagnetic susceptibility tensor, and they are opposite in sign.Alike the Gilbert damping, the magnetic inertia tensor isalso temperature dependent through the magnetic responsefunction and also magnetic moment dependent. Importantly,we ﬁnd that the intrinsic inertial damping is much smaller thanthe Gilbert damping, which corroborates the fact that magneticinertia was neglected in the early work on magnetizationdynamics [ 1–3,19]. This suggests, too, that the inﬂuence of magnetic inertia will be quite restricted, unless the real partof the susceptibility is much larger than the imaginary part.Another possibility to enhance the magnetic inertia wouldbe to use environmental inﬂuences to increase its extrinsiccontribution. Our theory based on the Dirac Hamiltonian leadsto exact expressions for both the intrinsic Gilbert and inertialdamping terms, thus providing a solid base for their evaluationwithin ab initio electronic structure calculations and giving suitable values that can be used in future LLG magnetizationdynamics simulations. ACKNOWLEDGMENTS We thank D. Thonig and A. Aperis for useful discussions. 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PhysRevLett.125.237203.pdf	Highly Anisotropic Magnetic Domain Wall Behavior in In-Plane Magnetic Films Xiaochao Zhou ,1,2Nicolas Vernier,2,3,Guillaume Agnus,2Sylvain Eimer ,2Weiwei Lin,1and Ya Zhai1,† 1School of Physics and Quantum Information Research Center, Southeast University, 211189 Nanjing, China 2Centre de Nanosciences et de Nanotechnologies, CNRS, Universit´ e Paris-Saclay, 91120 Palaiseau, France 3Laboratoire Lumi` ere, Mati` ere et Interfaces, Universit´ e Paris-Saclay, 91405 Orsay, France (Received 8 May 2020; revised 31 July 2020; accepted 13 October 2020; published 1 December 2020) We have studied the nucleation of magnetic domains and propagation of magnetic domain walls (DWs) induced by pulsed magnetic field in a ferromagnetic film with in-plane uniaxial anisotropy. In contrast toobserved behavior in films with out-of-plane anisotropy, the nucleated domains have a rectangular shape in which a pair of the opposite sides are perfectly linear DWs, while the other pair present zigzags. The field induced propagation of these two DW types are found to be different. The linear ones follow a creep lawidentical to what is usually observed in out-of-plane films, while the velocity of zigzag DWs depends linearly on the applied field amplitude down to very low field. This unexpected feature can be explained by the shape of the DW, and these results provide first experimental evidence of the applicability of the 1Dmodel in two-dimensional ferromagnetic thin films. DOI: 10.1103/PhysRevLett.125.237203 Introduction. —The propagation of magnetic domain walls (DWs) has been widely studied for out-of-plane magnetic thin films [1–6]. In such films, the absence of inversion symmetry breaking and related phenomena such as Dzyaloshinskii-Moriya interactions [7–11] result in highly isotropic DW propagation. As a result, when the applied field is sufficient to overcome pinning effects, theshape of magnetic domains is expected to become circular [3,4]. While such issues have been investigated in much detail in perpendicularly magnetized films, surprisingly very few works about DW propagation in in-plane full films have been reported to date. The main difficulty resides in the high DW velocities in such samples which renders their detection difficult with the typical field ofview of a longitudinal Kerr effect microscope. With the exception of GaMnAs, for which it has been possible to carry out a detailed study of the field dependence of DW velocity [12], it is generally necessary to infer the velocity from other means, such as measurements of the magnetic relaxation [13]and analyses of the laser spot polarization in nanostructured wires [14]. However, most of these have been limited to the one-dimensional limit of wall propa- gation as the wall extends across the width of the wire and can be considered as a rigid object along this dimension. This no longer holds in two-dimensional films in which the domain walls can deform, particularly in the vicinity of pinning defects. The dynamical behavior of domain walls in full in-plane magnetized fields therefore remains relatively unexplored. In particular, open questions include the role of short, high- amplitude field pulses, the processes by which nucleation takes place, how the wall propagates, and the ensuing domain shape. Here, we present a study of the dynamicbehavior of DW motion in in-plane magnetic films that go toward addressing some of these issues. Our experimentalsetup allows field pulses of durations as short as 1μs and of amplitudes up to 2 mT, making it possible to reach the fast velocity regime. The experimental results are analyzed inthe framework of a 1D model. Samples and experiments. —Sample fabrication and magnetic properties: The results we report have been obtained in films with the composition Si =SiO 2ðfew nm Þ= Tað2nmÞ=CoFeB ð30nmÞ=Tað1nmÞ. The films were grown at 300 K on Si(100) substrates with native oxide(spontaneous oxidation at room temperature) by a high vacuum dc sputtering system. No annealing was performed on the samples, which results in the CoFeB film retaining its amorphous as-grown state, with a roughness of around 1 nm. During the film growth, an in-plane magnetic fieldof around 1 mT was applied which induces a uniaxial anisotropy in the CoFeB layer. The target material is Co 60Fe20B20. In order to protect the magnetic layer from oxidation, a Ta layer was sputtered above the magnetic film. Several preliminary experiments have been performed to determine the magnetic properties of the film. The satu- ration magnetization Mswas measured by a vibrating sample magnetometer was found to be 9.6×105Am−1. An in-plane Kerr hysteresis loop using longitudinal mag- neto-optic Kerr effect has enabled us to check the coercive field as well as the anisotropy. The easy axis (EA) and hard axis (HA) have been identified (Fig. 1) by examining the hysteresis loops as a function of the angle between the longedge of sample and the applied magnetic field. The coercive field has been determined to be 0.40 mT along the EA. Moreover, Kerr loops measured along the HA werePHYSICAL REVIEW LETTERS 125, 237203 (2020) 0031-9007 =20=125(23) =237203(5) 237203-1 © 2020 American Physical Societyused to estimate the in-plane magnetic anisotropy (Fig. 1). The shape of the loop is consistent with an anisotropyenergy of the form E a¼−KaM2scos2φ, where φis the angle between the magnetization and the EA [15–19], with both directions being in the film plane. The anisotropy field μ0Hk¼2Ka=Mshas been found to be 3.5 mT, which is in agreement with the result of complementary ferromagnetic resonance measurements (see the second part in Supplemental Material [20]) from which we also extracted the Gilbert damping constant α¼0.0085 with an error within 5%. Magnetic domain wall velocity measurements: DW motion was investigated by a longitudinal Kerr microscope at room temperature [21–24]. In this setup, a parallel polarized light beam is directed toward the sample with an incidence angle of 45°, giving rise to a relatively large longitudinal Kerr rotation. The reflected beam was focusedon a CMOS camera, where the CMOS sensor plane and theobjective were slightly tilted with respect to the beam propagation axis [21,24] so that image plane of the film was in the plane of the sensor. The spatial resolution of thissetup is around 30μm. The coplanar magnetic pulse field was produced by a small coil of radius 17.5 mm centered on the sample. The field-of-view of the microscope is less than 10 mm, so that the field created by the coil is uniform within a precision of2% over the area studied. The inductance of the coil is between 4 and 40μH depending on the coil used, making it possible to create very short field pulses. A high voltagepulse generator was used, so that, with the coil in serial with a resistive charge of 50Ω, we could obtain current pulses up to 15 A, corresponding to a magnetic field of 2 mT. Thefastest rise time with the lowest inductance coil achievedwas 83 ns, making it possible to have pulses as short as1μs. The sample holder was made of plastic, i.e., an insulating material, so that no eddy currents were induced which can modify the characteristics of the magnetic field generated by the coil. For the work presented here, themagnetic field was always applied along the EA of thesample. DW velocities were measured with the usual stroboscopic approach and Kerr microscopy [1,2,12] (see Supplemental Material [20]). Results and discussion. —A typical example of DW motion is shown in Fig. 2in which Fig. 2(a) shows the nucleation and Fig. 2(b) shows the domains after propa- gation due to the second pulse. Figure 2(c) gives the difference between these two images, which allows the propagation distances to be more easily identified. The first notable observation concerns the shape of the nucleated domains [Fig. 2(a)], along with their shape after wall propagation [Fig. 2(b)]. In both of these cases, we observe similar highly anisotropic rectangular forms. While minimization of the magnetostatic energy can lead to rectangular domains [24,25] , it does not necessarily follow that such shapes remain metastable after wall propagation.In addition, we note that the boundary DW along the horizontal segment is almost straight, while the walls along the vertical boundaries exhibit zigzag structures [Fig. 2(d)]. This again can be explained by static energy minimization [24,26 –29]: along the vertical boundaries, because of the in-plane anisotropy, a straight vertical DW would meanhead-to-head or tail-to-tail DW (also called charged DW). To avoid the high energetic cost due to this kind of DW, zigzags appear [24,25,30] . Here, we have found that the zigzag angle β of the DW (Fig. 3) does not depend on the FIG. 1. Kerr hysteresis loops for magnetic in-plane field parallel to easy magnetization axis (EA) and hard magnetizationaxis (HA). FIG. 2. Typical sequence to measure velocity when the lengthof the pulsed field is much longer than the rise time. Starting froma saturated state, (a) and (b) show the full-view Kerr images afterthe application of the (a) first and (b) second pulse field. Thepulsed magnetic field (yellow arrow) was parallel to EA (reddash) with amplitude of 1 mT and length of 1.6μs. (c) shows the DW displacement during the second pulse Δt¼1.6μs. (d) presents the magnification of a rectangle domain with twotypes of DW (horizontal straight and vertical zigzag) being indicated. White (black) arrows denote the magnetization direc- tions outside (inside) the domain.PHYSICAL REVIEW LETTERS 125, 237203 (2020) 237203-2amplitude of the external pulse field and its value has been found to be approximately 22° ( /C61.5°), similar to the values found in other samples [30]. This can be explained by assuming some small movements induced by magnetostaticforces after the end of the magnetic pulse, so that βis defined during these movements and is independent of the applied pulse. The second notable result concerns the velocity, where an asymmetry is seen between the straight, horizontal DW and the zigzag, vertical DW, as shown in Fig. 2(c). The velocities for the two DW types are shown in Fig. 4. First, vertical zigzag DWs go faster than the horizontal straight DW. Second, the corresponding field dependence ofthe wall velocity is quite different for the two cases: forhorizontal straight DWs, we observe motion in the creep regime, which is usually observed in out-of-plane thin filmsand is described by the relationship [1,2] vðHÞ¼v 0exp/C20/C18Hp H/C191=4/C21 : ð1Þ The observed velocity range extends over almost 2 orders of magnitude. In general, pinning sites are commonly observed in magnetic films, so it is not surprising torecover such behavior in in-plane magnetized films.However, the behavior is quite different for the zigzagDW, where a linear velocity versus field relationship isfound where remarkably the intersect at zero field is at zerovelocity. While such behavior is consistent with the 1Dmodel [31], it is incompatible with the presence of pinning effects which would result in a finite pinning field for wall displacement. Analysis. —Below the Walker transition, the one- dimensional wall model predicts the following lineardependence of the wall velocity as a function of applied field: vðHÞ¼ γμ0HΔ α; ð2Þ where αis the Gilbert damping parameter, γthe gyromag- netic ratio, μ0the vacuum permittivity, Hthe applied field, andΔthe width of the DW, assuming a spatial profile of the form [31,32] , φ¼2arctan/C20 exp/C18x−x0 Δ/C19/C21 : ð3Þ Based on these relations, we can extract the wall width Δ from the experimental data. From a linear fit of the curve in Fig. 4, we find Δ¼17nm, which gives a domain wall width of πΔ¼53nm. This value is relatively small for an in-plane magnetized film, but it is of the right order ofmagnitude [24,25] , since other authors have also reported such narrow walls [14] and the nature of the DW can be difficult to ascertain [24,25] . We therefore consider this value as an “effective ”width and this is one more point for the validity of 1D model. Let us note that we have not beenable to view the Walker breakdown: using the threshold αM s=2of bulk material [31], we have found a typical value of 6 mT. Because our sample was a thin layer, in which theDW ’s driving field is more complicated to evaluate, the Walker breakdown might be different [33]. However, from this estimated value, we can think we did not reach thecritical Walker value. Now, the question is why does the 1D model apply for zigzag DW when it obviously does not for horizontalstraight DW? A possible explanation is that zigzag is a wayof inhibiting the pinning defect effect. Indeed, if the DWmeets a pinning defect, it can stay pinned at the veryposition of the defect while it keeps on moving on its sides. FIG. 3. Left panel: schematic of a zigzag wall as well as the definition of geometrical zigzag angle β; right panel: magnifi- cation of a single segment of zigzag wall with the intrinsic DWwidth Δ 0and effective DW width Δeffindicated by red and black dashed lines, respectively. Note that the effective DW width wasdefined parallel to the propagating direction of the DW markedby dashed arrow. FIG. 4. DW velocity as the function of pulse field for bothzigzag and straight wall denoted by v k(black open square) and v⊥(red open circle), respectively. The black solid line is the linear fit with the formula ν¼μH. The red dashed line is a guide to the eye. The inset shows the plot of Ln( ν)v sH−1=4for the straight wall with the linear fit (red solid line) using Eq. (1).PHYSICAL REVIEW LETTERS 125, 237203 (2020) 237203-3As a result, a quite big Vshape can be created. In this case, the sizes of the successive zigzags are defined by the defects and the zigzag pattern becomes highly irregular (see Supplemental Material [20], Fig. S5). But, this is true only at low field. Above a threshold field, pinning points become negligible, which means that the zigzag pattern can roughly get back to the regular figure defined by magnetostaticforces (here, the threshold would be around 0.7 mT). Now, a last question is how should we measure Δ? Indeed, for the horizontal straight DW, the width seems obvious, but the vertical zigzag ones? Should we use Δ effor Δ(see Fig. 3)? When calculating the velocity, one assumes φ½x−x0ðtÞ/C138, where φis the tilt angle of the local mag- netization, going from 0° to 180° along the wall and x0the position of the wall [25,31] . Through this method, the propagation velocity v¼dx0=dt is linked to ∂φ=∂t through ∂φ=∂t¼−vdφ=dx.A s dφ=dx is proportional to1=Δeff¼sinβ=Δ0, we expect DW velocity to be proportional to Δ0=sinβ, where Δ0is the “intrinsic width ” measured perpendicularly to the DW direction. Note that the xaxis can be chosen in any direction, it does not matter: for an infinite straight DW, the final result is the same. Indeed, if you translate such a DW over Δ0=sinβin the xdirection, whatever is the xdirection, starting from the same initial position, one gets the same final position (see Fig. 3). To check the Δ0=sinβdependency, using optical litho- graphy and ion beam etching (see Supplemental Material [20] for details), we have patterned wires from one of our 30 nm thick CoFeB samples [Fig. 5(a)]. The wires were narrow enough to avoid possible zigzag across their width, but wide enough to ensure that shape anisotropy remains negligible. Several sets of wires with different orientationswith respect to the easy axis were patterned to check the effect of the in-plane anisotropy. The variation of the wall velocity was verified at high fields ( >1.2mT), such that pinning is negligible and a 1D behavior according to Eq. (3)could be expected. Figures 5(b) and 5(c) show the nucleation and propagation on one set of wires. Quite surprisingly, the angle βwas not the same for all wires and did not appear to depend on the anisotropy axis of the film.This might be due to annealing that is conducted during the patterning process which results in suppressing the EA. As a result, we could plot the mobility v=μ 0Has a function of 1=sin. As the angle βwas reproducible for one wire, it has been possible to perform an average over several experi- ments in order to get an improved precision. As expected, aclear linear relationship has been obtained which is plottedin Fig. 5(d). In addition, in this graph, we have added a red point that represents the zigzag wall on the full film using the zigzag angle of 22° obtained above. In a wire, shapeanisotropy can change K effand consequently Δ[34]. But, as the wire width is very large, this change is negligible here, and, as expected, this point falls along the trendestablished by the data for the wires. Conclusion. —We have found a highly anisotropic dynamical behavior in an in-plane magnetized thin film of Ta =CoFeB =Ta. Using magnetic field pulses parallel to the easy plane, the shape of the domains nucleated by a pulse was rectangular. The limiting DWs of these rectan- gular domains were different according to the sides. Thetwo sides parallel to the easy magnetization axis were straight lines, while the two sides perpendicular to this axis showed a zigzag structure, as expected from magnetostaticsarguments to avoid charged DWs. Depending on the formof the wall, the propagation velocity was very different; creep motion was observed for straight walls, while zigzag walls propagated unimpeded with a linear velocity depend-ence on applied fields. We suggest that the possibility of creating zigzag at the blocking defects destroys the effect of the pinning. Finally, we have pointed out that the velocity isalso changed because of the tilting induced by the zigzag. We have shown that the velocity is proportional to the effective DW width, i.e., the width obtained when FIG. 5. (a) An optical image of the L-shaped microwires of Ta ð2nmÞ=CoFeB ð30nmÞ=Tað1nmÞstack with wire width of 100, 80, and50μm (from top to bottom). The EA has been marked by a red dashed line. The white parts are Au electrodes deposited on the top of the wires (not used in this work). (b) The initial DWs state in which DWs with the “slant ”angle β1,β2, and β3nucleated with a certain pulse in 100, 80, and 50μm wires, respectively. (c) A typical Kerr image of DWs in the wires after the application of a field pulse with an amplitude of 0.75 mT and length of 1.9μs. (d) The measured DW mobility μas the function of 1=sinβfor the three slant DWs in the wires. Error bar represents the standard deviation of the slant angle in the repeated experiments. The DW mobility for zigzag wallmeasured in the full film has also been displayed by the red solid square.PHYSICAL REVIEW LETTERS 125, 237203 (2020) 237203-4measuring it along the propagation direction. Let us add that some preliminary results with a permalloy film showthat the behavior seems to be the same: our results seemtypical of in-plane anisotropy thin films with an easy axis in the plane. This work is supported by the National Key Research and Development Program of China (GrantNo. 2017YFA0204800), the National Natural ScienceFoundation of China (No. 51571062), and the ChinaScholarship Council. W. 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PhysRevB.100.134431.pdf	PHYSICAL REVIEW B 100, 134431 (2019) Scattering theory of transport through disordered magnets Martin F. Jakobsen , Alireza Qaiumzadeh, and Arne Brataas Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (Received 2 August 2019; published 23 October 2019) We present a scattering theory of transport through noncollinear disordered magnetic insulators. For con- creteness, we study and compare the random ﬁeld model (RFM) and the random anisotropy model (RAM).The RFM and RAM are used to model random spin disorder systems and amorphous materials, respectively.We utilize the Landauer-Büttiker formalism to compute the transmission probability and spin conductance ofone-dimensional disordered spin chains. The RFM and the RAM both exhibit Anderson localization, whichmeans that the transmission probability and spin conductance decay exponentially with the system length. Wedeﬁne two localization lengths based on the transmission probability and the spin conductance, respectively.Next, we numerically determine the relationship between the localization lengths and the strength of the disorder.In the limit of weak disorder, we ﬁnd that the localization lengths obey power laws and determine the criticalexponents. Our results are expressed via the universal exchange length and are therefore expected to be general. DOI: 10.1103/PhysRevB.100.134431 I. INTRODUCTION In magnonics [ 1–8], the primary focus has recently been on the propagation of spin waves through various types of mag-netic insulators. A particular emphasis has been on orderedsystems, such as (anti)ferromagnets, and ferrimagnets. An advantage is that the spin current may suffer less Joule heating compared to electric currents, making insulator-magnonicsapplications potentially much more energy efﬁcient [ 3,9]. Numerous successful experiments have generated and manip-ulated spin currents using the spin-Hall effect and the inversespin-Hall effect [ 10]. A common experimental setup consists of sandwiching a magnetic insulator between two conductors and using the spin-Hall effect to generate a spin current in the left conductor that propagates through the magnetic insulatorand into the right conductor. The spin current in the rightconductor is converted into a charge current via the inversespin-Hall effect. This provides a useful method to measurethe spin current and infer the spin-transport properties of themagnetic insulator [ 11–25]. A class of materials that has recently attracted attention in the spintronics community is disordered magnetic insulators[26–31]. Notably, a recent experiment claimed that a spin current ﬂowing through a sample of amorphous yttrium-iron-garnet could travel tens of micrometers [ 32]. This distance is comparable to the spin current propagation length in a crystalline (anti)ferromagnet [ 25,33]. More generally, it is crucial to study disordered magnetic materials because almostall materials contain some degree of disorder, which willaffect the functional properties of magnonic devices. Whenthe disorder is sufﬁciently strong, the eigenstates becometrapped in a ﬁnite spatial region, completely suppressing thetransport properties. This phenomenon is known as Andersonlocalization, and the ﬁrst discussion of this phenomenon inmagnetic systems began in the 1960s [ 34–42]. Furthermore, it has been shown that, even with a small onset of disorder,the transport properties change from conductive to diffusive[29,43], which has important consequences for magnonics applications in low dimensions. The common sources of quenched disorder in magnetic insulators are (i) randomness due to anisotropies, local ﬁelds,and amorphous structure and (ii) frustration due to competinglong-range exchange interactions. In this paper we focus onmagnetic insulators with quenched disorder due to (i). Twomodels with these properties are the random ﬁeld model(RFM) and the random anisotropy model (RAM), wherethe disorder is caused by the competition between the ex-change interaction and the coupling to local random ﬁeldsand anisotropies, respectively. The RFM and RAM is used tomodel quenched spin disorder and amorphous magnets, re-spectively [ 44–49]. Experimental realizations of such systems are plentiful [ 50–54]. Furthermore, there are two types of RFM /RAM spin models. The ﬁrst is the Ising model, where the spins arescalars S i=±1 and are randomly pointing either parallel or antiparallel to each other in the ground state [ 55–62]. The second type is the Heisenberg model, where the spins arevectors S ithat in the ground state are pointing noncollinearly in random directions [ 63–68]. Because the ground state in the RFM /RAM Ising model is relatively simple, it can often be studied efﬁciently withanalytical methods. For example, one can either solve theequations of motion by a transfer matrix approach parallelingAnderson’s celebrated work on disordered fermionic systems[69,70] or one can use ﬁeld-theory methods, particularly the replica trick, replica symmetry breaking, and mean-ﬁeldtheory [ 54,71,72]. Although the RFM /RAM Ising models are analytically accessible, they are only simpliﬁed idealizationsof a real disordered magnet where the spins are noncollinear.In this work, we wish to focus on systems with noncollinearspins that are harder to describe analytically but exhibit morerealistic spin-wave dynamics. Disordered magnetic insulators with a noncollinear ground state are a notoriously difﬁcult system to describe. Due to their 2469-9950/2019/100(13)/134431(10) 134431-1 ©2019 American Physical SocietyJAKOBSEN, QAIUMZADEH, AND BRATAAS PHYSICAL REVIEW B 100, 134431 (2019) complexity, it is often useful to study the classical spin waves of the system. Our work is related to a recent study [ 29,73] in which the micromagnetic Landau-Lifshitz-Gilbert (LLG)equation was solved using a quasimonochromatic Gaussianwave packet as the initial condition. They found that the widthof the wave packet increases in time until it saturates aroundthe localization length of the system, which is a hallmarkof Anderson localization. In systems that exhibit Andersonlocalization, the localization length decreases as the systembecomes more disordered. However, the exact relationshipbetween the localization length and the strength of disorderis far from being well established in noncollinear disorderedmagnetic insulators. In this work, we attempt to shed somelight on these issues. The localization effect in spin models depends on the dimensionality of the system, similar to disordered fermionicsystems [ 74]. For fermionic systems in one dimension, there is Anderson localization; in two dimensions, the effect remainspresent but much weaker, while in three dimensions, there isthe possibility of both a localized and a delocalized phase. Thesame observations have been established for disordered mag-nets [ 50,75]. We focus on one-dimensional spin chains. With more computational time, this method can also be applied totwo- and three-dimensional systems. The numerical method that we develop is based on the Landauer-Büttiker formalism [ 76,77], which has proven to be extremely useful in studying the transport properties ofelectronic systems. To the best of our knowledge, such amethod has not previously been applied to the RFM /RAM Heisenberg model. In this paper, we investigate the effect ofAnderson localization on the spin-wave transport propertiesof a disordered magnetic insulator. To this end, we ﬁrstdetermine the relationship between the system size and thetransmission probability for different strengths of disorder andthen calculate the spin conductance. With this knowledge,we can investigate how the localization length of the systemscales with the strength of the disorder. In particular, wecalculate and compare the critical exponents of the RFM andthe RAM. These quantities provide us with direct insights intohow the transport properties of the spin waves are affected bythe localization phenomenon that is present due to quencheddisorder. We hope that this theoretical investigation may inspire an experimental investigation into the transport propertiesof disordered magnetic nanowires [ 78–82]. In particular, it would be interesting to compare the experimental relationshipbetween the localization length and strength of disorder to thecritical exponents that we determine in this work. The paper is organized as follows. In Sec. II, we introduce the RFM and the RAM Hamiltonians and discuss their groundstate. In Sec. IIIwe ﬁnd the linearized equations of motion, and derive expressions for the spin current and the spinconductance. Section IVcontains our numerical calculations of the scattering properties of the system. In Sec. IVwe summarize our results. II. THEORETICAL MODEL In this section we carefully introduce the model we are in- terested in studying. We start by presenting the HamiltoniansFIG. 1. Disordered magnet (blue) is sandwiched between two ferromagnets (red). In regions ( i)a n d( iii),ni=ˆz, while in region (ii)niis uniformly distributed on the unit sphere. Consequently, the spins in region ( ii) point in random directions, while the spins deep inside regions ( i)a n d( iii) point in the zdirection. The spins close to the two interfaces rotate similar to the spins in a domain wall. The length of the domain-wall-like region is illustrated and given by theexchange length l ex=√J/Kd. for the RFM and the RAM, and introduce the geometry. We conclude this section by presenting a method to calculate theclassical metastable states. A. Hamiltonian To investigate the transport properties of one-dimensional disordered noncollinear spin chains, we use the Hamiltonian Hκ=−J/summationdisplay iSi·Si+1−K/summationdisplay i(Si·ni)κ+1, (1) where κ=0 and κ=1 represent the RFM and the RAM, respectively. The dimensionless spins Siare attached to a one- dimensional lattice with lattice spacing d. The exchange interaction with J>0 attempts to align the spins. The terms proportional to Kencapsulate the quenched disorder of the system, and we choose K>0 without loss of generality. Each spinSiis coupled to a local random vector ni. The competition between the exchange and the random interactions in Eq. ( 1) results in a noncollinear disordered ground state. We use theparameter K/Jto characterize the strength of disorder. B. Geometry We consider a one-dimensional chain with Nlattice sites. The chain is split into three regions that we call ( i)t h el e f t lead, ( ii) the random region, and ( iii) the right lead; see Fig. 1. In regions ( i) and ( iii), we let the number of spins be equal toNLandNR, respectively. In addition, we let nipoint in the ˆzdirection. In region ( ii), we let the number of spins be equal toNrandandnito point in some random direction uniformly distributed on the unit sphere. Note that far away from therandom region (deep inside of the leads), the spins point inthe ˆzdirection, while in the random region, the spins are oriented randomly. In the regions close to the interface, thespins are rotating in a domain-wall-like fashion. The lengthof this domain-wall region is given by the exchange lengthl ex=√J/Kd. The scattering problem that we are interested in studying can now be realized by exciting coherent spin waves in theleft lead propagating towards the random region. As the spinwave approaches the random region, it will be scattered eitherback into the left lead (reﬂection) or into the right lead 134431-2SCATTERING THEORY OF TRANSPORT THROUGH … PHYSICAL REVIEW B 100, 134431 (2019) (transmission). We assume semi-inﬁnite leads such that NL andNR−→∞. C. Ground state Determining the true ground state of a disordered magnet (collinear or noncollinear) is a very challenging problem.The primary reason is that the randomness results in freeenergy with many nearly degenerate minima, separated byhigh energy barriers. The problem of determining the exactground state of disordered systems is its own research ﬁeld,and we do not wish to address that problem here [ 83–94]. However, due to the high energy barriers, the probability oftunneling between different metastable states is small. Hence,in an experiment, the disordered magnet becomes trapped ina state that may differ from the exact ground state when thesystem is cooled down, depending on the history. Thus, in thispaper, we study the transport properties of disordered magnetsaround classical metastable states. We can ﬁnd a classical metastable state of the system by treating the spins as classical vectors obeying the LLGequation of the form dS i dt=−γSi×Hκ i−λSi×/parenleftbig Si×Hκ i). (2) Here, the ﬁrst term with γ> 0 describes the spin Siprecess- ing around its instantaneous effective ﬁeld Hκ i=−δHκ/δSi, while the second term describes the damping towards thedirection of the instantaneous effective ﬁeld. The metastablestate is then obtained by specifying some arbitrary initialconﬁguration and allowing the spins to evolve according tothis equation for sufﬁciently long times t−→∞ . III. SCATTERING THEORY In this section we outline the theoretical approach that we will use to determine the transport properties of the RFM andthe RAM. We start by determining the linearized equations ofmotion, and formulate the scattering problem. Finally, we de-rive the expressions for the spin current and spin conductancein the linear response regime. A. Hamiltonian in terms of spin-wave operators To study the transport properties of the system, we can perform a Holstein-Primakoff expansion around one of themetastable states. Let us at each site ideﬁne a local coordinate system {ˆe x(i),ˆey(i),ˆez(i)}such that ˆ ez(i) is parallel to the spin at site iin the ground state. The spin operator in a low-lying excited state can then be written as Si=ˆez(i)Sz i+ˆex(i)Sx i+ˆey(i)Sy i. (3) We perform a Holstein-Primakoff transformation of the form Sx i≈/radicalbigg S 2(ai+a† i), (4a) Sy i≈−i/radicalbigg S 2(ai−a† i), (4b) Sz i=S−a† iai. (4c)In Eq. ( 4), we have only included the lowest-order terms because we are not interested in studying the interactionsbetween the spin waves. If we substitute Eqs. ( 3) and ( 4)i n t o Eq. ( 1) and introduce the notations ˆ e ±(i)=ˆex(i)±iˆey(i) and n± i=nx i±iny i, we obtain a Hamiltonian of the form Hκ=/summationdisplay ijAκ ija† iaj+Bκ ijaiaj+H.c., (5) where Aκ ij=δi,j/braceleftbigg JSˆez(i)ˆez(i+1)+1 2Knz i +κ/bracketleftbigg KS/parenleftbig nz i/parenrightbig2−1 2KSn− in+ i−1 2Knz i/bracketrightbigg/bracerightbigg −JS 2δi,j+1ˆe−(i)ˆe+(j), Bκ ij=−κKS 2(n− i)2δi,j−JS 2ˆe−(i)ˆe−(i+1)δi,j+1.(6) In the following, we will study the spin waves associated with the Hamiltonian of Eq. ( 5). B. Equations of motion The equations of motion for the spin-wave operators can now be calculated from the Heisenberg equation d dta± i=i ¯h[Hκ,a± i]. (7) For clarity, we reinstate the spin operators {Sx i,Sy i}using Eq. ( 4) and cast the equation of motion in the form ¯hdSx j dt=JS/braceleftbig ˆez(j)[ˆez(j+1)+ˆez(j−1)]Sy j −ˆey(j−1)ˆey(j)Sy j−1−ˆex(j−1)ˆey(j)Sx j−1 −ˆey(j)ˆey(j+1)Sy j+1−ˆey(j)ˆex(j+1)Sx j+1/bracerightbig +Knz jSy j+κ/braceleftbig 2KS/parenleftbig nz j/parenrightbig2Sy j−2KS/parenleftbig ny j/parenrightbig2Sy j −2KSnx jny jSx j−Knz jSy j/bracerightbig , (8) ¯hdSy j dt=JS/braceleftbig −ˆez(j)[ˆez(j+1)+ˆez(j−1)]Sx j +ˆex(j−1)ˆex(j)Sx j−1+ˆey(j−1)ˆex(j)Sy j−1 +ˆex(j)ˆex(j+1)Sx j+1+ˆex(j)ˆey(j+1)Sy j+1/bracerightbig −Knz jSx j−κ/braceleftbig 2KS/parenleftbig nz j/parenrightbig2Sx j −2KS/parenleftbig nx j/parenrightbig2Sy j−2KSnx jny jSy j+Knz jSy j/bracerightbig . (9) Equations ( 8) and ( 9) are identical to the linearized classical Landau-Lifshitz equations expressed in the local coordinatesystem {ˆe x(j),ˆey(j),ˆez(j)}. Since we are only interested in studying how the intrinsic disorder affects the transport prop-erties of the system, we have not included a Gilbert dampingterm. 134431-3JAKOBSEN, QAIUMZADEH, AND BRATAAS PHYSICAL REVIEW B 100, 134431 (2019) C. Scattering problem and solution Ansatz Equations ( 8) and ( 9) can be solved numerically in the classical regime, where we treat the spin operators as classicalvectors. The spin-wave solutions are the normal modes of thesystem and precess with the same frequency ω. Therefore, we can factorize out the time dependence of the spin operators ase −iωt. Deep inside the leads, the spins at neighboring sites are pointing in the zdirection; see Fig. 1. This considerably simpliﬁes the equations of motion in the leads: −i¯hωSx j=JS/parenleftbig 2Sy j−Sy j−1−Sy j+1/parenrightbig +K[1+κ(2S−1)]Sy j, −i¯hωSy j=JS/parenleftbig −2Sx j+Sx j−1+Sx j+1/parenrightbig −K[1+κ(2S−1)]Sx j. (10) The system behaves as a ferromagnet with an external ﬁeld or intrinsic anisotropy in the zdirection. The solutions are therefore circularly polarized plane waves traveling with aﬁxed wave number qand frequency ω. The dispersion relation can be determined by substituting the Ansätze Sj x=eiq jdand Sj y=−ieiq jdinto Eq. ( 10). The result is /epsilon1=¯hω=2JS(1−cosqd)+K[1+κ(2S−1)].(11) Let us now formulate the scattering problem. Deep inside the regions ( i) and ( iii)i nF i g . 1, we know that the solution must have the form Sx j=eiq jd+rxe−iq jd,Sy j=−i(eiq jd+rye−iq jd) (12) and Sx j=txeiq jd,Sy j=−itye−iq jd, (13) respectively. Inside region ( ii), we know that the spin com- ponents must satisfy Eqs. ( 8) and ( 9). Using the Ansätze as boundary conditions, we have found a ﬁnite set of algebraicequations that we can solve numerically to determine thereﬂection and transmission amplitudes {r x,ry,tx,ty}as func- tions of /epsilon1. D. Spin current and conductance Once we know the reﬂection and transmission amplitude, we can calculate the spin conductance of the disorderedmagnet utilizing the Landauer-Büttiker formalism in the linearresponse regime. In this section, we derive the expression forspin conductance. In the leads, the Hamiltonian in Eq. ( 5) simpliﬁes to H κ=/summationdisplay i{2JS+K[1+κ(2S−1)]}a† iai −JS(aja† j+1+a† jaj+1). (14) From the equation of motion, d dtNi=i ¯h[Ni,Hκ] =−iJS{(a† j+1aj−a† jaj+1)+(a† j−1aj−a† jaj−1)}, (15)FIG. 2. Disordered magnet (blue) is sandwiched between two ferromagnetic leads (red). The leads are connected to two spinreservoirs (green) with spin accumulations μ LandμR. The reservoirs are in thermodynamic equilibrium such that the magnon population is characterized by the Bose-Einstein distribution. A spin current isinduced when there is a nonzero spin bias δμ=μ L−μR. where Ni=a† iaiis the number operator, and we can extract the spin current from site jtoj+1a s Ij,j+1=iJS(a† j+1aj−a† jaj+1). (16) Now consider the situation in Fig. 2, where two reservoirs in thermodynamic equilibrium are attached to two leads witha scattering region between them. If the spin accumulation inthe left reservoir μ Lis greater than the spin accumulation in the right reservoir μR, the spin current in Eq. ( 16) will ﬂow from the left to the right reservoir. We deﬁne the operatorsα L,R(q) and βL,R(q) injecting and removing magnons with wave numbers qinto the leads, respectively. The relationship between these operators is given by the scattering matrix /parenleftbigg βL(q) βR(q)/parenrightbigg =/parenleftbigg rt/prime tr/prime/parenrightbigg/parenleftbigg αL(q) αR(q)/parenrightbigg , (17) where r(r/prime) and t(t/prime) are the reﬂection and transmission amplitudes, respectively, for a spin wave originating from theleft (right) lead. In the left lead, we can express a jas [95] aj=/integraldisplayπ/d 0dq 2π/d[eiq jdαL(q)+eiq jdβL(q)]. (18) If we substitute Eq. ( 18) and its complex conjugate into Eq. ( 16) and utilize that the leads are in thermal equilibrium with the reservoirs such that /angbracketleftα† L,R(q1)αL,R(q2)/angbracketright=2π dδ(q1− q2)fL,R(q1), we ﬁnd that /angbracketleftIj,j+1/angbracketright=1 2π/integraldisplay/epsilon1max /epsilon1mind/epsilon1T(/epsilon1)[fL(/epsilon1)−fR(/epsilon1)]. (19) In this expression, fL,R(/epsilon1) represents the Bose-Einstein dis- tributions in the left and right reservoirs, respectively, andT(/epsilon1)=\|t\| 2. The integration limits are obtained from Eq. ( 11). Assume that the spin accumulation in the left lead is μL=μ+δμand that the spin accumulation in the right lead is μR=μ, where δμ/μ /lessmuch1. We ﬁnd that, in the linear response, the spin conductance is given by G=1 2π/integraldisplay˜/epsilon1max ˜/epsilon1mind˜/epsilon1T(˜/epsilon1)/parenleftbigg −df d˜/epsilon1/parenrightbigg . (20) This result can also be derived using Green’s functions [ 96]. In Eq. ( 20), we are integrating over the dimensionless energies ˜/epsilon1=/epsilon1/J. Energies outside of the integration interval result in 134431-4SCATTERING THEORY OF TRANSPORT THROUGH … PHYSICAL REVIEW B 100, 134431 (2019) FIG. 3. Behaviors of /angbracketleftT/angbracketrightand/angbracketleftlnT/angbracketrightas a function of ˜ /epsilon1andK/J. the spin waves in Eqs. ( 12) and ( 13) becoming evanescent waves that do not contribute to the spin conductance. IV . RESULTS AND DISCUSSION For each realization of the system, we ﬁnd that rx=ry≡r and that tx=ty≡t, reﬂecting the fact that, inside the leads, the spin waves are circularly polarized. Furthermore, wedeﬁne R=\|r\| 2andT=\|t\|2as the reﬂection and transmission probabilities, respectively, and ﬁnd that R+T=1. Since R and Tdepend on the realization of the system, we must perform an ensemble average /angbracketleft.../angbracketrightto obtain physically mean- ingful quantities. In our calculations, we used 103different realizations for the random vectors ni.I nF i g . 3, we plotted /angbracketleftT/angbracketrightand/angbracketleftlnT/angbracketrightas a function of ˜ /epsilon1for different values of K/J and a ﬁxed system length L=Nrandd. In the remainder of this paper we set d=1 for convenience. A. Transmission probability As the system becomes more disordered, the transmission probability decreases for both the RFM and the RAM. How-ever, Fig. 3demonstrates that the quantitative behavior of the localization is signiﬁcantly different in the two models.In both models, as K/Jincreases, the maxima /angbracketleftT/angbracketright maxand /angbracketleftlnT/angbracketrightmaxshift towards higher ˜ /epsilon1, but in the RAM, this shift is greater than that in the RFM. In addition, the peak in thetransmission probability is wider in the RAM compared to theRFM for small K/J. Thus a broader range of spin waves can pass through the RAM compared to the RFM in the limit ofweak disorder. We can understand the difference in width from the Hamil- tonian in Eq. ( 1). In the RAM, the term causing disorder is (S i·ni)2; thus the spin Siwants to point either parallel or antiparallel to ni. The spin is also coupled to its neighbors through the exchange interaction. Therefore, in the RAM,whether the spin S ichooses to point parallel or antiparallel tonidepends on the neighboring spins. Meanwhile, in the RFM, the term causing disorder is Si·ni, and the spin wants to only point parallel to ni. The ability to select whether to point parallel or antiparallel to nileads to the spin chains in the RAM being less disordered than the spin chains in the FIG. 4. Length dependence of the relative variances RV G,R V lnG, RV T,a n dR V lnTfor the RFM and the RAM. The strength of disorder isK/J=0.4. RFM, which in turn leads to a broader peak in the transmission probability. B. Self-averaging In disordered systems, certain quantities are not self- averaging in the thermodynamic limit. This is well known indisordered fermionic systems and is expected to be a generalfeature of a broad spectrum of disordered systems [ 97]. A test to determine whether a quantity Ois self-averaging is to check whether the relative variance RV O=Var (O)//angbracketleftO/angbracketright2 vanishes (or is sufﬁciently small) in the limit L−→∞ .F o rt h e fermionic 1D Anderson model with on-site disorder, one ﬁndsthat the transmission probability and hence the conductanceare not self-averaging [ 98]. In two and three dimensions, one ﬁnds that the logarithms ln( T),ln(G) are self-averaging such that RV lnG∼L−D(D=2,3) [97,99]. In one dimension at ﬁnite temperature, one ﬁnds that ln Gis only marginally self-averaging because RV lnGdecays logarithmically with L [99–101]. As expected, we ﬁnd similar results in this work. Figure 4 shows that the relative variances RV Tand RV Gincrease with the length of the system. In addition, the relative variancesRV lnTand RV lnGdecrease with the length of the system. Hence, as the length of the system increases, the ﬂuctuationsinTand Gbecome much greater than the corresponding expectation values, meaning that they are not representativevariables in the thermodynamic limit. Therefore, we use{ln(T),ln(G)}, rather than {T,G}, to calculate the localization lengths of the system. C. Localization length In this work, it is natural to deﬁne two types of localization lengths. The ﬁrst is based on the maximum of the transmissionamplitude /angbracketleftlnT/angbracketright maxin Fig. 3. The second is based on the conductance /angbracketleftlnG/angbracketright. We refer to these localization lengths as ˜LlnTand˜LlnG, respectively. 134431-5JAKOBSEN, QAIUMZADEH, AND BRATAAS PHYSICAL REVIEW B 100, 134431 (2019) FIG. 5. Length dependence of /angbracketleftlnG/angbracketrightand/angbracketleftlnT/angbracketrightmaxfor the RFM can be approximated with a linear ﬁt. The strength of disorder isK/J=0.4 and the temperature is ˜T=0.05. In Fig. 5, we plotted /angbracketleftlnT/angbracketrightmaxand/angbracketleftlnG/angbracketrightas a function of the system length Lfor a ﬁxed K/Jand temperature ˜T= kT/J. We have performed a curve ﬁt with the functions /angbracketleftlnT/angbracketrightmax=L ˜LlnT+A (21) and /angbracketleftlnG/angbracketright=L ˜LlnG+B (22) such that the localization length can be extracted as the gradient of the straight lines in Fig. 5. In this particular case, we found a coefﬁcient of determination R2with the value R2=0.95 indicating a good ﬁt. To determine the localization lengths as a function of K/J, we performed straight line curve ﬁts for graphs such as those found in Fig. 5but with different K/Jand˜T. In all cases, we found that the coefﬁcient of determination was in the range (0.9,1) and that the aver-age coefﬁcient was /angbracketleftR 2/angbracketright=0.95, indicating reasonably good straight-line ﬁts. By then calculating the gradient of thesestraight lines, we can estimate the localization lengths as afunction of K/J. 1. Localization length from transmission In Fig. 6, we plot the localization length ˜LlnTand the 95%- conﬁdence interval for the RAM and RFM, respectively. Inboth cases, we have performed a curve ﬁt with the function ˜L lnT=η/parenleftbiggK J/parenrightbiggν +ξ. (23) The parameters (with conﬁdence intervals) are displayed in Table I. Similar to fermionic systems [ 102], we ﬁnd that the localization length decays monotonically as a power law aswe increase the strength of disorder. Our result can be mademore universal by introducing the exchange length such that ˜L lnT=η(lex)−2ν+ξ. (24) FIG. 6. Behavior of ˜LlnTas a function of K/Jfor the RAM and the RFM, respectively. The line represents the numerical ﬁt inEq. ( 23), the dashed lines represent the 95% conﬁdence interval, and the points with error bars represent the localization length calculated f r o mE q s .( 8)a n d( 9) with standard error. Note that, for weak disorder, the localization length is greater in the RAM than in the RFM. This is a consequenceof the fact that the spin chains are less disordered in the RAMcompared to the RFM, as we discussed at the end of Sec. IV A . 2. Localization length from conductance In Fig. 7,w ep l o t ˜LlnGas a function of K/Jfor different temperatures ˜T. There is an interval K/J≈(0.5,2) where the localization length increases for small ˜T. Furthermore, for sufﬁciently large ˜T, this interval vanishes such that the local- ization length decays monotonically for all K/J. This nontriv- ial behavior arises because there is a competition between thetemperature dependence of the broadening function −df/d˜/epsilon1 and the disorder dependence of the transmission probabilityT(˜/epsilon1)i nE q .( 20). As the temperature increases, the broadening function excites an increasing number of magnons, whichin turn leads to a greater conductance. Meanwhile, as thesystem becomes more disordered, the transmission probabil-ityT(˜/epsilon1) decreases, resulting in a smaller conductance. On the interval K/J≈(0.5,2), the increase in conductance due to temperature is greater than the decrease in conductancedue to disorder, which results in an increase in localizationlength. Furthermore, in this interval, the localization lengthis comparable to the lattice spacing d, which means that there TABLE I. Numerical values of the parameters in Eq. ( 23) for the RFM and the RAM. The brackets ( ...) give the 95% conﬁdence interval. RFM RAM η 1.3 (0.7, 2.0) 0.2 (0.1, 0.3) ν −1.2 (−1.4,−1.0) −2.2 (−2.4,−2.0) ξ 1.1 (0.3, 1.9) 1.6 (1.0, 2.1) 134431-6SCATTERING THEORY OF TRANSPORT THROUGH … PHYSICAL REVIEW B 100, 134431 (2019) FIG. 7. Temperature dependence of ˜LlnGfor strongly disordered magnetic insulators. may be complicated microscopic details of the model that may further enhance this effect. Due to the complicated temperature and disorder depen- dence, it is numerically challenging to determine a closedformula such as the one in Eq. ( 23) for the localization length ˜L lnG. However, in the weak-disorder limit K/J−→0, it is reasonable to assume that the localization length decays asa power law of the form L lnG∼/parenleftbiggK J/parenrightbiggγ =(lex)−2γ, (25) where γis the critical exponent. Figure 8shows the result of such a curve ﬁt for the RFM and the RAM for differenttemperatures. The corresponding critical exponents γ RFMand γRAMare given in Table II. In our simulations, we kept the temperature below the Curie temperature ˜T=1, where the temperature ﬂuctuations TABLE II. Numerical values of the critical exponent for the RFM and the RAM for different temperatures. The brackets ( ...) provide the 95% conﬁdence interval. ˜T γRFM γRAM 0.05 −3.8(−4.3,−3.4) −1.9(−2.3,−1.6) 0.1 −3.7(−4.2,−3.2) −1.7(−2.0,−1.4) 0.2 −3.6(−4.1,−3.1) −1.3(−1.6,−1.0) 0.5 −3.3(−3.8,−2.8) −0.7(−1.0,−0.4) FIG. 8. Temperature dependence of ˜LlnGin the limit of weak disorder K/J−→0. of the spins are negligible. For temperatures ˜T≈1, there will be additional temperature-induced disorder. This issuehas previously been investigated [ 73,103,104] by including a temperature-dependent stochastic ﬁeld in the effective ﬁeldH κ iin Eq. ( 2), and it was found that temperature ﬂuctuations shorten the localization length and enhance the Andersonlocalization. V . SUMMARY AND CONCLUSIONS In this paper, we have applied the Landauer-Büttiker for- malism to noncollinear disordered magnetic insulators. Wehave considered both amorphous magnets and magnets withspin disorder modeled by the RAM and the RFM, respec-tively. We calculated the self-averaging quantities /angbracketleftlnT/angbracketrightand /angbracketleftlnG/angbracketrightas a function of system length Lfor a broad range of disorder strengths K/J. Consistent with the literature, we found evidence for Anderson localization such that /angbracketleftlnT/angbracketright and/angbracketleftlnG/angbracketrightwere linear functions of the system length L.T h i s allowed us to deﬁne two localization lengths ˜L lnGand ˜LlnT based on the conductance and the maximum transmission probability, respectively. In the limit of weak disorder, thelocalization lengths obeyed power laws, and we calculated therelevant critical exponents. We expect our results to be generalbecause they are expressed through the universal exchangelength l ex. We found that the Anderson localization is more prominent in the RFM than in the RAM. The reason for this result isthat the competition between the exchange interaction and thedisorder term leads to more disordered spin chains in the RFM 134431-7JAKOBSEN, QAIUMZADEH, AND BRATAAS PHYSICAL REVIEW B 100, 134431 (2019) than in the RAM. The spin chains in the RAM are less disor- dered because the disorder arises from a random anisotropy,where the spin can point either parallel or antiparallel to theanisotropy with the same energy cost. Whether the spin pointsparallel or antiparallel to the anisotropy is determined bythe neighboring spins through the exchange interaction and,consequently, the conﬁguration with the least disorder will bechosen by the system. The results obtained here are valid in the limit of quenched disorder, i.e., ˜T/lessmuch1, where the random ﬁeld and anisotropy are temperature independent. If the temperature is close tothe Curie temperature of the system, one must include tem-perature ﬂuctuations in the Landau-Lifshitz equations. Sucheffects have been considered in other works, and it has beenshown that temperature ﬂuctuations decrease the localizationlength. To experimentally verify the critical exponents obtained in this paper, we propose a setup in which a disorderedmagnetic nanowire is sandwiched between two normal metals.Similar setups for ordered magnets have been considered inother works [ 11–25]. By applying a charge current in the left metal, the spin-Hall effect generates a spin current through thedisordered nanowire and into the right metal. This will giverise to a spin wave propagating through the hybrid structure and into the right metal, where the spin current is convertedinto a charge current via the inverse spin-Hall effect. Alternatively, we can instead sandwich a disordered mag- net between two ferromagnetic leads. We can excite spinwaves in the left ferromagnet by applying a microwave withthe ferromagnet resonance frequency. 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PhysRevB.76.104409.pdf	Spin-mixing conductances of thin magnetic ﬁlms from ﬁrst principles K. Carva1,and I. Turek2,† 1Department of Condensed Matter Physics, Charles University, Faculty of Mathematics and Physics, Ke Karlovu 5, CZ-12116 Prague 2, Czech Republic 2Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 22, CZ-61662 Brno, Czech Republic /H20849Received 23 April 2007; revised manuscript received 30 July 2007; published 10 September 2007 /H20850 We present a ﬁrst-principles theory of the spin-mixing conductance for a thin ferromagnetic ﬁlm embedded epitaxially between two nonmagnetic metallic electrodes. The complex spin-mixing conductance is formulatedas a linear response of the spin torque experienced by the ﬁlm due to the spin accumulation in one of theelectrodes. The derivation is based on nonequilibrium Green’s functions; the obtained result for the torqueresponse is in agreement with the response of spin ﬂuxes on both sides of the ferromagnet as well as withexpressions derived within the Landauer-Büttiker scattering theory. Numerical implementation of the devel-oped formalism employs the tight-binding linear mufﬁn-tin orbital method and calculations are performed forselected metallic and half-metallic ferromagnetic ﬁlms relevant for spintronics applications. The spin-mixingconductance of the Cu/Ni/Cu /H20849100/H20850system is found to exhibit pronounced oscillations as a function of Ni thickness; their period is explained by spin-resolved Fermi-surface properties of nickel. Investigated half-metallic ﬁlms include the full-Heusler Co 2MnSi compound and the diluted /H20849Ga,Mn /H20850As magnetic semiconduc- tors attached to nonmagnetic Cr /H20849100/H20850leads; the imaginary part of their spin-mixing conductance has a mag- nitude comparable to the real part. This unusual feature has been qualitatively explained in terms of a free-electron model. DOI: 10.1103/PhysRevB.76.104409 PACS number /H20849s/H20850: 75.70. /H11002i, 72.10.Bg, 72.25.Pn, 85.75. /H11002d I. INTRODUCTION Artiﬁcially prepared metallic magnetic multilayers and spin valves attract ongoing interest due to a unique interplaybetween their magnetic structure and transport properties, 1,2 especially in the current perpendicular to the planes /H20849CPP /H20850 geometry. This can be documented by the well-known giantmagnetoresistance /H20849GMR /H20850effect 3and by a more recent prediction4,5and realization6of the current-induced magne- tization switching /H20849CIMS /H20850. Subsequent research activities re- sulted, e.g., in a full control of the sign of both GMR andCIMS in spin valves containing two ferromagnetic layersseparated by a nonmagnetic metal 7or in a signiﬁcant reduc- tion of the critical current density necessary for CIMS asachieved in tunnel junctions containing a diluted ferromag-netic semiconductor. 8 The most successful phenomenological framework for quantitative understanding of both phenomena is the Valet-Fert model 9based on the linearized Boltzmann equation with a collision term accounting for the spin-ﬂip scattering. Thelatter mechanism provides an extension of the widely usedtwo-current series-resistor model 3and it has important con- sequences for layer thicknesses comparable to the so-calledspin-diffusion length. The description of the CPP transport incollinear spin structures within this scheme leads to a semi-classical concept of the spin accumulation in nonmagneticlayers, i.e., to a difference of effective chemical potentials/H20849Fermi levels /H20850for electrons in the two spin channels. A recent generalization of the Valet-Fert model to noncol- linear spin structures 10–12rests heavily on two additional properties of spin currents. First, the transverse /H20849perpendicu- lar to local exchange ﬁeld /H20850component of the spin current inside a ferromagnet becomes rapidly damped over a typicaldistance of a few interatomic spacings. 13,14This very shortmagnetic coherence length is a result of a large exchange splitting which leads to mostly destructive interference ef-fects due to all contributions of wave vectors on the twoFermi surfaces of the ferromagnetic metal. Consequently, thespin torque experienced by a ferromagnetic /H20849FM/H20850layer can be identiﬁed with the transverse spin current at its interfacewith a neighboring nonmagnetic /H20849NM /H20850layer. Second, the proper boundary conditions inevitable for a full solution ofthe diffusion equations must be formulated in terms of spin-mixing conductances of individual interfaces. 15The latter /H20849complex /H20850quantities together with the spin-resolved inter- face conductances provide a complete information on a lin-ear response of the currents and spin currents at an interfacedue to the bias and spin accumulation deep inside the neigh-boring materials. The magnetoelectronic circuit theory 15–17represents an- other ﬂexible approach to the transport properties of noncol-linear magnetic systems consisting of FM and NM elements/H20849nodes /H20850. This scheme is highly efﬁcient especially when di- mensions of individual nodes are smaller than the spin-diffusion lengths but bigger than the electron mean-freepaths of the corresponding materials. Within the developedformalism, the chemical potentials and spin accumulations of the nodes are contained in 2 /H110032 distribution matrices in the spin space while junctions among the nodes are featured bythe spin-resolved and spin-mixing conductances. The steady-state currents, spin currents, and spin torques in a device canbe obtained from applied voltages by solving a set of linearequations quite similar to Kirchhoff’s laws for usual elec-tronic circuits, see Ref. 17for a review. A truly microscopic /H20849quantum mechanical /H20850approach to all aspects of the GMR and CIMS seems to be prohibitivelycomplicated having in mind the large layer thicknesses andthe quality of interfaces in presently used multilayers andspin valves. A reasonable compromise between the accuracyPHYSICAL REVIEW B 76, 104409 /H208492007 /H20850 1098-0121/2007/76 /H2084910/H20850/104409 /H2084914/H20850 ©2007 The American Physical Society 104409-1and the complexity has been adopted by several authors in addressing the spin-polarized electronic and transport prop-erties of a single FM/NM interface 13,18with emphasis put on the conductances and their sensitivity, e.g., to interface alloy-ing. Since the traditional scheme for the transport, namely, the Landauer-Büttiker scattering theory, 17,19has been used in majority of papers, the effect of disorder was included by asupercell technique. 17,20 The spin-mixing conductances of FM ﬁlms of a ﬁnite thickness attached to two NM leads have been studied veryrecently for Co/Cu, Fe/Au, and Fe/Cr systems. 17,21It has been found that the thickness dependence of the real part ofthe spin-mixing conductance saturates very rapidly forthicker ﬁlms and that its value can be well approximated bythe Sharvin conductance of the NM electrode. This behavioris equivalent to the very short magnetic coherence length andit proves that the spin-mixing conductance is predominantlyan interface property. However, as suggested by several au-thors, this may not hold for other ferromagnets such as Ni/H20849Ref. 13/H20850or the diluted magnetic semiconductors /H20849Ga,Mn /H20850As, 11where the average exchange splitting is much smaller than in the 3 dtransition metals and the size of the Fermi surface is small which could enhance the coherencelength. 17 For systems investigated so far, the imaginary part of the spin-mixing conductance was found to be much smaller thanthe real part. 18,21This feature is closely linked to properties of the current-induced steady-state torques in noncollinearspin valves and it results in a very small torque componentperpendicular to the plane spanned by magnetization direc-tions of the two FM layers. 12The assumption of a negligible imaginary part of the spin-mixing conductance has also beenemployed in theoretical studies of angular magnetoresistanceof spin valves 22as well as of magnetization dynamics of thin FM ﬁlms;21its validity, however, has to be checked in each particular case. As indicated by several authors, this becomesespecially important for nonmetallic systems, such as sys-tems containing tunneling barriers 18or insulating FM parts.23 The purpose of the present paper is threefold. First, we consider a FM ﬁlm embedded between two NM metallicleads with epitaxial interfaces and give a general theoreticalformulation of the spin torque and the spin ﬂuxes due to thespin accumulation in one of the electrodes /H20849Sec. II /H20850. We use the language of nonequilibrium Green’s functions 19,24/H20849NGF /H20850 which in the present context is equivalent to the Landauer-Büttiker theory but it yields formulas that can be more easilyevaluated by means of standard Green’s-function techniques.Second, we describe brieﬂy a numerical implementationwithin the ﬁrst-principles tight-binding linear mufﬁn-tin or-bital /H20849TB-LMTO /H20850method 25,26/H20849Sec. III /H20850. The method has re- cently been combined with the coherent potentialapproximation 26–29/H20849CPA /H20850for charge CPP transport in disor- dered multilayers30and we employ the CPA here as well. Finally, we perform calculations and discuss the results for amodel Cu/Ni/Cu /H20849100/H20850system followed by a study of more complex magnetic ﬁlms with potential applicability in spin- tronics: a binary random alloy Ni 0.84Fe0.16 /H20849Permalloy, Py /H20850, the half-metallic ferromagnet Co 2MnSi, and the diluted mag- netic semiconductor /H20849Ga,Mn /H20850As/H20849Sec. IV /H20850.II. THEORY A. Response of the spin torque We consider a NM/FM/NM system with noninteracting electrons. Its effective one-electron Hamiltonian Hcan be written as H=H0+/H9253/H20849/H9268·n/H20850, /H208491/H20850 where H0represents a spin-independent part, /H9253is the ex- change splitting which is nonzero only inside an intermediateregion containing the FM ﬁlm and narrow parts of the twoadjacent semi-inﬁnite NM leads, the vector /H9268=/H20849/H9268x,/H9268y,/H9268z/H20850 denotes a vector of the Pauli matrices, and the unit vector n deﬁnes the direction of the exchange ﬁeld of the FM ﬁlm.The spin dependence of the Hamiltonian Hin Eq. /H208491/H20850implies that electrons with spin parallel and antiparallel to nexperi- ence Hamiltonians H ↑=H0+/H9253andH↓=H0−/H9253, respectively. We deﬁne the spin torque /H9270as the time derivative of the total spin magnetic moment. The latter operator is repre-sented by the Pauli matrices /H9268, so that /H9270=−i/H20851/H9268,H/H20852, /H208492/H20850 where atomic units /H20849/H6036=1/H20850are used. This deﬁnition of the spin torque differs formally from the usual deﬁnition based on the spin currents on both sides of the FM ﬁlm;4,31equiva- lence of both approaches for spin valves has been given by anumber of authors 32,33while their equivalence in the present case is proved in Sec. II B. The well-known algebraic rulesfor the Pauli matrices /H20849 /H9268·p/H20850/H20849/H9268·q/H20850=p·q+i/H20849p/H11003q/H20850·/H9268, /H20849/H9268·p/H20850/H9268=p+i/H9268/H11003p, /H208493/H20850 /H9268/H20849/H9268·q/H20850=q+iq/H11003/H9268, valid for arbitrary classical vectors pandq, yield an explicit form of the torque operator /H9270=2/H9253n/H11003/H9268. /H208494/H20850 This relation shows that the spin torque is a local operator nonzero only inside the intermediate region with a directionperpendicular to the exchange ﬁeld of the FM ﬁlm. The thermodynamic average of the spin torque /H9270for the NM/FM/NM system in a stationary nonequilibrium state isgiven by /H9270¯=1 2/H9266/H20885 −/H11009/H11009 Tr/H20853/H9270G/H11021/H20849E/H20850/H20854dE, /H208495/H20850 where G/H11021/H20849E/H20850is the lesser component of the NGF.19,24The latter quantity is related to the retarded and advanced Green’s functions Gr/H20849E/H20850andGa/H20849E/H20850by means of34 G/H11021/H20849E/H20850=Gr/H20849E/H20850/H9018/H11021/H20849E/H20850Ga/H20849E/H20850, Gr/H20849E/H20850=/H20851E−H−/H9018r/H20849E/H20850/H20852−1, /H208496/H20850K. CARV A AND I. TUREK PHYSICAL REVIEW B 76, 104409 /H208492007 /H20850 104409-2Ga/H20849E/H20850=/H20851E−H−/H9018a/H20849E/H20850/H20852−1, where /H9018/H11021/H20849E/H20850,/H9018r/H20849E/H20850, and/H9018a/H20849E/H20850denote the lesser, retarded, and advanced components of the self-energy, respectively. Note that all operators in Eqs. /H208495/H20850and /H208496/H20850are deﬁned in the Hilbert space of the intermediate region. The spin accumulation in the NM leads results in a change of the lesser self-energy /H9254/H9018/H11021/H20849E/H20850/H20849see below /H20850which induces the following ﬁrst-order change of the thermody- namic average /H208495/H20850: /H9254/H9270¯=1 2/H9266/H20885 −/H11009/H11009 Tr/H20853Ga/H20849E/H20850/H9270Gr/H20849E/H20850/H9254/H9018/H11021/H20849E/H20850/H20854dE. /H208497/H20850 The special form of the torque operator, Eq. /H208492/H20850, together with the expression for Gr,a/H20849E/H20850, Eq. /H208496/H20850, provide a relation Ga/H20849E/H20850/H9270Gr/H20849E/H20850=−i/H20851/H9268Gr/H20849E/H20850−Ga/H20849E/H20850/H9268/H20852+Ga/H20849E/H20850/H9268/H9003/H20849E/H20850Gr/H20849E/H20850, /H208498/H20850 where we introduced the usual abbreviation for the anti- Hermitean part of the self-energy, namely, /H9003/H20849E/H20850=i/H20851/H9018r/H20849E/H20850−/H9018a/H20849E/H20850/H20852. /H208499/H20850 In deriving Eq. /H208498/H20850, use was made of the fact that the self- energies of the unperturbed NM leads are spin independent,hence /H20851 /H9268,/H9018r,a/H20849E/H20850/H20852=0. The total self-energies can be written as sums of separate contributions due to the left /H20849L/H20850and the right /H20849R/H20850leads, /H9018x/H20849E/H20850=/H9018Lx/H20849E/H20850+/H9018Rx/H20849E/H20850,x=r,a,/H11021. /H2084910/H20850 For stationary nonequilibrium systems without spin accumu- lation, the lesser self-energies are given by /H9018L,R/H11021/H20849E/H20850=fL,R/H20849E/H20850/H9003L,R/H20849E/H20850, /H2084911/H20850 /H9003L,R/H20849E/H20850=i/H20851/H9018L,Rr/H20849E/H20850−/H9018L,Ra/H20849E/H20850/H20852, where the functions fL,R/H20849E/H20850refer to the Fermi-Dirac distri- butions of the two leads. Note that /H9003/H20849E/H20850=/H9003L/H20849E/H20850+/H9003R/H20849E/H20850. In the thermodynamic equilibrium, the distributions fL,R/H20849E/H20850coincide with the Fermi-Dirac distribution of the whole system. In the presence of spin accumulation in one of the leads /H20849L/H20850, the system is driven out of equilibrium by adding a spin-dependent shift to the Fermi energy of the lead characterized by a magnitude /H9254EL. This yields the ﬁrst-order change of the lesser self-energy in a form /H9254/H9018/H11021/H20849E/H20850=/H9254/H9018L/H11021/H20849E/H20850=f/H11032/H20849E/H20850/H20849/H9268·a/H20850/H9003L/H20849E/H20850/H9254EL, /H2084912/H20850 where f/H11032/H20849E/H20850means the derivative of the Fermi-Dirac distri- bution and ais a unit vector pointing in the direction of spin accumulation. The spin-dependent /H9254/H9018/H11021/H20849E/H20850according to Eq. /H2084912/H20850describes compactly a change of the left-lead Fermi en- ergy by − /H9254EL/H20849+/H9254EL/H20850for electrons with spin parallel /H20849anti- parallel /H20850to the spin accumulation direction a. For systems at zero temperature, which will be considered in the following, f/H11032/H20849E/H20850=−/H9254/H20849E−EF/H20850where EFis the Fermi energy. Substitution of Eqs. /H208498/H20850and /H2084912/H20850into Eq. /H208497/H20850provides a starting expres- sion for the corresponding response coefﬁcient CL:CL/H11013/H9254/H9270¯ /H9254EL=1 2/H9266Tr/H20851i/H20849/H9268Gr−Ga/H9268/H20850/H20849/H9268·a/H20850/H9003L −/H9268/H9003Gr/H20849/H9268·a/H20850/H9003LGa/H20852, /H2084913/H20850 where all omitted energy arguments equal the Fermi energy EF. In order to extract the dependence of the response coefﬁ- cient CLon orientation of the spin accumulation aand the magnetization direction n, the explicit structure of the Green’s functions Gr,aof the Hamiltonian /H208491/H20850with respect to the spin must be used, Gr,a=G↑r,a+G↓r,a 2+G↑r,a−G↓r,a 2/H20849/H9268·n/H20850, /H2084914/H20850 where the spin-resolved Green’s functions are deﬁned by Gsr,a/H20849E/H20850=/H20851E−Hs−/H9018r,a/H20849E/H20850/H20852−1,s=↑,↓. /H2084915/H20850 The substitution of Eq. /H2084914/H20850into Eq. /H2084913/H20850reduces its r.h.s. to a sum of terms of the form Tr /H20849/H9264X/H20850=trS/H20849/H9264/H20850tr/H20849X/H20850, where /H9264is a matrix in the spin indices only while Xis a matrix in the other /H20849site and orbital /H20850indices and where the symbols tr Sand tr denote the respective trace operations. Further steps em-ploy the rules /H208493/H20850and their consequences for trace relations: tr S/H20851/H9268/H20849/H9268·a/H20850/H20852=2a, trS/H20851/H9268/H20849/H9268·n/H20850/H20849/H9268·a/H20850/H20852=2in/H11003a, /H2084916/H20850 trS/H20851/H9268/H20849/H9268·n/H20850/H20849/H9268·a/H20850/H20849/H9268·n/H20850/H20852=4/H20849n·a/H20850n−2a. The resulting expression for CLfollows after a lengthy but straightforward manipulation: CL=D1a+D2a/H11003n−D3/H20849n·a/H20850n, /H2084917/H20850 where the prefactors D1,D2, and D3are given by D1=1 2/H9266tr/H20851i/H20849G↑r+G↓r−G↑a−G↓a/H20850/H9003L−/H9003G↑r/H9003LG↓a−/H9003G↓r/H9003LG↑a/H20852, D2=1 2/H9266tr/H20851/H20849G↑r−G↓r+G↑a−G↓a/H20850/H9003L+i/H20849/H9003G↑r/H9003LG↓a −/H9003G↓r/H9003LG↑a/H20850/H20852, /H2084918/H20850 D3=1 2/H9266tr/H20851/H9003/H20849G↑r−G↓r/H20850/H9003L/H20849G↑a−G↓a/H20850/H20852. The form of Eq. /H2084917/H20850can be simpliﬁed by using a general relation i/H20851Gr/H20849E/H20850−Ga/H20849E/H20850/H20852=Ga/H20849E/H20850/H9003/H20849E/H20850Gr/H20849E/H20850/H20849 19/H20850 that follows from Eqs. /H208496/H20850and /H208499/H20850. After inserting the spin- resolved counterparts of Eq. /H2084919/H20850,i/H20849Gsr−Gsa/H20850=Gsa/H9003Gsr/H20849s =↑,↓/H20850, in the expression /H2084918/H20850forD1one obtains D3=D1. The previous formula /H2084917/H20850for the response coefﬁcient CL can thus be rewritten in a form of a vector explicitly perpen- dicular to the vector n:SPIN-MIXING CONDUCTANCES OF THIN MAGNETIC … PHYSICAL REVIEW B 76, 104409 /H208492007 /H20850 104409-3CL=D1n/H11003/H20849a/H11003n/H20850+D2a/H11003n. /H2084920/H20850 The ﬁrst term in CLrefers to the torque component lying in the plane containing the two vectors nanda/H20849in-plane com- ponent /H20850, while the second term refers to the component per- pendicular to this plane /H20849out-of-plane component /H20850. A closer inspection of the real quantities D1andD2, Eq. /H2084918/H20850, reveals their simple relation to a single complex quantity—the spin-mixing conductance C Lmix: CLmix=1 2/H9266tr/H20851i/H20849G↑r−G↓a/H20850/H9003L−/H9003G↑r/H9003LG↓a/H20852, /H2084921/H20850 which yields D1=2R e CLmix,D2=2I m CLmix. /H2084922/H20850 Note that the complex response coefﬁcient CLmixis sometimes denoted as a “spin-pumping conductance.”21,35The formulas /H2084920/H20850–/H2084922/H20850represent the central result of this section. The real part of the CLmixis always positive, as can be shown from the identity D1=D3, see Eq. /H2084918/H20850and the text below Eq. /H2084919/H20850, and from the positive deﬁniteness of the operators /H9003L,R. Consequently, the in-plane torque component is always positive and the spin accumulation tends to alignthe magnetization direction ntoward the spin accumulation direction a. A recent theoretical study of Co/Cu/Co /H20849111/H20850 structures has predicted that the in-plane torque due to an applied spin-independent bias can change its sign with vary-ing thickness of the Co layer. 32This qualitative difference indicates that the phase coherence across a structure contain-ing two noncollinear magnetic layers gives rise to a morecomplex behavior than that due to noncollinearity betweenthe semiclassical spin accumulation and the exchange split-ting of a single FM ﬁlm. Another interesting change of thesign of the in-plane torque component has been predicted formagnetic tunnel junctions under a sufﬁciently high bias. 36A direct comparison of this phenomenon to results of thepresent linear-response theory seems to be impossible; more-over, the necessary high bias values can hardly be realized inmetallic systems studied here. B. Relation to spin currents It is well known that spin torques acting on FM layers of a magnetic multilayer are closely related to spin currentsﬂowing through the structure. The physical origin of thisrelation can be traced back to a general theorem for the timederivative of the spin /H20849magnetization /H20850density of a many- electron system which can be expressed by means of thedivergence of the spin-current tensor and the spin-torquedensity. 13,37Application of this theorem to stationary states gives the spin torque acting on an arbitrary spatial regionexactly equal to the ﬂux of the spin current across the surfaceof the region, i.e., equal to the surface integral of the spin-current tensor. In the present case, the spatial region com-prises naturally the FM ﬁlm and a few neighboring atomiclayers on its both sides; its surface is formed by two bound-ary planes parallel to the ﬁlm and located outside it. The spinﬂuxes on the two sides of the FM ﬁlm are deﬁned asJ L=−i/H9268/H20851/H9016L,H0/H20852=−i/H20851/H9268/H9016L,H/H20852, /H2084923/H20850 JR=−i/H9268/H20851/H9016R,H0/H20852=−i/H20851/H9268/H9016R,H/H20852, where the kinetic energy /H20849intersite hopping /H20850is contained in the spin-independent part H0of the Hamiltonian H, Eq. /H208491/H20850, and where the /H9016Land/H9016Rdenote projection operators on the respective leads including a few adjacent NM atomic layersof the intermediate region such that /H9016 L/H9253=/H9016R/H9253=0. Note that the projector operators /H9016Land/H9016Rproject on semi-inﬁnite regions with zero exchange splitting and that the operators−i/H20851/H9016 L,H0/H20852and − i/H20851/H9016R,H0/H20852in the deﬁnition of the spin ﬂuxes represent the usual particle ﬂuxes across the two boundary planes. The ﬁnal expression of the JL,Rin Eq. /H2084923/H20850 has a form of the time derivative due to the full HamiltonianH; this form is a consequence of the special choice of the projection operators /H9016 L,Rand it simpliﬁes the subsequent theoretical analysis considerably. In particular, the linear-response coefﬁcients for the spin ﬂuxes, deﬁned as K LL=/H9254J¯L /H9254EL,KLR=/H9254J¯R /H9254EL, /H2084924/H20850 can be obtained in a similar way as in Sec. II A for the response coefﬁcient CL, Eq. /H2084913/H20850. The explicit expressions are given by KLL=1 2/H9266Tr/H20851i/H20849/H9268Gr−Ga/H9268/H20850/H20849/H9268·a/H20850/H9003L−/H9268/H9003LGr/H20849/H9268·a/H20850/H9003LGa/H20852, /H2084925/H20850 KLR=−1 2/H9266Tr/H20851/H9268/H9003RGr/H20849/H9268·a/H20850/H9003LGa/H20852, where several simple properties of the projectors, such as /H20851/H9016L,/H9018r,a/H20849E/H20850/H20852=0 and /H9016L/H9003=/H9003L/H20849and similarly for /H9016R/H20850, were used. Note that KLL+KLR=CLwhich reﬂects the above men- tioned relation between the torque and the spin currents. The resulting formulas can be summarized as KLL=2R e CLL,mixn/H11003/H20849a/H11003n/H20850+2I m CLL,mixa/H11003n +/H20849C↑+C↓/H20850/H20849n·a/H20850n, /H2084926/H20850 KLR=2R e CLR,mixn/H11003/H20849a/H11003n/H20850+2I m CLR,mixa/H11003n −/H20849C↑+C↓/H20850/H20849n·a/H20850n, where we introduced complex coefﬁcients in analogy to the spin-mixing conductance, Eq. /H2084921/H20850, namely, CLL,mix=1 2/H9266tr/H20851i/H20849G↑r−G↓a/H20850/H9003L−/H9003LG↑r/H9003LG↓a/H20852, /H2084927/H20850 CLR,mix=−1 2/H9266tr/H20849/H9003RG↑r/H9003LG↓a/H20850, and the spin-resolved charge conductances /H20849in units of e2//H6036/H20850K. CARV A AND I. TUREK PHYSICAL REVIEW B 76, 104409 /H208492007 /H20850 104409-4Cs=1 2/H9266tr/H20849/H9003RGsr/H9003LGsa/H20850,s=↑,↓. /H2084928/H20850 Two comments to these results are now in order. First, the dependence of the spin ﬂuxes on the orientation of the spinaccumulation /H20849a/H20850and the magnetization /H20849n/H20850, Eq. /H2084926/H20850,i s more complicated than that of the spin torque, Eq. /H2084920/H20850. Sec- ond, an obvious relation C Lmix=CLL,mix+CLR,mixcan be proved from Eqs. /H2084921/H20850and /H2084927/H20850. However, this decomposition does not justify a direct interpretation of the quantities CLL,mixand CLR,mixas respective contributions to the total spin-mixing conductance due to the left and right spin ﬂuxes, since thelatter contain also terms proportional to the total charge con-ductance C ↑+C↓and parallel to the magnetization direction n, see Eq. /H2084926/H20850. Hence, the quantities CLL,mixandCLR,mixrefer only to the transverse components of the left and right spinﬂuxes with respect to the exchange ﬁeld of the FM ﬁlm. C. Relation to scattering theory The resulting dependences of the spin torque, Eq. /H2084920/H20850, and of the spin ﬂuxes, Eq. /H2084926/H20850, on the orientation of the spin accumulation and the magnetization are identical to thoseobtained within the Landauer-Büttiker scattering theory oftransport. 17In order to make the relation of this traditional tool to the NGF approach more explicit, we consider here thesimplest case, namely, a one-dimensional /H208491D/H20850system with one propagating mode in two identical NM leads. The conﬁguration space of the system is a real axis with positions denoted by a continuous variable x/H20849−/H11009/H11021x/H11021/H11009 /H20850. The spin-independent part of the Hamiltonian is given by H 0=−/H20849/H11509//H11509x/H208502/H20849we set the electron mass m=1/2 /H20850and the ex- change splitting /H9253/H20849x/H20850vanishes for x/H33355xLandx/H33356xR, where the points xLandxRdenote the boundaries between the NM leads and the intermediate region containing the FM part.The Fermi energy corresponds to a positive kinetic energy inthe leads, E F=k2with k/H110220. The spin-resolved retarded and advanced Green’s func- tions Gsr,a/H20849s=↑,↓/H20850at this real energy are constructed from two independent solutions /H92731s/H20849x/H20850and/H92732s/H20849x/H20850of the Schrödinger equation for the Hamiltonians H↑=H0+/H9253and H↓=H0−/H9253. The asymptotics of the two solutions are /H92731s/H20849x/H20850= exp /H20849ikx/H20850forx/H33356xR, /H2084929/H20850 /H92732s/H20849x/H20850= exp /H20849−ikx/H20850forx/H33355xL, and the retarded Green’s function is given by /H20855x/H20841Gsr/H20841x/H11032/H20856=Ws−1/H92731s/H20849x/H11022/H20850/H92732s/H20849x/H11021/H20850, /H2084930/H20850 where Ws=/H92732s/H20849x/H20850/H20851/H11509/H92731s/H20849x/H20850//H11509x/H20852−/H92731s/H20849x/H20850/H20851/H11509/H92732s/H20849x/H20850//H11509x/H20852denotes the /H20849xindependent /H20850Wronskian of the two solutions, while x/H11022=max /H20853x,x/H11032/H20854andx/H11021=min /H20853x,x/H11032/H20854. The asymptotic behavior of the solution /H92731s/H20849x/H20850forx /H33355xLis given by /H92731s/H20849x/H20850=ts−1/H20851exp/H20849ikx/H20850+rsexp/H20849−ikx/H20850/H20852, /H2084931/H20850 where we introduced the spin-resolved transmission /H20849ts/H20850and reﬂection /H20849rs/H20850coefﬁcients of the wave incoming from theleft. They satisfy the usual condition /H20841rs/H208412+/H20841ts/H208412=1 and their knowledge allows us to evaluate explicitly the Wronskian in Eq. /H2084930/H20850,Ws=2ikts−1, as well as the asymptotics of the solu- tion/H92732s/H20849x/H20850forx/H33356xR: /H92732s/H20849x/H20850=1 tsexp/H20849−ikx/H20850−rs tsexp/H20849ikx/H20850. /H2084932/H20850 These relations yield, e.g., the following elements of the re- tarded Green’s functions: /H20855xL/H20841Gsr/H20841xL/H20856=1 2ik/H208511+rsexp/H20849−2ikxL/H20850/H20852, /H20855xR/H20841Gsr/H20841xR/H20856=1 2ik/H208751−rsts tsexp/H208492ikxR/H20850/H20876, /H2084933/H20850 /H20855xL/H20841Gsr/H20841xR/H20856=ts 2ikexp/H20851ik/H20849xR−xL/H20850/H20852. Other elements, including those of the advanced Green’s functions, can be obtained with help of general identities /H20855x/H11032/H20841Gsr/H20841x/H20856=/H20855x/H20841Gsr/H20841x/H11032/H20856and /H20855x/H20841Gsa/H20841x/H11032/H20856=/H20855x/H20841Gsr/H20841x/H11032/H20856. Since the anti- Hermitean part of the self-energy /H9003=/H9003L+/H9003R, Eq. /H2084911/H20850,i si n this particular case given by38 /H20855x/H20841/H9003L/H20841x/H11032/H20856=2k/H9254/H20849x−xL/H20850/H9254/H20849x/H11032−xL/H20850, /H2084934/H20850 /H20855x/H20841/H9003R/H20841x/H11032/H20856=2k/H9254/H20849x−xR/H20850/H9254/H20849x/H11032−xR/H20850, the total spin-mixing conductance, Eq. /H2084921/H20850, and its left and right contributions due to the transverse spin currents, Eq./H2084927/H20850, are equal to C LL,mix=1 2/H9266/H208491−r↑r↓/H20850,CLR,mix=−1 2/H9266t↑t↓, /H2084935/H20850 CLmix=1 2/H9266/H208491−r↑r↓−t↑t↓/H20850. Note that the spin-resolved CPP conductances, Eq. /H2084928/H20850, re- duce within the present model to Cs=/H20841ts/H208412//H208492/H9266/H20850. The last result, Eq. /H2084935/H20850, proves an equivalence of the developed NGF approach to the existing Landauer-Büttiker formalism.17In particular, the left contribution CLL,mixwas identiﬁed with the spin-mixing conductance of a single NM/FM interface,15,16,18while the transmission term CLR,mix appeared naturally for FM ﬁlms of a ﬁnite thickness.21,22,31,35,39The presence of several propagating modes in the leads is taken simply into account by doublesummations over these channels in Eq. /H2084935/H20850, see References 16,17, and 21for details. The full equivalence of both approaches has been proven for particle currents due to spin-independent chemical poten-tials in the leads with any number of propagatingchannels; 40,41a similar general proof for spin currents due to the spin accumulation goes beyond the scope of the presentpaper. In the case of realistic multiorbital tight-bindingHamiltonians, a fundamental question concerns a possiblecontribution of evanescent states to the transport coefﬁcients.A study of a spinless case 42indicated that although the eva- nescent states do not contribute explicitly to the total con-SPIN-MIXING CONDUCTANCES OF THIN MAGNETIC … PHYSICAL REVIEW B 76, 104409 /H208492007 /H20850 104409-5ductance, their implicit effect has to be taken fully into ac- count, and that an improper treatment can cause serioustheoretical ﬂaws, such as a dependence of the calculated ﬂuxon the position chosen for its evaluation. While a subsequentpaper 43seems to disprove some of the conclusions of Ref. 42, the most recent studies of the spin-mixing conductance have revealed that evanescent states in FM leads cannot beignored in accurate calculations of single NM/FMinterfaces. 17,21 We think that detailed assessment of a role of the evanes- cent states for spin-mixing conductances is less important inthe present Green’s-function approach; instead, we show inthe Appendix that the derived transport coefﬁcients, Eqs./H2084921/H20850and /H2084927/H20850, of the NM/FM/NM systems do not depend on where boundaries between the leads and the intermediate re-gion are chosen. This invariance property represents a mostimportant feature both from a purely theoretical viewpointand for practical calculations. III. IMPLEMENTATION A. Translation into the tight-binding linear mufﬁn-tin orbital method The ab initio TB-LMTO method25,26,44proved to be an efﬁcient tool for electronic structure and transport calcula-tions. Various details of this approach for transport propertiesof bulk and layered systems can be found in the literatureboth for the Landauer-Büttiker formalism 17,20and for the Green’s-function point of view.30,45–48For this reason, we present below only the ﬁnal TB-LMTO expression for thespin-mixing conductance C Lmixcorresponding to Eq. /H2084921/H20850. The spin-mixing conductance per unit two-dimensional /H208492D/H20850cell is given by CLmix=1 2/H92661 N/H20648tr/H20851i/H20849g↑r−g↓a/H20850BL−/H20849BL+BR/H20850g↑rBLg↓a/H20852,/H2084936/H20850 where N/H20648refers to a large number of 2D cells in directions parallel to atomic layers and the trace is taken over the siteand orbital indices of the intermediate region. The quantities g srandgsa/H20849s=↑,↓/H20850denote spin-resolved auxiliary Green’s- function matrices calculated respectively at energies EF+i/H9257 and EF−i/H9257, where /H9257→0+. The spin-independent matrices BL,Rcorrespond to anti-Hermitean parts of self-energies of the NM leads.30,45In the principal-layer technique used here, the intermediate region consists of Nprincipal layers and the BLand the BRare localized in the ﬁrst and the Nth principal layer, respectively. Conversion from the atomic units to usualunits of conductance is achieved by a prefactor of e 2//H6036 which leads to replacement of 1/ /H208492/H9266/H20850in Eq. /H2084936/H20850by the conductance quantum e2/h. For epitaxial systems with perfect 2D translational sym- metry, the evaluation of Eq. /H2084936/H20850rests on the lattice Fourier transformation of the involved matrices. For FM ﬁlms withsubstitutional disorder, attached to nonrandom electrodes, theCPA is used for conﬁgurational averaging. 26,44The CPA- vertex corrections due to the second term in Eq. /H2084936/H20850are formulated and calculated according to Ref. 30.B. Computational details Details of the self-consistent electronic structure calcula- tions employing the TB-LMTO technique in the atomicsphere approximation /H20849ASA /H20850 were described elsewhere; 26,30,44the present results were based on the local spin-density approximation49/H20849LSDA /H20850to the density func- tional theory50with parametrization of the local exchange correlation potential according to Ref. 51and with a valence basis comprising s,p, and dorbitals. The systems treated in Sec. IV were derived from face-centered-cubic /H20849fcc/H20850and body-centered-cubic /H20849bcc/H20850lattices with neglected lattice re- laxations and with layer stacking along the /H20849100/H20850direction; in both cases one principal layer consisted of two neighbor-ing atomic layers. The energy arguments E F±i/H9257for the conductance calcu- lations contained an imaginary part of /H9257=10−7Ry. Evalua- tion of the trace in Eq. /H2084936/H20850used a uniform mesh in the 2D reciprocal space with densities equivalent to 6400 and 5000sampling k /H20648points in the full 2D Brillouin zone /H20849BZ/H20850of the 1/H110031 unit cell of the attached Cu /H20849100/H20850and Cr /H20849100/H20850leads, respectively. IV . RESULTS AND DISCUSSION A. Ni-based magnetic ﬁlms The present study of the fcc Cu/Ni/Cu /H20849100/H20850and Cu/Py/ Cu/H20849100/H20850systems is performed with a ﬁxed lattice having a lattice parameter between that of pure Ni /H208490.352 nm /H20850and pure Cu /H208490.361 nm /H20850; the actual value used in the calculations /H208490.355 nm /H20850is close to the experimental value of a Ni0.80Fe0.20Permalloy.52However, the reported results and the conclusions made are qualitatively fairly stable with re-spect to variations of the lattice parameter within the limitsset by pure nickel and copper. The spin-resolved conductances for the Cu/Ni/Cu /H20849100/H20850 system as functions of the Ni thickness are plotted in Fig. 1 /H20849top panel /H20850. They exhibit an expected behavior, namely, a nearly constant value for both spin channels with small os-cillations due to quantum-size effects. The majority conduc-tance is higher than the minority one due to the well-knownspin dependence of the potential difference at the Cu/Ni in-terface: the majority potentials of both species are nearlyidentical whereas the minority electrons feel a non-negligiblemismatch of the two potentials at the interface. For the samereason, the quantum-size effects are hardly visible in the ma-jority conductance but they are more pronounced in the mi-nority channel. Similar features were found in previous stud-ies of Cu/Ni and Cu/Co based magnetic multilayers; 45,53,54 they are responsible for the CPP-GMR phenomenon. The real and the imaginary parts of the spin-mixing con- ductance of the Cu/Ni/Cu /H20849100/H20850system, plotted in Fig. 1 /H20849bottom panel /H20850, exhibit a qualitatively different dependence on the Ni thickness as compared to the spin-resolved con-ductances. The behavior of C Lmixis characterized by pro- nounced oscillations with a period of 11 ML /H20849monolayers /H20850 and with a large amplitude that decays only slowly with in-creasing Ni thickness. The observed dependence is also dif-ferent from the dependences of C Lmixfound forK. CARV A AND I. TUREK PHYSICAL REVIEW B 76, 104409 /H208492007 /H20850 104409-6Cu/Co/Cu /H20849111/H20850and Au/Fe/Au /H20849100/H20850trilayers;17,21the oscil- lations found for the latter systems are smaller in magnitude and they are damped very rapidly being suppressed essen-tially for magnetic ﬁlm thicknesses greater than 15 ML. Notethat the Re C Lmixfor the Cu/Ni/Cu /H20849100/H20850system oscillates around a mean value that is close to the Sharvin conductance of the fcc Cu /H20849100/H20850lead, CSh=0.93 e2/h/H20849per one spin and one interface atom /H20850, while the mean value of the oscillating ImCLmixis appreciably smaller, in qualitative agreement with the other metallic systems. A quantitative theory of oscillations of the CPP transport properties was formulated a decade ago for the spin-resolvedconductances of magnetic multilayers with varying thicknessof a NM spacer, 55,56whereas the case of the spin-mixing conductance of a single magnetic ﬁlm with varying thicknesshas been worked out very recently. 17,21Both approaches em- phasize the role of the Fermi-surface /H20849FS/H20850properties of the thick layer, very much in the spirit of the theory of the os-cillatory interlayer exchange coupling in magneticmultilayers. 57,58 In the present case of Cu/Ni/Cu /H20849100/H20850, the unimportant oscillations of the conductances in each separate spin chan- nel together with the pronounced oscillations of the spin-mixing conductance /H20849see Fig. 1/H20850indicate that an origin of the latter has to be identiﬁed with stationary points of the differ- ence k i/H11036↑−kj/H11036↓as a function of the k/H20648point where iandjrunover the individual sheets of the fcc Ni FS in the two spin channels.17,21 The majority FS of fcc Ni is rather simple having only a single sheet /H20849topologically similar to the well-known FS of fcc Cu /H20850, whereas four sheets are encountered in the minority FS.59,60However, one of them, namely, the e6↓sheet,59is nearly parallel to the spin-up FS in large parts of the fcc BZ.The cross sections of these two FS sheets /H20849constructed from the bulk band structure of fcc Ni obtained using the presentTB-LMTO-ASA technique /H20850by the plane /H20849001/H20850are shown in Fig. 2. These two sheets give rise to a stationary point of the difference k /H11036↑−k/H11036↓; the corresponding k/H20648vector and the sta- tionary value of the difference, /H9004k/H11036, are marked in the ﬁgure as well. The resulting stationary value /H9004k/H11036/H110151.111 a−1, where adenotes the fcc lattice parameter, yields oscillations with a period /H9011=2/H9266//H9004k/H11036/H1101511.3 ML, in a very good agree- ment with the period observed in the calculated data, seebottom panel of Fig. 1. The large amplitudes of the oscilla- tions can qualitatively be understood in terms of the smallcurvatures of the two FS sheets at the stationary point. 55 The FS origin of the oscillations of CLmixin the pure Cu/Ni/Cu /H20849100/H20850system can be documented by their sensitiv- ity with respect to alloying in the magnetic ﬁlm. The thick- ness dependence of CLmixfor the Cu/Py/Cu /H20849100/H20850system /H20849where Py denotes a random fcc Ni 0.84Fe0.16alloy /H20850is free of any long-range oscillations, see bottom panel of Fig. 1. The randomly placed Fe impurities are very efﬁcient in suppress-ing the quantum interference effects in the Ni ﬁlms and theasymptotic values of the spin-mixing conductance are ob-tained already for Py ﬁlm thickness of about 12 ML. Since effects of alloying on FS properties can be very different in different parts of the alloy BZ 61and, for FM alloys, in the two spin channels,59,62we present the spin- resolved Bloch spectral functions /H20849BSFs /H20850of the fcc Ni0.84Fe0.16alloy in Fig. 3. The BSFs are evaluated for the k point in the center of the /H9003-Kline, i.e., close to the stationary point relevant for the oscillations of CLmixin the pure00.51C/cell (e2/h)Ni Ni 00.51 0 10 20 30Cmix/cell (e2/h) film thickness (ML )(Re) (Im)Ni NiPy Py FIG. 1. /H20849Color online /H20850The spin-resolved conductances of Cu/Ni/Cu /H20849100/H20850as functions of the Ni thickness /H20849top panel /H20850and the spin-mixing conductances of Cu/Ni/Cu /H20849100/H20850/H20849red symbols /H20850and Cu/Py/Cu /H20849100/H20850/H20849blue symbols /H20850as functions of the magnetic ﬁlm thickness /H20849bottom panel /H20850. The real and the imaginary parts of the complex spin-mixing conductances are denoted by the full and theempty symbols, respectively.012 0 1 2kya/π kxa/πΓ XWK FIG. 2. /H20849Color online /H20850Cross section of the Ni Fermi surfaces for majority /H20849full red curve /H20850and minority /H20849dashed blue curve /H20850spin channels in the /H20849001/H20850plane. The horizontal arrow denotes the posi- tion of the k/H20648vector in the /H20849100/H20850plane corresponding to the station- ary point of the difference k/H11036↑−k/H11036↓. The labels /H9003,X,W, and Krefer to special points of the fcc BZ.SPIN-MIXING CONDUCTANCES OF THIN MAGNETIC … PHYSICAL REVIEW B 76, 104409 /H208492007 /H20850 104409-7Cu/Ni/Cu /H20849100/H20850system, see Fig. 2. The spectral function for majority electrons near the alloy Fermi energy is character- ized by a narrow Lorentzian peak indicating a weak alloydisorder whereas a broad non-Lorentzian shape is observedin the minority channel slightly above the Fermi level. Thisbroad peak is closely related to a virtual Fe dbound state located in the same energy range in the minority density ofstates /H20849DOS /H20850of Ni-rich fcc NiFe alloys; 62the latter features prove a strong scattering regime which is responsible for theabsence of oscillations of C Lmixin the Cu/Py/Cu /H20849100/H20850system. B. Films of the Heusler compound Co 2MnSi The ferromagnetic full-Heusler compound Co 2MnSi with anL21structure represents an interesting system for spin- tronics due to its high Curie temperature of 985 K /H20849Ref. 63/H20850 and due to its half-metallicity that was predicted theoreticallya few years ago 64,65and that has very recently been con- ﬁrmed experimentally.66 The L21structure consists of four interpenetrating fcc sublattices with origins shifted to points /H208490,0,0 /H20850, /H20849a/4,a/4,a/4/H20850,/H20849a/2,a/2,a/2/H20850, and /H208493a/4,3a/4,3a/4/H20850, where adenotes the fcc lattice parameter. In the case of the Co2MnSi compound, these four sublattices are consecutively occupied by Co, Mn, Co, and Si atoms. The bulk self-consistent spin-polarized DOSs of the compound, calculatedby means of the TB-LMTO method for an experimental fcclattice /H20849a=0.565 nm /H20850, 63are presented in Fig. 4. The minority-spin DOS is characterized by a narrow band gap that is 0.43 eV wide and by the Fermi energy EFlocated only 0.05 eV below the bottom of the conduction band, in reason-able agreement with the experiment providing the width ofthe band gap of 0.35–0.40 eV and a very small energy sepa-ration of 0.01 eV between the E Fand the conduction band.66 The calculated total spin moment of 5 /H9262Bper formula unit as well as the ﬁnal DOS shapes agree well with existing full-potential results. 65 The CPP transport properties are studied for /H20849100/H20850ﬁlms of the Co 2MnSi compound embedded between nonmagnetic bcc Cr /H20849100/H20850leads, as motivated by prepared Cr/Co 2MnSi epitaxial interfaces.66All atoms are located at sites of anideal bcc lattice leading thus to a small /H20849/H110112% /H20850compression inside the Cr electrodes as compared to an equilibrium Cr bcc lattice parameter. The Co 2MnSi ﬁlms contain an even number of atomic layers with the pure Co layer and the MnSilayer neighboring the left and the right Cr lead, respectively. The resulting transport properties versus the Co 2MnSi thickness are plotted in Fig. 5. The spin-resolved conduc- tances /H20849top panel of Fig. 5/H20850correspond to a metallic and a tunneling regime in the two spin channels. The majority con-ductance is essentially constant with unimportant oscillationsdue to interference effects whereas the minority conductance50050100150 -0.3 -0.2 -0.1 0 0.1 0.2BSF(states /Ry) E-EF(Ry)k=(Γ+K)/2 FIG. 3. The spin-polarized Bloch spectral functions of the ran- dom fcc Ni 0.84Fe0.16alloy as functions of energy Efor a kpoint in the center of the /H9003-Kline of the BZ.804004080 -0.9 -0.6 -0.3 0 0.3DOS (states/Ry) E-EF(Ry) FIG. 4. Spin-polarized densities of states /H20849per formula unit /H20850of the Heusler compound Co 2MnSi as functions of energy E. -6-5-4-3-2-10log10[C/cell (e2/h)] 012 0 10 20 30Cmix/cell (e2/h) Co2MnSi thickness (ML)Re Im FIG. 5. CPP conductances of Cr/Co 2MnSi/Cr /H20849100/H20850as functions of the Co 2MnSi thickness: spin-resolved conductances on a loga- rithmic scale for majority /H20849↑/H20850and minority /H20849↓/H20850electrons /H20849top panel /H20850 and the real and the imaginary parts of the spin-mixing conductance/H20849bottom panel /H20850.K. CARV A AND I. TUREK PHYSICAL REVIEW B 76, 104409 /H208492007 /H20850 104409-8decreases exponentially with increasing ﬁlm thickness, de- spite the tiny energy separation between the EFand the bot- tom of the bulk minority-spin conduction band. The thickness dependence of the spin-mixing conductance /H20849bottom panel of Fig. 5/H20850exhibits a nearly constant value with superimposed small oscillations due to quantum-size effects,in analogy to the case of metallic FM ﬁlms with a strongexchange splitting, e.g., Cu/Co/Cu /H20849111/H20850 and Au/Fe/Au /H20849100/H20850; 17,21the real part of CLmixhas values only slightly smaller than the Sharvin conductance of the bcc Cr/H20849100/H20850lead, CSh=2.63 e2/h/H20849per one spin and two interface Cr atoms /H20850. However, the imaginary part of CLmixacquires values as high as one-half of the real part in the case ofCr/Co 2MnSi/Cr /H20849100/H20850. Such a high value of the imaginary part of the spin-mixing conductance has not been encoun- tered in metallic NM/FM and NM/FM/NM systems;17,18,21 this new feature deserves thus a more detailed study. The k/H20648-resolved contributions to the total imaginary part of the CLmixare shown in Fig. 6for a 20 ML thick ﬁlm of Co2MnSi. Note that due to the L21structure of the ﬁlm, the 2D BZ of the Cr/Co 2MnSi/Cr /H20849100/H20850system forms only one- half of the 2D BZ corresponding to the 1 /H110031 unit cell of the bcc Cr /H20849100/H20850electrode; moreover, the two BZs were rotated mutually by 45° for computational reasons. One can see thatdominating positive contributions originate in regions around the/H9003¯andM¯points; regions close to the X¯points are char- acterized by parts of the 2D BZ without propagating chan-nels in the Cr leads 17surrounded by small areas with slightly negative contributions. The rest of the k/H20648points yield posi- tive contributions of minor magnitudes. The sum of the nega-tive k /H20648-resolved contributions amounts only to 7% of the total Im CLmix. These facts distinguish the present system with a half-metallic ferromagnet from the studied metallicsystems 18,21where signiﬁcant cancellation of positive and negative contributions takes place that results in a small nettotal sum. The reduced destructive interference in theCr/Co 2MnSi/Cr /H20849100/H20850system is explained qualitatively in Sec. IV D.C. Films of (Ga,Mn)As diluted magnetic semiconductor The diluted ferromagnetic semiconductors, such as Mn- doped GaAs,67,68represent systems with full spin polariza- tion of electron states at the Fermi energy and a small aver-age exchange splitting; this particular combination of spin-dependent properties might lead to unexpected CPP transportcharacteristics including the spin-mixing conductance. The bulk electronic structure obtained within the LSDA for the Ga 1−xMn xAs alloy using the TB-LMTO-CPA tech- nique is shown in Fig. 7in terms of spin-polarized DOSs. The computational treatment of the system relies on the samefour fcc sublattices as in Sec. IV B with consecutive occu-pancies by the atomic species as follows: Ga 1−xMn x, As, Vac 1, and Vac 2, where the latter two symbols denote so- called empty atomic spheres introduced for reasons of goodspace ﬁlling. The experimental fcc lattice parameter /H20849a =0.565 nm /H20850of the parent GaAs semiconductor is used throughout the whole interval of Mn concentration studied /H208490.03/H33355x/H333550.10 /H20850. Further details on the bulk electronic struc- tures can be found in Refs. 69and70. The resulting DOSs exhibit clear half-metallic features with the Fermi energy in-side the minority-spin band gap; the separation between theE Fand the top of the minority-spin valence band is appre- ciably smaller as compared to the width of the minority-spinband gap as well as to the width of the unoccupied part of themajority-spin valence band /H20849see Fig. 7/H20850. For examination of the CPP transport properties of the /H20849Ga,Mn /H20850As alloy, its ﬁlms in /H20849100/H20850stacking direction are chosen and attached to bcc Cr /H20849100/H20850electrodes, motivated by the small lattice mismatch between the two systems. Theideal bcc structure used for these systems corresponds againto a 2% compression of the bcc Cr lattice; the /H20849Ga,Mn /H20850As ﬁlms contain an odd number of atomic layers, terminated onboth sides by As layers. The spin-resolved CPP conductances of the Cr//H20849Ga,Mn /H20850As/Cr /H20849100/H20850systems exhibit a tunneling regime for the minority-spin channel due to the half-metallic nature of the bulk /H20849Ga,Mn /H20850As, whereas an Ohmic regime is encoun- tered in the majority-spin channel for ﬁlm thicknesses bigger −101234 FIG. 6. /H20849Color online /H20850k/H20648-resolved contributions to the imagi- nary part of the spin-mixing conductance of the systemCr/Co 2MnSi/Cr /H20849100/H20850with a 20 ML thick Co 2MnSi ﬁlm. The square displays one-quarter of the full 2D BZ and it is equivalent to two irreducible BZs; the /H9003¯point is located in the lower left corner, theM¯point lies in the upper right corner, and the two remaining corners correspond to the X¯points.201001020 -0.6 -0.3 0 0.3DOS (states/Ry) E-EF(Ry) FIG. 7. /H20849Color online /H20850Spin-polarized densities of states /H20849per formula unit /H20850of the bulk diluted /H20849Ga1−xMn x/H20850As ferromagnetic semi- conductor as functions of energy E: for x=0.03 /H20849full red curves /H20850and x=0.10 /H20849dashed blue curves /H20850.SPIN-MIXING CONDUCTANCES OF THIN MAGNETIC … PHYSICAL REVIEW B 76, 104409 /H208492007 /H20850 104409-9than 15 ML.30The latter fact proves a very strong intrinsic disorder in the /H20849Ga,Mn /H20850As ﬁlms despite the small content of Mn impurities. The dependences of the spin-mixing conductance on the /H20849Ga,Mn /H20850As thickness are presented in Fig. 8for two Mn concentrations. Similar dependences have been found for allother concentrations; they reveal rapid convergence of C Lmix with increasing ﬁlm thickness. The magnitude of the imagi- nary part is again comparable to the real part as in the case ofthe Co 2MnSi ﬁlms. There are, however, two striking differ- ences between the two half-metallic systems: the signs ofImC Lmixare different, and the magnitudes of CLmixare signiﬁ- cantly smaller in the /H20849Ga,Mn /H20850As case than in the Co 2MnSi case, see Figs. 5and8. Note that small spin-mixing conduc- tances with comparable magnitudes of real and imaginaryparts were reported for interfaces of doped nonmagneticsemiconductors /H20849InAs /H20850and metallic ferromagnets /H20849Fe/H20850, 71 which has been explained by a very small FS of the doped semiconductor.17 In order to get understanding of the present results, asymptotic values of the spin-mixing conductance, obtainedfor a large ﬁxed /H20849Ga,Mn /H20850As thickness, are plotted in Fig. 9 as functions of the Mn content x. One can see that both the real and the imaginary parts exhibit simple concentrationtrends, namely, their magnitudes increase monotonically withincreasing x. These trends indicate that an origin of the small values of C Lmixmight be closely related to the number of holes in the majority-spin valence band of the bulk/H20849Ga,Mn /H20850As, see Fig. 7, or, equivalently, to its majority FS. 72 This interpretation is also supported by a quantitative analy- sis of a simple free-electron model carried out in the nextsection. D. Free-electron model The results obtained in Secs. IV B and IV C for spin- mixing conductances in the presence of qualitatively differ-ent half-metallic FM ﬁlms deserve a unifying theoretical pic-ture capable to explain roughly the reported properties. Forthis reason, we consider here a very simple free-electronmodel of the NM/FM/NM system that has been frequently quoted in the literature, mainly in a context of metallicsystems. 13,21 Since the minority-spin channel is in a tunneling regime both for Co 2MnSi and /H20849Ga,Mn /H20850As ﬁlms, the contribution of transmitted electrons, CLR,mixin Eq. /H2084927/H20850, becomes negligible for thicker ﬁlms and the problem can thus be reduced to the contribution of reﬂected electrons, CLL,mixin Eq. /H2084927/H20850, evalu- ated for a single NM/FM interface of two semi-inﬁnite parts. Let us denote the spin-resolved values of the constant potential in the FM half-metal as Us/H20849s=↑,↓/H20850, while the constant potential inside the NM metal is set zero, i.e., it is taken as a reference value for one-electron energies. Theminority-spin potential is always repulsive, U ↓/H110220, whereas both signs of the majority-spin potential U↑are allowed in the range U↑/H11021U↓. The reﬂection coefﬁcients obtained at a NM/FM interface in a 1D system /H20849Sec. II C /H20850for electrons with a kinetic energy E/H110220 in the NM metal are given by rs/H20849E/H20850=1 Us/H208512E−Us−2/H20881E/H20849E−Us/H20850/H20852,s=↑,↓, /H2084937/H20850 where /H20881E/H20849E−Us/H20850/H11013i/H20881E/H20849Us−E/H20850has to be used for E/H11021Us. The electron energy at the studied NM/FM interface con- tains a kinetic contribution due to the electron motion in twodirections parallel to the interface; this leads to a variation ofthe energy Eof the perpendicular motion in the range 0 /H33355E/H33355E Fwhere the positive Fermi energy EFrepresents an- other parameter of the model. The latter is further con-strained by E F/H33355U↓owing to the assumed half-metallicity of the FM part. Integration of CLL,mix, Eq. /H2084935/H20850, over the k/H20648vec- tors yields the resulting spin-mixing conductance in a form CLmix CSh=1 EF/H20885 0EF /H208511−r↑/H20849E/H20850r↓/H20849E/H20850/H20852dE, /H2084938/H20850 where CShdenotes the Sharvin conductance of the NM metal per one spin channel. In order to make the model appropriate for the Cr/Co 2MnSi MnSi system /H20849Sec. IV B /H20850, we identify the bot- tom of the minority-spin band with the Fermi energy, U↓-0.300.30.6 0 10 20 30Cmix/cell (e2/h) (Ga,Mn)As thickness (ML)(Re) (Im)x = 0.05x = 0.08 x = 0.05 x = 0.08 FIG. 8. The spin-mixing conductances of Cr//H20849Ga1−xMn x/H20850As/Cr /H20849100/H20850forx=0.05 and x=0.08 as functions of the/H20849Ga,Mn /H20850As thickness: the real part /H20849full symbols /H20850and the imagi- nary part /H20849empty symbols /H20850.-0.300.30.6 0 0.05 0.1Cmix/cell (e2/h) Mn concentration xRe Im FIG. 9. Concentration dependence of the spin-mixing conduc- tance of Cr/ /H20849Ga1−xMn x/H20850As/Cr /H20849100/H20850for a ﬁxed /H20849Ga,Mn /H20850As thick- ness of 21 ML: the real part /H20849full symbols /H20850and the imaginary part /H20849empty symbols /H20850.K. CARV A AND I. TUREK PHYSICAL REVIEW B 76, 104409 /H208492007 /H20850 104409-10/H11013EF, see Fig. 4, and treat the attractive majority-spin poten- tialU↑/H110210 as an independent variable. The model is appli- cable to the Cr/ /H20849Ga,Mn /H20850As system as well; however, one has to take into account the negative effective electron mass at the top of the valence band of bulk /H20849Ga,Mn /H20850As, i.e., a particle-hole symmetry of transport properties must be em-ployed in this case. The restriction to U ↓/H11013EFcan again be used, see Fig. 7, while the majority-spin potential acquires now positive values with the limit U↑→EFcorresponding naturally to x→0/H20849undoped GaAs /H20850. The reﬂection coefﬁcients, Eq. /H2084937/H20850, in the case of U↑ /H110210/H20849Cr/Co 2MnSi /H20850, shown in Fig. 10for a particular nega- tive value of U↑, are real in the majority channel and com- plex in the minority channel. Consequently, they do not leadto any destructive interference effects in the Im C Lmix, Eq. /H2084938/H20850, and the latter can reach positive values non-negligible with respect to the Re CLmix, see Fig. 11, in qualitative agree- ment with data in the bottom panel of Fig. 5. Note that CLmix/CSh→1+i/H9266/4 for U↑→−/H11009, hence Im CLmixcan be as high as 78% of Re CLmixin the present model. Moreover, the real part of CLmixcomes out slightly smaller than the Sharvinconductance of the NM metal, see Fig. 11, reproducing thus another feature of the results of Sec. IV B. The case of positive U↑/H20849Cr//H20849Ga,Mn /H20850As/H20850leads to com- plex reﬂection coefﬁcients in both spin channels. In the limit ofU↑→EF/H20849diluted case, x→0/H20850, the spin-mixing conduc- tance tends to zero, the magnitudes of the Re CLmixand the ImCLmixbecome comparable, and the sign of the latter is negative, see Fig. 11. The magnitudes, signs, and concentra- tion trends of the data points in Fig. 9are thus semiquanti- tatively explained by the adopted simple free-electron model. V . CONCLUSIONS We have developed a nonequilibrium Green’s-function approach to the linear response of the spin torque and thespin ﬂuxes at a ferromagnetic thin ﬁlm due to the spin accu-mulation in one of adjacent nonmagnetic electrodes. We havesketched an equivalence of the developed scheme to the stan-dard Landauer /H20849scattering theory /H20850formulation of the spin- mixing conductance and have given a proof of invariance ofthe response coefﬁcients with respect to the choice of bound-aries between the semi-inﬁnite leads and the intermediateregion. The theory was implemented on an ab initio level using the TB-LMTO method; application to several ex-amples yields results that partly disprove general conclusionsdrawn from previous studies of other systems. In contrast to metallic ferromagnets with a large exchange splitting /H20849Fe, Co /H20850where strong damping of the transverse spin current takes place, the weak splitting of nickel and theparticular shape of its Fermi surfaces are responsible for pro-nounced long-range oscillations of the calculated spin-mixing conductance of Cu/Ni/Cu /H20849100/H20850system as a function of the ﬁlm thickness. The period of the oscillations can be quantitatively described by existing theories. This oscillatorybehavior proves that the transverse component of the spincurrent in the nickel ﬁlm is not absorbed within a few atomiclayers near the interface but it survives over appreciablylonger distances. Half-metallic ferromagnetic ﬁlms exhibit spin-mixing conductances with magnitudes of the imaginary parts com-parable to the real parts; this has been demonstrated both fora strongly polarized system, namely, the full-Heusler com-pound Co 2MnSi, and for diluted ferromagnetic semiconduc- tors /H20849Ga,Mn /H20850As. This property as well as other features of the spin-mixing conductance /H20849magnitude, sign of the imagi- nary part, concentration dependence /H20850have been explained within a free-electron model of the spin-polarized metal/half-metal interface. The non-negligible imaginary part of thespin-mixing conductance can give rise to big perpendicular/H20849out-of-plane /H20850spin-transfer torques in spin valves containing half-metallic ferromagnetic layers. The obtained results canalso be important for studies of magnetization dynamics ofthin ﬁlms in contact with nonmagnetic electrodes, since theimaginary part of the spin-mixing conductance is directlyrelated to the effective gyromagnetic ratio entering theLandau-Lifshitz-Gilbert equation of motion as discussed inRefs. 21,35, and 39. The present Green’s-function formulation of the linear re- sponse is inevitable for application of effective-medium-101 0 0.5 1r(E) E/EF(Re) (Im) FIG. 10. /H20849Color online /H20850Spin-resolved reﬂection coefﬁcients in the free-electron model as functions of energy E: in the spin- ↑chan- nel for the potential U↑=−EF/H20849black curve /H20850and in the spin- ↓chan- nel for the potential U↓=EF/H20849red line, real part; blue curve, imagi- nary part /H20850. -0.500.51 -5 -3 -1 1Cmix/CSh U/ EFRe Im FIG. 11. The spin-mixing conductance in the free-electron model as a function of the majority-spin potential U↑/H20849full curve, real part; dashed curve, imaginary part /H20850.SPIN-MIXING CONDUCTANCES OF THIN MAGNETIC … PHYSICAL REVIEW B 76, 104409 /H208492007 /H20850 104409-11theories of substitutional disorder, such as the coherent po- tential approximation, and it will be used in future systematicinvestigations, e.g., of effects of interdiffusion at the Cu/Niinterface or of antisite atoms in the half-metallic ferromag-netic compounds and diluted ferromagnetic semiconductors. ACKNOWLEDGMENTS The authors acknowledge the ﬁnancial support provided by the Ministry of Education of the Czech Republic /H20849No. MSM0021620834 /H20850, the Academy of Sciences of the Czech Republic /H20849No. A V0Z20410507, No. KAN400100653 /H20850, and the Grant Agency of the Academy of Sciences of the CzechRepublic /H20849No. A100100616 /H20850. APPENDIX: INV ARIANCE PROPERTY OF THE SPIN- MIXING CONDUCTANCE Let us consider the thermodynamic average of a one- particle quantity Q/H20849e.g., a component of the spin torque or of the spin current /H20850in a stationary nonequilibrium state, Q¯=1 2/H9266/H20885 −/H11009/H11009 Tr/H20853QGr/H20849E/H20850/H9018/H11021/H20849E/H20850Ga/H20849E/H20850/H20854dE, /H20849A1/H20850 where Qis an operator nonzero only inside the intermediate region, denoted here by I, the trace refers to the Hilbert space of Iand the self-energy /H9018/H11021/H20849E/H20850is given by Eqs. /H2084910/H20850 and /H2084911/H20850. In contrast to the usual case of scalar Fermi-Dirac functions fL,R/H20849E/H20850, the presence of spin accumulation in the leads requires to include spin-dependent distributions. This means that fL,R/H20849E/H20850in Eq. /H2084911/H20850must be understood as opera- tors acting only on spin indices; for a given lead /H20849L,R/H20850and a given energy E, this operator is uniquely speciﬁed by a Hermitian 2 /H110032 matrix. We assume that fL/H20849E/H20850commutes with the Hamiltonian Hinside the left lead, so that /H20851fL/H20849E/H20850,/H9018Lr,a/H20849E/H20850/H20852=0, and similarly for the right lead; however, the operators fL,R/H20849E/H20850do not in general commute with H inside the intermediate region Iand with the operator Q. Let us prove that the resulting Q¯, Eq. /H20849A1/H20850, does not depend on positions of interfaces L/IandI/R/H20849provided that Qre- mains localized in I/H20850. It is implicitly assumed that matrix elements of Hare short ranged, i.e., His a tight-binding Hamiltonian. The total value Q¯can easily be decomposed in two con- tributions according to Eqs. /H2084910/H20850and /H2084911/H20850,Q¯=Q¯L+Q¯R, where Q¯L,R=1 2/H9266/H20885 −/H11009/H11009 Tr/H20853QGr/H20849E/H20850fL,R/H20849E/H20850/H9003L,R/H20849E/H20850Ga/H20849E/H20850/H20854dE. /H20849A2/H20850 Since the propagators Gr,a/H20849E/H20850refer to the whole inﬁnite sys- tem while the operator Qis localized in the interior of Iand the operators /H9003L,Rare localized in narrow regions at the respective interfaces, it is obvious that the contribution Q¯R does not depend on the position of the L/Iinterface and vice versa. Let us investigate the dependence, e.g., of Q¯L,o nt h e position of the L/Iinterface. Let us move the interface to-ward the left, which results in a modiﬁed lead L˜/H20666Land a region /H9011of a ﬁnite thickness such that /H9011=L\L˜,L=L˜/H33371/H9011. The original intermediate region Iis thus modiﬁed to an extended region I˜=/H9011/H33371I. An explicit expression of Q¯L, Eq. /H20849A2/H20850, in terms of the left self-energies is given by Q¯L=i 2/H9266/H20885 −/H11009/H11009 Tr/H20853QGr/H20849E/H20850fL/H20849E/H20850/H20851/H9018Lr/H20849E/H20850−/H9018La/H20849E/H20850/H20852Ga/H20849E/H20850/H20854dE, /H20849A3/H20850 where the trace refers to the original intermediate region I. Let us consider the original lead Ldecoupled from the rest of the system; its Green’s function projected on the re-gion/H9011will be denoted G Lr,a/H20849E/H20850. It holds /H20851fL/H20849E/H20850,GLr,a/H20849E/H20850/H20852=0. The self-energy of the original left lead can be expressed as /H9018Lr,a/H20849E/H20850=tGLr,a/H20849E/H20850t†, /H20849A4/H20850 where tdenotes that part of the Hamiltonian Hthat describes hoppings from /H9011toIwhile t†describes hoppings from Ito /H9011; these /H20849spin-independent /H20850hoppings satisfy /H20851fL/H20849E/H20850,t/H20852=0. We assume for simplicity that /H9011is thick enough so that no matrix elements of Hcouple L˜toI. Substitution of Eq. /H20849A4/H20850 in Eq. /H20849A3/H20850yields Q¯L=i 2/H9266/H20885 −/H11009/H11009 Tr/H20853QGr/H20849E/H20850tfL/H20849E/H20850/H20851GLr/H20849E/H20850−GLa/H20849E/H20850/H20852t†Ga/H20849E/H20850/H20854dE. /H20849A5/H20850 Let us further denote by /H9018˜Lr,a/H20849E/H20850the self-energy of the modi- ﬁed lead L˜. Since GLr,a/H20849E/H20850refers to the Green’s function of L=L˜/H33371/H9011, i.e., of a ﬁnite region /H9011attached to the semi- inﬁnite lead L˜, the following relation holds: GLr/H20849E/H20850−GLa/H20849E/H20850=GLr/H20849E/H20850/H20851/H9018˜Lr/H20849E/H20850−/H9018˜La/H20849E/H20850/H20852GLa/H20849E/H20850,/H20849A6/H20850 which represents an analogy to Eqs. /H208499/H20850and /H2084919/H20850. The use of Eq. /H20849A6/H20850in Eq. /H20849A5/H20850leads to Q¯L=i 2/H9266/H20885 −/H11009/H11009 Tr/H20853QGr/H20849E/H20850tGLr/H20849E/H20850fL/H20849E/H20850/H20851/H9018˜Lr/H20849E/H20850 −/H9018˜La/H20849E/H20850/H20852GLa/H20849E/H20850t†Ga/H20849E/H20850/H20854dE. /H20849A7/H20850 Finally, let us take into account the Dyson equation for a coupling of the isolated left lead L=L˜/H33371/H9011to the rest of the whole system, I/H33371R, by using the hoppings tandt†as a perturbation. Since the operator Qis localized inside the re- gionIwhile the self-energy /H9018˜Lr,a/H20849E/H20850is localized in /H9011, one can replace the products Gr/H20849E/H20850tGLr/H20849E/H20850andGLa/H20849E/H20850t†Ga/H20849E/H20850in Eq. /H20849A7/H20850by the perturbed Green’s functions G˜r/H20849E/H20850and G˜a/H20849E/H20850, respectively. Here, the G˜r,a/H20849E/H20850denote propagators of the coupled inﬁnite system, projected on the extended inter- mediate region I˜, in contrast to their projections Gr,a/H20849E/H20850on the original region I. This replacement yields a modiﬁed formula for Q¯L,K. CARV A AND I. TUREK PHYSICAL REVIEW B 76, 104409 /H208492007 /H20850 104409-12Q¯L=i 2/H9266/H20885 −/H11009/H11009 Tr/H20853QG˜r/H20849E/H20850fL/H20849E/H20850/H20851/H9018˜Lr/H20849E/H20850−/H9018˜La/H20849E/H20850/H20852G˜a/H20849E/H20850/H20854dE, /H20849A8/H20850 where the trace is taken over the extended region I˜. A com- parison of Eq. /H20849A8/H20850and Eq. /H20849A3/H20850proves insensitivity of thecontribution Q¯Lto the position of the L/Iinterface. This completes a proof of the invariance of the thermodynamic average Q¯, Eq. /H20849A1/H20850, with respect to the L/I/Rpartitioning. 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PhysRevB.96.214410.pdf	PHYSICAL REVIEW B 96, 214410 (2017) Spin-torque diode frequency tuning via soft exchange pinning of both magnetic layers A. A. Khudorozhkov, P. N. Skirdkov, and K. A. Zvezdin* Moscow Institute of Physics and Technology (State University), 141700 Dolgoprudny, Russia; Russian Quantum Center, 121353 Skolkovo, Moscow, Russia; and Prokhorov General Physics Institute, Russian Academy of Sciences, 119991 Moscow, Russia P. M. Vetoshko Moscow Institute of Physics and Technology (State University), 141700 Dolgoprudny, Russia; Russian Quantum Center, 121353 Skolkovo, Moscow, Russia; and Kotel’nikov Institute of Radio-engineering and Electronics (IRE), Russian Academy of Sciences, 125009 Moscow, Russia A. F. Popkov Moscow Institute of Physics and Technology (State University), 141700 Dolgoprudny, Russia and National Research University of Electronic Technology (MIET), 124498 Zelenograd, Moscow, Russia (Received 31 August 2017; revised manuscript received 30 October 2017; published 8 December 2017) A spin-torque diode, which is a magnetic tunnel junction with magnetic layers softly pinned at some tilt to each other, is proposed. The resonance operating frequency of such a dual exchange-pinned spin-torque diodecan be signiﬁcantly higher (up to 9.5 GHz) than that of a traditional free layer spin-torque diode, and, at the sametime, the sensitivity remains rather high. Using micromagnetic modeling we show that the maximum microwavesensitivity of the considered diode is reached at the bias current densities slightly below the self-sustainedoscillations initiating. The dependence of the resonance frequency and the sensitivity on the angle betweenpinning exchange ﬁelds is presented. Thus, a way of designing spin-torque diode with a given resonance responsefrequency in the microwave region in the absence of an external magnetic ﬁeld is proposed. DOI: 10.1103/PhysRevB.96.214410 I. INTRODUCTION Currently there is much interest in spintronic devices which deal not only with electrons’ charge but also with their spins.These devices are expected to surpass conventional electronicdevices showing better characteristics and providing new func-tional capabilities [ 1–3]. Among these devices are new types of memory [ 4–6], nanogenerators [ 7–10], microwave detectors [11–16], magnetic ﬁeld detectors [ 17], etc. In particular, one of the most promising spintronic devices is a spin-torquediode based on the magnetic tunnel junction (MTJ) [ 18,19] and spin-transfer-torque effect [ 20–25]. It was shown that the application of a radiofrequency (RF) alternating current(ac) can cause the generation of a direct-current (dc) voltageacross the structure interface leading to the spin-diode effect[11–14]. Thus, a spin-torque diode can be used for detection of radiofrequency signals [ 12–15] and even for harvesting their energy [ 14]. As a consequence, a spin-torque diode should have a high sensitivity, which is the ratio of the incident powerto the output dc voltage. Despite the fact that the sensitivityof a spin-torque diode reached only 1 .4m V/mW in the ﬁrst report [ 11], the continuous efforts on optimization of the MTJ allowed us to demonstrate the spin-torque diode sensitivityof 12 000 mV /mW [ 26] under tilted magnetic ﬁeld applied and even 75 400 mV /mW [ 27] under zero-bias magnetic ﬁeld. These results far exceed the sensitivity of the state-of-the-artsemiconductor Schottky diode detectors (3800 mV /mW) [ 26]. However, such high resonant sensitivity has not yet beenachieved at resonant frequencies higher than 2–4 GHz. For *konstantin.zvezdin@gmail.comsome applications it is desirable to increase the frequency ofthe microwave signal [ 28,29]. In our work we propose a spin-torque diode with magnetic layers softly pinned at some tilt to each other. The possi-bility of controlling the operating frequency and resonantcharacteristics of such a spin-torque diode by changing thetilt angle between magnetizations of ferromagnetic layersusing the soft exchange pinning of both magnetic layerswith additional antiferromagnetic layers (AFM) which havedifferent temperatures of superparamagnetic blocking T Bis considered. This can be done by using AFM layers withdifferent Neel temperature T N[30], for example, IrMn 3(TN= 690 K, TB=540 K) and FeMn ( TN=510 K, TB=450 K) or the same material but with different thickness [ 31,32]. In this case it is possible to conduct two-step annealing with differenttemperatures and different ﬁeld directions to manufacturedevice with predetermined tilt angle between AFM pinning. The strong impact of dc bias on the sensitivity of studied system is observed and the dependence of the sensitivity on theAFM pinning angle is obtained. The relation between the AFMpinning angle and angle between mean magnetizations in FMlayers is found, and the key role of magnetostatic interactionis proved. II. CONSIDERED SYSTEM AND METHOD The studied structure is a ﬁve-layer nanopillar with a diameter of 140 nm composed of an MTJ with two ferro-magnetic layers, FM 1and FM 2, separated by a tunneling barrier MgO and located between two antiferromagnets, AFM 1 and AFM 2(Fig. 1). Both FM layers are softly pinned by antiferromagnetic layers (AFM 1and AFM 2) with exchange 2469-9950/2017/96(21)/214410(5) 214410-1 ©2017 American Physical SocietyA. A. KHUDOROZHKOV et al. PHYSICAL REVIEW B 96, 214410 (2017) FIG. 1. (a) Top view and (b) side view of the studied structure. Studied nanopillar structure contains an MTJ with two FM layers separated by a MgO tunnel barrier. Both FM layers are softly pinned with AFM layers with exchange bias ﬁelds tilted to each other at anangleϕ. bias ﬁelds tilted to each other at an angle ϕ. The thicknesses of FM 1and FM 2layers have been chosen to be 2 and 6 nm, respectively, and MgO layer is 1 nm thick. In thiscase, the effective antiferromagnetic interlayer exchange ﬁeldsfor the FM 1and FM 2layers can be estimated as 500 and 167 Oe, respectively. It is worth noting that these pinningexchange ﬁelds do not prevent magnetization oscillations withamplitudes sufﬁcient to create overall considerable spin-diodemicrowave sensitivity. Moreover, the magnetization directiondoes not match the direction of pinning (this fact is analyzed indetail below). The electric current is applied along the verticalaxis perpendicular to the plane of layers (current perpendicularto the plane (CPP) geometry). The magnetization dynamics in the both FM layer is described by the Landau-Lifshitz-Gilbert (LLG) equation withan additional term responsible for the spin transfer [ 20,21]: ˙M i=−γMi×Hi eff+Ti STT+α MS(Mi×˙Mi), (1) where Miis the magnetization vector of the FM i,γis the gyromagnetic ratio, αis the Gilbert damping constant, MSis the saturation magnetization, and Hi effis the corresponding to each layer effective ﬁeld consisting of a magnetostaticﬁeld, an exchange ﬁeld, an anisotropy ﬁeld, and an effec-tive antiferromagnetic interlayer exchange ﬁeld. The spintransfer torque T i STTis represented by two components: a Slonczewski torque (ST) Ti ST=−γf(θ)jaj MSMi×(Mi× mj) and a ﬁeldlike torque (FLT) Ti FLT=−γjbj(Mi×mj), where mjis a normalized vector along the local magne- tization direction of opposite layer and jis the current density along the zdirection. The angular dependence of the Slonczewski torque is represented [ 20,33,34]b yf(θ)= 2/Lambda12/[(/Lambda12+1)+(/Lambda12−1) cos θ], where θis the angle be- tween local magnetizations of the two layers and the /Lambda12 parameter characterizes the spacer layer. The ST amplitude is given by aj=¯hP/2heM S, where Pis the spin polarization of the current, his the thickness of the free layer, and e>0i s the charge of the electron. The amplitude of the FLT is givenbybj=ξCPPaj, where ξCPPcan be larger than 0.4 in the case of an asymmetric magnetic tunnel junction [ 35]. It is worth noting that all the details of the torque calculation in modelingpresented below are similar with widespread micromagneticsolvers (e.g., MuMax [ 34]). To investigate the effect of microwave signal rectiﬁcation we have performed a series of numerical integration of theLLG equation ( 1) using our micromagnetic ﬁnite-difference code SpinPM based on the fourth-order Runge-Kutta methodwith an adaptive timestep control for the time integration andam e s hs i z e2 .5×2.5×hnm 3, where his the corresponding layer’s thickness. The FM magnetic parameters used in themodeling are as follows: the saturation magnetization M S= 920 emu /cm3, the exchange constant A=1.3×10−6erg/cm, the Gilbert damping factor α=0.01, and the bulk anisotropy is neglected. These parameters are typical for permalloy [ 36]. The spin polarization of the current is chosen to be P=0.4 [35,36] and the parameter /Lambda12≈2.33, which corresponds to MTJ with TMR 160% [ 37]. It is worth noting that the reliability of the used method was proved by the fact that several resultspredicted by our solver for close systems were observedexperimentally later [ 35,36,38]. For the correct processing of simulation results, let us assume that the spin diode is connected to a transmissionline and the microwave signal is incident onto it. Thenthe current density ﬂowing through the MTJ would be j= j 0+j1cos (2πf t), where j0is the bias current density (dc) andj1is the amplitude of the microwave current density (ac) with the frequency f. Due to the telegraph equations, taking into account the impedance matching and assuming that thetransmission line is short, the connection between the incidentmicrowave power P inand the power consumed by the spin diodeP0is the following: Pin=P0(R+Z0)2 4RZ 0, (2) where Ris the time-average spin-diode resistance and Z0is the transmission line impedance. The power consumed by the spindiode P 0=/angbracketleftI2(t)R(t)/angbracketright, where I=jS,Sis the diode cross- section area, and the brackets /angbracketleft.../angbracketrightdenote time averaging. The time-average voltage across the structure could be estimatedas/angbracketleft/Delta1V/angbracketright=/angbracketleftI(t)R(t)/angbracketright. In case of uniformly magnetized FM layers, the dependence of the MTJ resistance Ron the angleθbetween FM layers’ magnetizations could be derived asR(θ)=¯R 1+ρcosθ, where ¯R−1=R−1 ↑↑+R−1 ↑↓ 2,ρ=R↑↓−R↑↑ R↑↓+R↑↑, and R↑↑andR↑↓are the resistance of the MTJ in parallel ( θ=0o) and antiparallel ( θ=180o) states correspondingly [ 13]. If the magnetization distribution in FM layers is not uniform, thenthe spatial averaging should be performed. Following previouswork [ 37], we can estimate the spin-diode sensitivity: ε=/angbracketleft/Delta1V/angbracketright Pin=A/angbracketleftbigcos(ωt) 1+ρcos(θ(t))/angbracketrightbig /angbracketleftbigcos2(ωt) 1+ρcos(θ(t))/angbracketrightbig, (3) where A=1 j1S4RZ 0 (R+Z0)2andθ(t) is determined in accordance with an LLG equation solution. Here the brackets /angbracketleft.../angbracketrightdenote the time and space averaging. We consider the following parameters of the MTJ in our simulations: cross-section area S=πr2=1.54×104nm2, 214410-2SPIN-TORQUE DIODE FREQUENCY TUNING VIA SOFT . . . PHYSICAL REVIEW B 96, 214410 (2017) FIG. 2. The dependence of the critical current and corresponding voltage on the angle θbetween the mean magnetizations of the FM layers. Inset: The dependence of the angle θbetween the mean magnetizations of the FM layers on the AFM pinning angle ϕ. transmission line impedance Z0=50/Omega1, the average diode resistance R=415/Omega1, and magnetoresistance/Delta1R R↑↑=R↑↓−R↑↑ R↑↑= 160%. Using it one can ﬁnd ρ=0.44 and /Delta1R=369/Omega1. III. MODELLING RESULTS AND DISCUSSION First, let us analyze equilibrium states of both FM layers. Micromagnetic modeling demonstrates that both layers havequasiuniform magnetization distribution; however, the magne-tizations are not collinear to the direction of the AFM pinningat equilibrium, since, besides the effective antiferromagneticinterlayer exchange ﬁeld, which tends to set magnetizationalong the AFM pinning direction, there is the magnetostaticeffective ﬁeld, which favors an antiparallel magnetizationconﬁguration. The relation between the AFM pinning angleϕand resulting angle θbetween mean magnetizations in FM 1 and FM 2is presented in Fig. 2(inset). While the angle between the mean magnetization θonly decreases approximately to 110◦, the AFM pinning angle ϕdecreases down to 20◦.T h i s means that in the considered case, the magnetostatic effect hasa signiﬁcant impact on the system. At the same time, the θ(ϕ) dependence is close to linear for ϕhigher than 60 ◦. Below this point the magnetization distribution is becoming less uniformand nonlinear effects play a signiﬁcant role. Further, we investigate the excitation by dc only (in this casej 1=0). Modelling demonstrates that in a wide range of the AFM pinning angle ϕ(and corresponding angle θ) there is a critical current jcr, at which the system switches into the autooscillation mode with both FM layers oscillating. Thedependence of this critical current and corresponding voltageon the the angle θbetween the mean magnetizations of the FM layers is presented in Fig. 2. Below we investigate all the dependencies on the angle ϕ, since it can be chosen during the annealing in fabrication of the structure. At the sametime the real angle between magnetizations θcan be easily identiﬁed using θ(ϕ) dependence from the inset in Fig. 2. To analyze which of the torques is mostly responsible for magnetization excitation we considered their action onthe system separately. For ϕ=80 ◦,j0=9.5×106A/cm2 (which is higher than the critical current), and j1=0, we have FIG. 3. Spin diode sensitivity depending on microwave current frequency for different values of dc bias currents. AFM pinning angle ϕ=120◦. performed two calculations, considering ﬁeldlike torque only (aj=0,bj/negationslash=0) and Slonczewski torque only ( aj/negationslash=0,bj= 0). In the ﬁrst case the transient oscillations subsided to zero ina short time, while in the second case the autooscillations wereobserved. This demonstrates that the Slonczewski torque playsa leading role in magnetization excitation in the consideredstructure. Now let us consider the action of ac with and without bias dc on the example of the case AFM pinning angle ϕ=120 ◦. Here and below, theac current density amplitude was chosenasj 1=104A/cm2, which corresponds to the power of the incident microwave signal Pin=10−9W. The sensitivity of the spin diode was calculated using Eq. ( 3) for varied frequencies in the range 9–9.5 GHz for different dc bias currents (belowthe critical one, which is j cr=6.52×106A/cm2in this case). The simulation results are presented in Fig. 3. It can be seen that the maximum resonant sensitivity is observed when the dcbias current is slightly below the critical value ( j 0=0.99jcr) and the damping is nearly compensated by the spin torque.These results prove that for considered diode dc current evenless than critical one can signiﬁcantly improve the sensitivity(from 20 to 1200 mV /mW in the considered case). Next we investigate the sensitivity of the MTJ at zero bias current for different AFM pinning angle ϕ(from 70 ◦ to 160◦). The results are represented in the inset in Fig. 4. The resonant sensitivity rises when the angle ϕdecreases, achieving maximum at 60◦. At angles about 160◦the resonance peak vanishes. At the same time, the maximum value ofsensitivity in this case does not exceed 40 mV /mW. As a ﬁnal step of the simulation, for each AFM pinning angleϕthe dc bias current which is 99% from the critical value has been applied and the sensitivity dependence on microwavesignal frequency has been calculated using the formula ( 3). The results are shown in Fig. 4. The resonance frequency gradually changes as the angle ϕbetween exchange bias ﬁelds varies. The maximum sensitivity of 1670 mV /mW is reached at the angleϕof about 125 ◦. As one can see from these results the sensitivity remains rather high down to the angle ϕ=70◦.A t the same time it is well known [ 39,40] that the Slonczewski spin-transfer torque (which determines the effect in consideredcase) becomes ineffective at exciting steady-state magneticoscillations when the magnetic misalignment is close to 90 ◦, 214410-3A. A. KHUDOROZHKOV et al. PHYSICAL REVIEW B 96, 214410 (2017) FIG. 4. The dependence of the sensitivity on ac frequency for different AFM pinning angle ϕin case of 0 .99jcrbias current. Inset: The dependence of the sensitivity on ac frequency for different AFMpinning angle ϕin case of only ac excitation. and therefore the critical current density should diverge as 1/cosθto large magnitudes near this angle. However, there is no contradiction with our results. The considered range ofAFM pinning angle ϕfrom 70 ◦to 160◦corresponds to the angleθbetween the average magnetizations of the FM layers from 141◦to 171◦(see inset in Fig. 2). Thereby the value of θ=90◦is simply not achieved, and the angle θremains always greater than this value. This also explains the fact that we donot observe the inversion of the critical current sign in Fig. 2. These results proves the possibility of production consideredspin diodes with predetermined in wide range (from 8.5 to9.5 GHz) resonant frequency by changing angle ϕduring the annealing without the signiﬁcant loss of their performance. It is important to note that the experimental implementation of the reliably operating spin diode with tilted soft exchange bi-asing would require signiﬁcant efforts toward stack design andoptimization. For example, exchange biasing to the antiferro-magnet can increase the effective damping for a magnetic layer[41,42]. This leads to an increase of the critical current and the corresponding voltage. However, this issue could be addressedin the following ways. Even a slight increase of the thicknessof FM layers would result in strong decrease of damping(proportional [ 41]t ot h e h −2) induced by antiferromagnetic biasing without inﬂuencing signiﬁcantly the magnetizationdynamics but only moving the position of the resonance. Forinstance, if we increase the thickness of the FM 1from 2 nm only to 2.9 nm and 4.3 nm, then the effective damping al-ready decreases down to α(2.9n m ) ≈0.049 and α(4.3n m ) ≈ 0.027, respectively. The critical currents in this case will bej cr(2.9n m ) ≈2.7×107A/cm2andjcr(4.3n m ) ≈1.56× 107A/cm2and corresponding voltage V(2.9n m ) ≈1.7 V and V(4.3n m ) ≈1 V (see Fig. 5), which are still reasonable [ 43]. The effective damping was recalculated to the thickness usingexperimental results from work [ 41]. On the other hand, in order to avoid the effect of the growth of magnetic damping due to the exchange pinning ofthe magnetic layer on the rough surface of the antiferromag-net, one can use synthetic antiferromagnet AFM 1/FM 0/Ru (<1n m )/FM 1. In this case, there is a sufﬁciently strong FIG. 5. The dependence of the critical current and corresponding voltage on the effective damping α. Red points represent modeling results, the blue point corresponds to FM 2with thickness h=6 nm, and green points represent the cases of thickness corresponding to voltages 1 V and 1.7 V , respectively. exchange bias acting on the FM 1layer. Together with that, the linewidth of this magnetic layer, and therefore the effectivedamping, will be small [ 44,45]. Moreover, since the rigid ﬁxation of the FM layers magnetization is not required forthe operation of considered system, it is possible to use morecomplex designs of antiferromagnetically coupled soft bias[46] instead of the AFM layer. In this case the effective damping in both FM layers almost does not differ from thedamping value for a magnetic layer in isolation. IV . SUMMARY AND CONCLUSIONS In summary, the MTJ spin-torque diode with both FM layers softly pinned at different tilt angles by AFMs having differentNeel temperatures is proposed. The spin-torque diode effectin such a system and the impact of the dc bias is consideredby means of micromagnetic modeling. We demonstrate thatthe resonance operating frequency of the spin diode withbilateral tilted soft exchange pinning can be signiﬁcantlyhigher than that of a traditional spin-torque diode with onepinned layer. We also demonstrate that such a system hassensitivity comparable to the semiconductors in the wide range8.5–9.5 GHz. Moreover, it is possible to tune the resonantfrequency of the diode in this wide frequency range duringthe manufacturing of the device (by ﬁtting ϕduring the annealing) without signiﬁcant loss of the sensitivity. On theother hand, using of voltage-controlled anisotropy instead ofbias exchange pinning allows us to change the angle betweenthe equilibrium of magnetizations of the magnetic layersdynamically and therefore to tune the frequency during theoperation. The key role of the magnetostatic interaction forthe considered system is demonstrated. It is worth mentioningthat all results were obtained in the absence of an externalmagnetic ﬁeld. The proposed approach can be useful in theengineering of spin diodes for practical applications. ACKNOWLEDGMENT The authors acknowledge A. K. Zvezdin for the fruitful discussion. 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PhysRevB.96.014407.pdf	PHYSICAL REVIEW B 96, 014407 (2017) Dynamics and inertia of a skyrmion in chiral magnets and interfaces: A linear response approach based on magnon excitations Shi-Zeng Lin Theoretical Division, T-4 and CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Received 8 March 2017; published 6 July 2017) Taking all the magnon modes into account, we derive the skyrmion dynamics in response to a weak external drive. A skyrmion has rotational symmetry, and the magnon modes can be characterized by an angular momentum.For a weak distortion of a skyrmion, only the magnon modes with an angular momentum \|m\|=1 govern the dynamics of skyrmion topological center. The skyrmion inertia is determined by the magnon modes in thecontinuum spectrum. For a skyrmion driven by a magnetic ﬁeld gradient or by a spin transfer torque generatedby a current, the dynamical response is practically instantaneous. This justiﬁes the rigid skyrmion approximationused in Thiele’s collective coordinate approach. For a skyrmion driven by a spin Hall torque, the torque couplesto the skyrmion motion through the magnons in the continuum and damping; therefore the skyrmion dynamicsshows sizable inertia in this case. The trajectory of a skyrmion is an ellipse for an ac drive of spin Hall torque. DOI: 10.1103/PhysRevB.96.014407 I. INTRODUCTION A magnetic skyrmion in magnets is swirling spin texture that behaves as a particle with a long lifetime [ 1–3] due to the topological protection. Skyrmions were observed experimen-tally recently in chiral magnets where the inversion symmetryis broken [ 4,5]. For their unique topological properties and long lifetime, skyrmions have attracted considerable interestas possible information carriers. Skyrmions can be drivenby various external ﬁelds, such as an electric current [ 6–8], a magnetic/electric ﬁeld gradient [ 9,10], a thermal gradient [11–13], a magnon current [ 14], etc. The ability to manipulate skyrmions with an electric current is especially attractivebecause this implies immediately that skyrmions can be usedin spintronic devices [ 15,16]. Moreover, the threshold current to make skyrmions mobile is weak, thanks to the smooth spintexture and the Magnus force associated with skyrmions. For applications, it is crucial to understand the dynamics of skyrmions in response to an external drive. The equationof motion of a rigid skyrmion in two dimensions (in thex-yplane) was obtained by Thiele a long time ago [ 17]. It has the following form: G T×v+DTv=FT, (1) where vis the skyrmion velocity. The ﬁrst term describes gyromotion with the gyrovector GTperpendicular to the x-yplane, and the second term is the damping. Here DT/lessmuch GT, and the skyrmion moves almost perpendicular to the external force FT. In Thiele’s collective coordinate approach, the skyrmion texture S(r)i sa s s u m e dt ob er i g i d ,a n di tm o v e sa s a whole, S[r−R(t)], with R(t) representing the translational motion of a skyrmion. It has been demonstrated in numericalsimulations that Thiele’s equation of motion correctly capturesthe skyrmion dynamics driven by a spin transfer torque inducedby a dc current [ 18–20]. Nevertheless, how to justify the rigid skyrmion approximation is still unknown. A skyrmion is a collective excitation of spin texture and thus has internal degrees of freedom [ 14,21]. In the presence of an external drive, the skyrmion can be deformedwhen the magnon modes are excited by the external drive.In the continuum approximation where the system preservesthe translational symmetry, there is a Goldstone mode in thecharacteristic deformations of the skyrmion corresponding to the translational motion of a skyrmion [ 22–24]. Thiele’s equation of motion includes only the Goldstone mode. Inprinciple, the deformations associated with other magnonmodes can also be involved in the motion of a skyrmion. To gobeyond Thiele’s approach, one needs to take all the magnonexcitations into account. One can introduce corrections toThiele’s equation of motion, such as mass and gyrodamping,and then ﬁt the generalized Thiele’s equation of motion to theskyrmion trajectory obtained from direct simulation of thespin dynamics based on the Landau-Lifshitz-Gilbert equation.In this way the parameters in Thiele’s equation of motion canbe extracted [ 25–27]. Similar to Thiele’s original equation of motion, the generalized Thiele’s equation is not justiﬁed. Atheory to describe the skyrmion dynamics by treating all themagnon modes on an equal footing thus is required. In this work, we present a linear theory for the skyrmion dynamics by taking all the magnon modes into account.Speciﬁcally, we consider a small oscillation of a skyrmionsubjected to a weak oscillating magnetic ﬁeld gradient, spintransfer torque, and spin Hall torque. To do that, we ﬁrstdeﬁne the skyrmion center as its topological charge center.We then express the dynamics of the skyrmion center in termsof the magnon modes. We ﬁnd that only the magnon modeswith an angular momentum \|m\|=1 are responsible for the dynamics of the skyrmion center. The retardation or inertia ofskyrmions is due to the magnon modes in the continuum. For askyrmion driven by a ﬁeld gradient or spin transfer torque, theinertia is negligible, which justiﬁes the Thiele’s rigid skyrmionapproximation. For a spin Hall torque, the motion of skyrmionsoriginates from the coupling between the magnon continuumand the current. The inertia is signiﬁcant in this case. Theinertia can be quantiﬁed by measuring the phase shift betweenthe skyrmion velocity and the driving ﬁeld. II. MODEL AND BLOCH SKYRMION SOLUTION We consider the following Hamiltonian density for spins S(r) in two-dimensional space [ 3]: H=Jex 2/summationdisplay μ=x,y(∂μS)2+DS·∇×S−B·S, (2) 2469-9950/2017/96(1)/014407(10) 014407-1 ©2017 American Physical SocietySHI-ZENG LIN PHYSICAL REVIEW B 96, 014407 (2017) 02468 1 0-10123 r(r) r(r)B=0.8 FIG. 1. Proﬁles of θ(r)a n d ∂rθ(r)f o ras k y r m i o ni nt h e ferromagnetic background. The inset shows a schematic view of a skyrmion. Here B=0.8. which successfully captures many experimental observations in chiral magnets. Here Sis a unit vector representing the spin direction, Jexis the exchange interaction, Dis the Dzyaloshinskii-Moriya (DM) interaction [ 28–30], and B= Bˆz, with the unit vector ˆzbeing the external magnetic ﬁeld perpendicular to the plane. We have neglected the weakdipolar interaction. The skyrmion size is much bigger thanthe spin-lattice constant, and this justiﬁes the continuumapproximation in Eq. ( 2). We renormalize the length in units ofJ ex/Dand energy density and Bin units of D2/Jex. Then Eq. ( 2) takes a dimensionless form. We will use dimensionless quantities in the following derivations. The Hamiltonian ( 2) supports the Bloch skyrmion solution. The results for the Néelskyrmions will be discussed in Sec. VI. We focus on a single skyrmion in the ferromagnetic back- ground, which is a metastable state for B> 0.55 [14,21,31]. Because of the rotational symmetry, it is more convenient towork in the polar coordinate r=(r, φ). The skyrmion solution isS 0=(cosϕsinθ,sinϕsinθ,cosθ), with ϕ=φ+π/2 andθ(r) being a function of only r. The stationary skyrmion solution is obtained by minimizing Hwith respect to θ, and we have equation for θ(r): cos(2θ)+1 2rsin(2θ)+Brsinθ−(∂rθ+1)−r∂2 rθ=0. (3) We solve Eq. ( 3) using the relaxation method to ﬁnd θ(r). The skyrmion structure and the results for θ(r) and∂rθ(r)a t B=0.8 are displayed in Fig. 1. III. EIGENMODE ANALYSIS We calculate the eigenmodes of a skyrmion in the ferro- magnetic background, following the approach in Ref. [ 14]. We introduce a local coordinate system with the local zaxis along the spin direction S. The spin representation in the laboratory coordinate and the local coordinate are related bythe following subsequent rotation operations in the laboratoryframe: rotation along the zaxis by φ 0, rotation along the yaxisbyθ, and rotation along the zaxis by ϕ. We choose φ0=π/2. Then the spin in the local coordinate L=(LX,LY,LZ)i s related to that in the laboratory frame Saccording to S=ˆOL, where [ 21] ˆO=⎛ ⎝−sinϕ−cosϕcosθcosϕsinθ cosϕ−sinϕcosθ sinϕsinθ 0s i n θ cosθ⎞ ⎠. (4) In the stationary state ¯LX=¯LY=0 and ¯LZ=1. We consider small deviations Lfrom the stationary state and introduce the magnon ﬁelds ψ=LX+iLY√ 2,ψ∗=LX−iLY√ 2, (5) andLZ=1−ψψ∗, with \|ψ\|/lessmuch1. Expanding the Hamilto- nian to second order in ψ, we obtain Hψ=1 2ˆψ†Hψˆψ, (6) Hψ=(−∇2+V0)I0−2σz/parenleftbiggcosθ r2−sinθ r/parenrightbigg i∂φ+V1σx, (7) withσi(i=x, y, z ) being the Pauli matrices and I0being the unit matrix. Here ˆψ†=(ψ∗,ψ), (8) V0=1+3 cos(2 θ) 4r2−3s i n ( 2 θ) 2r+Bzcosθ−∂rθ−1 2(∂rθ)2, (9) V1=sin2θ 2r2+sin(2θ) 2r−∂rθ−1 2(∂rθ)2. (10) Equations ( 6)–(10) were ﬁrst derived in Ref. [ 14]. Note that with the deﬁnition of the magnon wave function in Eq. ( 5), there is a minus sign in front of σzin Eq. ( 7), which is different from the results in Ref. [ 14]. The eigenmodes are determined by the equation −iσz∂tˆψ=Hψˆψ. (11) This equation for magnons has the form of the Schrödinger equation in the presence of a centrosymmetric potential.We can introduce an angular momentum mwithψ= ψ m(r,t)e x p (imφ) to classify the eigenmodes. The two com- ponents in ˆψare related by complex conjugation because the magnetic moment Sis real. This indicates that the matrix equation in Eq. ( 11) is redundant. Indeed, Hψ has particle-hole symmetry, Hψ=σxKHψKσx, with K being the complex-conjugate operator. This means that if exp[i(ωt+mφ)] ˆηm, with ˆη† m≡(η∗ 1,η∗ 2), solves Eq. ( 11), then exp[−i(ωt+mφ)]σxKˆηmalso solves Eq. ( 11). Then ˆψcan be obtained by linear superposition of the two symmetry-relatedsolutions, ˆψ m=bexp[i(ωt+mφ)] ˆηm+b∗exp[−i(ωt+mφ)]σxKˆηm. (12) Forˆψmin Eq. ( 12), its two components are complex conjugate to each other. Here ˆ ηis determined by the eigenvalue 014407-2DYNAMICS AND INERTIA OF A SKYRMION IN CHIRAL . . . PHYSICAL REVIEW B 96, 014407 (2017) problem ωmσzˆηm=Hψˆηm. (13) To solve Eq. ( 13), we use the Bessel functions Jm(kr)a sab a s i s to represent the matrix Hψas detailed in Appendix A.F r o m ˆηm, we know Lmfrom Eq. ( 5). Then we obtain the eigenmodes ˜Smin the laboratory frame through rotation, ˜Sm=ˆOLm. In the high-energy limit ω/greatermuchωg, with ωg=Bbeing the magnon gap, we can use magnon momentum kto label the magnon mode. The eigenfrequency in this limit is ωm(k)= k2+ωg, and the eigenmodes are ˆ η† m=(1,0)Jm(kr). When ω is comparable to ωg, the momentum kis not a good quantum number because the presence of the skyrmion breaks thetranslational invariance for magnons. In the continuum limit adopted in Eq. ( 2), which is valid when the skyrmion size is much bigger than the spin-latticeconstant, the mode corresponding to the translational motionof a skyrmion is a Goldstone mode. It has \|m\|=1, and the magnon mode in the laboratory frame is ˜S \|m\|=1,j=1=im∂xS0−∂yS0 √ 2/radicalBig/integraltext dr2(∂xS0)2. (14) The imaginary and real parts of ˜S\|m\|=1correspond to the translation motion along the xandydirections, respectively. IV . LINEAR RESPONSE TO EXTERNAL DRIVE We proceed to calculate the response of skyrmions to external drive in terms of the eigenmodes. We consider theLandau-Lifshitz-Gilbert equation of motion for S, ∂ tS=−S×Heff+αS×∂tS+/Gamma1, (15) where αis the Gilbert damping, Heff≡−δH/δSis the effective magnetic ﬁeld, and /Gamma1is the torque due to external ﬁelds. Here the time is in units of Jex/(γD2), with γbeing the gyromagnetic ratio. We study the linear response of a skyrmionto a weak torque, \|/Gamma1\|/lessmuch1. We will work in the laboratory frame here. We consider a small oscillation of the skyrmioncenter around the equilibrium position. The linear responseS=S 0+˜Sis governed by ∂t˜S(t)=ˆHS˜S+αS0×∂t˜S+/Gamma1. (16) Here ˆHSand˜Sare connected to Hψand ˆψthrough rotation, Eq. ( 4). Expanding ˜Sin terms of the eigenmodes, we have ˜S(t)=Re⎡ ⎣exp(iωt)/summationdisplay m,jam,j˜Sm,j⎤ ⎦. (17) The index jlabels the modes at a given m. Substituting Eq. ( 17) into Eq. ( 16) and projecting into the ˜Sm,jmode, we obtain am,j(ω−ωm,j)+iω/summationdisplay j/primeαm,j,j/primeam,j/prime=Fm,j, (18) where Fm,jrepresents the coupling between the external torque and the eigenmodes, Fm,j(ω)=−i/integraldisplay dr2˜S† m,j·/Gamma1, (19)andαm,j,j/primeis the damping coefﬁcient for different modes, αm,j,j/prime=iα/integraldisplay dr2˜S† m,j·(S0×˜Sm,j/prime) =i2πα/integraldisplay drr[L∗ Y;m,j(r)LX;m,j/prime(r) −L∗ X;m,j(r)LY;m,j/prime(r)]. (20) As shown in Appendix B, the diagonal elements \|αm,j,j\| are much larger than the off-diagonal elements \|αm,j,j/prime\|.F o r α/lessmuch1, we can neglect the off-diagonal elements of αm,j,j/primeand takeαm,j,j/prime=αm,jδj,j/prime, with δj,j/primebeing the Kronecker delta function. For ωm,j/greatermuchωg,w eh a v e αm,j=1. We then obtain amin the frequency domain, am,j(ω)=Fm,j ω−ωm,j+iαm,jω. (21) To obtain the equation of motion for a skyrmion as a particle, we need to deﬁne its center. We use the deﬁnitionbased on its topological charge density R(t)=/integraltext dr 2rS·(∂xS×∂yS)/integraltext dr2S·(∂xS×∂yS)=/integraltext dr2rS·(∂xS×∂yS) 4πNs, (22) where Ns=−1 is the skyrmion topological charge for the skyrmion shown in Fig. 1, which is invariant with respect to small perturbations. Taking the time derivative of R(t) and expanding in terms of the eigenmodes, we obtain R(ω)=/summationdisplay m,jam,j(ω)Wm,j(ω), (23) with Wm,j=1 4πNs/integraldisplay dr2r[˜Sm,k·(∂xS0×∂yS0) +S0·(∂x˜Sm,k×∂yS0)+S0·(∂xS0×∂y˜Sm,k)] =ζm,j(ˆx+imˆy)δ\|m\|,1 (24) and ζm,j=1 4Ns/integraldisplay drr[imL X;m,j∂rθ−∂r(LY;m,jsinθ)]δ\|m\|,1. (25) One important observation is that only the modes with \|m\|=1 couple to the skyrmion center motion. The lowest modewith\|m\|=1 is the Goldstone mode corresponding to the translational motion of a rigid skyrmion. Other modes with\|m\|=1 lie in the magnon continuum. For a rigid skyrmion, the response to external drive is instantaneous; that is, the inertiaof a skyrmion is absent. Therefore the inertia of a skyrmion iscontributed by the excitation of the magnon continuum. Notethat the linear analysis in Eq. ( 16)i sv a l i dw h e n R(ω) is smaller than the skyrmion size. An alternative deﬁnition of the skyrmion center that is more relevant to experiments is based on the out-of-plane componentof the spin, R /prime=/integraltext dr2r(Sz−1)/integraltext dr2(Sz−1). (26) 014407-3SHI-ZENG LIN PHYSICAL REVIEW B 96, 014407 (2017) The equation of motion for R/primeto the linear order in perturbation has the same expressions as those in Eqs. ( 23) and ( 24), except forζm,j, ζm,j=π/integraltext dr2(S0,z−1)/integraldisplay drr2LY;m,jsinθδ\|m\|,1. (27) In the present work, we use the skyrmion center deﬁned in Eq. ( 22). The external torque /Gamma1is proportional to some control parameters P, such as the electric current density or magnetic ﬁeld gradient. The response of velocity v=iωR(ω)t oPcan be expressed in a matrix form: /parenleftbigg vx vy/parenrightbigg =/parenleftbigg χ11χ12 χ21χ22/parenrightbigg/parenleftbigg Px Py/parenrightbigg . (28) The response is isotropic under the spatial rotation of Pand v, which requires χ22=χ11andχ12=−χ21.H e r e χijcan be complex, and the skyrmion trajectory is generally an ellipse.The skyrmion inertia manifests itself in the phase shift betweenthe drive P(ω) and velocity v(ω). We can deﬁne a longitudinal phase shift /Theta1 /bardbland transverse phase shift /Theta1⊥, tan/Theta1/bardbl≡Im[χ11] Re[χ11],tan/Theta1⊥≡Im[χ21] Re[χ21]. (29) To cast Eq. ( 28) into the standard Thiele form, we can introduce a generalized force ffrom Pand rewrite Eq. ( 28)a s /parenleftbigg D+iωm G−iωA −G+iωAD +iωm/parenrightbigg/parenleftbigg vx vy/parenrightbigg =/parenleftbigg fx fy/parenrightbigg . (30) HereD∝αis the damping, mis the mass, Gis the gyrocou- pling, and A∝αis the gyrodamping. All those quantities are frequency dependent. The quantities on the left-hand side ofEq. ( 30) depend on the deﬁnition of the generalized force f on the right-hand side. In our discussion, we will focus onEq. ( 28) since it captures fully the dynamics of a skyrmion. In the limit of ω=0, only the Goldstone modes with \|m\|=1 and j=1 contribute to the summation in Eq. ( 23). In this limit, the rigid skyrmion approximation employed inThiele’s collective coordinate approach becomes exact. Fromthe Goldstone modes in Eq. ( 14), we obtain α \|m\|=1,j=1= −mα/κ andW\|m=1\|,j=1=−(miˆx−ˆy)/√ 8πκ, where the skyrmion form factor κis κ=/integraldisplay dr2(∂μS0)2/(4π), (31) withμ=x, y.H e r e κis of the order of unity, κ∼1. The equation of motion becomes v(ω)=/summationdisplay m=±1,j=1iFm,j(−miˆx+ˆy)√ 8πκ(1−imα/κ ). (32) V . APPLICATIONS In the following, we calculate the response matrix χijfor a skyrmion driven by a magnetic ﬁeld gradient, spin transfertorque, and spin Hall torque separately. J Heavy metal Chiral magnet B x z Chiral magnet J B B Chiral magnet (b) (a) (c) FIG. 2. Schematic view of a skyrmion in chiral magnets subjected to different drives: (a) a linear magnetic ﬁeld gradient, (b) spin transfer torque, and (c) spin Hall torque. The crosses in (c) represent the spinaccumulation and spin current normal to the interface due to the spin Hall effect. A. Magnetic ﬁeld gradient We ﬁrst consider the skyrmion motion driven by a magnetic ﬁeld gradient as shown in Fig. 2(a). This has been studied by numerical simulations recently [ 32,33]. We assume that the ﬁeld is along the zaxis and the gradient is along the xdirection. The total magnetic ﬁeld is BT=Bˆz+/Delta1(x) Brcosφˆz.T h e magnetic ﬁeld in the ﬁrst term stabilizes a stationary skyrmionstructure, and that in the second term drives the skyrmion intomotion. The unit of the ﬁeld gradient is D 3/(J2 exMs), with Ms being the saturation ﬁeld. The torque is then given by /Gamma1=−S0×/parenleftbig /Delta1(x) Brcosφˆz/parenrightbig . (33) The coupling coefﬁcient Fm,jis Fm,j=−i/Delta1(x) Bπδ\|m\|,1/integraldisplay drr2LX;m,jsinθ. (34) HereFm,jis nonzero only for \|m\|=1. In the rigid skyrmion approximation, or ω→0, we have F\|m\|=1,j=1=iπ/Delta1(x) B√ 8πκ/integraldisplay drr(sinθ)2. (35) The equation of motion is v(ω)=−/Delta1(x) B(αˆx+κˆy) 4(α2+κ2)/integraldisplay drr(sinθ)2. (36) Forα/lessmuch1, the skyrmion moves almost perpendicular to the magnetic ﬁeld gradient. There is an additional velocitycomponent antiparallel to the ﬁeld gradient due to the damping. The external drive in Eq. ( 28) is the ﬁeld gradient P μ=/Delta1(μ) B. To include all the magnon modes, we calculate numerically the response matrix χij, and the results are shown in Fig. 3.A tω=0,χijis real, and Thiele’s equation of motion in Eq. ( 36) is recovered. We can see that Im[ χij]i s much smaller than Re[ χij] because χijis mainly contributed by the Goldstone modes. The phase shift near the magnongap is /Theta1 /bardbl∼10−3and/Theta1⊥∼10−4. The response is almost instantaneous, and the inertia is weak. As shown in Fig. 6, the magnon modes associated with the Goldstone modeswith\|m\|=1,j=1 are localized in the skyrmion, while the extended modes with j>1 have only small weight around the skyrmion. Therefore \|F \|m\|=1,j=1\|/greatermuch\|F\|m\|=1,j>1\|, and\|W\|m\|=1,j=1\|/greatermuch\| W\|m\|=1,j>1\|. B. Spin transfer torque H e r ew es t u d yt h es k y r m i o nm o t i o nd r i v e nb yas p i nt r a n s f e r torque as shown in Fig. 2(b). We inject an electric current 014407-4DYNAMICS AND INERTIA OF A SKYRMION IN CHIRAL . . . PHYSICAL REVIEW B 96, 014407 (2017) -0.0955-0.0954-0.0953(a) (b) (c) (d)Re[χ11] -0.00015-0.00010-0.000050.000000.000050.000100.00015Im[χ11] 0 . 00 . 51 . 01 . 52 . 02 . 53 . 0-0.57273-0.57270-0.57267 Re[χ21] ω0.0 0.5 1.0 1.5 2.0 2.5 3.00.000000.000040.000080.00012 Im[χ21] ω FIG. 3. Real and imaginary parts of χ11andχ21in the case of a skyrmion driven by a magnetic ﬁeld gradient. Here B=0.8a n d α=0.2. into to a chiral magnet. The electric current is polarized automatically by the skyrmion spin texture because of Hund’scoupling, which results in a spin current. For a nonuniform spintexture, the polarization of the spin current changes in space.To conserve magnetic moment, there is a transfer of magneticmoment between the spin current and localized spin texture,which generates a torque acting on the localized moments. Thespin transfer torque can be expressed as [ 34,35] /Gamma1=[∂ xS0−βS0×(∂xS0)]Jx. (37) The current has units of 2 eD/(¯hPs) and is assumed to be along the xdirection, J=Jxˆx. The dimensionless parameter Psis the spin-polarization factor. In the ﬁrst approximation, the direction of spin current polarization is always parallelto the local magnetization vector. This contribution, calledadiabatic spin transfer torque, is described by the ﬁrst term onthe right-hand side. The second term, proportional to β/lessmuch1, describes the nonadiabatic and dissipative spin transfer torque,originating from the spatial mistracking of polarization of spincurrent and localized magnetization [ 36]. We then compute F m,j=−iJx(τSTT m,j+βτN m,j), with τSTT m,j=−π/integraldisplay dr[−imL∗ X;m,jsin(θ)+rL∗ Y;m,j∂rθ]δ\|m\|,1δj,1, (38) τN m,j=−π/integraldisplay dr[imL∗ Y;m,jsin(θ)+rL∗ X;m,j∂rθ]δ\|m\|,1.(39) The coupling coefﬁcient Fm,jvanishes for \|m\|/negationslash=1. Further- more, for the adiabatic spin transfer torque τSTT m,j, the current couples with the translational mode ∂xS0. Therefore τSTT m,jis nonzero only for the translational mode with j=1. The inertia of a skyrmion arises from the contribution of the nonadiabaticspin transfer torque τ N m,jand damping. For a rigid skyrmion valid when ω→0, we have F\|m=1\|,j=1=Jx/parenleftBigg −m√ 2πκ+i/radicalbigg 2π κβ/parenrightBigg . (40)-1.0080-1.0076-1.0072-1.0068-1.0064-1.0060Re[χ11](a) (b) (c) (d)-5.0x100.05.0x101.0x10 Im[χ11] 0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.08288-0.08284-0.08280-0.08276-0.08272Re[χ21] ω0.0 0.5 1.0 1.5 2.0 2.5 3.001x102x103x104x10Im[χ21] ω FIG. 4. Real and imaginary parts of χ11andχ21in the case of a skyrmion driven by a spin transfer torque. Here B=0.8,α=0.1, andβ=0.2. The equation of motion becomes v(ω)=/bracketleftbigg −κ2+αβ α2+κ2ˆx+(α−β)κ α2+κ2ˆy/bracketrightbigg Jx(ω), (41) which reproduces the well-known Thiele equation of motion, Eq. ( 1). It also describes the fullequation of motion when β=0 because the adiabatic spin transfer couples only with the translational mode. In real materials, α/lessmuch1 and β/lessmuch1, the skyrmion moves almost antiparallel to current, but thereis a transverse motion due to the damping. We can deﬁne aHall angle for the skyrmion motion tan θ H≡\|vy\|/\|vx\|=\|α− β\|κ/(κ2+αβ). Forα=β,E q .( 15) is a Galilean invariant for a rigid skyrmion, and the skyrmion moves exactly antiparallelto the current. Forβ> 0 and to deal with the skyrmion distortion, we compute numerically χ 11andχ21atB=0.8 by taking the modes with j>1 into account, and the results are displayed in Fig. 4.H e r e Pin Eq. ( 28)i sP=J.W eh a v eR e [ χ11]≈ −(κ2+αβ)/(α2+κ2), Re[χ21]≈(α−β)κ/(α2+κ2). The phase shift near the magnon gap is /Theta1/bardbl∼10−9and/Theta1⊥∼10−7. The contributions from other modes are negligible, and thedynamics of a skyrmion is described by Thiele’s equation ( 41). Therefore the response of a skyrmion to a spin transfer torqueis instantaneous, and the inertia of a skyrmion is negligiblecompared to the viscous and Magnus forces. This justiﬁed therigid skyrmion approximation used in Thiele’s approach in thiscase. C. Spin Hall torque We consider a bilayer system with a chiral magnet atop a heavy metal, such as Ta and Pd, as depicted in Fig. 2(c).W e then apply a current in the heavy metal. Because of the spinHall effect, there is spin current normal to the interface, whichgenerates a torque in the chiral magnet. This is called the spinHall torque and is given by [ 37–40] /Gamma1=S 0×[S0×(ˆz×J)]. (42) 014407-5SHI-ZENG LIN PHYSICAL REVIEW B 96, 014407 (2017) The current is in units of 2 D2ed/(Jex¯hθSH), with θSHbeing the spin Hall angle and dbeing the ﬁlm thickness. For a current along the xdirection, J=Jxˆx, the coupling coefﬁcient Fm,j becomes Fm,j=−iπJxδ\|m\|,1/integraldisplay drr(L∗ Y;m,jcosθ−imL∗ X;m,j).(43) In the rigid skyrmion approximation, or ω→0, F\|m\|=1,j=1=−i/integraldisplay dr2⎡ ⎣−im∂xS0−∂yS0 √ 2/radicalBig/integraltext dr2(∂xS0)2⎤ ⎦·/Gamma1.(44) We obtain F\|m\|=1,j=1=0 because the skyrmion has rotational symmetry. Therefore the rigid skyrmion does not couple tothe spin Hall torque when the damping is neglected. Thetorque couples to the skyrmion motion through excitation ofthe magnon modes in the continuum and the weak off-diagonaldamping term in Eq. ( 20). Therefore the spin Hall torque is less efﬁcient in driving the skyrmion compared to that of spin trans-fer torque. Here we provide an estimate on the current densityrequired in order to achieve the same velocity for the cases withspin transfer torque and spin Hall torque. For the spin transfertorque, the velocity is almost frequency independent for agiven current; however, for the spin Hall torque, the velocityis maximal at a given current when the frequency is near themagnon gap. Taking the optimal velocity for the spin Halltorque, to attain the same velocity, the required current densityfor both torques is about J SHEθSHJex/JSTTdDP s∼103, where JSHE(JSTT) is the current density in the case of spin Hall (transfer) torque. Therefore JSHEis larger than JSTTby several orders of magnitude for materials with θSHJex/dDP s∼1. This is consistent with the experimental observations that theskyrmion velocity at a given current in the case of spin transfertorque is much bigger than that in the case of spin Hall torque[8,41], although different pinning strengths in these systems may also partially account for the difference. In the present case, Pin Eq. ( 28)i sP=J.The numerical results of χ ijare presented in Fig. 5. Since χ21∼iχ11,t h e trajectory is roughly a circle. One prominent feature is that -0.0002-0.00010.00000.00010.0002(a) (b) (c) (d)Re[χ11] -0.0004-0.0003-0.0002-0.00010.0000Im[χ11] 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00000.00010.00020.00030.0004Re[χ21] ω0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.0002-0.00010.00000.00010.0002Im[χ21] ω FIG. 5. Real and imaginary parts of χ11andχ21in the case of a skyrmion driven by a spin Hall torque. Here B=0.8a n dα=0.2.χij(ω) develops a resonancelike feature around the magnon gap,ω≈ωg. This can be understood as follows. For a given angular momentum \|m\|=1, the density of state of the magnon, ρ(ω)=1/√ω−ωg, diverges at ωg. The equation of motion of a skyrmion including all magnon excitations is R(ω)=/summationdisplay \|m\|=1,jFm,jWm,j ω−ωm,j+iαm,jω ≈F(ωg)W(ωg)/integraldisplay+∞ ωgdωρ(ω)1 ω−ωm,j+iα(ωg)ω =2F(ωg)W(ωg)1/radicalbigω−ωg+iα(ωg)ω, (45) where F(ωg),W(ωg), and α(ωg) are the corresponding quantities evaluated at the magnon gap. From Eq. ( 45), it is clear that χijdevelops peaks around ω≈ωg, in agreement with the results in Fig. 5. VI. NÉEL SKYRMION Skyrmions can also be stabilized at interfaces of het- erostructure, where the inversion symmetry is broken explic-itly [ 15,42,43]. The skyrmion at the interface has a helicity of zero and is called the Néel skyrmion. The effectiveHamiltonian supporting Néel skyrmions in two dimensions,(x, y), can be written as [ 39,44] H n=Jex 2/summationdisplay μ=x,y(∂μn)2+D[nz(∇·n)−(n·∇)nz]−B·n. (46) The Bloch skyrmion described by Eq. ( 2) can be obtained by a global rotation of spin associated with the Néel skyrmionalong the magnetic ﬁeld direction by π/2, i.e., S=ˆO nn, with the spin rotation operator ˆOn=⎛ ⎝0−10 10 000 1⎞ ⎠. (47) One can verify that H ncan be obtained from Hin Eq. ( 2)b yt h e same global rotation of spins. Therefore the eigenfrequenciesof the internal modes for the Bloch and Néel skyrmions areidentical, and the eigenmodes for two skyrmion textures arerelated by a spin rotation. One can also connect the equation of motion for the Bloch skyrmion to that of the Néel skyrmion by the same spinrotation. For a Bloch skyrmion driven by a magnetic ﬁeldgradient with polarization perpendicular to the layer, spintransfer torque, and spin Hall torque, the equation of motionfor spins is ∂ tS=−S×Heff(S)+αS×∂tS+(JSTT·∇)S −βS×(JSTT·∇)S+S×[S×(ˆz×JSHE)].(48) After the spin rotation, the equation of motion for spins associated with a Néel skyrmion has the same form as Eq. ( 48) if we replace the effective ﬁeld by Heff(n)≡−δHn/δnand JSHE←ˆO−1 nJSHE. Therefore the results derived for the Bloch skyrmions can be applied directly to the Néel skyrmions. 014407-6DYNAMICS AND INERTIA OF A SKYRMION IN CHIRAL . . . PHYSICAL REVIEW B 96, 014407 (2017) VII. DISCUSSION AND CONCLUSIONS The current can generate a magnetic ﬁeld gradient which affects the skyrmion trajectory. The magnetic ﬁeld gradientdepends on the sample geometry. In our linear approximation,we need to superpose the trajectory due to the current onthe trajectory due to the induced ﬁeld gradient. Meanwhile,the skyrmion motion distorts magnetic ﬁeld outside of themagnetic plate, which contributes to the skyrmion kineticenergy. In the adiabatic approximation, a moving skyrmioninduces an electric ﬁeld ∇×E=(1/c)˙B=−(v·∇)B/cac- cording to Faraday’s law. As a result its kinetic energy increases as/integraltext d 2rE2/(8π)=Mem˙R2, which determines the electromagnetic part of the skyrmion mass tensor Mem. Such an inertia effect has a relativistic origin and is proportional to(v/c) 2. Since the maximal skyrmion velocity, 102m/s[19], is much smaller than the light velocity, 108m/s, the correction to skyrmion dynamics due to the electric ﬁeld outside themagnetic plate is negligible. Let us compare our results on the inertia of a skyrmion to those in the literature. The mass of a skyrmion bubblewas calculated and measured in Refs. [ 27,45], which found that the mass term is important to reproduce the skyrmiontrajectory. In the calculations [ 45], the authors calculated the mass from the edge modes of the bubble. In experiments [ 27], the mass was obtained by ﬁtting the experimentally measured trajectory determined from the spin polarization to the equation of motion of a skyrmion as a particle with a mass term,assuming a harmonic potential for the skyrmion bubble. InRefs. [ 27,45], the skyrmion bubbles were stabilized by the dipolar interactions, and the magnon excitations were expectedto be quite different from those of a skyrmion in chiral magnets.This may be the origin of the difference between our resultsand those in Refs. [ 27,45]. This also suggests that skyrmions in chiral magnets are more rigid and are advantageous forapplications. The inertia of a skyrmion in chiral magnets wasstudied recently in Ref. [ 26]. They calculated numerically the trajectory of a skyrmion from the Landau-Lifshitz-Gilbertequation with thermal noise, where the skyrmion diffuses in the sample. They then ﬁt the trajectory to the equation of motion for a skyrmion as a particle according to Eq. ( 30), from which they extracted the mass and gyrocoupling coefﬁcient. Theyfound that the inertia is important to describe the skyrmiondiffusion. For a skyrmion driven by a spin transfer torque,they found the response of a skyrmion to current is almostinstantaneous, consistent with our results. The importance ofnonlinearity in their calculations is unclear. The dynamics of skyrmions depends on its internal modes. The analysis here is applicable for skyrmions stabilized bythe DM interaction, where the helicity of skyrmions is nota Goldstone mode. Skyrmions can also exist in inversion-symmetric magnets with competing interactions [ 46–49]. Due to the preservation of the inversion symmetry, the helicityof skyrmions is also a Goldstone mode, in addition to themode associated with translational motion. The dynamicscan be very different, as has been demonstrated recently inRefs. [ 48,50]. Skyrmions with higher topological charge can also be stabilized [ 51]. The internal modes are different for skyrmions with higher topological charge, and the dynamicsof these skyrmions requires a separate study.To conclude, we have developed a linear theory to calculate the equation of motion of a skyrmion by taking all the magnonmodes into account. We calculate the skyrmion velocity inresponse to external drives, such as a magnetic ﬁeld gradient,spin transfer torque, and spin Hall torque. The skyrmiondynamics is governed only by the magnon modes with anangular momentum \|m\|=1. The inertia of a skyrmion is contributed by the magnon continuum. For a skyrmion drivenby a magnetic ﬁeld gradient or spin transfer torque, thedynamics is dominated by the Goldstone modes correspond-ing to the translational motion of a skyrmion because theeigenfunction of the Goldstone modes has maximal weightaround the skyrmion center, while the modes in the magnoncontinuum have very little weight around the skyrmion. Theresponse of a skyrmion is instantaneous, and our calculationsjustify the rigid skyrmion approximation employed in Thiele’scollective coordinate approach. In the case of a spin Halltorque, the skyrmion motion couples to the torque throughthe magnons in the continuum and the Gilbert damping.Since the magnon density of state diverges around the magnongap, there are resonances in the response functions around themagnon gap. The trajectory of a skyrmion is an ellipse for askyrmion driven by a spin Hall torque at ﬁnite frequencies.The inertia of a skyrmion can be quantiﬁed in experimentsby measuring the phase shift between the external drive andvelocity. For applications, it is desirable for skyrmions torespond to an external drive without delay or retardation evenat high frequencies. This can be achieved by driving a skyrmionwith a spin transfer torque. Our results establish the connectionbetween the skyrmion dynamics and its magnon spectrum andshed new light on the skyrmion dynamics. ACKNOWLEDGMENTS The author is indebted to L. N. Bulaevskii, A. Saxena, C. D. Batista, and S. Hayami for helpful discussions. Thiswork was carried out under the auspices of US DOE ContractNo. DE-AC52-06NA25396 through the LDRD program. APPENDIX A: NUMERICAL METHOD FOR THE EIGENMODES ANALYSIS We present the details of the numerical evaluation of the eigenmodes in Eq. ( 13). We introduce an orthogonal complete basis to expand ηmin this basis. In the polar coordinate, a natural choice is the Bessel functions of the ﬁrst kind. Todetermine the proper Bessel functions, we ﬁrst check ˆ η min the r/lessmuch1 limit in Eq. ( 13), which can be solved analytically, ˆηm=/parenleftbigg c−r\|m−1\| c+r\|m+1\|/parenrightbigg . (A1) From the asymptotic behavior of ˆ ηm, we choose the following basis: \|pm,i/angbracketright=√ 2 RcJm(km−1,i)Jm−1/parenleftbigg km−1,ir Rc/parenrightbigg exp(imφ)/parenleftbigg 1 0/parenrightbigg , (A2) \|hm,i/angbracketright=√ 2 RcJm+2(km+1,i)Jm+1/parenleftbigg km+1,ir Rc/parenrightbigg exp(imφ)/parenleftbigg 0 1/parenrightbigg , (A3) 014407-7SHI-ZENG LIN PHYSICAL REVIEW B 96, 014407 (2017) 0 1 02 03 04 05 0-0.4-0.3-0.2-0.10.0 01 0 2 0 3 0 4 0 5 00.00.10.20.30.40.5 0 5 10 15 20 25 300.00.30.60.91.21.51.8(b) (a)Re[LX,m=1, j] rj=1 j=2 j=3 j=4 j=5 Here Im[ LX,m=1, j]=0 Im[LY,m=1, j] rj=1 j=2 j=3 j=4 j=5 Here Re[ LY,m=1, j]=0ωm=1,j j FIG. 6. Five lowest magnon wave functions with m=1i nt h e rotated frame. LX;m=1,jis real, and LY;m=1,jis imaginary. The inset shows the eigenfrequency. Here B=0.8. where we have used the box normalization, with Rcbeing the radius of the box, and km,iis theith zero of the Bessel function Jm(k). Then the matrix element of Hψis ˆH(m) 11;ij=/angbracketleftpm,i\|Hψ\|pm,j/angbracketright,ˆH(m) 12;ij=/angbracketleftpm,i\|Hψ\|hm,j/angbracketright, ˆH(m) 21;ij=/angbracketlefthm,i\|Hψ\|pm,j/angbracketright,ˆH(m) 22;ij=/angbracketlefthm,i\|Hψ\|hm,j/angbracketright.(A4) By diagonalizing the matrix σzHψ, we obtain the eigenfre- quencies and eigenmodes. We take Rc=50 and truncate the Bessel series at imax=100. The results for the eigenmodes and eigenfrequencies with m=1 in the rotated frame are shown in Fig. 6. There is only one bound state, with ωm=1,j=1=0 corresponding to the translational motion of skyrmion. Thewave function of the bound state has large weight around theskyrmion center, while the magnon modes in the continuumhave little weight around the skyrmion. FIG. 7. Results of ln( \|αm=−1,j,j/prime\|/α)f o r1 /lessorequalslantj, j/prime/lessorequalslant51 in the frequency region 0 /lessorequalslantωm,j/lessorequalslant11 obtained at B=0.8. 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PhysRevB.93.104418.pdf	PHYSICAL REVIEW B 93, 104418 (2016) Cloaking the magnons Mehrdad Elyasi, Charanjit S. Bhatia, Cheng-Wei Qiu, and Hyunsoo Yang* Department of Electrical and Computer Engineering, National University of Singapore, 117576 Singapore (Received 28 July 2015; revised manuscript received 24 December 2015; published 21 March 2016) We propose two approaches to cloak the spin waves (magnons) by investigating magnetization dynamics. One approach is based on a spatially inhomogeneous anisotropic magnetic moment tensor. The other mechanismis using a spatially inhomogeneous anisotropic gyromagnetic factor tensor and an inhomogeneous externalmagnetic ﬁeld. For both approaches, the damping tensor is also inhomogeneous and anisotropic. The magneticcharacteristic functions of the magnetic materials have been theoretically derived for both mechanisms. Anonmagnetic core, which prevents magnons from entering and consequently distorts the spin-wave propagation,can be cloaked by a structured magnetic shell to redirect the spin wave around the core using the above designmechanisms. We discuss the feasibility of the proposed mechanisms in an ensemble of quantum dot moleculesand magnetic semiconductors. The proposed approaches shed light on transformation magnonics, and can beutilized for future spin-wave lenses, concentrators, low backscattering waveguides, and ultimately quantumcomputing. DOI: 10.1103/PhysRevB.93.104418 I. INTRODUCTION Invisibility cloaking of different types of waves has been pursued intensively in the last decade [ 1–13]. Different mechanisms for decreasing the scattering of objects withelectromagnetic (EM) waves have been proposed and inves-tigated [ 1,3,6,7,13]. Among those, the space transformation method intuitively provides a promising avenue to achieve theinvisibility by designing a particular shell to hide the core.The Maxwell’s equations are invariant under such a spacetransformation, leading to materials in the shell region beinginhomogeneous and anisotropic [ 6,11]. This method or its simpliﬁed approximation has been demonstrated theoreticallyand experimentally for microwave EM waves by using EMmetamaterials [ 3,6,11–13]. It has also been revealed that the two-dimensional (2D) Schr ¨odinger equation can be invariant under space transformation by using inhomogeneous andanisotropic effective mass and magnetic potential in the shellarea [ 9]. Although it has been suggested that the perfect invariance for general three-dimensional (3D) elastodynamicwaves is not possible [ 5,14], the transformation based cloaking of such waves has been demonstrated for special cases. Ithas been also proven that the acoustic wave equations remaininvariant in 2D and 3D by having a speciﬁc inhomogeneity andanisotropy in the mass density and the bulk moduli [ 8,15,16]. Recently, there have been demonstrations of bilayer cloaks fortemperature, dc current, and magnetostatic ﬁelds by utilizingbulk homogeneous materials [ 4,10,17–19]. The conception of invisibility is not always reducing the wave scattering. It canalso be referred to as hiding a feature of an object or space.As an example, it was proposed that in bilayer graphene, theconﬁned states in the barrier can be cloaked due to chiralitymismatch with the continuum states, for the normal andoblique incident currents [ 20]. Another example is cloaking polarizable discrete systems using an anomalous resonancetechnique [ 7]. *eleyang@nus.edu.sgAnother important type of wave that has been the subject studied for decades, is spin wave with magnons as itsquanta [ 21]. The features that makes spin waves interesting for applications are their wave vectors as small as a fewnanometers and frequencies of tens of GHz. As it is possibleto engineer different dispersions in magnetic lattices, themagnonics has thereby attracted a lot of attention as apromising candidate with low energy consumption and highthroughput computation, which may possibly go beyondphotonic or even electron devices. There have been manytheoretical and experimental demonstrations of different typesof passive and active magnonic systems and crystals, suchas transistors, interferometers, waveguides, and logic gates[21–24]. In addition, due to the high nonlinearity in mag- netization dynamics, spin waves have become a base forstudying different phenomena such as time reversal and Bosecondensation of magnons as bosonic waves [ 25,26]. As the transformation optics enabled alternative possibilities in photonics, the transformation magnon-ics can introduce alternative approaches in magnon-ics. Despite that a negative refraction index for spinwaves, and graded index systems for the manipula-tion of spin wave propagation [ 27] have been reported, there has not been a uniﬁed account of transformation magnon-ics for more sophisticated applications such as magnonicinvisibility cloaks. As an alternative to the transformationtechniques for controlling waves, recent proposals and demon-strations of topologically protected edge modes for photons,phonons, and magnons can be utilized to cloak the defects inthe edges [ 28–31]. In this work, we investigate the invariance in the governing equations of the magnetization dynamics under the spacetransformation. The objective is to reduce the scattering ofa nonmagnetic core (blocking the magnons) by designing themagnetic characteristics in a surrounding shell, such as themagnetic moment, gyromagnetic factor, exchange constant,Gilbert damping, and external magnetic ﬁeld. The spatial proﬁles of these parameters in the shell area can be designed such that the magnetization dynamics is rendered maximallyinvariant under the space transformation. 2469-9950/2016/93(10)/104418(9) 104418-1 ©2016 American Physical SocietyELY ASI, BHATIA, QIU, AND Y ANG PHYSICAL REVIEW B 93, 104418 (2016) II. CLOAKING SHELL DESIGNS BASED ON TRANSFORMATION MAGNONICS A. Governing equations of the magnons and possible cloaking approaches The Landau-Lifshitz-Gilbert (LLG) equation governs the magnetization dynamics. We can write the total magnetization /vectorM(/vectorρ)a s/vectorM(/vectorρ)=/vectorM0(/vectorρ)+/vectorMd(/vectorρ), where /vectorρis the coordination vector, /vectorM0(/vectorρ) is the static part of the magnetization, and /vectorMd(/vectorρ) is the dynamic part of the magnetization. For media withisotropic magnetic moments, under the assumption of zero temperature, \|/vectorM(/vectorρ)\|is temporally constant, implying /vectorM 0(/vectorρ)· /vectorMd(/vectorρ)=0. More generally to account for possible anisotropy in the moment, we can write /vectorM(/vectorρ)=¯Ms(/vectorρ)[/vectorm0(/vectorρ)+/vectormd(/vectorρ)], where we deﬁne ¯Ms(/vectorρ) as the moment tensor. /vectorm0(/vectorρ) and/vectormd(/vectorρ) are the mathematical vectors [ /vectorm0(/vectorρ)+/vectormd(/vectorρ) is a unit vector] that determine the static [ /vectorM0(/vectorρ)] and dynamic [ /vectorMd(/vectorρ)] part of the magnetization, respectively, based on ¯Ms(/vectorρ)[/vectorM0(/vectorρ)= ¯Ms(/vectorρ)/vectorm0(/vectorρ) and /vectorMd(/vectorρ)=¯Ms(/vectorρ)/vectormd(/vectorρ)]. If the moment is isotropic [ ¯Ms(/vectorρ) has equivalent diagonal and zero off-diagonal components] at position /vectorρ,/vectorM0(/vectorρ)/bardbl/vectorm0(/vectorρ) and /vectorMd(/vectorρ)/bardbl /vectormd(/vectorρ), while in the case of anisotropic moment (e.g., if ¯Ms(/vectorρ) has nonequivalent nonzero diagonal and zero off-diagonal components), /vectorM0(/vectorρ)[/vectorMd(/vectorρ)] is not necessarily parallel to /vectorm0(/vectorρ)[/vectormd(/vectorρ)]. It must be noted that the vectors /vectorm0(/vectorρ) and /vectormd(/vectorρ) are not physical parameters and should not be confused with the normalized vectors /vectorM0(/vectorρ)//bardbl/vectorM0(/vectorρ)+/vectorMd(/vectorρ)/bardbland /vectorMd(/vectorρ)//bardbl/vectorM0(/vectorρ)+/vectorMd(/vectorρ)/bardbl, respectively. The LLG equation can be written as ¯Ms(/vectorρ)∂/vectorm(/vectorρ) ∂t=¯γ(/vectorρ)[¯Ms(/vectorρ)/vectorm(/vectorρ)×/vectorH(/vectorρ)] +¯α(/vectorρ)¯γ(/vectorρ)/braceleftbig¯M−1 s(/vectorρ)/vectorM0(/vectorρ) ×[¯Ms(/vectorρ)/vectorm(/vectorρ)×/vectorH(/vectorρ)]/bracerightbig , (1) where ¯ γ(/vectorρ) is the gyromagnetic factor tensor and /vectorH(/vectorρ) is the magnetic ﬁeld that can be written as /vectorH(/vectorρ)= /vectorHext(/vectorρ)+/vectorHex(/vectorρ)+/vectorHm(/vectorρ), while /vectorHext(/vectorρ) is the external dc ﬁeld, and /vectorHex(/vectorρ) and /vectorHm(/vectorρ) are the exchange and dipolar ﬁelds induced by /vectorM(/vectorρ), respectively. In the equilibrium, /vectorM0(/vectorρ)×/vectorH0(/vectorρ)=0, where /vectorH0(/vectorρ) is the static part of /vectorH(/vectorρ). /vectorH0,ex(/vectorhd,ex) and /vectorH0,m(/vectorhd,m) are the exchange and dipo- lar ﬁelds arisen from /vectorM0(/vectorρ)[/vectorMd(/vectorρ)], respectively. /vectorm(/vectorρ)= /vectorm0(/vectorρ)+/vectormd(/vectorρ), and ¯ α(/vectorρ) is the Gilbert damping constant tensor. As a proof-of-concept demonstration, we focus on a cylin- drical magnonic cloak. In order to assume dynamic invariancein the out-of-plane direction (i.e., zdirection), the thickness of the thin ﬁlm should be small such that the excitationfrequency does not induce modes with out-of-plane wavenumbers. The cylindrical coordinates rϕzare transformed into r /primeϕ/primez/prime, where r/prime=g(r),ϕ/prime=ϕ, andz/prime=z, indicating that the mapping only occurs to the radial axis. The transformationfunction should satisfy the boundary conditions of g(0)=c andg(b)=b, where cis the core radius, and bis the outer shell radius. Under such a transformation, we can obtain /vectorM d(/vectorρ)= T/vectorM/prime d(/vectorρ/prime), where Tis a diagonal matrix with components FIG. 1. (a) The structural dimensions, ﬁelds, and spin-wave propagation direction, the respective Cartesian coordinate axes, /vectorM0and/vectorHextoutside the shell, and an example vector plot of /vectorMd(/vectorρ) [the color map represents the amplitude of /vectorMd(/vectorρ),/bardbl/vectorMd(/vectorρ)/bardbl], where a cylindrical core is perfectly cloaked from propagatingmagnons (outside the shell, /vectorM d(/vectorρ)=sin[2π/(100 nm) ×x]ˆx+ cos[2π/(100 nm) ×x]ˆz)o f/vectorq⊥/vectorM0type. Orepresents the origin of the structure. The vector plot is magniﬁed in the dashedbox. (b) Schematic of magnetization precession in anisotropic ¯ γ. (c) Schematic of magnetization precession in anisotropic ¯M s.I n (b) and (c), the distance between the red and blue curves indicates theamplitude of the gyromagnetic factor, while the black dashed curve indicates the trajectory of the magnetization /vectorM. A thicker /vectorMindicates a higher moment in (c). ofTrr=∂g(r) ∂r=g/prime(r),Tϕϕ=r/prime r=g(r) r, and Tzz=1[6,11]. There are two types of spin waves in terms of propagation, /vectorq/bardbl/vectorM0and/vectorq⊥/vectorM0, where /vectorqis the wave vector. Figure 1(a) shows the schematic vector plot of /vectorMd(/vectorρ), where a cylindrical core is perfectly cloaked from propagating magnons of /vectorq⊥/vectorM0 type. Equation ( 1) indicates that the magnetization dynamic has dual characteristics, which provide two degrees of freedomfor controlling its inertia. One is the magnetic moment whichis the manifestation of carrier and orbital spin populationin a preferential direction in the Hilbert space ( /vectorM) and is determined by the moment tensor ¯M s(/vectorρ). The other one is the factor which determines the modiﬁcation type andstrength of gyration under an application of magnetic ﬁelds(¯γ). Conceptually, with any change in ¯M sor ¯γ, we can modify the pointwise dynamics, as it can be inferred from Eq. ( 1), and as schematically shown in Figs. 1(b) and1(c). Figure 1(b) shows an isotropic magnetization vector ( /vectorM) precessing in an anisotropic ¯ γ. Figure 1(c) shows an anisotropic magnetization vector ( /vectorM) precessing in an isotropic ¯ γ. However, we should note that there are two strong nonlocal correlations in magnetic systems, exchange and dipolar interactions [ /vectorHex(/vectorρ) and/vectorHm(/vectorρ)], which determine the dispersion relation of the spin waves for given ¯Msand ¯γ. First, we demonstrate a cloaking mechanism based on ¯ γ. Subsequently, we utilize ¯Msto cloak the cylindrical core. Finally, we discuss the feasibility of theproposed methods. 104418-2CLOAKING THE MAGNONS PHYSICAL REVIEW B 93, 104418 (2016) Based on the linear perturbation of Eq. ( 1), we can write the dynamic part of the magnetization as ¯Ms(/vectorρ)∂/vectormd(/vectorρ) ∂t=(/vector/Pi1+/vector/Omega1), /vector/Pi1(/vectorρ)=¯γ(/vectorρ)[/vectorM0(/vectorρ)×/vectorhd(/vectorρ) +¯Ms(/vectorρ)/vectormd(/vectorρ)×/vectorH0(/vectorρ)], /vector/Omega1(/vectorρ)=¯α(/vectorρ)¯γ(/vectorρ)/bracketleftbig¯M−1 s(/vectorρ)/vectorM0(/vectorρ)×¯γ−1(/vectorρ)/vector/Pi1(/vectorρ)/bracketrightbig , (2) where /vectorhd(/vectorρ)=/vectorhd,ex(/vectorρ)+/vectorhd,m(/vectorρ). Here, we consider the case of the propagating magnons of type /vectorq⊥/vectorM0(similar discussions apply for /vectorq/bardbl/vectorM0), while outside the cloaking area (r>b ) we assume homogeneous and isotropic magnetic moments ¯Ms(/vectorρ)=Ms,0I(Iis the identity matrix), and a gyromagnetic factor ¯ γ(/vectorρ)=γ0I, where /vectorM0=Ms,0ˆy.W e assume W,L→∞ andd/W/lessmuch1, where Wis the width, Lis the length, and dis the thickness of the structure [refer to Fig. 1(a)]. Under this assumption, the dynamic part of the magnetization has the form /vectorMd(/vectorρ)=/vectorMd,0(/vectorρ)e−ikxx(kxis the wave number in the xdirection), and the dynamic demag- netization tensor ¯Ndhas only two nonzero components in thexyzcoordination system, Nd,xx=− [1−(1−e−kxd)]/kxd andNd,zz=− 1−Nd,xx [32,33]. The dynamic dipolar ﬁeld is related to /vectorMd(/vectorρ)a s/vectorhd,m(/vectorρ)=¯Nd/vectorMd(/vectorρ). In addition, the static demagnetization tensor ¯N0has only one nonzero com- ponent Nd,zz=− 1, while /vectorH0,m(/vectorρ)=¯N0/vectorM0(/vectorρ). The isotropic exchange ﬁeld in the continuum limit can be written as /vectorHex(/vectorρ)=/Lambda1∇2[/vectorM0(/vectorρ)+/vectorMd(/vectorρ)], where /Lambda1=A/2πM2 s,0and Ais the exchange constant. /vectorhd,ex(/vectorρ)=−/Lambda1k2 x/vectorMd(/vectorρ), while the homogeneity of /vectorM0implies /vectorH0,ex(/vectorρ)=0. B. Cloaking shell designs based on anisotropic inhomogeneous ¯γ In order to achieve cloaking, Eq. ( 2) should remain invariant if we rewrite /vectorMd(/vectorρ)a s T/vectorM/prime d(/vectorρ/prime). The con- ditions for Eq. ( 2) to remain invariant in the trans- formed space are T−1/vector/Pi1(/vectorρ/prime)=/vector/Pi1/primeandT−1/vector/Omega1(/vectorρ/prime)=/vector/Omega1/prime, where /vector/Pi1/prime=¯γ/prime(/vectorρ)[/vectorM/prime 0(/vectorρ/prime)×/vectorh/prime d(/vectorρ/prime)+Ms,0/vectorm/prime d(/vectorρ/prime)×/vectorH/prime 0(/vectorρ/prime)] and/vector/Omega1/prime=¯α/prime(/vectorρ/prime)¯γ/prime(/vectorρ/prime)[1 Ms,0/vectorM/prime 0(/vectorρ/prime)×(γ−1 0/vector/Pi1/prime)] (values with prime refer to the transformed space). In the ¯ γbased design, we assume that the moment is isotropic and ¯M/prime s(/vectorρ/prime) can be replaced by the scalar M/prime s,0(/vectorρ/prime). The invariant conditions to satisfy T−1/vector/Pi1(/vectorρ/prime)=/vector/Pi1/primeandT−1/vector/Omega1(/vectorρ/prime)=/vector/Omega1/primeare derived to be H/prime 0,ϕ(/vectorρ/prime)=TrrH0,ϕ(/vectorρ/prime),M/prime 0,ϕ(/vectorρ/prime)=TrrM0,ϕ(/vectorρ/prime), H/prime 0,r(/vectorρ/prime)=TϕϕH0,r(/vectorρ/prime),M/prime 0,r(/vectorρ/prime)=TϕϕM0,r(/vectorρ/prime), γ/prime rr(/vectorρ/prime)=γ01 T2rr,γ/prime ϕϕ(/vectorρ/prime)=γ01 T2ϕϕ,γ/prime zz=γ0, α/prime rr(ϕϕ)(/vectorρ/prime)=α0,α/prime zz(/vectorρ/prime) =α0[M0,r(/vectorρ/prime)]2+[M0,ϕ(/vectorρ/prime)]2 T2ϕϕ[M0,r(/vectorρ/prime)]2+T2rr[M0,ϕ(/vectorρ/prime)]2. (3) γ/prime ii(α/prime ii),i=r,ϕ, andz, are the diagonal components of ¯ γ/prime (¯α/prime).α0is the value of the homogeneous Gilbert dampingconstant outside the cloaking area ( r>b ). The steps to achieve the invariant conditions presented in Eq. ( 3)a r e explained in Appendix. It should be mentioned that theconditions in Eq. ( 3) satisfy the invariant conditions in all three directions of the cylindrical coordination system, if /vectorh d(/vectorρ)= T/vectorh/prime d(/vectorρ/prime) [note that /vectorMd(/vectorρ)=T/vectorM/prime d(/vectorρ/prime)]. This condition holds, because /vectorhd(/vectorρ) can be written as a linear function of /vectorMd(/vectorρ) [/vectorhd(/vectorρ)=(¯Nd−/Lambda1k2 x)/vectorMd(/vectorρ), where ¯Nddoes not have a spatial functionality] under the assumptions given above [ ¯Ms(/vectorρ)= Ms,0I,W,L →∞,d/W /lessmuch1, and /vectorMd(/vectorρ)=/vectorMd,0(/vectorρ)e−ikxx]. The functionality of M/prime 0,ϕ(r)(/vectorρ/prime)i nE q .( 3) will modify the static ﬁeld H0,ϕ(r)(/vectorρ/prime) through the exchange and dipolar ﬁelds, which will contradict the other assumptions whichled us to the established invariance conditions. To cancel theredundant static exchange and dipolar ﬁelds, H 0,ϕ(r)(/vectorρ/prime) should be modiﬁed as H0,ϕ(r)(/vectorρ/prime)=H/prime 0,ϕ(r)(/vectorρ/prime)+/braceleftbigg −/bracketleftbigg/integraldisplay V¯G(/vectorρ/prime,/vectorτ)/vectorM/prime 0(/vectorτ)dτ/bracketrightbigg −/Lambda1∇2[/vectorM/prime 0(/vectorρ/prime)]/bracerightbigg ·ˆϕ(ˆr), (4) where ¯G(/vectorρ/prime,/vectorτ) is the dipolar Green function tensor in the cylin- drical coordination system. Therefore, the perfect cloaking ofmagnons which are governed by Eq. ( 2) can be achieved for the aforementioned assumptions, if Eqs. ( 3) and ( 4) are satisﬁed. We name this method the ¯ γmechanism, contrasting the ¯M s mechanism that will be described later. It can be inferred from Eq. ( 3) that in the ¯ γmechanism, ¯M/prime s(/vectorρ/prime)=M/prime s(/vectorρ/prime)I, where M/prime s(/vectorρ/prime)=/bardbl/vectorM/prime 0(/vectorρ/prime)/bardblis not homogeneous but is isotropic. C. Cloaking shell designs based on anisotropic inhomogeneous ¯Ms For the ¯ γmechanism, we investigated the conditions that should be held for Eq. ( 2) to remain invariant under the space transformation required for cloaking [ T−1/vector/Pi1(/vectorρ/prime)=/vector/Pi1/prime andT−1/vector/Omega1(/vectorρ/prime)=/vector/Omega1/prime]. However, the magnetization is a vector ﬁeld whose magnitude can be anisotropic, enabling anotherapproach for rendering Eq. ( 2) invariant. In the transformed space, the dynamic part of the magnetization should be /vectorM /prime d(/vectorρ/prime)=T−1/vectorMd(/vectorρ), which can be achieved if ¯M/prime s(/vectorρ/prime)=Ms,0T−1. (5) The conditions to achieve invariance in Eq. ( 2) with the assumption of Eq. ( 5), become (refer to Appendix for details) H/prime 0,ϕ(/vectorρ/prime)=H0,ϕ(/vectorρ/prime),M/prime 0,ϕ(/vectorρ/prime)=M0,ϕ(/vectorρ/prime), m/prime 0,ϕ(/vectorρ/prime)=Tϕϕm0,ϕ(/vectorρ/prime), H/prime 0,r(/vectorρ/prime)=H0,r(/vectorρ/prime),M/prime 0,r(/vectorρ/prime)=M0,r(/vectorρ/prime), m/prime 0,r(/vectorρ/prime)=Trrm0,r(/vectorρ/prime), α/prime rr(/vectorρ/prime)=α0 Tϕϕ,α/prime ϕϕ(/vectorρ/prime)=α0 Trr, α/prime zz(/vectorρ/prime)=α0[M0,r(/vectorρ/prime)]2+[M0,ϕ(/vectorρ/prime)]2 Trr[M0,r(/vectorρ/prime)]2+Tϕϕ[M0,ϕ(/vectorρ/prime)]2.(6) It should be noted that the gyromagnetic factor is assumed to be isotropic for the ¯Msmechanism, ¯ γ/prime(/vectorρ/prime)=γ0I. In contrast 104418-3ELY ASI, BHATIA, QIU, AND Y ANG PHYSICAL REVIEW B 93, 104418 (2016) FIG. 2. The magnetization in the zdirection ( Mz) after 2.2 ns of an excitation with /vectorhmw=1×sin[2π/(50 GHz) ×t]ˆxOe atx=800 nm. (a) No cylindrical core ( b=0a n d c=0). (b) Cylindrical core but no cloaking shell ( b=50 nm and c=50 nm). (c) Cylindrical core with the shell designed for the ¯ γmechanism ( b=100 nm and c=50 nm). (d) Cylindrical core with the shell designed for the ¯Msmechanism (b=100 nm and c=50 nm). The dashed boxes represent the shadow region used for calculation of Mz,sr(x) plotted in Fig. 3(a). The inner circle shows the boundary of the core. The larger circle shows the outer boundary of the shell. to the ¯ γmechanism, M/prime 0,r(ϕ)(/vectorρ/prime)=M0,r(ϕ)(/vectorρ/prime) holds for the ¯Msmechanism [comparing Eqs. ( 3) and ( 6)]; therefore there is no modiﬁcation in the static exchange or dipolar ﬁeld[H 0,ϕ(r)(/vectorρ/prime)=H/prime 0,ϕ(r)(/vectorρ/prime)]. III. NUMERICAL DEMONSTRATION OF THE CLOAKING MECHANISMS In order to demonstrate the functionality of the proposed magnonic cloak, we have developed an in-house code to solvethe LLG equation [Eq. ( 1)] for the anisotropic ¯ γ,¯M s, and ¯α[34]. We assume the structure dimensions in Fig. 1(a) to be W=420 nm, L=820 nm, c=50 nm, b=100 nm, d=3.8n m , xc=550 nm, and yc=210 nm ( xcandycare the center positions of the cylindrical core along the xand ydirection, respectively, with respect to the origin). The transformation function g(r)=c+r(b−c)/bis employed. The meshing cells are cubic and have the dimensions of2n m ×2n m ×3.8 nm. In the nontransformed area, we assume /vectorm 0/bardblˆy,/vectorHext=10 000 ˆyOe,Ms,0=8×105A/m, A=0.5×10−11J/m, and γ0=2.2×105Hz/(A m). The microwave excitation is applied as /vectorhmw=1×sin(ωmwt)ˆxOe, atx=800 nm. The microwave excitation frequency was set as ωmw=2π×50×109rad/s. We apply matched layers (ML) of 4 nm width in all four in-plane boundaries. In the ML areaα 0=1, while α0=0.01 for the rest of the structure. Figure 2shows the snapshot of the magnetization in the zdirection ( Mz)a tt=2.2 ns (well before the wave reaches the ML layer at x=0, in which case the reﬂection drives its adjacent magnetization dynamics unstable). The dashedlines at x=800 nm represent the microwave excitation lines, inducing spin waves propagating in the –xdirection. The horizontal lines in Fig. 2separate regions of each graph that have a speciﬁc color code shown on their right side. Figure 2(a) demonstrates the spin-wave conﬁguration when there is nocylindrical core ( b=0 and c=0). It can be observed that due to the ﬁnite width ( W=420 nm), in addition to thepropagating magnons in the –xdirection, standing spin waves are formed across the ydirection ( k y/negationslash=0). Figure 2(b) shows a case with a cylindrical core while no cloaking mechanism wasapplied ( b=50 nm and c=50 nm). The shadowing of the core in the spin-wave conﬁguration can be clearly observedin Fig. 2(b). Figures 2(c) and2(d) show the cases where the ¯γmechanism and ¯M smechanism were applied, respectively, withb=100 nm and c=50 nm. Both Figs. 2(c) and 2(d) demonstrate the reduction of the shadow of the core incomparison with Fig. 2(b). The values of M z,sr(x)=1 20 nm/integraltextyc+10 nm yc−10 nmMz(x,y)dyare shown in Figs. 3(a) and3(b) for 0/lessorequalslantx/lessorequalslant420 nm and 0 /lessorequalslantx/lessorequalslant 820 nm, respectively. Especially, Fig. 3(a) shows a quantitative comparison of Mzin the shadowing region (the dashed boxes FIG. 3. The variation of average Mz[Mz,sr(x)= 1 20 nm/integraltextyc+10 nm yc−10 nmMz(x,y)dy]f o r( a )0 /lessorequalslantx/lessorequalslant420 nm (the shadow region of the core), and (b) 0 /lessorequalslantx/lessorequalslant820 nm. The variation of average Mz[Mz,T(x)=1 420 nm/integraltext420 nm 0n mMz(x,y)dy]f o r( c )0 /lessorequalslantx/lessorequalslant420 nm and (d) 0 /lessorequalslantx/lessorequalslant820 nm. 104418-4CLOAKING THE MAGNONS PHYSICAL REVIEW B 93, 104418 (2016) in Fig. 2) for all of the four cases in Fig. 2. It can be seen that the spin waves are suppressed after the propagation throughthe core where no cloaking mechanism is applied, whileboth the cloaking mechanisms have the values of M z,sr(x) close to that of no core. Despite this signature of cloaking(reduction in the core shadow) shown in Figs. (2) and 3(a),Fig. 3(c) shows that the total average of M zin the ydirection [Mz,T(x)=1 420 nm/integraltext420 nm 0n mMz(x,y)dy]f o r0 /lessorequalslantx/lessorequalslant420 nm has almost the same value for all four cases. The reason isthat the energy exchange between the standing spin wave intheydirection and the propagating spin wave in the xdirection provides a nonlinear route for the magnon population to passthrough such cores. This can also be justiﬁed by the results in Figs. 3(b) and3(d) that show M z,sr(x) andMz,T(x)f o rt h e whole range of x, respectively, with similar amplitudes for all four cases. If such standing spin waves in the ydirection are omitted from the system by expanding the width of themagnetic structure ( W→∞ ), we can expect to observe higher reﬂection and shadow of the core with no cloaking mechanism,and more clear cloaking for both the ¯ γmechanism and ¯M s mechanism. The main reason behind the imperfection of the cloaking for both of the mechanisms [refer to Figs. 2(c) and 2(d)]i s that due to the existence of the waveform in the ydirection, the assumption of /vectorhd(/vectorρ)=T/vectorh/prime d(/vectorρ/prime), which was the basis for derivation of the material properties in the shell, is no longer perfectly satisﬁed. /vectorhd(/vectorρ)=T/vectorh/prime d(/vectorρ/prime) holds if only one of the /vectorq⊥/vectorM0or/vectorq/bardbl/vectorM0modes exists. Other reasons for the cloaking imperfections observed in Figs. 2(c) and2(d) are using cubic (aligned with the Cartesian axes) and limited number of cells inthe shell for the simulations. The difference between the resultsof the ¯ γmechanism and ¯M smechanism, corresponding to Figs. 2(c) and2(d), respectively, originates from the degree of vulnerability of the methods with respect to the discrepancy of /vectorhd(/vectorρ)=T/vectorh/prime d(/vectorρ/prime) from perfection due to mixing of the /vectorq⊥/vectorM0 and/vectorq/bardbl/vectorM0modes. For the ¯Msmechanism, mixing affects the dynamic dipolar ﬁeld [ /vectorh/prime d,m(/vectorρ/prime)] leading to distortion and inac- curacy of Eq. ( 6), while for the ¯ γmechanism, in addition to the distortion induced by /vectorh/prime d,m(/vectorρ/prime), Eq. ( 3) no longer holds exactly. Therefore, more distortion for the ¯ γmechanism in comparison with the ¯Msmechanism in the presence of mode mixing is expected, as inferred by comparing Figs. 2(c) and2(d). IV . SPIN METAMATERIALS FOR TRANSFORMATION MAGNONICS A. Physical feasibility of the ¯γmechanism Figures 4(a)–4(c) show the material properties [γrr/γϕϕ,Ms(/vectorρ), and αzz/α0] and the direction of the external ﬁeld /vectorHext(/vectorρ)f o rt h e ¯ γmechanism based on Eq. ( 3). Figures 4(e) and4(f)showMs,rr/Ms,ϕϕandαzz/α0for the ¯Ms mechanism based on Eqs. ( 5) and ( 6). Magnon cloaking by the ¯γmechanism or the ¯Msmechanism cannot be achieved in metallic ferromagnets as the large anisotropy in ¯ γor¯Msis not possible. In addition, the anisotropy and tuning range of ¯ γis limited in the magnetic molecules or magnetic semiconductorsfor realizing the ¯ γmechanism [refer to Fig. 4(a)][35,36]. However, high anisotropy and large tuning range of ¯ γis possible in quantum dot molecules (QDMs) [refer to the box FIG. 4. Spatial pattern of (a) γrr/γϕϕ,( b )Ms(/vectorρ) (color map) and /vectorHext(/vectorρ) (cone plot), and (c) αzz/α0for the ¯ γmechanism. (d) The schematic of the ensemble of quantum dot molecules is in the left section. The schematic of an individual quantum dot molecule, itsCartesian coordination, the electric ﬁeld ( E z,q), and magnetic ﬁeld (/vectorBq) are in the right section. Spatial pattern of (e) Ms,rr/Ms,ϕϕ,a n d (f)αzz/α0,f o rt h e ¯Msmechanism. (g) Example schematic of a rutile crystalline structure consisting of metal and oxygen sites, as well as an interstitial impurity and oxygen vacancy positions. The right panel is an example of the octahedral oxygen coordination of an interstitial impurity. /vectorEiis the external electric ﬁeld on the supercell iusing Cartesian coordinates. Gray spheres represent metal (e.g., Sn, Hf, Ti,etc.), red ones are O, green spheres are interstitial impurities (e.g., transition metals such as V), and the hollow sphere is an oxygen vacancy V O. (h) The schematic of the charge rings σ1(2,3,4)and the orbital moment directions /vectorv1(2,3,4), for the four possible independent oxygen octahedral coordination of the interstitial impurities. (i) The schematic of the interaction of two adjacent interstitial impurity sites(AandB) with the nearby V O, the resulting charge ring hybridization, and orbital moments. /vectorvA,/vectorvB,a n d/vectorvTare the orbital moments of site A, site B, and the hybridization, respectively. It is assumed that VO is closer to the site Athan to site B. In the right panel, the orbital moment vectors are shown with the same origin. In (a–c) and (e–f), the white regions in the center are hollow and no value is assigned tothem. in Fig. 4(d) for a stacked quantum dot molecule schematic] [37–40]. Spin states in quantum dots (QDs) or QDMs are the main candidates for quantum computing [ 41]. It has been demonstrated theoretically and experimentally that the Lande 104418-5ELY ASI, BHATIA, QIU, AND Y ANG PHYSICAL REVIEW B 93, 104418 (2016) gfactor or ¯ γin our notation, can be tuned in a large range (including zero crossing) in quantum wells (QWs), QDs, andQDMs with an electric ﬁeld. However, for QDMs, the effectof electric ﬁeld is much richer on tuning both the amplitudeand the anisotropy of the hole-spin Lande gfactor. There are rich crossings and anticrossings for the ground, the excitonic,and the charged excitonic states [ 37–39,42–48]. The crossings and anticrossings in QDMs happen due to bonding andantibonding of electron and hole wave functions betweenthe two QDs in a typical stacked QDM [refer to Fig. 4(d)] which occur by changing the electric ﬁeld [ 37,39,42]. In the right part of Fig. 4(d), a schematic of a QDM, its respective Cartesian coordination, the applied electric ﬁeld in the z direction ( E z,q, where qis the number of the QDM in the QDM ensemble), and the magnetic ﬁeld /vectorBqare depicted. It has been demonstrated theoretically that for a QDM in Fig. 4(d), the axes of the Lande gfactor ( ¯ γ) ellipsoid are along the zdirection, ˆxq+ˆyq, and ˆxq−ˆyq(xqandyq are the local Cartesian directions for the QDM number q). Based on this information about QDMs, we propose utilizingthe ¯γmechanism for an ensemble of QDMs as depicted schematically in the left part of Fig. 4(d). If we assume (ˆx q+ˆyq)/bardblˆrand ( ˆxq−ˆyq)/bardblˆϕ,t h ev a l u eo f γzzcan be tuned for more than 100%, while γrrandγϕϕcan be tuned for up to 800% with varying Ez,q[39]. However, γrrandγzzare spatially constant in the shell, while varying Ez,qfor tuning γϕϕwill change γrrandγzzas well. To overcome this issue, we can utilize the local angle ( θq)o f ˆxq+ˆyqwith respect to ˆr as another variable. In addition, the application of a spatiallyfunctionalized strain (adjacent piezoelectric layers) or dopingcan be used as other tuning factors for achieving the desired ¯ γ at the position of QDM number q[refer to Fig. 4(a)]. To achieve the required M s(/vectorρ) conﬁguration for the ¯ γmechanism [refer to Fig.4(b)], the density of the QDMs in the cloaking shell should be spatially functionalized. However, the inhomogeneousdistribution of QDMs in the shell causes inhomogeneityof distance between the QDMs which directly affects therespective exchange mechanisms. To compensate for thiseffect, a spatially functionalized electric ﬁeld can be appliedin the semiconductor regions between the QDMs to tune theexchange strength [ 49,50]. The spin lifetime in QDs can be up to the order of µs[51], and due to the atomiclike behavior of QDs, the phenomenological Gilbert damping and the requiredα zz/α0conﬁguration [refer to Fig. 4(c)] can be ignored. B. Physical feasibility of the ¯Msmechanism There are theoretical and experimental demonstrations for anisotropies in both the moment and exchange interaction inmagnetic semiconductors [ 52–62]. Those anisotropies stem from the spin-orbit interactions (SOIs) in the materials that lackinversion symmetry. In bulk semiconductors with wurtzite orzinc blende crystalline structures, the antisymmetric part of theanisotropic exchange of the localized electrons is dominatedby Dzyaloshinskii-Moriya interaction (DMI) [ 63,64] which is the ﬁrst-order perturbation in SOI of Rashba [ 65] and Dyakonov-Kachorovskii [ 66] types. There have been exper- imental demonstrations of anisotropic exchange by showinganisotropic dephasing in bulk GaN and impurity-bound elec-trons in n-doped ZnO [ 54,55].There have been theoretical and experimental demonstra- tions of magnetic orderings in semiconductors [ 52,56,60,67– 83]. Such magnetic orderings have been achieved due to the presence of carrier doping, cation vacancy, cation substitu-tion, anion vacancy, anion substitution, interstitial impurities,structural strain, and the combination of them. Although awide range of doping and defect gives rise to local spins ororbital moments, not all of them form a long-range magneticorder. The exchange interaction between local moments isgoverned by several mechanisms, such as double exchangeand superexchange. In addition to large magnetic orderings,the anisotropy in moments has been demonstrated in severaloxides such as substituted ZnO, V-doped SnO 2,H f O 2, and TiO 2,a sw e l la sL i 2(Li1−xFex)N [52,53,56,60,69,83,84]. The anisotropic moment arises due to lifting the degeneracyin orbital interactions, and mixing of molecular orbitalssurrounding the point defects induced by oxygen vacancies,cation vacancies and interstitial or substitution impurities.Molecular orbitals surrounding the point defects mix withthe nearest neighbors and next-nearest neighbors (sourceof magnetic ordering), enabling the possibility of differentanisotropy patterns based on the respective position of theimpurities (depending on the SOI strength of the defect). In order to be more speciﬁc, we propose a system of interstitial impurities and oxygen vacancies in a metal oxide.Figure 4(g) shows a 2 ×2×1 supercell of rutile crystalline structure consisting of metal sites and oxygen sites, hosting two interstitial impurities and an oxygen vacancy, for example.The surrounding oxygen octahedral of the interstitial imposesa crystal ﬁeld on the impurity and possibly splits the degeneratebands based on the symmetry rules. If the bonding molecularorbitals induced by impurities are ﬁlled with carriers, anorbital moment can be generated. In the presence of spin-orbit coupling, the impurity induced spin moment alignswith the orbital moment and if an exchange mechanismexists, both the orbital moment and spin moment can giverise to a macroscopic ferromagnetic order. The presence ofoxygen vacancy ( V O) provides electrons, and if its defect state overlaps with the impurity induced bands, there couldbe both orbital moments and long-range exchange interaction[52,56,60,68,73,75,76,79,80,82]. Therefore, it is important to choose the host and interstitial metals in order to have therequired interactions. The amplitude and the direction of theelectric ﬁeld /vectorE ion the supercell ican determine the respective conﬁguration of the impurities and defects. In the rutile structure, there are four independent octahedral sites for interstitial impurities as indicated in Fig. 4(h). Figure 4(h) shows a simpliﬁed demonstration of the molecular orbitals as charge rings σ1(2,3,4)that give rise to orbital moment vectors /vectorv1(2,3,4). The coexistence of orbital moment and high SOI results in anisotropic moments. In order to control theaxis of anisotropy, there is a need for at least two interactinginterstitial impurities. Figure 4(i) shows a possible route to control the anisotropy axis in the entire three dimensions.If the distance of the V Oto the interstitial impurity in site Ais less than its distance to site B, the carrier density in the charge ring of Awill be higher than that of B; therefore the orbital magnetization in Awill be higher than that in B (\|/vectorvA\|>\|/vectorvB\|). Hybridization of AandBcharge rings results in a net orbital moment /vectorvT. It can be inferred from Figs. 4(h) 104418-6CLOAKING THE MAGNONS PHYSICAL REVIEW B 93, 104418 (2016) and4(i)that the direction of /vectorvTcan be tuned by placing the impurities in different octahedral sites and placing the oxygen vacancy in different oxygen sites. Spatial functionalization ofthe temperature [ 56], impurity concentration, charge doping, and oxygen vacancy can be utilized for tuning the amplitude ofthe moment as is needed in addition to the anisotropy directionto achieve the desired ¯M s. The proposed method of transformation magnonics for spin-wave cloaking might be very challenging to be realizedexperimentally, as it requires spatially varying anisotropic ¯γor¯M swith precision in the nanometer scale. However, the proposed mechanisms may ﬁnd plausible applicationsin simpler transformation designs such as magnon lenses,concentrators, bending waveguides, and ultimately spin basedquantum computing. V . CONCLUSION We have proposed two transformation magnonics based ap- proaches for cloaking of a cylindrical nonmagnetic core. The ¯ γmechanism imposes an inhomogeneous anisotropic gyromag- netic tensor in the cloaking shell, while the ¯Msmechanism is based on inhomogeneous and anisotropic magnetic moments.We show that the wave front of the incident spin wave remainsinvariant after propagating through the shell for both themechanisms, indicating that the nonmagnetic cylindrical corehas been invisible towards the incident magnons. We discussthe feasibility of the ¯ γmechanism in the ensemble of quantum dot molecules. We also propose functionalized defects inmagnetic oxides for the feasibility of the ¯M smechanism. The reported design mechanism of transformation magnonicsfor manipulating magnons in magnetic semiconductors orquantum dot ensembles paves an alternative way for realizingadvanced functionalities such as magnonic cloaking, lensing,and concentrations, etc. ACKNOWLEDGMENTS This work was supported by the Singapore Ministry of Education Academic Research Fund Tier 1 (Grant No. R-263-000-A46-112). APPENDIX For the ¯ γmechanism, T−1/vector/Pi1(/vectorρ/prime)=/vector/Pi1/primeresults in three coupled equations, which with a straightforward algebraic investigation determine the invariant conditions for different spatial coordinates of /vectorH/prime 0(/vectorρ/prime) and /vectorM/prime 0(/vectorρ/prime) based on the matrix components of T. Such invariance is achieved using the ¯ γ/prime(/vectorρ/prime) tensor as a degree of freedom, as shown in Eq. ( 3). To be more speciﬁc, the T−1/vector/Pi1(/vectorρ/prime)=/vector/Pi1/primein the zdirection leads to [note that /vectorhd(/vectorρ)=T/vectorh/prime d(/vectorρ/prime) and /vectorMd(/vectorρ)=T/vectorM/prime d(/vectorρ/prime)] γ0 Tzz[(M0,rTϕϕh/prime d,ϕ−M0,ϕTrrh/prime d,r)+(TrrM/prime d,rH0,ϕ−TϕϕM/prime d,ϕH0,r)]ˆz =γ/prime zz[(M/prime 0,rh/prime d,ϕ−M/prime 0,ϕh/prime d,r)+(M/prime d,rH/prime 0,ϕ−M/prime d,ϕH/prime 0,r)]ˆz. (A1) WithTzz=1, Eq. ( A1) can be satisﬁed if γ/prime zz=γ0,H/prime 0,ϕ(/vectorρ/prime)=TrrH0,ϕ(/vectorρ/prime),M/prime 0,ϕ(/vectorρ/prime)=TrrM0,ϕ(/vectorρ/prime),H/prime 0,r(/vectorρ/prime)=TϕϕH0,r(/vectorρ/prime), andM/prime 0,r(/vectorρ/prime)=TϕϕM0,r(/vectorρ/prime). In the presence of the latter conditions and noting that M0,z(/vectorρ/prime)=0 andH0,z(/vectorρ/prime)=0, the invariance ofT−1/vector/Pi1(/vectorρ/prime)=/vector/Pi1/primein the ϕandrdirections can be achieved if γ0 Trr/bracketleftbigg/parenleftbigg −M/prime 0,ϕ TrrTzzh/prime d,z/parenrightbigg +/parenleftbigg TzzM/prime d,zH/prime 0,ϕ Trr/parenrightbigg/bracketrightbigg ˆr=γ/prime rr[(−M/prime 0,ϕh/prime d,z)+(M/prime d,zH/prime 0,ϕ)]ˆr, γ0 Tϕϕ/bracketleftbigg/parenleftbiggM/prime 0,r TϕϕTzzh/prime d,z/parenrightbigg +/parenleftbigg −TzzM/prime d,zH/prime 0,r Tϕϕ/parenrightbigg/bracketrightbigg ˆϕ=γ/prime ϕϕ[(M/prime 0,rh/prime d,z)+(M/prime d,zH/prime 0,r)] ˆϕ. (A2) Equation ( A2) can be satisﬁed if γ/prime rr(/vectorρ/prime)=γ01 T2rrandγ/prime ϕϕ(/vectorρ/prime)=γ01 T2ϕϕ, respectively. By using the invariance conditions that satisfy T−1/vector/Pi1(/vectorρ/prime)=/vector/Pi1/prime, the constraints on the components of the Gilbert damping tensor ¯ α/primeare determined by considering T−1/vector/Omega1(/vectorρ/prime)=/vector/Omega1/primewith the following similar procedures [see Eq. ( 3)f o rt h e ¯ α/primecomponents]. For the ¯Msmechanism, /vector/Pi1(/vectorρ/prime)=/vector/Pi1/primeshould be satisﬁed instead of T−1/vector/Pi1(/vectorρ/prime)=/vector/Pi1/prime, because the assumption in Eq. ( 5)f o rt h e ¯M/prime stensor renders the left part of Eq. ( 2) invariant. To satisfy /vector/Pi1(/vectorρ/prime)=/vector/Pi1/prime, the condition of Eq. ( 5) is only required, and the values of the bias ﬁeld and the static part of the magnetization in the prime and the physical space remain equivalent, /vectorH/prime 0(/vectorρ/prime)=/vectorH0(/vectorρ/prime) and/vectorM/prime 0(/vectorρ/prime)=/vectorM0(/vectorρ/prime). 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PhysRevB.77.134407.pdf	Electron transport driven by nonequilibrium magnetic textures Yaroslav Tserkovnyak and Matthew Mecklenburg Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA /H20849Received 25 January 2008; published 2 April 2008 /H20850 Spin-polarized electron transport driven by inhomogeneous magnetic dynamics is discussed in the limit of a large exchange coupling. Electron spins rigidly following the time-dependent magnetic proﬁle experiencespin-dependent ﬁctitious electric and magnetic ﬁelds. We show that the electric ﬁeld acquires important cor-rections due to spin dephasing, when one relaxes the spin-projection approximation. Furthermore, spin-ﬂipscattering between the spin bands needs to be taken into account in order to calculate voltages and spinaccumulations induced by the magnetic dynamics. A phenomenological approach based on the Onsager reci-procity principle is developed, which allows us to capture the effect of spin dephasing and make a connectionto the well studied problem of current-driven magnetic dynamics. A number of results that recently appeared inthe literature are related and generalized. DOI: 10.1103/PhysRevB.77.134407 PACS number /H20849s/H20850: 72.15.Gd, 72.25.Ba, 75.47. /H11002m, 75.75. /H11001a Interest in magnetic heterostructures,1which was initially fueled by the discovery of the giant magnetoresistance and, adecade later, by the current-induced switching in spin valvesand related systems, has more recently spilled over intocurrent-driven phenomena in magnetic bulk, individual mag-netic ﬁlms, and nanowires. 2Particular attention was given to the problems of current-driven Doppler shift of spin waves,magnetic instabilities, and domain-wall motion. The latterhas also enjoyed a very vibrant experimental activity, whichis in part motivated by a promising application potential inspintronics. The past year 3–6saw a revival of interest in the inverse effect of electromotive forces induced by the time- dependent magnetization, which were previously studied invarious physical contexts /H20849see, e.g., Refs. 7–9/H20850. In this paper, we will exploit the reciprocal relation between the two phe-nomena, which will allow us to understand important spin-dephasing corrections to the electromotive force. Such cor-rections were ﬁrst mentioned in Ref. 4and the Onsager principle in the present context was invoked in Ref. 5. Ref- erence 3reported the magnetically induced electromotive forces as a manifestation of the position-dependent Berry-phase accumulation, and Ref. 6considered these forces act- ing on semiclassical wave packet motion, mainly reproduc-ing results from Ref. 9. For completeness, it should also be mentioned that an earlier paper 10already contains some seminal phenomenological insights related to the problem ofthe electric response to the magnetic domain-wall dynamics. In the following, we start by recalling how most of the results recently discussed in the literature can be captured byan SU /H208492/H20850gauge transformation together with the projection of spins on the magnetic direction. 9The corrections due to the remaining transverse spin dynamics are governed by spindephasing, which have already been studied for the recipro-cal process of current-driven magnetic dynamics, 2and can be translated to the current problem by the Onsager principle.We will develop a phenomenological framework, that willallow us to relate and generalize the more specialized cases,which were recently studied using different methods. 3–6Fi- nally, we will derive spin-charge diffusion equations, ac-counting for spin-ﬂip scattering and respecting local chargeneutrality, which is necessary in order to relate the micro-scopic electromotive forces to measurable quantities, such as an induced voltage and spin accumulation. Most of our analysis will pertain to the following time- dependent Hamiltonian: H/H20849t/H20850=p 2 2m+/H9004xc 2/H9268ˆ·m/H20849r,t/H20850+Vc/H20849r,t/H20850+H/H9268. /H208491/H20850 Here, H/H9268is the contribution due to spin-relaxation processes, which will be characterized by a Bloch-type T1spin ﬂipping and T2spin dephasing, and Vc/H20849r,t/H20850stands for a Hartree charging potential, which will be taken into account onlyinsofar as enforcing local charge neutrality. /H9004 xcis the ferro- magnetic exchange band splitting, /H9268ˆis the vector of Pauli spin matrices, and mstands for the local magnetization di- rection unit vector, so that the magnetization is given by M =Mm. The exchange ﬁeld /H9004xcmmay in practice be provided by localized magnetic dorbitals /H20849as in the so-called s-d model /H20850or it may self-consistently be governed by the itiner- ant electron spin density /H20849as in the Stoner model or local spin-density approximation /H20850.2We will ﬁrst perform a micro- scopic calculation for the idealized Hamiltonian Eq. /H208491/H20850ne- glecting H/H9268and subsequently utilize the Onsager theorem to capture the spin-dephasing corrections. The spin-ﬂip scatter-ing will be included phenomenologically in the ﬁnal diffu-sion equation. By disregarding H /H9268, we can perform an SU /H208492/H20850gauge transformation by rotating mto point along the zaxis for all randt.9,11This is conveniently achieved by the Hermitian spin-rotation matrix Uˆ=/H9268ˆ·n/H20849such that Uˆ=Uˆ†=Uˆ−1/H20850, where nis the unit vector n/H11008m+z. It is easy to see that Uˆ/H20849/H9268ˆ·m/H20850Uˆ=/H9268ˆz/H20849since Uˆcorresponds to a /H9266-angle spin rota- tion around n/H20850. By applying this gauge transformation to the spinor wave function, we get for the transformed Hamil-tonian we get H /H11032/H20849t/H20850=1 2m/H20849p−Â/H208502+Vˆ+/H9004xc 2/H9268ˆz+Vc, /H208492/H20850 where the SU /H208492/H20850vector potential is given byPHYSICAL REVIEW B 77, 134407 /H208492008 /H20850 1098-0121/2008/77 /H2084913/H20850/134407 /H208494/H20850 ©2008 The American Physical Society 134407-1Aˆi=i/H6036Uˆ/H11612iUˆ=−/H6036/H9268ˆ·/H20849n/H11003/H11612in/H20850 and the SU /H208492/H20850ordinary potential is Vˆ=−i/H6036Uˆ/H11509tUˆ=/H6036/H9268ˆ·/H20849n /H11003/H11509tn/H20850/H20849setting the particle charge and speed of light to unity /H20850.p=−i/H6036/H11633is the canonical momentum. If the exchange ﬁeld /H9004xc is large and the magnetic texture is sufﬁciently smooth and slow, we can project the ﬁctitious potentials on the zaxis as Vˆ→V/H9268ˆz, where V =/H6036z·/H20849n/H11003/H11509tn/H20850=/H6036sin2/H20849/H9258/2/H20850/H11509t/H9278, and similarly for the vector potential, Ai=−/H6036z·/H20849n/H11003/H11612in/H20850=−/H6036sin2/H20849/H9258/2/H20850/H11612i/H9278, where /H20849/H9258,/H9278/H20850 are the spherical angles parametrizing m. We thus get for the effective electric ﬁeld4,6,9 E=−/H11509tA−/H11633V=/H6036 2sin/H9258/H20851/H20849/H11509t/H9258/H20850/H20849/H11633/H9278/H20850−/H20849/H11509t/H9278/H20850/H20849/H11633/H9258/H20850/H20852,/H208493/H20850 or, written in the explicitly spin rotationally invariant form, Ei=/H6036 2m·/H20849/H11509tm/H11003/H11612im/H20850. /H208494/H20850 The effective magnetic ﬁeld is6,9,12 B=/H11633/H11003A=/H6036 2sin/H9258/H20849/H11633/H9278/H20850/H11003/H20849/H11633/H9258/H20850=/H6036 4/H9280ijkmi/H20849/H11633mk/H11003/H11633mj/H20850, /H208495/H20850 where /H9280ijkis the antisymmetric Levi–Civita tensor, and a summation over repeated indices is implied. The total forceon a spin- ↑/H20849↓/H20850electron moving with velocity vis thus given by F ↑/H20849↓/H20850=/H11006/H20849E+v/H11003B/H20850. /H208496/H20850 Equations /H208493/H20850and /H208495/H20850were ﬁrst derived in Ref. 9and recently rederived within a semiclassical wave-packetanalysis. 6The gauge-transformation-based approach9puts the result into a broader perspective, allowing us, for ex-ample, to consider the effect of the magnetic ﬁeld /H20851Eq. /H208495/H20850/H20852 on the quantum transport corrections, such as a weak local-ization, as well as to include spin-independent electron-electron interactions, which would not modify ﬁctitiousﬁelds /H20851Eqs. /H208493/H20850and /H208495/H20850/H20852. Note, in particular, that the magnetic ﬁeld /H20851Eq. /H208495/H20850/H20852can in practice be quite large: For example, for a static magnetic variation on the scale of 10 nm, the corre-sponding ﬁctitious ﬁeld is of the order of 10 T. It is mostconvenient to estimate the strength of the electric ﬁeld /H20851Eq. /H208493/H20850/H20852by the characteristic voltage it induces over a region where the magnetization direction ﬂips its direction: /H6036 /H9275/e, where /H9275is the frequency of the magnetic dynamics. In the following, we will concentrate on the semiclassical spin andcharge diffusion generated by the effective electric ﬁeld E.I n order to make a closer connection to the experimentally rel-evant quantities, we will need to take into account spin re-laxation and also enforce local charge neutrality for electrondiffusion. The role of spin relaxation can be twofold. First of all, spin accumulation, which will generally be generated by thespin-dependent force /H20851Eq. /H208496/H20850/H20852, will relax, characterized by the longitudinal spin-ﬂip time T 1. There is also another more subtle effect, which is due to the dephasing of electron spinsfollowing a dynamic magnetic proﬁle, since the exchange ﬁeld/H9004xcis not inﬁnite and spins do not perfectly align with the local magnetization. Hence, there is generally a ﬁnitespin misalignment, which dephases with a characteristic timeT 2. This gives corrections to the results obtained by a rigid projection of spins on the local magnetization direction. Wewill see that such corrections turn out to be important for thecurrents generated by magnetic dynamics, in the same sensethat analogous corrections are crucial for understandingcurrent-induced magnetic motion. 2 Let us now take a step aside by recalling the general ex- pression for the dynamics of an isotropic ferromagnet wellbelow the Curie temperature: 2 /H11509tM=−/H9253M/H11003Heff+/H9251 MM/H11003/H11509tM +/H6036/H20849/H9268↑−/H9268↓/H20850 2S/H20849/H11612i/H9262/H20850/H208731−/H9252 MM/H11003/H20874/H11612iM, /H208497/H20850 which is valid for spatially smooth magnetic proﬁles /H20849the so-called adiabatic approximation /H20850and weak currents. Here, /H9251is the Gilbert damping constant, /H9252is another dimension- less phenomenological parameter whose physical meaningwill be discussed later, /H9262is the electrochemical potential, /H9268s is the spin- sconductivity /H20849along the local magnetization di- rection m/H20850relating particle currents to /H11612/H9262,Sis the equilib- rium spin density of the ferromagnet along m, and/H9253=M /Sis the gyromagnetic ratio. Recall that the effective ﬁeld Heffis the quantity deﬁned to be thermodynamically conjugate to the magnetization: Heff=/H11509MF/H20849note the sign difference from the standard deﬁnition /H20850, where Fis the free energy and /H11509M stands for the functional derivative. The other thermody- namic variable we will consider is the electron density /H9267/H20849r,t/H20850, whose thermodynamically conjugate counterpart is the electrochemical potential /H9262=/H11509/H9267F. Suppose we perturb the electron density with respect to an equilibrium with some static magnetic texture and uniformchemical potential and consider the ensuing magnetic re-sponse. Equation /H208497/H20850then describes the nonequilibrium cou- pling of the magnetization dynamics to the electron density’sthermodynamic conjugate, which is slightly out of equilib-rium. The Onsager reciprocity principle 13allows us to imme- diately write down the response of the electron density to asmall modulation of the effective ﬁeld H effwith respect to an equilibrium conﬁguration. To simplify things, let us for amoment disregard Gilbert damping /H9251in Eq. /H208497/H20850and return to include it later on. An electric response to a magnetic pertur-bation then becomes 14 /H11509t/H9267=−/H9253/H6036/H20849/H9268↑−/H9268↓/H20850 2/H11612i/H20853Heff·/H20851/H208491+/H9252m/H11003/H20850/H11612im/H20852/H20854. /H208498/H20850 By comparing Eq. /H208498/H20850with the continuity equation /H11509t/H9267=−/H11612iji, we can identify the particle current as ji=/H9253/H6036/H20849/H9268↑−/H9268↓/H20850 2Heff·/H20851/H208491+/H9252m/H11003/H20850/H11612im/H20852. /H208499/H20850 Since for each spin species, js=/H9268sFs, where Fsis the effec- tive force, we ﬁnally get for the latter F↑,↓=/H11006F,15whereYAROSLAV TSERKOVNYAK AND MATTHEW MECKLENBURG PHYSICAL REVIEW B 77, 134407 /H208492008 /H20850 134407-2Fi=/H6036 2/H20849m/H11003/H11509tm/H20850/H20851/H208491+/H9252m/H11003/H20850/H11612im/H20852 =/H6036 2/H20851m·/H20849/H11509tm/H11003/H11612im/H20850+/H9252/H20849/H11509tm·/H11612im/H20850/H20852, /H2084910/H20850 after inverting the magnetic equation of motion /H20851Eq. /H208497/H20850/H20852in order to express the effective ﬁeld Heffin terms of the mag- netization dynamics m/H20849r,t/H20850./H20849Note that since the currents themselves are now generated by the magnetization dynam-ics, we can neglect their backaction on the magnetic responsewhen inverting the equation of motion to express H effin terms of m, since it would give rise to higher-order terms. /H20850 Equation /H2084910/H20850is a key result of this paper. It is also easy to show that taking into account Gilbert damping /H9251has no con- sequences for the ﬁnal result /H20851Eq. /H2084910/H20850/H20852 /H20851after rewriting Eq. /H208497/H20850in the Landau–Lifshitz form, in order to eliminate the /H11509t term on the right-hand side and thus make the equation suit- able for the Onsager theorem /H20852. This is not surprising, since the physics of the Gilbert damping /H9251does not have to be related to the magnetization–particle-density coupling thatdetermines the force /H20851Eq. /H2084910/H20850/H20852. 2 Physically, the /H9252correction in Eq. /H2084910/H20850is related to a slight spin misalignment of electrons propagating through aninhomogeneous magnetic texture with the local direction ofthe magnetization m. In the limit of /H9004 xc→/H11009, this misalign- ment vanishes and so should /H9252, reducing the result, Eq. /H2084910/H20850, to Eq. /H208494/H20850. Indeed, a microscopic derivation of Eq. /H208497/H20850shows /H9252/H11011/H6036/T2/H9004xc, where T2is the characteristic transverse spin relaxation time.2The/H9252term in Eq. /H2084910/H20850can thus be viewed as a correction to the topological structure of the electrontransport rigidly projected on the magnetic texture, due to theremaining transverse spin dynamics and dephasing. Such a /H9252 correction was ﬁrst reported in Ref. 4, which used a very different and more technical language and did not beneﬁtfrom the reciprocity relation with the current-driven mag-netic dynamics /H20851Eq. /H208497/H20850/H20852. Our phenomenological derivation of Eq. /H2084910/H20850based on the Onsager theorem provides a much simpler framework for studying these subtle spin-dephasingeffects. Let us now discuss the measurable consequences of Eq. /H2084910/H20850in two simple scenarios sketched in Fig. 1. Consider a nontrivial one-dimensional magnetic proﬁle along the xaxis, such as a magnetic domain wall in a narrow wire, with neg-ligible transverse spin inhomogeneities. First, let us look at asteady rotation of the entire one-dimensional texture aroundthexaxis with a constant frequency /H9275. Then, /H11509tm=/H9275x/H11003m and /H9004V=−/H20885dxF x=−/H6036/H9275 2x/H20885/H20849dm+/H9252m/H11003dm/H20850. /H2084911/H20850 In the absence of spin dephasing /H9252, this result can be easily understood by transforming into the rotating frame of refer-ence: By the Larmor theorem, this corresponds to a ﬁctitious ﬁeld along the xaxis: H /H11032=−/H20849/H6036/H9275/2/H20850/H9268ˆx. For spins up /H20849down /H20850projected on the local magnetization direction, this corresponds to the potential V=/H11007/H20849/H6036/H9275/2/H20850x·m.I ti s equally straightforward to interpret this result in terms of therate of the Berry-phase accumulation by spins adiabaticallyfollowing the steady exchange ﬁeld precession, 3,16which is proportional to the position-dependent solid angle enclosedby spin precession. The /H9252term in Eq. /H2084911/H20850gives a correction to these idealistic considerations, which depends on the ge-ometry of the magnetic texture. Next, we consider the volt-age induced by a rigid translation of a one-dimensional mag-netic texture m/H20849x− vt/H20850along the xdirection with velocity v. The corresponding force Fx=−/H6036 2/H9252v/H20849/H11509xm/H208502/H2084912/H20850 is then entirely determined by the /H9252term, which drags spins down along the direction of the magnetic texture motion andspins up in the opposite direction. This is analogous to thecurrent-driven domain-wall velocity in one dimension,which, for smooth walls and low currents, is proportional to /H9252.2 Finally, we need to include spin-ﬂip relaxation time T1 and derive spin-charge diffusion equations, enforcing local charge neutrality. Assuming diffusive transport, the force/H20851Eq. /H2084910/H20850/H20852can now be added as a contribution to the gradient of the effective electrochemical potential. The diffusionequation for spin- sparticles is then given by /H20849 /H11509t−Ds/H116122/H20850/H9267s+/H9268s/H20849s/H11633·F−/H116122Vc/H20850=/H9267−s /H9270−s−/H9267s /H9270s, /H2084913/H20850 where /H9267sis the nonequilibrium /H20849spin- s/H20850particle density, Dsis the diffusion coefﬁcient, and /H9270sis the spin-ﬂip time. Recall that the conductivity is related to the density of states Nsby the Einstein’s relation: /H9268s=NsDs.Vcis the electric potential, which has to be found self-consistently by enforcing localcharge neutrality. Note that the equilibrium considerationsrequire that /H9270s//H9270−s=Ns/N−s. We should also stress that the force /H20851Eq. /H2084910/H20850/H20852may have a ﬁnite curl, so that we cannot generally describe it by a ﬁctitious potential. After straight-forward manipulations, we can decouple the diffusion equa- FIG. 1. /H20849Color online /H20850Two simple scenarios for voltage genera- tion by the magnetic dynamics: /H208491/H20850Magnetic texture m/H20849x,t/H20850, such as a domain wall along the xaxis, is steadily rotating around the x axis and /H208492/H20850the same texture rigidly sliding along the xaxis. In the former case, the force Fxacting on electrons is proportional to the frequency of rotation /H9275, with the dominant term having a purely geometric meaning in terms of the position-dependent Berry-phaseaccumulation rate. /H20849An alternative physical explanation can also be provided by transforming to the rotating frame of reference andapplying the Larmor theorem. /H20850In the case of the sliding dynamics, the leading contribution to the magnetically induced force is pro-portional to the spin-dephasing rate /H20849parametrized by /H9252/H20850and the “curvature” of the texture proﬁle /H20849/H11509xm/H208502.ELECTRON TRANSPORT DRIVEN BY NONEQUILIBRIUM … PHYSICAL REVIEW B 77, 134407 /H208492008 /H20850 134407-3tion for the spin accumulation /H9262/H9268/H20849deﬁned as the difference between the spin-up and spin-down electrochemical poten-tials, divided by 2 /H20850from the average electrochemical poten- tial /H9262as follows: /H20849/H11509t+/H9270−1−D/H116122/H20850/H9262/H9268=−D/H11633·F, −/H116122/H9262=P/H20849/H116122/H9262/H9268−/H11633·F/H20850. /H2084914/H20850 Here, P=/H20849/H9268↑−/H9268↓/H20850//H20849/H9268↑+/H9268↓/H20850is the conductivity polarization, D=/H20849D↑+D↓/H20850/2−P/H20849D↑−D↓/H20850/2 is the effective spin-diffusion constant, and /H9270−1=/H9270↑−1+/H9270↓−1is the characteristic T1−1rate for spin ﬂipping. If /H11633/H11003F=0, we can integrate the second equa- tion to express the electrochemical potential gradient interms of the force Fand the spin accumulation gradient as follows:/H11633 /H9262=P/H20849F−/H11633/H9262/H9268/H20850, /H2084915/H20850 assuming the appropriate boundary conditions. According to Eqs. /H2084914/H20850, the spin accumulation decays in the absence of the force Fon the scale of the spin-diffusion length /H9261sd=/H20881D/H9270. Away from the dynamic magnetic texture /H20849on the scale of /H9261sd/H20850, the generated electrochemical potential /H20851Eq. /H2084915/H20850/H20852will then be determined simply by integrating the force F.I n general, however, especially when /H11633/H11003F/HS110050, one has to revert to Eqs. /H2084914/H20850. In summary, we theoretically studied electron transport generated by a dynamic magnetization texture. We repro-duced and generalized the results that recently appeared inliterature, 3,4,6revealing an intricate connection with the theory of the current-induced magnetization dynamics.2We expect that in practice it is considerably simpler to solve thisreciprocal problem, especially for including subtle correc-tions to the topological Berry-phase structure of spins as-sumed to rigidly follow the time-dependent magnetic proﬁle. 1D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 /H208492007 /H20850, and references therein. 2Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, J. Magn. Magn. Mater. 320, 1282 /H208492008 /H20850, and references therein. 3S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601 /H208492007 /H20850. 4R. A. Duine, Phys. Rev. B 77, 014409 /H208492008 /H20850. 5W. M. Saslow, Phys. Rev. B 76, 184434 /H208492007 /H20850. 6S. A. Yang, D. Xiao, and Q. Niu, arXiv:0709.1117 /H20849unpublished /H20850. 7A. Stern, Phys. Rev. Lett. 68, 1022 /H208491992 /H20850. 8M. 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Lifshitz, Statistical Physics, Part 1 , Course of Theoretical Physics Vol. 5, 3rd ed. /H20849Pergamon, Ox- ford, 1980 /H20850. 14Note that the Onsager reciprocity relations are somewhat special in the present case: since one of the thermodynamicquantities /H20849namely, the magnetization /H20850is odd under time rever- sal, the matrix that couples the rates of the relaxation of theinvolved quantities to their thermodynamic conjugates is anti-symmetric, which is subject also to changing the sign of themagnetic ﬁeld and the equilibrium magnetization /H20849Ref. 13/H20850. Hence, the overall minus sign in Eq. /H208498/H20850and the sign change in front of /H9252. 15Strictly speaking, the analysis based on Eq. /H208497/H20850can only capture the total particle current /H20851Eq. /H208499/H20850/H20852. In order to rigorously calcu- late the spin-resolved forces, we would have to explicitly takeinto account one more thermodynamic variable, namely, thenonequilibrium spin accumulation /H20849which is proportional to the chemical potential mismatch between the up and down electronsalong the local magnetization /H20850and consider its action on the magnetic dynamics. The latter program would push us too muchoff track, and we choose not to pursue it here. We only wish tonote that the /H9252term describing the spin-dephasing correction to the ﬁctitious ﬁeld /H20851Eq. /H208494/H20850/H20852may in general become different for the two spin species when the ferromagnetic exchange energy iscomparable to the Fermi energy. 16M. V. Berry, Proc. R. Soc. London, Ser. A 392,4 5 /H208491984 /H20850.YAROSLAV TSERKOVNYAK AND MATTHEW MECKLENBURG PHYSICAL REVIEW B 77, 134407 /H208492008 /H20850 134407-4
PhysRevB.81.184523.pdf	Critical tunneling currents in quantum Hall superﬂuids: Pseudospin-transfer torque theory Jung-Jung Su1,2and Allan H. MacDonald1 1Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA 2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA /H20849Received 17 January 2010; revised manuscript received 22 April 2010; published 20 May 2010 /H20850 At total ﬁlling factor /H9263=1 quantum Hall bilayers can have an ordered ground state with spontaneous interlayer phase coherence. The ordered state is signaled experimentally by dramatically enhanced interlayertunnel conductances at low-bias voltages; at larger bias voltages interlayer currents are similar to those of thedisordered state. We associate this change in behavior with the existence of a critical current beyond whichstatic interlayer phase differences cannot be maintained, and examine the dependence of this critical current onsample geometry, phase stiffness, and the coherent tunneling energy density. Our analysis is based in part onanalogies between coherent bilayer behavior and spin-transfer torque physics in metallic ferromagnets. Com-parison with recent experiments suggests that disorder can dramatically suppress critical currents. DOI: 10.1103/PhysRevB.81.184523 PACS number /H20849s/H20850: 71.35. /H11002y, 71.10. /H11002w, 73.21. /H11002b, 73.22.Gk I. INTRODUCTION At Landau level ﬁlling factor /H9263=1 bilayer two- dimensional /H208492D/H20850electron systems in the quantum Hall re- gime can have broken-symmetry ground states1–4with spon- taneous interlayer phase coherence. These ordered states canbe viewed as excitonic superﬂuids 5,6or as XY pseudospin ferromagnets7in which the pseudospin is formed from the two-valued which layer quantum degree of freedom. The most robust experimental signature of these states, a vastlyenhanced interlayer tunnel conductance 8–11at small-bias voltages, is still poorly understood from a quantitative pointof view. Two types of ideas, which differ most essentially in how the bias voltage is introduced in the theory, have been ex-plored in an effort to understand the height and width of thetunnel conductance peak. In one approach 12–15the bias volt- age is introduced as an effective magnetic ﬁeld, uniformacross the bilayer, which induces pseudospin precession around the zˆaxis, driving the interlayer phase difference at a steady rate and inducing a purely oscillating interlayer cur-rent. When the microscopic interlayer tunneling amplitude istreated as a perturbation, thermal and disorder ﬂuctuations ofthe condensate are then responsible added to V3MOD begin for a ﬁnite dc conductance peak. We refer to this type oftheory below as the weak-coupling theory of the tunnelinganomaly. Weak-coupling theories predict 12–15splitting of the zero-voltage tunnel conductance peak into separate ﬁnitevoltage peaks in the presence of a magnetic ﬁeld componentparallel to the two-dimensional layers, an effect that is not 16 seen experimentally. The second type of transport theory17–21 is formulated in terms of local chemical potentials of fermi- onic quasiparticles which may be altered by the ordered-statecondensate but are still responsible for charge conduction. Inthis type of theory the tunnel conductance is ﬁnite even inthe absence of disorder and thermal ﬂuctuations becausecharge has to be driven between normal-metal source anddrain contacts. The resistance generally depends 18,21on how the fermionic degrees of freedom which carry charge be-tween leads are inﬂuenced by order parameter and disorder conﬁgurations and cannot be described in terms of conden-sate dynamics alone. In the second approach the width involtage of the conductance peak is simply equal to the prod-uct of this resistance and the maximum current betweensource and drain at which the order parameter can maintain atime-independent steady-state value. In this paper we expand on the second type of theory of the conductance peak, using pseudospin-transfer torqueideas 22,23borrowed from recent ferromagnetic-metal spin- tronics literature to model the inﬂuence of the transport cur-rent on the pseudospin magnetization. We ﬁnd that the criti-cal current depends, in general, on details of the samplegeometry and on how disorder and localization physics in-ﬂuence transport inside the system. Generally speaking,however, the critical current is proportional to system areaand to the interlayer tunneling amplitude when the conden-sate’s Josephson length is larger than the system perimeterand proportional to the system perimeter and to the squareroot of the interlayer tunneling amplitude when it is shorter. Our paper is organized as follows. In Sec. IIwe introduce the pseudospin-transfer torque theory of order-parameter dy-namics in a bilayer quantum Hall ferromagnet. We use this theory in Sec. IIIto discuss critical current values from a qualitative point of view. In Sec. IVwe report on a series of numerical studies which take into account the two-dimensional nature of the systems of interest and the edgedominated current paths typical of strong magnetic ﬁelds.The pseudospin-transfer torque theory enables us to assessthe inﬂuence of sample geometry on critical currents. Finallyin Sec. Vwe discuss the signiﬁcance of our ﬁndings in rela- tion to recent experiments. We ﬁnd that experimental criticalcurrents are several orders of magnitude smaller than theo-retical ones and argue that vortexlike disorder-induced pseu-dospin textures must be largely responsible for this discrep-ancy. We propose experimental studies which can test ourideas and thereby achieve progress toward a quantitativetheory of the spontaneous coherence tunneling anomaly.PHYSICAL REVIEW B 81, 184523 /H208492010 /H20850 1098-0121/2010/81 /H2084918/H20850/184523 /H2084911/H20850 ©2010 The American Physical Society 184523-1II. PSEUDOSPIN MAGNETISM AND THE LANDAU- LIFSHITZ-SLONCZEWSKI EQUATION In the lattice model24of/H9263=1 bilayer systems, the local which layer degree of freedom can be expressed as a pseu- dospin deﬁned by the operator S/H6023i=1 2/H20858 /H9268,/H9268/H11032ai,/H9268†/H9270/H6023/H9268,/H9268/H11032ai,/H9268/H11032, /H208491/H20850 where iis the site index, /H9268is the layer index, and /H9270/H6023is the Pauli matrix vector. The pseudospin Hamiltonian has theform 24 Hint=1 2/H20858 i,j/H208492Hi,j−Fi,jS/H20850SizSjz−Fi,jD/H20849SixSjx+SiySjy/H20850, /H208492/H20850 where Hi,j=/H20855i,/H9268;j,/H9268/H11032/H20841Vcol/H20841i,/H9268;j/H9268/H11032/H20856is the direct Coulomb interaction associated with the zˆpseudospin component /H20849i.e., with charge transfer between layers /H20850,Fi,jS =/H20855i,/H9268;j,/H9268/H20841Vcol/H20841i,/H9268;j/H9268/H20856is the exchange interaction between orbitals in the same layer, and Fi,jD=/H20855i,/H9268;j,/H9268¯/H20841Vcol/H20841i,/H9268;j/H9268¯/H20856is the exchange interaction between orbitals located in different layers. Since Hi,jis generally larger than Fi,jS, the classical ground state is an easy-plane pseudospin ferromagnet with a hard zˆaxis. In the limit of smooth textures the pseudospin energy functional has the form7 E/H20851m/H6023/H20852=/H20885d2r/H20877/H9252/H20849mz/H208502+1 2/H9267s/H20851/H20841/H11612/H6023mx/H208412+/H20841/H11612/H6023my/H208412/H20852−1 2/H9004tnmx/H20878, /H208493/H20850 where m/H6023=/H20853mx,my,mz/H20854is the local pseudospin direction. The parameters which appear in this expression are the aniso-tropy parameter /H9252/H110220, the pseudospin stiffness /H20849or equiva- lently the exciton superﬂuid density /H20850/H9267s, the splitting between symmetric and antisymmetric single-particle bilayer statesdue to interlayer tunneling /H9004 t, and the pseudospin density n. The mean-ﬁeld-theory pseudospin ferromagnet1consists of a full Landau level of electrons in identical phase coherentbilayer states. It follows that the mean-ﬁeld-theory pseu-dospin density nis equal to the full Landau-level density, /H208492 /H9266l2/H20850−1./H20849Here l=/H20849/H6036c/eB/H208501/2, where Bis the magnetic ﬁeld strength and lis the magnetic length. /H20850Mean-ﬁeld theory can also used7to ﬁnd explicit expressions for /H9267sand/H9252. In prac- tice the values of these three parameters are modiﬁed25by quantum and thermal ﬂuctuations. The fourth parameter /H9004tis exponentially sensitive to the tunnel barrier between layers.Parameters values can also be inﬂuenced by disorder onlength scales shorter than those on which this coarse-grainedcontinuum theory is applied; disorder on longer length scaleswould have to be treated explicitly as we mention in thediscussion section. The upshot is that the numerical values ofthe parameters in Eq. /H208493/H20850are usually not accurately known and likely vary substantially from sample to sample. It isworth noting that nmust vanish at ﬁnite temperatures when /H9004 t→0 since two-dimensional systems cannot support spon- taneous long-range phase order. Among all continuum modelparameters the value of /H9267s, which is typically /H1101110−4eV, is likely the most reliably known.Since the which layer pseudospin and the true electron spin have identical quantum-mechanical descriptions, we canborrow from the ferromagnetic-metal spintronicsliterature 22,23and use the Landau-Lifshitz-Slonczewski /H20849LLS /H20850equation to describe how the semiclassical pseudospin dynamics is inﬂuenced by a transport current dm/H6023 dt=m/H6023/H11003H/H6023eff−/H20849j/H6023·/H11509r/H6023/H20850m/H6023 n−/H9251/H20873m/H6023/H11003dm/H6023 dt/H20874. /H208494/H20850 /H20849j/H6023is the number current density for electrons. /H20850The second term on the right-hand side of Eq. /H208494/H20850captures the transport current effect. Its role in these equations is similar to the roleplayed by the leads in the pioneering analysis of coherentbilayer tunneling by Wen and Zee 17who argued that a term should be added to the global condensate equation of motionto account for the contribution of transport currents to thedifference in population between top and bottom layers.Equation /H208494/H20850describes how a transport current alters the con- densate equation of motion locally. Its justiﬁcation for thebilayer quantum Hall case is discussed in more detail below.The essential validity of this expression has been veriﬁed bycountless experiments. In the ﬁrst term on the right-hand side H /H6023eff=/H208492//H6036n/H20850/H20849/H9254E/H20851m/H6023/H20852//H9254m/H6023/H20850/H20849 5/H20850 describes precessional pseudospin dynamics in an effective magnetic ﬁeld which is deﬁned by the energy functional. Thethird term accounts for damping of the collective motion dueto coupling to its environment, in the present case the Fermisea of quasiparticles. The damping term in Eq. /H208494/H20850has the standard isotropic form used in the magnetism literature. Amicroscopic theory 26of damping in quantum Hall bilayers makes it clear that the damping is actually quite anisotropic.We return to this point below. Magnetic order in the metallic ferromagnets to which this equation is normally applied is extremely robust, justifyingthe assumption that the spin-density magnitude is essentiallyunchanged even when the system is driven from equilibriumby a transport current. The only relevant degree of freedomis the spin-density direction, whose dynamics is described byEq. /H208494/H20850. We expect that the pseudospin-transfer torque de- scription of bilayer quantum Hall systems will be most reli-able when the order is most ﬁrmly established, that is faraway from the phase boundary 27that separates ordered and disordered states. The transport-current /H20849Slonczewski23/H20850term in Eq. /H208494/H20850can be understood in several different ways. In the spintronicsliterature this term is normally motivated by an appeal tototal spin conservation and referred to as the spin-transfertorque. The idea is that when a ferromagnet’s spin-polarizedquasiparticles carry a transport current through a spatial re-gion with a noncollinear magnetization, they violate spinconservation. The collective magnetization must thereforecompensate by rotating at a constant rate which is propor-tional to the fermion drift velocity. The form we use for thepseudospin-transfer torque assumes that each component ofthe pseudospin current is locally equal to the number currenttimes the corresponding pseudospin direction cosine. Thisproperty does not hold locally in a microscopic theory, but isJUNG-JUNG SU AND ALLAN H. MACDONALD PHYSICAL REVIEW B 81, 184523 /H208492010 /H20850 184523-2expected to be valid in the smooth pseudospin texture limit we address. /H20849In metals an additional phenomenological fac- tor is required to account for the difference in drift velocitybetween majority and minority spin electrons. /H20850 Since pseudospin is not conserved in bilayer quantum Hall systems, as we can see from Eq. /H208492/H20850or Eq. /H208493/H20850, this argument does not apply directly. If we appeal to a mean-ﬁeld description of the pseudospin ferromagnet, however, wecan obtain the same result by the following argument. Mean-ﬁeld quasiparticles satisfy a single-particle Hamiltonian for aparticle in a magnetic ﬁeld which experiences both a scalarpotential and a pseudospin-dependent potential that can beinterpreted as a pseudospin effective magnetic ﬁeld. Follow-ing standard textbook derivations it is possible 28to derive an expression for the time dependence of the contribution of asingle quasiparticle to any component of the pseudospin den-sity. The expression contains a pseudospin precession termand an additional term which is the divergence of the currentof that component of pseudospin. Summing over all quasi-particle states we obtain a precession term that depends onthe conﬁguration of the order parameter, and hence on thepseudospin ﬁeld to which it gives rise, even in the absence ofa current. We obtain the additional current-driven term on theright-hand side of Eq. /H208494/H20850when the pseudospin currents are nonzero and space dependent. The additional term in theequation of motion can also be viewed 29as a consequence of an altered relationship between pseudospin-dependent effec-tive magnetic ﬁeld and pseudospin polarization direction forquasiparticles that carry a current. Equation /H208494/H20850should, in principle, also include a current-related damping term 30 which we ignore in the present paper. When quantum Hall bilayer pseudospin ferromagnets are tilted far from their easy plane, order tends31to be destroyed. For that reason we are often most interested in the limit inwhich m zis much less than 1. It is therefore convenient to express the pseudospin direction in terms of the azimuthalangle /H9278, which is the interlayer phase difference, and mz which is proportional to the layer polarization. In terms of these variables the LLS equations take the form m˙z=/H20877−2 n/H6036/H9267sm/H110362/H11612/H60232/H9278+/H9004t /H6036m/H11036sin/H9278/H20878/H20853−/H20849v/H6023ps·/H11612/H6023/H20850mz/H20854 +/H9251zm/H110362/H9278˙, /H9278˙=mz/H208772 n/H6036/H9267s/H20873/H20841/H11612/H6023/H9278/H208412+2 m/H110364/H20841/H11612/H6023mz/H208412+2mz m/H110362/H11612/H60232mz/H20874−4 n/H6036/H9252 −/H9004t /H60361 m/H11036cos/H9278/H20878/H20853−/H20849v/H6023ps·/H11612/H6023/H20850/H9278/H20854−/H9251/H9278 m/H110362m˙z. /H208496/H20850 In Eq. /H208496/H20850we have written j/H6023/n, which has units of velocity, as the pseudospin velocity v/H6023ps. These equations do not on their own provide a closed description of pseudospin dynamics in the presence of elec-trical bias potentials and need to be supplemented by atheory which speciﬁes the spatial dependence of the pseu-dospin current. In general this quantity depends 21on the or- der parameter conﬁguration as well as on the contact geom-etry and external current or voltage biases. The transporttheory and the pseudospin dynamics theory are therefore not independent. In the present paper we study voltage-biasedHall bars with source and drain leads at opposite ends andwith a variety of shapes and sizes. We assume, as a simpli-ﬁcation, that the current distribution is deﬁned by a localconductivity tensor with a large Hall angle. Given these sim-pliﬁcations, we are able to explicitly evaluate the maximumcurrent at which time-independent order parameters are pos-sible. Because collective tunneling no longer contributesstrongly to the dc interlayer current when the interlayerphase is time dependent, the interlayer conductance mecha-nism changes qualitatively when this maximum current isexceeded. We therefore associate this current with the experi-mental critical current. III. APPROXIMATE CRITICAL CURRENTS In this section we discuss approximate upper bounds on the critical current which are helpful in interpreting the nu-merical results described in the following section. We use a simpliﬁed version of the static limit of the m˙ zLLS equation /H20851Eq. /H208496/H20850/H20852in which mzis assumed to be small 0=−/H9267s /H6036/H11612/H60232/H9278+1 2/H9004tn /H6036sin/H9278−1 2j/H6023·/H11612/H6023mz. /H208497/H20850 The three terms on the right-hand side can be identiﬁed as contributions to the time dependence of mz/H20849or equivalently of the exciton density /H20850due, respectively, to the divergence of the exciton supercurrent, coherent condensate tunneling, and the divergence of the zˆ/H20849layer antisymmetric or counterﬂow /H20850 fermionic pseudospin current. The last contribution would beviewed as a spin-transfer torque in the analogous equationsfor an easy-plane anisotropy ferromagnetic metal. We start our qualitative discussion of critical currents by identifying some relevant length scales. First the lengthscale, /H9261= /H208812/H9267s /H9004tn, /H208498/H20850 often referred to as the Josephson length because of the simi- larity between these equations and those which describe Jo-sephson junctions, emerges from balancing ﬁrst and secondterms. In this paper we assume that the pseudospin magneti- zation direction departs from the xˆ-yˆplane only over a small region close to the source and drain contacts whose spatialextent is small compared to the Josephson length. /H20849This issue is addressed again in the discussion section. /H20850If this is cor- rect, we can separate length scales by identifying a regionclose to the contact which is small enough that we can ignorecoherent tunneling by setting /H9004 tnto 0 and large enough that we can assume that mzis close to zero in the rest of the system /H20849see Fig. 1/H20850. When /H9004tn→0, Eq. /H208497/H20850simply expresses the conservation of the sum of the excitonic and quasiparticlecounterﬂow currents. Integration of Eq. /H208497/H20850over the area close to the contact then simply describes conversion of qua-siparticle counterﬂow current into condensate counterﬂowcurrent. The total counterﬂow current emerging from the areanear the source contact is half of the total current ﬂowingCRITICAL TUNNELING CURRENTS IN QUANTUM HALL … PHYSICAL REVIEW B 81, 184523 /H208492010 /H20850 184523-3into the system since mz=/H110061 in the contact and mz→0 far away from the contact. In a tunnel geometry experiment thesame counterﬂow current is generated near source and draincontacts located at opposite dies of the sample, so the totalcounterﬂow supercurrent injected intothe system is equal to the total number current ﬂowing through the system. With this separation of length scales the quasiparticle /H20849pseudospin-transfer torque /H20850term can be dropped in the re- maining portion A ˜of the total system area A. For static so- lutions of the LLS equations the condensate must satisfy an elliptic sine-Gordon equation inside A˜ /H92612/H11612/H60232/H9278− sin/H9278=0 . /H208499/H20850 When /H9261→/H11009/H20849/H9004tn→0/H20850, this equation states that /H11612/H60232/H9278is zero. It then follows from Green’s theorem that no net counterﬂow supercurrent can ﬂow into the area A˜. In the tunnel geometry that means the time-independent order parameter values can-not be maintained in the presence of a transport current un-less/H9004 tn/HS110050. The maximum tunneling current that can ﬂow through the system is particularly simple to determine in thesmall /H9004 tn, large /H9261limit. When /H9261is much larger than the system size the angle /H9278cannot vary substantially over the system area. With this simpliﬁcation the elliptic sine-Gordon equation can be integrated over the area A˜to obtain /H9267s/H20885 P/H11612/H6023/H9278·nˆ=/H9267sA˜ /H92612sin/H20849/H9278/H20850, /H2084910/H20850 where Pis the perimeter of A˜andnˆis proportional to the outward normal. The left-hand side of Eq. /H2084910/H20850is the net supercurrent which ﬂows out of the region A˜from its bound- aries near the source and drain contacts, identiﬁed above asthe number current ﬂowing through the system. Since themaximum value of /H20841sin/H20849 /H9278/H20850/H20841is 1, it follows that the maximum current consistent with a time-independent order parameterin this case is I Bc=eA˜/H9267s /H6036/H92612=eA˜/H9004tn 2/H6036. /H2084911/H20850 Since the critical current in the small /H9004tnlimit is propor- tional to the area of the system we will refer to this quantityas the bulk critical current, as suggested by the notation usedabove. For larger /H9004 tn,/H9261is no longer larger than the system size and it is not possible to maintain the maximum value ofsin/H20849 /H9278/H20850across the system. The LLS equation critical current in this regime depends on geometric details and we have notbeen able to obtain rigorous bounds. We can make a roughestimate by following an argument along the following lines.The elliptic sine-Gordon equation is very similar to the regu-lar sine-Gordon equation in which second-order derivativeswith respect to time and position appear with opposite signs.This 1+1 dimensional sine-Gordon equation appears as theEuler-Lagrange equation of motion of a system with a La-grangian with a kinetic-energy term proportional to /H9267s/H20851/H11509t/H9278/H20849x,t/H20850/H208522and a potential-energy term proportional /H9004tncos/H20851/H9278/H20849x,t/H20850/H20852. Since total energy /H20849integrated over position x/H20850is conserved by this dynamics it follows that the variation in the typical value of /H9267s/H20849/H11509t/H9278/H208502along the space-time bound- ary cannot be larger than /H11011/H9004tn. When this energy- conservation condition is mapped from the regular sine-Gordon equation to the /H20849imaginary-time /H20850elliptic sine- Gordon equation we can conclude that the typical value of /H11612 /H6023/H9278·nˆalong the boundary of A˜near the source contact can- not differ from the typical value of /H11612/H6023/H9278·nˆalong the boundary near the drain contact by more than /H11011/H9004tn. It follows that the current ﬂowing through the system from source to drainshould not be much larger than I Ec/H11011eW/H9267s /H6036/H9261, /H2084912/H20850 where Wis the length of the contract region or the width of a Hall bar assumed to be contacted at its edges. We will refer toIEcas the edge critical current, since it is limited by the length of one edge. For stronger interlayer coupling thencritical current is expected to vary as /H20849/H9004 tn/H208501/2once /H9261is smaller than the Hall bar length. Finally we note that becauseof hot-spot effects in transport with large Hall angles thepseudospin-transfer torque will act at the sample corners.Since the supercurrent is converted into coherent pseudospinprecession over the length scale /H9261, the effective size of the contact region will be /H11011/H9261when the Hall angle is large and /H9261 is smaller than W. We therefore estimate that the critical current is close to I Cc/H11011e/H9267s /H6036, /H2084913/H20850 independent of /H9004tn, under these circumstances. We refer to this last critical current as the corner critical current. In the FIG. 1. /H20849Color online /H20850Separation of transport length scales in quantum Hall superﬂuid transport. This theory is intended to applywhen the ordered state is well established and Hall angles are largebecause of the developing /H9263=1 quantum Hall effect. At large Hall angles current enters and leaves the samples at the hot-spot corners, even when the source and drain contacts /H20849gray /H20850fully cover the ends of a Hall bar. When order is well established, the pseudospin ori-entation of the transport electrons achieves alignment with the con-densate within a relatively small fraction of the sample area close tosource and drain /H20849solid yellow /H20850. In these areas pseudospin-transfer torques convert transport currents into condensate counterﬂow su-percurrents. When source and drain are connected to opposite layersa net supercurrent is injected into the interior region of the sample.In the remaining sample area /H20849shaded blue /H20850collective interlayer tunneling can act as a sink for the counterﬂow supercurrent.JUNG-JUNG SU AND ALLAN H. MACDONALD PHYSICAL REVIEW B 81, 184523 /H208492010 /H20850 184523-4following section we compare numerical LLS critical cur- rents with these rough estimates. IV . MODEL CALCULATIONS The numerical calculations we describe below are similar to those carried out in micromagnetic descriptions offerromagnetic-metal spin-transfer torque physics, but are ap-plied here to pseudospin-transfer physics in condensed bilay-ers. We divide the system area into pixels within which thepseudospin magnetization is assumed to be constant and re-write Eq. /H208496/H20850in an appropriate decretized form /H20849see below /H20850. We then integrate over time numerically using time steps thatare small compared to typical evolution rates and letting m /H6023/H20849t+/H9254t/H20850=m/H6023/H20849t/H20850+m/H6023˙/H11003/H9254t. By tracking m/H6023/H20849t+/H9254t/H20850−m/H6023/H20849t/H20850,w ec a n distinguish circumstances under which a steady state isreached from circumstances under which the magnetizationis dynamic. In these calculations, the values used for /H9252and /H9267sare taken from previous mean-ﬁeld theory estimates.7Ex- cept where noted the pixel size was chosen to be 10 l/H1100310l, where lis the magnetic length. The discrete version of the spin-transfer torque term in Eq. /H208496/H20850is m˙/H9251/H20841ST=/H20858 kIk nApixel/H20851mk,/H9251−/H20849m/H6023k·m/H6023/H20850m/H9251/H20852, /H2084914/H20850 where klabels the four neighboring sites, Ais the pixel area, and/H9251labels components of the magnetization orientation in the pixel of interest. Ikis the quasiparticle number current ﬂow from neighbor site kinto this pixel. In this paper we estimate values of Ikby solving the resistor network model obtained by discretizing a continuum classical linear trans-port model with a local conductivity that includes a Hallcomponent. The current and voltage distribution is deter-mined by setting /H20853I/H20854 i, the sum of the currents into pixel i from all neighbors jto zero Ii=/H20858 jIji=0 . /H2084915/H20850 Iijcan be expressed in terms of conductances /H20853/H9268/H20854ikand local voltages by Iji=/H20858 jk/H9268jki/H20849Vk−Vi/H20850, /H2084916/H20850 where jvaries over all the nearest-neighbors i, and k=jwith /H9268jki=/H9268Lijcaptures the longitudinal current or kequals a di- agonally displaced second neighbor of jand a near neighbor ofiwith/H9268jki=/H9268Hto capture the Hall current. We thus reach a matrix equation 0= /H20851G/H20852/H20853V/H20854, /H2084917/H20850 where /H20851G/H20852ij=/H20858k/H9268ikj=−/H20851G/H20852jiand /H20853V/H20854i=Vi. Given the source and drain contact voltages, we can solve for the internal volt-age and the interpixel current and therefore the variation insource and drain currents across the contacts. The local Hallconductivity was set to /H9268H=e2/h, which is close to the ap- propriate value for /H9263=1 whether or not the Hall plateau isfully formed. Note that this is where the quantum Hall phys- ics comes into play in our calculation. The longitudinal con-ductivity was set to /H9268L= 0.05 exp /H20849−mz2/W/H20850/H20849m/H6023·m/H6023L/H20850/H20849m/H6023·m/H6023R/H20850e2/h, /H2084918/H20850 where m/H6023L,Ris the magnetization orientation to the left and the right of a boundary separating two pixels. The Hall angleused in these calculations was therefore tan −1/H2084920/H20850over the largest part of the sample in which the pseudospin magneti-zation is close to collinear and planar. The results we reporton are not sensitive to the Hall angle, provided that it islarge. The role of disorder is considered phenomenologicallyas the constant in /H9268Lat the current stage. The current which ﬂows into a pixel is assumed to have the same pseudospin polarization as the pixel from which itis incident. Currents entering or exiting from the contacts areassumed to have m z=1 for top-layer contacts and mz=−1 for bottom-layer contacts. For instance, the drag geometrywould then correspond to m z=1 on both ends while the tun- neling geometry corresponds to mz=1 on the one end and mz=−1 on the other end. In this way the pseudospin-transfer torque and the Landau-Lifshitz precessional torque acting oneach pixel’s pseudospin can be evaluated. Note that thepseudospin-magnetization conﬁguration will change the con-ductivities between pixels and therefore change the currentpath. A. Critical current identiﬁcation As mentioned in the previous section, we identify the critical current as the circuit-current value above which atime-independent solution no longer exists. When the exter-nal current is small, the Gilbert damping term in the LLSequation relaxes the pseudospin magnetization into time-independent conﬁgurations. The behavior of the pseudospinas the current increases is partly analogous to the behavior ofa damped pendulum driven by an increasingly strong torque. We identify the critical current numerically by slowly in- creasing the driving voltage /H20849by /H9254Vper time step /H20850and moni- toring the change in magnetization. To be more explicit, weexamine /H9254m/H11013/H20858 i/H20841m/H6023i/H20849V+/H9254V/H20850−m/H6023i/H20849V/H20850/H20841/N. /H2084919/H20850 The sum in Eq. /H2084919/H20850is over all pixels. The critical current can be deﬁned precisely as the current above which /H9254mre- mains ﬁnite when /H9254V→0. In practice we choose a suitably small value of /H9254Vand examine the current or voltage depen- dence of /H9254m. We ﬁnd that /H9254mincreases dramatically and begins oscillating at a voltage we identify as the critical volt-age /H20849see Fig. 2/H20850. An alternative but more laborious method of obtaining critical currents is to sequentially examine the dy-namics of the pseudospin magnetization at a series of ﬁxedvalues of the applied voltage V. If the applied voltage is below its critical value, /H9254mwill approach zero exponentially at large times. If the applied voltage is above its criticalvalue, /H9254mwill not approach zero and usually exhibits an oscillatory time dependence. In our calculations we used theﬁrst approach to determine an approximate value of the criti-CRITICAL TUNNELING CURRENTS IN QUANTUM HALL … PHYSICAL REVIEW B 81, 184523 /H208492010 /H20850 184523-5cal voltage /H20849and hence the critical current /H20850and the second method to reﬁne accuracy. B.Icvs/H9004t It is useful to start by brieﬂy discussing the single-pixel limit of the calculation, which should apply approximately tothe case in which the Josephson length is longer than the system size. The steady limit of the LLS equation for the zˆ component of the pseudospin is 0=1 2/H9004tnApixel /H6036sin/H9278−I 2e/H20849mz,L−mz,R/H20850, /H2084920/H20850 where Iis the charge current ﬂowing through the system and mz,Land mz,Rare the zcomponent of pseudospin for the source and drain leads at the left and right ends of thesample. For the drag geometry /H20849current entering and exiting from the same layer /H20850m z,L=mz,R, there is no spin-torque term in the single-pixel calculation, and the steady-state equationcan be satisﬁed by setting sin /H9278=0. For the tunneling geom- etrymz,L=−mz,R, the maximum pseudospin torque that can be compensated by the tunneling term is obtained by settingsin /H9278→1. We therefore obtain Ic=IBc=e 2/H6036/H9004tnApixel. /H2084921/H20850 This gives a linear dependence of Icon the single-particle tunneling strength /H9004t. The numerical procedures described above accurately reproduce this simple result. As explained previously and discussed more fully later, we believe that the pseudospin-transfer torque in most quan-tum Hall superﬂuid experiments acts in a small fraction ofthe system area. We therefore need to perform calculationswith many pixels in order to represent a typical measure-ment. Figure 3shows numerical critical current results for a ﬁxed sample geometry as a function of /H9004 t. In this ﬁgure I0 =ICc=e/H9267s//H6036/H110118 nA is the unit of current and E0=e2//H9280lis theunit of energy. Typical quantum Hall superﬂuid experiments are performed at a magnetic ﬁeld of roughly 2.1 T for whichthe magnetic length is 17.65 nm which gives E 0=6.4 meV. In all the calculations reported on here we used a pixel areaA pixel=10/H1100310l2and the mean-ﬁeld theory estimate7/H20849/H9267s /H112290.005 E0/H20850for the pseudospin stiffness. The calculations in Fig. 3are for 40 pixels in the width Wdirection and 50 pixels in the current Ldirection. For this system size, the Josephson length is approximately equal to Lwhen 2/H9266l2/H9004tn/H110114/H1100310−7E0. The numerical results illustrated in Fig.3show the crossovers from /H9004tdependence, to /H9004t1/2de- pendence, to saturation as /H9004tincreases that was anticipated in our qualitative discussion. At small /H9004tnthe critical current is reduced by a small fraction compared to the single-pixelresult in accord with the Fig. 1. We now examine the ingredients which enter these nu- merical results in greater detail. According to the schematicFig.1, the pseudospin-transfer torque acts only near the hot spots at which current enters and exits the sample. Figure 4 shows a typical numerical results for the spatial distributionof the pseudospin-transfer torque. In the present model, thearea of the region in which transport current is converted intosupercurrent depends on the pixel size and the pseudospinstiffness and anisotropy coefﬁcients. In the tunneling geom-etry, counterﬂow supercurrent is generated near both sourceand drain hot spots and ﬂows diagonally toward the center ofthe sample. In Fig. 5we show a typical supercurrent distri- bution for the tunneling geometry case. The correspondingdistribution for an equivalent drag experiment /H20849both contacts connected to the same layer /H20850is illustrated in Fig. 6. Since the elliptic sine-Gordon equation applies locally when thepseudospin-transfer torque is negligible, it follows fromGreen’s theorem and Eq. /H208497/H20850that in a steady state the total/CID303m/CID108N Voltage(/CID541V)Vc0.91 0.92 0.93 0.940.010.020.030.040.050.06 0.9 FIG. 2. Magnetization change per time step /H9254m/H11003Nvs applied voltage in a 500 l/H11003400lsystem for 2 /H9266l2n/H9004t=10−5E0. This curve was obtained by changing the bias voltage by /H9254V=2.5/H1100310−5/H9262V at each time step. E0=e2//H9280l, the energy unit used in all our calcu- lations has a typical experimental value /H110117 meV. /H9254mdevelops large amplitude oscillation when the voltage exceeds the criticalvoltage V c.1E-8 1E-7 1E-6 1E-50.010.1110 Numerical IBc IEc ICc (L= 500 l,W= 400 l)I/I0 2/CID83l2n/CID39t/E0 FIG. 3. /H20849Color online /H20850Critical current vs 2 /H9266/H51292n/H9004tin a 400 l /H11003500lsystem. The dashed-dotted /H20849red/H20850, long-dashed /H20849blue /H20850, and short-dashed /H20849black /H20850curves plot the values of the bulk /H20849IBc/H20850, edge /H20849IEc/H20850, and corner /H20849ICc/H20850limited critical currents discussed in the text for this sample geometry. The square dots plot LLS equation criticalcurrents at a series of /H9004 tnvalues. All the calculations in this paper were performed using pseudospin stiffness /H20849exciton superﬂuid den- sity/H20850/H9267s=0.005 E0, where E0=e2//H9280lis the energy unit. Currents are in units of I0=IEc=e/H9267s//H6036. For the value of /H9267sused in these calcula- tions I0/H112298n A .JUNG-JUNG SU AND ALLAN H. MACDONALD PHYSICAL REVIEW B 81, 184523 /H208492010 /H20850 184523-6counterﬂow current injected into the interior of the sample /H20849which for the tunnel geometry equals the total charge cur- rent ﬂowing through the system /H20850must match the area inte- gration of /H208491/2/H20850/H20849/H9004tn//H6036/H20850sin/H9278. For a drag-geometry experi- ment the same integral should vanish. We now examine steady-state condensate conﬁgurations at currents near the critical current for both small n/H9004tand large n/H9004tlimits. In the former case our expectation that sin /H9278 should be nearly constant except near the hot spots is con- ﬁrmed in Fig. 7. Collective tunneling acts sinks the injected counterﬂow supercurrent at a rate that is nearly constantacross the sample area. The large n/H9004 tcase is more complex. /H9278varies over a large range and sin /H20849/H9278/H20850changes sign in dif- ferent parts of the sample. Near the sample corners /H9278varies rapidly with position because of the large counterﬂow super-currents. In the interior of the sample the phase changes lessrapidly because the ﬂow pattern spreads and because coher-ent tunneling is providing the required current sink. The criti-cal current in the large n/H9004 tlimit depends in a complex way on the geometry of the sample and on the spatial distributionof the pseudospin-transfer torques. Nevertheless, the critical-current saturation we ﬁnd in our numerical studies suggeststhat once the Josephson length is smaller compared to boththe width and the length of a Hall bar sample with a largeHall angle, it is no longer relevant to the critical-currentvalue /H20849Fig.8/H20850. C.Icvs system geometry Finally, we brieﬂy discuss critical-current dependence on Hall bar dimensions at ﬁxed n/H9004t. In Fig. 9we plot Icvs Hall bar length in a series of model samples with a ﬁxed single-pixel width W=10land single-particle tunneling amplitude 2 /H9266l2n/H9004t=10−6E0. For these parameters the Wis much smaller than the Josephson length. The critical current in-creases linearly with sample length and is therefore propor-tional to sample area until it saturates at L/H110111000 l. The length at which the current saturates is a bit longer than theJosephson length and the value of the critical current is ac-cordingly somewhat larger than the one-dimensional /H208491D/H20850spintorque(E0) y/lx/l FIG. 4. /H20849Color online /H20850Spatial distribution of the zˆcomponent of the pseudospin-transfer torque in a system with 500 /H11003400l2area, 2/H9266l2n/H9004t=10−6E0, and I/Ic=0.29. The pseudospin-transfer torque in the model studied here acts mainly in the hot-spot pixels. E0andl are deﬁned as in previous ﬁgures. y/4 0l x/40l FIG. 5. /H20849Color online /H20850Supercurrent distribution in system with area 500 /H11003400l2,2/H9266l2n/H9004t=10−6E0, and I/Ic=0.29. This plot is for a tunneling geometry in which the source is a top-layer contact andthe drain is a bottom-layer contact. Supercurrents are generated nearboth hot spots and ﬂow diagonally toward the sample center. E 0and lare deﬁned as in previous ﬁgures. y/4 0l x/40l FIG. 6. /H20849Color online /H20850Supercurrent distribution in system with area 500 /H11003400l2,2/H9266l2n/H9004t=10−6E0. This plot is for the drag- geometry case in which the source and drain are both top-layercontacts, but other parameters of the calculation are identical tothose used in the preceding tunnel-geometry ﬁgure. E 0andlare deﬁned as in previous ﬁgures. /CID186(rad) y/lx/l FIG. 7. /H20849Color online /H20850Pseudospin phase distribution in a system with area 500 /H11003400l2,2/H9266l2n/H9004t=10−8E0, and I/Ic=0.75. The Jo- sephson length at this value of /H9004tis/H110112500 l. Note that /H9278is roughly constant through the system and that its value is close to /H9266/2 be- cause Iis close to Ic. The units used here are the same as in previ- ous ﬁgures.CRITICAL TUNNELING CURRENTS IN QUANTUM HALL … PHYSICAL REVIEW B 81, 184523 /H208492010 /H20850 184523-7estimate21IEc. The large Lbehavior is, however, consistent with the expectation that the critical current should not in-crease with system length once Lis substantially larger than the Josephson length /H9261. In Fig. 10we plot I cvs system width Wwith the length ﬁxed at 500 land the same /H9004tas above. The range of Wcovered is limited somewhat by numerical practicalities and goes from a width much smaller than theJosephson length to a width which is somewhat larger. Overthis range deviations from the 1D model in which the criticalcurrent is proportional to Hall bar width are small. V . DISCUSSION AND CONCLUSIONS We start our discussion by commenting brieﬂy on some essential differences between the critical current of a Joseph-son junction and critical currents for coherent bilayer tunnel-ing. In a Josephson junction, current can ﬂow without dissi-pation across a thin insulating layer that separates twosuperconductors. The difference in condensate phase acrossthe junction /H9278Jis normally zero in equilibrium but can be driven to a nonzero steady-state value when biased by cur-rent ﬂow IJin the circuit in which the junction is placed. For thick insulating layers the current is related to the phase dif-ference by I J=IJcsin/H20849/H9278J/H20850, /H2084922/H20850 where IJcis the junction’s critical current. Equation /H2084922/H20850 should be compared with Eq. /H208497/H20850. The most obvious differ- ence is the appearance of the lateral 2D coordinate in thecoherent bilayer case. In the Josephson-junction case, thelateral dependence of /H9278Jusually plays no role unless an ex- ternal magnetic ﬁeld is present. In the coherent bilayer case,on the other hand, lateral translational invariance is alwaysbroken because the pseudospin-transfer torques that ulti-mately drive the coherent tunneling current do not act uni-formly across the system. A closer comparison is possible in the special case in which the pseudospin stiffness is large enough to inhibit lat-eral variation in /H9278. Integration of Eq. /H208497/H20850over the area then yields for the coherent bilayer IBL=IBcsin/H20849/H9278/H20850, /H2084923/H20850 where IBcis the bulk critical current discussed in the body of the paper and IBL=e/H20885 P/H20875/H9267s /H6036/H11612/H6023/H9278+mz 2j/H6023mz/H20876·nˆ /H2084924/H20850 is the injected counterﬂow current. The most essential differ- ence between tunneling in coherent quantum Hall bilayersand Josephson junctions lies in the difference in physicalcontent between the bias currents I JandIBL. In the case of a Josephson junction it is a bulk dissipationless supercurrentwhich ﬂows perpendicular to the plane of the junction. In thecase of a coherent quantum Hall bilayer, it is the counterﬂow /H20849zˆ/H20850component of the quasiparticle current. The counterﬂow component of the quasiparticle current is normally fully con-verted to condensate counterﬂow supercurrents bypseudospin-transfer torques, as we have discussed at length.The voltage drop across a Josephson junction can be mea-sured and vanishes below the critical current. Because fermi- /CID186(rad) y/lx/l FIG. 8. /H20849Color online /H20850Phase distribution of pseudospin in a system with area 500 /H11003400l2,2/H9266l2n/H9004t=1.2/H1100310−5E0, and I/Ic =0.97. The Josephson length at this value of /H9004tis/H1101170l. Note that a steady state is reached even though /H9278varies considerably across the sample and has values larger than /H9266/2. The units used here are the same as in previous ﬁgures. 10 100 1000 100001E-30.010.11 L/lNumerical IBc IEc ICc (/CID79=250.7 l,W=1 0 l)I/I0 FIG. 9. /H20849Color online /H20850Critical current vs system length Lfor a narrow Hall bar with width W=10l. The single-particle tunneling amplitude 2 /H9266l2n/H9004t=10−6/H20849E0/H20850corresponding to /H9261/H11011250l.10 1000.11 Numerical IBc IEc ICc (/CID79=250.7 l,L= 500 l)I/I0 W/l FIG. 10. /H20849Color online /H20850Critical current vs system width Wfor length L=500 l. The single-particle tunneling amplitude 2 /H9266l2n/H9004t =10−6/H20849E0/H20850, corresponding to /H9261/H11011250l.JUNG-JUNG SU AND ALLAN H. MACDONALD PHYSICAL REVIEW B 81, 184523 /H208492010 /H20850 184523-8onic quasiparticle transport, transverse counterﬂow supercur- rents, and collective interlayer tunneling are unavoidablyintertwined in the coherent bilayer tunneling case, there is nocorresponding measurable voltage which vanishes below thecritical current. Instead, the critical current is marked experi-mentally by an abrupt increase in measured resistances. Next we draw attention to the signiﬁcance of the qualita- tive difference in experiment between drag-geometry andtunnel-geometry transport measurements. If our theoreticalpicture is correct there should be no qualitative differencebetween voltage measurements in these two geometries atcurrents below the critical current, once coherence is wellestablished. /H20849Important differences between these measuring geometries can occur in the interesting regime close to thephase boundary where interlayer coherence may be estab-lished over a small fraction of the sample area. 32/H20850 Pseudospin-transfer torques in tunnel and drag geometry ex-periments with similar total circuit currents give rise to simi- lar counterﬂow supercurrents, as seen in Figs. 5and6. Ex- citonic condensates have a maximum local /H20849critical /H20850 supercurrent density that they can support, which in the caseof quantum Hall bilayer exciton condensates has beenestimated 33to be /H110111A m−1. For a typical sample width of 10−4m this corresponds to a critical current in the milliam- pere regime, orders of magnitude larger than the current lev-els typically employed in quantum Hall experiments. Evenaccounting for possible corrections due to disorder, localcritical currents are unlikely to be approached experimentallyfor either contact geometry. /H20849If they were, we would expect more similarity between the two experiments. /H20850The critical currents which are important for typical tunneling experi-ments are not local critical currents, but global critical cur-rents which set an upper limit on the rate at which injectedsupercurrent can be sunk by collective tunneling. Since thereis no net injected supercurrent in the drag-geometry experi-ment, only local limits apply. In the tunnel-geometry experi-ment, the global limit must be satisﬁed, setting a critical-current scale which can be orders of magnitude smaller if /H9004 t is small. Although the relationship anticipated here between tunneling-geometry and drag-geometry transport has notbeen speciﬁcally tested experimentally, it appears to us that itis clearly consistent with published data. We now turn to a comparison of our theory with experi- ment, and, in particular, with the recent experiments of Tie-mann et al. 34who have systematically studied the depen- dence of the tunneling critical currents in their samples ontemperature and layer separation. The data of Tiemann et al. appear to be broadly consistent with that reported inearlier 8,16but focus more on tunneling anomalies in the re- gion of the phase diagram far from the coherent-state phaseboundary. Tiemann et al. ﬁnd /H20849i/H20850that the critical tunneling currents in their samples are proportional to system area, /H20849ii/H20850 that their typical value is /H1101110 nA mm −2, and that they satu- rate upon moving away from the coherent-state phaseboundary 27either by lowering temperature or by decreasing the ratio of layer separation to magnetic length. Tiemann et al. ’s ﬁnding that the critical current is propor- tional to area would be consistent with our analysis if theJosephson length was comparable to or longer than the/H1101110 mm length of their long-thin Hall bars. When in thebulk-limited critical-current regime, the critical current should be given by I Bc IBc A=e/H9004˜t 4/H9266l2/H6036= 3.1 B/H20851T/H20852/H11003/H9004˜t/H20851eV/H20852/H11003104Am m−2,/H2084925/H20850 where /H9004˜t=2/H9266l2/H9004tnis the interlayer tunneling amplitude suit- ably renormalized by quantum and thermal ﬂuctuations andBis the magnetic ﬁeld strength at /H9263=1. Inserting the mag- netic ﬁeld strength, experimental critical currents can be re- covered by setting /H9004˜t→10−13eV. Therein lies the rub. Al- though values for /H9004˜ton this scale do imply Josephson lengths that are on the millimeter length scale and thereforeconsistent with bulk-limited critical currents in the samplesstudied by Tiemann et al. , they are four or more orders of magnitude smaller than values expected on the basis of the-oretical estimates. Suspicion that there is a fundamental dis-crepancy is established more convincingly, perhaps, by ob- serving that these critical currents also correspond to /H9004 ˜t values several orders of magnitude smaller than seemingly reliable experimental estimates35based on the interlayer tun- neling conductance in similar samples at zero ﬁeld. The keyissue then in comparing critical-current theories with experi-ment appears to be explaining why they are so small. We expect on general grounds that the experimental value of/H9004 tn/H11013/H9004˜t/2/H9266l2should be renormalized downward by both thermal and quantum ﬂuctuations and by disorder. Indeedaccording to the familiar Mermin-Wagner theorem, the orderparameter nmust vanish for /H9004 t→0 at ﬁnite temperatures. The importance of thermal ﬂuctuations is strongly inﬂuencedbyk BT//H9267s. If mean-ﬁeld theory estimates7of/H9267scan be trusted, the value of /H9267sin typical experimental samples should be /H110113/H1100310−5eV and kBT//H9267sshould therefore be less than 0.1 at the lowest measurement temperatures. At theselow-temperatures thermal ﬂuctuations alone appear to beinsufﬁcient 36to explain the discrepancy even though /H9004t/kBT is certainly small. The experimental ﬁnding that the criticalcurrent saturates at low temperatures supports this conclu-sion. Similarly, theoretical estimates that quantum ﬂuctuationcorrections to the order parameter are not 25large well away from the transition boundary are consistent with the experi-mental ﬁnding of saturating critical currents in this regime. Itappears that an explanation for the small critical currentsmust be found in the disorder physics of quantum Hall su-perﬂuids. The analysis of tunneling critical currents presented in this paper can accommodate disorder implicitly through itsinﬂuence on the parameters /H9267sand/H9004tn. Including disorder effects through renormalized coupling constants would beadequate if the characteristic length scales for disorder phys-ics are smaller than characteristic length scales like /H9261which are relevant for pseudospin-transfer torques. In quantum Hallsuperﬂuids disorder may play a more essential role by nucle-ating charged merons 37/H20849vortices /H20850. As we have explained, provided that the pseudospin-transfer torque acts only closeto the source and drain, the critical current is proportional tothe integral of /H9004 tnover the area of the sample. This integral is reduced to zero by a single undistorted meron located atCRITICAL TUNNELING CURRENTS IN QUANTUM HALL … PHYSICAL REVIEW B 81, 184523 /H208492010 /H20850 184523-9the center of a large sample. It has, in fact, often12–14,38–40 been recognized that disorder-induced vortices might play an essential role in many of the transport anomalies associated with bilayer coherence. It seems likely to us that this type ofphysics is very likely responsible for the small critical cur-rents seen in experiment, but that existing theory is unable toaccount for this effect quantitatively. The current status ofthe subject calls for a detailed analysis of how they inﬂuencecritical currents. On the experimental side, the importance ofcomplex disorder-related pseudospin textures for critical cur-rent values could be reduced and the essential physics whichlimits critical currents revealed more clearly by studyingsamples with much larger bare values of /H9004 t. Although we have attributed the substantial quantitative disagreement between pseudospin-transfer torque theory andexperiment to disorder-induced pseudospin textures and havesuggested a strategy for achieving more quantitative tests ofour theory, it is appropriate to step back and reconsider othertheoretical pictures that might be relevant to coherent bilayertunneling experiments under some circumstances. For ex-ample, the version of the pseudospin-transfer torque theoryapplied here is based on the simplest possible assumption forthe local pseudospin polarization of the transport current,namely, that the pseudospin current polarization simply fol-lows the pseudospin density polarization. This assumption iscertainly not generically correct, but its replacement requiresmore detailed knowledge of fermionic quasiparticle transportbehavior. One approach is to assume that the quantum Halleffect establishes edge state transport and use experimentalvoltage-probe data to infer 41the length scale on which the pseudospin polarization of injected electrons is relaxed, andtherefore the length scale over which the pseudospin-transfertorque acts. The advantage of this approach is that experi-mental data could be used to obtain the spatial distribution of pseudospin-transfer torques. Examination of coherent bilayertransport data suggests 36that the torques sometimes act along most of the perimeter of the system, as assumed in aprevious 18attempt to understand coherent-bilayer tunneling data, and sometimes close to the source and drain the con-tacts as assumed here. The current model is most appropriatewell away from the coherent-state phase boundary as wehave discussed. The present version of the pseudospin-transport torque theory does not account for thermal or quantum ﬂuctuationsof the condensate, which are unimportant in metallic ferro-magnets but might sometimes be important in coherent bi-layers. Weak-coupling theories 12–15of bilayer tunneling do account for ﬂuctuations, but treat /H9004tperturbatively. These theories cannot account for the existence of a critical currentsand in practice assume that each layer has a separate well-deﬁned local chemical potential. It is clear from publishedtransport data that this assumption is not always valid, inparticular, that it is not valid in the portion of the phasediagram far away from the phase boundary on which thepresent paper focuses. Experimentally 34the critical-current value decreases as the phase boundary of the coherent state isapproached by increasing either temperature or the effectivelayer separation d/l. It would be interesting to attempt quan- titative tests of the predictions of weak-coupling theories inthe portion of the phase diagram close to the phase boundary. ACKNOWLEDGMENTS The authors acknowledge essential contributions by T. Pereg-Barnea to initial stages of this work and insightsgained from valuable discussions with A. Balatsky, J. P.Eisenstein, W. Dietsche, A. D. K. Finck, Y. Joglekar, D.Pesin, L. Radzihovsky, L. Tiemann, and K. von Klitzing. 1H. A. Fertig, Phys. Rev. B 40, 1087 /H208491989 /H20850. 2A. H. MacDonald, P. M. Platzman, and G. S. Boebinger, Phys. Rev. Lett. 65, 775 /H208491990 /H20850. 3X.-G. Wen and A. Zee, Phys. Rev. Lett. 69, 1811 /H208491992 /H20850. 4J. P. Eisenstein and A. H. MacDonald, Nature /H20849London /H20850432, 691 /H208492004 /H20850. 5A. H. MacDonald and E. H. Rezayi, Phys. Rev. B 42, 3224 /H208491990 /H20850. 6A. H. MacDonald, Physica B 298, 129 /H208492001 /H20850. 7K. Moon, H. Mori, K. Yang, S. M. Girvin, A. H. MacDonald, L. Zheng, D. 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MacDonald, and N. Ma, Phys. Rev. Lett. 86, 1829 /H208492001 /H20850. 13L. Balents and L. Radzihovsky, Phys. Rev. Lett. 86, 1825 /H208492001 /H20850. 14M. M. Fogler and F. Wilczek, Phys. Rev. Lett. 86, 1833 /H208492001 /H20850. 15R. L. Jack, D. K. K. Lee, and N. R. Cooper, Phys. Rev. Lett. 93, 126803 /H208492004 /H20850;Phys. Rev. B 71, 085310 /H208492005 /H20850; O. Ros and D. Lee, ibid.,81, 075115 /H208492010 /H20850. 16I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 87, 036803 /H208492001 /H20850.Instead of splitting the zero- bias conductance peak into two ﬁnite-bias peaks, a parallel ﬁeldgradually decreases its height. Small features appear in the tailsof these peaks near the voltages at which the conductances peaksare expected in the weak-coupling theory. 17X. G. Wen and A. Zee, Phys. Rev. B 47, 2265 /H208491993 /H20850. 18E. Rossi, A. S. Nunez, and A. H. MacDonald, Phys. Rev. Lett. 95, 266804 /H208492005 /H20850. 19R. Khomeriki, L. 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West, Phys. Rev. Lett. 100, 096801 /H208492008 /H20850; P. Giudici, K. Mu- raki, N. Kumada, Y. Hirayama, and T. Fujisawa, ibid. 100, 106803 /H208492008 /H20850; A. Finck, J. Eisenstein, L. Pfeiffer, and K. West, ibid. 104, 016801 /H208492010 /H20850. 28Strictly speaking the statements are true only if the potential terms in the Schrodinger equation are local, a property not sat-isﬁed by exchange potentials. This limitation is irrelevant sinceour goal is to describe systems with pseudospin textures that aresmooth on microscopic length scales.29A. S. Nunez and A. H. MacDonald, Solid State Commun. 139, 31/H208492006 /H20850. 30See, for example, P. M. Haney, R. A. Duine, A. S. Nunez, and A. H. MacDonald, J. Magn. Magn. Mater. 320, 1300 /H208492008 /H20850. 31A. R. Champagne, A. D. K. Finck, J. P. Eisenstein, L. N. Pfe- iffer, and K. W. West, Phys. Rev. B 78, 205310 /H208492008 /H20850. 32A. Stern and B. I. Halperin, Phys. Rev. Lett. 88, 106801 /H208492002 /H20850. 33M. Abolfath, A. H. 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PhysRevB.102.104416.pdf	PHYSICAL REVIEW B 102, 104416 (2020) Topological energy barrier for skyrmion lattice formation in MnSi A. W. D. Leishman,1R. M. Menezes ,2,3G. Longbons,1E. D. Bauer,4M. Janoschek,4,5D. Honecker,6L. DeBeer-Schmitt,7 J. S. White ,8A. Sokolova ,9M. V . Miloševi ´c,2,10and M. R. Eskildsen1,* 1Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA 2Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium 3Departamento de Física, Universidade Federal de Pernambuco, Cidade Universitária, 50670-901 Recife-PE, Brazil 4Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 5Laboratory for Neutron and Muon Instrumentation, Paul Scherrer Institute, CH-5232 Villigen, Switzerland 6Institut Laue-Langevin, 71 avenue des Martyrs, CS 20156, F-38042 Grenoble cedex 9, France 7Large Scale Structures Group, Neutron Sciences Directorate, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 8Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut, CH-5232 Villigen, Switzerland 9Australian Centre for Neutron Scattering, Australian Nuclear Science and Technology Organization, NSW 2234, Australia 10NANOlab Center of Excellence, University of Antwerp, B-2020 Antwerp, Belgium (Received 14 May 2020; revised 25 August 2020; accepted 1 September 2020; published 14 September 2020) We report the direct measurement of the topological skyrmion energy barrier through a hysteresis of the skyrmion lattice in the chiral magnet MnSi. Measurements were made using small-angle neutron scatteringwith a custom-built resistive coil to allow for high-precision minor hysteresis loops. The experimental data wereanalyzed using an adapted Preisach model to quantify the energy barrier for skyrmion formation and corroboratedby the minimum-energy path analysis based on atomistic spin simulations. We reveal that the skyrmion latticein MnSi forms from the conical phase progressively in small domains, each of which consisting of hundreds ofskyrmions, and with an activation barrier of several eV . DOI: 10.1103/PhysRevB.102.104416 I. INTRODUCTION Magnetic skyrmions are topological spin structures that have garnered much attention as they show promise as bitsin next generation memory devices [ 1]. A key ingredient for their stabilization is broken inversion symmetry, eitherin the underlying crystal lattice of bulk magnetic materialsor in the interfaces of thin ﬁlm heterostructures. This bro- ken symmetry, combined with a strong spin-orbit coupling, produces an antisymmetric exchange interaction known asthe Dzyaloshinskii-Moriya interaction (DMI) [ 2,3]. More re- cently there have also been reports of skyrmions stabilized bymagnetic frustration [ 4,5]. In chiral helimagnets such as MnSi and FeGe, the DMI competes with the exchange interaction to produce three dis- tinct magnetic phases below the Curie temperature, including the skyrmion lattice (SkL) hosting A phase [ 6–8]. The A phase is bounded by ﬁrst order transitions to the paramagneticphase on the high temperature side and to the conical phase,where the spins precess around a helix with its axis parallelto the applied ﬁeld, in all other directions of the ﬁeld-temperature phase diagram [ 7]. Due to the skyrmions’ inherent topological structure, there is an energy barrier for both the creation and destructionof the SkL from any nontopological phase (e.g., the coni-cal, helical, or ﬁeld-polarized ferromagnetic phases). As aresult, both the conical and the SkL phases are bistable as *eskildsen@nd.edulocal minima in the free energy over a ﬁnite region of pa-rameter space, giving rise to phenomena such as quenchmetastability and ﬁeld history dependence [ 9–14]. Unique skyrmionic spin structures have even been predicted to bebistable with each other in certain thin ﬁlm systems [ 15]. The metastability gives rise to activated behavior reportedfor Fe 1−xCoxSi [16] and Zn-doped Cu 2OSeO 3[17], and the activation barrier for the destruction of a metastable SkL inthe latter compound was previously determined from time-dependent measurements [ 18]. Similarly, the activation barrier for single skyrmions in magnetic thin ﬁlms have been pre-dicted from theoretical calculations [ 19–22]. It is the inherent stability provided by the topological energy barrier that makesskyrmions promising candidates for memory applications, andunderstanding the nature of this barrier is the key to the devel-opment of new skyrmion-based devices. In spite of this need, acomplete description of the nucleation mechanism of the SkLin chiral magnets has not yet been fully established. Here we report direct evidence of the skyrmions’ topologi- cal energy barrier through a measurement of hysteresis in theSkL-conical phase transition in MnSi, using small-angle neu- tron scattering (SANS) [ 23]. Importantly, these measurements were performed on the equilibrium SkL phase rather thanmetastable conﬁgurations as discussed above. The existenceof hysteresis is direct evidence of the bistability of the SkL andconical phases. We further employ a minimum-energy pathanalysis, based on an atomistic spin model, to both understandand quantify the nature of this barrier and the microscopicdynamics of the phase transition itself. The combined datashow unambiguously that it is energetically favorable for the 2469-9950/2020/102(10)/104416(9) 104416-1 ©2020 American Physical SocietyA. W. D. LEISHMAN et al. PHYSICAL REVIEW B 102, 104416 (2020) SkL phase to form progressively, in domains consisting of hundreds of skyrmions. II. EXPERIMENTAL DETAILS Initial, exploratory small-angle neutron scattering mea- surements were performed on the CG-2 General PurposeSANS instrument [ 24] at the High Flux Isotope Reactor at Oak Ridge National Laboratory, and the D33 instrument atInstitut Laue-Langevin [ 25]. Systematic SANS measurements of the SkL hysteresis were carried out at the SANS-I instru-ment at the Paul Scherrer Institute (PSI) (neutron wavelengthand bandwidth: λ=0.6n m , /Delta1λ/λ =10%) and the Bilby instrument [ 26] at the Australian Nuclear Science and Tech- nology Organization (ANSTO) ( λ=0.5n m , /Delta1λ/λ =10%). From the beam collimation and the neutron wavelength andbandwidth, one can estimate the experimental resolution [ 27]: σ 2 R=4π2(δθ/λ )2+q2(/Delta1λ/λ )2, (1) σ2 L=(qλ/2π)2σ2 R. (2) Here,σ2 Randσ2 Lare radial and longitudinal resolution respec- tively, δθis the standard deviation of the beam divergence, andqis the magnitude of the scattering vector. The 3.2×2.0×1.3m m3MnSi single crystal used for the SANS measurements was cut from a larger crystal grown bythe Bridgman-Stockbarger method followed by a one weekannealing at 900 ◦Cin vacuum. The parent crystal has pre- viously been well characterized conﬁrming its high quality.Speciﬁcally, different pieces of the same crystal were investi-gated by AC magnetic susceptibility and electrical resistivitymeasurements [ 28,29]. This conﬁrmed that the phase diagram agrees well with those reported in literature [ 6](T c=29 K), and yielded a residual resistivity ratio (RRR) of 87 (deﬁnedas the ratio of the electrical resistivity at 300 and 2 K). Thisis comparable to samples used in previous neutron scatteringstudies on the SkL in MnSi. Further pieces were also charac-terized by resonant ultra sound measurements and energy dis-persive x-ray spectroscopy, with the latter conﬁrming the cor-rect stoichiometry [ 29,30]. Finally, an earlier SANS study of inﬂuence of uniaxial strain on the SkL has been carried out onparts of the same crystal [ 28]. For the SANS experiments, the MnSi crystal was aligned with the [110] direction parallel toboth the applied ﬁeld and the incident neutron beam, such thatonly one SkL orientation was energetically favorable, withSkL vector parallel to the crystallographic [1 ¯10] direction. At the beginning of each SANS experiment, temperature sweeps (26–32 K) and ﬁeld sweeps (130–240 mT) were per-formed to locate the A phase boundaries. The main SANSresults consist of hysteresis loops, obtained by sweeping theﬁeld between the SkL and conical phases at constant tem-peratures. For these loops, temperatures were selected whichcorrespond to the maximal observed scattered intensity of theSkL (28.1 K), and to a 50% reduction of this intensity on thewarmer (28.4 K) and cooler (27.8 K) sides of the A phase. Forthe major loops, the ﬁeld was swept between 130 and 240 mTusing the superconducting cryomagnet. This traverses the en-tire A phase, with both ﬁeld endpoints well within the conicalphase. For the minor hysteresis loops a resistive coil was usedto supplement to the superconducting magnet, and achieve ahigher precision of the magnetic ﬁeld. A Cernox sensor and a nichrome heater were mounted in direct contact with thesample disk, allowing an independent temperature control ofthe sample to within ±10 mK throughout the minor loops. Prior to the minor hysteresis loops the sample was heated toa temperature above the A phase, and then ﬁeld-cooled tothe center of the upper phase transition in a constant ﬁeldof 205 mT. III. EXPERIMENTAL RESULTS A typical SkL diffraction pattern is given in Fig. 1(a).T h i s shows the sum of the scattered intensity as the SkL is rotatedabout the vertical axis to satisfy the Bragg condition for eachof the six peaks. Bragg peaks associated with the conicalphase are not visible in this geometry, and therefore do notcontribute to the scattering. Figure 1(b) shows the angular de- pendence of the intensity of a single peak, as both the sampleand applied ﬁeld are rotated together through the Bragg con-dition. The rotation is performed perpendicular to the Ewaldsphere, eliminating the need for a Lorentz correction of theangular peak width. This so-called rocking curve is well ﬁttedby a Lorentzian line shape, indicating that it is dominated byspatial or temporal ﬂuctuations of the SkL rather than exper-imental resolution [ 23]. We believe it unlikely that these ﬂuc- tuations are temporal like those associated with critical ﬂuctu-ations observed above T c[31–33], but rather are a result of a ﬁnite skyrmion correlation length along the ﬁeld direction dueto crystal mosaicity as reported in other studies of MnSi [ 6]. Full width half maximum (FWHM) rocking curve widths, /Delta1ω, obtained from the Lorentzian ﬁts, are shown in Fig. 1(c). Here the horizontal axis is the integrated intensity, wherethe maximal value corresponds to being fully in the SkLphase and zero corresponds to being fully in the conicalphase. The lowest intensity where complete rocking curvemeasurements are feasible is roughly one tenth of the maximalintensity. From the widths one can estimate the longitu-dinal correlation length ζ L=2(qSkL/Delta1ω)−1, where qSkL= (0.388±0.002) nm−1is the magnitude of the SkL scattering vector. As the rocking curve widths greatly exceed σR/qSkL= 0.3◦FWHM obtained from Eq. ( 2), corrections to /Delta1ωdue to the experimental resolution are negligible. The measuredwidths yield values of ζ Lranging from 130 nm to 90 nm, indicating a reduction of the average SkL domain length alongthe ﬁeld direction by the introduction of conical phase regionswithin the sample. Similarly, the lateral correlation lengthζ Rcan be estimated from FWHM of the Bragg peak in the radial direction within the detector plane, /Delta1qR. Fits of the radial intensity yields /Delta1qR∼6.3×10−2nm−1fully within the SkL phase, increasing to ∼6.9×10−2nm−1upon entering the conical phase [apart from a re-scaling of the vertical axis, thebehavior is near identical to that of the rocking curve widths inFig.1(c)]. Correcting for the comparatively poorer resolution within the detector plane yields ζ R=2(/Delta1q2 R−σ2 R)−1/2, with σR=4.8×10−2nm−1obtained from Eq. ( 1). From this, one ﬁnds a lateral correlation length decreasing from 50 to 40 nmupon leaving the SkL phase. Together, these results suggestthat the phase transition proceeds locally, with nanoscaleregions transitioning independently over a range of appliedﬁelds. 104416-2TOPOLOGICAL ENERGY BARRIER FOR SKYRMION … PHYSICAL REVIEW B 102, 104416 (2020) FIG. 1. (a) Diffraction pattern of the SkL of MnSi at H= 195 mT. This is a sum of measurements at different rocking angles,with peaks on the horizontal axis appearing fainter as they were, on average, further from the Bragg condition. Background scattering near the detector center ( q=0) is masked off. (b) Rocking curve at H=205 mT, midway along the upper SkL-conical phase transition. The curve is ﬁt to a Lorentzian distribution with a width /Delta1ω= 2.44 ◦±0.04◦FWHM. (c) Widths, obtained from Lorentzian ﬁts to the rocking curves, along the upper SkL-conical phase transition for both increasing and decreasing ﬁeld sweeps.The total integrated Bragg peak intensity is proportional to the number of scatterers (skyrmions) in the system [ 23], and therefore the fraction of the sample volume within theSkL phase. Within the detector plane integration is performedby summing counts in the pixels spanning a Bragg peak.Integration along the third dimension of reciprocal space isobtained from rocking curves. However, as the applied ﬁeld H is increased into the conical phase and the scattered intensityfrom the SkL vanishes, the rocking curve widths only change modestly as seen in Fig. 1(c). The SkL volume fraction is thus taken to be proportional to the rocking curve peak intensityfor studies of hysteresis associated with the SkL-conical phasetransition. While it is possible to make corrections for the sys-tematic variation in the rocking curve width in Fig. 1(c),t h i si s a comparatively small effect and does not inﬂuence the analy-sis of the data in a signiﬁcant manner as we shall discuss later. Figures 2(a) and2(b) shows respectively a major and a minor hysteresis loop at T=28.1 K. In both cases, the intensity was normalized by the maximal observed intensity,which corresponds to the entire sample being in the SkLphase. In the major hysteresis loop, the ﬁeld was swept from130 to 240 mT and back. Both end points are well inside theconical phase, and this loop covers the entire A phase. Here, aclear separation of the two sweep directions is observed, withthe SkL volume fraction lagging in the direction the ﬁeld ischanging. Importantly, thermal relaxation times in MnSi aremuch shorter than the SANS count times at the measurementtemperatures [ 12], and do not contribute to the hysteresis. To conﬁrm hysteretic behavior, a series of minor loops were measured, each of which was centered on the high ﬁeldphase transition into the conical state. Prior to each minorloop, the sample was cooled from the paramagnetic state tothe measurement temperature in a constant ﬁeld (205 mT),followed by a reduction of the ﬁeld to the starting point. Fromhere, minor hysteresis loops were recorded by raising the ﬁeldto partially leave the SkL phase and then decreasing it to reenter. An example of a minor loop is show in Fig. 2(b). The minor loops show a clear nesting, quantiﬁed by the looparea which grows superlinearly as the loops become longer asshown in Fig. 2(c). Here the horizontal axis is the effective ﬁeld sweep range /Delta1H eff, deﬁned as the separation between the two crossing points of the different ﬁeld sweep directionsillustrated in the Fig. 2(b) inset. Values for /Delta1H effand the loop area were determined by ﬁts to the data described below, and the area was found to grows as a power law ∝/Delta1H1.45±0.1 eff. To quantify the activation barrier for skyrmion formation and destruction, the SANS hysteresis loops are analyzed usingan adapted Preisach model. This is suitable for transitionsin bistable systems, where two phases coexist as local freeenergy minima over some range of the external ﬁeld [ 34]. In the region of bistability, the free energy Fis assumed to be linearly proportional to the magnetic ﬁeld B: F(B,T,...)=F(B c,T,...)∓(X−X0/2)(B−Bc).(3) Here, Xis an order parameter with dimensions of a mag- netic moment, used to distinguish the conical ( X=0) and skyrmion ( X=X0) phases. The sign of the second term in Eq. ( 3) correspond to respectively the lower ( −) and upper ( +) transition between the SkL and conical phases. 104416-3A. W. D. LEISHMAN et al. PHYSICAL REVIEW B 102, 104416 (2020) FIG. 2. (a) Major hysteresis loop for T=28 K recorded at PSI. (b) Minor hysteresis loop at the same temperature, centered around205 mT and with a ﬁeld sweep range of 33 mT. Symbols are the same as in (a). Bottom left inset: expanded view of the central part of the loop. Top right inset: Schematic showing ﬁeld sweep directionand effective sweep range /Delta1H eff. Curves in (a) and (b) are ﬁts to an adapted Preisach model described in the text. (c) Area of hysteresis loops as a function of the effective sweep range.FIG. 3. Behavior of an individual Preisach unit. (a) Free energy for different values of the applied ﬁeld. Black curves correspond to ﬁelds where the conical and SkL phases have the same energy.Red (blue) curves indicate the location of the phase transition for increasing (decreasing) ﬁeld. (b) Hysteretic response of the order parameter. The Preisach free energy as a function of applied ﬁeld is shown in Fig. 3(a). A similar picture was previously proposed to describe temperature-quenched metastable SkL phases inMnSi [ 12]. The low- and high-ﬁeld transitions are treated indepen- dently, with each one governed by a pair of parameters: Thecritical ﬁeld ( B c1/c2) where the two phases have the same free energy, and the height of the activation barrier ( Ba1/a2) that inhibits the transition. As the external magnetic ﬁeld isincreased from zero and approaches the lower conical-to-SkLphase transition, the conical state free energy increases and theSkL state free energy decreases. At B=B c1+Ba1, the conical phase minimum vanishes and the system transitions to theskyrmion phase. For decreasing ﬁelds, the transition occursatB=B c1−Ba1. Similarly, the upper SkL-to-conical transi- tion occurs at B=Bc2±Ba2, where the situation is reversed and the conical and SkL free energies respectively decreaseand increase with increasing ﬁeld. The Preisach model is aninherently zero-temperature model, and a transition betweenthe states only occur when one minima disappears and thesystem is no longer bistable. This is appropriate for the SkLas reported activation barriers are much greater than k BT[18] forT/lessorequalslantTc. Preisach free energy curves produce perfectly rectangu- lar hysteresis loops, centered around Bcand with width 2Ba, as shown in Fig. 3(b). Rounded loops are obtained by considering the sample to be composed of microscopic,independently-acting, “Preisach units,” each with its own 104416-4TOPOLOGICAL ENERGY BARRIER FOR SKYRMION … PHYSICAL REVIEW B 102, 104416 (2020) TABLE I. Preisach parameters obtained from ﬁts to major hysteresis loops. Uncertainties indicate the one sigma conﬁdence interval provided by the ﬁtting algorithm. Facility T(K) Hc1(mT) σc1(mT) Ha1(mT) Hc2(mT) σc2(mT) Ha2(mT) ANSTO 27.8 188 ±81 9 ±2 1.1 ±0.3 211 ±31 4 ±1 1.0 ±0.2 PSI 28.1 155.3 ±0.2 12.5 ±0.2 0.94 ±0.14 204.4 ±0.2 9.5 ±0.2 0.96 ±0.12 ANSTO 28.1 160.2 ±0.5 13.7 ±0.3 1.1 ±0.2 212.5 ±0.4 12.0 ±0.3 0.8 ±0.2 ANSTO 28.4 168 ±92 1 ±3 1.0 ±0.3 200 ±61 9 ±2 0.7 ±0.3 Bc1/c2andBa1/a2. Since the magnetization is approximately linear across both the upper and lower ﬁeld phase tran-sitions [ 7], we express B c1/c2and Ba1/a2in terms of the corresponding applied ﬁelds Hc1/c2andHa1/a2. To model the SANS hysteresis loops, Preisach units are assumed to followa Gaussian distribution in both critical and activation ﬁelds.These distributions are characterized by their mean values( Hc1/c2,Ha1/a2) and standard deviations ( σc1/c2,σa1/a2). A ﬁt to the PSI major hysteresis loop for T=28.1Ki s shown in Fig. 2(a), and the resulting parameter values are summarized in Table I. Values of σa1/a2converge to zero during the ﬁt, and this parameter was therefore eliminated.Differences between the ﬁt and the data near the maximumSkL volume fraction are due to the Gaussian Preisach dis-tribution used. A skewed distribution, introducing additionaldegrees of freedom, could improve the overall ﬁt. However,the values of Ha, which is the principal variable of interest, would most likely remain unchanged as they depend on thewidth of the hysteresis (separation of up- and down-sweeps)at half SkL volume fraction, where the current ﬁts are verygood. Finally, rescaling the data to account for the changingrocking curve width previously discussed only effects thePreisach ﬁts minimally. Speciﬁcally, Hc1/Hc2are shifted by ∼2% in opposite directions to increase the width of the SkL phase, σc1/σc2are both reduced by ∼5%, and Ha1/Ha2remain within uncertainty of the values in Table I. Also included in Table Iare results of ﬁts to the major loops recorded at ANSTO at three different temperatures. Thedifference in the ﬁtted values of Hc1/c2at 28.1 K may be attributed to variations in the remnant ﬁeld of the cryomagnetsused, supported by the similar separation between the upperand lower transitions for the PSI and ANSTO results. Thelarger uncertainty on Hc1/Hc2and greater values of σc1/σc2 at 27.8 and 28.4 K are due to the weaker scattering at these temperatures. Importantly, the least affected parameters arethe two activation ﬁelds, which remains consistent and withmodest uncertainties across all the measurements. As the two transitions are treated independently some Preisach units could, in principle, return to the conical phasebefore others have entered the SkL phase. At 28.1 K, wherethe separation of the transition ﬁelds is much greater thanσ c1/c2, this rarely occurs. However, at 27.8 and 28.4 K the transitions overlap signiﬁcantly, preventing the intensity fromreaching the maximum at 28.1 K, which is reﬂected inthe increased values of σ c1/σc2. More importantly, the good agreement between Ha1andHa2supports a topological origin for the activation barrier which should be similar for bothphase transitions. Further support for this conclusion comesfrom the comparable values of the activation ﬁelds at differenttemperatures. This indicates that the ﬁnite temperature rangeof the A phase is not due to a signiﬁcant reduction of the activation barrier, but rather a convergence of the two criticalﬁelds as the energy separation between the conical and SkLphases is reduced. While applying the Preisach model does not require prior knowledge about the nature of individual units, it isnonetheless relevant to consider their nature. In the originalapplication to ferromagnetic hysteresis, magnetic domains be-have sufﬁciently independent to be treated as Preisach units.By analogy, we anticipate that in the present case they corre-spond to microscopic SkL domains, within which the cascadeof individual skyrmion formation occurs much faster thanthe measurement time. In this way, each domain experiencesthe phase transition quasi-instantaneously and independent ofother domains. This is consistent with the longitudinal andlateral correlation lengths discussed previously, providing acharacteristic length scale for the SkL domains of the order100 and 50 nm, respectively. In such a scenario, variationsof the local magnetic ﬁeld due to crystal inhomogeneitiesand demagnetization effects give rise to a range of differenttransition ﬁelds and therefore a nonzero σ c. It is likely that both the distribution of SkL domains throughout the sample as well as their sizes depend on theﬁeld and temperature history, which may affect the activationbarriers observed in the SANS experiments. To explore thispossibility Preisach model ﬁts were performed on the minorhysteresis loops, where the initial conﬁguration was obtainedby a ﬁeld cooling to the midpoint of the SkL-conical transi-tion. In contrast, the major loop has a starting point entirelywithin the conical phase. The results of the minor loop ﬁtsare summarized in Table II. While the values of Hc2agree with those obtained from the major loop, Ha2is reduced signiﬁcantly, conﬁrming that the barrier to create or destroySkL domains depends on the ﬁeld history. We return to thispoint later. IV . THEORETICAL MODELING To complement the SANS data, atomistic spin dynam- ics simulations were performed to investigate the transition TABLE II. Preisach parameters obtained from minor hysteresis loops at T=28.1 K (PSI). /Delta1Heff(mT) Hc2(mT) σc2(mT) Ha2(mT) 5.5±1.0 203.6 ±0.2 11.5 ±0.2 0.18 ±0.05 15±2 204.8 ±0.1 10.5 ±0.1 0.16 ±0.05 23±5 205.1 ±0.2 10.5 ±0.1 0.25 ±0.04 104416-5A. W. D. LEISHMAN et al. PHYSICAL REVIEW B 102, 104416 (2020) FIG. 4. Unit cell of the B20-structure of MnSi showing only the location of the Manganese atoms. The magnetic ﬁeld Bis applied along the [001] direction. between the conical and SkL states using a homemade simula- tion code [ 21]a sw e l la st h e SPIRIT package [ 35]. The extended Heisenberg Hamiltonian that describes the system of classicalspins can be written as H=−J/summationdisplay /angbracketlefti,j/angbracketrightni·nj−/summationdisplay /angbracketlefti,j/angbracketrightDij·(ni×nj)−/summationdisplay iμB·ni,(4) where μiis the magnetic moment of the ithatomic site with \|μi\|=μ, and ni=μi/μis the ith spin orientation. Here J represents the ﬁrst-neighbours exchange stiffness, Dijis the DMI vector, Bis the perpendicular external magnetic ﬁeld, and/angbracketlefti,j/angbracketrightdenotes pairs of nearest-neighbour spins iand j. For the simulations we adopt parameters J=1 meV and D=0.18J, which are reasonable values for MnSi [ 36,37]. Although intrinsic exchange and cubic anisotropies [ 38]m a y deﬁne a preferential direction for the spin rotation in MnSiat zero ﬁeld, such high-order contributions are much weakerthan the energy terms in Eq. ( 4) and are therefore neglected in the calculations. Similarly, the small contribution from adipolar interaction is also not included [ 39,40]. The dynamics of the spin system is governed by the Landau-Lifshitz-Gilbertequation ∂n i ∂t=−γ (1+α2)μi/bracketleftbig ni×Beff i+αni×/parenleftbig ni×Beff i/parenrightbig/bracketrightbig ,(5) where γis the electron gyromagnetic ratio, αis the damping parameter and Beff i=−∂H/∂niis the effective ﬁeld. The MnSi crystal, shown in Fig. 4, consists of a B20 structure (space-group P2 13) with four Mn atoms and four Si atoms located at the 4( a)-type sites of the simple-cubic unit cell with position coordinates ( u,u,u), (0.5+u,0.5−u,−u), (−u,0.5+u,0.5−u), and (0 .5−u,−u,0.5+u), where uMn=0.137 and uSi=0.845 [ 41]. For the simulations, only Mn magnetic moments are considered. The spin dynamicssimulations were performed in a mesh of N×√ 3N×N unit cells with N=26, and the SkL state consists of two skyrmion tubes located at respectively the center and corners.The choice of Nwas veriﬁed to minimize the SkL energy. Periodic boundary conditions are considered along thethree dimensions. To obtain the ground state of the spinmodel, the energy of the considered states are minimized fordifferent values of the applied ﬁeld B/bardbl[001]. The choiceof ﬁeld direction parallel to one of the unit cell main axes ensures that skyrmions form as uniform tubes within thesimulation box. However, the direction of the applied ﬁeldis not expected to have much impact on the energetics aslong as a high-symmetry direction of the crystal is chosen.Figure 5(a) shows the energy obtained in the simulations for the ﬁeld-polarized ferromagnetic, conical and SkLstates, from where the ground state is found to be conicalforμB<0.007Jand 0 .018J<μB<0.028J, SkL for 0.007J<μB<0.018J, and ﬁeld-polarized ferromagnetic forμB>0.028J. Next, the transition between conical and SkL states is considered. At the critical ﬁelds μB c1=0.007JandμBc2= 0.018J, both states have approximately the same energy. The activation barrier between the two states can be calculated bythe geodesic nudged elastic band (GNEB) method [ 21,43] and a climbing image method [ 44], allowing a precise determi- nation of the highest energy saddle point along the minimalenergy path connecting the two states. Here, the reactioncoordinate deﬁnes the normalized (geodesic) displacementalong the formation path. Figure 5(b) shows the activation barrier calculated between the two states in both critical ﬁelds.From this one ﬁnds that it is energetically favorable to breakthe conical state locally in different stages, nucleating theskyrmions individually instead of the whole lattice at once(see also animated data in Ref. [ 42]). Figure 5(c) shows the topological charge, given by [ 1] Q=1 4π/integraldisplay n·/parenleftbigg∂n ∂x×∂n ∂y/parenrightbigg dxdy, (6) calculated along the formation path for each xylayer of the sample for B=Bc2. Notice that the tube of the ﬁrst skyrmion is formed gradually, layer-by-layer, in a conicalbackground and the average topological charge approachesQ=1, giving rise to the ﬁrst elongated maximum in the minimal energy path. This is consistent with previous workssuggesting that skyrmions are nucleated or annihilated bythe formation and subsequent motion of Bloch points (mag-netic monopoles) [ 45–47]. After that, the second skyrmion is formed in a similar way, after which the average topologicalcharge approaches Q=2 and the transition is complete. Ener- getically equivalent paths were obtained for the ﬁrst skyrmionnucleating either at the center or the corners. As recognized previously, the transitions between the SkL and conical states are not expected to occur in a spatiallyhomogeneous fashion. As a result, the average energy perspin necessary to nucleate a single skyrmion depends on thelateral size of the domains. An estimation of the activationbarrier can be obtained by comparing the energy separation/Delta1E a=\|ESkL−ECon\|of the SkL and conical states near the critical ﬁeld, due to an activation ﬁeld Baequivalent to the one obtained from the SANS experiments. Adjusting for the dif-ference between the transition ﬁelds obtained experimentallyand from the simulations one ﬁnds B a≈(Bc2−Bc1)/50≈ 2×10−4J/μ, and from there /Delta1Ea≈10−5J. This value is roughly two orders of magnitude smaller than the activationenergy calculated in the GNEB simulation where the SkL wasformed in two steps. Therefore, to nucleate one skyrmion witha 100 times smaller activation ﬁeld in the simulations we need 104416-6TOPOLOGICAL ENERGY BARRIER FOR SKYRMION … PHYSICAL REVIEW B 102, 104416 (2020) FIG. 5. (a) Energy per spin vs applied ﬁeld for each state. The ground state is indicated by the colored shading with blue for the conical (Con) state, red for the SkL and green for the ﬁeld-polarized ferromagnetic (FM) state. (b) Minimal energy path between conical and SkL states forμB=0.007JandμB=0.018J. (c) Topological charge as a function of the reaction coordinate for μB=0.018J. (d) Spin conﬁgurations in aN×√ 3N×2Nmesh along the formation path for μB=0.018J, as indicated in (b) (see also animated data in Ref. [ 42]). to consider a phase transition that occurs in 100 times as many steps as previously. This is exactly equivalent to using a 100times larger simulation box, as the activation energy is givenby the number of skyrmion nucleations per area. Consideringthe SkL periodicity of 19 nm in MnSi [ 6], this corresponds to skyrmion domains of order ∼0.05μm 2. This is the same order of magnitude as the correlation length determined directlyfrom the SANS rocking curve widths. As the formation barrier for the individual skyrmions along the reaction coordinate are all roughly the same height [seeFig.5(b)], once the system has sufﬁcient energy to overcome the initial barrier skyrmions will continue to nucleate until de-fects or demagnetization makes it energetically unfavorable.This limits the size of the SkL domains, and we speculatethat this mechanism is responsible for the discrete Preisachunits observed in the SANS measurements. In contrast, thechange of SkL volume fraction for the minor hysteresis loopsis due to the expansion /reduction of already present domains formed during the ﬁeld cooling. This results in a smaller ac-tivation barrier, which persists since the crystal never reachesa fully saturated conical or SkL phase throughout the minorloop. Spatially resolved measurements would be required toconﬁrm this picture.The topological energy barrier for each skyrmion can be estimated by multiplying /Delta1E aby the number of spins within a SkL unit cell, and increasing the length of the skyrmions inthe simulations to the thickness of the single crystal used inthe SANS experiments. Using the above relationship betweenB aandJ/μwithμ=0.4μB[48], this yields /Delta1Ea≈7e V per skyrmion. By the nature in which it was obtained, theactivation energy above should be considered as an estimaterather than an exact value. Taking into account that /Delta1E ascales linearly with the sample thickness, our estimate for MnSiis roughly 3–4 times greater than the ∼1.6 eV reported for zinc-substituted Cu 2OSeO 3[18]. This difference may be due to the higher temperature ( ∼53 K versus ∼28 K) and lower ﬁelds ( ∼25 mT versus ∼180 mT) at which the A phase exists in Cu 2OSeO 3. V . CONCLUSION In summary, we presented the ﬁrst direct observation of the hysteresis in the formation and destruction of the skyrmionlattice in MnSi. The measured hysteresis proves that theskyrmion lattice and the conical phase are bistable over aﬁnite range of parameters, with a ﬁnite topological activation 104416-7A. W. D. LEISHMAN et al. PHYSICAL REVIEW B 102, 104416 (2020) barrier inhibiting the phase transition in either direction. This observation validates the topological stability of skyrmions.Comparing the experimental data to the results of atomisticspin simulations indicates that the skyrmion lattice is formedprogressively in smaller domains, containing hundreds ofskyrmions, with an activation barrier of several eV /mm for a single skyrmion. Our results advance the understanding of the nucleation mechanism of the SkL in chiral magnets, and we expect thatour ﬁndings will instigate further measurements of topologi-cal energy barriers between different (chiral) magnetic states.Such studies are key to understanding the evolution of mag-netic states in bulk and ultrathin materials and will establishdeﬁnitively the feasibility of high-density devices based ontopological spin structures. ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Ofﬁce of Basic Energy Sciences, under AwardNo. DE-SC0005051 (A.W.D.L., G.L., M.R.E.), the Re- search Foundation - Flanders (FWO-Vlaanderen) (R.M.M.,M.V .M.), and Brazilian Agencies FACEPE, CAPES andCNPq (R.M.M.). M.J. was supported by the LANL DirectedResearch and Development (LDRD) program via the DirectedResearch (DR) project “A New Approach to Mesoscale Func-tionality: Emergent Tunable Superlattices (20150082DR).”E.D.B. was supported by the U.S. Department of Energy,Ofﬁce of Basic Energy Sciences, Division of Materials Sci-ence and Engineering, under project “Quantum Fluctuationsin Narrow-Band Systems.” A portion of this research usedresources at the High Flux Isotope Reactor, a DOE Of-ﬁce of Science User Facility operated by the Oak RidgeNational Laboratory. Part of this work is based on experi-ments performed at the Swiss spallation neutron source SINQ,Paul Scherrer Institute, Villigen, Switzerland. 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PhysRevB.84.104445.pdf	PHYSICAL REVIEW B 84, 104445 (2011) Stability of precessing domain walls in ferromagnetic nanowires Yan Gou,1Arseni Goussev,1,2J. M. Robbins,1and Valeriy Slastikov1 1School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom 2Max Planck Institute for the Physics of Complex Systems, N ¨othnitzer Straße 38, D-01187 Dresden, Germany (Received 28 June 2011; published 29 September 2011) We show that a recently reported precessing solution of Landau-Lifshitz-Gilbert equations in ferromagnetic nanowires is stable under small perturbations of the initial data, applied ﬁeld, and anisotropy constant. Linearstability is established analytically, while nonlinear stability is veriﬁed numerically. DOI: 10.1103/PhysRevB.84.104445 PACS number(s): 75 .75.−c, 75.78.Fg I. INTRODUCTION The manipulation and control of magnetic domain walls (DWs) in ferromagnetic nanowires has recently become asubject of intense experimental and theoretical research. Therapidly growing interest in the physics of the DW motion canbe mainly explained by a promising possibility of using DWsas the basis for next-generation memory and logic devices. 1–5 However, in order to realize such devices in practice it is essential to be able to position individual DWs preciselyalong magnetic nanowires. Generally, this can be achievedby either applying external magnetic ﬁeld to the nanowire, orby generating pulses of spin-polarized electric current. Thecurrent study is concerned with the former approach. Even though the physics of magnetic DW motion under the inﬂuence of external magnetic ﬁelds has been studiedfor more than half a century, 6–9the current understanding of the problem is far from complete and many new phenomenahave been discovered only recently. 10–14In particular, a new regime has been reported13,14in which rigid proﬁle DWs travel along a thin, cylindrically symmetric nanowire with theirmagnetization orientation precessing around the propagationaxis. In this paper, we address the stability of the propagationof such precessing DWs with respect to perturbations of theinitial magnetization proﬁle, some anisotropy properties of thenanowire, and applied magnetic ﬁeld. Let m(x)=[cosθ(x),sinθ(x) cosφ(x),sinθ(x)s i nφ(x)] denote the magnetization along a one-dimensional wire. Witheasy magnetization axis along ˆxand hard axis along ˆy,t h e micromagnetic energy is given by 15 E(m)=1 2/integraldisplay/bracketleftbig Am/prime2+K1/parenleftbig 1−m2 1/parenrightbig +K2m2 2/bracketrightbig dx =1 2/integraldisplay [Aθ/prime2+sin2θ(Aφ/prime2+K1+K2cos2φ)]dx, (1) where Ais the exchange constant and K1andK2are the anistropy constants. Here and in what follows, integrals aretaken between −∞ and∞(for the sake of brevity, the limits of integration will be omitted). We consider here the case of uniaxial anisotropy, K 2=0. Minimizers of Esubject to the boundary conditions lim x→±∞m(x)=± ˆx, (2)describe optimal proﬁles for a domain wall separating two magnetic domains with opposite orientation. The optimalproﬁles satisfy the Euler-Lagrange equation m×H=0, (3) where H=−δE δm=Am/prime/prime+K1(m·ˆx)ˆx=−e0m+e1n+e2p.(4) Here, m,n=∂m/∂θ, and p=m×nform an orthonormal frame, and the components of Hin this frame are given by e0=Aθ/prime2+sin2θ(K1+Aφ/prime2), e1=Aθ/prime/prime−1 2sin 2θ(K1+Aφ/prime2), (5) e2=Asinθφ/prime/prime+2Acosθθ/primeφ/prime. In terms of these components, the energy ( 1) (with K2=0) is given by E(m)=1 2/integraldisplay e0dx, (6) and the Euler-Lagrange equation becomes e1=e2=0. While the energy Eis invariant under translations along and rotations about the xaxis, the optimal proﬁles cannot be so invariant (because of the boundary conditions). Instead,the optimal proﬁles form a two-parameter family obtained byapplying translations, denoted T(s), and rotations, denoted R(σ), to a given optimal proﬁle m ∗. We denote the family by T(s)R(σ)m∗. In polar coordinates, T(s)R(σ)m∗is given by φ(x)=σ(the optimal proﬁle lies in a ﬁxed half-plane), and θ(x)=θ∗[(x−s)/d0], where d0=√A/K 1and θ∗(ξ)=2t a n−1(e−ξ). (7) It is clear that θ∗(ξ) satisﬁes θ/prime ∗=− sinθ∗, sinθ∗(ξ)=sechξ. (8) The dynamics of the magnetization in the presence of an applied magnetic ﬁeld is described by the Landau-Lifschitz-Gilbert equation, 16which for convenience, we write in the equivalent Landau-Lifschitz (LL) form: ˙m=m×(H+Ha)−αm×[m×(H+Ha)]. (9) Here,α> 0 is the damping parameter, and we take the applied ﬁeld to lie along ˆx, Ha=H1(t)ˆx. (10) 104445-1 1098-0121/2011/84(10)/104445(7) ©2011 American Physical SocietyGOU, GOUSSEV , ROBBINS, AND SLASTIKOV PHYSICAL REVIEW B 84, 104445 (2011) In polar coordinates, the LL equation is given by ˙θ=αe1−e2−αH 1sinθ, (11) sinθ˙φ=e1+αe2−H1sinθ. (12) The precessing solution is a time-dependent translation and rotation of an optimal proﬁle, which we write asT[x 0(t)]R[φ0(t)]m∗. The center x0(t) and orientation φ0(t)o f the domain wall for the precessing solution evolve accordingto ˙x 0=−αd0H1,˙φ0=−H1. (13) It was shown13,14thatT(x0)R(φ0)m∗satisﬁes the LL equation. It is important to note that the precessing solution is funda- mentally different from the so-called Walker solution.8Indeed, the latter is deﬁned only for K2>0 (the fully anisotropic case) and a time-independent H1less than the breakdown ﬁeldHW=αK 2/2. The Walker solution is given by m(x,t)= [cosθW(x,t),sinθW(x,t) cosφW,sinθW(x,t)s i nφW] with θW(x,t)=θ∗[γ−1(x−VWt)], (14) sin 2φW=H1/HW, (15) and VW=γ(α+α−1)d0H1, (16) γ=/parenleftbiggK1 K1+K2cos2φW/parenrightbigg1 2 . (17) Equations ( 14)–(17) describe a DW traveling with a con- stant velocity VWwhose magnitude cannot exceed γ(α+ α−1)d0HW; note that VWdoes not depend linearly on the applied ﬁeld H1. In contrast, the velocity ˙x0of the precessing solution is proportional to H1, and can be arbitrarily large. Also, while for the Walker solution the plane of the DWremains ﬁxed, for the precessing solution, it rotates aboutthe nanowire at a rate proportional to H 1. Finally, for the Walker solution, the DW proﬁle contracts ( γ< 1) in response to the applied ﬁeld, whereas for the precessing solution theDW proﬁle propagates without distortion. In this paper, we consider the stability of the precessing solution. We establish linear stability with respect to pertur- bations of the initial optimal proﬁle (Sec. II), small hard-axis anisotropy (Sec. III), and small transverse applied magnetic ﬁeld (Sec. IV); speciﬁcally, we show, to leading order in the perturbation parameter, that up to translation and rotation,the perturbed solution converges to the precessing solution (inthe case of perturbed initial conditions) or stays close to it forall times (for small hard-axis anisotropy and small transversemagnetic ﬁeld). The argument is based on considerations ofenergy, and depends on the fact that for all t, the precessing solution belongs to the family of global minimizers. Theanalytic argument establishes only linear stability. Nonlinearstability is veriﬁed numerically for all three cases in Sec. V. For convenience, we choose units so that A=K 1=1. II. PERTURBED INITIAL PROFILE Letm/epsilon1(x,t) denote the solution of the LL equation with initial condition m∗+/epsilon1μ, a perturbation of an optimal proﬁle.LetT[x/epsilon1(t)]R[φ/epsilon1(t)]m∗denote the optimal proﬁle that, at time t, is closest to m/epsilon1; that is, the quantity \|\|m/epsilon1−T(s)R(σ)m∗\|\|2=/integraldisplay [m/epsilon1(x,t)−R(σ)m∗(x−s)]2dx (18) is minimized for s=x/epsilon1(t) andσ=φ/epsilon1(t). Then the following conditions must hold /integraldisplay m/epsilon1·/braceleftbigg T[x/epsilon1(t)]R[φ/epsilon1(t)]∂m∗ ∂x/bracerightbigg dx=0, (19) /integraldisplay m/epsilon1·{ˆx×T[x/epsilon1(t)]R[φ/epsilon1(t)]m∗}dx=0. It is clear that x/epsilon1(t)=x0(t)+O(/epsilon1) andφ/epsilon1(t)=φ0(t)+O(/epsilon1), but we shall not explicitly calculate the O(/epsilon1) corrections produced by the perturbation. Rather, our approach is to showthat to leading order O(/epsilon1 2),\|\|m/epsilon1−T(x/epsilon1)R(φ/epsilon1)m∗\|\|2decays to zero with t. This will imply that the precessing solution is linearly stable under perturbations of initial conditions up totranslations and rotations. Letθ /epsilon1(x,t) andφ/epsilon1(x,t) denote the spherical coordinates of m/epsilon1(x,t). We expand these in an asymptotic series, θ/epsilon1(x,t)=θ∗[x−x/epsilon1(t)]+/epsilon1θ1[x−x/epsilon1(t),t]+··· , (20) φ/epsilon1(x,t)=φ∗(t)+/epsilon1φ1[x−x/epsilon1(t),t]+··· , where the correction terms θ1(ξ,t),φ1(ξ,t), etc. are expressed in a reference frame moving with the domain wall. Then toleading order O(/epsilon1 2), \|\|m/epsilon1−T(x/epsilon1)R(φ/epsilon1)m∗\|\|2=/epsilon12/integraldisplay (θ2 1+sin2θ∗φ2 1)dξ =/epsilon12/angbracketleftθ1\|θ1/angbracketright+/epsilon12/angbracketleftsinθ∗φ1\|sinθ∗φ1/angbracketright, (21) where for later convenience we have introduced Dirac notation, expressing the integral in Eq. ( 21) in terms of inner products. It is straightforward to show that the conditions ( 19)i m p l y (using θ/prime ∗=− sinθ∗) that /angbracketleftsinθ∗\|θ1/angbracketright=/angbracketleft sinθ∗\|sinθ∗φ1/angbracketright=0, (22) which expresses the fact that the perturbations described by θ1 andφ1are orthogonal to inﬁnitesimal translations (described by sin θ∗) along and rotations about ˆx. Since the difference between m/epsilon1andT(x/epsilon1)R(φ/epsilon1)m∗is O(/epsilon1), the difference in their energies is O( /epsilon12)[ a sT(x/epsilon1)R(φ/epsilon1)m∗ satisﬁes the Euler-Lagrange equation ( 3)], and is given to leading order by the second variation of Eabout m∗, /Delta1E/epsilon1=E(m/epsilon1)−E[T(x/epsilon1)R(φ/epsilon1)m∗] =E(m/epsilon1)−E(m∗)=/epsilon12 2/integraldisplay f0dξ, (23) where f0=θ/prime 12+cos 2θ∗θ2 1+sin2θ∗φ/prime 12. Using the relations Eq. ( 8) and performing some integrations by parts, we can write /integraldisplay f0dξ=/angbracketleftθ1\|H\|θ1/angbracketright+/angbracketleft sinθ∗φ1\|H\|sinθ∗φ1/angbracketright, (24) 104445-2STABILITY OF PRECESSING DOMAIN W ALLS IN ... PHYSICAL REVIEW B 84, 104445 (2011) whereHis the Schr ¨odinger operator −d2/dξ2+V(ξ) with the potential given by V(ξ)=1−2 sech2ξ. (25) V(ξ) is a particular case of the P ¨oschl-Teller potential, for which the spectrum of His known.17Hhas two eigen- states, namely, sin θ∗(ξ)=sechξwith eigenvalue λ0=0 and cosθ∗(ξ)=tanhξwith eigenvalue λ1=1, and its continuous spectrum is bounded below by λ=1. This is consistent with the fact that the optimal proﬁles are global minimizers of E [subject to the boundary conditions Eq. ( 2)], which implies that the second variation of Eabout m∗is positive for variations transverse to translations and rotations of m∗. It follows that, for any (smooth) square-integrable function f(ξ) orthogonal to sinθ∗,w eh a v e /angbracketleftf\|Hj+1\|f/angbracketright/greaterorequalslant/angbracketleftf\|Hj\|f/angbracketright (26) forj/greaterorequalslant0 (we will make use of this for j=0 and j=1). In particular, since θ1and sin θ∗φ1are orthogonal to sin θ∗[cf. Eq. ( 22)], it follows that /angbracketleftθ1\|H\|θ1/angbracketright/greaterorequalslant/angbracketleftθ1\|θ1/angbracketright, (27) /angbracketleftsinθ∗φ1\|H\|sinθ∗φ1/angbracketright/greaterorequalslant/angbracketleftsinθ∗φ1\|sinθ∗φ1/angbracketright. (28) Therefore, from the preceding Eqs. ( 27)–(28) and Eqs. ( 21) and ( 23)–(24), we get, to leading order O( /epsilon12) that \|\|m/epsilon1−T(x/epsilon1)R(φ/epsilon1)m∗\|\|2/lessorequalslant2/Delta1E/epsilon1. (29) Below we show that, to leading order O( /epsilon12), for small enough H1(it turns out that \|H1\|<1/2 is sufﬁcient), we have the inequality d dt/Delta1E/epsilon1/lessorequalslant−γ/Delta1E /epsilon1 (30) for some γ> 0. Taking Eq. ( 30) as given, it follows from the Gronwall inequality that /Delta1E/epsilon1/lessorequalslant1 2C/epsilon12e−γt(31) for some C> 0 (which depends only on the form of the initial perturbation). From Eq. ( 29), it follows that \|\|m/epsilon1−T(x/epsilon1)R(φ/epsilon1)m∗\|\|2/lessorequalslantC/epsilon12e−γt. (32) The result ( 32) shows that, to O( /epsilon12),m/epsilon1converges to an optimal proﬁle with respect to the L2norm. In fact, with a small extension of the argument, we can also show that, to O( /epsilon12), m/epsilon1converges to an optimal proﬁle uniformly (that is, with respect to the L∞norm). Indeed, making use of the preceding estimates, one can obtain a bound on \|\|m/prime /epsilon1−T(x/epsilon1)R(φ/epsilon1)m/prime ∗\|\|, theL2norm of the difference in the spatial derivatives of the perturbed solution and the optimal proﬁle. To O( /epsilon12), \|\|m/prime /epsilon1−T(x/epsilon1)R(φ/epsilon1)m/prime ∗\|\|2 =/epsilon12(/angbracketleftθ/prime 1\|θ/prime 1/angbracketright+/angbracketleft sinθ∗φ/prime 1\|sinθ∗φ/prime 1/angbracketright+/angbracketleft sinθ∗θ1\|sinθ∗θ1/angbracketright) /lessorequalslant/epsilon12(3(/angbracketleftθ1\|H\|θ1/angbracketright+/angbracketleft sinθ∗φ1\|H\|sinθ∗φ1/angbracketright) /lessorequalslant6/epsilon12/Delta1E/epsilon1. (33) Arguing as in Eqs. ( 29)–(32), we may conclude that \|\|m/prime /epsilon1− T(x/epsilon1)R(φ/epsilon1)m/prime ∗\|\|decays exponentially with t. Thus m/epsilon1con- verges to an optimal proﬁle with respect to the Sobolev H1norm (where \|\|f\|\|2 H1=\| \|f\|\|2+\| \|f/prime\|\|2). It is a standard result that this implies that the convergence is also uniform [again,to O(/epsilon1 2)]. It remains to establish Eq. ( 30). From Eq. ( 9), we have that for any solution m(x,t) of the LL equation, d dtE(m)=−/integraldisplay H·˙mdx =/integraldisplay (m×H)·Hadx −α/integraldisplay (m×H)2+(m×H)·(m×Ha)dx =−α/integraldisplay/parenleftbig e2 1+e2 2+H1sinθe1/parenrightbig dx, (34) where e1ande2are given by Eq. ( 5), and we have used the fact that the term ( m×H)·Havanishes on integration. Substituting the perturbed solution m/epsilon1into Eq. ( 34) and noting that the E[T(x/epsilon1)R(φ/epsilon1)m∗]=E(m∗) does not vary in time, we obtain after some straightforward manipulation that d dt/Delta1E/epsilon1=−α/epsilon12[/angbracketleftθ1\|H2\|θ1/angbracketright +/angbracketleftsin2θ∗φ1\|H2\|sinθ∗φ1/angbracketright+H1F] (35) to leading O( /epsilon12), where F=/integraldisplay/parenleftbig cosθ∗f0+cosθ∗sin2θ∗θ2 1/parenrightbig dξ. (36) For the ﬁrst two terms on the right-hand side of Eq. ( 35), we have, from Eqs. ( 26) and ( 23)–(24), /angbracketleftθ1\|H2\|θ1/angbracketright+/angbracketleft sinθ∗φ1\|H2\|sinθ∗φ1/angbracketright /greaterorequalslant/angbracketleftθ1\|H\|θ1/angbracketright+/angbracketleft sinθ∗φ1\|H\|sinθ∗φ1/angbracketright=2 /epsilon12/Delta1E/epsilon1.(37) The term H1Fin Eq. ( 35) is not necessarily positive, as H1 can have arbitrary sign. But for sufﬁciently small \|H1\|,i ti s smaller in magnitude than the preceding two terms. Indeed,we have, again using Eqs. ( 26) and ( 23)–(24), \|F\|/lessorequalslant/integraldisplay/parenleftbig \|f 0\|+θ12/parenrightbig dξ/lessorequalslant2 /epsilon12/Delta1E/epsilon1+/angbracketleftθ1\|θ1/angbracketright /lessorequalslant2 /epsilon12/Delta1E/epsilon1+/angbracketleftθ1\|H\|θ1/angbracketright/lessorequalslant4 /epsilon12/Delta1E/epsilon1. (38) Substituting Eqs. ( 37) and ( 38) into Eq. ( 35), we get d dt/Delta1E/epsilon1/lessorequalslant−2α(1−2\|H1\|)/Delta1E/epsilon1, (39) from which the required estimate ( 30) follows for \|H1\|<1/2. It is to be expected that the stability of the precessing solution depends on the applied ﬁeld not being too large.Indeed, it is easily shown that, for H 1>1 (respectively, H1< −1), the static, uniform solution m=− ˆx(respectively, m= +ˆx) becomes linearly unstable. As the precessing solution is nearly uniform away from the domain wall, one would expectit to be similarly unstable for \|H 1\|>1. The numerical results of Sec. VA bear this out. Finally, we remark that the stability criterion obtained here, namely, \|H1\|<1/2, is certainly not optimal. 104445-3GOU, GOUSSEV , ROBBINS, AND SLASTIKOV PHYSICAL REVIEW B 84, 104445 (2011) III. SMALL HARD-AXIS ANISOTROPY Next we assume that the hard-axis anisotropy is small but nonvanishing, taking K2=/epsilon1> 0. Let m/epsilon1(x,t) denote the solution of the LL equation with initial condition m/epsilon1(x,0)= m∗(x). As above, let T[x/epsilon1(t)]R[φ/epsilon1(t)]m∗denote the translated and rotated optimal proﬁle closest to m/epsilon1at time t. Adapting the argument of the preceding section, we show below that, toleading order O( /epsilon1 2), \|\|m/epsilon1−T(x/epsilon1)R(φ/epsilon1)m∗\|\|2/lessorequalslantC2/epsilon12for all t>0, (40) for some constant C2>0. In contrast to the preceding result, Eq. ( 32), for perturbed initial conditions, here we do not expect m/epsilon1to converge to T(x/epsilon1)R(φ/epsilon1)m∗. Indeed, while an explicit analytic solution of the LL equation is not available for smallK 2(the Walker solution is valid only for K2>2\|H1\|/α), it is easily veriﬁed that there are no exact solutions of the formT[x /epsilon1(t)]R[φ/epsilon1(t)]m∗. The result, Eq. ( 40), demonstrates that, through linear order in /epsilon1, the solution for K2=/epsilon1remains close to the precessing solution, up to translation and rotation. To proceed, let /Delta1E/epsilon1denote, as above, the difference in the uniaxial micromagnetic energy, i.e., the energy given byEq. ( 1) with K 2=0, between m/epsilon1andT(x/epsilon1)R(φ/epsilon1)m∗. Then, as in Eq. ( 29), we have \|\|m/epsilon1−T(x/epsilon1)R(φ/epsilon1)m∗\|\|2/lessorequalslant2/Delta1E/epsilon1. (41) SinceE[T(x/epsilon1)R(φ/epsilon1)m∗]=E(m∗) is constant in time, we have d dt/Delta1E/epsilon1=d dtE(m/epsilon1). (42) The hard-axis anisotropy affects the rate of change of the uniaxial energy through additional terms in ˙m. Indeed, for any solution m(x,t) of the LL equation, we have d dtE(m)=d dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle K2=0E(m)+G(m), (43) where d/dt\|K2=0E(m) denotes the rate of change when K2= 0, as given by Eq. ( 34), and G(m)=−/epsilon1/integraldisplay R(m·ˆy)[m×H(m)]·ˆydx +/epsilon1α/integraldisplay [m×H(m)]·(m×ˆy)(m·ˆy)dx.(44) Taking m=m/epsilon1, we recall from the preceding section [c.f. Eq. ( 30)] that, for \|H1\|<1/2, d dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle K2=0E(m/epsilon1)/lessorequalslant−γ/Delta1E /epsilon1 (45) for some γ> 0. Below we show that there exists constants C1,γ1withγ1<γ such that \|G(m/epsilon1)\|/lessorequalslantγ1/Delta1E/epsilon1+C1/epsilon12. (46) Taking Eq. ( 46) as given and substituting it along with Eq. ( 45) into Eqs. ( 42)–(43), we get that d dt/Delta1E/epsilon1/lessorequalslant−(γ−γ1)/Delta1E/epsilon1+C1/epsilon12. (47) From Gronwall’s equality, it follows that /Delta1E/epsilon1/lessorequalslantC1 γ−γ1/epsilon12, (48)which together with Eq. ( 41) yields the required result ( 40). It remains to show Eq. ( 46). Substituting the asymptotic ex- pansion Eq. ( 20), we obtain after straightforward calculations that, to leading order O( /epsilon12), G(m/epsilon1)=−/epsilon12cos2φ∗(t) ×/integraldisplay (sin4θ∗φ/prime 1+4/3αsin3θ∗θ/prime 1)dξ. (49) This can be estimated using the elementary inequality 2\|ab\|/lessorequalslantβa2+b2 β, (50) which holds for any β> 0. Indeed, recalling Eqs. ( 8), (23), and ( 27), and using integration by parts where necessary, we have that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay sin4θ∗φ/prime 1dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantβ 2/integraldisplay sin2θ∗φ/prime 12dξ+1 2β/integraldisplay sin6θ∗dξ /lessorequalslantβ /epsilon12/Delta1E/epsilon1+8 15β, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay sin 3θ∗θ/prime 1dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantβ 2/integraldisplay θ/prime 12dξ+1 2β/integraldisplay sin6θ∗dξ /lessorequalslantβ /epsilon12/Delta1E/epsilon1+8 15β. (51) From Eqs. ( 49)–(51), it is clear that β,γ1, andC1can be chosen so that Eq. ( 46) is satisﬁed. IV . SMALL TRANSVERSE APPLIED FIELD Suppose the applied magnetic ﬁeld has a small transverse component, so that Ha=H1ˆx+H2ˆy, where H2=/epsilon1h2(x) (52) (h2depends on xbut not t). For simplicity, let K2=0. Let m/epsilon1(x,t) denote the solution of the LL equation with initial condition m/epsilon1(x,0)=m∗(x). As above, let T[x/epsilon1(t)]R[φ/epsilon1(t)]m∗ denote the translated and rotated optimal proﬁle closest to m/epsilon1 at time t. We ﬁrst note that, unless h2vanishes as x→± ∞ ,m/epsilon1will not remain close to T[x/epsilon1(t)]R[φ/epsilon1(t)]m∗. For example, if h2 is constant, then away from the domain wall, m/epsilon1will relax to one of the local minimizers of the homogeneous energyK 1(1−m2 1)−Ha·m, and these do not lie along ±ˆxforH2/negationslash= 0. It follows that \|\|m/epsilon1−T[x/epsilon1(t)]R[φ/epsilon1(t)]m∗\|\|will diverge with time. Physically, this divergence is spurious. It stems from the fact that we are taking the wire to be of inﬁnite extent. Oneway to resolve the issue, of course, would be to take the wireto be of ﬁnite length. However, one would then no longer havean explicit analytic solution of the LL equation. Here, we shall take a simpler approach, and assume that the transverse ﬁeld h 2(x) approaches zero as xapproaches ±∞ . In fact, for technical reasons, it will be convenient to assumethat the integral of h 2 2+h/prime 22, i.e., the squared Sobolev norm \|\|h2\|\|H1, is ﬁnite. Then without loss of generality, we may assume \|\|h2\|\|2 H1=/integraldisplay/parenleftbig h2 2+h/prime 22/parenrightbig dξ=1. (53) 104445-4STABILITY OF PRECESSING DOMAIN W ALLS IN ... PHYSICAL REVIEW B 84, 104445 (2011) Under this assumption, the main result of this section is that m/epsilon1 stays close to an optimal proﬁle up to translation and rotation. That is, for some C1>0, \|\|m/epsilon1−T(x/epsilon1)R(φ/epsilon1)m∗\|\|2/lessorequalslantC1/epsilon12. (54) The demonstration proceeds as in the preceding section, so we will discuss only the points at which the present case isdifferent. The main difference is that, in place of Eq. ( 49), we get (by considering the LL equation with H 2/negationslash=0 rather than K2/negationslash=0) the following expression for G(m/epsilon1) to leading order O(/epsilon12): G(m/epsilon1)=/epsilon12/bracketleftbigg αcosφ∗(t)/integraldisplay cosθ∗(θ/prime/prime 1−cos 2θ∗θ1)h2dξ −αsinφ∗(t)/integraldisplay sinθ∗(φ/prime/prime 1−2 cosθ∗φ/prime 1)h2dξ −sinφ∗(t)/integraldisplay (θ/prime/prime 1−cos 2θ∗θ1)h2dξ −cosφ∗(t)/integraldisplay sinθ∗cosθ∗(φ/prime/prime 1−2 cosθ∗φ/prime 1)h2dξ/bracketrightbigg . (55) After some straightforward manipulations including integra- tion by parts and making use of the inequality ( 50), one can show that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay cosθ ∗(θ/prime/prime 1−cos 2θ∗θ1)h2dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantβ 2/bardblθ1/bardbl2 H1+1 2β, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay sinθ ∗(φ/prime/prime 1−2 cosθ∗φ/prime 1)h2dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantβ 2\|\|sinθ∗φ/prime 1\|\|2+1 2β, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay (θ/prime/prime 1−cos 2θ∗θ1)h2dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantβ 2/bardblθ1/bardbl2 H1+1 2β, (56) /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay sinθ∗cosθ∗(φ/prime/prime 1−2 cosθ∗φ/prime 1)h2dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle /lessorequalslantβ 2\|\|sinθ∗φ/prime 1\|\|2+1 2β. From Eqs. ( 23), (24), and ( 27), it follows that /integraldisplay/parenleftbig θ/prime 12+sin2θ∗φ/prime 12/parenrightbig dξ/lessorequalslant4 /epsilon12/Delta1E/epsilon1, (57) and/integraldisplay θ2 1dξ/lessorequalslant2 /epsilon12/Delta1E/epsilon1. (58) Substituting Eqs. ( 56)–(58) into Eq. ( 55), we get that \|G(m/epsilon1)\|/lessorequalslant(1+α)/parenleftbigg 3β/Delta1E /epsilon1+1 β/epsilon12/parenrightbigg . (59) This estimate is of the same form as Eq. ( 46), and the argument given there, with βchosen appropriately, establishes Eq. ( 54). V . NUMERICAL STUDIES In the preceding Secs. II–IVwe have shown that the precessing solution is linearly stable; to leading order O( /epsilon1), a perturbed solution either approaches or stays close to theprecessing solution up to a translation and rotation, accordingto whether the perturbation is to the initial conditions or tothe anistropy and transverse applied magnetic ﬁeld in the LL equation. Here, we present numerical results that verifynonlinear stability for the precessing solution under smallperturbations. To this end, we investigate the energy, /Delta1E /epsilon1= E(m/epsilon1)−E(m∗), of the numerically computed perturbed DW m/epsilon1(x,t) relative to the minimum energy E(m∗) of an optimal proﬁle, as a function of time t. Throughout, Eis taken to be the uniaxial micromagnetic energy given by Eq. ( 1)w i t h K2=0. As in the preceding sections, we choose units so that A= K1=1. In these units, E(m∗)=2. In typical ferromagnetic microstructures, the value of the Gilbert damping parameterαis known to lie between 0.04 and 0.22 (see, e.g., Ref. 18 and references within), so we take α=0.1 throughout our numerical study. A. Perturbed initial proﬁle We ﬁrst investigate the evolution of a DW m/epsilon1(x,t) from an initial perturbation of an optimal proﬁle. We take the initialcondition in polar coordinates to be given by θ /epsilon1(x,0)=θ∗/parenleftbiggx 1+/epsilon11/parenrightbigg ,φ /epsilon1(x)=φ0+/epsilon12x, (60) which corresponds to stretching the unperturbed proﬁle along and twisting it around the axis of the nanowire. The appliedﬁeld is directed along the nanowire, H a=H1ˆx, and we take K2=0. Figure 1shows the dependence of the relative energy /Delta1E/epsilon1 on time tfor different values of the applied ﬁeld H1. The ﬁgure presents 13 curves corresponding, from top to bottom, to H1 varying from −1.2 to 0 at the increment of 0 .1. With the initial condition given by Eq. ( 60), we take /epsilon11=0.1 and/epsilon12=π/50. Figure 1clearly indicates that /Delta1E/epsilon1(t) decays exponentially for weak applied ﬁelds, \|H1\|/lessorequalslant1/2, in accord with the analytic result ( 31). However, for \|H1\|∼1, deviations from exponential decay are evident, and the precessing solutionappears to become unstable for \|H 1\|/greaterorsimilar1. 0 50 100 150 20010−810−610−410−2100 tΔE/epsilon1 FIG. 1. (Color online) Relative energy /Delta1E/epsilon1(t) of the perturbed DW for 13 different values of the applied ﬁeld H1.S e et e x tf o r discussion. 104445-5GOU, GOUSSEV , ROBBINS, AND SLASTIKOV PHYSICAL REVIEW B 84, 104445 (2011) 0 10 20 30 40 50 60 70 8000.0010.0020.0030.0040.0050.0060.0070.0080.0090.01 tΔE/epsilon1 FIG. 2. (Color online) Relative energy /Delta1E/epsilon1(t) of the perturbed DW for ﬁve different values of the hard-axis anisotropy constant K2. See text for discussion. B. Small hard-axis anisotropy We consider next the evolution of a DW from an optimal proﬁle at t=0 when the hard-axis anisotropy K2is nonvan- ishing. We ﬁx H1=− 0.5. Figure 2shows the dependence of the relative energy /Delta1E/epsilon1 on time tfor different values of K2. The ﬁgure presents ﬁve curves corresponding, from top to bottom, to K2varying from 0 .1t o0 .02 at the decrement of 0 .02. (The blue and red colorings alternate to make adjacent curves more easilydistinguishable.) It is evident that the relative energy remainssmall, verifying the linear analysis of Sec. III. Figure 3shows the maximum value of the relative energy /Delta1E /epsilon1(over the interval 0 /lessorequalslantt/lessorequalslant80) as a function of K2. Red squares represent numerically computed values. The 0 0.02 0.04 0.06 0.08 0.100.0010.0020.0030.0040.0050.0060.0070.0080.0090.01 K2max(Δ E/epsilon1) FIG. 3. (Color online) Maximum value of the relative energy /Delta1E/epsilon1 of the perturbed DW as a function of the hard-axis anisotropy K2. Numerically computed values are represented by (red) squares. The(black) solid curve is a parabola, max( /Delta1E /epsilon1)=CKK2 2withCK= 1.3207, ﬁtted by the method of least squares through the data points withK2/lessorequalslant0.04.0 50 100 150 200 250 300 350 40010−310−210−1100 tΔE/epsilon1 FIG. 4. (Color online) Relative energy /Delta1E/epsilon1(t) of the perturbed DW for ﬁve different values of the transverse ﬁeld amplitude ¯H2.S e e text for discussion. black solid curve is the parabola CKK2 2, with CK=1.3207 ﬁtted by the method of least squares through the data pointswithK 2/lessorequalslant0.04. We obtain convincing conﬁrmation of the leading-order analytical result ( 48). For larger values of K2, we see departures from quadratic dependence; for sufﬁcientlylarge values of K 2(not shown), the Walker solution was recovered. C. Small transverse applied ﬁeld Finally, we address the stability of the precessing solution under an applied magnetic ﬁeld, Ha=H1ˆx+H2ˆy, with a 0 0.02 0.04 0.06 0.08 0.100.20.40.60.81 ¯H2max(Δ E/epsilon1) FIG. 5. (Color online) Maximum value of the relative energy /Delta1E/epsilon1 of the perturbed DW as a function of the amplitude of the transverse applied ﬁeld, ¯H2. Numerically computed values are represented by (red) squares. The (black) solid curve is a parabola, max( /Delta1E/epsilon1)= CH¯H2 2withCH=99.6586, ﬁtted by the method of least squares through the data points with ¯H2/lessorequalslant0.04. 104445-6STABILITY OF PRECESSING DOMAIN W ALLS IN ... PHYSICAL REVIEW B 84, 104445 (2011) small transverse component, H2(x). As discussed in Sec. IV, we want H2(x) to vanish as x→± ∞ . Here, we take H2(x)=¯H2w(x), (61) where w(x) is equal to one inside the window 0 /lessorequalslantx/lessorequalslant20 and vanishes outside [the argument of Sec. IVis easily modiﬁed to establish the linear stability result ( 48) in this case]. We consider the evolution of a DW given at t=0 by the optimal proﬁle m∗centered at x=0. We take H1=− 0.5, so that in the absence of the transverse ﬁeld, the DW velocity is positive[cf. Eq. ( 13)] and the DW crosses the window. We take K 2=0. Figure 4shows the dependence of the relative energy /Delta1E/epsilon1on time tfor different values of the transverse ﬁeld amplitude ¯H2. The ﬁgure presents ﬁve curves corresponding, from top to bottom, to ¯H2varying from 0 .1t o0 .02 at the decrement of 0 .02. (The blue and red colorings alternate to make adjacent curves more easily distinguishable.) Therelative energy /Delta1E /epsilon1(t) is presented over the time interval 0/lessorequalslantt/lessorequalslant400, which, for small values of ¯H2, is sufﬁcient for the DW to traverse the spatial window 0 /lessorequalslantx/lessorequalslant20 [cf. Eq. ( 13)]. The results conﬁrm that the relative energy of the perturbedmagnetization proﬁle remains small for small values of ¯H 2,i n accord with the leading-order results of Sec. IV. Figure 5shows the maximum value of the relative energy /Delta1E/epsilon1(over the interval 0 /lessorequalslantt/lessorequalslant400) as a function of ¯H2. Red squares represent numerically computed values. Theblack solid curve corresponds to the parabola C H¯H2 2withCH=99.6586 ﬁtted by the method of least squares through the data points with ¯H2/lessorequalslant0.04. The ﬁgure provides a conﬁrmation of the leading-order analytical result of Sec. IV that the maximum relative energy depends quadratically on ¯H2 for small ¯H2. Deviations from the parabolic dependence can be seen for ¯H2/greaterorsimilar0.08. VI. CONCLUSIONS The precessing solution is a new, recently reported ex- act solution of the Landau-Lifschitz-Gilbert equation. Itdescribes the evolution of a magnetic domain wall in aone-dimensional wire with uniaxial anisotropy subject to aspatially uniform but time-varying applied magnetic ﬁeldalong the wire. We have analyzed the stability of theprecessing solution. We have proved linear stability withrespect to small perturbations of the initial conditions aswell as to small hard-axis anisotropy and small transverseapplied ﬁelds, provided the applied magnetic ﬁeld along thewire is not too large. We have also carried out numericalcalculations that conﬁrm full nonlinear stability under theseperturbations. Numerical calculations suggest that, for sufﬁciently large perturbations and applied longitudinal ﬁelds, the precessingsolution becomes unstable, and new stable solutions appear. Itwould be interesting to analyze these bifurcations and studythese new regimes for DW motion. 1D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn, Science 309, 1688 (2005). 2R. P. Cowburn, Nature (London) 448, 544 (2007). 3S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 4M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin,Science 320, 209 (2008). 5L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin, Science 330, 1810 (2010). 6L. D. Landau and E. M. Lifshitz, Phys. Zeitsch. Sowietunion 8, 153 (1935). 7T. L. Gilbert, Phys. Rev. 100, 1243 (1955); IEEE Trans. Magn. 40, 3443 (2004). 8N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974). 9A .M .K o s e v i c h ,B .A .I v a n o v ,a n dA .S .K o v a l e v , Phys. Rep. 194, 117 (1990).10M. C. Hickey, Phys. Rev. B 78, 180412(R) (2008). 11X. R. Wang, P. Yan, and J. Lu, Europhys. Lett. 86, 67001 (2009). 12X. R. Wang, P. Yan, J. Lu, and C. He, Ann. Phys. (NY) 324, 1815 (2009). 13Z. Z. Sun and J. Schliemann, Phys. Rev. Lett. 104, 037206 (2010). 14A. Goussev, J. M. Robbins, and V . Slastikov, Phys. Rev. Lett. 104, 147202 (2010). 15V . V . Slastikov and C. Sonnenberg, IMA J. Appl. Math. (2011), doi:10.1093/imamat/hxr019 . 16A. Hubert and R. Sch ¨afer, Magnetic Domains: Analysis of Magnetic Microstructures (Springer, Berlin, 1998). 17P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I (McGraw-Hill, New York, 1953). 18Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 104445-7
PhysRevLett.111.217203.pdf	Domain Wall Tilting in the Presence of the Dzyaloshinskii-Moriya Interaction in Out-of-Plane Magnetized Magnetic Nanotracks O. Boulle,1,S. Rohart,2L. D. Buda-Prejbeanu,1E. Jue ´,1I. M. Miron,1S. Pizzini,3J. Vogel,3G. Gaudin,1and A. Thiaville2 1SPINTEC, CEA/CNRS/UJF/INPG, INAC, 38054 Grenoble Cedex 9, France 2Laboratoire Physique des Solides, Universite ´Paris-Sud, CNRS UMR 8502, 91405 Orsay, France 3Institut Ne ´el, CNRS and UJF, 25 avenue des Martyrs, B.P. 166, 38042 Grenoble Cedex 9, France (Received 4 April 2013; revised manuscript received 1 August 2013; published 20 November 2013) We show that the Dzyaloshinskii-Moriya interaction (DMI) can lead to a tilting of the domain wall (DW) surface in perpendicularly magnetized magnetic nanotracks when DW dynamics are driven by aneasy axis magnetic ﬁeld or a spin polarized current. The DW tilting affects the DW dynamics for largeDMI, and the tilting relaxation time can be very large as it scales with the square of the track width. The results are well explained by an extended collective coordinate model where DMI and DW tilting are included. We propose a simple way to estimate the DMI in magnetic multilayers by measuring thedependence of the DW tilt angle on a transverse static magnetic ﬁeld. These results shed light on thecurrent induced DW tilting observed recently in Co=Nimultilayers with structural inversion asymmetry. DOI: 10.1103/PhysRevLett.111.217203 PACS numbers: 75.70.Tj, 75.60.Ch, 75.78. /C0n, 85.75. /C0d The effect of structural inversion asymmetry (SIA) on the magnetic and electronic transport properties at inter- faces of low dimensional magnetic ﬁlms is currentlyattracting growing attention. In the presence of spin-orbitcoupling, SIA leads to an additional term in the exchangeinteraction, namely the Dzyaloshinskii-Moriya interaction(DMI) [ 1,2], which tends to make the magnetization rotate around a local characteristic vector D. This can destabilize the uniformly magnetized states leading to novel chiral magnetic orders, such as spin spirals [ 3]. Novel out-of- equilibrium transport phenomena have also been demon-strated, such as current induced spin-orbit torques inducedby the Rashba spin-orbit coupling and/or the spin Halleffect, leading to current induced magnetization reversal[4–6]. A recent striking example of the impact of SIA in ultrathin magnetic ﬁlms is the current induced domain wallmotion (CIDM) in perpendicularly magnetized nanotracks. This was ﬁrst outlined by Miron et al. who reported very efﬁcient CIDM in asymmetric Pt=Coð0:6n mÞ=Aloxide (AlOx) multilayers, whereas symmetric Pt=Co=Ptmulti- layers showed no effects [ 7,8]. The high perpendicular anisotropy in this material leads to narrow domain walls(DWs) ( /C245n m ), so that in typical experiments, the nano- track width ( /C24100 nm ) is much larger than the DW width. Thus, it is expected that the magnetization rotates parallel to the DW surface (Bloch DWs) to minimize the magneto- static energy. Whereas these experiments were ﬁrst inter-preted in terms of a high nonadiabatic torque induced bythe Rashba spin-orbit coupling, it was recently proposedthat the high efﬁciency arises from two key features result-ing from SIA and the high spin-orbit coupling in thismaterial [ 9]: First, the change of the DW equilibrium structure from Bloch to Ne ´el induced by the DMI. This leads to chiral DWs where the DW magnetization rotates perpendicular to the DW surface with a unique sense ofrotation [ 10,11]. Second, a large Slonczewski-like spin- orbit torque (SOT) which is maximal in the Ne ´el conﬁgu- ration [ 4,12,13]. Recent CIDM experimental results in Pt=Co=Ni[14] and Pt=CoFe=MgO multilayers [ 15] with SIA seem to support this scheme. In this Letter, we show that SIA not only affects the DW dynamics through a change of the internal DW structurebut also through a modiﬁcation of its geometrical shape. Inperpendicular magnetized nanotracks, the DW surface is expected to be perpendicular to the nanotrack axis to minimize the DW energy. However, in the presence of DMI, when driving the DW dynamics, micromagneticsreveals that a large DMI can lead to a sizable tilting ofthe DW surface which can strongly affect the DW dynam-ics. This DW tilting is a dynamical effect which occurswhatever the driving mechanism, e.g., an external mag-netic ﬁeld or a spin polarized current, and thus, is intrinsi- cally different from the previously reported current induced DW tilting [ 16–19]. The results are well explained using an analytical model based on a Lagrangian approachwhere the DMI and the DW tilting are included. We alsoshow that the DW tilting can be controlled using a statictransverse magnetic ﬁeld, providing a simple way to mea-sure the DMI. Our results shed light on the unexplainedcurrent induced DW tilting observed in Co=Niasymmetric multilayer nanotracks [ 20] and are in agreement with the presence of DMI in these samples [ 14]. We consider a magnetic ultrathin ﬁlm grown on a sub- strate with a capping layer in a different material so that theinversion symmetry is broken along the vertical axis ( z). The magnetization is supposed oriented out-of-plane witha strong perpendicular anisotropy. In addition to the stan-dard micromagnetic energy density which includes the exchange, anisotropy, Zeeman and demagnetizing energy, we add the following DMI that reads in a continuousPRL 111, 217203 (2013) PHYSICAL REVIEW LETTERSweek ending 22 NOVEMBER 2013 0031-9007 =13=111(21) =217203(5) 217203-1 /C2112013 American Physical Societyform [ 9]EDM¼D½mzð@mx=@xÞ/C0mxð@mz=@xÞþðx!yÞ/C138. This form corresponds to a sample isotropic in the plane,where the Dzyaloshinskii vector for any in-plane directionuisDz/C2uwithDa uniform constant, originating from the symmetry breaking at the zsurface. Micromagnetic simulations are based on the Landau-Lifschitz-Gilbertequation @m @t¼/C0/C130 /C220Ms/C14E /C14m/C2mþ/C11m/C2@m @t /C0/C130HSOJm/C2ðm/C2uyÞ; (1) where /C130¼/C220j/C13jwith/C13the gyromagnetic ratio, Ethe energy density and Msthe saturation magnetization. We assume that the injection of a current density Jin the nanotrack leads to a Slonczewski-like torque /C0/C130HSOJm/C2ðm/C2uyÞ[4,6,21]. To simplify, we do not consider the effect of the adiabatic and nonadiabatic spin transfer torque nor the ﬁeldlike part of the SOT [ 19,21,22]. In the following, we consider sufﬁciently large values of D (D> 0:12 mJ =m2for our simulation parameters) so that the Ne ´el conﬁguration is stable at equilibrium [ 9]. 2D micromagnetic simulations are performed using modiﬁedhomemade micromagnetic solvers [ 9,23,24]. The follow- ing parameters have been used [ 8]: exchange parameter A¼10 /C011J=m, saturation magnetization Ms¼ 1:09/C2106A=m, uniaxial anisotropy constant K¼ 1:25/C2106J=m3, Gilbert damping parameter /C11¼0:5, thickness of magnetic layer tm¼0:6n m . The DW tilting induced by the DMI can simply be introduced by considering the effect of a static in-planemagnetic ﬁeld H ytransverse to the magnetic track [Fig 1(b)]. In the presence of Hy, the Zeeman interaction leads to a rotation of the DW magnetization away from the Ne´el conﬁguration. To recover the Ne ´el conﬁgurationenergetically favored by the DMI, the DW surface tilts by an angle /C31at the cost of a higher DW energy due to the larger DW surface. Figure 1(b) shows the resulting DW tilting for /C220Hy¼100 mT and a large value D¼ 2m J=m2. The tilt angle as a function of HyandDis plotted on Figs. 1(c)and1(d). As expected, the DW tilting increases with Hyand, for a ﬁxed Hy, increases with D. The tilt angle can be roughly estimated from energetic considerations assuming that the DW always stays in aNe´el conﬁguration with an energy per unit surface /C27 0 (large Dlimit). On the one hand, for a DW tilted by an angle /C31, the DW surface and, thus, the total energy are increased by a factor 1=cos/C31; on the other hand, the Zeeman energy per unit surface scales as /C27Zsin/C31with /C27Z¼/C0/C25/C22 0HyMs/C1(/C1is the DW width). This leads to a total DW energy EDW/C25wtmð/C270/C0/C27Zsin/C31Þ=cos/C31, where wis the track width. The minimization of this energy leads to sin/C31¼/C27Z=/C270. Importantly, the slope of the DW tilting as a function of Hyon Fig. 1(c) depends directly on the value of D. This provides a direct way to measure D, from the dependence of the DW equilibrium tilt angle on Hy. In the presence of DMI, a tilting of the DW surface can also be induced dynamically by applying an easyaxis external magnetic ﬁeld H z. The magnetization distri- bution in the track for different magnetic ﬁelds and 0120306090DW tilt (deg) D (mJ/m2)0 100 2000204060 D=2 mJ/m2 D=1 mJ/m2 D= 0.6 mJ/m2DW tilt (deg) µ0Hy (mT)ψψψψ χxy (a) )d( )c((b) DWz z 1 -1Hy D=2 mJ/m2 D=1 mJ/m2 D= 0.6 mJ/m2ψψψψ xx )c( mz 1 -1 FIG. 1 (color online). (a) Schematic of the tilted DW. (b) Micromagnetic conﬁguration of a 100-nm-wide track with D¼2m J=m2and a transverse magnetic ﬁeld /C220Hy¼100 mT . (c) DW tilt angle as a function of /C220Hyfor several values of Dand (d) as a function of Dfor/C220Hy¼100 mT . Dots are the results of micromagnetic simulations, whereas the continuous lines are the results of the collective coordinates model described later. 01230102030 w=75 nm w=100 nm w=150 nmDW tilt (deg) time (ns)µ0H=100 mT(a) (c) 0 100 200 300 4000200400600800 µ0Hz (mT)DW velocity (m/s)(b) (d) z 1 -1200mT 300mT100mT 010203040 2 mJ/m2 1 mJ/m2 0.6 mJ/m2 0 100 200 3000500 vn (m/s) µ0Hz (T) w=75 nm w=100 nm w=150 nmµ0H=100 mT mz 1 -1200mT 300mT100mT 0 DW tilt (deg) FIG. 2 (color online). Dynamics of the DW driven by an external magnetic ﬁeld Hzfor a 100-nm-wide nanotrack. (a) Magnetization pattern of the DW for different values of Hz withD¼2m J=m2. Tilt angle (b) and velocity (d) of the DW as a function of Hzfor different values of D. The inset in (d) shows the DW velocity vnin the direction perpendicular to the DW surface ( vn¼vcos/C31). (c) Time dependence of the tilt angle for/C220Hz¼100 mT applied at t¼0and different track widths wforD¼2m J=m2. In (b)–(d), the results of the micromagnetic simulation (resp. CCM) are plotted in colored dots (resp. con- tinuous lines). The dashed (resp. continuous) black line in (d) (resp. (d), inset) corresponds to the prediction of the standard(q, c) model for D¼2m J=m2.PRL 111, 217203 (2013) PHYSICAL REVIEW LETTERSweek ending 22 NOVEMBER 2013 217203-2D¼2m J=m2[Fig. 2(a)] reveals that the DW tilts signiﬁ- cantly in the steady state regime when driven by Hz.A s shown on Fig. 2(b), the steady-state tilt angle rapidly increases with HzandD, although a saturation is observed for large Hz. Figure 2(d) shows the DW velocity valong the track direction as a function of Hzfor different values ofD. As expected, the DMI leads to an increase of the Walker ﬁeld [ 9]. For large values of Hz, the DW velocity signiﬁcantly deviates from the expected linearity as D increases. This deviation is the result of the DW tilting: the propagation of the tilted DW at a velocity vnnormal to its surface leads to a velocity v¼vn=cos/C31along the track direction. When considering vninstead of v[Fig. 2(d), inset], a linear scaling is obtained and the velocity in thesteady state regime does not depend on D. Thus, the DW tilting does not affect the DW velocity perpendicular to theDW surface. The time dependence of the DW tilt angle isshown on Fig. 2(c)for several values of the track width w when applying /C22 0Hz¼100 mT att¼0[25]. To describe the dynamics of tilted DWs induced by the DMI, we consider an extended collective coordinate model (CCM) [ 26] where the DW is described by three variables: its position in the track q, the DW magnetization angle c, and the tilt angle of the DW surface /C31[cf. Fig. 1(a)]. The DW proﬁle is described by the following ansatz for theazimuthal /C18and polar angle ’[with the deﬁnition m¼ ðsin/C18cos’;sin/C18sin’;cos/C18Þ]:’ðx; y; tÞ¼ cðtÞ/C0/C25=2and /C18¼2 arctan fexp½ðxcos/C31þysin/C31/C0qcos/C31Þ=/C1/C138g[/C1¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A=ðK/C0/C220M2s=2Þp is the DW width]. The effect of the DMI on the DW proﬁle and DW dynamics is taken intoaccount by an additional term in the DW energy (see below) [ 9]. To derive the dynamical equations, a Lagrangian approach is considered [ 27–29]. The Landau- Lifschitz-Gilbert equation can be derived by writing theLagrange-Rayleigh equations for the Lagrangian L¼Eþ ðM s=/C13Þ’_/C18sin/C18with Ethe micromagnetic energy density. The effects of the damping and SOT areincluded by considering the dissipative function F¼ /C11M s=ð2/C13Þ½dm=dt/C0ð/C130=/C11ÞHSOJm/C2uy/C1382. The Lagrange-Rayleigh equations then lead to the fol- lowing CCM equations: _cþ/C11cos/C31 /C1_q¼/C130Hzþ/C25 2/C130HSOJsinc; (2) _qcos/C31 /C1/C0/C11_c¼/C130Hk 2sin2ðc/C0/C31Þþ/C25D/C13 0 2/C220Ms/C1cosðc/C0/C31Þ /C0/C25 2/C130Hysinc; (3) _/C31/C11/C22 0Ms/C1/C252 6/C130/C18 tan2/C31þ/C18w /C25/C1/C1921 cos2/C31/C19 ¼/C0/C27tan/C31þ/C25Dcosðc/C0/C31Þþ/C220HkMs/C1sin2 ðc/C0/C31Þ; (4)where /C27is the wall energy per unit area with /C27¼4ﬃﬃﬃﬃﬃﬃﬃ AKp þ/C25Dsinðc/C0/C31Þþ/C220HkMs/C1sin2ðc/C0/C31Þþ /C25/C1MsHycosc, with Hkthe DW demagnetizing ﬁeld. These equations can be easily generalized to include the effects of the spin transfer torque as well as nonconstant /C1 [19,29,30]. Assuming that /C11w/C29/C1, these equations lead to a typical time scale for the tilting to settle /C28¼ /C11/C22 0Msw2=ð6/C27/C130/C1Þ. The w2dependence is explained by the time to reverse the spins in the nanotrack surface sweptby the DW when the tilting takes place. On the other hand,the magnetization angle in the DW frame relaxes on ashorter time scale /C28 /C8¼ð1þ/C112Þ=½/C11/C13ð/C25D=ð2Ms/C1Þ/C0 HkÞ/C138which does not depend on w. In the steady state regime ( _/C31¼0,_c¼0) and for Hy¼0, the tilt angle is directly related to the DW velocity vas tan/C31¼2Ms /C13/C27vcos/C31; (5) with the DW velocity v¼ð/C130/C1=/C11cos/C31ÞðHzþ ð/C25=2ÞHSOJsincÞ. This points to the dynamical origin of the DW tilting. Another physical picture can be obtainedfrom the expression of the Lagrangian integrated over the nanotrack L DW=ðtmwÞ¼/C27=cos/C31/C02Msð/C8þ/C31Þ_q=/C13, where /C8is the magnetization angle in the DW frame ( /C8¼ c/C0/C31). The ﬁrst term is the DW internal energy propor- tional to the DW surface which scales as 1=cos/C31. The second term can be seen as a kinetic potential [ 27], which contrary to a kinetic energy, is linear in the DW velocityand the DW angle. For the ﬁeld driven case in the steadystate regime, /C8is deﬁned only by the in-plane torques due toH z,D, andHk[see Eqs. ( 2) and ( 3)] and does not depend on/C31. The tilt angle in the steady state regime can thus be deduced from the minimization of LDWwith/C31at ﬁxed /C8, which leads to Eq. ( 5). Thus the tilt angle is the result of a balance between the gain in the kinetic potential resultingfrom the DW tilting and the cost in the increased DWenergy due to the larger surface. We now compare the predictions of this model with the results of the micromagnetic simulations. The continuous lines on Figs. 1(c)and1(d)show the DW tilting induced by H ypredicted by the CCM, whereas the DW tilt angle, time dependence, and DW velocity driven by Hzare plotted in continuous lines on Figs. 2(b)–2(d): a general good agree- ment is obtained with the micromagnetic simulations despite the simplicity of the model. We also plotted theresults of the standard ( q, c) model on Fig. 2(d) (dashed line). The model does not reproduce the nonlinear increaseof the DW velocity, but a good agreement is obtained whenconsidering the DW velocity in the direction perpendicularto its surface v n(inset). Thus, the DW tilting does not affect DW velocity perpendicular to its surface when driven by Hz. We now consider the current driven DW dynamics in- duced by the Slonczewski-like spin-orbit torque in the presence of a large DMI. This torque is expected for samples with SIA such as Pt=Co=AlOx trilayersPRL 111, 217203 (2013) PHYSICAL REVIEW LETTERSweek ending 22 NOVEMBER 2013 217203-3[4,8,22]. It may arise from the spin Hall effect due to the current ﬂowing in the nonmagnetic layer and/or from theRashba spin-orbit interaction [ 4,6,31]. It leads to an effec- tive easy-axis magnetic ﬁeld on the DW H SOJproportional tosinc[see Eq. ( 2)], which thus, is maximal for a Ne ´el DW conﬁguration ( c¼/C6/C25=2). The ﬁeld HSOcan be very large/C240:07 T=ð1012A=m2ÞinPt=Co=AlOx [21] (see also Refs. [ 5,15,32,33]). Similar to the action of Hy, the SOT tends to rotate the DW magnetization along the ydirection away from the Ne ´el conﬁguration, providing an additional source for the DW tilting. The results of micromagnetic simulations of the DW dy- namics driven by SOT with /C220HSO¼0:1T=ð1012A=m2Þ are shown in Fig. 3. When injecting a current in the track, a fast DW motion is observed against the electron ﬂowand the velocity increases with JandD[see Fig. 3(d)]. At the same time, a signiﬁcant tilting of the DW occurs[see Fig. 3(a) forD¼2m J=m 2], which increases with J andD[Fig. 3(b)]. The DW velocity and the tilting predicted by the CCM are shown on Figs. 3(b) and3(d), continuous lines. An excellent agreement is obtained with the micro-magnetic simulations except at higher current densities for the tilt angle due to the onset of a more complex DW structure [see Fig. 3(a) forJ¼2:5/C210 12A=m2]. The DW velocity in the direction perpendicular to the DWsurface v nforD¼2m J=m2is plotted on Fig. 3(d), inset. Contrary to the ﬁeld driven case, the standard ( q,c) modelstrongly overestimates the DW velocity (continuous line). The DW tilting leads to an additional rotation of the DW angle caway from /C25=2where the torque is maximal. Thus, the DW tilting leads to a large decrease of the DW velocity.This clearly illustrates the importance of the DW tilting on the CIDM for large DMIs. Figure 3(c) shows the time dependence of the tilting for a current step of J¼0:25/C2 10 12A=m2applied at t¼0. The CCM (continuous lines) reproduces well the time scale for the tilting to take place which scales as w2. Experimentally, Ryu et al. recently reported fast current induced DW motion associated with a signiﬁcant DW tiltingin perpendicularly magnetized ( Pt=Co=Ni=Co=TaN) nanotracks with SIA [ 14,20]. By studying the dependence of the current induced DW velocity on an in-plane longitu-dinal magnetic ﬁeld, they present evidence of chiral DWsdriven by the Slonczewski-like SOT in agreement with the presence of DMI. The DW tilting is reversed for up (down) and down (up) DW which is well explained by Ne ´el DWs pointing in opposite directions due to the DMI. From the longitudinal magnetic ﬁeld required to suppress the CIDM and using the magnetic and transport parameters ofRef. [ 14,20], one can deduce a DMI of D¼0:8m J=m 2 forA¼1/C210/C011J=m. Using this value, micromagnetic simulations predict a steady state tilt angle of about 18/C14for J¼1/C21012A=m2close to the one measured experimen- tally (/C2420/C14)[19]. Note that smaller additional contributions may arise from the anomalous Hall effect and the Oersted ﬁeld [ 19], and DW pinning may also affect the results. Thus, our model accounts for the DW tilting reported by Ryu et al. To conclude, we have shown that the DMI can lead to a large tilting of the DW surface in perpendicularly magne- tized nanotracks when DW dynamics are driven by an easy axis magnetic ﬁeld or a spin polarized current. The DWtilting strongly affects the DW dynamics for large DMI, and the tilting relaxation time can be very large as it scales with the square of the track width. We propose a simple wayto estimate the DMI in magnetic multilayers by measuringthe dependence of the DW tilt angle on a transverse static magnetic ﬁeld. Our results propose an explanation for the current-induced DW tilting observed in perpendicularlymagnetized Co=Nimultilayers with SIA [ 20] where chiral effects were reported [ 14] and are in agreement with the DMI scenario in these samples. This work was supported by project Agence Nationale de la Recherche, Project No. ANR 11 BS10 008ESPERADO. olivier.boulle@cea.fr [1] I. E. Dzyaloshinskii, Sov. Phys. JETP 5, 1259 (1957). [2] T. Moriya, Phys. Rev. 120, 91 (1960) . [3] M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch,0204060D=2 mJ/m2 D=1 mJ/m2 D=0.6 mJ/m2DW tilt (deg) 01230200400600 J (1012 A/m2)DW velocity (m/s)(b) (d) z 1 -11.5x1012 2.5x10120.5x1012(a) 0123450481216 w=75 nm w=100 nm w=150 nmDW tilt (deg) time (ns)(c) 0450 vn (m/s) J (1012 A/m2) mz 1 -11.5x1012 2.5x10120.5x1012 0123 FIG. 3 (color online). Dynamics of the DW driven by the spin- orbit torque [ /C220HSO¼0:1T=ð1012A=m2Þ] for a 100-nm-wide nanotrack. The results of the micromagnetic simulation (CCM) are plotted in colored dots (continuous lines). (a) Magnetization pattern of the DW for D¼2m J=m2and different values of J. The white arrow indicates the current direction. (b),(d) Tilt angle (b) and velocity (d) of the DW as a function of Jfor different values of D. The inset in (d) shows the DW velocity in the direction perpendicular to the DW surface vnforD¼2m J=m2. The continuous black line is the result of the standard ( q,c) model. (c) Time dependence of the DW tilt angle for differenttrack widths wfor a current of density 0:25/C210 12A=m2 applied at t¼0(D¼2m J=m2).PRL 111, 217203 (2013) PHYSICAL REVIEW LETTERSweek ending 22 NOVEMBER 2013 217203-4S. Blu ¨gel, and R. Wiesendanger, Nature (London) 447, 190 (2007) . [4] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl,and P. Gambardella, Nature (London) 476, 189 (2011) . [5] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012) . [6] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012) . [7] I. M. Miron, P.-J. Zermatten, G. Gaudin, S. Auffret, B. Rodmacq, and A. Schuhl, Phys. Rev. Lett. 102, 137202 (2009) . [8] I. M. Miron et al. ,Nat. Mater. 10, 419 (2011) . [9] A. Thiaville, S. Rohart, E. Jue ´, V. Cros, and A. Fert, Europhys. 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[19] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.111.217203 for adiscussion on other mechanisms which can lead to DW tilting, as well as the effect of the spin transfer torque and ﬁeld like SOT on the DW tilting. [20] K.-S. Ryu, L. Thomas, S.-H. Yang, and S. S. P. Parkin, Appl. Phys. Express 5, 093006 (2012) . [21] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Blu ¨gel, S. Auffret, O. Boulle, G. Gaudin, and P. Gambardella, Nat. Nanotechnol. 8, 587 (2013) . [22] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella, Nat.Mater. 9, 230 (2010). [23] H. Szambolics, J. C. Toussaint, A. Marty, I. M. Miron, and L. D. Buda-Prejbeanu, J. Magn. Magn. Mater. 321, 1912 (2009) . [24] S. Rohart and A. Thiaville, arXiv:1310.0666v1 . [25] See also the corresponding movie for w¼100 nm in the Supplemental Material [ 19]. [26] A. Malozemoff and J. 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PhysRevB.49.10417.pdf	PHYSICAL REVIEW B VOLUME 49,NUMBER 15 15APRIL1994-I Shubnikov —deHaasoscillations underhot-electron conditions inSi/Si,„Ge„heterostructures G.Stoger,G.Brunthaler, andG.Bauer InstitutfiirHalbleiterphysik, Universitiit Linz,A404-0Linz,Austria K.Ismail' andB.S.Meyerson IBMThomasJ.WatsonResearch Center, Yorktomn Heights,¹mYork10598 J.LutzandF.Kuchar InstitutfiirPhysik,Montanuniversitiit Leoben, A8700-Loeben, Austria (Received 27May1993;revisedmanuscript received 17November 1993) Theenergy-loss rateofhotcarriers inseveralmodulation-doped Si/Sil„Ge„heterostructures has beenstudied. TheOhmicproperties oftheSi/Si,„Ge„samples, whichweregrownbyultrahigh- vacuum chemical-vapor deposition, werestudied byHalleffect,conductivity, Shubnikov —deHaas,and quantum Halleffectmeasurements. Forthesamples withmobilities ranging from1.3X10to1.3X10' cm/VsatT=2Ktheratioofthetransport timetothesingle-particle scattering timeincreases from2.4 to7.7.Thisresultclearlyindicates thechangefromdominant short-range toratherlong-range scatter- ingmechanisms inthehigherqualitySi/Sil„Ge„heterostructures. Thedependence oftheenergy-loss rate(PE)onelectron temperature (T,)wasobtained fromthedampingoftheShubnikov —deHaasoscil- lationswithappliedelectric fieldupto5V/cm.Intheelectron temperature rangefrom1.6to7K,the functional dependence ofPEdoesnotchange whenthemobilityofthesamples isvariedbyafactorof 10,andthusPE(T,)isunaffected bythenatureoftheelastic-scattering mechanisms withintheselimits. Inthiselectron temperature rangethedominant energy-loss mechanism isduetoacoustic-phonon scattering viadeformation-potential coupling. Foradeformation-potential coupling constantof9eV, takingstaticscreening intoaccount, aquantitative agreement between experimental andcalculated valuesoftheenergy-loss rateisobtained without anyfitparameter. I.INTRODUCTION Recent advances inepitaxial growth techniques of Si/Si,„Ge„heterostructures haveresulted inhigh- mobility two-dimensional electron (2DEG) andhole gases.'Theseheterostructures areofconsiderable im- portance forpossible deviceapplications. Heterobipolar (HBT}aswellastwo-dimensional electron-gas field-effect transistors (TEGFET's) wererealized bygroupsatIBM (Refs.5-7)andDaimler Benz.'Sincethecarriermobil- itiesinmodulation-doped Si/Si,„Ge„samples available exceedthoseofSi—metal-oxide semiconductor (MOS}de- vicesbyfar,studiesofcarriertransport whichwerepre- viouslyrestricted mainlytotheGaAs/Ga,,AI„Assys- temcanbeperformed nowadays too. Itisthepurposeofthispapertoreporthot-carrier studies inseveralmodulation-doped n-typeSi/Si,„Ge„ heterostructures atlowtemperature. Inordertoachieve atwo-dimensional electron channel, theconduction-band edgeofSihastobelowered withrespecttothatof Si,„Ge„.''"Thetypicalwayofachieving thisrequire- mentisbygrowing arelaxedSi,Gealloyorsuperlat- ticebufferonwhichtheSilayerisunderbiaxialtensile strain.TheGeconcentration inthebuffer,thesequence oflayersinthebuffer,anditstotalthickness determine thedensityofthreading dislocations, whichresultfrom therelaxation process duetotheSi-Gelatticeconstant mismatch (ha/a =4%%uo), anddiminish theelectron mo- bility.Wehavemeasured theenergy-loss rateinsamples whereboththebufferthickness andthethickness ofthe nominally undoped Si,„Ge„spacer layerwerechanged inordertoachieve atransition fromcomparatively low (1.3X10cm/Vs)tohigh-mobility (1.3X10cm/Vs) behavior. Theoretically thelow-temperature electron mobilities anddensities inmodulation-doped Si/Sio7Geo3heterostructures werestudied bySternand Laux' asafunction ofspacerthickness. Gold'has pointedoutthat,duetothehighvalueofthelongitudinal mass(m,—=mt=0.92mo), inhigh-quality Si/Si,„Ge„ structures, interface scattering shouldnotbeofconsider- ableimportance inSiquantum wells(QW's)widerthan 60A.High-quality Si/Si,„Ge„heterostructures have mobilities whicharesuperior toeventhebestSi MOSFET's reported sofar.Si/Si,„Ge„heterostruc- turespossessthepotential toreplace SiMOSstructures forcertainapplications inthefuture.Insuchdevices, duetotheirsmallcharacteristic lengths, theappliedvolt- ageinevitably causeshot-electron phenomena tooccur. Consequently itisdesirable tounderstand thenon-Ohmic transport inSi/Sii„Ge„heterostructures. Inthispaperwereportastudyoftheenergy-loss rate inthreemodulation-doped Si/Si&„Cze„samples. The experimental resultsarecompared withcalculations of theenergy-loss rateduetoacoustic-phonon scattering in- cludingstaticscreening effects.Inaddition, thepresent resultsobtained fortheSi/Si&Ge„heterostructures are compared toprevious experimental andtheoretical data 0163-1829/94/49(15)/10417(9)/$06. 00 4910417 1994TheAmerican Physical Society 10418 G.STOGER etal. 49 II.EXPERIMENTAL DETAILS ThethreeSi/Si,„Ge„heterostructures investigated in thisstudyweregrownat550Conhigh-resistivity p substrates usinganultrahigh-vacuum chemical-vapor deposition (UHV-CVD) system. Figure 1showsasketch ofthesample crosssection together withthe conduction-band diagram. A500-nm-thick, Si/Si&„Ge, superlattice (SL)bufferwithanaverageGecontentof0.3 isgrowntoserveasastrainreliefmultilayer. Abovethe SLeitherapartlyrelaxed 50-nmorfullyrelaxed1500- nm-thick Sip7Gep3alloybufferisgrownwhichdeter- minesthestrainstatusofthesubsequent layers.Then followsthe10-nrn-wide tensilystrained Silayer,andan either4-or15-nm-wide Sip76ep3 spacerlayer.The M1,M2(nm) n 4n+ 4n 4n+M19(nm) SiCap 2 Sip8Gep2Supp&y 4 Sip7Gep3 4 Sip8Gep2Supply 4Surface Sip7Gep3Spacer 15forSiMOSstructures. Inordertodetermine therelevant energy-loss mecha- nismsofhotcarriers atlowlatticetemperatures, the determination oftheirelectron temperature asafunction ofappliedelectric fieldisacommonly usedtechnique. At liquid-helium temperatures thisdependence canbeob- tainedfromthemeasurement ofthedamping oftheam- plitudesofShubnikov —deHaasoscillations inrelatively smallmagnetic fields. Thistechnique wasoriginally usedforthree- dimensional' electron systems, andwaslateradopted for studies oftwo-dimensional electron systems in GaAs/Ga, „Al„As(Refs.15—19)andSiMOSFET's. Furthermore, inhigh-mobility GaAs/Ga& „Al„As het- erostructures aquantum correction totheconductivity, whichisduetoelectron-electron interaction, wasob- servedinthemagnetic-field rangeofthenonoscillatory magnetoresistance. Thesuppression oftheelectron- electron interaction inthehot-electron regimehasbeen usedtodetermine theelectron temperature T,aswell.phosphorus-doped supplylayer(SipsGep2ND=4X10" cm)isfollowed byasequence ofthreecaplayers:a Sip7Gep3layer,asecond 4-nm-thick (ND=2X10's cm)supplylayer,andontopa2-nm-thick Silayer. Thesecondsupplylayerisusedtoprevent depletion by thesurfacepotential.' SincetheSichannel experiences tensilestraininthe structures, thesixfold-degenerate levelsaresplitintodou- blydegenerate A2andfourfold-degenerate h4states.The lowest-lying statesarethe62states,whichareoriented withtheirmainaxisperpendicular tothe(001)surface. Thelongitudinal massisgivenbym,=0.92mp,andthe transversal in-plane massbym,=0.19mp.Forthissitua- tionthetwo-dimensional electron gasisconfined ina triangular-shaped potential well.'Forthethreesamples investigated (whicharelabeledMl,M2,andM19)all electrons areinthelowestsubband, asdeduced from Fourier analysisoftheShubnikov —deHaasoscillations. ThecarriersarelocatedonthesideoftheSilayerwhich isclosertothesupplylayer.Theconduction-band offset isknowntobeabout120—150meV;theexactvaluede- pendsontheamountofstrainrelaxation inthebufferlay- ers.Themaindifferences between samplesM1andM2 ontheonehand,andM19ontheother,arethe thicknesses ofthebufferandspacerlayers,whichhave beenincreased from50to1500nmandfrom4to15nm, respectively (Fig.1).Thisreduces theamountofcarrier transfer intothe2DEGchannel from9.8X10" cm (Ml)toabout4.7X10" cm(M19)andincreases the low-temperature mobility fromabout1.3X10(M2)to l.3X10cm/Vs(M19). Thesampleparameters are listedinTableI. Aspointed outbyNelsonetal.andXieetal.,the low-temperature mobility inthe2Dchannels grownon thickbuffers isnotlimited bythedensityofthreading dislocations fordensities lessthan10cm.Inhigh- mobility UHV-CVD samples comparable toM19,the threading-dislocation density wasdetermined tobeon theorderof10cm(Ref.23). Thetransport measurements wereperformed onlitho- graphically definedHallbarswithachannel widthofei- ther140(M1,M2)or75pm(M19)andachannel length of450(Ml,M2)or1500iMm(M19). Thevoltagedrop wasmeasured withvoltageprobes200or750pmapart. OhmiccontactsofanAuSballoywereusedforallsam- ples.ForsamplesM1andM19thecontacts werean- nealedat300'C,whereas forsampleM2theywerenot. SiChanne& 10 l I Sip+ep3Buffer 1500 Ef ETABLEI.Carrierconcentration, Hallmobility, Fermiener- gy,Dingletemperature, andscattering timesafterillumination atT=2Kforlowelectricfields. 500 Sj/SjGe SL 500Sample M2 M19 SiSubstrate FIG.1.Sketchofthesample structure, withlayer thicknesses giveninnmforsamplesM1,M2,andM19aswell astheconduction-band diagram.n(10'cm) I(cm'rV s) EF(meV) TD(K) ~,(ps) ~,(ps) ~,/z,0.98 1.43X10 6.2 1.9 0.64 1~55 2.40.88 1.31X10 5.6 2.5 0.49 1.42 2.90.47 1.32X10 3.0 0.67 1.8 14.3 7.9 49 SHUBNIKOV-de HAASOSCILLATIONS UNDERHOT-... 10419 LateritwasfoundthatsamplesM1andM19weresensi- tivetoband-gap illumination atlowtemperatures and showed persistent conductivity effects, whereas sample M2didnotexhibitanydetectable sensitivity toillumina- tion. Hall-efFect investigations wereperformed inabathcry- ostatat4.2K,andinacontinuous-flow cryostat with variable-temperature controlatmagnetic fieldsoftypical- fy0.35T. Shubnikov-de Haas(SdH)oscillations wererecorded underconstant-current conditions foramagnetic fieldap- pliedperpendicular totheplaneofthe2DEG. Increas- inglattice orelectron temperature damps the Shubnikov-de Haasoscillations through thermal broadening, i.e.,through changing theratioA'co,/ksT, (Ace,isthecyclotron energy), whichcrucially determines theoscillatory magnetoresistance. HencetheSdHam- plitudes dependexplicitly ontheelectron temperature T, butnotonthelatticetemperature TL.Theelectron tem- peratures athigherinputpoweraredetermined bycom- paringtheratiooftheresulting amplitudes totheampli- tudeobtained underOhmicconditions (atT,=2K)with theoryaccording totheprocedure outlined byBauerand Kahlert.'Itisassumed thattheDingletemperature TD remains constant anddoesnotchangewithelectric field. UnderOhmicconditions, wefoundnodependence ofthe Dingletemperature onlatticetemperature intherange fromTI=1.6-4.2K. Forcomparison, thedamping oftheSdHoscillations wasmeasured atahigherlatticetemperature of4.2K andlowinputpower(i.e.,intheOhmicregime). These calibration dataat4.2Kareingoodagreement withthe electron temperatures evaluated fromthedamping ofthe SdHoscillations athigherinputpower.Indeed,thiscali- brationprocedure reliesonthefactthattheamplitudes oftheSdHoscillations dependontheelectron tempera- turewhichappears intheelectron distribution function. Duetotheresistivity oscillations, theelectric fieldbe- tweenthevoltage probesisnotconstant, butforsmall SdHoscillations thedeviations areonlyinthepercent range.TheSdHmeasurements underhot-electron condi- tionswereperformed atalatticetemperature ofabout2 K.Thecurrents wereoftheorderof0.1-100pA,corre- sponding toelectric fieldsintherangefrom1mV/cm to 5V/cm.Forlowerelectric fieldsnochangeinthedamp- ingoftheSdHoscillations wasobserved. Fortheevaluation ofthedataithastobeconsidered thatthecalculation oftheenergy-loss rateusedbelowis basedontheassumption ofaconstant 2Ddensityof states.TheSdHoscillations occurinaregimewherethe magnetic fieldaltersthe2Ddensityofstatesandhence alsothescattering ratesofthecarriers. Inordertokeep thisdisturbance small,themagnetic fieldfortheevalua- tionoftheSdHoscillations hastoberestricted to suSciently smallvaluessothathR/Risoftheorder ofafewpercent. III.OHMIC TRANSPORT Thesamples werecharacterized byHall-effect andcon- ductivity measurements. Theresulting temperaturedependences ofcarrierconcentration andHallmobility forthelow-mobility sampleMlandhigh-mobility sam- pleM19areshowninFig.2.Theroom-temperature mobilities wereabout1200and2000cm/Vs,respective- ly.Atlowtemperatures, bothsamples displayed inFig.2 showapersistent increaseofelectron mobility afteril- lumination withaGaAslight-emitting diode(LED) (A,=950nm).Thereisadifference between theheating andcooling curvesfortemperatures uptoabout100K. Forhighertemperatures thepersistent ormetastable effectspresent inmobility aswellasincarrierconcentra- tionvanish. TheHalldataforsampleM2arecompara- bletothoseofsampleM1,buttheydonotshowanysen- sitivitytoband-gap illumination. Duetothesensitivity 1.4 CD 1.2 0 ~%~~I 0.8 00.6M1~ OOOOO~~— ~~0 EL 0.4dddd gJgkam- 0.2 10 T(K)100300 105 M19 kp ~sill ~l~~l104— 1000 10 T(K)100300 FIG.2.Carrierconcentration (a)andHallmobility (b)vs temperature ofsamplesM1andM19duringcoolingfromroom temperature to4or1.8Kbeforeillumination (fullsymbols), and duringwarming upafterillumination (opensymbols). 10420 G.STOGER etal. 49 3000 ( ~2500 2000 ~A 1500 O1000 ~50OIII[III II=311=214000 12000 10000e 8000 6000 tv) 4000 2000 IIIIIIIIIII» IIIIsII~IIIII0 246810121416 Magnetic Field(T) FIG.3.SdHoscillations andquantum Halleffectforsample M2atT=1.8K.Spinsplitstates(f/$)ofthesecondLandau leveln=2andvalleysplitting(+/—)forn=1areclearlyob- servable. Hallplateaus areindicated forfillingfactorsv=2,4, and8.ofsamplesM1andM19toband-gap radiation, alltrans- portmeasurements wererecorded afterillumination at lowtemperatures. Asanexplanation forthesensitivity ofthesesamples toillumination withband-gap radiation, wesuggest a process whereionizedimpurities, locatedclosetothe2D electrons, areneutralized. During illumination, freecar- riersareexcitedintohigh-energy statesabovetheband offsetbetween SiGeandSilayers.Fromtheresomecar- rierswillentertheSiGebarrierlayerandwillbeableto neutralize localdefects. Asimilarchargetransfer across heterojunctions wasobserved inAlGa,,As/GaAs quantum wellsduringillumination. Inordertocharacterize thesamples further, theresis- tivityandHalleffectofthelow-mobility samplesM1and M2weremeasured atT=1.8Kuptomagnetic fieldsof 16T.TheseresultsareshownforsampleM2atT=1.8 KinFig.3.Forfillingfactorsbetweenv=4and8the spinsplitting isclear1yresolved inp„.Forhighermag- neticfieldspshowstheresultsoftheliftingofthetwo- foldvalleydegeneracy attheoddfillingfactorv=3.For magnetic fields8(4T,thernagnetoresistance oscilla- tionscorrespond tofillingfactorswhicharemultiples of 4(sinceneitherthetwofold spinsplitting northetwofold valleysplitting isresolved). Fromthevaluesofthev=4 and8integerquantum Hallplateaus, oneconcludes that theamountofparallelconduction isnegligible. Thecar- rierconcentrations evaluated fromtheSdHoscillations are,withinexperimental error,identical tothoseobtained fromtheHallmeasurements onallsamples, whichfur- therprovesthatparallel conduction isnegligible atlow temperatures. TheDingletemperature TDisdeduced fromthedamp- ingoftheSdHoscillations asafunctionofmagnetic field. Forthelow-mobility samplesM1andM2weobtained TD=1.9and2.5K,andforthehigh-mobility sample M19avalueofTD=0.67K(TableI). Theratiooftransport relaxation time~,tothesingle- particle relaxation timer,(orquantum lifetimer)de-ducedfromtheDingletemperature Tzyieldsinforma- tionaboutthedominant scattering mechanism.'' Thetime~,describes aLorentzian broadening ofthe Landau levelsduetoscattering ofelectrons. Thetwo characteristic timesarecalculated fromp,=ex.,/m,and A/2m',=k&TD, respectively, whereeistheelementary charge, m,=0.19moistheeffective transverse massof conduction-band electrons, andA'isPlanck's constant. Thesingle-particle orquantum 1ifetime isdetermined by thetotalscattering rate —=fW(k,k')d8,1 T5 whereW(k,k')isthetransition rateforscattering from wavevectorkintok'and8denotes thescattering angle. Intherelaxation timeansatzforsolvingtheBoltzmann transport equation, thetransport relaxation timeisgiven by —=fW(k,k')(1—cos8)d8,1 i.e.,itisweighted bythescattering angle.Coleridge, Stoner, andFletcher andDasSarmaandStern have pointed outthatforashort-range scattering potential theratior,/r,isclosetounity,whereas forlong- rangescattering thisratioismuchlarger. Indeed, for remote impurity scattering inmodulation-doped GaAs/Ga,,Al,Asstructures, thisratiowasfoundtobe 10ormore. ForsampleMlwedetermined aratior,/r,=2.4, whileforsampleM2weobtained avalueofr,/7;=2.9. Fromtheseratiosweconclude thatinsamplesM1and M2,wherethespacerthickness is4nrn,remoteimpurity scattering definitely doesnotlimitthemobility. Accord- ingtoSternandLaux'aratior,/~,ofabout6—7isex- pectedfortheseparticular samples. Consequently other scattering mechanisms limitthemobility, suchasunin- tentional impurities inthe2DEGchannel orthespacer layer,orscattering duetointerface roughness. Forthehigh-mobility sampleM19theratior,/r,is about8.Foraspacerlayerof15nm,r,/r,valueswell above10wouldbeexpected fromthecalculations inRef. 12.Basedonthesecalculations onecannotdrawthecon- clusionthatthemobility inthehigh-mobility sample is limited byremote impurity scattering only.The inhuence ofinterface roughness onthemobility isnotyet wellestablished.'Goldhascalculated valuesoftheelec- tronmobility limited byinterface roughness in Si/Si,Gerectangular quantum wellsforseveral we11 widthsranging from40to120A.'Heobtained limiting valuesforpofabout10crn/Vsforthe40-Awells,and about10can/Vsforthe60-Awells.However, thesere- sultsarenotapplicable foradirectcomparison withthe dataonone-sided modulation-doped Si/Sil„Gestruc- tures.Gold'sresultwasobtained assuming asymmetric, sinusoidal wavefunctionofthegroundstatewhichvan- ishesattheboundaries. Inatriangular potential well, theprobability offindingtheelectron inthelowestsub- bandclosetotheinterface ishighercompared tothatofa squarewellwithinfinitely highpotential boundaries. 49 SHUBNIKOVMe HAASOSCILLATIONS UNDERHOT-... 10421 IV.NON-OHMIC TRANSPORT Thedamping oftheamplitudes oftheShubnikov —de Haasoscillations withincreasing electric fieldwasusedto determine theelectron temperature. Forthesemeasure- mentsthelatticetemperature waskeptconstant, i.e.,the samples werekeptimmersed insuperQuid heliumata bathtemperature ofT=1.8KforsampleM1,T=2.1K forsampleM2,andT=1.55KforsampleM19. Theelectric fieldsemployed aresosmall(lessthan2 V/cm, 5V/cm,and300mV/cm forthethreesamples Ml,M2,andM19,respectively) thatthereisnoappre- ciablepopulation ofthesecondelectric subband atall, i.e.,aFourier analysisoftheSdHdatarevealsthateven atthemaximum electric fieldsonlythelowestelectron subband isoccupied. Atmuchhigherfields(F)8V/cm) anincrease inthemobility wasobserved, whichisattri- butedtoapopulation ofthesecondsubband, similarto resultsofinvestigations onSiMOSstructures. Thedataoftheexperimentally obtained energy-loss rateasafunctionofT,forthethreesamples areshown inFig.4,andaredenoted bytheopensymbols. Insteady state,thelossrateperelectron PEequalstheinputpower whichisgivenbyPE=U/RN,=epF, whereRisthe sampleresistance, Uthevoltagedrop,andN,thetotal numberofelectrons. Themobilitypvarieswithapplied electric fieldFbyroughly15%(measured atzeromag- neticfieldatTI=2K}andhastobeconsidered forthe evaluation. Itisworthpointing outthatP,(T,)isnearlyidentical forallthreesamples, despitethefactorof10difference in electron mobility andthusalsointhepowerinputfora 1013 S10-14 S4givenelectric field.Thesedataprovethattheenergy-loss rateisvirtually unaffected bytheelasticscattering mech- anisms, asshouldbethecaseinamodelwherethehot- carrierdistribution function isaFermidistribution func- tionwithT,replacing TI.Tothebestofourknowledge, thedependence ofPE(T,}hasnotbeeninvestigated for suchawiderangeof2DEG mobilities in GaAs/Ga, „Al„AsorSiMOSdevices. InFig.4experimental dataonPE(T,)obtained bythe samemethod onSiMOSstructures byNeugebauer and Landwehr areshownaswell.TheirsampleP77M2PP hadasheetelectron densityofn=3.7X10' cmand anelectron mobilityof@=4000 cm/Vs(Ref.30)at Tz=1.7K.Thereisanapparent difference between ex- perimental resultsforSi/Si,„Ge„heterostructures and theSiMOSdeviceswhichwillbediscussed inSec.VI. V.ENERGY-LOSS RATE:ACOUSTIC PHONONS Theenergy-loss rateduetoacoustic-phonon scattering intwo-dimensional systems, e.g.,inSiMOSstructures andinGaAs/Ga& „Al„As heterostructures, hasbeen treated byseveralauthors.'''''ApartfromManion etal.,'whousedaselfconsistently calculated wave function, allauthors havesofarassumed thevariational ansatzofFang,Howard, andSternfortheirwavefunc- tioninthelowestsubband, asdiscussed inRef.32. Forthecaseof(001)Si/Sio7Geo ~heterostructures the useofthevariational wavefunction withanodeatthein- terfaceiswelljustified duetothefactthatthelongitudi- nalmassm,=m,=0.92moissolargethatthesmall penetration ofthewavefunction ofthelowestelectric subband intotheSiQ7Ge03barrier willnotmodifythat energy-loss ratesubstantially. Therelevant Simassis largerbyaboutafactorofabout14thanthatoftheelec- tronsinaGaAsquantum well. Thenetaverage energy-loss rateperelectron iscalcu- latedfromtheenergygainedbythephonons fromthe hotelectrons. Usingthisapproach, wewrite 10-'5 DC cn10-160 C410-17 10-18-----Shinba—-—Si-MOS I II~ ~III~IIIIIII~I~~IIIIIIItIII~IIIwheref(E,T,)=Iexp[(E E~)/k~T,]+1—j'isthe Fermi-Dirac distribution foranelectron temperature T, different fromthelatticetemperature. Thenetenergy- lossrateofanelectron withenergyEisexpressed as 1234567 T(K) E+8—E0dO, (4) FIG.4.Energy-loss ratePEvselectron temperature T, determined foralatticetemperature ofTL=1.8Kforsample M1(circles), 2.1KforsampleM2(squares), and1.55Kfor sampleM19(triangles). ThedataforaSiMOSdevicefrom Ref.20(fulldiamonds) areindicated forcomparison (TL=1.7 K).Theenergy-loss ratesduetoacoustic-phonon scattering for theparameters ofsamplesM1(solidline),M2(dottedline),and M19(dash-dotted line),andfortheSiMOSstructure (large dash-dotted line)according toourcalculations andforparame- terssimilartosamplesM1andM2according tocalculations by Shinbaetal.(Ref.31)(dashed line)arealsoshown.XN[1f(E+iticoq)]— forphononabsorption, and E(8)=fdq,ficoq~I(q, )~P(q„~,q,)(N+1) X[1f(Efico)]e(E ficoq)———(5)withthedifferential energychange duetoscattering of the2Delectrons withwavevectorkexpressed as E+(~)=fdq,+coq II(q,)IP(q~q,} 10422 6.STOGER etaI,. 49 forphonon emission. Sincethedifferential scattering crosssections dependonthephonon energy,thephonon energy ficowasnottakenoutsideoftheintegral inthe expressions forE+(8)andE(8)[Eqs.(5)and(6)],in contrast toRef.17. InEqs.(3)—(6),8isthescattering angle,and E=Ak/2mistheelectron kineticenergy, withI' theelectron in-plane effective massm,andk=~k~.The phonon components parallel andnormaltothehetero- planearedenoted byq„~andq„respectively. ~I(q,)~is theformfactorwhichdescribes thecoupling oftheelec- tronwavefunction tothezcomponent ofthescattered phonon,P(q,q,)isthescattering probability factor, N=[exp(fuu~ /kttTL)—1]'thephonon occupation number, ande(E)theunit-step function. Thephonon energy %coisgivenbyfiutqforlongitudinal-acoustic modes. HereuListhelongitudinal soundvelocity, and q—(q2+q2)1/2 Thescattering probability factorfortheacoustic- phonon scattering viadeformation-potential coupling P(q„„,q,)(Ref.17)wascalculated usingcL=1.68X10" N/mforthelongitudinal elasticconstant, andtaking thestaticscreening factorwhichwasgivenbyHirakawa andSakaki according tothetreatment originally de- rivedbyPriceforthescreening ofthedeformation- potential interaction intherandom-phase approximation. Thedeformation-potential constant fortheinteraction withlongitudinal-acoustic (LA)modeswastakentobe D=9eVwhichisthesamevalueasusedbySternand Laux' forcalculating theOhmic mobility in Si/Si&„Ge„heterostructures. Forthelongitudinal soundvelocity anaverage valueuL=8.8X10m/swas used. Thevariational parameter b(whichisinversely corre- latedwiththeextension ofthewavefunction) wascal- culated usinga=11.7forthedielectric constant, and determined tobe9.3X10,8.8X10, and7.3X10 rn forthethreesamplesM1,M2,andM19,respectively. FortheLA-phonon scattering intheheterostructure, weassumethatthephonon energies arethoseofthebulk material whichformsthewells,i.e.,intheSi/Si,„Ge„ caseweusethebulkSiphonon modes. Thisassumption isinagreement withcalculations previously published for theenergy-loss rateinSiMOSstructures and GaAs/Ga, „Al„As heterostructures. Wewanttopoint outthatintheSi-based heterostructures theacoustic mismatch oftheLA-phonon modes across the Si/Si,„Ge„interface ismuchlessthanfortheSiMOS structures containing anamorphous oxide. InFig.4ourcalculated energy-loss ratesforcarrier concentrations andlatticetemperatures corresponding to thethreesamples areshown.Furthermore weperformed thesamecalculation fortheenergy-loss ratewithparam- eterscorresponding toaSiMOSdeviceusedbyNeu- gebauer andLandwehr. Inaddition, calculated databy Shinbaetal.'fortheenergy-loss rateinaSiMOSde- viceareincluded. ThedatatakenfromRef.31arebased onaealeulation comparable tooursforacarrierconcen- trationof1.1X10'cmandTI=2K,alsotakingstat- icscreening effectsintoaccount. Theseparameters aregp-8 gp-9 0.1 1 Te-Tl(K)10 FIG.5.Energyrelaxation time~,forSi/Si&„Ge„hetero- structures M1(opencircles)andM19(triangles) vsT,-TLasde- ducedfromtheexperimentally determined energy-loss rate. Fulllines:~,determined fromthecalculated energy-loss rate. closetothoseforsamplesMlandM2.Theresultsof Shinbaetal.'forPE(T, )agreequitewellwiththe presentones. Comparison ofthecalculations forsamplesM1and M19,whichdifferbyafactorof2incarrierconcentra- tion,showsthatforagivenelectron temperature theloss rateincreases withdecreasing carrier concentration, whichisduepartlytotheweaker screening effects. On theotherhand,varyingthelatticetemperature andkeep- ingthecarrierconcentration constant affectsthelossrate substantially onlyatelectron temperatures inthevicinity ofTL.Thisbehavior isdemonstrated bytheresultsof thecalculations forsamplesM1andM2whichhave quitesimilarcarrierconcentration, butforwhichthe damping oftheSdHoscillations wasevaluated atlattice temperatures of1.8and2.1K.Thecalculated lossrates forthesedifferent latticetemperatures almostcoincide forT,)3K. Itisnotintended toproduce afitoftheenergy-loss ratevsT,withpowerlaws.Theselawsdepend onap- proxirnations whichareneverfulfilled overanextended temperature range.However, thestandard definition of anenergyrelaxation time"r,"(see,e.g.,Ref.18)isused inordertoobtainanestimateofitsorderofmagnitude, despitethefactthatitactually cannotbedinedoverthe entirerangeofelectron temperatures investigated here fortheSi/Si,„Ge„heterostructures. Theenergyrelaxa- tiontimeisdetermined from k(T,Tt)PF'.— 6EF Figure 5shows~,asafunctionofT,-TLforsamples MlandM19(whichdifferincarrierconcentration and thusinFermienergybyafactorof2)together withthe calculations basedonEqs.(3)—(6)fortheenergy-los rate. Figure 5showsthat~,isabout4—2.5ordersofmagni- tudelargerthanthetransport timer,(TableI)inthe electron temperature rangefrom1.6to7K.Evenfor small(T,Tt)theenergyrelax-ation timecannotbeap- proximated byaconstant value. SHUBNIKOV-dc HAASOSCILLATIONS UNDERHOT-... 10423 VI.DISCUSSION Theenergy-loss rateinSi/Si&Ge„heterostructures wasdetermined experimentally andcompared withcalcu- latedenergy-loss ratesduetoacoustic-phonon scattering including staticscreening inFig.4.Thedataweretaken on20electron gaseswithelectron mobilities from 1.3X10to1.3X10cm/Vs.Asexpected fromasim- pleelectron temperature model consideration, the energy-loss rateturnsouttobeessentially independent of theelasticscattering mechanisms, sincetheexperimental- lydetermined lossratesforbothhigh-andlow-mobility samples showthesamefunctional dependence. Theex- perimental dataforsamplesM1andM19coincide al- mostexactly withthecalculated energy-loss data.%e wanttopointoutthatthisagreement isachieved without anyfittingparameter. ForsampleM2(Tz=2.1K)the calculation doesnotdifferfromthatforsampleM1 (TL=1.8K)forelectron temperatures above3K(Fig. 4).However, theexperimental datadifferslightly from thecalculated ones.Previous calculations oftheenergy- lossratebyShinbaetal.'forSiMOSdevicesarein goodagreement withthepresentcalculated datashown inFig.4.Similartothecalculation presented inSec.V fortheenergy-loss rateinSi/Si,„Ge„heterostructures, Shinbaetal.'assumed thatinSiMOSstructures the two-dimensional hotcarriers losetheirexcessenergy through interaction withbulklike acoustic phonons insil- icon. Furthermore, Fig.4showsthattheexperimentally determined energy-loss rateinaSiMOSdevice(p,=4000 cm/Vs)(Ref.36)atagivenelectron temperature ac- cordingtoNeugebauer andLandwehr isaboutanorder ofmagnitude belowthatdetermined fortheSi/Si&„Ge„ heterostructures. Thisresultisquiteastonishing. Al- thoughourcalculated energy-loss rateforthisSiMOS deviceissomewhat reduced byscreening duetoitshigher carrierconcentration of3.7X10'cmincomparison to thatofthepresentSi/Si,„Ge„samples (1X10' and 4.7X10"cm),thecalculated P,(T,)valuesforthisSi MOSdeviceareconsiderably higherthantheexperimen- taldata(seeFig.4).Inaddition, weevaluated thePE(T,) dependence fromdatagivenbyHonlein andLandwehr foraSiMOSdevicewithacarrierconcentration of 2.5X10' cm2andamobility of7000cm/Vsat Tz=2.0K.Theenergy-loss ratederived fromtheirex- perimental datashowsalmostexactlythesamefunc- tionaldependence astheonetakenfromRef.20 (displayed inFig.4).ThusforSiMOSFET's thereisan apparent discrepancy between thecalculated energy-loss rate(ourcalculation aswellasthatofShinbaetal.) andtheexperimentally derived values.'Incontrast, asshownabove,thereisaperfectagreement oftheory andexperiment forSi/Si& Creheterostructures. Inthe following possible originsforthisdifference between Si MOSdevicesandSi/Si,Geheterostructures aredis- cussed. Toaccount forthelowerenergy-loss rateintheSi MOSdevice incomparison toboththelow-andhigh- mobility Si/Si&Geheterostructure, onewouldhavetodecrease thevalueofthedeformation-potential constant inthecalculation substantially, i.e.,toD=3.5eVwhich isunrealistically small.Another possibility toexplain Pz(T, )wouldbetoassumethatacoustic-phonon modes attheinterface totheSioxidediffersubstantially from thoseinbulkSi,orthatlong-wavelength phonons areal- teredbytheproximity ofthesurface. %'enotethatthe longitudinal acoustic-phonon modes inSi,Ge„are quitesimilartothoseinSi,whereas thisisnotthecase fortheacoustic modesinamorphous Si02. Thereisafurtherpuzzlewiththequantitative explana- tionofhot-electron transport inSiMOSdevices at liquid-helium temperatures. Manion eta/.'haveargued thatexperimental dataforthehot-electron mobility inSi MOSdevices giveevidence thatthecurrent theoriesof scattering inquasi-2D systems donotaccount properly forthedeformation-potential interaction atinterfaces sinceanomalously largedeformation-potential constants werenecessary tofitthemobility data.Thisisinstark contrast totheenergy-loss data,whichrequiretoosmall adeformation-potential constant. Uptonow,this discrepancy hasnotbeensolved. IntheSi/Si,„Ge„samples, wedidnotobserve any significant dependence ofT,onmagnetic field8evalu- atedinthemagnetic-field rangebelow2.5or1.5Teither forthelow-orhigh-mobility samples. For GaAs/Ga, „Al„As heterojunctions, Leadley etal.'re- portedadependence T,(B)whichwasinterpreted asevi- denceofcyclotron phonon emission. Theseauthors re- portedthatthecyclotron phonon emission isimportant fork~T,&fico,/A,,where A,isafactorbetween 2and3. Ifasimilarcondition werevalidforSi/Si,„Ge„hetero- structures, theeffectshouldbeobserved forSdHoscilla- tionsbelow0.6Tinoursamples. However, forthe Si/Si,„Ge„samples investigated theDingletempera- tureistoolargeeveninthehigh-mobility sampleM19, andthusthemagnetoresistance doesnotshowoscilla- tionsbelow8=0.6TatTL=1.55K.Forhigherelec- trontemperatures thecondition k~T,&%co,/1,cannotbe fulfilled sincetheSdHoscillations vanish inthecorre- sponding low-magnetic-field range.InSiMOSstructures cyclotron phonon emission wasobserved byChallisand Kentbyathermal detection oftheemitted phonon in- tensityasafunctionofgatevoltage. Apartfromtheexcellent agreement between calculated andexperimentally observed dependences PE(T, )forthe Si/Si,„Ge„heterostructures, wehaverecently obtained furthersupport fortheuseoftheelectron temperature modelattheselowlatticetemperatures andcomparative- lysmallappliedelectric fields.Fromexperiments onthe weaklocalization effectonsampleM2,thetemperature dependence ofthephasecoherence timewasdeduced, andfoundtobeintherangeofabout 1psfortempera- turesbetween 1.8and4.2K.Thisphasecoherence timeisdetermined byinelastic electron-electron- scattering processes. Itsvalueiscomparable tothetrans- port(momentum relaxation) time,andbothareabout twotothreeordersofmagnitude smallerthantheenergy relaxation time.Therefore theuseofaFermidistribu- tionfunction withanelectron temperature higherthan 10424 G.STOGER etal. 49 thelatticetemperature isappropriate. Besides thehigh-field transport properties, thereare apparent differences intheOhmictransport inSiMOS devices andSi/Si,„Ge,heterostructures. Aspointed outbyGold,'thelimiting scattering mechanism forthe Ohmicmobilityofthe2DEGintheSichannel isdueto interface charges. Thehighestelectron mobility(4X10 cm/Vs)reported sofarforSiMOSdevicescorresponds toabout10'interface charged percm.Naturally, inter- facecharges arelessimportant inSi/Si,„Ge,hetero- structures. Asalready pointed out,remote impurity scattering isthedominant scattering mechanism inhigh- mobility Si/Si,„Ge„samples (p)10'cm/Vs, r,/r,=10)according toSternandLaux.' Thefactorof10variation intheOhmicmobilities in thepresent Si/SiQ7Geo3samples resultsbothfrom different bufferlayerthicknesses andthereduction ofre- moteimpurity scattering duetodifferent spacerlayer thicknesses. Nelsoneta/.haveshownthatthechange ofthebufferlayerthickness isaccompanied byadrastic increaseofthecorrelation lengthofsurface roughness fromabout50to1000A,whencomparing samplesM1 andM2ontheonehand,andM19ontheother.Similar phenomena werereported byXieetal.Ourexperimen- talenergy-loss ratedataonthethreeSi/Si,„Ge,sam- plesnowprovethattheinterface roughness doesnot influence thefunctional dependence ofPz(T, )inthe rangeofelectric fieldsinvestigated.VII.CONCLUSION Theenergy-loss rateforhotelectrons inSi/Si,„Ge„ modulation-doped heterostructures wasobtained asa functionoftheelectron temperature intherangefrom 1.6to7K.ThedampingofShubnikov —deHaasoscilla- tionamplitudes withappliedelectric fieldwasusedtoex- perimentally determine theelectron temperature. These experiments werecompared withcalculations ofthe energy-loss rateforcarrierscattering byacoustic pho- nonsviathedeformation-potential coupling, takingstatic screening oftheelectron-phonon interaction intoac- count.Thecalculations oftheenergy-loss ratesrepro- ducetheobserved dependence Pz(T, )inallthreesam- pleswithout anyfitparameter, despitethefactthattheir mobilities varyfrom1.3X10to1.3X10cm/Vs. Theseresultsdemonstrate thattheenergy-loss ratedoes notdepend ontheelastic-scattering processes (remote ionized impurity scattering andinterface roughness scattering) whichlimittheelectron mobility intherange ofelectron temperatures investigated. ACKNOWLEDGMENTS Thisworkwassupported byProject No.Gz 601.528/2-26/92 of"Bundesministerium fiirWissen- schaftundForschung", Vienna. WethankM.Helmfor acriticalreadingofthemanuscript. Present address: Dept.ofElectronics, FacultyofEngineering, CairoUniversity, Egypt. 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PhysRevB.85.224401.pdf	PHYSICAL REVIEW B 85, 224401 (2012) Bloch point structure in a magnetic nanosphere Oleksandr V . Pylypovskyi,1Denis D. Sheka,1,2,and Yuri Gaididei2 1Taras Shevchenko National University of Kiev, 01601 Kiev, Ukraine 2Institute for Theoretical Physics, 03143 Kiev, Ukraine (Received 20 December 2011; revised manuscript received 1 March 2012; published 1 June 2012) A Bloch point singularity can form a metastable state in a magnetic nanosphere. We classify possible types of Bloch points and analytically derive the shape of the magnetization distribution for different Bloch points.We show that an external gradient ﬁeld can stabilize the Bloch point: The shape of the Bloch point becomesradially dependent. We compute the magnetization structure of the nanosphere, which is in good agreement withperformed spin-lattice simulations. DOI: 10.1103/PhysRevB.85.224401 PACS number(s): 75 .10.Hk, 75 .40.Mg, 05 .45.−a I. INTRODUCTION Topological singularities are widely recognized as a key for understanding the behavior of a wide variety of condensed- matter systems. Linear topological singularities, such as dislocations, disclinations, and vortices play a crucial rolein low-dimensional phase transitions, 1crystalline ordering on curved surfaces,2rotating trapped Bose-Einstein condensates,3 etc. Recent advances in microstructuring technology havemade it possible to fabricate various nanoparticles with well-prescribed geometry. Much recent research in this ﬁeld hasfocused on the statics and dynamics of topological singularitiesin nanoscale conﬁned systems: Essentially, inhomogeneousstates can be realized in magnetic nanoparticles 4–6and ferro- electric nanoparticles.7As a result of the competition between exchange and magnetic dipole-dipole interactions, the groundstate of magnetic disks with sizes larger than some tens of nanometers is a ﬂux-closure vortex state. Besides linear singularities, there also exist so-called point singularities, such as monopoles, Bloch points, andboojums. For example, hedgehog (monopole) singularitiesplay a crucial role in the behavior of matter near quantumphase transitions that are seen in a variety of experimentallyrelevant two-dimensional antiferromagnets, 8boojums are rel- evant in superﬂuid He-3,9and Bloch points along with Bloch lines are principal in the understanding of magnetic bubbledynamics. 4,10 The concept of point singularities in magnetism was introduced by Feldtkeller,11who considered different mag- netization distributions around the singularity and proposedthe ﬁrst estimations of the Bloch point shape. Later, D ¨oring 12 studied how magnetostatic energy governed the Bloch point structure by selecting the rotation angle inside the Blochpoint. Bloch point singularities were directly observed inyttrium iron garnet crystals. 13During the last decade, Bloch points were also studied by micromagnetic simulations innanowires, 14in bubble materials,15and in disk-shaped16,17 and astroid-shaped nanodots.18The ultrafast switching of the vortex core magnetization opens doors to consider the vortexstate nanoparticles as promising candidates for magneticelements of storage devices. There are different scenariosof the switching process: (i) The symmetric or so-calledpunch-through core reversal takes place under the action ofa dc magnetic ﬁeld applied perpendicularly to the magnetplane. 9,16,19,20This reversal process, as a rule, is mediated bythe creation of two Bloch points.16However, the single Bloch point scenario was also mentioned in Thiaville et al.16(ii) The switching, under the action of different in-plane ac magneticﬁelds or by spin-polarized currents, 21–26is accompanied by the temporary creation and annihilation of the vortex-antivortexpair. The latter is accompanied by Bloch point creation. 17 The purpose of the current paper is to study the magneti- zation structure of the Bloch point in the spherical nanosizedparticle. As opposed to bubble ﬁlms where the static Blochpoint results from the transition between Bloch lines 4,10and vortex nanodots where the Bloch point dynamically appearsduring the vortex core switching process, 16,21the nanosphere is a natural geometry where the Bloch point forms a metastablestatic conﬁguration. Such a singularity is, in some respect, the only stable singularity in the ferromagnet. 16We consider different types of Bloch points and classify them in terms ofvortex parameters. The conventional magnetization distribu-tion in the Bloch point is generalized for the radially dependentone. Such radial distribution becomes important for the Blochpoint nanosphere under the action of a nonhomogeneousmagnetic ﬁeld. We show that a radial gradient ﬁeld can stabilizethe Bloch point and we compute the magnetization structure,which is in good agreement with performed spin-latticesimulations. The paper is organized as follows. In Sec. II, we describe the model and present the classiﬁcation of different Blochpoint types (Sec. II A). The energetic analysis and the Bloch structure is analyzed in Sec. II B. In order to stabilize the Bloch point inside the nanosphere, we consider the inﬂuence ofan external gradient ﬁeld on the magnetization structure. TheBloch point solution becomes radially dependent: We calculatethe magnetization structure analytically in Sec. III. In Sec. IV, we study the Bloch point structure numerically, in particular,the problem of stability. We discuss our results in Sec. V.I n the Appendix, we analyze the Bloch point structure under theinﬂuence of weak ﬁelds using the linearized equations. II. THE MODEL AND THE BLOCH POINT SOLUTIONS Let us consider the classical isotropic ferromagnetic sphere of radius R. The continuum dynamics of the magnetization can be described in terms of the magnetization unit vector m=M/MS=(sin/Theta1cos/Phi1,sin/Theta1sin/Phi1,cos/Theta1), where /Theta1 and/Phi1are, in general, functions of the coordinates and the 224401-1 1098-0121/2012/85(22)/224401(8) ©2012 American Physical SocietyPYLYPOVSKYI, SHEKA, AND GAIDIDEI PHYSICAL REVIEW B 85, 224401 (2012) time, and MSis the saturation magnetization. The total energy Eof such a sphere, normalized by 4 πM2 SVwithV=4 3πR3, reads E=Eex+Ef+Ems. (1a) The ﬁrst term in Eq. (1a) is the dimensionless exchange energy, Eex=3 8πε/integraldisplay dr[(∇/Theta1)2+sin2/Theta1(∇/Phi1)2], (1b) withε=/lscript2/R2being the reduced exchange length, /lscript=√ A/4πM2 Sbeing the exchange length, Abeing the exchange constant, and r=(x,y,z )/Rbeing the reduced radius vector. The second term determines the interaction with externalmagnetic ﬁeld H, E f=−3 4π/integraldisplay dr(m·h), (1c) where h=H/4πMSis a reduced external ﬁeld. We will discuss the inﬂuence of the external ﬁeld later, see Sec. III. The last term determines the reduced magnetostatic energy, Ems=−3 8π/integraldisplay dr(m·hms), (1d) where hms=Hms/4πMSis a reduced magnetostatic ﬁeld Hms. Magnetostatic ﬁeld hmssatisﬁes the Maxwell magne- tostatic equations,4,5 ∇×hms=0,∇·hms=4πλ, (2) which can be solved using a magnetostatic potential hms= −∇ψ. The sources of ﬁeld hmsare magnetostatic charges: volume charges λ≡− (∇·m)/4πand surface ones σ≡(m· n)/4πwithnbeing the external normal. The magnetostatic potential inside the sample reads ψ(r)=/integraldisplay Vdr/primeλ(r/prime) \|r−r/prime\|+/integraldisplay SdS/primeσ(r/prime) \|r−r/prime\|(3a) ≡1 4π/integraldisplay Vdr/prime(m(r/prime)·∇r/prime)1 \|r−r/prime\|. (3b) The equilibrium magnetization conﬁguration is determined by minimization of the energy functional ( II), which leads tothe following set of equations: ε∇2m=∇ψ,∇2ψ=∇·m. (4) A. Classiﬁcation of singularities Let us start the Bloch point as a particular solution of Eq.(4). In the exchange approach, the simplest hedgehog-type Bloch point is characterized by the magnetization distributionof the form m=r/rwith a singularity at the origin. Using a spherical frame of reference for the radius vector rwith the polar angle ϑand the azimuthal one ϕ, one can describe the magnetization angles of such a Bloch point as follows: /Theta1=ϑ and/Phi1=ϕ. The energy of the Bloch point in the exchange approach reads 12 Eex 0=3ε, Eex 0=4πAR. (5) This interaction is invariant with respect to the joint rotation of all magnetization vectors, which gives the possibilityfor considering a family of solutions with different rotationangles. 11,12 We consider the following singular magnetization distribution: /Theta1(ϑ)=pϑ+π(1−p)/2,/Phi1 (ϕ)=qϕ+γ, (6) p,q=± 1, which describes a three-parameter Bloch point. We refer to the parameter q=± 1 as the vorticity of the Bloch point andp=± 1 as its polarity using the conventional symbols for magnetic vortices. The last parameter γdescribes the azimuthal rotational angle of the Bloch point.11,12 We refer to the micromagnetic singularity (6)as BPp q. For example, the hedgehog-type Bloch point is a vortexBloch point with positive polarity ( p=1,q=1,γ=0). The schematic of magnetization distribution in different types ofBloch points is presented in Fig. 1. The analogy between Bloch point and vortices comes from the vortex polarity switchingprocess under the action of a dc perpendicular magnetic ﬁeld. 16 A single Bloch point can be imagined as a composite of twovortices with opposite polarities: Such a singularity can appearin three-dimensional (3D) Euclidean space during the vortexpolarity switching process in antiferromagnets. 8,27Due to the equivalence of two face surfaces of the nanodot, the vortex (a)p=q= 1 (b) p=−1q= 1 (c) Vortex switching(d)p=1 q=−1 (e) p=q=−1 (f) Antivortex switching FIG. 1. Schematic of different types of Bloch points. Magnetization distribution in azimuthal vortex Bloch points in a sphere, see (a) and (b), and both Bloch points in the axial part of the cylinder-shaped sample during the vortex polarity switching process, see (c). The same for the azimuthal antivortex Bloch points, see (d) and (e), and both singularities in the axial part of the astroid-shaped sample during the switching, see (f ). During the switching process shown in (c) and (f), two Bloch points move along the disk axis in opposite directions and ﬁnally annihilate halfway. 224401-2BLOCH POINT STRUCTURE IN A MAGNETIC NANOSPHERE PHYSICAL REVIEW B 85, 224401 (2012) polarity switching is mediated, as a rule, by the creation of two Bloch points (symmetric or so-called punch-throughmechanism), see Fig. 1(c). They are injected from the ends of the nanodot and annihilate on its axis. 16All four distributions for different signs of pandqcan be observed during symmetrical Bloch points’ injection in the polarity switchingprocess of vortices 16[Fig. 1(c)] and antivortices18[Fig. 1(f)]. Topological properties of the Bloch point can be described by the topological (Pontryagin) index, Q=1 4π/integraldisplay sin/Theta1(r)d/Theta1(r)d/Phi1(r)=pq. (7) Different Bloch point distributions with equal Qare topolog- ically equivalent: e.g., BP−1 −1can be obtained from BP1 1by simultaneous rotation of all magnetization vectors by πin the vertical plane, and BP1 −1transforms to BP−1 1by rotation byπ/2 in the vertical plane. Note that similar topological notations were introduced by Malozemoff and Slonzewski10 for magnetic bubbles.28 B. Magnetization structure of Bloch points The strongest exchange interaction is isotropic, hence, the exchange energy takes the same values for any rotationangle γ. Such degeneracy is removed under the account of magnetostatic interaction. It is worth noting that the problemof stray ﬁeld inﬂuence on the Bloch point energetics has along story. Feldtkeller, in his pioneer paper, 11used a so-called pole avoidance principle, see, e.g., Ref. 29: The magnetostatic charge tries to avoid any sort of volume or surface charge. Inthis way, he calculated the angle γfrom the condition of the total volume magnetostatic charge/integraltext λ(r)dr=0, where λ(r) is the charge density. For the Bloch point given by ansatz (6),i t has a form λ(r)=− [psin 2ϑ+cosγ(cos2ϑ+1)]/4πrand leads to the rotation angle, γF=arccos/parenleftbigg −p 2/parenrightbigg =/braceleftbigg120◦,p=+ 1, 60◦,p =− 1.(8) It is interesting to note that the same value γFalso corresponds to the absence of the total surface charge/integraltext σ(r)dS=0 where the surface charge density is σ(r)=(pcos2ϑ+ cosγsin2ϑ)/4π. Another approach was put forward by D ¨oring,12who determined the equilibrium angle of γby minimizing the energy, Ems D=3 8π/integraldisplay Vdr(hms)2, (9) and obtained γD=arccos/parenleftbigg −11 29/parenrightbigg ≈112.3◦. (10) However, one has to emphasize that the equilibrium angle (10) minimizes only the inner part of the magnetostatic energy because the integration in Eq. (9)is carried over the sample volume V, while the outer part of the stray ﬁeld is ignored. Note that a similar approach was used in a quite recent paper30 where a magnetization contraction was taken into account. The aim of this section is to ﬁnd the equilibrium rotation angle which minimizes the total magnetostatic energy. In orderto derive the magnetostatic energy of Bloch points (6),w eﬁ r s t calculate magnetostatic potential (3b) using an expansion of 1/\|r−r/prime\|over the spherical harmonics, 1 \|r−r/prime\|=1 r>∞/summationdisplay l=0l/summationdisplay m=−l4π 2l+1/parenleftbiggr< r>/parenrightbiggl Ylm(ϑ,ϕ)Y⋆ lm(ϑ/prime,ϕ/prime), withr<=min(r,r/prime) andr>=max(r,r/prime), which results in ψp q=1(r)=pπr+π 3(9r−8) cos γ+πr(p−cosγ) cos2ϑ, (11a) ψp q=−1(r)=pπr (1+cos2ϑ)+πrcos(2ϕ+γ)s i n2ϑ. (11b) Simple calculations show that the magnetostatic energy of the antivortex Bloch point does not depend on γandEms q=−1= 7/30≈0.23. In contrast to this, the vortex Bloch point energy depends on rotation angle γand has the form Emsp q=1(γ)=1 30(7+4pcosγ+4 cos 2 γ). (12) The equilibrium value of rotation angle γ0corresponds to the minimum of the energy (12).I tg i v e s γ0=arccos/parenleftBig −p 4/parenrightBig ≈/braceleftbigg105◦,p=+ 1, 76◦,p =− 1.(13) Let us compare Bloch point energies (12) for the above- mentioned approaches: The energy of the Feldtkeller11Bloch point is Emsp q=1(γF)=0.1, for the D ¨oring12Bloch point, one hasEms(γD)≈0.088, and the result by El ´ıas and Verga30is Ems 1(γEV)≈0.089. The minimal energy has a Bloch point with rotation angle γ0,s e eE q . (13), Emsp q=1(γ0)=1 12≈0.083. (14) In order to verify our results, we performed numerical spin-lattice simulations, see details in Sec. IV. We compare analytical dependence Emsp=1 q=1(γ), see Eq. (12), with the dis- crete energy (24), extracted from simulations, see Fig. 2.B o t h dependencies are matched in maximum at γ=0. Comparison can be provided by calculating the energy gain /Delta1E(γ)= Ems max−Ems(γ) for different rotation angles γ. According to the 00.10.20.30.40.5 -150 -100 -50 0 50 100 150Ems(γ) Angle γ,d e g0.050.1 80 100 120γ0γDγF FIG. 2. (Color online) The Bloch point energy vs the rotation angle for BP1 1: analytical result (12) (solid curve) and simulations (symbols). Simulations parameters: sphere diameter 2 R=35a0, exchange length /lscript=3.95a0, and damping parameter η=0.5. 224401-3PYLYPOVSKYI, SHEKA, AND GAIDIDEI PHYSICAL REVIEW B 85, 224401 (2012) simulation results, the energy gains for the above-mentioned angles read /Delta1E(γF)≈0.446,/Delta1 E(γD)≈0.460,/Delta1 E(γ0)≈0.465. The maximum energy gain takes place for γ0, which corre- sponds to the energy minimum in good agreement with ouranalytical result (13). III. THE BLOCH POINTS IN AN EXTERNAL FIELD The Bloch point does not form a ground state of a magnetic sphere. It corresponds to the saddle point (sphaleron) of theenergy functional. 31This brings up the question: How to stabilize the Bloch point? In this section, we show that one wayto achieve this goal is to apply a magnetic ﬁeld, which has thesame symmetry as the hedgehog Bloch point with m=r/r, i.e., a radially symmetric magnetic gradient magnetic ﬁeld inthe form h=br. (15) Under the action of the space-dependent magnetic ﬁeld (15), the magnetization distribution also becomes space dependent. We take into account possible dependence by thefollowing radial Bloch point ansatz: /Theta1(ϑ)=pϑ+π(1−p)/2,/Phi1 (r,ϕ)=qϕ+γ(r), (16) with a radially dependent parameter γ(r) in comparison with Eq.(6). The form of this ansatz will be justiﬁed by numerical simulations in Sec. IV. Inserting Eq. (16) into Eq. (1b) for the exchange energy of such magnetization distributions, we get E ex=3ε+ε/integraldisplay1 0/parenleftbiggdγ dr/parenrightbigg2 r2dr. (17a) The magnetostatical potential of the Bloch point (16) reads ψp=1 q=1(r)=−4π 3/integraldisplay1 r[1+2 cosγ(r/prime)]dr/prime −4π 33 cos2ϑ−1 r3/integraldisplayr 0r/prime3[cosγ(r/prime)−1]dr/prime. Here and below, we consider the case of BP1 1only. The magnetostatic energy of such a Bloch point has the form Ems=1 10/integraldisplay1 0r2[7+4 cos γ(r)+4 cos 2 γ(r)]dr. (17b) From Eq. (1c), we obtain that the Bloch point interaction with a magnetic ﬁeld can be expressed as follows: Ef=− 2b/integraldisplay1 0r3cosγ(r)dr. (17c) By minimizing the total energy δE/δγ=0, we obtain that the equilibrium distribution γ(r) is a solution of the following nonlinear differential equation: εd2γ dr2+2ε rdγ dr+1 5sinγ+2 5sin 2γ−brsinγ=0 (18)-1-0.9-0.8-0.7-0.6 0 0.25 0.5 0.75 1gε(r) Distance rAnalytics b=0.05 b=0.80 FIG. 3. (Color online) Reduced rotation angle gε(r), see Eq. (20) for different ﬁeld intensities and ε=0.05: analytical result (A1) (solid curve) and numerical solution of Eq. (18) (dashed curves). augmented by boundary conditions of the form dγ dr/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=0=dγ dr/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=1=0. (19) In the case of weak ﬁelds, one can linearize Eq. (18) in the vicinity of spatially uniform solution (13) and obtain that γ(r)≈γ0+bgε(r),\|b\|/lessmuch 1. (20) An explicit form of the function gε(r) is calculated in the Appendix. The comparison with the numerical solution ofEq.(18) shows quite good agreement up to relatively strong ﬁelds ( b/lessorsimilar1), see Fig. 3. Another limiting case is realized in the case of strong magnetic ﬁelds when the Bloch point magnetization is parallelto the external ﬁeld. In this case, the rotation angle is γ= 0( m o d π). To describe the behavior of the Bloch point in a critical region b≈b cwhere the spatially nonuniform distribution transforms to the spatially uniform one, we use a variationalapproach with a two-harmonics trial function γ(r)≈α 0+ α1cosπr. Near the critical point, α0,α1/lessmuch1. We expand the total energy in a Taylor series up to the fourth order withrespect to α 0and to the second order with respect to α1.B y excluding α1and keeping terms not higher than α4 0, we get E(γ)≈E0+p2(b,ε)α2 0+p4(b,ε)α4 0. (21) The energy (21), as a function of α0, has a double-well shape [p2(b,ε)<0] for b<b cwith the critical magnetic ﬁeld bc given by bc(ε)≈1.8−21.6ε+/radicalbig 0.4−20.2ε+467ε2. (22) In the critical region, when 0 <bc(ε)−b/lessmuchbc(ε), α0(b)≈a(ε)√bc(ε)−b. (23) Forb>b c,p 2>0, and the function Eq. (21) has a minimum forα0=0. It corresponds to γ=0. Numerical integration of Eq.(18) forε=0.05 shows that the phase transition occurs when bc≈1.47, see Fig. 4. It agrees well with the value bc(0.05)≈1.46 obtained from Eq. (22). The critical behavior predicted by Eq. (23) is also conﬁrmed by our numerical simulations [see Fig. 4(a)]. 224401-4BLOCH POINT STRUCTURE IN A MAGNETIC NANOSPHERE PHYSICAL REVIEW B 85, 224401 (2012) 00.40.81.21.6 0 0.25 0.5 0.75 1 1.25 1.5 1.75Rotation angle γ Field b00.10.2 1.4 1.45 1.5γ(0) γ(1) γth(0) γth(1) bc Reduced exchange lenght ε Field bNonhomogeneous 00.020.040.060.080.1 1.4 1.6 1.8 2 2.2 2.4Hedgehog(a) (b) FIG. 4. (Color online) Bloch point under the action of the gradient ﬁeld. (a) Critical behavior where the rotation angle vs ﬁeld intensity b near the critical ﬁeld bc=1.47 from the numerical solution of Eq. (18) (blue curves) and theoretical estimation by Eq. (23) (red curves) with ε=0.05. Solid lines correspond to rotation angle γ(0), and the dashed line corresponds to γ(1). (b) Phase diagram for solutions of Eq. (18). The upper (hedgehog) phase corresponds to the solution γ=0, and the lower (nonhomogeneous) one corresponds to the radially dependent Bloch point with γ(r). Dashed lines correspond to the analytical result for the critical ﬁeld bc≈1.46 for ε=0.05, see text. IV . NUMERICAL STUDY OF THE BLOCH POINT STRUCTURE In order to check analytical results about Bloch point structure, we performed simulations using the in-house de-veloped spin-lattice simulator SLaSi (Ref. 32) that solves the Landau-Lifshitz-Gilbert equation in terms of spins, dS n dt=−1 ¯h/bracketleftbigg Sn×∂H ∂Sn/bracketrightbigg −η S/bracketleftbigg Sn×dSn dt/bracketrightbigg , where His a lattice Hamiltonian of the classical ferromagnet, H=−J 2/summationdisplay (n,δ)Sn·Sn+δ+2μBH/summationdisplay nSn +2μ2 B/summationdisplay n/negationslash=k/bracketleftbigg(Sn·Sk) rnk3−3(Sn·rnk)(Sk·rnk) rnk5/bracketrightbigg .(24) Here, Snis a classical spin vector with ﬁxed length Sin units of action on the site nof a three-dimensional cubic lattice with lattice constant a0,Jis the exchange integral, μBis the Bohr magneton, rnkis the radius vector between nth and kth nodes, ηis a damping parameter, His the external magnetic ﬁeld, andδruns over six nearest neighbors. Integration is performed by the modiﬁed fourth- and ﬁfth-order Runge-Kutta-Fehlbergmethod (RKF45) and free spins on the surface of the sample. 33 Numerically, we checked the Bloch point structure, given by the radially dependent ansatz (16) by modeling a spherically shaped sample with diameter 2 R=35a0(such a sample consists of 24 464 nodes with nonzero spin) and exchangelength /lscript=3.95a 0(ε=0.05). In order to stabilize the Bloch point, we applied the gradient magnetic ﬁeld with b=1.0. By modeling the overdamped dynamics, we observed thatthe Bloch point structure quickly relaxed to the state similarto the one given by Eq. (16): The polar Bloch point angle /Theta1(r) does not deviate from ϑwithin an accuracy of 0 .099 rad. The azimuthal angle is also well described by Eq. (16) with the radially dependent rotation angle γ(r), see Fig. 5. Simulations were performed for crystallographic directions[111] ( ϑ=π/4) and [110] ( ϑ≈π/2, the plane is shifted by z=− 0.5a 0from the origin). From Fig. 5, one can see thatnumerical data are well conﬁrmed by analytical curve γ(r), calculated as a numerical solution of Eq. (18). To validate our theory, we also performed a direct stability check. Numerically, we checked the stability of the Blochpoint against the shift in its position. We start simulationswith the Bloch point state using ansatz function (16), which is shifted along the ˆzaxis by /Delta1z=− 2a 0. We also apply γ(r,t= 0)=3◦in order to break the symmetry. For rapid relaxation, in most of the simulations, we used the overdamped regime(the damping parameter η=0.5). We checked the shift in the Bloch point by controlling the total spin projections: Only forthe Bloch point, situated at the sample origin, is the total spinS tot x=Stot y=Stot z=0. The temporal evolution of the initially shifted Bloch point is presented in Fig. 6for the Bloch point sample with 2 R= 35a0(24 456 nodes) in an applied ﬁeld with b=1, see also the Supplemental Material.34Originally, the Bloch point was shifted down from the origin, which corresponds to Stot z>0, see inset (a). During the evolution, a number of magnons aregenerated, inset (b). After quick damping of the oscillations,the micromagnetic singularity goes to the sample origin, see 0.60.70.80.911.11.21.3 0 0.2 0.4 0.6 0.8 1Rotation angle γ(r) Distance rTheory Data ( ϑ=π/2) Data ( ϑ=π/4) FIG. 5. (Color online) Radial dependence of rotation angle γin a spherical particle. Line: numerical integration of Eq. (18). Symbols: SLASI simulations for crystallographic directions [110] and [111]. Parameters are the same as in Fig. 2. 224401-5PYLYPOVSKYI, SHEKA, AND GAIDIDEI PHYSICAL REVIEW B 85, 224401 (2012) -0.0500.050.10.150.20.250.3 0 100 200 300 400 500 600 700 800 900Total spin Stot z/Su, tot z Timet,ω−1 0Dynamics of Stot zand plane z=−0.5a0 (a)(b) (c) (d)Plane y=−0.5a0 (a)t=ω−1 0 (b)t=2 0 0 ω−1 0 (c)t=4 0 0 ω−1 0(d)t=9 0 0 ω−1 0 -1-0.500.51 FIG. 6. (Color online) Dynamics of total spin along the zaxis of the sample. The Bloch point is initially shifted by /Delta1z=− 2a0from the center of the sample. The insets show magnetization distribution in z=− 0.5a0andy=− 0.5a0planes for different times. The color bar indicates Sz,nfor different lattice nodes. The applied ﬁeld amplitude b=1 and the other parameters are the same as in Fig. 2. inset (d). The relaxation process consists of two parts: (i) The rotation angle γ(r) changes its value from the initially uniform one to the ﬁnal nonhomogeneous state during a time τγ≈ 500ω−1 0. (ii) The relaxation of the Stot zcomponent of the total spin of the sample took approximately the same time. Duringall simulation times, \|S tot x\|≈\|Stot y\|/lessorsimilar10−11. V . CONCLUSION To summarize, we studied the magnetization structure of the Bloch point. Despite the fact that the Bloch point, as asimplest 3D topological singularity, was studied for a long timefrom the pioneering papers by Feldtkeller 11and D ¨oring,12for ar e v i e w ,s e ea l s oR e f s . 4and 10, the problem of the Bloch point structure still causes discussions.16,30,35The point is that the strongest exchange interaction depends only on relativedirection of neighboring magnetic moments due to the isotropyof exchange. Therefore, it does not determine the value ofrotation angle γ. This rotation angle, which is determined by the magnetostatic interaction, is most questionable: Itsvalue is equal to 120 ◦according to Feldtkeller,11to 112 .3◦ following D ¨oring,12and to 113◦following El ´ıas and Verga.30 We analyzed the origin of all these results and calculated the equilibrium value, about 105◦,s e eE q . (12), which minimizes the total magnetostatic energy, not only a part of it. The next problem appears in the modeling of the Bloch point. It was discussed by Thiaville et al.16that the modeling of singularity is mesh dependent within the continuum descrip-tion of micromagnetism. In particular, a mesh-friction effectand a strong mesh dependence of the switching ﬁeld during theBloch point mediated vortex switching process was detectedusing OOMMF micromagnetic simulations.16The reason is that micromagnetic simulators consider the numerically dis-cretized Landau-Lifshitz equation, which is valid in continuumtheory. Since the Bloch point appears as a singularity ofcontinuum theory, it is always located between mesh pointsand causes the mesh-dependent effects and, therefore, may beinsufﬁcient for describing near-ﬁeld Bloch point distribution.In contrast to this, spin-lattice simulations are free from theseshortages. From the beginning, we considered discrete spins,located on the cubic lattice, and their dynamics was governedby the discrete versions of the Landau-Lifshitz equations. Thelattice Hamiltonian allows us to calculate the discrete energyof the Bloch point similar to the atomiclike calculations by Reinhardt. 36 Using the in-house developed spin-lattice SLaSi (Ref. 32) simulator, we modeled the Bloch point state nanosphere andchecked our analytical predictions about the Bloch pointstructure. We stabilized the singularity inside the sphericalparticle by applied gradient magnetic ﬁeld. The ﬁeld causesadditional radial dependence of rotation angle γ(r)i nt h e Bloch point structure. ACKNOWLEDGMENTS The authors acknowledge computing time on the high- performance computing cluster of the National TarasShevchenko University of Kyiv 37and the SKIT-3 Computing Cluster of the Glushkov Institute of Cybernetic of NAS ofUkraine. 38This work was supported by the Grant of the President of Ukraine Grant No. F35/538-2011. We thankV . Kravchuk for helpful discussions. APPENDIX: BLOCH POINT STRUCTURE IN A WEAK FIELD Here, we consider the magnetization structure of a Bloch point under the action of a weak magnetic ﬁeld. One hasto linearize Eq. (18) on the background of the unperturbed rotation angle γ 0,s e eE q . (20), which can be presented as follows: γ(r)≈γ0+bgε(r),g ε(r)=2√ 5ε 3f(λr),λ=1 2/radicalbigg 3 ε. Here, the function f(ξ) satisﬁes the linearized version of Eq.(18), d2f dξ2+2 ξdf dξ−f=ξ, which can be easily integrated f(ξ)=Cλsinhξ ξ+2coshξ−1 ξ−ξ, (A1) Cλ=λ2−2λsinhλ+2 cosh λ−2 λcoshλ−sinhλ. The graphics of the gε(r)f o rε=0.05 is presented in Fig. 3 together with the numerical solution of Eq. (18) by the shooting 224401-6BLOCH POINT STRUCTURE IN A MAGNETIC NANOSPHERE PHYSICAL REVIEW B 85, 224401 (2012) -1.2-1-0.8-0.6-0.4-0.20 0 0.25 0.5 0.75 1gε(0)andgε(1) ε=2/R2gε(0) gε(1) FIG. 7. (Color online) Reduced rotation angle gεvs reduced exchange length εatr=0 (solid curve) and r=1 (dashed curve).method. Despite the limitation of our analysis by the case of a weak ﬁeld \|b\|/lessmuch 1, the function gε(r) provides a good approximation for the solution of nonlinear Eq. (18) up to very strong ﬁelds b/lessorequalslant1 with a relative error of \|[γ(r)num− γ(r)theor]/γ(r)num\|/lessorequalslant0.04. The rotation angle in the Bloch point is essentially inﬂuenced by the exchange parameter ε, see Fig. 7.I n the limiting case of a small particle ( ε/greatermuch1), the role of exchange is dominant, which results in the constant angleg ∞=−√ 15/4≈− 0.97. In the opposite case ε/lessmuch1, the role of the magnetostatic interaction is enhanced, and this leads to a nonhomogeneous rotational angle distribution. In the limiting case,g0(0)=0 andg0(1)=−/radicalbig 5/3≈− 1.3. Such a limiting case is realized in typical soft nanomagnets sized in some tensof nanometers. sheka@univ.net.ua 1Phase Transitions and Critical Phenomena , edited by C. Domb and M. S. Green (Academic, New York, 1983), V ol. 7, p. 328;Bond-Orientational Order in Condensed Matter Systems , edited by K. J. Strandburg (Springer, Berlin, 1991), p. 388; F. Yonezawaand T. Ninomiya, in Topological Disorder in Condensed Matter , Springer Series in Solid-State Sciences V ol. 46 (Springer, Berlin,1983), p. 253. 2M. J. Bowick and L. Giomi, Adv. Phys. 58, 449 (2009). 3A. L. Fetter, Rev. Mod. Phys. 81, 647 (2009). 4A. Hubert and R. Sch ¨afer, Magnetic Domains: The Analysis of Magnetic Microstructures (Springer-Verlag, Berlin, 1998). 5J. St ¨ohr and H. C. 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PhysRevB.96.224431.pdf	PHYSICAL REVIEW B 96, 224431 (2017) Systematic motion of magnetic domain walls in notched nanowires under ultrashort current pulses A. Pivano and V . O. Dolocan* Aix-Marseille Université, CNRS, IM2NP UMR7334, F-13397 Marseille Cedex 20, France (Received 20 June 2017; revised manuscript received 8 November 2017; published 27 December 2017) The precise manipulation of transverse magnetic domain walls in ﬁnite/inﬁnite nanowires with artiﬁcial defects under the inﬂuence of very short spin-polarized current pulses is investigated. We show that for a classical 3 d ferromagnet material like nickel, the exact positioning of the domain walls at room temperature is possibleonly for pulses with very short rise and fall time that move the domain wall reliably to nearest neighboringpinning position. The inﬂuence of the shape of the current pulse and of the transient effects on the phase diagramcurrent-pulse length are discussed. We show that large transient effects appear even when α=β, below a critical value, due to the domain wall distortion caused by the current pulse shape and the presence of the notches.The transient effects can oppose or amplify the spin-transfer torque (STT), depending on the ratio β/α.T h i s enlarges the physical comprehension of the domain wall (DW) motion under STT and opens the route to the DWdisplacement in both directions with unipolar currents. DOI: 10.1103/PhysRevB.96.224431 I. INTRODUCTION Current induced magnetic domain wall motion (CIDWM) in nanowires or nanostrips is a highly active research ﬁeld[1,2] with applications in high-density and ultrafast nonvolatile data-storage devices like the racetrack memory [ 3] or for logic devices [ 4]. In the racetrack memory, the data processing is based on the controlled displacement between precisedistinct positions of the domain walls (DWs) due to thetransfer of angular momentum (spin-transfer torque) from aspin-polarized electric current. To achieve precise positioningof DW, artiﬁcial constrictions, or others, patterned geometricaltraps are usually used, which create an attractive pinningpotential for the DW. Different types of traps were studiedin cylindrical or ﬂat/strip nanowires [ 5–8], along with the possible interaction between the DWs [ 9,10]. In some cases, depending on the pinning potential, the DW displacementbetween pinning sites can display a chaotic behavior [ 11]o ra stochastic resonance [ 12] under harmonic excitation. The required currents for STT based DW movement are usually high ( ∼1A/μm 2), which limits the applicability due to Joule heating. To displace accurately the DWs between thepinning sites, the current density should be kept at relativelylow values and/or very short current pulses should be applied.Experimentally, it was observed that an efﬁcient DW motionis reached for pulses in the nanosecond regime [ 13] and that the resonant excitation of the DW by a short train of currentpulses decreases the depinning current [ 14]. More recently, the effect of the temporal and spatial shape of the current pulse washighlighted [ 15–17]. It was shown that a fast changing current with an ultrashort pulse rise time decreases the critical currentdensity due to the dependence of the DW motion on the timederivative of the current [ 18] and leads to high DW velocity [19]. Another aspect of the CIDWM under short pulses is the existence of important transient effects related with theDW inertialike behavior [ 13,20–23] due to deformation of the wall. The consequences are a delayed response at the currentonset and at the end of the current pulse. The theoretical *voicu.dolocan@im2np.frand experimental results show that the distance traveled bya DW is almost proportional to the current pulse length andthat the transient motion depends on the variation of thegeneralized angle of the wall, the wall width, and the ratio ofthe damping ( α) and nonadiabatic ( β) parameters [ 20,21]. For very short pulses (one nanosecond), the transient displacementis comparable with the steady-state motion [ 24]. A DW that propagates without deformation should display no inertia [ 25] like in cylindrical nanowires [ 26] or in certain perpendicular magnetic anisotropy systems [ 27]. The absence of inertia will allow a fast response to external forces while the transientDW displacement after the application of the pulse limits theapplication in fast devices. In the main time, it was recentlydemonstrated that inertialike behavior of a DW can be also anadvantage when ultrashort optical pulses are used [ 28] with applications in the optical recording. Moreover, in systemswith strong spin-orbit coupling where additional contributionsfrom spin-Hall effect complicate the DW dynamics, a DWtunable inertia was proposed [ 29]. Before studying more complex systems, the inﬂuence of the transient effects onthe systematic DW movement under ultrashort spin-polarizedcurrent pulses should be completely understood in a classical3dferromagnet like nickel, which is the aim of this paper. In this paper, we address the systematic motion of a magnetic transverse DW between ﬁxed artiﬁcial constrictions(notches), when submitted to a series of ultrashort spin-polarized current pulses (transient regime) at low and roomtemperature. The artiﬁcial constrictions are situated at regularpositions in a ﬂat ﬁnite or inﬁnite nanowire with only in-plane(shape) anisotropy like in a classical 3 dferromagnet. We determine the inﬂuence of the current pulse shape (rise andfall time) on the motion of the DW. We show that, even at zerotemperature, there is a transition region between the differentbands in the current-pulse time phase diagram, each bandcorresponding to the positioning of the DW at a well-deﬁnednotch. Our results show that, at room temperature, the precisepositioning can be achieved only by very short pulses with veryshort rise and fall times that displace the DW by only one notchat a time. Therefore, to move a DW several notches reliably, asequence of very short pulses should be used. By examiningthe inﬂuence of the damping and nonadiabatic parameters, we 2469-9950/2017/96(22)/224431(10) 224431-1 ©2017 American Physical SocietyA. PIV ANO AND V . O. DOLOCAN PHYSICAL REVIEW B 96, 224431 (2017) t(ps )0je(A/ μm2)trts tf tz100 300 500 700 900 x (nm)00.170.34E (eV) 100 300 500 700 900 x (nm)00.130.26E (eV) (a) (b)(c) (d)-1 1 mxy x FIG. 1. (a) Simulated structure: planar nanowire with ten symmetric double notches. The equilibrium position of a pinned DW is shown in the ﬁnite case. (b) Deﬁnition of current pulse with its temporal characteristics. Two successive current pulses with opposite polarity are shown. The normalized potential pinning energy for ﬁnite and inﬁnite cases as determined by micromagnetic simulations (symbols) are shown in (c)and (d), respectively. The line is a ﬁt as described in the text. show that when β=0 the transient effects (automotion) of the DW are very large and oppose the STT, being observed inthe phase diagrams as predicted [ 20]. The transient effects are related with the change in the DW structure that is due to acombination of factors: pinning potential of the notches whichinduces a sufﬁcient variation of the DW angle [ 30], position of the DW inside the potential well (different restoring force), lowdamping parameter, and shape of the current pulse. Contraryto expectations, the transient effects also appear when α=β, below a critical value. For β>α , the transient effects can oppose or amplify the STT, thus explaining the oscillatory DWdepinning at higher currents observed experimentally [ 13]. This brings new physical insight into CIDWM under STT andpaves the way for systematically displacing DW in nanowiresin both directions using only unipolar current pulses. This article is organized as follows. In Sec. II, we present the micromagnetic and the stochastic 1D model used tocalculate the pulsed DW dynamics. In Sec. III, we compute and investigate the phase diagram of the DW dynamicsfor a ﬁnite and inﬁnite nanostrip at T=0 K and room temperature. Discussion and concluding remarks are presentedin Sec. IV. II. MODEL We study numerically the systematic motion of a pinned transverse domain wall in a ﬁnite or inﬁnite Ni nanostrip withsymmetric rectangular notches. The ﬁnite strip has a lengthL x=1μm ,ac r o s ss e c t i o no f Ly×Lz=60×5n m2, and has ten rectangular symmetric double notches separated by80 nm. The results presented below are for notch dimensionsof 20×9×5n m 3. The variation of length and depth of the notches does not inﬂuence much the physics of phase diagramspresented in Sec. III. The notch depth inﬂuences the depinning current as the potential barrier increases, while the notch lengthinﬂuences lightly the depinning current and mostly the slopeof the potential wells.Figure 1(a) shows the equilibrium position of a head-to- head transverse DW in the notched nanostrip using the pa-rameters of nickel: saturation magnetization M s=477 kA /m, exchange stiffness parameter A=1.05×10−11J/m, spin polarization P=0.7, and damping parameter α=0.05. The DW is moved by a series of spin polarized current pulsesapplied along the xaxis. The geometry of the current pulse is described in Fig. 1(b), with t r,ts,tf, andtzthe rise, stable, fall time, and zero-current time, respectively. The nonadiabaticparameter is set to β=2α, if not speciﬁed otherwise. The DW dynamics was computed using 3D micromagnetic simulations with the MUMAX3 package [ 31] and with the one-dimensional DW model [ 32,33]. In both cases, the mag- netization dynamics is determined from the Landau-Lifschitz-Gilbert (LLG) equation with adiabatic and nonadiabatic spin-transfer torques [ 34]: ˙M=−γ 0M×Heff+α(M×˙M)−(u·∇)M +βM×(u·∇)M, (1) where γ0is the gyromagnetic ratio, u=jePμB/eM sis the spin drift velocity, Pthe spin polarization of conductions electrons, μBthe Bohr magneton, and jethe applied current density. No additional exotic torques (like the ones due to the spin-Hall orRashba effect) were considered. For the micromagnetic computations, the strip was dis- cretized into a mesh with a cell size of 2 ×3×2.5n m 3, inferior to the exchange length ( ∼5 nm). The DW dynamics in a ﬁnite wire is compared with the one of an inﬁnity longwire where the magnetic charges at the ends of the nanostripare compensated. The average position of the DW center ( X) is extracted for each simulation (in the axial xdirection) along with the azimuthal angle ( ψ) of magnetization in the yz plane. No magnetocrystalline anisotropy is considered; theshape anisotropy insures that the easy axis is in-plane. Theeffect of the temperature is studied both micromagneticallyand with the 1D model. The 1D model of the DW (collectivecoordinates Xandψ) supposes that the DW is rigid and gives a 224431-2SYSTEMATIC MOTION OF MAGNETIC DOMAIN W ALLS IN . . . PHYSICAL REVIEW B 96, 224431 (2017) quasiquantitative understanding of the motion of the DW. The Langevin equations of motion of the DW [ 2,35] are detailed in the Supplemental Material [ 36]. The pinning potential energy is determined from quasistatic micromagnetic simulations and is shown in Figs. 1(c) and 1(d) for the ﬁnite case and the inﬁnite case, respectively. The pinning potential determined by ﬁtting the micromagneticresults is harmonic inside the notches and sinusoidal betweenthem: V p(x) =/braceleftbigg1 2ki(x+xi)2, forxi−L/lessorequalslantx/lessorequalslantxi+L, V0cos(2πf x+φi,i+1),forxi+L<x/lessorequalslantxi+1−L, (2) withkithe stiffness constant and xithe DW stable equilibrium position of the site i, andφi,i+1the phase between the isite and its nearest neighbor. V0andfcorrespond respectively to the effective height of the potential and its spatial frequency. Forthe ﬁnite case, the stiffness constant varies from 7 .07×10 −5 N/mt o6.77×10−5N/m, when moving from the center of the nanowire to its ends. In the inﬁnite case, the stiffness constantis equal to 7 .16×10 −5N/m for all the pinning sites and L= 16.5 nm. The expression [Eq. ( 2)] was used in the equations of the 1D model through the pinning ﬁeld Hpincluded in Heff. The pinning energy is controlled by the dimensions and distance between the notches. A clear difference is observedbetween the two potentials due to edge dipolar energy. In theﬁnite strip, the depinning ﬁeld decreases from 39 Oe, in thecentral wells, to 26 Oe when the DW is closed to the two endsof the strip. This is due to the attractive interaction betweenthe DW and magnetic surface charges located at the sides [ 37]. As a result, the potential wells are asymmetric in energy alongthe strip and their energy minima decrease when the distancebetween the notches and the ends of the strip is reduced. Incontrast, for the inﬁnite case, each well has the same depinningﬁeld and energy barriers. III. RESULTS Our analysis of the DW dynamics begins with the study of the differences between a ﬁnite and an inﬁnite nanostrip at T= 0 K. Afterwards, the inﬂuence of the pulse shape is discussedand the particularities of the DW motion at room temperature.The last subsection details the results when the damping andnonadiabatic parameters are varied and their inﬂuence on thetransient displacement. A. Phase diagrams at T=0K To characterize the systematic motion of the DW between the notches, we computed point-by-point phase diagrams forall systems with the 1D DW model for a large range ofpulse duration and current amplitude. We compare our 1Dresults with phase diagrams computed micromagnetically onless points than the 1D calculus. A similar micromagneticcomputation will require an enormous execution time. Thecontrol parameters are the amplitude, the duration, and theshape of the current pulses. The range of the current amplitude(/lessorequalslant10 A/μm 2) is chosen to have only viscous motion (no FIG. 2. Phase diagram for a DW at T=0 K in a ﬁnite (a) and inﬁnite (b) nanostrip in the parameter space stable-time–currentamplitude with t r=tf=5p s a n d tz=10 ns. The total time of the periodic pulses is 350 ns. The micromagnetic results (scattered symbols) are compared to the 1D model (colored regions). Thediagrams show only a few bands due to the ﬁnite size of the wire or due to the number of notches used. The upper right region is due to the ﬁnite size of the nanowire or of the simulated window(inﬁnite case). precession) for the pulse duration used ( /lessorsimilar1.5 ns). The pulse duration range is selected to be on the same order of magnitudewith access or reading/writing time in possible magneticmemories based on DW. The phase diagram, in the parameter space stable time– current amplitude, which characterizes the DW dynamicsin the ﬁnite nanostrip at T=0 K, is shown in Fig. 2(a). The diagram represents 200 ×400 point-by-point integration with a fourth order Runge-Kutta scheme. The micromagneticresults (scattered symbols) are compared with the 1D results(colored regions). The DW is initialized in the left centralwell and a series of periodic bipolar current pulses are appliedduring 350 ns to move periodically forth and back the DWbetween two desired notches. The pulse characteristics aret r=tf=5 ps and tz=10 ns. The inﬂuence of trandtf is discussed below. The value of tz(10 ns) was chosen to correspond to the return to equilibrium time of the DW atroom temperature. The t zcan be reduced to 3 ns for T=0K , without a change of the phase diagrams. In Fig. 2(a), several regions are visible, each region corresponding to one stateof the DW oscillation. The ﬁrst state which appears is the 224431-3A. PIV ANO AND V . O. DOLOCAN PHYSICAL REVIEW B 96, 224431 (2017) pinned state, noted state zero in the micromagnetic simulation, and corresponds to the DW being pinned in the initial notch.After the pulse ends, the DW behaves like a damped harmonicoscillator. The state zero is observed for all t s, when the external current jeis inferior to the depinning current 2 .31 A/μm2. This state is also observed at higher currents (until 10 A /μm2), when the stable time tsis low (between 0 ps and 55 ps). The diagram displays other bands, where the DW oscillatesperiodically between the same two potential wells, whichcan be next neighbors (noted as band 1) or not, until theﬁfth state that corresponds to the periodic oscillation betweenthe initial notch and the fourth notch to the right (band4) hopping the three notches in between. The number ofbands is given by the considered ﬁnite size of the nanostrip.The second state (next-neighbors notches noted band 1) isobserved up to t s=0.96 ns at je=2.3308 A /μm2, while the other bands continue above ts=1 ns. Thus the DW can cross several notches back and forth for a given current pulsecharacteristic. We observe that the interband transitions arecharacterized by an unintended state (state u), where the DW oscillation does not take place between the desired positions.This transition is more pronounced between the last two bands.The micromagnetic results, which are superimposed on the1D results, give quantitatively the same results until the fourthstate, after which a small shift appears in the t sandjevalues, but the bands are qualitatively the same. The upper right region, which corresponds to an unwanted state, is due to the ﬁnite size of the nanowire; here the DW is pinned at the nanowireend. In the inﬁnite case, the DW is initialized in the ﬁrst well from the left. A phase diagram similar to the ﬁnite casei ss h o w ni nF i g . 2(b). This diagram contains three more bands than the ﬁnite case, which correspond to additionalstates where the DW oscillates between two notches, startingfrom the initial one, separated by four intermediate notches(band 5) until six intermediate notches (band 7). To havea better visibility over these new bands we computed theDW dynamics for t sup to 1.5 ns on 300 ×400 points. We observe that the bands 1 and 2 exist until ts=1.045 ns and ts=1.27 ns, respectively, at je=2.3308 A /μm2, while the other bands continue above 1.5 ns. The interband transition(band u) is observed between the bands zero and one around j e=3A/μm2andts=250 ps. For the superior bands, the interband transition at boundaries is quasi-nonexistent, whichshows that the inﬁnite case is more stable than the ﬁnite case. Inboth cases, the 1D model gives quantitatively the same resultsas the micromagnetic simulations in the three ﬁrst bands andquasiquantitatively in the others (a shift in values is visible).As for the ﬁnite case, the upper right region appears due tothe ﬁniteness of the simulated length of the nanowire, even ifthe end charges are suppressed. If a longer simulated windowis considered, other superior bands will follow as seen forexample in Fig. 5or Fig. 6. The inﬂuence of the pulse shape is detailed in Fig. 3for the inﬁnite wire as calculated with the 1D model. In panel (a),the pulse shape is symmetric as t r=tf=5 ps and tz=10 ns, while in panels (b) and (c) the pulse shape is asymmetric,t r/negationslash=tf, with tzis kept constant at 10 ns. The values of trand tfwere varied between 5 ps and 300 ps. In panel (b), the case withtr=5 ps and tf=300 ps is shown, while in panel (c), 1 5 10 je(A/μm2)0.30.91.5tto t(ns) 1 5.5 10 je(A/μm2)0.30.91.5tto t(ns) 1 5.5 10 je(A/μm2)0.30.91.5tto t(ns) (a) (b) (c) u 12 0 34567 FIG. 3. Inﬂuence of the pulse shape on the phase diagram for a DW at T=0 K in an inﬁnite nanostrip. The parameter space is the total time ( tr+ts+tf) vs current amplitude. (a) Symmetric pulse withtr=tf=5p sa n d tz=10 ns; asymmetric pulse with (b) tr=5 ps,tf=300 ps, and tz=10 ns and (c) tr=300 ps, tf=5p s ,a n d tz=10 ns. The upper right region is due to the ﬁnite size of the simulated window. the one with tr=300 ps and tf=5 ps. In the three panels, the total time ( ttot=tr+ts+tf) is shown starting from 300 ps, to be able to compare the diagrams evolution with the pulseshape. We observe that the ﬁrst depinning current dependsmainly on the rise time as depicted in panels (a) and (b),where only the fall time is varied. In these cases, the ﬁrstdepinning current is the same and equal to 2 .31 A/μm 2.T h e inﬂuence of the tfis an offset of the bands along the total time axis; therefore, if tfis decreased the second band is shifted to shorter times and almost disappears from the shown phasediagram. The inﬂuence of the rise time manifests itself alsoas an offset of the bands to larger times, but also to largercurrents and therefore a higher ﬁrst depinning current equalto 2.69 A/μm 2. The ﬁrst depinning current for a total time of 0.3 ns is 2 .78 A/μm2for the symmetric pulse, rising to 4.54 A/μm2[panel (b)] and 5 .03 A/μm2[panel (c)] in the asymmetric case. The micromagnetic calculation (not shown) 224431-4SYSTEMATIC MOTION OF MAGNETIC DOMAIN W ALLS IN . . . PHYSICAL REVIEW B 96, 224431 (2017) 2.5 6.25 10 je(A/μm2)00.51ts(ns) 0.010.511.01 tto t(ns) 2.5 6.25 10 je(A/μm2)00.751.5ts(ns) 0.010.761.51 tto t(ns) 2.5 6.25 10 je(A/μm2)00.751.5ts(ns) 0.31.051.8 tto t(ns) 2.5 6.25 10 je(A/μm2)00.751.5ts(ns) 0.31.051.8 tto t(ns) (a) (b) (c) (d) FIG. 4. Probability of DW motion in different bands for the ﬁnite strip (a) and the inﬁnite strip (b)–(d) at T=293 K. (a), (b) A symmetric pulse is applied with the characteristics tr=tf=5 ps, after an initial and ﬁnal tzof 10 ns. (c) An asymmetric pulse is applied with pulse shape tr=5p s ,tf=300 ps, after an initial and ﬁnal tzof 10 ns. (d) An asymmetric pulse is applied with pulse shape tr=300 ps, tf=5 ps, after an initial and ﬁnal tzof 10 ns. The damping parameter is taken as α=0.02 and β=2α. The band pockets, which appear in the panels (b) to (d) on the left, correspond to the band −1. The dotted lines are guide to the eyes and represent the band’s edges. gives similar results as the ones shown in Fig. 2, meaning a small offset of the bands compared with the ones calculatedwith the 1D model starting from the fourth band. The dependence of the depinning current on the rise time was deduced from the linearized equation of motion in the1D approximation, as the force on the DW can be written as[15,18] ¨X=−˙X τd−1 mdE dX+β ατdu+1+αβ 1+α2˙u, (3) where m=2αSμ 0Msτd /Delta1γ 0is the DW mass, τd=1+α2 αγ0Hkis the damping time, with Hkthe anisotropy ﬁeld, /Delta1is the DW width, andEis the pinning energy. The force on the wall depends on the current and its derivative; therefore, a shorter rise timeincreases the derivative term which leads to a decreasing ofthe depinning current and vice versa. For the present results,the damping time is 0.27 ns (0.68 ns for α=0.02); therefore, the DW is in the transient regime for the pulse duration used. B. Temperature dependence The temperature inﬂuences the systematic motion of the DW by modifying the DW relaxation in a potential wellafter an applied current pulse. The oscillations during the DWrelaxation could be sustained by the thermal noise, which couldlead to a jump to the wrong well while the following pulseoccurs, or on the contrary, the thermal noise could counter theeffect of the current pulse and the DW could stay pinned in the nondesired potential well. To carry out this study, we computed the DW dynamics atT=293 K for ﬁnite/inﬁnite nanostrip and different pulse shapes. The results are shown in Fig. 4. In all cases, panels (a) to (d), the DW ﬁrst oscillates freely (relaxation) in thepresence of the thermal noise in its initial well during 10 nsand afterwards a current pulse is applied to push the DW toanother well (corresponding to one of the bands in Figs. 2 and 3), followed by a DW relaxation during another 10 ns. The damping parameter αis taken as 0.02, lower than the one at T=0K[ 38], and the nonadiabatic parameter is taken asβ=2α=0.04. The same phase diagrams, as presented in Figs. 2and3, were recalculated with α=0.02 and β=0.04 atT=0 K and these bands, from one to four, are indicated by dotted lines in Fig. 4(a) for the ﬁnite nanostrip, while the bands from one to seven are indicated in panels (b) to (d) for theinﬁnite nanostrip. Starting from the fourth band, the shape ofthe bands changes showing a reentrant transition [except panel(c)], and the phase diagrams from panels (b) to (d) show bandpockets to the left corresponding to negative DW displacementof one notch (noted as band −1) even though the STT pushes the DW in the positive direction. These features are discussedin Sec. III C . The stochastic motion of the DW was computed for a number of bands with the stochastic 1D model, and only ona certain number of points micromagnetically. Figure 4(a) 224431-5A. PIV ANO AND V . O. DOLOCAN PHYSICAL REVIEW B 96, 224431 (2017) shows the probability of the DW motion in the ﬁrst four bands for a ﬁnite nanowire, when a symmetric current pulse(t r=tf=5 ps) is applied after a relaxation time of 10 ns. At the end of the pulse, the DW is relaxed another 10 nsbefore its position is considered acquired. A certain number ofrealizations was computed for a quarter of the phase diagrampoints of each band: 2700 realizations for each point shownfrom the ﬁrst band and 500 realizations for each point of thesuperior bands. The maximum of probability (100%) for theprecise positioning of the DW to the nearest notch is found onlyfor 0.76% of the ﬁrst band’s calculated points (17 points), while on 32.18% of the points the probability is superior to 95%. The maximum of probability decreases rapidly with increasing theband number, being 98 .6% (for three points) for the second band, 67 .4% for the third band, and 71% for the fourth band. These results are to be compared with panel (b), where thesame pulse is applied in an inﬁnite nanowire and the samenumber of realizations were computed for each band. Themaximum probabilities are similar for the ﬁrst two bands, forsimilar band point number density, indicating that at roomtemperature only few current pulse characteristics give 100%probability of precise positioning. The points in the ﬁrst band,that correspond to 100% probability of desired motion, appearfor an applied current superior to 7.7 A /μm 2and a pulse length between 100 ps and 130 ps for the ﬁnite strip and superior to8.1 A/μm 2and a pulse length between 90 ps and 120 ps for the inﬁnite case, respectively. For superior bands, starting with the fourth, the maximum of probability and band point numberdensity are increased in the inﬁnite nanostrip compared to theﬁnite case, as in the latter the potential barrier is weaker forthe more distant notches [see Fig. 1(c)]. Micromagnetically, we computed the probability for the phase diagram pointscorresponding to the 100% values found with the 1D model(which appear only in the ﬁrst band) on 200 realizations/point.These probabilities vary between 92% and 97%. The smalldifference between the probability calculations of the micro-magnetic and 1D model is attributed to the small shift in thephase diagrams that was shown to exist between the two. Figures 4(c) and4(d) show the probability of DW motion when an asymmetric current pulse is applied after 10 ns initialand ﬁnal relaxation. The pulse shape is t r=5p s ,tf=300 ps in panel (c) and tr=300 ps, tf=5 ps in panel (d). The maximum of probability in the ﬁrst band of panel (c) is96.77%, with only 2 .31% of calculated band points having a probability superior to 95% (from 2700 realizations). For thesuperior bands, the maximum of probability diminishes withonly 2 .93% of points having a probability superior to 90% in the second band and 2 .32% of points having a probability superior to 80% in the third band. For panel (d), the maximumof probability in the ﬁrst band is found to be 98 .07%, with 7.69% of points having a probability superior to 95%. The maximum of probability diminishes faster in the superiorbands as compared with the panel (c) results, with less pointshaving maximum band probability. For example, only 0 .79% of points have a probability superior to 90% in the secondband and 0 .54% of points have a probability superior to 80% in the third band. The difference in probability between thetwo asymmetric pulses is due to the fact that the long risetime of the pulse [panel (d)] produces the shift up and rightof the bands compared with panel (c); therefore, more pointswith maximum probability are found in the ﬁrst band (as these points are usually close to the band center) and lower pointswith maximum probability in the superior bands as there areless points on these bands. These results suggest that to achieve a well deﬁned DW positioning at room temperature by STT alone, individualpulses should be applied with very short rise and fall time. C. Inﬂuence of αandβon the DW dynamics The inﬂuence of the damping parameter αand of the nonadiabatic parameter βon the phase diagram for an inﬁnite strip at T=0 K is detailed in Fig. 5. The damping parameter αwas varied between 0.02 and 0.05, which correspond to the zero and room temperature values [ 38]. The nonadiabatic parameter was varied between zero and 2 α. The case with β=2αis shown in Fig. 2(b) and Fig. 4(b) forα=0.05 and 0.02, respectively. The computations with β=0 are presented in panels (a) and (b) of Fig. 5, while the ones with β=αare displayed in panels (c) and (d). We observe that for α=0.05, when βis diminished from 2 αto zero, important changes appear in the phase diagram only when βis less than α.I n this case [panel (a)], the results obtained with the 1D model(colored regions) exhibit only the ﬁrst band (band +1) with large pockets of negative numbered bands. In all cases, theDW is pushed initially by the current pulse in the positivedirection (to the right), so a negative band expresses a DWposition at the end of the pulse to the left of the initialnotch, in the opposite direction of the STT. The micromagneticcomputations conﬁrm this behavior, which was predicted andobserved before [ 13,20,21,27]. Due to the pinning potential, the DW deforms and can change its internal structure givingrise to a transient motion associated with DW inertia. Thetransient DW movement is proportional to the variation of thegeneralized angle of the wall: δX=−/Delta1 α/parenleftbigg 1−β α/parenrightbigg δψ, (4) withψthe azimuthal angle of the wall. For β=0, the transient motion is increased for large DW width or smalldamping parameter. This effect is displayed in Fig. 5(b) for α=0.02, as compared with panel (a), where an increased number of negative bands are visible. For the ultrashortcurrent pulses used, which are comparable with the DWdamping time τ d, the transient effects dominate at low damping parameter. A discrepancy is found between the micromagnetic results (symbols) and the 1D results in the upper right quadrant (highcurrent, longer pulse) for the phase diagrams with β=0. This discrepancy is due to the large angle variation at largecurrent and longer pulse that leads to the transformation of thetransverse DW and the creation of an antivortex close to theinitial notches before depinning [ 39] (see ﬁgure and movie in the Supplemental Material [ 36]). DW velocity boosting was predicted through antivortex generation at a singular notch in ananostrip [ 40]. The antivortex disappear quickly after the end of the current pulse, and in certain cases can reverse theorientation of the magnetization at the center of the transverseDW leading to DW motion in the opposite direction. Theantivortex does not appear in all the computed micromagnetic 224431-6SYSTEMATIC MOTION OF MAGNETIC DOMAIN W ALLS IN . . . PHYSICAL REVIEW B 96, 224431 (2017) je(A/μm2)1 5.5 10ts(ns) 00.751.5ts(ns) 00.751.5 (c) (d)(a) (b) je(A/μm2)1 5.5 10 -3-2-1u 12 0 3456789 10200 300 400 X (nm)-15-10-50510ψ (deg) 200 300 400 X (nm)-15-10-50510ψ(deg) 0 2.5 5 t (ns)-15010ψ (deg)Without notches With notches(e) (f) (h)0 1 2 t (ns)-90010ψ(deg) (g) FIG. 5. Inﬂuence of the damping parameter αand nonadiabatic parameter βon the phase diagram for an inﬁnite strip at T=0K :( a ) α=0.05 and β=0, (b) α=0.02 and β=0, (c) α=β=0.05, and (d) α=β=0.02. A sequence of symmetric pulses with the same characteristics as in Fig. 2are applied. The scattered symbols represent the micromagnetic results, while the colored regions are calculations with the 1D model. Trajectories in the phase space ( X,ψ ) corresponding to the −1 band in panels (b) [(d)]: (e) ts=250 ps, je=3.4A/μm2, α=0.02,β=0 and (f) ts=300 ps, je=3.6A/μm2,α=β=0.02. The scattered symbols and the full line represent the micromagnetic and the 1D results, respectively. The dotted part of the full line indicates the applied pulse duration. (g) DW angle variation for α=0.02,β=0 for several points in the panel (b). (h) Comparison of the DW angular variation for α=β=0.02 and ts=300 ps, je=3.6A/μm2for the inﬁnite strip with notches [band −1 in panel (d)] and strip without notches. results in the upper right quadrant of these phase diagrams. In the 1D model, the antivortex nucleation is not taken intoaccount. As is obvious from Eq. ( 4), forβ=αthe transient motion is somehow blocked and the DW travels rigidly. This seemsto be the case for β=α=0.05, as computed in the phase diagram shown in Fig. 5(c). However, when β=α=0.02, the negative bands are still present at low currents [panel (d)]without the apparition of an antivortex. This effect was veriﬁedmicromagnetically on a number of points (empty symbols),which compare very well with the 1D results. The pocketform of the −1 band is respected with a small shift, even a −2 band was observed micromagnetically corresponding to thesmall−2 pocket inside the −1 band. The DW motion in the caseβ=αcan be well explained analyzing the 1D equations of motion of the DW: (1+α 2)˙X=−αγ/Delta1 2μ0MsS∂E ∂X+γ/Delta1 2Hksin 2ψ+(1+αβ)u, (1+α2)˙ψ=−γ 2μ0MsS∂E ∂X−γα 2Hksin 2ψ+β−α /Delta1u, (5)where Sis the section of the wire, γthe gyromagnetic ratio, and Hkthe DW demagnetizing ﬁeld. Eis the pinning potential energy which is assumed parabolic inside the notch.The DW width variation is given by /Delta1(t)=/Delta1[/Psi1(t)]= π/radicalBig 2A μ0MS2sin2ψ+μ0MSHk. From the above equations, one can notice that the azimuthal DW angle ψand DW position X depend on the pinning potential (restoring force) created bythe symmetric notches. When a current pulse is applied, theDW is ﬁrst compressed and distorts on the potential barriermoving in the direction of the STT, while the azimuthalangle decreases in some cases below −10 ◦(dotted line), as is displayed in Fig. 5(h) (videos and additional ﬁgures in the Supplemental Material [ 36]). Initially, the STT pushes the DW in the positive direction, resulting in a positive DWvelocity and a negative restoring force (negative DW angle).If the DW does not have enough velocity to surpass thepotential barrier in this direction, it goes down the potentialwell towards the center of the notch and the restoring forcestill stays negative, while the velocity and angle continue todecrease with the velocity becoming negative. When the DWstarts to mount the potential well in the other direction, therestoring force becomes positive so the velocity and DW angle 224431-7A. PIV ANO AND V . O. DOLOCAN PHYSICAL REVIEW B 96, 224431 (2017) start to increase [Fig. 5(f)]. When the current pulse ends, the velocity decreases abruptly (increases in absolute value) andψincreases lightly due to the short fall time. At this moment, the DW position inside the potential well is important as itimposes a restoring force in the direction of movement or inthe opposite direction. However, the velocity and the DW angleare equally important for the jump to the previous notches tohappen (see Supplemental Material [ 36]). The jump occurs therefore due to a combination of factors, even if the restoringforce is positive or negative. The presence of the notch pinningpotential induces the DW distortion and therefore the transientmotion, as for the same pulse parameters no observable anglevariation is determined for a perfect strip. As the variation ofthe DW angle is directly related to the variation in position,the automotion is possible in this case triggered by the pinninginduced DW distortion, restoring potential force and smallnessof the damping parameter. In Fig. 5(g), several angular variations are presented corresponding to the −1 and −2 bands from panel (b). We observe that the amplitude of angular variation is directlyproportional with the spatial displacement. When an antivortexappears, the DW angle rotates out of plane and the DW positiondoes not correspond anymore to the 1D results and the motionopposite to the STT can be completely blocked. In general,we observe the automotion in all the cases when the DWangle increases above 10 ◦(in absolute values) during or after the current pulse and the maximum DW velocity is close or superior to 400 m /s (details in the Supplemental Material [36]). Exactly at the boundary between the −1 band and the zero band (pinned state), a small increase in the azimuthalangle of 0 .2 ◦and of the DW velocity by 5 m /s at the end of the current pulse, between two points in different bands, is enoughto promote a DW jump to a previous notch. The transient effects also appear when β>α , as depicted in Fig. 4forβ=2α=0.04. A low value of the damping parameter αis required to obtain observable consequences. A particularity of the case β>α is that the transient effects oppose or amplify the STT, as negative bands are determinedand a reentrant transition is seen at higher currents and pulselength in panels (b) and (d). The transient effects depend onthe pulse shape (see Supplemental Material [ 36]), as for the asymmetric pulse with t r=5p s ,tf=300 ps, the band −1i s barely visible and no reentrant transition of the bands is seen,while the −1 band increases when the pulse is symmetric and continues to increase, with even a second −1 band appearing at larger t sfor the asymmetric pulse with tr=300 ps, tf=5p s . The fall time tfplays an important role in the value of the DW velocity and DW angle at the end of the current pulse, as a shortt fleads to a very high DW velocity and a higher DW angle at the end of the pulse increasing the impact of transient effectsand inducing a DW depinning. Contrary to earlier beliefs [ 13], the DW depinning does not necessarily result from a large DWangle or the DW position at the end of the pulse. A maximumDW angle and DW velocity large enough during the currentpulse sufﬁce to ensure for example the jump to the previousnotch, even if the angle is not that large (and at its maximum)at the pulse end, as observed in the case of the asymmetricpulse with t r=5p s ,tf=300 ps (see Supplemental Material [36]). This is purely a transient effect due to a combined action of the pinning potential, low αand pulse shape. FIG. 6. Inﬂuence of the pulse raise time tron the phase diagram for a DW at T=0 K in an inﬁnite nanostrip for different parameters αandβ. The parameter space is the raise time vs current amplitude. In all cases, ts=tf=5p sa n d tz=10 ns. (a) α=0.02 and β= 0.04, (b) α=β=0.02, and (c) α=0.05 and β=0.1. In (a) the micromagnetic results (scattered symbols) are compared to the 1D model (colored regions), while in (b) and (c) only 1D model resultsare shown. At high currents, the micromagnetic results give several unintended states [the empty scattered symbol region to the right of (a)]. The inﬂuence of the rise time tron the phase diagrams is presented in Fig. 6forts=tf=5 ps and several parameters αandβ. The rise time is varied between 0 and 1.5 ns (case of a very asymmetric pulse). A ﬁrst observation is that the ﬁrstdepinning current is increased as the force term that depends onthe current derivative in Eq. ( 3) diminishes. The ﬁrst depinning current actually oscillates with t rdue to resonant effects, as the resonant frequency of the potential well is around 1.75GHz. This resonant effect is more important for the bandswith negative numbers for low α, as is depicted in panels (a) and (b) for α=0.02 and β=0.4 or 0.2, respectively. The 1D model gives good quantitative results as compared with themicromagnetic calculations, as shown in panel (a), although a 224431-8SYSTEMATIC MOTION OF MAGNETIC DOMAIN W ALLS IN . . . PHYSICAL REVIEW B 96, 224431 (2017) small shift is again found at high currents and more unwanted states. When β/lessorequalslantα, the positive bands are shifted to very high current, with only negative bands remaining for β=0 (image not shown). The phase diagram for α=0.05 and β=0.1, also shown in panel (c), still does not present negative bands butthe same oscillation of the ﬁrst depinning current and of theinterband transitions. The inﬂuence on the phase diagram is less dramatic when varying the fall time t f(images shown in the Supplemental Material [ 36]). For α=0.02 and β>α no negative bands appear and the positive bands are shifted to larger currents.When β/lessorequalslantα=0.02, only a small pocket of the −1 band appears and the positive bands are shifted as compared withFig. 5(d). IV . DISCUSSION AND CONCLUSION The presence of artiﬁcial constrictions in a nanostrip inﬂuences drastically the movement of a magnetic DW.When ultrashort current pulses are applied, the DW canexhibit an important distortion at the notch depending onthe pulse characteristics. Thereby, the DW displays inertialikeeffects, which can have dramatic consequences on its transientdisplacement. These effects depend largely on the dampingparameter αand on the nonadiabatic parameter β.F o rβ<α , these effects generally oppose the STT effect after the pulseend and DW motion in the direction of the electric currentis possible. If β>α , these effects oppose or amplify the STT effect and jumps to the left or the right notches arepossible after the end of the pulse. The transient effects in thiscase depend on the pulse characteristics. This could constituteanother way of experimentally comparing the two parametersαandβfor a ferromagnetic material. At room temperature, the jump probability to the desired notches decreases with increasing band number, each bandnumber corresponding to a positioning to desired notches inthe direction of the STT. Maximum positioning probabilityis reached only for very short rise and fall time to thenearest neighbors notches only. Therefore, to shift reliablythe DWs between notches, current pulses correspondingto displacement from one symmetric notch to the closestneighbors should be used. The shape of the current pulseinﬂuences the depinning current and shifts the bands. Thephase diagram for the case of two domain walls situatedat different symmetric notches (images not shown), that aredisplaced by the same current pulse in the same direction, isvery similar with the ones presented in Sec. III, but the bands are narrower and the interband depinning is larger between the ﬁrst bands.The main drawback of the classical DW displacement under STT alone compared with more exotic torques (of spin-orbitorigin) is the high current values necessary. The currentinduces Joule heating in the nanostrip that can largely increasethe temperature and could even destroy the ferromagneticstate. The increase in temperature is even more important atthe constrictions in the nanostrip. Several theoretical studieswere dedicated to Joule heating in nanowires [ 41–44], usually considering a standard Py nanostrip on a Si /SiO 2substrate. We evaluated the temperature increase for our Ni strips ondifferent substrates like pure Si, SiO 2,o rN i 3Si4membranes for a current pulse length of 1 ns. On pure Si, consideringan inﬁnite 3D substrate [ 41], the temperature increase is negligible being of 4 K for j=5A/μm 2(17 K for j= 10 A/μm2). However, on SiO 2substrate of 300 nm thickness [42], the temperature increase is larger, being 26 K for j=5A/μm2(103 K for j=10 A/μm2). If 100 nm thick Ni3Si4membranes are used [ 15,42], the temperature increase is of 41 K for j=5A/μm2(163 K for j=10 A/μm2). The variation in temperature depends on the material parametersof the substrate (like thermal conductivity) and on theconductivity of the nanostrip. In the above estimations, thebulk Ni room temperature conductivity was used ( σ −1= 7.3μ/Omega1cm) [ 45]. If the dimensions are reduced to nanometers [46], the conductivity of Ni can vary drastically (a factor four) and the temperature increase can be more important, but in the main time the length of the current pulse that amounts to the maximum probability at room temperature isaround 0.1 ns, which reduces signiﬁcantly the temperatureincrease. 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PhysRevE.101.012205.pdf	PHYSICAL REVIEW E 101, 012205 (2020) Magnon-induced chaos in an optical PT-symmetric resonator Wen-Ling Xu, Xiao-Fei Liu, Yang Sun, Yong-Pan Gao, Tie-Jun Wang ,and Chuan Wang School of Science and the State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China (Received 30 June 2019; published 10 January 2020) Optomagnonics supports optical modes with high-quality optical factors and strong photon-magnon interac- tion on the scale of micrometers. These novel features provide an effective way to modulate the electromagneticﬁeld in optical microcavities. Here in this work, we studied the magnon-induced chaos in an optomagnonicalcavity under the condition of parity-time symmetry, and the chaotic behaviors of electromagnetic ﬁeld couldbe observed under ultralow thresholds. Even more, the existence optomagnetic interaction makes this chaoticphenomenon controllable through modulating the external ﬁeld. This research will enrich the study of lightmatter interaction in the microcavity and provide a theoretical guidance for random number state generation andthe realization of the chaotic encryption of information on chips. DOI: 10.1103/PhysRevE.101.012205 I. INTRODUCTION The achieving of strong interaction between photons and solid state is an important issue in the ﬁeld of nanophotonicsand quantum information science. On the micrometer scale,optical microcavities [ 1], including the silica microsphere and the silicon-based microresonators, localize the opticalﬁeld with large intensity due to the high-quality factor andsmall mode volume which could enhance the Purcell factor[2,3] between the photons and the solid atoms. Along with the increment of the optical ﬁeld in these microresonators,the nonlinear interaction between the mechanical (magnetic)ﬁeld and the optical ﬁeld is triggered, which are deﬁned asthe optomechanics and optomagnonics, respectively. Thesenonlinear effects are caused by the radiation pressure effectand the optomagnetic effect. During recent studies, optome-chanics [ 4–12] and optomagnonics [ 13–16] have become two important approaches to realize the interaction between hybridmodes in optical microresonators. Usually, there is one modein the electromagnetic degree of freedom and another mode inthe mechanical or magnetic degree of freedom. To be speciﬁc, in an optomagnonic resonator, both the magnons and photons are strongly localized in the resonator.This feature also ensures that there are strong interactionsbetween magnons and photons in it. In addition, as magnonsor spin waves are unique platforms for the realization oflong-lifetime quantum memories [ 17,18] and quantum state transfer, the study of the optomagnonical cavity has attractedmuch attention. Recently, studies on the interaction betweenphotons and magnons in optomagnonical cavity have ex-plored, for example, the efﬁcient magneto-optical couplingin whispering-gallery-mode (WGM) resonators [ 15,19–21]. Also, the pronounced nonreciprocity and asymmetry in thesideband signals have been observed [ 22]. In addition, the electromagnetic cavity-mediated phonon-magnon interaction wangtiejun@bupt.edu.cnhas been investigated [ 23]. Moreover, the nonlinear dynamics such as the chaotic motions of cavity optomagnonics systems[24,25] have attracted more attention. On the other hand, the photonic molecule [ 26]i sa ni m p o r - tant concept in the coupled cavities system, which has beenused since 1998 in electromagnetically interacting opticalmicrocavities. The photonic molecule is constructed by a clus-ter of coupled optical microcavities. When individual opticalmicrocavities are brought into close proximity, their opticalmodes interact and then form the photonic molecules, andtheir modes are usually called supermodes [ 27]. Several basic issues in quantum physics have been explored in the studyof optical microresonators, for example, the parity-time ( PT) symmetry [ 28–31]. The concept of PT symmetry originated from the description of the properties of the Hamiltonianwhich has real eigenvalue even it contains an imaginary part.It was ﬁrst mentioned theoretically in Ref. [ 28] and demon- strated experimentally in the higher-order harmonic oscillatorscheme [ 32]. Recently, this concept has been extended to the research ﬁeld of micro- or nano-optics [ 29,33–36].PT symmetry can be practically used in various applications,such as in optical isolators [ 29] and high-resolution sensing [37]. Although people have done in-depth research on PT- symmetry systems, studies on PT-symmetric optomagnoni- cal systems are still rare. In this paper, we propose a PT-symmetric optomagnonical system and investigate the chaotic behaviors of the ﬁeld,which has been widely studied in various hybrid microcavitysystems [ 38–40]. Yttrium iron garnet (YIG) spheres [ 41–44] are usually regarded as magnon resonators, but their potentialas high- Q( a sh i g ha s3 ×10 6) optical WGM microresonators is always overlooked [ 15,45]. In our proposal, we excite the optical mode in the YIG sphere and ﬁrst focus on thenonlinear characteristics of optical ﬁeld affected by magneticmaterials; when the intensity of the optical ﬁeld approachesthe chaotic threshold, the nonlinear system is accompanied bychaotic motion [ 46,47]. To analyze the chaotic behavior, the Lyapunov exponent [ 48] is numerically calculated and phase 2470-0045/2020/101(1)/012205(8) 012205-1 ©2020 American Physical SocietyXU, LIU, SUN, GAO, W ANG, AND W ANG PHYSICAL REVIEW E 101, 012205 (2020) γ κJa1 YIG a2 ActiveSin Sout(a) WGM H yz x magnon modeoptical mode(b) YIG G FIG. 1. Schematic diagram of an optomagnonical system. (a) WGM PT-symmetric system including a YIG sphere a1coupled with an active cavity a2with coupling strength J. Here Sinand Soutrepresent input and output, respectively. The frequency and amplitude of the driving ﬁeld are ωdand/Omega1d;γrepresents the decay rate of optical mode in a1andκrepresents the gain rate of cavity a2. (b) Optomagnonic cavity with homogeneous magnetization along thezaxis (an external magnetic ﬁeld His added along the zaxis). The arrows represent the magnons with components in all directions. There is a localized optical mode with circular polarization in the y-zplane. The dotted line indicates the homogeneous magnon mode couples to the optical mode with strength G. diagrams of the motion equation of photons are displayed. The nonlinear coefﬁcient of photons is determined by the couplingstrength and magnon, and thus the generation of chaos canbe controlled by adjusting coupling strength or magnon. It isworth noting that magnon can be modulated by manipulatingthe external magnetic ﬁeld [ 16], which provides us a reliable method for effectively controlling the chaotic behavior. Com-pared with normal coupled cavities, PT-symmetric cavities have strong local ﬁeld even under the circumstance of weakdriving. Therefore, the generation of chaos only requiresan ultralow threshold. Our research provides a method forgenerating random numbers and realizing secret informationprocessing [ 49,50]. II. MODEL AND DYNAMICAL EQUATIONS The PT-symmetric optomagnonical system is shown in Fig. 1(a). The system shows a photonic molecule structure which consists of a YIG sphere coupled withan Er3+/Yb3+-doped microsphere [ 51] (there are also many other Er3+/Yb3+-doped microresonator [ 52,53] structures have been experimentally proven to be achievable). These twomicrospheres are coupled through an evanescent ﬁeld withcoupling strength J. The YIG sphere coupled to a ﬁber-taper is a passive cavity which supports a homogeneous Kittelmagnon mode (i.e., the ferromagnetic resonance mode) [ 54] tuned at the resonant frequency /Omega1=5 MHz and an optical WGM. The Er 3+/Yb3+-doped microsphere which supports an optical WGM is denoted as the active cavity. The magneticpolarons that exist in the YIG sphere are pumped by theexternal bias magnetic ﬁeld [ 55]. The Hamiltonian of our system can be described as follows [ 39]: ˆH=ˆH c+ˆHm+ˆHI ˆHc=¯h/Delta1c(ˆa† 1ˆa1+ˆa† 2ˆa2)−¯hJ(ˆa† 1ˆa2+ˆa1ˆa† 2) +i¯h/Omega1d(ˆa† 1−ˆa1)( 1 ) ˆHm=¯h/Omega1ˆSz ˆHI=− ¯hGˆSxˆa† 1ˆa1, where ˆ a† 1(ˆa1) and ˆ a† 2(ˆa2) are the creation (annihilation) operators for optical modes of the YIG sphere and the activecavity, respectively. These two modes have the same resonantfrequency, denoted as ω c. The passive cavity is driven by the ﬁeld with frequency ωdand amplitude /Omega1d.Gis the coupling strength of optomagnonic. Considering the systemin a frame rotating with ω d, the detuning of the frequency can be expressed as /Delta1c=ωc−ωd. In our proposed system, the YIG sphere is magnetized by an external bias magneticﬁeld along zaxis and can be used to control the precession with frequency /Omega1. In theory, /Delta1 cis the beat frequency formed by the interaction of the cavity and the input photon, and itresonates with magnons when /Delta1 c=/Omega1. In other words, under the action of the cavity, the pump light can resonate with themagnetons. Here we assume a magnetic system with spinS=(S x,Sy,Sz) which is dimensionless. The coupling term between the magnon and optical mode in the YIG sphereis shown in ˆH I. As shown in Fig. 1(b), we assume that the optical ﬁeld couples only to the xcomponent of the macrospin Swith coupling coefﬁcient Gwhich is a constant depended by material [ 39] for convenience. Other parameters of this system are (γ,κ,G,/Omega1,Sx,/Delta1c,/Omega1d)=(1 MHz, 0 .4γ,5×102Hz, 5M H z ,1 ×106,/Omega1,1×102γ) throughout this paper. To explore the nonlinear dynamics of PT-symmetric op- tomagnonical system, we focus on derivation of the coupledsemiclassical Langevin equations of motion from Eq. ( 1) under the classical limit by considering the loss γand gain κof two cavities, respectively. The intrinsic spin Gilbert damping [ 56] of the YIG sphere is about 10 −4[43] which can be omitted in our scheme. In the case of mean-ﬁeldapproximation, the equations of motion can be expressed asfollows: ˙S x=−/Omega1Sy ˙Sy=/Omega1Sx+Ga∗ 1a1Sz ˙Sz=−Ga∗ 1a1Sy (2) ˙a1=(−i/Delta1c−γ/2)a1+iJa 2+iGS xa1+/Omega1d ˙a2=(−i/Delta1c+κ/2)a2+iJa 1. 012205-2MAGNON-INDUCED CHAOS IN AN OPTICAL … PHYSICAL REVIEW E 101, 012205 (2020) Based on Eq. ( 2), the detuning eigenfrequencies can be obtained as /Delta1ω=i(κ−γ)/4+/radicalbig 16J2−(κ+γ)2/4. Here /Delta1ω is a complex number whose real part represents the frequency difference of the supermode and the imaginarypart represents the linewidth. When Jis enhanced to be comparable to the optical linewidth γ, the supermode ap- pears around the center resonant frequency with splitting width /Delta1ω=/radicalbig 16J2−(κ+γ)2/4 due to the tunneling ef- fect between loss and gain cavities. Equation ( 2)s h o w s that the intracavity ﬁeld and the magnon mode would ef-fect each other during the evolution via the optomagnonicalinteraction. In our system, when /Delta1ω=0, i.e., J=(γ+ κ)/4, it approaches the exceptional point, where eigenvalues and corresponding eigenstates of the system coalesce. WhenJ>(γ+κ)/4, the system exhibits PT-symmetric charac- teristics, which will signiﬁcantly affect the nonlinear dy-namics. According to Eq. ( 2), the evolutionary trajectory of the system depends on the initial conditions. To describethis dependence, we introduce time-dependent perturbation /vectorδ=(δS x,δSy,δSz,δa1r,δa1i,δa2r,δa2i), where δajrandδaji (j=1,2) represent the real and imaginary parts of the corre- sponding perturbations, respectively. By ignoring the effects of high-order perturbations, we can obtain /vectorδ=Mδ[57] with coefﬁcient matrix Mwhich is used to describe the divergence of nearby trajectories in the phase space. Each value in thematrix Mcan be regarded as an inﬁnitesimally perturbation to the initial condition, M=⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣0 −/Omega1 00 0 0 0 /Omega1 0 G/parenleftbig a 2 1r+a2 1i/parenrightbig 2Ga1rSz 2Ga1iSz 00 0 −G/parenleftbig a2 1r+a2 1i/parenrightbig 0 −2Ga1rSy −2Ga1iSy 00 −Ga1i 00 −γ/2( /Delta1c−GSx)0 −J Ga1r 00 ( −/Delta1c+GSx) −γ/2 J 0 00 0 0 −J κ/2/Delta1c 00 0 J 0 −/Delta1cκ/2⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦. III. RESULT AND DISCUSSION There is usually a chaotic region in nonlinear dynamical systems. One of the parameters that may affect the generationof chaos is the coupling strength J, which can be tuned by adjusting the distance of the two resonators. To study the inﬂu- ence of the coupling strength, we numerically solved Eq. ( 2) under different coupling intensities. We deﬁne I a=\|a1\|2as the optical intensity in cavity a1and the corresponding power spectrum S(ω) can be obtained by performing fast Fourier transform of Ia, i.e., S(ω)∝\|/integraltext+∞ −∞Iae−iωtdt\|, which can be directly measured [ 58]. Based on numerically results, the opti- cal trajectories are presented in phase space, and the evolution of the optical intensity Iaand the logarithm of power spectrum LnS(ω) are illustrated in Fig. 2. For a ﬁxed driving amplitude /Omega1d=1×102γ, the nonlinearity of intracavity ﬁeld will be enhanced when the strength Jis increased. We then turn to the time interval 8 →12μs where the system tends to stabilize. The system works in a state under the PT broken situation at weak coupling J=0.1γ. The intensity of the optical ﬁeld in cavity a1is small and the cavity mode can be periodically modulated. The trajectory in phase space is regular as shownin Fig. 2(a). On the other hand, when Jis increased to 3 γ, the system is at the state of PT symmetry, and the intensity of the optical ﬁeld in YIG sphere is enhanced. The trajectory presents more complicated behavior in phase space and the power spectrum Ln S(ω) is larger due to enormous energy accumulation. In addition, chaotic motions of the optical ﬁeld become more obvious if we continue to increase the strength Jto 8γ. The intensity of optical ﬁeld in the cavity a 1is so strong that the optical trajectories in phase space are much more complicated and the corresponding power distribution characterizes the emergence of chaotic motion.To give quantitative description of the inﬂuence of J,t h e Lyapunov exponent with different J/γis studied, as shown in Fig. 3. We deﬁne the parameter of perturbation δIa= \|a1+δa1\|2−\|a1\|2, whose logarithmic slope versus time tis regarded as the Lyapunov exponent [ 48]. When Lyapunov exponent is positive, it means that the disturbance is exponen-tially increasing and the system is sensitive to the initial con-ditions; in other words, it is approaching the chaotic region.Figure 3presents that in the case of a given component of the macrospin S x=1×106, the intensity of the electromagnetic ﬁeld in cavity a1is enhanced as the increment of J/γ.I t clearly shows that the chaotic motion appears when one tunesthe coupling strength Jso that the system is at the state of PT symmetry. It also should to be noticed that when J/γ∼6, the Lyapunov exponent is unstable. The reason is that theevolution of the system shows a certain randomness when thesystem is in a chaotic state. Therefore, even though the Lya-punov index is different, the chaotic properties of the systemare still the same in this region. Moreover, in order to know if the chaos correspond to high or low intensity spin waves, we deﬁne a parameter η=S x/Sto compare the value Sxwith S, where Sis the total spin expressed as S=√ S2 x+S2 y+S2 z. A ss h o w ni nF i g . 4, because of the ﬂuctuation phenomenon, one can observe that ηexhibits periodical changing over time t(note that here we only give an example by J=8γto illustrate the problem). Figure 4indicates that the direction of the spin is resonant and it also proves that our waves areindeed spin waves. Then it is reasonable to use S xto study the nature of spin waves. As discussed earlier, as part of the PT-symmetric opto- magnonical system, the nature of the magnon in the systemwill inevitably affect the generation of chaos in the system. Tostudy this inﬂuence, we investigate the variation of dynamical 012205-3XU, LIU, SUN, GAO, W ANG, AND W ANG PHYSICAL REVIEW E 101, 012205 (2020) 8 9 10 11 12150200250300350(b) 0 1 02 03 04 045678(c) 8 9 10 11 12100150200250300350(e) 0 1 02 03 04 045678(f) 1048 9 10 11 120200400600800(h) 0 1 02 03 04 056789(i)l ll 0123-4-20241011 (g) FIG. 2. Dependence of system dynamics on coupling strength J. Panels (a), (d), and (g) are optical trajectories in phase space; panels (b), (e), and (h) are the evolution of optical intensity Ia; and panels (c), (f), and (i) are corresponding power spectrum Ln S(ω) changed over ω//Omega1 for different strength J. [(a)–(c)] J=0.1γ. [(d)–(f)] J=3γ. [(g), (h), and (i)] J=8γ. The decay rate of YIG is γ=1M H z ,t h eg a i n of active cavity is κ=0.4γ, the coupling strength between magnon mode and optical mode is G=5×102Hz, the precession frequency is /Omega1=5M H z ,t h e xcomponent of the macrospin SisSx=1×106, the detuning is /Delta1c=/Omega1, and the driving amplitude is /Omega1d=1×102γ. 02468 1 0-0.500.511.522.533.5106 0.3 0.35 0.4-1-0.280.38 FIG. 3. Dependence of Lyapunov exponent on coupling strength J. Lyapunov exponent in cavity a1varies with different J/γ.T h e inset corresponds to the Lyapunov exponent for J=0.3γ,J= 0.35γ,a n d J=0.4γ. Other system parameters are the same as in Fig. 2.8 9 10 11 12-1-0.500.51 FIG. 4. Parameter ηdeﬁned as Sx/Sversus time twith J=8/γ. Here Sis the total spin with S=√ S2 x+S2 y+S2 z. Because of the ﬂuctuation phenomenon, ηexhibits periodical change over time t. Other system parameters are the same as in Fig. 2. 012205-4MAGNON-INDUCED CHAOS IN AN OPTICAL … PHYSICAL REVIEW E 101, 012205 (2020) 0 500 1000 1500 2000-5-3-1135109 (a) 8 9 10 11 122004006008001000(b) 02468 1 056789(c) 8 9 10 11 1203006009001200 (e) 02468 1 056789(f) 8 9 10 11 12100200300400(h) 02468 1 045678(i)l ll0 400 800 1200 1600-2-10121010 (d) FIG. 5. Dependence of PT-symmetric system dynamics on magnon. Panels (a), (d), and (g) are optical trajectories in the phase space; panels (b), (e), and (h) are the evolution of optical intensity Ia; panels (c), (f), and (i) are corresponding power spectrum Ln S(ω) changed over ω//Omega1 for different Sx.( a ) Sx=1×103.( b ) Sx=1×105.( c ) Sx=1×106. The coupling strength J=2γbetween the two cavities is ﬁxed, and other system parameters are the same as in Fig. 2. behavior of system under different component of the magnon, as shown in Fig. 5. We present the dIa/dt-Iaphase diagram in Figs. 5(a),5(d) and5(g); the time evolution of the optical in- tensity Iain Figs. 5(b),5(e) and5(h); and the power spectrum in Figs. 5(c),5(f) and5(i). To ensure that the system works under the PT-symmetric conditions, we set the coupling strength as J=2γ. Then, by gradually increasing the strength of the magnon, the dynamics of our system could be observed.When the component is S x=1×103as shown in Figs. 5(a), 5(b) and 5(c), we can ﬁnd the orbits in phase space show chaotic behavior, while there are only three main spectrums in the frequency domain. When the component Sxis increased to 1×105, the system is still chaotic, while amplitudes of side- bands have been greatly enhanced. It can be explained that theinteractions between the photons and magnons are enhancedwith the component of magnon increasing. Meanwhile, theoptical nonlinearity of the system is strengthened because itis easier to exchange energy between photons and magnonsunder such strong interactions. Then the four-wave mixingprocess is more intensive, and the number of sidebands of thesystem gets larger. In order to verify our assumption, we in-crease S xto 1×106, then the optical trajectories in the phase space are more complicated, and the chaotic phenomenon ofthe system is quite signiﬁcant. On the other hand, the numberof sidebands in the system continues increasing, which alsoconﬁrms our previous assumption. Since the magnons can becontrolled by an external bias magnetic ﬁeld, the generation ofthe sidebands can be effectively controlled by the additionalfar ﬁeld. Previously, the effect of coupling strength is discussed based on whether the system is under PT-symmetric condi- tions. It is still required to discuss the performance of the PT- symmetric operation or the coupling strength on the chaosgeneration. To further classify the effects of PT-symmetric operation and the coupling strength, the Lyapunov exponentis calculated in the passive-active (or PT-symmetric) cavi- ties coupled photonic molecule and passive-passive cavitiescoupled photonic molecule with different driving amplitude.We compare these two models in Fig. 6under strong cou- pling J=3γ. It is obvious that the driving threshold of chaos under the PT-symmetric condition is much lower than the passive-passive coupled condition. The threshold value 012205-5XU, LIU, SUN, GAO, W ANG, AND W ANG PHYSICAL REVIEW E 101, 012205 (2020) 0 200 400 600 800 1000-101234567106 (0.5, 7.516e4) (600, 1.379e6) FIG. 6. Dependence of Lyapunov exponent on driving amplitude /Omega1d. The red line indicates the effect of the driving amplitude on the Lyapunov exponent in the passive-passive system. When /Omega1d= 6×102γ, Lyapunov exponent is positive and chaos will appear. The blue line indicates the effect of the driving amplitude on Lyapunov exponent in the PT-symmetric system. When /Omega1d=0.5γ, Lyapunov exponent is positive. System parameters are κ=− 0.4γ,J=3γand other parameters are the same as in Fig. 2. /Omega1d=0.5γof the PT-symmetric condition is even lower than the dissipation of the cavity. Thus, we can concludethat the PT symmetry dominates the generation of chaos in our scheme. This conclusion also enriches the novelty of thePT-symmetry system. IV . DISCUSSION The experimental value of the coupling strength between the magnon and photon is weak, but to our knowledge,there are several ways to enhance this interaction. First, thecoupling strength Gcan be increased by the coupling the optical modes with the magnetic textures. If we can increasethe mode volume of magnon or reduce the mode volume of theoptical mode, then the coupling strength can obtain increased.It has been demonstrated in Ref. [ 59]. Second, the interaction can be effectively enhanced through pump-probe technology,which has been demonstrated experimentally [ 15]. Moreover, the coupling strength of our article can be reached by optimalmode matching, and it has been theoretically discussed inRef. [ 60]. Moreover, because of the coupling between the electric ﬁeld and the magnetization in Faraday-active materials, theelectromagnetic energy is modiﬁed and expressed as [ 61]: ˆH MO=− iθFλn 2π/epsilon10/epsilon1 2/integraldisplay drm(r,t)·[E∗(r,t)×E(r,t)],(3) where m(r,t) is the magnetization in the sample. θFis the Faraday rotation per unit length λnand the prefactorθFλn 2π∼ 4×10−5in YIG. /epsilon10and/epsilon1(∼5 in YIG) are the vacuum and relative permittivity, respectively. Since m(r,t)i sr e - lated to the local spin operator which, in general, cannot bewritten as a linear combination of bosonic modes. In ourproposal we consider the homogeneous Kittel mode where all spins precess in phase and can be replaced by a precessingmacrospin, but, for example, in Ref. [ 59], the authors consid- ered spin wave excitations on top of a possibly nonuniformstatic ground state m 0(r) andδm(r,t)=m(r,t)−m0(r). In the case of \|δm\|/lessmuch 1, the harmonic oscillators corresponding to the magnon modes can be used to express these terms,and by quantizing the spin wave, they obtained the couplingHamiltonian linearized in the spin ﬂuctuations, ˆH MO=/summationdisplay αβγGαβγˆa† αˆaβˆbγ+H.c.,(4) where ˆ aandˆbrepresent photon and magnon operators, respec- tively. Gαβγis the optomagnonic coupling. This Hamiltonian is the same in form as our scheme but has a different physicalmeaning. Moreover, Eq. ( 4) is still nonlinear, since it contains interacting terms. Actually, as long as the system is nonlinear,the chaotic motion is expected. It can be demonstrated byour methods, for example, calculating the optical trajectoriesin the phase space and corresponding Lyapunov exponent.Moreover, according to our knowledge, if there is no nonlin-earity in the system, then no chaos will occur in the systemregardless of the form of magnetic excitations. In summary, we have studied the chaotic behavior of the electromagnonic ﬁeld in the optomagnetical photonicmolecule under the PT-symmetric condition and discussed the effective controlling method of the chaotic phenomenon.First, the effects of PT symmetry by adjusting the coupling strength between the passive and active cavities are presented.It is found that the electromagnetic ﬁeld exhibits chaoticbehavior even with weak optical drive. On the other hand,we also investigated inﬂuence of the strength of magnonsin the passive optical microcavity. The result indicates thatthe strength of magnons is related to the four-wave mixingprocess, and then we can modulate the external bias magneticﬁeld to control the generation of sidebands in the system.Even more, when the spin waves are strong enough, thesidebands of the system will become complicated, and thechaotic phenomenon of the system will be more obvious.Finally, we compared our scheme with passive-passive cou-pled cavities and found that the threshold of PT-symmetric scheme is lower, which is conducive to the development ofsecret communications. The study proposes an achievableapproach to control the nonlinear dynamics of the system,especially the generation of chaos, which paves the way formany important applications, such as the chaotic encryptionof information and random numbers generation. ACKNOWLEDGMENTS The authors gratefully acknowledge the Project funded by the Ministry of Science and Technology of the People’sRepublic of China (2016YFA0301304), the National NaturalScience Foundation of China through Grants No. 61622103and No. 61671083, the Fok Ying-Tong Education Foundationfor Young Teachers in the Higher Education Institutions ofChina (Grant No. 151063), the Fundamental Research Fundsfor the Central Universities. 012205-6MAGNON-INDUCED CHAOS IN AN OPTICAL … PHYSICAL REVIEW E 101, 012205 (2020) [1] K. J. Vahala, Nature 424,839(2003 ). [2] E. M. Purcell, Phys. Rev. 69,37(1946 ). [3] S. Horoche and D. Kleppner, Phys. Today 42,24(1989 ). [4] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86,1391 (2014 ). [ 5 ] T .J .K i p p e n b e r ga n dK .J .V a h a l a , Science 321,1172 (2008 ). [ 6 ] W .P .B o w e na n dG .J .M i l b u r n , Quantum Optomechanics (CRC press, 2015). [7] J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, Nat. 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PhysRevApplied.12.064004.pdf	PHYSICAL REVIEW APPLIED 12,064004 (2019) Limited Stochastic Current for Energy-Optimized Switching of Spin-Transfer-Torque Magnetic Random-Access Memory Eunchong Baek, Indra Purnama, and Chun-Yeol You* Department of Emerging Materials Science, Daegu Gyeongbuk Institute of Science and Technology, Daegu, Korea (Received 5 August 2019; revised manuscript received 26 September 2019; published 3 December 2019) The switching of spin-transfer-torque magnetic random-access memory (STT MRAM) in the simple macrospin model is determined by the amplitude and pulse duration of the applied current, and it requires a current that is higher than a critical current, Ic. However, this critical current misses one fundamental physical issue for the commercialization of STT MRAM; the so-called nonswitching probability ( PNS)o r write soft-error rate (WSER), which is inﬂuenced by the stochastic nature of the switching process at ﬁnite temperature. Herein, we propose a limited stochastic switching (LSS) current, which is another deﬁnition for the critical current with the PNSincorporated. The deﬁnition of the LSS current and the analytical expressions are obtained by solving the Fokker-Planck equation with a given speciﬁc PNSvalue. Most importantly, by using the LSS current and optimizing it together with the related pulse-duration time, we ﬁnd the optimum combination of current amplitude and pulse duration, which may reduce the energy consumption of the STT MRAM by up to 75%. DOI: 10.1103/PhysRevApplied.12.064004 I. INTRODUCTION In recent years, the spin-transfer-torque magnetic random-access memory (STT MRAM) has been actively studied for high-density nonvolatile memory applica- tions [ 1–14]. A typical STT MRAM structure with per- pendicular magnetization is shown in Fig. 1(a), where a tunneling barrier layer is sandwiched between two ferro- magnetic layers that are referred to as the free layer (FL) and pinned layer (PL). The device operates by applying current to switch the magnetization states of the FL in the STT MRAM between the low-resistance state, where the magnetization of the FL and the PL are parallel to each other, and the high-resistance state where the two ferromagnetic layers have antiparallel magnetization. The switching is mediated by the spin-transfer-torque eﬀect [3,15,16], whereby the conduction electrons of the injected current exchange their angular momentum with the local magnetization of the FL. For application purposes, the quality of STT MRAM is thus determined by the speed as well as the energy eﬃciency of the reading and writing processes. These factors can then be evaluated by noting the switching current, the switching time and the thermal stability of the STT MRAM. Several advances have been made to reduce the minimum current and consequently improve the energy eﬃciency of the STT MRAM, such as by using materials with perpendicular anisotropy [ 17,18], enhancing the second-order anisotropy [ 19], engineering *cyyou@dgist.ac.krthe size of the magnetic tunnel junction (MTJ) [ 20,21], introducing tilting PL magnetization direction [ 22], intro- ducing lateral asymmetry of junction shape [ 23], or by introducing a strong spin-scattering layer [ 24,25]. Conven- tionally, the critical current ( Ic) that is needed to perform the FL switching can be obtained by solving the Landau- Lifshitz-Gilbert (LLG) equation with a simple macrospin model [ 15,26,27]. However, this method does not give us other important physical information such as the writing error rate (WER), which is the rate where the FL mag- netization does not switch even when the applied current amplitude is higher than Ic. To explain this inconsistency, a new deﬁnition of critical current is needed, which takes into account the stochastic nature of the FL switching dynamics. In this work, we propose an alternative deﬁnition of switching current, which is termed as the limited stochastic switching (LSS) current. The LSS current, iLSS, is derived by solving the Fokker-Planck equation (FPE), which in turn gives us the relation between the applied LSS current and the error rate of both the writing and the reading pro- cess. We show that there is an exponential relation between the applied LSS current and the error rate of the STT MRAM, which gives us a better control over the opera- tion error of the STT MRAM. In addition, by calculating the energy consumption using the LSS current and the cor- responding pulse-duration time, we show that the most energy-eﬃcient STT MRAM operation can be achieved when iLSS≈2( I=2Ic), whereby doing so reduces the energy cost by up to 75%. 2331-7019/19/12(6)/064004(8) 064004-1 © 2019 American Physical SocietyBAEK, PURNAMA, and YOU PHYS. REV. APPLIED 12,064004 (2019) (a) (b) FIG. 1. (a) Schematic of the perpen- dicular STT MRAM. Current is applied across the device to switch the magneti-zation of the free layer by spin-transfer torque. (b) Conventional switching dia- gram with applied current and pulseduration, t p. The magnetization is not switched in the white region. Icis the critical current for switching in a simplemacrospin model. II. MODEL In this Paper, we consider a typical perpendicular STT MRAM structure as shown in Fig. 1(a). The schematic ﬁgure shows the application of current to the device in order to switch the FL magnetization from the up to the down state. We assume that the magnetization of the PL does not change when the device is under operation, which can be achieved by introducing an antiferromagnet layer underneath the PL to induce exchange bias between the PL and the antiferromagnet layer. The conventional switching diagram obtained by using the LLG equation is shown in Fig.1(b). In this diagram, the conventional critical current Icis shown by the solid line, which sharply separates the “switched” and the “not-switched” regions. We can then predict whether the STT MRAM switches at a given cur- rent amplitude and pulse duration by comparing the values to the critical-current line in the diagram. However, as mentioned previously, using this phase diagram as well as Icdoes not give us the error rate of the device at the ﬁnite temperature. Instead of using the LLG equation to produce the error rate, the FPE is used in this work to describe the switching dynamics statistically. Several advances have been made by using the FPE to analyze the stochastic dynamics of spin-torque-induced switching [ 28–33]. For instance, Mat- sumoto et al. have devised an alternative way to deﬁne the thermal stability of a nanomagnet under the inﬂuence of current [ 33], while Moon et al. have uncovered the rela- tionship between the damping constant and the switching- time distribution of a ferromagnetic nanostructure [ 32]. By solving the FPE, we can obtain the magnetization switch- ing probability under thermal ﬂuctuation as well as under the application of current. Here it is assumed that the FL has a perpendicular uniaxial anisotropy without external magnetic ﬁeld. Following the previous work by Butler et al. [29], the FPE for time-dependent probability distri- bution of magnetization as a function of the polar angleθ=cos−1(mz)of the FL is given by ∂ρ(θ ,τ) ∂τ=−1 sinθ∂ ∂θ/bracketleftbigg sin2θ(i−cosθ)ρ(θ ,τ) −1 2/Delta1sinθ∂ρ(θ ,τ) ∂θ/bracketrightbigg .( 1 ) Here, the ﬁeld-like term of the STT and azimuthal angle dependence are not considered. ρ(θ,τ)is the prob- ability of the magnetization pointing at θat the normal- ized time τ=(t/t0)=[αμ0γHeﬀ k/(1+α2)]t, while i= (I/Ic)=(η/planckover2pi1/2αeμ0Heﬀ kM sV)Iis the normalized current, αis the magnetic damping constant, Heﬀ k=(2Keﬀ u/μ0M s) is eﬀective anisotropy ﬁeld, ηis the spin polarization of PL, Vis the volume of the FL and M sis the saturation magnetization at T=0K ./Delta1=(Keﬀ uV/kBT)is the ther- mal stability factor, Keﬀ u=Ku−(1/2)μ0M2 sis the eﬀec- tive uniaxial magnetic anisotropy constant while kBis the Boltzmann constant, Tis the absolute temperature, and Kuis the ﬁrst-order uniaxial magnetic anisotropy constant. We consider a 40-nm-diameter Co 20Fe60B20/MgO perpen- dicular MTJ system, where α=0.027, μ0Heﬀ k=340 mT, M s=1.58 T, thickness is 1 nm, and corresponding /Delta1= 43 for T=300 K [ 34]. In this dimension, the macrospin model is considered over the domain-wall model [ 35–38] as the macrospin model has been shown to match well with the experimental data [ 34]. For larger or smaller nanos- tructures, the accuracy of our model can be improved by using the appropriate parameterization for the shape anisotropy and the thermal stability. For those material and nanostructure parameters, we have 1 τ=0.62 ns and 1i=88.02μA. III. RESULTS AND DISCUSSIONS A. Nonswitching probability The analytical solution of Eq. (1)can be obtained by using the ansatz solution ρ(θ,τ)=[2/W(τ)]e x p 064004-2LIMITED STOCHASTIC CURRENT FOR. . . PHYS. REV. APPLIED 12,064004 (2019) {−[θ2/W(τ)]}in the limit where θis small. Then the nonswitching probability ( PNS), i.e., the error rate of the writing process, is given by [ 29] PNS(i,τ)=1−exp/braceleftbigg/parenleftBigπ 2/parenrightBig2 (i−1)/Delta1 1−iexp[2(i−1)τ]/bracerightbigg .( 2 ) This PNSis obtained by integrating ρ(θ,τ)fromθ=0t o θ=(π/2). We assume that the magnetization switching occurred when θ>( π / 2). Figure 2shows the error rate of the switching, PNS, of the STT MRAM with /Delta1=60 as a function of iandτp, pulse-duration time in the unit of the normalized time τ. Compared to the conventional switching diagram shown in Fig. 1(b), here we can see that the device has a ﬁnite nonzero PNS, even with a current that is higher than the conventional critical current Ic(i>1). In addition, it is also possible to calculate the error rate of the reading process. In the reading process, magnetiza- tion switching is not desired. Therefore, the error rate of the reading process is the switching probability ( PS). By using the aforementioned method, the PSis expressed by PS(i,τp)=1−PNS =exp/braceleftbigg/parenleftBigπ 2/parenrightBig2 (i−1)/Delta1 1−iexp[2(i−1)τ p]/bracerightbigg .( 3 ) Since, for reading events, ( i−1) is negative, we can sim- plify the equation by considering i×exp[2(i−1)τ p]/lessmuch1. PS(i,τp)=exp/bracketleftbigg/parenleftBigπ 2/parenrightBig2 (i−1)/Delta1/bracketrightbigg .( 4 )In this case, surprisingly the PSis independent of τpand only proportional to the exponential of iand/Delta1. It implies that PSalways has a ﬁnite nonzero value, and we can ﬁnd the proper reading current to satisfy a speciﬁc PSvalue. We discuss more details later. B. The limited stochastic switching current With the above discussion in mind, it is clear that know- ing the conventional critical current Icalone is not enough for optimizing the STT MRAM working condition. Previ- ously, as shown in Fig. 1(b), the critical-current line Icin the conventional switching diagram is supposed to sepa- rate the region where a switching occurs or otherwise with100% certainty. However, as shown in Fig. 2, the switch- ing probability is not 100% even when i>1(I>I c). Hence, we introduce a new deﬁnition of the LSS current for writ- ing and reading, which considers the PNSor PSobtained by Eqs. (2)and(4), respectively. Let us consider the LSS current for the writing event ﬁrst. In this case, since the nonswitching events are undesirable only during the writing event, we assume i>1. Then, in the limit of i×exp[(i−1)τ p]/greatermuch1, Eq.(2)becomes PNS(i,τp)≈1−exp/braceleftbigg −/parenleftBigπ 2/parenrightBig2(i−1)/Delta1 iexp[−2(i−1)τ p]/bracerightbigg . (5) By performing Taylor series expansion at i=2 and noting that exp[ −2(i−1)τ p]/lessmuch1, we obtain PNS(i,τp)≈/parenleftBigπ 2/parenrightBig2(i−1)/Delta1 iexp[−2(i−1)τ p], (6) FIG. 2. Three-dimensional sur- face plot with color map show- ing the nonswitching probability, PNSas a function of the normal- ized applied current, i, and pulse duration, τp. The thermal stability factor here is 60 and switching is assumed at θ=(π/2). 064004-3BAEK, PURNAMA, and YOU PHYS. REV. APPLIED 12,064004 (2019) ln/bracketleftBigg PNS /Delta1/parenleftbigg2 π/parenrightbigg2/bracketrightBigg =ln/parenleftbigg 1−1 i/parenrightbigg −2(i−1)τ p,( 7 ) iw LSS(PNS,τp)≈1+2l n/bracketleftbig (2√e//Delta1)( 2/π)2PNS/bracketrightbig −4τp+1.( 8 ) By using the above deﬁnition, iw LSSgives us the information of how much current is needed for the device to operate with a speciﬁc nonswitching probability value at a given pulse duration. The LSS current is proportional to the nat- ural logarithm of PNSand inversely proportional to τp. Figure 3(a) shows iw LSSas a function of τpin a logarithm scale for various values of PNSwith/Delta1=60. The result shows that depending on the PNS,iw LSScan be more thandouble. The dotted line shows the conventional critical- current line obtained by solving the LLG equation. For solving the LLG equation, we assume the same MTJ model for the FPE, then the critical time for switching is given byτc=({−(1/2)(i+1)ln(1−z0)+(1/2)(i−1)ln(1+ z0)+ln[1−(z0/i)]}/i2−1), where z0is the initial value of cos θ[29,39]. If we take the inverse function of this and let the critical time be a variable, we obtain the critical- current line for a given critical time, which is a given pulse duration. The dotted line is the numerical results of those inverse functions. Since it does not consider the nonswitch- ing events, it has a smaller value for the same τpwhen compared to the LSS current. For both methods, iw LSSis saturated to 1 at a large enough τp, as shown in Fig. 3(a). Figure 3(b) shows iw LSSincreases linearly with exponential (a) (b) (c) FIG. 3. (a) Limited stochastic switching current for the writing process, iw LSS, as a function of logarithmic τpfor various PNS.T h e dotted line shows the result of the LLG equation, which does not consider the stochastic eﬀect. (b) iw LSSas a function of logarithmic PNSfor various τp. (c) Limited stochastic switching current for the reading process, ir LSS, as a function of logarithmic PSforτp=5, 10, and 50. The thermal stability factor here is 60. 064004-4LIMITED STOCHASTIC CURRENT FOR. . . PHYS. REV. APPLIED 12,064004 (2019) decrease of PNS. Depending on τp, the linear increase of iw LSShas a diﬀerent slope. For the reading event, the LSS current, which takes into account the error rate PS,can be obtained from Eq. (4)as ir LSS(PS)≈1+1 /Delta1/parenleftbigg2 π/parenrightbigg2 lnPS.( 9 ) The LSS current for the reading process is also found to be independent of τpas we already mention, it is inversely proportional to /Delta1and is proportional to the natural log- arithm of Ps. Figure 3(c) shows ir LSSwith/Delta1=60. The dotted line is the result of Eq. (9)withτp=∈ fty, while the red, blue, and purple solid lines are numerical solu- tions of Eq. (3)forτp=5, 10, and 50, respectively. The red and blue line has a ﬁnite diﬀerence from the dotted line at a large value of ln PS. However, when ln PSbecomes smaller, there is no diﬀerence for the τp=10 (blue) line, theτp=50 (purple) line, and the τp=∈ fty(dotted) line, which shows that at low reading current, the error rate is not aﬀected by the pulse-duration time. C. Energy consumption of the STT MRAM under the LSS current Up to this point, we discuss the analytical solution of the LSS current at both the writing and the reading pro- cesses. Equation (8)shows that device operation with low PNSrequires large amplitude of pulse current or long pulse duration. However, this has a negative impact for energy-eﬃcient operation where the energy consumption of the device is a product of the power and the time.Furthermore, long pulse duration implies slower operat- ing time of the devices. Therefore, we should optimize how the current is applied by considering not only PNSbut also the energy consumption of the device. If we assume that the resistance of device ( R) is constant and set by 30 k/Omega1[34], the energy consumption of each current pulse (E) can then be written as E=P×t=RI2t=E0i2τp, where E0=RI2 ct0=0.14 pJ for the 40-nm-diameter Co20Fe60B20/MgO perpendicular MTJ system [ 34]. From Eq.(2), we can obtain τpas a function of ifor a given PNSas τp(PNS,i) =ln{1+(π/2)2[/Delta1/ln(1−PNS)](1−i)}−lni 2(i−1). (10) Then, Eas a function of i,PNSis given as E(PNS,i)=E0i2τp(PNS,i) =E0i2/parenleftBigg ln{1+(π/2)2[/Delta1/ln(1−PNS)](1−i)}−lni 2(i−1)/parenrightBigg . (11) Now, the energy consumption includes stochastic eﬀect and is related to iand PNS. Since E0and/Delta1are material and device parameters, if we set a speciﬁc PNSvalue, Eis directly related only to the applied current, i. Figure 4(a) shows Eas a function of ifor various PNSand/Delta1.A s shown in Fig. 4(a),f o r PNS=1×10−8and 1 ×10−10, Eincreases sharply at i≈1 because the pulse-duration (a) (b) FIG. 4. (a) Energy consumption of the STT MRAM, E(pJ) as a function of i. for various PNSand/Delta1. Inset in (a) is the enlargement of i=1.3–2.7 for better comparison of a minimum point. (b) E(pJ) as a function of the nonswitching probability, PNSfor i=2, /Delta1=60 and 100. 064004-5BAEK, PURNAMA, and YOU PHYS. REV. APPLIED 12,064004 (2019) timeτpthat is needed to maintain the speciﬁc error rate increases sharply at low current, which in turn results in the sharp increase of energy usage. Most interestingly, E decreases as iincreases and it has a minimum point near i–2. At this point, the energy per write cycle is approx- imately down to just 1/4 of the energy expenditure near i–1. This decrease in the energy expenditure, E, can be attributed to the sharp decrease of the pulse duration τp that is needed to maintain the error rate PNS. Beyond this point, Estarts to increase again as the contribution from the i2term starts to overcome the contribution from τp.A s a function of PNS,Edoes not have a minimum point and only increase as PNSdecreases [Fig. 4(b)]. The analytical solution of iwhere Eis minimum can be obtained from Eq. (11) with the same approxima- tion for Eqs. (5)–(8). If we set and assume that q≡ (π/2)2[/Delta1/ln(1−PNS)]/greatermuch1, Eq. (11) becomes E(PNS,i)≈E0i2 2(i−1)ln/bracketleftbigg −q/parenleftbigg 1−1 i/parenrightbigg/bracketrightbigg . (12) By performing Taylor series expansion at i=2, we obtain E(PNS,i)≈E0i2 4(i−1)/bracketleftBig 2l n/parenleftBig −q 2e/parenrightBig +i/bracketrightBig . (13) This result also supports the validity of the Taylor series expansion at i=2 throughout this Paper, because Eis a minimum near i=2 regardless of PNSas shown inFig.4(a).F r o mE q . (13), we obtain iEminas iEmin=1 4/braceleftbigg/radicalbigg/bracketleftBig 2l n/parenleftBig −q 2e/parenrightBig +1/bracketrightBig/bracketleftBig 2l n/parenleftBig −q 2e/parenrightBig +9/bracketrightBig −2l n/parenleftBig −q 2e/parenrightBig +3/bracerightBig . (14) Here, iEminis an energy minimizing LSS current with a given PNSvalue that is obtained by using the same approxi- mation as Eq. (8). With the energy-minimization condition satisﬁed, now iEminis only a function of PNSinstead of τp.I nE q . (14), we can simplify the 2 ln[ −(q/2e)] term by using the proper approximation and also by performing Puiseux series expansion [ 40]a t PNS=0a s 2l n/parenleftBig −q 2e/parenrightBig ≈2/braceleftbigg ln/bracketleftbigg/parenleftBigπ 2/parenrightBig2/Delta1 2e/bracketrightbigg −lnPNS−PNS 2/bracerightbigg ≈2/braceleftbigg ln/bracketleftbigg/parenleftBigπ 2/parenrightBig2/Delta1 2e/bracketrightbigg −lnPNS/bracerightbigg . (15) In Fig. 5(a), the solid lines show the results of Eq. (14) after replacing 2 ln[ −(q/2e)] by the expression in Eq. (15) for various /Delta1. As shown in Fig. 5(a),iapproaches 2 at low PNSregardless of /Delta1. Similar to iEmin, it is also possible to get the analytical solution of energy-minimizing pulse duration, τEmin.A si n the case of current, the energy-minimizing pulse duration, τEminis determined by the PNS. By combining Eq. (10) with Eq. (14), we obtain τEminas a function of PNSas (a) (b) [Numerical solutions of eq. (11)] FIG. 5. (a) Limited stochastic switching current, which minimizes the energy, iEminas a function of logarithmic PNSfor various /Delta1. The solid lines show the results of Eq. (3.5) and the dotted line shows the numerical solutions of Eq. (3.2) for /Delta1=40. (b) Pulse duration at the minimum energy, τEminas a function of logarithmic PNSfor various /Delta1. By following these two ﬁgures, if PNSis given, as e to f (iEmin,τEmin)can be determined. 064004-6LIMITED STOCHASTIC CURRENT FOR. . . PHYS. REV. APPLIED 12,064004 (2019) τEmin(PNS)=1 4+2l n [−(q/2e)]+1/radicalbig {2l n [−(q/2e)]+1}{2l n [−(q/2e)]+9}−2l n [−(q/2e)]−1. (16) Figure (b) shows τEminincreases as PNSdecreases while iEminapproaches 2. Then, by using iEmintogether with the related τEmin, we ﬁnd the optimum combination of current amplitude and pulse duration for the most energy-eﬃcient operation of the STT MRAM. IV . CONCLUSION In conclusion, we formalize an alternative perspective on the minimum current that is needed to operate STT MRAM, which is termed as LSS current. By using the LSS current formalization, we obtain a better control over the error rate during the operation of the STT MRAM. For instance, to obtain a low write soft-error rate of 1×10−7[41], we can apply the LSS current of 0.16 mA (I=2Ic) with pulse duration of 6.2 ns ( t=10t0). More- over, we showed that the most beneﬁcial condition for STT MRAM operation is to apply current that is around twice that of Ic.At iLSS≈2(I≈2Ic), the energy cost per cycle is approximately 1/4 of that when iLSS≈1(I≈Ic). Also, we obtain the analytical expressions of LSS current that minimizes the energy cost together with the correspond- ing pulse-duration conditions. Both are dependent on PNS only. In this study, all relevant material parameters are rep- resented in /Delta1and Ic. Material parameterwise, the LSS current is weakly dependent on /Delta1as shown in Fig. 5(a), and it should be mentioned that the LSS current is scaled by Ic. For a given PNS, those analytical expressions gave us the most energy-optimized combination. This study hence- forth hopes to bring a better understanding of the relation between the applied current and the error rate as well as to further improve the performance of next-generation STT MRAM. ACKNOWLEDGMENTS This work is supported by the National Research Foundation (NRF) of Korea (2015M3D1A1070465, 2017R1A2B3002621, 2017H1D3A1A01013754). [1] J. C. Slonczewski, Current-driven excitation of magnetic multilayers, J. Magn. Magn. Mater. 159, L1 (1996). [2] L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, P h y s .R e v .B 54, 9353 (1996). [ 3 ] E .B .M y e r s ,D .C .R a l p h ,J .A .K a t i n e ,R .N .L o u i e ,a n d R. A. 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PhysRevB.89.134427.pdf	PHYSICAL REVIEW B 89, 134427 (2014) Switching ﬁeld distributions with spin transfer torques in perpendicularly magnetized spin-valve nanopillars D. B. Gopman,1,D. Bedau,2S. Mangin,3E. E. Fullerton,4J. A. Katine,2and A. D. Kent1 1Department of Physics, New York University, New York, New York 10003, USA 2HGST San Jose Research Center, San Jose, California 95135, USA 3Institut Jean Lamour, Universit ´e de Lorraine, UMR CNRS 7198 Nancy, France 4CMRR, University of California at San Diego, La Jolla, California 92093, USA (Received 6 December 2013; revised manuscript received 7 April 2014; published 28 April 2014) We present switching ﬁeld distributions of spin-transfer-assisted magnetization reversal in perpendicularly magnetized Co/Ni multilayer spin-valve nanopillars at room temperature. Switching ﬁeld measurements of theCo/Ni free layer of spin-valve nanopillars with a 50 nm ×300 nm ellipse cross section were conducted as a function of current. The validity of a model that assumes a spin-current-dependent effective barrier for thermallyactivated reversal is tested by measuring switching ﬁeld distributions under applied direct currents. We showthat the switching ﬁeld distributions deviate signiﬁcantly from the double exponential shape predicted by theeffective barrier model, beginning at applied currents as low as half of the zero ﬁeld critical current. Barrierheights extracted from switching ﬁeld distributions for currents below this threshold are a monotonic functionof the current. However, the thermally induced switching model breaks down for currents exceeding the criticalthreshold. DOI: 10.1103/PhysRevB.89.134427 PACS number(s): 85 .75.Bb,75.60.Jk,75.76.+j I. INTRODUCTION Spin-transfer driven magnetization reversal is of great fundamental interest and has a direct impact on magneticinformation storage technologies [ 1]. Nanostructures with perpendicular magnetic anisotropy are of particular impor-tance to storage applications [ 2–4]. The all-perpendicular geometry yields reduced critical currents I c, high uniaxial symmetry, and high spin-torque switching efﬁciency, i.e., asmall ratio of the critical current to energy barrier I c/U[5–7]. Implementation of competitive spin-transfer devices requireslow critical currents while maintaining sufﬁcient thermalstability to suppress thermally activated switching betweenmagnetization conﬁgurations. Understanding the thermal stability of a nanomagnet under spin-transfer torque (STT) is of critical importance topredicting device performance, particularly in the subthresholdcurrent drive regime in which devices can still switch bythermal activation. In the absence of STT, the probability that atﬁnite temperature a nanomagnet’s direction of magnetizationswitches in an applied magnetic ﬁeld is expected to followa simple model of thermal activation over an energy barrier[8,9]. A widely used recent model predicts that spin-transfer torques lead to a spin-current-dependent effective energybarrier for thermally assisted transitions [ 10,11]. This model predicts that a nanomagnet under STT reaches a new steadystate that corresponds to an equilibrium distribution overmagnetic conﬁgurations with an effective potential energylandscape that is modiﬁed by the current. The predictionsfrom this model were investigated numerically using Fokker-Planck calculations [ 12,13] and empirically using dwell-time measurements of in-plane magnetized nanopillar devices [ 14]. Recent spin-torque switching studies in perpendicularly magnetized nanopillar spin valves have applied this model. daniel.gopman@physics.nyu.eduExperimentally obtained energy barrier heights were shownto be much lower than the uniaxial barrier height determinedby the entire magnetic free layer volume [ 15]. Nevertheless, the switching appears well described by thermally overcom-ing a single energy barrier, whose height is related to anexcited magnetic subvolume in the free layer element [ 16]. Standard measurements probing the effects of spin torques onswitching—current-ﬁeld state diagrams and measurements ofthe Stoner-Wohlfarth astroid—also appear to agree with thesimple effective barrier model [ 6,17–19]. In order to further test the validity of this effective model under STT, it is important to probe the thermal switchingbehavior of a spin-torque-driven nanomagnet. We focus onthe magnetization reversal characteristics of the Co-Ni freelayer (FL) element in all metallic spin-valve (SV) nanopillarswith a perpendicularly magnetized polarizing reference layer(RL) composed of Co-Ni and Co-Pt. Spin valves withboth the polarizer and the free layer having perpendicularmagnetic anisotropy (PMA) are a uniaxial model system, inwhich all of the contributions (internal and external ﬁelds,anisotropy axis, and spin-current axis) are nearly alignedperpendicular to the ﬁlm plane. Co-Ni multilayered ﬁlmsshow high PMA, signiﬁcant spin polarization, and low Gilbertdamping compared to other PMA systems (Co/Pt, Co/Pd,FePt) [ 20–24]. Furthermore, the all-metal system allows us to generate current densities higher than those possible inmagnetic tunnel junctions [ 25]. In this paper we focus on the inﬂuence of STT on thermally assisted reversal. After a brief description of the spin valvesstudied here, we demonstrate a variety of methods to probe thethermally activated reversal characteristic of a nanomagnet.We begin by introducing standard measurements probing theeffects of spin torques on switching—the current-ﬁeld statediagram. We then present measurements of the coercivityversus ﬁeld-sweep rate under several direct currents to probechanges in the spin-current-dependent effective energy barrierheight. Finally, we focus on statistical measurements of the 1098-0121/2014/89(13)/134427(7) 134427-1 ©2014 American Physical SocietyGOPMAN, BEDAU, MANGIN, FULLERTON, KATINE, AND KENT PHYSICAL REVIEW B 89, 134427 (2014) switching ﬁeld under ﬁnite direct currents. Using a switching ﬁeld model for thermal activation over a single energy barrier,we extract the effective barrier height from switching ﬁelddistribution measurements at each applied current in orderto monitor the evolution of the effective barrier height withcurrent. We have obtained over 5000 switching events ateach applied current, which allows us to sample a relativelylarge number of the statistically rare events at the distributiontails. These statistically rare events are indicators of wheredeviations from an equilibrium (effective barrier) model ﬁrstemerge. We use a Gauss quantile plot of the switching ﬁelddistributions to highlight the data at the distribution tailsand demonstrate deviations of our data from the model forthe top one percent of the switching probability for currentsbelow the zero-ﬁeld critical switching current I c. Furthermore, signiﬁcant deviations from the equilibrium model emerge afterexceeding this current threshold. II. DEVICE FABRICATION AND ELECTRICAL MEASUREMENTS The Co/Ni nanopillars studied here are part of an all-perpendicular spin-valve device. Details on materi-als and sample preparation have been reported previ-ously [ 6]. The magnetic multilayered structure consists of a Pt(3)/[Co(0.25)/Pt(0.52)] ×4/Co(0.25)/[Ni(0.6)/Co(0.1)] ×2 hard reference layer and a [Co(0.1)/Ni(0.6)] ×2/Co(0.2)/Pt(3) free layer separated by a 4 nm Cu spacer layer and patternedinto 50 ×300 nm 2ellipse-shaped nanopillars by a process that combines e-beam and optical lithography. Figure 1(a) presents a scanning electron micrograph of a representative50×300 nm 2ellipse. Measurements were taken at room temperature and with ﬁelds applied within 3 deg of the freelayer easy axis. The reference layer magnetization switchesfor an applied ﬁeld close to 1 T. Since no ﬁelds greater than0.3 T are applied during the measurements, the reference layeris expected to remain ﬁxed and pointing along the direction ofnegative magnetic ﬁelds, unless otherwise speciﬁed. The magnetization of the free layer is probed indirectly with four-probe measurements of the spin-valve magnetoresistance.Figure 1(b) portrays the measurement geometry and circuit FIG. 1. (Color online) (a) Scanning electron micrograph of a patterned 50 ×300 nm2ellipse-shaped spin-valve nanopillar and (b) magnetotransport measurement setup and deﬁnition of currentdirection.diagram for measurements of the free layer magnetization orientation. The differential resistance was measured using asmall amplitude ac current ( I ac=100μA rms) at a frequency of 10 kHz and lock-in ampliﬁer, with a 300 μs time constant. Lock-in data was acquired at a rate of 1 kHz. To avoidsigniﬁcant spin-transfer effects, this ac current was designed tobe lower than the room temperature, zero-ﬁeld critical current,I c≈5m A /greatermuchIac. For experiments under constant dc current, the positive current direction corresponds to electrons ﬂowingfrom the ﬁxed polarizing layer (unidirectional arrow) to thefree layer (bidirectional arrow). III. STATE DIAGRAM: EFFECT OF SPIN-TRANSFER TORQUES ON COERCIVITY The current-ﬁeld state diagram investigates the stability of different spin-valve states under STT from electric currentsas well as applied magnetic ﬁelds. This diagram revealsthe regions of applied ﬁelds and currents that exhibit onlyan antiparallel (AP) or parallel (P) state as well as bistableregions where either AP or P states can be stabilized. Figure 2 illustrates the state diagram of one of our 50 ×300 nm 2SV devices alongside the method for generating the diagram froma series of ﬁeld hysteresis loops under many different appliedcurrents. Figure 2(a)presents a resistance versus perpendicular FIG. 2. (Color online) Experimental state diagram of a 50×300 nm2ellipse spin-valve device: (a) Red and blue curves show the increasing and decreasing branches of the resistance versus perpendicular applied ﬁeld hysteresis loop. ( Ris the resistance deviations from R0=3.218/Omega1.) (b) We subtract the resistances of the decreasing branch from the increasing branch Rdiff(H)=Rdec(H)−Rinc(H). (c) A series of resistance difference traces from hysteresis loops at applied currents \|Idc\|/lessorequalslant15 mA is used to generate an interpolated density map to determine the state diagram. This density map corresponds to the states available to thespin-valve device: Green regions indicate only one state (antiparallel or parallel), which orange regions represent an area of bistability. Vertical arrows illustrate the magnetization orientations of the twolayers in each region. 134427-2SWITCHING FIELD DISTRIBUTIONS WITH SPIN . . . PHYSICAL REVIEW B 89, 134427 (2014) applied ﬁeld hysteresis loop with a decreasing ﬁeld (AP →P) and an increasing ﬁeld (P →AP) branch. The resistance difference between the AP →P and P →AP branches at each applied ﬁeld value is plotted in Fig. 2(b), revealing ﬁeld ranges with a low resistance difference, indicating that only one state(AP or P) is stable. There is also a region with a substantialresistance difference, indicating a region of bistability. The setof resistance difference traces obtained from ﬁeld hysteresisloops at a series of currents is interpolated into a densitymap that is presented as the state diagram in Fig. 2(c).A trend line can be seen along the perimeter between the orange(bistable) region and the green (AP or P) regions. The lineardependence of the switching currents with applied ﬁeld ina region surrounding zero ﬁeld can be understood within amodiﬁed N ´eel-Brown law in which the spin current modiﬁes the effective barrier separating AP and P states. This result isconsistent with ﬁnite temperature calculations of the current-ﬁeld evolution of the state diagram [ 26]. It is noteworthy that the effective barrier model is only approximately valid, and itsrange of applicability is conﬁned to the high barrier regime,which can be modiﬁed by the current amplitude. Critical tothis study is the effective barrier height before switching. Wecan estimate the barrier height by considering the measurementtime (1 ms) and an approximate attempt time (1 ns). The barrierbefore switching can be estimated from these two time scales:ln (10 −3/10−9)∼10/greatermuch1. Thus, for the currents used in our study, it appears reasonable to use this approximate model. The critical switching current at room temperature and zero ﬁeld, Ic(H=0 ) ,i s5m Af o rA P →P and −7m A for P →AP. The sudden increase in slope dIc/dH for ﬁelds\|μ0H\|/greaterorsimilar100 mT cannot be understood by a modiﬁed N´eel-Brown law, but tilts of the applied ﬁeld relative to the uniaxial axis and higher-order terms in the uniaxial potentialenergy landscape (e.g., sin 2nθ,n/greaterorequalslant2) may be important for the origin of the deviations from the predicted lineardependence [ 17]. IV . FIELD-SWEEP RATE MEASUREMENTS We can probe changes in the thermal stability of a nanomagnet under STT through variable ﬁeld-sweep ratemeasurements. The evolution of the mean switching ﬁeldversus ﬁeld-sweep rate is sensitive to the nanomagnet’s thermalstability factor ξ=E 0/kBT, where E0is the barrier height at zero applied ﬁeld, kBis the Boltzmann constant, and T=300 K for our measurements. We assume an Arrhenius- type law for thermal activation /Gamma1(H)=/Gamma10exp(−ξεη), where ε=(1−H/H c0) andη=1.5[27–29] determine the scaling of the thermal stability with ﬁeld, Hc0is the switching ﬁeld at zero temperature and we assume /Gamma10=1 GHz. Then, the cumulative probability that the nanomagnet does not switchunder a magnetic ﬁeld ramped linearly in time ( dH/dt =v= const.) from zero up to a ﬁeld Hhas the form of a double exponential: P NS(H)=exp/bracketleftbigg −/integraldisplayH 0/Gamma1(H/prime)/v dH/prime/bracketrightbigg . (1) This thermal activation expression models the experimentally obtained switching ﬁeld distributions that we will introducefurther below, but also yields an approximate expression for FIG. 3. (Color online) (a) Dependence of the mean switching ﬁeld (μ0H)o ft h eA P →P transition on ﬁeld-sweep rate ( dH/dt ) for a second 50 ×300 nm2ellipse spin-valve device of the same composition with somewhat different characteristics subjected to several applied dc currents. (b) Evolution of the extracted effective thermal stability parameter ( ξ) versus dc current ( Idc) obtained from best-ﬁt lines [Eq. ( 2)] to data in (a). Inset to (b) showing the state diagram for this second device. the mean switching ﬁeld for a given ﬁeld ramping rate v: H(v)∼=Hc0/braceleftbigg 1−/bracketleftbigg ξln/parenleftbigg/Gamma10Hc0 ηvξεη−1/parenrightbigg/bracketrightbigg1/η/bracerightbigg . (2) We use the above expression to ﬁt the evolution of the mean switching ﬁeld with sweep rate for a series of direct currentsin order to determine the evolution of the thermal stabilityparameter ξwith current. We have conducted statistical switching ﬁeld measurements under STT of multiple spin-valve devices, presenting similar behavior although with varying thermal stability. To betterhighlight the features general to these devices, we willpresent the results in this section on variable ﬁeld-sweep ratemeasurements on a second spin-valve device of the same sizeand composition with somewhat different characteristics thanthe device studied in Fig. 2. Figure 3(a) shows the evolution of the mean switching ﬁeld for the AP →P transition versus the logarithm of the ﬁeld-sweep rate along with best-ﬁt trendlinesﬁtting the data from Eq. ( 2). From these trend lines, we have extracted the thermal stability ξfor each applied current. The 134427-3GOPMAN, BEDAU, MANGIN, FULLERTON, KATINE, AND KENT PHYSICAL REVIEW B 89, 134427 (2014) evolution of the thermal stability versus applied current is plotted in Fig. 3(b), with the state diagram for this device provided in the inset as comparison to the device studied inFig. 2. Applying a linear ﬁt to the data set consistent with an effective barrier model [ 10], we extrapolate a zero-temperature critical switching current I c0=9 mA. This value is consistent with the closing of the bistable region of the state diagramplotted in the inset. In Fig. 3we present results on the mean switching ﬁeld of an AP →P transition that agrees with the predictions of an effective barrier model under STT. We will proceed furtherto offer a more rigorous test to the model through statisticalmeasurements distributions under STT. We will investigate theentire switching ﬁeld distribution to test the extent to which thedata agrees with the switching ﬁeld model at the distributiontails. In the following section we demonstrate that plotting theswitching ﬁeld distributions on a Gauss quantile scale permitsus to better assess the quality of ﬁt to our statistical data at therare events comprising the tails of the distributions. V . QUANTILE SCALE PLOTTING OF THE SWITCHING FIELD DISTRIBUTIONS Figure 4shows the distributions of switching ﬁelds for over 5000 switching ﬁeld events, AP →P, under direct currents of 2.5 mA [Figs. 4(b) and4(d)] and 5 mA [Figs. 4(a) and4(c)], for the SV device ﬁrst introduced in Fig. 2. The switching ﬁeld distributions in the top half [Figs. 4(a) and 4(b)]a r e plotted on linear axes for both applied ﬁeld μ0Hand for the cumulative nonswitching probability PNS, while in the bottom half [Figs. 4(c) and4(d)]w ep l o t PNSon a Gaussian quantile scale. This rescaling permits us to qualitatively assess (a) thedouble-exponential character of thermal activation [Eq. ( 1)] giving rise to the asymmetric shape of the distribution (e.g.,non-Gauss) and (b) the quality of ﬁtting the rare events at the FIG. 4. (Color online) Switching ﬁeld distributions for AP →P transition under constant dc currents [(b) and (d) 2.5 mA and (a) and (c) 5 mA]. (a) and (b) Plotted on a linear yscale. (c) and (d) Plotted on a Gauss quantile yscale rescaled by Y=√ 2i n v e r f( 2 y−1) to magnify the data sets at the tails of the distributions. This plotting scheme highlights deviations of the data from the thermal activationmodel at the distribution tails.tail of our distribution by stretching the yaxis around where the tails of the distribution is condensed on a linear scale. We maptheyaxis representing P NSonto a Gauss-quantile scale using the following rescaling of the yaxis: Y=√ 2i n v e r f( 2 y−1), (3) in which inverf is the inverse error function. For a normal (Gaussian) distribution, the data will be on a line whose slopeis equal to the inverse of σ, the standard deviation of the mean. The symmetric shape of a Gauss distribution is inconsistentwith switching ﬁeld distributions, in which thermal activationskews the distribution toward lower ﬁelds. This is why ourswitching ﬁeld data curves away from an imaginary tangentline at the median ( P NS=0.5) due to the double-exponential character of thermal activation. Figures 4(a) and4(c) clearly show that the switching ﬁeld distribution under a 5 mA current (open blue triangles) deviatessharply from the thermal activation model (dashed red line).The deviations appear over a sufﬁciently large region of thedistribution that it is visible even on the linearly scaled axis inFig. 4(a). However, when we compare the distributions under a 2.5 mA current in Figs. 4(b) and4(d), the deviations are too subtle to ascertain from the linearly scaled plot in the top rightcorner. Once we plot our data on the Gauss quantile-scaledyaxis, disagreement between data and model at the tails of this distribution becomes clear. This result shows that a deepstatistical survey of the switching ﬁeld reveals problems athigh current density and the gradual onset of deviations fromour model at the distribution tails. VI. TESTING THE MODEL: SWITCHING FIELD DISTRIBUTIONS UNDER CONSTANT DC CURRENTS In this section we will test the scope of the “modiﬁed barrier” model by conducting deep statistical measurements ofthe switching ﬁeld over a wide range of dc currents. Assumingthermal activation over a single energy barrier, at ﬁxed tem-perature the cumulative probability to remain in a metastablemagnetization state under ﬁnite ﬁeld μ 0His given by the double-exponential expression in Eq. ( 1). Our previous switch- ing ﬁeld studies taken in zero dc current were consistent withthe single barrier model and serve as a baseline from which tocompare switching distributions with ﬁnite dc currents [ 30,31]. In order to test the effective barrier model, we will permit thethermal stability and zero-temperature coercive ﬁeld to varyin Eq. ( 1), as spin currents may modify the thermal stability ξ as well as the effective anisotropy ﬁeld H c0for switching. Figure 5shows the distributions of switching ﬁelds for 5000 switching events, both AP →P and P →AP, under direct currents of ±5,±2.5, and 0 mA for the ﬁrst device shown in Fig. 2. Curves to the right of zero magnetic ﬁeld correspond to P →AP transitions and to the left correspond to the AP →P transitions. The switching ﬁeld distributions for the 5000 events at each current are plotted on a Gaussianquantile scale, introduced previously in Sec. Vto assess the quality of ﬁtting the rare events at the tails of our distributions. For the single barrier model we apply to each data set, we observe qualitatively good agreement of the ﬁts to themajority of our measured switching ﬁeld data. Furthermore,the centers of the distributions are shifted according to theapplied currents. This behavior is consistent with the evolution 134427-4SWITCHING FIELD DISTRIBUTIONS WITH SPIN . . . PHYSICAL REVIEW B 89, 134427 (2014) FIG. 5. (Color online) Switching ﬁeld distributions for 5000 events under direct currents ( Idc=0,±2.5,±5 mA) plotted on a Gauss quantile scale. AP →P( P→AP) distributions fall to the left (right) of the imaginary line at μ0H=0. of the switching ﬁeld with applied current presented earlier in the state diagram boundaries. However, we note that the ﬁrst1% of AP →P transitions for I dc=2.5 and 5 mA occur with lower probability than the model would predict, extrapolatingoutward from the median. We note that deviations from themodel at 5 mA also coincide with the onset of switching forthe AP →P transition at zero applied ﬁeld (see Fig. 2). We extract the best-ﬁt thermal stability factor for each switching ﬁeld distribution, by ﬁtting the thermal activationmodel to our data assuming a constant prefactor /Gamma1 0=1 GHz andv=100 mT /s, the linear sweep rate of the applied external magnetic ﬁeld. The best-ﬁt parameter ξis shown as a function of applied current in Fig. 6. The dissimilarity between ξfor the AP →P and P →AP transitions at zero direct current is a common feature in perpendicularly magnetized SV devicesand is related to an asymmetry caused by the polarizer dipoleﬁeld [ 30]. We note that the trend for small currents is a FIG. 6. (Color online) Effective thermal stability ξversus direct current Idcfor\|Idc\|<9 mA. Thermal stability ξ=E0/kBTfor AP→P( P→AP) transitions are referenced to 300 K and plotted as red circles (blue squares). Hollow symbols represent extracted barrier heights for switching distributions that do not agree well withthe model.monotonic decrease in ξfor increasingly positive (negative) currents for the AP →P( P→AP) transition, which is in good agreement with the effective barrier picture. On the otherhand, as the current becomes increasingly negative (positive),the thermal stability for the AP →P( P→AP) transition levels off. This is in contrast to the effective barrier model,which would predict a steadily increasing ξwith current [ 10]. Additionally, the deviation in our barrier height ξscaling with current appears to be consistent with the deviations in thelinearity of the barrier height or the deviations in the evolutionof the switching rate with current as calculated by Taniguchiet al. [32,33]. While the current densities considered in this study fall below the thresholds considered by Taniguchi et al. (greater than 80% of the critical current I c), it is possible that the onset of these deviations occurs for lower currentdensities in perpendicularly magnetized elements. We denotethese deviations from the model with hollow symbols in Fig. 6 to contrast with the switching ﬁeld distributions that closelyfollow the monotonic trend line. We also apply hollow symbolsforξvalues extracted from switching ﬁeld distributions that appreciatively deviate from the best model ﬁt. As we willdiscuss below, this is also a consideration due to changes inthe shape of the underlying switching ﬁeld distribution for thehigher current densities. For sufﬁciently high current densities, the switching ﬁeld distributions show clear deviations from the thermal distri-bution model, whose double-exponential shape is evident inFig. 5. Figure 7(a) illustrates the switching ﬁeld distributions for large negative applied currents of −9,−11,−11.3, and −11.5 mA. At higher negative currents ( I dc/lessorequalslant−11 mA) the switching ﬁeld distribution develops a kink (compared withI dc=− 9 mA), which could indicate a crossover between different competing reversal modes. These competing modesmay involve excitation of the polarizer layer and are likelyassociated with the precessional modes typically seen at theedges of the bistable region of the state diagram, as in Fig. 2(a). Figure 7(b) illustrates the switching ﬁeld distributions for large positive applied currents of +5,+5.5,+6, and +7 mA. At 5 mA, the shape of the switching ﬁeld distribution ﬁrst losesits curvature and the switching rate ( −dP NS/dH ) exceeds the thermal distribution model (dashed line) for ﬁelds belowthe median. Moreover, the switching rate for ﬁelds above themedian is lower than predicted. This results in apparently lineardistributions at 5.5, 6, and 7 mA when plotted on a Gaussquantile scale (compared to solid line), indicating that theswitching rates are symmetric for a given deviation from themedian switching ﬁeld /Delta1=\|H−H 0\|. In order to test whether any thermal process with multiple pathways can describe the data, we calculated switching ﬁeld distributions assuming that multiple switching pathways are available for thermal activation. As it has been seen thatspin-transfer torques can redistribute energy across ﬂuctuationmodes in a nanomagnet [ 34,35], competing ﬂuctuation modes could be the origin of a distribution of switching pathways. Webegin with a Gaussian distribution of switching rates /Gamma1 i, each with their own energy barrier ξi. However, the energy barrier mainly determines the extent to which the PNScurve bends above the knee, which makes it impossible for a distributionof these switching pathways to create a Gaussian switchingﬁeld distribution. 134427-5GOPMAN, BEDAU, MANGIN, FULLERTON, KATINE, AND KENT PHYSICAL REVIEW B 89, 134427 (2014) − 200 − 175 − 150 − 125 µ0H(mT).01%1%25%75%99%99.99%PNSAP→P(a) IDC(mA) = -9 -11 -11.3 -11.5 0 10 20 30 40 50 60 70 µ0H(mT)AP→P(b) IDC(mA) = 5 5.5 6 7 FIG. 7. (Color online) Deviation of the switching ﬁeld distributions from the predicted double-exponential shape at high currents. Switching distributions plotted on a Gauss quantile scale for (a) Idc=− 9,−11,−11.3,−11.5m Ar e v e a lak i n kf o r Idc/lessorequalslant−11 mA and (b) Idc= 5,5.5,6,7 mA become apparently linear on a Gauss quantile scale. Broken lines in (a) and (b) represent the best-ﬁt thermal activation curve to the data. Solid lines in (b) represent the best-ﬁt Normal distribution to the data. While the origin of the change in the shape of the switching ﬁeld distributions is unclear, several factors may play a role athigh currents. The nearly symmetric distributions in Fig. 7(b) exhibit a lower switching rate for the low ﬁeld, high- P NSevents than predicted by a thermal activation model, which mayindicate that spin-transfer torques suppress the ﬂuctuationsthat would result in thermal switching. Another possibilityis that the large current densities exceeding 10 11A/m2may be driving the nanomagnet into an intermediate regime be-tween thermally assisted reversal and deterministic (ballistic)switching in which neither a deterministic switching model nora thermally assisted switching model is valid [ 15]. Switching ﬁeld distributions at these higher currents may be reﬂectingthis intermediate regime. Recent work by Taniguchi et al. demonstrates that the switching rate, attempt frequency, and the scaling are indi- vidually sensitive to the magnitude of applied current and/or magnetic ﬁeld (through ﬁeld-induced changes in the criticalcurrent) [ 32,36]. While the threshold at which these effects begin to strongly inﬂuence the dynamics are quite close to thecritical current in the Taniguchi work (focusing on dynamicsin an in-plane magnetized element), it may extend to lowercurrents and ﬁelds in a perpendicularly magnetized element,for which the switching dynamic is qualitatively different thanin the in-plane geometry. The double exponential behavior [Eq. ( 1)] is limited to a regime where the overall switching rate is relativelysmall. Although experiments are conducted in the long-time(1 ms) regime, rare events at the tails of the switchingﬁeld distribution may reveal dynamics outside the scope ofthe double exponential model. However, the breakdown ismost signiﬁcant in the high damping regime (e.g., when thespin-transfer torque and the damping are effectively collinearand parallel to the initial state and the overall switching rateshould be suppressed). The origin for the behavior that deviatesfrom the single barrier/double exponential model remainsunclear.In conclusion, we have tested the effective barrier model for spin-transfer-assisted thermally activated reversal of spin-valve nanopillars with perpendicular magnetization. Althoughthe effective temperature model catches the salient featuresof the average switching behavior seen in state diagrammeasurements, a closer investigation demonstrates a gradualdeviation at the tails of a nanomagnet’s switching distributionunder increasing spin-transfer torques. This shows that deepstatistical measurements of the switching ﬁeld combined withpresentation on a Gauss quantile scale can reveal the onsetand degree of deviation of the switching distributions from thethermal activation model. Clear deviations from the thermalswitching model become apparent at currents exceeding thezero-ﬁeld switching current. In particular, the switching rateat the distribution tails is signiﬁcantly reduced, which mayindicate a suppression of the thermal activation by the STT. The deviations suggest a sudden change in the switching process and a breakdown in the validity of a model of thermally inducedswitching. This could have signiﬁcant impact on magneticmemory cells in which the bit write error rate in the tailsmay deviate signiﬁcantly from the effective barrier model.Also, we demonstrate that the barrier height versus currentdependence levels off for large current values. The origin ofthis as well as the effect of switching out of a dynamic stateis not well understood. Nevertheless, our results demonstratethe need for additional investigations on the thermal stabilityof a nanomagnet under large spin-transfer torques. ACKNOWLEDGMENTS This research was supported at NYU by NSF Grants No. DMR-1006575 and No. NSF-DMR-1309202, as well as thePartner University Fund (PUF) of the Embassy of France. Re-search at UL supported by ANR-10-BLANC-1005 “Friends,”the European Project (OP2M FP7-IOF-2011-298060), and theRegion Lorraine. Work at UCSD supported by NSF Grant No.DMR-1008654. 134427-6SWITCHING FIELD DISTRIBUTIONS WITH SPIN . . . PHYSICAL REVIEW B 89, 134427 (2014) [1] J. A. Katine and E. E. Fullerton, J. Magn. Magn. Mater. 320, 1217 (2008 ). [ 2 ] A .D .K e n t , Nat. Mater. 9,699(2010 ). [3] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno,Nat. Mater. 9,721(2010 ). [4] A. Brataas, A. D. 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PhysRevLett.101.066601.pdf	Spin-Torque Shot Noise in Magnetic Tunnel Junctions A. L. Chudnovskiy,1J. Swiebodzinski,1and A. Kamenev2 1I. Institut fu ¨r Theoretische Physik, Universita ¨t Hamburg, Jungiusstrasse 9, D-20335 Hamburg, Germany 2Department of Physics, University of Minnesota, Minneapolis, Minnesota 55455, USA (Received 27 March 2008; published 8 August 2008) A spin polarized current may transfer angular momentum to a ferromagnet, resulting in a spin-torque phenomenon. At the same time the shot noise, associated with the current, leads to a nonequilibrium stochastic force acting on the ferromagnet. We derive a stochastic version of the Landau-Lifshitz-Gilbert equation for a magnetization of a ‘‘free’’ ferromagnetic layer in contact with a ‘‘ﬁxed’’ ferromagnet. Wesolve the corresponding Fokker-Planck equation and show that the nonequilibrium noise yields to a nonmonotonic dependence of the precession spectrum linewidth on the current. DOI: 10.1103/PhysRevLett.101.066601 PACS numbers: 72.25. /.0255b, 75.70.Cn, 75.75.+a Magnetization dynamics of a ferromagnet under the inﬂuence of a spin polarized current is a subject of inten-sive investigations (for recent reviews see Refs. [ 1,2]). It was realized [ 3,4] that the spin current may transfer the angular momentum to the ferromagnet, resulting in atorque acting on its magnetization direction. In the caseof a small ferromagnetic domain the torque may lead to a rotation of the magnetization as a whole, rather than to an excitation of spin waves. This phenomenon, allowing foran electronic manipulation of the magnetization, has prom-ise for a number of potential applications. The effect has been recently observed [ 5–9] in a setup, where the spin-torque magnitude and direction are tuned tocompensate exactly the dissipation force acting on themagnetization of the ‘‘free’’ ferromagnetic layer. This leads to an undamped precession which is detected through the induced microwave radiation. Both the spectral widthand the generated power exhibit a strong dependence onthe current ﬂowing through the interface of the two ferro-magnets. It was shown later [ 10,11] that the equilibrium thermal noise, ﬁrst introduced in dynamics of micromag-nets by Brown [ 12], may partially account for the observed linewidth. On the other hand, since the experiments are performed under nonequilibrium conditions (spin current strongenough to balance the dissipation), one needs to addressnonthermal sources of noise as well. The most essential ofthem is the spin shot noise associated with the discreteness of the spin passing through the interface. The spin shotnoise accompanies every spin-torque experiment, such as,for example, current-induced magnetization switching or current-induced domain wall motion. The role of spin shot noise in those experiments remains largely unexplored yet.In this Letter we present a general treatment of the spinshot noise in magnetic tunnel junctions (MTJ). We usethe stochastic generalization of Landau-Lifshitz-Gilbert(LLG) equation and derive the correlator of the Langevinforces that systematically takes into account both thermaland nonequilibrium sources of noise. This equation may serve as a base for theoretical investigations of a broad range of experiments with electric spin manipulation.The effect of spin shot noise may be accounted for by adding a ﬂuctuating part to the spin current vector in the Slonczewski’s torque term of the LLG equation I s!Is/.0135 /.0014Is/.0133t/.0134. The resulting stochastic LLG equation for the unit vector m/.0136M=Min the direction of the magnetization M takes the form dm dt/.0136/.0255/.0013/.0137m/.0002Heff/.0138/.0135/.0011/.0133/.0018/.0134/.0020 m/.0002dm dt/.0021 /.0135/.0013 MV/.0137m/.0002/.0137/.0133Is/.0135/.0014Is/.0134/.0002m/.0138/.0138: (1) Here/.0013is gyromagnetic ratio, Heff/.0136/.0255@F=@ Mis the effective magnetic ﬁeld, which includes both an external ﬁeld and magnetic anisotropy, and Vis a volume of the free ferromagnet. Gilbert damping /.0011/.0133/.0018/.0134is renormalized by the coupling to the ﬁxed ferromagnet [ 2,13] and is thus dependent on a relative orientation angle /.0018of the ﬁxed and free ferromagnets. One could expect that the ﬂuctuating part of the spin current vector /.0014Is/.0133t/.0134is preferentially directed along the spin polarization of the incoming electron ﬂux, i.e., along Is. This is notthe case, however, due to the quantum nature of the effect. Indeed, each spin-ﬂip event transfers exactlyone@unit of the angular momentum to the free ferromag- net. Because of the uncertainty principle, the direction ofan ensuing magnetization rotation is completely random.As a result, the ﬂuctuating part of the spin torque must havean isotropic correlator h/.0014Is;i/.0014Is;ji/.01362D/.0133/.0018/.0134/.0014ij/.0014/.0133t/.0255t0/.0134; (2) where the variance D/.0133/.0018/.0134does not depend on the Cartesian indexesi;j/.0136x;y;z , but may depend on the mutual ori- entation of the two ferromagnets. Because of the isotropy of the stochastic torque, it can be equally well represented by a ﬂuctuating magnetic ﬁeld Heff!Heff/.0135h/.0133t/.0134, instead of the ﬂuctuating spin ﬂux /.0014Is/.0133t/.0134. In the latter case the correlator of the stochastic ﬁelds reads as hhi/.0133t/.0134hj/.0133t0/.0134i /.0136 2D/.0133MV/.0134/.02552/.0014ij/.0014/.0133t/.0255t0/.0134. This type of nonequilibrium noise was considered in Ref. [ 14] in the context of normal-ferromagnetic-normal (NFN) structures.PRL 101, 066601 (2008) PHYSICAL REVIEW LETTERSweek ending 8 AUGUST 2008 0031-9007 =08=101(6)=066601(4) 066601-1 ©2008 The American Physical SocietyHere we consider a model of a MTJ (Ref. [ 15]). Magnetization dynamics of the free ferromagnet is de-scribed using Holstein-Primakoff (HP) parametrization [16] of its total spin operator by deriving semiclassical equations of motion for HP bosons. We employ Keldyshformalism to allow for nonequilibrium conditions, i.e.,voltage bias between the two ferromagnets [ 17–19]. After integrating out the fermionic degrees of freedom inthe second order in both tunneling and spin-ﬂip processes,we obtain an effective action for HP bosons, which en-capsulates deterministic forces (external magnetic ﬁeld and spin-torque) along with the stochastic term. The latter encodes both equilibrium and nonequilibrium noisecomponents. The result of the program, outlined above, is the follow- ing noise correlator D/.0133/.0018/.0134/.0136MV /.0013/.00110T/.0135@ 2Isf/.0133/.0018/.0134coth/.0018eV 2T/.0019 ; (3) where/.00110is a bare Gilbert damping of an isolated grain and Vis a voltage bias between the two ferromagnets. The nonequilibrium part of the noise is proportional to the spin-ﬂip current I sf/.0133/.0018/.0134. The latter counts the total number of spin ﬂips irrespective of the direction of the ensuing mag-netization change (as opposed to the spin current I sasso- ciated with a directed ﬂow of the angular momentum). Inthe MTJ setup we found for the spin-ﬂip conductance dIsf/.0133/.0018/.0134 dV/.0136@ 4e/.0020 GPsin2/.0018/.0018 2/.0019 /.0135GAPcos2/.0018/.0018 2/.0019/.0021 ; GP/.0136G/.0135/.0135/.0135G/.0255/.0255;GAP/.0136G/.0135/.0255/.0135G/.0255/.0135;(4) where we adopted notations of Ref. [ 15] for the partial conductances G/.0027/.00270between the spin-polarized bands of the two ferromagnets. Here GP=APstay for electric con- ductances in parallel (P) and antiparallel (AP) conﬁgura- tions, with GP/.0021GAP. The electric conductance of the MTJ in an arbitrary orientation is given by dIe/.0133/.0018/.0134=dV/.0136 GPcos2/.0133/.0018=2/.0134/.0135GAPsin2/.0133/.0018=2/.0134. Notice that the spin shot noise is minimal for the P orientation and maximal forthe AP one—exactly opposite to the charge current and thecharge shot noise. In the same notations the spin conduc-tance is [ 15] dIs dV/.0136@ 4e/.0133G/.0135/.0135/.0255G/.0255/.0135/.0135G/.0135/.0255/.0255G/.0255/.0255/.0134; (5) where the angular dependence is already explicitly taken into account in Eq. ( 1). A nonpolarized current, i.e., G/.0135/.0027/.0136G/.0255/.0027, does not exert deterministic spin torque, nevertheless it still induces the shot noise torque on the ferromagnet. This effect was recently discussed in the context of NFN structures [ 14]. The same calculation also leads to a renormalization of the Gilbert damping coefﬁcient in the LLG equation ( 1) /.0011/.0133/.0018/.0134/.0136/.00110/.0135@/.0013 eMV/.0018dIsf/.0133/.0018/.0134 dV/.0019 ; (6)where the spin-ﬂip conductance is given by Eq. ( 4). Such an enhancement of the dissipation due to the spin transportbetween the two ferromagnets is discussed in Refs. [ 2,13]. In compliance with the ﬂuctuation-dissipation theorem, the equilibrium ( V!0) noise correlator is given by D/.0136 TMV/.0011/.0133/.0018/.0134=/.0013[12]. Because of the renormalization, the damping is minimal in the P orientation. The angulardependence of the damping term selects a unique angle,where the spin torque compensates dissipation. Upon in-creasing the spin current above the critical one, such anangle gradually increases from zero to /.0025. Below we restrict ourselves to a setup, where both the magnetic ﬁeld and the spin current are directed along the z axis speciﬁed by the magnetization direction of the ﬁxedferromagnet. Transforming Eq. ( 1) to the spherical coor- dinates [ 20,21], one obtains two coupled Langevin equa- tions for the relative angle /.0018/.0133t/.0134and the azimuthal angle /.0030/.0133t/.0134 _/.0018/.0136/.0255sin/.0018Tz/.0133/.0018/.0134/.0135cot/.0018~D/.0133/.0018/.0134/.0135/.0024/.0018/.0133t/.0134; (7) _/.0030/.0136/.0010/.0133/.0018/.0134/.0135csc/.0018/.0024/.0030/.0133t/.0134; (8) Tz/.0133/.0018/.0134/.0136/.00130/.0133/.0011/.0133/.0018/.0134Hz/.0135Is=/.0133MV/.0134/.0134;/.0010/.0136/.0013Hz/.0255/.0011Tz; where/.00130/.0136/.0013=/.01331/.0135/.00112/.0134and~D/.0133/.0018/.0134/.0136D/.0133/.0018/.0134/.0013/.00130=/.0133MV/.01342. Here/.0024/.0018/.0133t/.0134and/.0024/.0030/.0133t/.0134are two uncorrelated random noises with the correlators h/.0024/.0018/.0133t/.0134/.0024/.0018/.0133t0/.0134i /.0136 h/.0024/.0030/.0133t/.0134/.0024/.0030/.0133t0/.0134i /.01362~D/.0133/.0018/.0134/.0014/.0133t/.0255t0/.0134:(9) Equations ( 1) and ( 7) and ( 8) should be interpreted in the sense of retarded regularization, or Ito calculus [ 20]. Let us ﬁrst analyze deterministic dynamics described by Eqs. ( 7) and ( 8) with~D!0. For a strong enough (and negative for positive Hz) spin current the condition Tz/.0133/.0018/.0134/.01360may be satisﬁed for a certain angle /.0022/.0018. Notice that it is the angular dependence of the enhanced Gilbertdamping [ 2,13], which is responsible for the angle selec- tivity. In such a case Eq. ( 8) describes a stable undamped precession with the frequency /.0010/.0133/.0022/.0018/.0134/.0136/.0013H z. The intensity of the induced microwave radiation [ 5–9] is given by the square of the oscillating magnetic moment, i.e., propor-tional tosin 2/.0022/.0018. To analyze effects of the noise we shall assume that /.0011/.0133/.0022/.0018/.0134/.00281, allowing for time scale separation [ 22]. The fast variable is the azimuthal angle /.0030/.0133t/.0134, while the angle /.0018/.0133t/.0134is the slow one. For a ﬁxed slow variable /.0018, Eqs. ( 8) and ( 9) lead to the Lorentzian shape of the emitted micro- wave power spectrum S/.0133!;/.0018/.0134/sin2/.00182csc2/.0018~D/.0133/.0018/.0134 /.0137!/.0255/.0010/.0133/.0018/.0134/.01382/.0135/.0137csc2/.0018~D/.0133/.0018/.0134/.01382:(10) To compare with the observed power spectrum this expres- sion should be averaged over the stationary probability distribution P/.0133/.0018/.0134of the slow degree of freedom S/.0133!/.0134/.0136Rsin/.0018d/.0018P/.0133/.0018/.0134S/.0133!;/.0018/.0134. The distribution function P/.0133/.0018;t/.0134PRL 101, 066601 (2008) PHYSICAL REVIEW LETTERSweek ending 8 AUGUST 2008 066601-2obeys the Fokker-Planck (FP) equation which follows from Eqs. ( 7) and ( 9)[12,20] _P/.01361 sin/.0018@/.0018/.0137sin2/.0018Tz/.0133/.0018/.0134P/.0135sin/.0018@/.0018/.0133~D/.0133/.0018/.0134P/.0134/.0138:(11) The stationary solution of Eq. ( 11) is given by P/.0133/.0018/.0134/.01361 Z~D/.0133/.0018/.0134exp/.0026 /.0255Z/.0018 0sin/.00180d/.00180Tz/.0133/.00180/.0134 ~D/.0133/.00180/.0134/.0027 ; (12) where constant Zis chosen to satisfy the normalization conditionR/.0025 0sin/.0018d/.0018P/.0133/.0018/.0134/.01361. For a weak noise the sta- tionary distribution function has a sharp maximum close to the angle /.0022/.0018, where the deterministic spin torque compen- sates the dissipation. Figure 1shows the calculated spectral linewidth at the half maximum as a function of the applied voltage. Theorigin of the nonmonotonic dependence may be under- stood by inspection of Eq. ( 10). The initial decline is due to the geometric factor csc 2/.0018in the width of the Lorentzian. As the voltage increases, so does the anglewhere the distribution function exhibits the maximum.Because of csc/.0018, coming from the noise term on the right-hand side (r.h.s.) of Eq. ( 8), the amplitude of the phase noise decreases leading to a narrowing of the powerspectrum. The initial decrease of the spectral width was discussed in Refs. [ 10,11] in terms of the equilibrium thermal noise [ 12]. The subsequent increase of the spectral width at yet larger voltages was observed in a number of experiments[6–9] and, to the best of our knowledge, remained unex- plained. It naturally comes about due to the nonequilibriumcomponent of the noise. Indeed the noise correlator ( 3) grows with the applied voltage due to the growth of thespin-ﬂip current. The dependence of the spin-ﬂip current I sf/.0133/.0018/.0134on the bias is faster than the linear because of the in- crease of the spin-ﬂip conductance, Eq. ( 4), with the op- eration angle /.0022/.0018on top of the overall proportionality of Isfto the biasV. As a result, for eV*2Tthe noise variance D grows rapidly, leading to the broadening of the powerspectrum, cf. Eq. ( 10). Another consequence of the nonequilibrium noise is the saturation of the spectral linewidth at small temperatures[9]. Since the devices are always operated at currents larger than the critical one, the noise intensity ( 3) is ﬁnite D/.0133/.0018/.0134/.0136 /.0133@=2/.0134I sf/.0133/.0018/.0134even atT!0. Thus decreasing the tempera- ture one should observe saturation of the linewidth at T/.0024 eV, provided the induced damping [the second term on the r.h.s. of Eq. ( 6)] is larger than the bare one. In the remainder of the Letter we outline derivation of Eqs. ( 1)–(6). The MTJ is modeled by the two itinerant ferromagnets, whose majority ( /.0027/.0136/.0135 ) and minority (/.0027/.0136/.0255 ) bands are described by the operators cy k/.0027;ck/.0027 for the ﬁxed ferromagnet and dy l/.0027;dl/.0027for the free layer. We found it convenient to work in the instantaneous refer- ence frame, where the magnetization of the free layer points in the zdirection. The corresponding Hamiltonian is H0/.0136X k;/.0027/.0015k/.0027cy k/.0027ck/.0027/.0135X l/.0027/.0133/.0015l/.0255JSz/.0027/.0134dy l/.0027dl/.0027/.0255/.0013S/.0001H /.0255J/.0133S/.0135s/.0255/.0135S/.0255s/.0135/.0134/.0135/.0020X kl;/.0027/.00270W/.0027/.00270 klcy k/.0027dl/.00270/.0135H:c:/.0021 : Here the spin-dependent tunneling matrix elements are W/.0027/.00270 kl/.0136h/.0027j/.00270iW, where the spin-transformation matrix ish/.0027j/.0027i/.0136e/.0255i/.0027/.0030=2cos/.0018=2andh/.0027j/.00270i/.0136ei/.0027/.0030=2sin/.0018=2, ands/.01361 2P l/.0027/.00270dy l/.0027~/.0027/.0027/.00270dl/.00270is the spin of itinerant elec- trons, while s/.0006/.0136sx/.0006isy. We have explicitly accounted for the interactions of the itinerant electrons in the free layer with its total spinS/.0136MV=/.0013. To make the latter a dynamical variable we use HP parametrization [ 16] Sz/.0136S/.0255byb;S/.0255/.0136by/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 2S/.0255bybp ;S/.0135/.0136/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 2S/.0255bybp b; whereby;bare usual bosonic operators. Next we write the corresponding action in terms of complex fermionic and bosonic ﬁelds ck/.0027/.0133t/.0134,dl/.0027/.0133t/.0134,b/.0133t/.0134, where the time variable runs along the closed Keldyshcontour [ 17–19]. We then transform to two-component vector notations in terms of symmetric (classical ‘‘cl’’) and antisymmetric (quantum ‘‘ q’’) combinations of the forward and backward propagating ﬁelds. One shouldkeep in mind that the distribution functions of the cand dfermions have a relative shift of the chemical potentials byeV. The fermionic ﬁelds may be integrated out exactly and the remaining bosonic effective action expanded to thesecond order in the tunneling amplitude Wand to the ﬁrst and second orders in the spin-ﬂip processes S /.0006s/.0007. The corresponding processes are represented by the diagramsof Fig. 2. The approximations are justiﬁed by the weakness of tunneling and by S/.0029@.∆ω FIG. 1 (color online). Calculated linewidth of the microwave spectral power versus applied voltage in units of tempera- ture. Inset: Calculated microwave power (solid line) and the average precession angle (dashed line) versus voltage.Parameters: MV=/.0133@/.0013/.0134/.013610,@/.0013H z/.01363K,T/.01361K,/.00110/.0136 0:01. Conductances in units e2=h:GP/.01360:181,GAP/.01360:019. dIs=dV/.01360:01e.PRL 101, 066601 (2008) PHYSICAL REVIEW LETTERSweek ending 8 AUGUST 2008 066601-3The resulting action for the complex bosonic ﬁelds bcl/.0133t/.0134;bq/.0133t/.0134takes the form A/.0136A0/.0135A1/.0135A2where the subscript indicates the order in spin-ﬂips processes. Here the bare action is [ 23] A0/.0136Z dt/.0022bq/.0133t/.0134/.0133i@tbcl/.0133t/.0134/.0135/.0013/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 S=2p H/.0135/.0134/.0135c:c:(13) The ﬁrst order correction in spin-ﬂip amplitude is repre- sented by the diagram of Fig. 2(a). This is a virtual tran- sition into an opposite spin band with a subsequenttunneling out of the free layer into the ‘‘correct’’ spinband of the ﬁxed magnet. The latter process is possible for/.0018/.02220;/.0025, due to a ﬁnite W /.0027/.00270 kl. The net result is trans- ferring angular momentum @to the total spin of the free layer, i.e., the deterministic spin torque [ 3,4]. The corre- sponding contribution to the action is A1/.0136i/.0129/.0129/.0129/.0129/.0129/.0129 2SpZ dt/.0022bq/.0133t/.0134Issin/.0018e/.0255i/.0030/.0135c:c:; (14) whereIsis given by Eq. ( 5) withG/.0027/.00270/.01364/.0025e2 hjWj2/.0023/.0027c/.0023/.00270 d and/.0023/.0027 c;dare densities of states of the two ferromagnets in the/.0027band. The second order processes in spin ﬂips are depicted by the diagram of Fig. 2(b). These are real (i.e., golden rule) processes, whose matrix elements include thespin ﬂips. They lead to dissipation as well as ﬂuctuations.The corresponding action is A2/.0136Z dt/.0020 /.0011/.0133/.0018/.0134/.0133/.0022bq@tbcl/.0255/.0022bcl@tbq/.0134/.01352i SD/.0133/.0018/.0134/.0022bqbq/.0021 ; (15) where D/.0133/.0018/.0134and/.0011/.0133/.0018/.0134are given by Eqs. ( 3), (4), and ( 6). One then decouples the last term on the r.h.s. of Eq. ( 15) by means of the complex Hubbard-Stratonovich ﬁeld/.0014I /.0135/.0133t/.0134/.0136Is;x/.0135iIs;y. The remaining action is linear in bq/.0133t/.0134and/.0022bq/.0133t/.0134. It thus constitutes the resolution of func- tional/.0014functions of the ﬁrst order Langevin equations on bcl/.0133t/.0134/.0136/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 MV=/.01332/.0013q /.0134m/.0135/.0133t/.0134and its complex conjugate.Those are nothing but m/.0006components of Eq. ( 1), with the noise intensity given by Eqs. ( 2)–(4)[23]. We are grateful to P. Crowell, A. Levchenko, D. Pfannkuche, and V . Kagalovsky for useful discussions. A. C. and J. S. acknowledge ﬁnancial support from DFGthrough Sonderforschungsbereich 508 and Sonderforsch-ungsbereich 668. A. K. was supported by the NSF GrantNo. DMR-0405212 and by the A. P. Sloan Foundation. [1] D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008). [2] Y . Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). [3] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [4] L. Berger, Phys. Rev. B 54, 9353 (1996). [5] S. I. Kiselev et al. , Nature (London) 425, 380 (2003). [6] W. H. Rippard et al. , Phys. Rev. Lett. 92, 027201 (2004). [7] W. H. Rippard, M. R. Pufall, and S. E. Russek, Phys. Rev. B 74, 224409 (2006). [8] Q. Mistral et al. , Appl. Phys. Lett. 88, 192507 (2006). [9] J. C. Sankey et al. , Phys. Rev. B 72, 224427 (2005). [10] V . Tiberkevich, J. V. Kim, and A. N. Slavin, arXiv:0709.4553. [11] J. V . Kim, V . Tiberkevich, and A. N. Slavin, Phys. Rev. Lett. 100, 017207 (2008). [12] W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963). [13] Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). [14] J. Foros, A. Brataas, Y . Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 95, 016601 (2005). [15] J. C. Slonczewski and J. Z. Sun, J. Magn. Magn. Mater. 310, 169 (2007). [16] T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940). [17] L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964); [Sov. Phys. JETP 20, 1018 (1965)]. [18] A. Kamenev, in Nanophysics: Coherence and Transport , edited by H. Bouchiat et al. (Elsevier, Amsterdam, 2005), pp. 177–246. [19] R. A. Duine, A. S. Nunez, J. Sinova, and A. H. MacDonald Phys. Rev. B 75, 214420 (2007). [20] N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 2001). [21] The second term on the r.h.s. of Eq. ( 7) appears due to Stratonovich regularization of the variable change, which leads to an additional deterministic term in the FP equa-tion [ 12]. To keep the Ito regularization from now on we have explicitly included this term in Eq. ( 7). [22] The sufﬁcient conditions for the time scale separation imply/.0011/.00281as well as /.0013I s=MV/.0028/.0010; however, close to the out-of-equilibrium steady state the two conditions coincide. [23] In the instantaneous reference frame mz/.01361andm/.0006/.01360, while_mz/.01360and_m/.0006/.02220. Therefore all the terms in the action/.0024bcl/.0133t/.0134/m/.0135/.0133t/.0134do not contribute to the equations of motion and are omitted. We thus keep only terms/.0024@ tbclalong with those which contain only bq. FIG. 2. Diagrams for spin-ﬂip processes: (a) the ﬁrst order, describing the spin torque, and (b) the second order, describing the spin shot noise along with the enhanced damping. Solid (dashed) lines denote electronic propagators in the free (ﬁxed)layers. Bold dashed lines are propagators of HP bosons (spin ﬂips). Tunneling vertices are denoted by circles with crosses.PRL 101, 066601 (2008) PHYSICAL REVIEW LETTERSweek ending 8 AUGUST 2008 066601-4
PhysRevB.87.024402.pdf	PHYSICAL REVIEW B 87, 024402 (2013) Magnetic texture-induced thermal Hall effects Kevin A. van Hoogdalem,1Yaroslav Tserkovnyak,2and Daniel Loss1 1Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland 2Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA (Received 8 August 2012; revised manuscript received 14 October 2012; published 2 January 2013) Magnetic excitations in ferromagnetic systems with a noncollinear ground-state magnetization experience a ﬁctitious magnetic ﬁeld due to the equilibrium magnetic texture. Here, we investigate how such ﬁctitious ﬁeldslead to thermal Hall effects in two-dimensional insulating magnets in which the magnetic texture is caused byspin-orbit interaction. Besides the well-known geometric texture contribution to the ﬁctitious magnetic ﬁeld insuch systems, there exists also an equally important contribution due to the original spin-orbit term in the freeenergy. We consider the different possible ground states in the phase diagram of a two-dimensional ferromagnetwith spin-orbit interaction: the spiral state and the skyrmion lattice, and ﬁnd that thermal Hall effects can occurin certain domain walls as well as the skyrmion lattice. DOI: 10.1103/PhysRevB.87.024402 PACS number(s): 75 .30.Ds, 75 .47.−m, 66.70.−f I. INTRODUCTION Traditionally, the role of information carrier in spin and electronic devices is taken by the spin or the charge,respectively, of the conduction electrons in the system. Inrecent years, however, there has been an increasing awarenessthat spin excitations in insulating magnets (either magnonsor spinons) may be better suited for this task. The reasonbehind this is that these excitations are not subject to Jouleheating. Therefore, the energy associated with the transport ofa single unit of information carried by a magnon (or spinon)current can be much lower in such insulating systems. 1The quasiparticle bosonic nature of the magnons, furthermore,allows in principle to essentially eliminate losses due toscattering or contact impedances at low temperatures. 2 Creation of a magnon current has been shown to be possible by means of the spin Seebeck effect,3the spin Hall effect,4 and with high spatial accuracy by means of laser-controlledlocal temperature gradients. 5The resulting spin current can be measured utilizing the inverse spin Hall effect.4,6It has been shown that the magnon current can propagate overdistances of several centimeters in yttrium iron garnet (YIG). 4 It has recently been shown that it is theoretically possibleto implement the analogs of different electronic componentsusing insulating magnets. 2,7 Hall effects for magnon currents are of interest both from a fundamental point of view as well as from the point ofview of applications. Even though the physical magnetic ﬁelddoes not directly couple to the orbital motion of neutralmagnons, certain kinds of spin-orbit interactions can leadto Hall phenomena similar to those of a charged particlein a magnetic ﬁeld. Mechanisms that have been shown togive rise to nonzero Hall conductances in certain insulatingmagnets include coupling of spin chirality to a magnetic ﬁeld 8 and Dzyaloshinskii-Moriya (DM) interaction.9Of interest for applications is the fact that Hall effects in insulating magnetsallow one to control the magnon spin current. Recently, Katsura et al. predicted 8a nonzero thermal Hall conductivity (see Fig. 1) for the Heisenberg model on the kagome lattice. The ﬁnite conductivity originates from thefact that the coupling of spin chirality to an applied magneticﬁeld leads to a ﬁctitious magnetic ﬂux for the magnonsin the speciﬁc case of the kagome lattice. Later, Onose et al. measured 9the thermal Hall effect in the pyrochlore ferromagnet Lu 2V2O7. In this experiment, the combination of DM interaction and the pyrochlore structure leads to the ﬁnitethermal Hall conductivity. In those previous studies, the thermal Hall effect was considered using a quantum-mechanical lattice model as starting point. The symmetry of the underlying lattice played a crucial role. We take a different approach and considerinsulating ferromagnets with a noncollinear ground-statemagnetic texture, which we model using a phenomenologicaldescription. It is well known that the effect of the presenceof a noncollinear ground state on the elementary excitationsin a ferromagnet can be captured by introducing a ﬁctitious electromagnetic potential in the equation of motion for the magnons. 10,11Spin-orbit interactions generally also contribute non-Abelian gauge ﬁelds into the magnetic Hamiltonian.12 Furthermore, nonlinearized gauge ﬁelds for Dzyaloshinskii-Moriya interaction were derived in Ref. 13using the CP 1 representation. There are, correspondingly, two contributions to the ﬁctitious electromagnetic potential. The ﬁrst one depends only on the equilibrium magnetic texture; the second depends on the form of the free energy that gives rise to thenoncollinear ground state in the ﬁrst place, i.e., the contributionto the free energy due to spin-orbit interaction. Since bothcontributions are determined by the spin-orbit interaction, theywill generally be of similar magnitude. It has been shown that the ﬁctitious electromagnetic potential couples the motion of magnetic texture and thatof heat currents. 14This coupling reveals itself through local cooling by magnetic texture dynamics14and thermally induced motion of magnetic textures such as domain walls.15,16 This work is organized as follows. In Sec. IIwe introduce our system and derive the ﬁctitious electromagnetic vectorpotential that acts on the magnons, which turns out to includean often-overlooked contribution. In Sec. IIIwe derive the relevant ground-state properties of the different ground statesin the phase diagram of an insulating ferromagnet with nonzeroDM interaction. In Sec. IVwe calculate the band structure of one of the ground states, the triangular skyrmion lattice, andcalculate its thermal Hall conductivity. 024402-1 1098-0121/2013/87(2)/024402(7) ©2013 American Physical SocietyV AN HOOGDALEM, TSERKOVNY AK, AND LOSS PHYSICAL REVIEW B 87, 024402 (2013) FIG. 1. (Color online) Pictorial representation of the thermal spin Hall effect. A temperature difference /Delta1Tapplied to a sample leads to a ﬁnite heat current. Since the heat current is carried by the magnons in the system, the ﬁctitious magnetic ﬁeld that magnons experiencedue to a nontrivial magnetic ground state will lead to a ﬁnite thermal Hall conductivity. II. MAGNONS IN THE PRESENCE OF MAGNETIC TEXTURE We consider a two-dimensional nonitinerant ferromagnet in the x-yplane with spatially varying and time-dependent spin density sm(r,t). The spin density is related to the magnetization M(r,t)a ssm(r,t)=M(r,t)/γ, where γis the gyromagnetic ratio ( γ< 0 for electrons). The magnitude sof the spin density is assumed to be constant, and m(r,t)i sau n i t vector. The system is described by the Lagrangian10,11 L=/integraldisplay d2r[D(m)·˙m−F(m,∂jm)]. (1) Here D=s¯h(nD×m)/(1+m·nD) is the vector potential corresponding to the Wess-Zumino action with an arbitraryn Dpointing along the Dirac string. F(m,∂jm) is the magnetic free energy density of the system, which we assume to be ofthe form (here j=x,y; double indices are summed over) F(m,∂ jm)=Js 2(∂jm)2−Msm·H+sF/Gamma1(m,∂jm). (2) HereJis the strength of the exchange interaction, Ms=γs is the saturation magnetization, Hthe external magnetic ﬁeld (which we will always assume to be in the zdirection), and F/Gamma1(m,∂jm) describes terms due to broken symmetries. For isotropic ferromagnets in the exchange approximation, theleading-order terms in the free energy are quadratic in thetexture [ﬁrst term in Eq. (1)]. Breaking inversion symmetry by spin-orbit interactions, while still retaining isotropy in thex-yplane, allows us to construct terms that are ﬁrst order in texture. These terms are given by F /Gamma1(m,∂jm)=/Gamma1Rmz∇·m+/Gamma1DMm·(∇×m). (3) We deﬁned ∇=∂xx+∂yy, where x,yare unit vectors. The ﬁrst term is due to structural inversion symmetry breaking andhence is anisotropic in the zdirection. Such terms occur in systems with ﬁnite Rashba spin-orbit interaction 17or on the surface of a topological insulator.18The second term describes DM interaction,19which originates from the breaking of bulk inversion symmetry and is therefore isotropic. We note thatthe two terms in Eq. (3)are equivalent (up to an irrelevant boundary term) under a simple rotation around the zaxis inspin space. Since such a rotation does not have any additional effect on the equation of motion for the magnetization, Eq. (4), we can always absorb the term proportional to /Gamma1 Rin the term proportional to /Gamma1DM. We will therefore put /Gamma1Rto zero in the remainder of this work. For simplicity, we have ignored a term−κm 2 zthat would describe easy axis anisotropy, and a term −Msm·Hm/2, where Hmdescribes the magnetic stray ﬁeld, in Eq. (2). Substitution of Eq. (1)in the Euler-Lagrange equation leads to the Landau-Lifshitz equation s¯h˙m−m×δmF(m,∂jm)=0, (4) where F(m,∂jm) is the total magnetic free energy of the system. We split the magnetization min a static equilibrium magnetization m0and small fast oscillations δm(spin waves) around the equilibrium magnetization. To lowest order in δm the two are orthogonal. In a textured magnet m0=m0(r), which makes ﬁnding the elementary excitations a nontrivialtask. To circumvent this issue we introduce a coordinatetransformation m /prime(r)=ˆR(r)m(r), where ˆR(r) is such that the new equilibrium magnetization m/prime 0is constant and parallel to thezaxis. In this coordinate frame the spin waves are in the x-yplane. The 3 ×3m a t r i x ˆRdescribes a local rotation over an angle πaround the axis deﬁned by the unit vector n= [z+m0]/[2 cos( θ/2)]. Here, θis the polar angle of m0, andzis a unit vector. Using Rodrigues’ rotation formula, we ﬁnd ˆR=2nnT−ˆ1. The effect of the transformation to the new coordinate system is that we have to use thecovariant form of the differential operators, ∂ μ→(∂μ+ˆAμ), with ˆAμ=ˆR−1(∂μˆR), in the Landau-Lifshitz equation. The subscript μdescribes both time ( μ=0) and space ( μ=1,2) coordinates. In the new coordinate system, the Landau-Lifshitz equation for the free energy Eq. (2)becomes i¯h˙m+=[J(∇/i+A)2+ϕ]m+. (5) Here,m±=(δm/prime x±iδm/prime y)/√ 2 describes circular spin waves in the rotated frame of reference. Furthermore, ϕ=m0· H/s+¯h[ˆR−1(∂tˆR)]\|12, and the components of the vector potential Aare given by Aj=ˆAj\|12. The skew-symmetric matrices ˆAjare here deﬁned as ˆAj=ˆR(∂j−ζˆIj)ˆR.I nt h e latter equation we deﬁned ζ=/Gamma1DM/J, and the matrices ˆIx=⎛ ⎜⎝00 0 00 −1 01 0⎞ ⎟⎠ and ˆIy=⎛ ⎜⎝001 000 −100⎞ ⎟⎠.(6) We see that the components Ajof the ﬁctitious magnetic vector potential consist of two contributions. The ﬁrst comes fromthe exchange interaction in the presence of magnetic texture;the second (texture-independent) part originates from the DMinteraction term in the free energy. While it may be temptingto neglect the latter contribution, we will show here that it hasimportant consequences. Indeed, typically both contributionswill be of the same order of magnitude. This is because themagnetic texture itself is caused by the DM interaction, andwill therefore manifest itself on length scales J//Gamma1 DM. We can quantize Eq. (5)by introducing the bosonic creation operator b†∝m−. This quantization works since m/prime +and 024402-2MAGNETIC TEXTURE-INDUCED THERMAL HALL EFFECTS PHYSICAL REVIEW B 87, 024402 (2013) m/prime −satisfy approximate bosonic commutation relations in the limit of small deviations from equilibrium. After quantization,Eq. (5)can be interpreted as the von Neumann equation belonging to the Hamiltonian H=/integraldisplay d 2rb†[J(∇/i+A)2+ϕ]b. (7) Therefore, the elementary excitations of the system behave as noninteracting bosonic quasiparticles. The effect of thesmoothly varying equilibrium magnetization is captured bythe inclusion of a ﬁctitious magnetic vector potential Aand electric potential ϕ. In the derivation of Eq. (5)we have assumed that the length of a typical wave packet is much smaller than thespatial extension over which the magnetic texture varies. Wewill refer to this as the adiabatic approximation. 20Using this assumption, we have neglected terms in Eq. (5)that are higher order in texture. Such terms, which become important at lowerwave vectors, lead to two distinct effects. 20Firstly, a term −J[(ˆAx\|13)2+(ˆAx\|23)2+(ˆAy\|13)2+(ˆAy\|23)2]/2, which is quadratic in magnetic texture, has to be added to the ﬁctitiouselectric potential ϕin Eq. (5)at low wave vectors. Secondly, at low wave vectors one needs to add to the right-hand side ofEq.(5)at e r m J[(ˆA x\|13+iˆAx\|23)2+(ˆAy\|13+iˆAy\|23)2]m−, which introduces a ﬁnite ellipticity of the magnons. III. TEXTURED GROUND STATES In this section we present the different possible ground states for systems with free energy given by Eq. (2)(with /Gamma1R=0) as a function of the external magnetic ﬁeld H=Hz. We also present the ﬁctitious magnetic vector potentials thatﬁnd their origin in these textured ground states. It has beenshown 13,21that as the magnetic ﬁeld Hincreases from zero, the ground state of a two-dimensional ferromagnet with spin-orbit interaction changes from a spiral state for H<H c1,t oa skyrmion lattice state for magnetic ﬁelds Hc1<H<H c2, and ﬁnally to the ferromagnetic ground state for H>H c2.B o t h critical ﬁelds Hc1andHc2are typically of the order /Gamma12 DM/J (see Refs. 13and 22). This last observation, in combination with the adiabatic assumption and the fact that the equilibriummagnetization is time independent, allows us to neglect theﬁctitious electric potential ϕin Eq. (7). Since the ferromagnetic ground state has no magnetic texture, it is of no interest for ourpurposes. In this section we will therefore derive the propertiesof the spiral and skyrmion lattice ground state. A. Spiral state Following Ref. 21we will derive the properties of the spiral ground state m0(r) of a two-dimensional ferromagnet with DM interaction. We write m0(r) in the following form: m0(r)=cosξsinθx+sinξsinθy+cosθz. (8) For the spiral state, θ=θ(y) andξis a constant. With these constraints, the free energy becomes a functional that dependsonly on θ(y) and ∂ yθ(y). Minimizing this functional with respect to θ(y) gives the following differential equation, ∂2 yθ+αsinθ=0, (9)where we deﬁned α=−γH/J . Equation (9)is the equation of motion for the mathematical pendulum. The general solutionis given in implicit form by the expression /integraldisplay θ(y) 0dθ 21/radicalbig 1−m2sin2θ/2=1 2βy, (10) where m=4α/(2α+C) and β=√ 2α+C. Alternatively, we can write θ(y)=2φ(βy/2,m), where φ(u,m)i st h e amplitude of the Jacobi elliptic function. The constant Cis the ﬁrst constant of integration. To determine it, we use the factthatθ(y) is a periodic function with period y 0. By integrating the inverse of the ﬁrst integral ∂yθof Eq. (9)over one period we can determine y0as y0=/integraldisplay2π 0dθ1√ 2αcosθ+C. (11) To ﬁx C, we minimize the average free energy (1/y0)/integraltexty0 0F(θ,∂yθ), which leads to the following implicit expression for C: /integraldisplay2π 0dθ√ 2αcosθ+C=2πζ. (12) The minimization of the average free energy also ﬁxes cos ξ= 1. From this we see that the ground state is a spiral state whosestructure locally resembles a Bloch domain wall, as is expectedfor the DM interaction. 21We also note that in the case of zero magnetic ﬁeld ( α=0) the spiral state is described by a simple sinusoid with period y0=2π/ζ, whereas for ﬁnite magnetic ﬁeld the mirror symmetry with respect to the x-yplane is broken. Equation (12) also puts a constraint on the maximum value of Hfor which the spiral state is stable. Some general observations can be made with regard to the ﬁctitious magnetic vector potential due to the spiral groundstate. For the ground state Eq. (8)withθ=θ(y) and ξ constant, the ﬁctitious vector potential is A(y)=ζsinθ(y)x. This potential is caused solely by the DM contribution to A; the geometric texture contribution is zero everywhere. The z component of the ﬁctitious magnetic ﬁeld that the magnons ex-perience is given by B z(y)=∇× A\|z=−ζ∂yθ(y) cosθ(y). It is easily seen that the total ﬁctitious magnetic ﬂux over oneperiod of the spiral /angbracketleftB z/angbracketright=/integraltexty0 0dyBz(y)=0. The ﬁctitious magnetic ﬁeld Bz(y) has been plotted in Fig. 2for different magnitudes of the applied magnetic ﬁeld H=Hz. Transport in the presence of a magnetic ﬁeld that is spatially varying inone direction and has zero average has been studied extensively(see Ref. 23for a recent review). It is well known that these systems do not display a ﬁnite Hall conductivity. However,such magnetic ﬁelds have been predicted to inﬂuence thelongitudinal conductance, due to the presence of localizedsnake orbits at energies that are low compared to the cyclotronfrequency associated with the amplitude of the magneticﬁeld. 24,25From our analysis it is also seen that one-dimensional textures can give rise to a nonzero average ﬁctitious magneticﬂux for certain domain walls, since these consist of only halfa period of the spiral. Hence, such domain walls will displaythe thermal Hall effect. Lastly, we note that a proper statistical mechanical descrip- tion of the spiral phase in three dimensions (or less) requires theinclusion of leading-order nonlinearities in the free energy. 26 024402-3V AN HOOGDALEM, TSERKOVNY AK, AND LOSS PHYSICAL REVIEW B 87, 024402 (2013) FIG. 2. (Color online) Fictitious magnetic ﬁeld due to the spiral ground state. Parameters are ζ=70μm−1,J/(kB/epsilon12)=63 K, and the interatomic spacing is taken to be /epsilon1=4.5˚A( s e eR e f . 22). To make the connection to electromagnetism, we note that a ﬁctitious ﬁeldBz=2πζ/y 0(0)≈5×1015m−2acting on a spin wave gives rise to the same magnetic length as a¯h eζ2≈3 T magnetic ﬁeld acting on a free electron. The role of those nonlinearities in the thermal Hall physics is yet to be understood. B. Skyrmion lattice For magnetic ﬁelds Hc1<H<H c2the ground state of the two-dimensional ferromagnet with DM interactionis a skyrmion lattice. 22This triangular lattice has basis vectors a1=axanda2=(a/2)x+(a√ 3/2)y, and contains skyrmions with radius R. The size of a single unit cell is (√ 3/2)a2, where a=2R. The magnetization m0(r) of a single skyrmion of radius Rcentered at the origin is parametrized in polar coordinates ( ρ,φ)b yE q . (8)withθ=θ(ρ) and boundary conditions θ(0)=πandθ(R)=0. (13) Furthermore, ξ=Nφ−π/2, where Nis the charge of the skyrmion. We will assume N=1 throughout. The magne- tization proﬁle can in principle be determined numericallyby minimizing the free energy with the aforementionedboundary conditions. However, for simplicity we will assumea linear dependence θ(ρ)=π(1−ρ/R ) for our analysis of the texture-induced thermal Hall effect. In polar coordinates the ﬁctitious magnetic vector potential A(r) due to a single skyrmion centered at the origin is given by (here 0 /lessorequalslantρ/lessorequalslantRandφis a unit vector) A(r)=/bracketleftbiggcosθ(ρ)−1 ρ−ζcosθ(ρ)/bracketrightbigg φ. (14) Thezcomponent of the ﬁctitious magnetic ﬁeld for this vector potential is given by Bz(ρ)=ρ−1∂ρ(ρAφ). It follows that the total ﬂux through a unit cell is /angbracketleftBz/angbracketright=2π/integraltextR 0dρρB z=4π. This means that each unit cell contains two magnetic ﬂuxquanta. The nonzero average ﬂux is caused by the texturecontribution to A(r); the DM-interaction contribution averages to zero. From the fact that the average magnetic ﬂux isnonzero, it follows that the skyrmion lattice has a nonzero Hallconductivity. One might then be inclined to take the averagevalue of the ﬁctitious magnetic ﬁeld and ignore the spatialdependence when calculating the thermal Hall conductivity of the skyrmion lattice. However, we will show shortly that thespatial variation of the ﬁctitious magnetic ﬁeld is substantial,so that we should take both contributions into account in ouranalysis. To illustrate this point, let us consider the situation in which R=π/ζ. In that case B z(ρ)=ζ2cosθ(ρ). The spatial variation is therefore large enough that the ﬁctitious ﬁeldswitches from a negative minimum at ρ=0t oap o s i t i v e maximum at ρ=R. Such large variations have been shown to have a signiﬁcant inﬂuence on the band structure of magneticlattices. 27 For what follows, it will be convenient to formally split the ﬁctitious magnetic vector potential in two parts, A(r)= A0(r)+A/prime(r), where A0(r) describes the contribution from the homogeneous nonzero average ﬁctitious magnetic ﬂux, andA /prime(r) the periodic contribution with zero average (we work in the Landau gauge) A0(r)=−B0yx, (15) A/prime(r)=/summationdisplay τ,η[Ax(τ,η)x+Ay(τ,η)y]ei(τk1+ηk2)·r. Here,B0=8π/(√ 3a2) is the average ﬁctitious magnetic ﬁeld, andk1=(2π/a)(x−y/√ 3) and k2=(2π/a)(2/√ 3)yare the basis vectors of the reciprocal lattice, such that the periodicpart of the ﬁctitious vector potential satisﬁes A /prime(r+a1)= A/prime(r+a2)=A/prime(r). Such spatially varying magnetic ﬁelds are known to give rise to a ﬁnite Hall conductivity, even in theabsence of a nonzero average. 28 IV . THERMAL HALL CONDUCTIVITY OF THE SKYRMION LATTICE Since the magnetic excitations of the skyrmion lattice can be described by a free bosonic Hamiltonian with a spatiallyvarying ﬁctitious magnetic ﬁeld with on average two magneticﬂux quanta per unit cell and the same symmetry as theskyrmion lattice, the eigenstates of the skyrmion lattice aremagnetic Bloch states. In Sec. IV A we will determine the excitation spectrum and explicit form of these states. InSec. IV B we will show how the thermal Hall conductivity of the skyrmion lattice is determined by the Berry curvatureof these magnetic Bloch states. A. Diagonalization To ﬁnd the elementary excitations of the skyrmion lattice, we need to diagonalize the Hamiltonian Hin Eq. (7)with the ﬁctitious magnetic vector potential given in Eq. (14).W ed o this by numerically diagonalizing the matrix that results fromrewriting Hin the basis of the Landau levels that describe excitations with the appropriate symmetry in the presenceof the ﬁctitious magnetic vector potential A 0(r) only. Our derivation follows that of Ref. 27, with the difference that we consider the case with two ﬂux quanta instead of one ﬂuxquantum per unit cell. The eigenstates of a free system of dimensions L×L with only a homogeneous magnetic ﬁeld B 0zand without any 024402-4MAGNETIC TEXTURE-INDUCED THERMAL HALL EFFECTS PHYSICAL REVIEW B 87, 024402 (2013) underlying symmetries are given by ψnkx(r)=Nn√ Le−ikxxϕn/parenleftbig B1 2 0y+B−1 2 0kx/parenrightbig , (16) where Nn=1√ 2nn!(B0 π)1 4andϕn(x)=e−x2/2Hn(x), with Hn(x) thenth Hermite polynomial. The corresponding energies are En=2JB 0(n+1/2). To account for the presence of the triangular lattice and the fact that every unit cell containstwo ﬂux quanta, we need to ﬁnd the most general linearcombination of eigenstates that satisﬁes ˆM a1ψnmk(r)=eik1aψnmk(r), ˆMa2ψnmk(r)=eik2aψnmk(r). (17) Here,k1andk2are deﬁned such that (2 π/a)k=k1k1+k2k2. Furthermore, kis restricted to lie within the ﬁrst Brillouin zone. We will discuss the origin of the quantum number m later. We have to work with magnetic translation operators ˆMa1,2since the canonical momentum is no longer a good quantum number in the presence of the vector potentialA 0(r). These magnetic translation operators are deﬁned as ˆMa1=ˆTa1and ˆMa2=exp[−i(4π/a)x]ˆTa2, where ˆTa1,2are the usual translation operators. The appropriate eigenstates arethen given by ψ nmk(r)=∞/summationdisplay l=−∞(−1)(l+m 2)(l+m 2−1)e−i(l+m 2)(k1 2−k2)a ×ψn,−k1−(l+m 2)4π a. (18) The quantum number m, which in our case can take values 0 or 1, accounts for the fact that in the presence of a natural numberpof ﬂux quanta per unit cell, each magnetic band will split up inpsubbands. These subbands are degenerate for a constant magnetic ﬁeld but will generally split for a spatially varyingmagnetic ﬁeld, as we will see later. The set of wave functionsdeﬁned in Eq. (18) constitutes a complete orthonormal basis with triangular symmetry. The eigenfunctions are chosen insuch a way that perturbations in the ﬁctitious magnetic vectorpotential that are periodic in the triangular lattice are diagonalin the momenta k 1andk2. We are now in a position to calculate the matrix elements ofHwith respect to the basis deﬁned by the eigenstates in Eq.(18). We rewrite H=H0+H1+H2, where the subscript denotes the order in which A/prime(r) occurs in the respective term. The matrix elements of H0are then trivially given by (we have suppressed the kdependence of the eigenstates in our notation) /angbracketleftn/prime,m/prime\|H0\|n,m/angbracketright=2JB 0(n+1/2)δn,n/primeδm,m/prime. (19) The matrix elements of H1are given by /angbracketleftn/prime,m/prime\|H1\|n,m/angbracketrightn/prime/greaterorequalslantn =J/summationdisplay τ,ηδ(mod2) m/prime−m,τB(τ,η) ×/bracketleftbigg Ln/prime−n n(zτη)−/parenleftbiggn+n/prime zτηLn/prime−n n(zτη)−2n/prime zτηLn/prime−n n−1(zτη)/parenrightbigg/bracketrightbigg ×(−1)mηGn/primen(τ,η) (20)and the matrix elements of H2by /angbracketleftn/prime,m/prime\|H2\|n,m/angbracketrightn/prime/greaterorequalslantn =J/summationdisplay τ/prime,η/prime,τ,ηδ(mod2) m/prime−m,τ/prime+τ ×[Ax(τ/prime,η/prime)Ax(τ,η)+Ay(τ/prime,η/prime)Ay(τ,η)] ×(−1)m(η/prime+η)Gn/primen(τ/prime+τ,η/prime+η). (21) We deﬁned the function Gn/primen(τ,η)=/parenleftbiggn! n/prime!/parenrightbigg1/2 (/radicalbig 2/B0π)n/prime−n/bracketleftbigg i2η−τ√ 3a−τ a/bracketrightbiggn/prime−n ×e−zτη/2eπiτη/ 2eiηk 1a/2eiτ(k2a+π)/2. (22) Furthermore, we deﬁned zτη=(2π/√ 3)(τ2−τη+η2). The function Lα n(x) is the associated Laguerre polynomial. The function δ(mod2) i,j is deﬁned as δ(mod2) i,j=1 when i=j(mod2), andδ(mod2) i,j=0 otherwise. The ﬁrst ten subbands of the band structure of the skyrmion lattice with parameters 2 JB 0/kB≈ 50 mK, R=45 nm, and ζ=70μm−1(similar values to those found in Ref. 22)a r eg i v e ni nF i g . 3. In our numerical calculation we used the fact that the coupling between twoband decays superexponentially [to be precise, it decays as√ (n!/n/prime!)], so that only a limited number of bands have to be taken into account. It is seen that the inclusion of thespatially varying ﬁctitious magnetic ﬁeld has a pronouncedeffect, leading both to different splittings of the differentsubbands, as well as substantial broadening of the subbands.From Fig. 3it is seen that the typical level splitting between magnetic subbands is 50 mK, which sets the temperature scaleon which the system is in the quantum Hall regime. Systemswith larger ratio /Gamma1 2 DM/Jwill display quantum Hall behavior at higher temperatures. We note that ﬁnite Gilbert damping α will broaden the different magnetic subbands by an amount(/Delta1ω/ω )=2α. Eventually this will destroy the visibility of individual subbands. However, since the Gilbert damping isaround α∼10 −3in a range of different materials, this only becomes problematic at high magnetic subbands. FIG. 3. (Color online) Band structure of the skyrmion lattice with parameters R=45 nm, ζ=70μm−1,a n d2 JB 0/kB≈50 mK. The labels on the horizontal axis denote ( k1,k2), with the wave vectors normalized to 2 π/a. 024402-5V AN HOOGDALEM, TSERKOVNY AK, AND LOSS PHYSICAL REVIEW B 87, 024402 (2013) We note that within our model we do not ﬁnd the expected Goldstone modes associated with the skyrmion lattice.29We argue that this is due to our adiabatic assumption, which breaksdown for the smallest wave vectors. Assuming a quadraticdispersion for the magnons, we can estimate the magnitude\|k m\|of the characteristic wave vector of the magnons that make up the lowest magnetic subband as J\|km\|2=JB 0, which leads to a typical magnon wavelength λm∼a. The wave vector \|km\|increases for higher subbands. Since the accuracy of our model increases with increasing wave vector, our descriptionimproves for higher magnetic subbands. In the next section we will investigate the effect of the ﬁnite bandwidth of the magnetic subbands on the thermal Hallconductivity of the skyrmion lattice. B. Thermal Hall conductivity It is well known30that the semiclassical dynamics of a wave packet in the basis of the magnetic Bloch states unk(r)= e−ik·rψnk(r) is described by ˙r=∂kEn(k)−˙k×/Omega1n(k) and ¯ h˙k=0. (23) We have assumed here that there are no electric ﬁelds present and that the states unk(r) are the eigenstates of the Hamiltonian H, including the ﬁctitious magnetic vector potential A(r). /Omega1n(k) is the Berry curvature of the nth magnetic Bloch band. Since we consider a two-dimensional system, only its z component is relevant. It is given by /Omega1n(k)=2Im/bracketleftbigg/angbracketleftbiggunk(r) ∂kx/vextendsingle/vextendsingle/vextendsingle/vextendsingleu nk(r) ∂ky/angbracketrightbigg/bracketrightbigg . (24) For the skyrmion lattice, the magnetic Bloch states are given by unk(r)=e−ik·r/summationdisplay n/prime,m/primecn n/primem/primekψn/primem/primek(r). (25) The weights cn n/primem/primekfollow from the diagonalization performed in Sec. IV A . It should be noted that every completely ﬁlled subband carries a total Berry curvature that is a multiple of 2 π, in accordance with the quantization of the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) invariant. 31 Matsumotu and Murakami have shown32that the thermal Hall conductivity for a system described by Eq. (23) is given by κxy=k2 BT ¯hV/summationdisplay n,kc2(ρnk)/Omega1n(k). (26) Here c2(ρ)=(1+ρ)(log1+ρ ρ)2−(logρ)2−2Li 2(−ρ) describes the effect of the thermal distribution of the magnons, and ρ(/epsilon1)=(expβ(/epsilon1−μ)−1)−1is the Bose-Einstein distribution function. Figure 4shows the Berry curvature for the two highest magnetic subbands in Fig. 3. Together with the band structure given in Fig. 3, the Berry curvature completely determines the thermal Hall conductivity, as can be seen from Eq. (26). The position and width of the magnetic subbands determine atwhich energies states become available for thermal transport;the Berry curvature determines the extent to which these statescontribute to the thermal Hall conductivity.FIG. 4. (Color online) Berry curvature of the two highest mag- netic subbands in Fig. 3in a single Brillouin zone. The subband corresponding to the top ﬁgure does not carry a net curvature; thebottom ﬁgure carries 2 π. We have shown then that the spatially varying ﬁctitious magnetic ﬁeld gives rise to a nontrivial structure of the Berrycurvature as well as a broadening of the magnetic subbands.As follows from Eq. (26), the combination of these two effects may be studied experimentally by measuring κ xyas a function of temperature. As the temperature increases, the thermaldistribution of the bosonic magnons broadens, which enablesthe higher bands to contribute to thermal transport. Alternatively, one can probe the chiral subgap edge states (above the bands whose total TKNN number is odd) directlyby microwave means. The ﬁrst magnetic subband correspondsto several gigahertz, which falls within the scope of standardmicrowave techniques. Excitation of some of the low-energymodes of the skyrmion lattice by microwave radiation hasbeen performed 33and analyzed.34The magnonic reﬂectionless waveguide modes are analogous to those found in photoniccrystals 35and can provide intriguing spintronics applications. Magnonic edge states have recently been proposed to exist inYIG without any equilibrium spin texture but with an array ofFe-ﬁlled pillars, in which ellipticity of magnons due to dipolarinteractions can give rise to magnonic bands with a nonzeroTKNN invariant. 36 V . CONCLUSIONS We have studied how ﬁctitious magnetic ﬁelds, which are caused by a textured equilibrium magnetization, lead tothermal Hall effects in two-dimensional insulating magnetsin which the nontrivial equilibrium magnetization is causedby spin-orbit interaction. We have given a general expressionfor the ﬁctitious magnetic vector potential and found that itconsists of two contributions: a geometric texture contributionand a contribution due to the original spin-orbit term in the freeenergy. We have shown that both contributions are generallyof the same order of magnitude. We have derived the relevant properties of the two ground states of interest to us (the spiral state and the skyrmion lattice 024402-6MAGNETIC TEXTURE-INDUCED THERMAL HALL EFFECTS PHYSICAL REVIEW B 87, 024402 (2013) state) in the phase diagram of a two-dimensional nonitinerant ferromagnet with nonzero Dzyaloshinskii-Moriya interaction.We have found that a system which has the spiral state asmagnetic ground state does not have a ﬁnite thermal Hallconductivity. However, we predicted that certain domain wallstructures do display thermal Hall effects. We have numerically diagonalized the Hamiltonian describ- ing the triangular skyrmion lattice. We found that due to thespatially varying ﬁctitious magnetic vector potential, the exci-tation spectrum consists of broadened magnetic subbands. Wehave calculated the Berry curvature of the magnetic subbandsand showed that the Berry curvature in combination with theexcitation spectrum completely determines the thermal Hallconductivity of the skyrmion lattice. At present, we are onlyable to capture the contribution to the thermal Hall conductivity from higher magnetic subbands, as well as thermal- ormicrowave transport through the associated edge states. Inorder to properly describe the lowest subbands, our modelhas to be amended to capture nonadiabatic magnon-transporteffects, in the way described at the end of Sec. II. ACKNOWLEDGMENTS This work has been supported by the Swiss NSF, the NCCR Nanoscience Basel (K.v.H. and D.L.), the NSF under GrantNo. DMR-0840965, and DARPA (Y .T.). K.v.H. is grateful forhospitality at the University of California, Los Angeles, wherepart of this work has been carried out. 1B. Trauzettel, P. Simon, and D. Loss, Phys. Rev. Lett. 101, 017202 (2008). 2F. Meier and D. Loss, Phys. Rev. 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PhysRevApplied.11.054040.pdf	PHYSICAL REVIEW APPLIED 11,054040 (2019) Correction of Phase Errors in a Spin-Wave Transmission Line by Nonadiabatic Parametric Pumping Roman Verba,1,Mario Carpentieri,2Yu-Jin Chen,3Ilya N. Krivorotov,3Giovanni Finocchio,4 Vasil Tiberkevich,5and Andrei Slavin5 1Institute of Magnetism, Kyiv 03142, Ukraine 2Department of Electrical and Information Engineering, Politecnico di Bari, I-70125 Bari, Italy 3Department of Physics and Astronomy, University of California, Irvine, California 92697, USA 4Department of Electronic Engineering, Industrial Chemistry and Engineering, University of Messina, I-98166 Messina, Italy 5Department of Physics, Oakland University, Rochester, Michigan 48309, USA (Received 12 January 2019; revised manuscript received 18 March 2019; published 15 May 2019) It is shown that phase errors in a microwave spin-wave transmission line can be corrected by subjecting the signal-carrying propagating spin wave to the action of a localized nonadiabatic parametric pumping, having the localization length smaller than the spin-wave wavelength. In such a transmission line thephase-transmission characteristic has a “steplike” shape containing ﬂat “stabilization plateaus” separated by intervals of size π. Within the “plateau” regions the phase of the output spin wave is practically constant in a rather wide range of phases of the input spin wave. This eﬀect can be used in magnonic logic devicesfor the correction of phase errors of up to ±0.25π. It is also proved that this phase-stabilization eﬀect is stable against the variations of the spin-wave amplitude and is present across the amplitude range of the stable spin-wave propagation. DOI: 10.1103/PhysRevApplied.11.054040 I. INTRODUCTION Spin waves (SWs) propagating in nanoscale ferromag- netic waveguides are considered to be promising candi- dates for applications in a new generation of digital- and analog-signal-processing devices [ 1–5]. Recently, several novel concepts of magnonic logic elements and circuits have been proposed [ 6–12]. In magnonic logic, a digi- tal signal can be coded via SW amplitude [ 6,13–15]o r SW phase [ 2,16]. Obviously, in the case of phase-coded magnonic logic devices the spin-wave phase should be well deﬁned, and should not ﬂuctuate substantially in thecourse of the spin-wave propagation. This property of the SW phase stability is also crucial for amplitude-coded magnonic logic devices. Indeed, these devices often use SW interference for information processing and the phase relations between several processed SWs should be well deﬁned for correct device operation [ 13,15,17,18]. For example, the result of the interference of two SWs having phases ϕ 1andϕ2and similar amplitudes is propor- tional to cos[ (ϕ1−ϕ2)/2]. Deviation of the phase diﬀer- ence by 0.55 πcould be enough for incorrect interpretation of the interference result—instead of 1 in the ideal case ϕ1−ϕ2=0( o r0i f ϕ1−ϕ2=π), the resulting signal verrv@ukr.netbecomes less than 2 /3 (greater than 1 /3), which is com- monly interpreted as indeterminate in amplitude-coded logic [ 15]. Deviation of the phase diﬀerence by 0.8 πleads to a wrong result—logic “0” instead of “1” and vice versa. There are several reasons for the SW phase deviation in a magnonic circle. The ﬁrst is the deviation of the SW waveguide length due to lithographic misprints. This devi-ation can occur, for example, due to a misposition of thewaveguide bends and a spread of bend shapes in a circuit. Small length deviations, not exceeding 1 nm at each bend, can accumulate over a magnonic circle and, for exemplarySWs of 100-nm wavelength, could reach critical values(corresponding to 0.55 πphase shift) after passing several tens of such bends. Similarly, a misprint of the waveguide width leads to a change of SW dispersion and, thus, SW wave number at a ﬁxed frequency, which is another sourceof random phase accumulation. Finally, stability of the SW phase can be violated by thermal ﬂuctuations and the phase deviations leading, eventually, to a signal-processing error can accumulate in the course of the SW propagation in a magnonic circuit. Therefore, the timely correction of these phase errors is very important for stable and error-free operation of magnonic logic circuits. In this work, we demonstrate that the problem of phase- error correction can be solved by application of localized parametric pumping, i.e., by using the interaction of a 2331-7019/19/11(5)/054040(7) 054040-1 © 2019 American Physical SocietyROMAN VERBA et al. PHYS. REV. APPLIED 11,054040 (2019) propagating SW with a localized microwave magnetic ﬁeld (external or internal) of approximately double the SW fre- quency. Parametric pumping is a well-known method for excitation and ampliﬁcation of SWs [ 19–22]. It is also known that parametric interaction becomes phase sensitive in the case of so-called “nonadiabatic” localized pump- ing, having a localization length (or other characteristic length of the spatial variation) that is smaller than the SW wavelength [ 23–25]. In our current work we calculate the phase-transmission characteristics for a SW transmission line containing a region where nonadiabatic parametric pumping is acting and show that the phase-transmission characteristics of such a line demonstrate “stabilization plateaus,” within which the phase of the output SW signal is more or less constant in a rather wide range of phases of the input SW signal. Thus, the phase ﬂuctuation of the SW signal acquired in the course of its propagation can be corrected. It is important that these phase-stabilization plateaus are separated by intervals of size almost exactly equal to π, which is thus perfectly suitable for phase-coded magnonic logic and/or signal processing. II. THEORY A sketch of the considered SW transmission line is shown in Fig. 1. It is a ferromagnetic nanowire of width wy and thickness h. The SWs of the frequency ωk, propagat- ing in the +xdirection, are excited by the excitation gate or, in an integrated magnonic circuit, come from a preced- ing SW signal-processing device. The parametric pumping gate of length Lpis placed on the propagation path of the SWs. The parametric pumping can be created by a microwave magnetic ﬁeld with polarization parallel to the direction of static magnetization of the nanowire [ 20,21], by the microwave electric ﬁeld via various magnetoelec- tric eﬀects [ 22], or by other means. The phase-stabilization eﬀects discussed below do not depend on the nature of the pumping and are also independent of the direction of static magnetization of the nanowire. To be speciﬁc with the coeﬃcients used in our calculations, we consider the case of parametric pumping FIG. 1. A sketch of the considered magnonic transmission line showing regions of the SW excitation and the localization of theparametric pumping. The reference points A–Dused in the in the micromagnetic simulations are also shown.produced by a microwave voltage via the voltage- controlled magnetic anisotropy eﬀect (VCMA) [ 26,27], which is the most eﬃcient and convenient method for applications at the nanoscale. In this case the pumping gate consists of a strip of a normal metal separated by a dielectric layer from the conductive ferromagnetic material of the nanowire. The application of a microwave voltage of frequency ωpto the gate results in oscillations of the perpendicular magnetic anisotropy at the ferromagnetic- dielectric interface with the same frequency [ 28,29]. It has been shown that these oscillations of anisotropy can cou- ple parametrically to the SWs propagating in the nanowire, both in the case of the in-plane and out-of-plane static magnetization direction of the nanowire [ 22,29,30]. In the former case the coupling is stronger, and demonstrates no limits with respect to the SW wave number [ 30], so here we consider only this case of the in-plane static magnetization, as shown in Fig. 1. In the parametric process of ﬁrst order, the pumping is coupled to a pair of SWs having wave vectors kand k/prime.T h e eﬃciency of the parametric interaction is proportional to the(k+k/prime)th Fourier harmonic bp,k+k/primeof the spatial dis- tribution of the eﬀective pumping ﬁeld bp(x). Therefore, in the case of weakly localized quasiuniform pumping, when kLp/greatermuch1, only the SWs with opposite wave vectors, k/prime=− k, interact eﬃciently with the pumping, which is a consequence of the momentum conservation law (the case of “adiabatic pumping”). In contrast, when the pumping localization length L becomes smaller than the SW wavelength (or if the pump- ing is spatially nonuniform with the characteristic length comparable to the SW wavelength), not only can the con- trapropagating SWs ( k/prime=− k) interact with the localized pumping ﬁeld, but also other SWs ( k/prime/negationslash=− k), in partic- ular copropagating SWs, can do so as well. This is the case of “nonadiabatic parametric pumping,” as described in Ref. [ 23]. It should be noted that the parametric interaction has maximum eﬃciency when the resonance condition ωp= ωk+ωk/primeis satisﬁed. This condition severely limits the number of SWs that can eﬃciently interact with the pump- ing. In the simple, and most common case, when the pump- ing frequency is twice as large as the SW frequency, ωp= 2ωk, the only SWs eﬃciently interacting with the nonadia- batic pumping are the above-mentioned contrapropagating SWs having the same modulus of SW wave vectors kand −k; however, the nonadiabatic term results in additional coupling of these SWs with themselves (that is the lim- iting case of the coupling of copropagating SWs, when approaching exact parametric resonance). The SW dynamics under a localized parametric pump- ing is convenient to study using Bloembergen’s system of equations. For the case of nonadiabatic pumping, it has been generalized in Ref. [ 23] and, neglecting the higher-order nonlinear SW interactions, it can be 054040-2CORRECTION OF PHASE ERRORS IN A SPIN-WAVE. . . PHYS. REV. APPLIED 11,054040 (2019) written as /parenleftbigg∂ ∂t+v∂ ∂x+/Gamma1/parenrightbigg a1=Vb0e−iψa∗ 2+Vb2ke−iψa∗ 1, /parenleftbigg∂ ∂t−v∂ ∂x+/Gamma1/parenrightbigg a∗ 2=Vb0eiψa1+Vb2keiψa2.(1) This system describes the evolution of the envelope ampli- tudes a1(x,t)and a2(x,t)of the two SW wave pack- ets, having carrier wave vectors kand−k, respectively. In our problem, a1describes the envelope amplitude of the incident SW, which propagates toward the pumping region, and a2is the envelope amplitude of the idler SW, which is counterpropagating to a1, and appears in the pumping region as a result of the parametric interaction. The relation of envelope amplitudes to the real magne- tization amplitudes is given by the equation m1,2(x,t)= [mka1,2(x,t)exp(±ikx−iωkt)+c.c.], where mkdescribes the vector structure (ellipticity) of a particular SW. In Eq.(1),vand/Gamma1are the group velocity and the damp- ing rate of the SWs, Vis the eﬃciency of the paramet- ric coupling, ψis the phase of the pumping, and bk= (1/Lp)/integraltextLp/2 −Lp/2bp(x)eikxdxis the Fourier harmonic of the eﬀective ﬁeld of pumping with the spatial proﬁle bp(x). The fact that pumping is nonadiabatic is reﬂected by the last term in the equations, which describes the parametric coupling of the copropagating SWs ( k/prime=k). In the case of quasiuniform adiabatic pumping this term is naturally absent, since b2k→0. The value α=\|b2k/b0\|describes the strength of the nonadiabatic term relative to the adia- batic one, and is called “the degree of nonadiabaticity of the pumping.” In our particular case of the in-plane static magnetiza- tion and VCMA-induced pumping, the eﬃciency of the parametric coupling is given by V=γ\|mk,z/4mk,y\|, the pumping ﬁeld is bp=2βE/hM swithβbeing the magne- toelectric coeﬃcient, Eis the amplitude of the microwave electric ﬁeld applied to the pumping gate [ 30], and the pumping Fourier harmonics bkare given by the expression bk=bpsinc(kLp/2)≡bpsin(kLp/2)/(kLp/2). The pumping phase ψis deﬁned in such a way that the applied microwave electric ﬁeld is E(t)=Esin(ωpt+ ψ), with ωp=2ωk(exact parametric resonance). The real dynamic magnetization, corresponding to the steady propagating SW of the envelope amplitude a1=\|a1\|e−iϕ, ismz(x,t)=2mz,ksin(ωkt+ϕ−kx), where ϕis the SW phase. Note that the point x=0i sa s s u m e dt ob ea tt h e center of the pumping gate, as shown in Fig. 1and its position obviously aﬀects the deﬁnitions of the phases ϕ andψ. For other cases of the parametric pumping source and other directions of the static magnetization, the only diﬀerences in Eq. (1)come from the diﬀerent values of the parametric coupling eﬃciency V[19,21,22], and the relations of the phases ψandϕto the real time proﬁlesof the dynamic magnetization and applied pumping signal (microwave magnetic or electric ﬁeld). For further analysis it is convenient to introduce new real variables A1±and A2,±,a s a1=e−iψ/2(A1++iA1−) and a2=e−iψ/2(A2+−iA2−), which is possible if the pumping is harmonic, i.e., if the pumping phase is time independent, ψ/negationslash=ψ(t). This operation, in fact, is a decom- position of a harmonic wave with an arbitrary phase into two partial waves, sine and cosine. Then, Eq. (1)is transformed to [ 23] /parenleftbigg∂ ∂t+v∂ ∂x+/Gamma1∓Vb2k/parenrightbigg A1±=Vb0A2±, /parenleftbigg∂ ∂t−v∂ ∂x+/Gamma1∓Vb2k/parenrightbigg A2±=Vb0A1±.(2) As one can see, the pairs of partial waves (A1+,A2+)and (A1−,A2−)evolve independently and are connected only by the boundary conditions. The action of the nonadiabatic term Vb2kresults in diﬀerent eﬀective damping for partial waves: eﬀective damping for the “in-phase” partial waves (A1+,A2+)is decreased, while the “out-of-phase” SWs (A1−,A2−)acquire an additional damping term. Thus, the partial waves evolve diﬀerently under the action of pump- ing, since the pumping pumps energy more eﬀectively into the “in-phase” partial waves. To ﬁnd a steady-state solution of the transmission prob- lem, we consider a stationary regime, setting ∂Ai/∂t=0. Equation (2)should be accompanied by a boundary con- dition a1(−Lp/2)=A0e−iϕ0, which describes the incom- ing SW with amplitude A0and arbitrary phase ϕ0,a n d a2(Lp/2)=0, meaning that no idler wave is incident to the pumping region. Then, the envelope amplitude of the output SW aout=a1(Lp/2)can be found to be aout=A0e−iψ/2/bracketleftbigg cos/parenleftbigg ϕ0−ψ 2/parenrightbigg K+ −is i n/parenleftbigg ϕ0−ψ 2/parenrightbigg K−/bracketrightbigg ,( 3 ) where K±=/bracketleftBigg cos(κ±Lp)+˜/Gamma1± vκ±sin(κ±Lp)/bracketrightBigg−1 (4) are the ampliﬁcation rates for partial waves, ˜/Gamma1±=/Gamma1∓ Vb2k,a n dκ2 ±=(Vb0)2−˜/Gamma12 ±. As usual [ 21], the parametric pumping results in a partial ampliﬁcation of the incident SW, until the pumping ampli- tude reaches a certain threshold, at which a spontaneousgeneration of SWs takes place (the threshold of genera- tion is determined from the condition K +→∞ ). Due to the nonadiabatic term, the ampliﬁcation rates of the par- tial waves are diﬀerent, resulting in the dependence of the 054040-3ROMAN VERBA et al. PHYS. REV. APPLIED 11,054040 (2019) output SW amplitude on its phase [ 23,24]. Simultaneously, this means that the ratio between the amplitudes of the partial waves A1+and A1−changes within the pumping region, and is diﬀerent at the end of the pumping gate com- pared to that at the gate entrance. Thus, the phase of the incident SW a1changes during the propagation through the pumping gate. Since the “in-phase” partial wave A1+ grows faster (or decays slower) than the “out-of-phase” partial wave, the phase of the incident wave approaches the phase of the “in-phase” partial wave, which is ﬁxed by the phase of pumping to an accuracy of an integer multiple ofπ:ϕ(x)→ψ/2+πn,n∈Z. The phase-transmission characteristics are obtained from Eq. (3)simply as ϕout=− Arg(aout). In the case of adiabatic pumping, when b2k=0, the phase-transmission characteristic is a simple straight line, ϕout=ϕ0[Fig. 2(a)]. Recall that the SW phase ϕwas introduced as a phase of the SW envelope, so the propagation phase shift kLpis not taken into account in Fig. 2. This leads to a simple vertical shift of all the curves. In contrast, as soon as the pumping becomes nonadia- batic, the SW phase-transmission characteristics become (a) (b) FIG. 2. (a) SW phase-transmission characteristics ϕout=f(ϕ0) and (b) ampliﬁcation rates \|K\|= f(ϕ0)of a parametric pump- ing gate for diﬀerent degrees of the pumping nonadiabaticity α=\|b2k/b0\|. The pumping length Lp=0.1v//Gamma1, the pumping strength is 90% of the SW generation threshold, and the pumpingphase ψ=0.nonlinear. They demonstrate pronounced plateaus near the values ϕ=0,π, which are the phases of the “in-phase” partial wave (since it is assumed that the pumping phase ψ=0). Within these plateaus, the output SW phase is almost constant in a wide range of the input SW phases, i.e., nonadiabatic parametric pumping demonstrates the eﬀect of SW phase stabilization. Importantly, the SW phase-stabilization plateaus are separated by phase inter- vals of size π, which perfectly matches the needs of the phase-coded magnonic logic, as under this approach the logic state “0” and the logic state “1” are coded by the SWs with a phase diﬀerence of π. Stabilization plateaus become wider and more ﬂat with an increase in the degree α=\|b2k/b0\|of the pump- ing “nonadiabaticity” [Fig. 2(a)]. A similar enhancement of the phase-stabilization properties is observed with an increase of the pumping strength, when this strength approaches the threshold of the parametric SW genera- tion. In a limiting case, when K+/greatermuchK−(which means that the pumping amplitude is close to the threshold or that the length of the pumping region is suﬃciently large), the phase-transmission characteristic becomes almost a steplike function. In should be noted that the pumping nonadiabaticity also results in the dependence of the output SW ampli- tude on the input SW phase, as shown in Fig. 2(b). When the phase stabilization becomes better, the variations of the SW amplitude also increase. Large variations of the SW amplitude, naturally, are not acceptable in SW pro- cessing devices, which limits the achievable ranges of the possible phase-error corrections in practice. Usually, about 10–15% of the SW amplitude variation can be considered acceptable, which deﬁnes the practical limits of the possi- ble phase-error-correction interval as being about ±0.25π. Additional improvements can be achieved by placing a phase-insensitive amplitude-stabilization device after the phase stabilizer, which could use a nonlinear regime of the SW interaction with adiabatic pumping [ 31] or other nonlinear phenomena. At the same time, a certain degree of SW amplitude vari- ation can even be useful. When the SW phase is close to ϕ0=π/2, this means that the phase error is large and the interpretation of the SW phase as being the closest value to 0o rπmay be incorrect. In the case of such large values of the phase errors it is often recommended to start the signal processing again. The above proposed phase-stabilization device indicates such large phase errors by a signiﬁcant reduction of the amplitude of the output SW. In summary, by using the proposed phase-stabilization device small and moderate phase errors can be corrected, while the pres- ence of large phase errors can clearly be determined andindicated. Finally, we note that Fig. 2illustrates the case when the phase stabilization is accompanied by ampliﬁcation of the processed SWs. Often, this ampliﬁcation is desirable 054040-4CORRECTION OF PHASE ERRORS IN A SPIN-WAVE. . . PHYS. REV. APPLIED 11,054040 (2019) in magnonic circuits to compensate for propagation and processing losses, but sometimes a regime of no ampli- ﬁcation [ K(0,π)≈1] for in-phase waves needs to be realized. Fortunately, it is easy to vary the SW ampliﬁca- tion rate by choice of the pumping amplitude and length. For example, the case of no ampliﬁcation requires either a suﬃciently long parametric pumping gate or enhanced magnetic damping within the parametric gate, so that the “out-of-phase” partial SWs decay signiﬁcantly. The pump- ing nonadiabaticity in the case of a relatively long pumping gate can be realized by creating a spatially nonuniform pumping (e.g., a pumping gate consisting of several ﬁngers having diﬀerent polarities and/or strengths of the applied voltage). If the averaged pumping signal is nonzero (e.g., if ﬁngers of opposite polarity are of unequal length), the SW dynamics is described by the same Eq. (1)in which the nonadiabatic term b2kbecomes large if 2 k≈2π/P, where Pis the period of the ﬁngers array. However, even in the case of zero averaged pumping one should expect the phase-stabilization eﬀect to occur. In this case only the nonadiabatic term remains and one arrives at the limiting case of the parametric interaction of copropagating waves, when the idler wave is equivalent to the signal wave. In the case of copropagating waves, parametric pumping also can amplify waves (but cannot excite them) [ 32,33], and the nonadiabatic term is still phase sensitive; thus, one should expect qualitatively the same eﬀect. III. MICROMAGNETIC SIMULATIONS To conﬁrm our theoretical predictions about the SW phase stabilization we perform a series of micromagnetic simulations using the GPMagnet solver [ 34,35]. In our simulations the SWs are excited linearly by a microwave magnetic ﬁeld applied at the excitation gate of length Le=50 nm. The excitation frequency is 6.49 GHz, which corresponds to a SW wavelength of 210 nm. Microwave parametric pumping in the form of modulation of the perpendicular anisotropy /Delta1K⊥=bpMssin(ωpt)at a fre- quency ωp/(2π)=12.98 GHz is applied at the pumping gate of length Lp=50 nm, separated from the excita- tion gate by a distance LAB=250 nm. The corresponding degree of pumping nonadiabaticity in this case is α=0.67. To avoid the mistakes in the output SW phase determina- tion due to the presence of the idler SW, the phase of the output SW is calculated at the point D. The SW phase at the end of the gate is retrieved by subtraction of the prop- agation phase accumulation kLCD, where LCD=250 nm. The following material parameters of the Fe/MgO struc- ture (common for VCMA experiments [ 36]) are used: saturation magnetization μ0Ms=2.1 T, exchange length λex=3.4 nm, surface perpendicular anisotropy energy Ks=1.36 mJ /m2, and eﬀective Gilbert damping (includ- ing nonuniform broadening for a given SW frequency)αG=0.02. The nanowire thickness is set to h=1 nm, the width is w=20 nm, and the bias magnetic ﬁeld is absent. Simulations performed with no incident SW and a ﬁnite temperature of 1 K give a threshold of parametric exci- tation equal to bp,th=130 mT. It is somewhat smaller than the threshold of 169 mT calculated using the analyt- ical equation Eq. (4)(from the condition K+→∞ ). We believe that the discrepancy is caused by the dispersion of SW group velocity. In the simulations of the SW phase-transmission charac- teristics in the presence of an incident SW we set the pump- ing strength to bp=100 mT, which is 77% of the SW generation threshold. Thermal ﬂuctuations are switched oﬀ to speed up the simulations—since we work suﬃciently away from the threshold we do not expect a signiﬁcant growth of thermal ﬂuctuations under the parametric pump- ing gate. The simulated phase-transmission characteristic for small-amplitude (linear) SWs that are excited by a 1-mT excitation ﬁeld are shown in Fig. 3(blue dots). The ﬁgure shows deﬁnite phase-stabilization plateaus and matches well with the phase characteristics obtained in the analytical calculation (solid line) for a pumping strength equal to 77% of the theoretical SW generation threshold. We believe that the small upshift of the simulated phase characteristic is also related to the dispersion of the SW group velocity. We also verify how phase-transmission characteristics change with the SW amplitude, when diﬀerent nonlinear SW interactions become important. For this purpose we perform simulations for larger excitation ﬁelds, at 10 mT and 30 mT. For the excitation ﬁeld of 30 mT the SW amplitude reaches a value of My/Ms≈0.15, which is def- initely beyond the range where the excited SWs can be considered small amplitude (or linear) and in which our FIG. 3. Phase-transmission characteristics of a VCMA para- metric pumping gate for diﬀerent incident SW amplitudes cre-ated by diﬀerent excitation ﬁelds B e(symbols, micromagnetic simulations; solid line, theoretical curve for linear SWs). 054040-5ROMAN VERBA et al. PHYS. REV. APPLIED 11,054040 (2019) analytical theory is valid. From Fig. 3one can see that the phase-stabilization eﬀect is still present in the case of the large-amplitude nonlinear SWs, and the sizes and slope of the phase-stabilization plateaus are almost the same, as in the linear case. The only diﬀerence is a downshift of these plateaus, which is a consequence of the nonlinear SW phase accumulation. Thus, nonadiabatic parametric pump- ing can be used for phase-error correction of both linear and nonlinear SWs. IV . SUMMARY In summary, we demonstrate that the interaction of a propagating SW with localized nonadiabatic parametric pumping leads to a shift of the SW phase, in addition to a simple propagation phase accumulation kLp.A sa result, the SW phase-transmission characteristics become nonlinear, demonstrating a “steplike” shape. They contain pronounced ﬂat “stabilization plateaus,” within which the output SW phase is almost constant in a certain range of phases of the input SW. The phase-stabilization eﬀect becomes more pronounced with the increased level of thepumping nonadiabaticity and when the pumping strength approaches the threshold of the parametric SW generation (but it should not exceed the threshold). Our ﬁndings open a way for the implementation of phase-error corrections in magnonic logic circuits. The range of possible phase-error corrections is limited mainly by the phase dependence of the output SW amplitude and is about ±0.25πfor both linear and nonlinear SWs. ACKNOWLEDGMENTS This work was supported in part by Grants No. EFMA- 1641989, No. ECCS-1708982, and No. DMR-1610146 from the U.S. NSF, and by the DARPA M3IC grant under Contract No. W911-17-C-0031. R.V. acknowledges support from the Ministry of Education and Science of Ukraine (Project No. 0118U004007). I.N.K. acknowledges support by the Army Research Oﬃce through Grant No. W911NF-16-1-0472 and by the Defense Threat Reduc- tion Agency through Grant No. HDTRA1-16-1-0025. 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PhysRevB.78.024417.pdf	Current ﬂuctuations and magnetization dynamics symmetry in spin-torque-induced magnetization switching Xiaobin Wang, Wenzhong Zhu, and Dimitar Dimitrov Seagate Technology, 7801 Computer Avenue South, Bloomington, Minnesota 55435, USA /H20849Received 19 May 2008; revised manuscript received 26 June 2008; published 18 July 2008 /H20850 We studied spin-torque-induced magnetization switching with current ﬂuctuations using models and experi- ment measurements. It was found that the efﬁciency of accelerating magnetization reversal by noisy currentstrongly depends on the symmetry of magnetization dynamics. The efﬁciency of noisy current on acceleratingmagnetization switching is quite different for a magnetic thin-ﬁlm element without rotational symmetry and auniaxial anisotropy magnetic element with rotational symmetry. The study reveals that interactions betweenmagnetization dynamics symmetry and system ﬂuctuations are critical for predicting the switching behavior ofspin-torque excited magnetic system with noise. DOI: 10.1103/PhysRevB.78.024417 PACS number /H20849s/H20850: 85.75. /H11002d I. INTRODUCTION Magnetization switching under spin-torque current1,2has received increasing attention as a novel magnetizationswitching mechanism. 3Polarized current induced spin- torque provides a path to switch ferromagnetic order param-eter without using external magnetic ﬁeld. One example is apromising new mechanism for the write operation of nano-magnetic memory element. 4The obstacles that prevent wide application of spin-torque current induced magnetizationswitching are high critical switching current and switchingcurrent variations. For a magnetic thin-ﬁlm element in nano-magnet structures, the critical spin-torque switching currentmagnitude is dominated by the out-of-plane demagnetizationfactor. 5This is quite different from the case of magnetic switching driven by magnetic ﬁeld /H20849where the critical switching magnetic-ﬁeld magnitude is determined by the dif-ference between lateral demagnetization factors /H20850. The funda- mental physics behind this is the broken rotational symmetryof the spin-torque excited thin-ﬁlm magnetization dynamics.Understanding this is very important for the study and designof magnetic structures with efﬁcient spin-torque switching. For magnetic nanostructures, ﬂuctuations are inevitable. It is well known that random thermal ﬂuctuations at ﬁnite tem-perature lower critical switching current, and the criticalswitching current magnitude strongly depends on the tem-perature and the measurement time scale. Recently, supple-menting dc with a noisy current component has been pro-posed to reduce critical switching current. 6The idea is that the current ﬂuctuations were combined to thermal ﬂuctua-tions to help spin-torque-induced magnetization switching.Theoretical analysis in Ref. 6shows that for a magnetic el- ement with coercivity of 50 mT /H20849500 Oe /H20850and magnetic mo- ment of 10 −17Am2/H2084910−14emu /H20850, the current induced mag- netization switching time can be reduced drastically /H20849orders of magnitude /H20850by a modest level of externally generated cur- rent noise in the order of 10−20C2/s. While the prediction in Ref. 6is based on a uniaxial anisotropy magnetic element with rotational symmetry magnetization dynamics, this rota-tional symmetry is usually broken for thin-ﬁlm element inreal spin valve or magnetic tunneling junction /H20849MTJ /H20850device. As discussed above, magnetization dynamics symmetryplays important roles in spin-torque-induced magnetization switching, even for deterministic magnetic systems withoutﬂuctuations. Thus, it is important to explore magnetizationdynamics symmetry in spin-torque excited magnetic systemwith ﬂuctuations. In this paper, we measure and model noisycurrent effects on spin-torque magnetization switching for aMTJ thin ﬁlm. Our study shows that the efﬁciency of accel-erating magnetization reversal by noisy current strongly de-pends on the symmetry of magnetization dynamics. The ef-ﬁciency of noisy current on accelerating magnetizationswitching is quite different for a magnetic thin ﬁlm withoutrotational symmetry and a uniaxial anisotropy magnetic ele-ment with rotational symmetry. Our study reveals that under-standing the interaction between magnetization dynamicssymmetry and system ﬂuctuations is important for exploringmagnetic structures with efﬁcient spin-torque-induced mag-netization switching. II. EXPERIMENT MEASUREMENT To study MTJ switching under noisy current in a wide time range, a special probing assembly was used, which cov-ered from dc up to gigahertz range. It included a Tektronix-Sony 710 arbitrary waveform generator /H20849AWG710 /H20850, a pico- probe microwave probe, a Keithley 2400 source meter, and anoise generator 7110 /H20849ASIG /H20850. The AWG710 allowed pulse duration to vary from larger tha n1st oa s short as 250 ps. The bandwidth of picoprobe was from dc to 40 GHz. Thebandwidth of noise generator can reach 1.5 GHz. After eachpulse was applied, the device resistance was measured byusing Keithley 2400 source meter. The measurement proce-dure for determining the critical switching current for a par-ticular current duration were as follows: /H208491/H20850a dc was applied through the device, the current was sufﬁcient to consistentlyset the device into the antiparallel /H20849or parallel /H20850state; /H208492/H20850an opposite polarity current pulse with both signal and noisycomponents at a certain pulse duration was applied throughthe device, and the device resistance was measured; and /H208493/H20850 The applied pulse amplitude was increased until the devicestate was changed. This procedure was repeated 100 times toget switching current statistical ensembles.PHYSICAL REVIEW B 78, 024417 /H208492008 /H20850 1098-0121/2008/78 /H208492/H20850/024417 /H208495/H20850 ©2008 The American Physical Society 024417-1Figure 1shows the resistance versus current for the MTJ device in the measurement. Figure 2shows examples of measured critical switching current versus dc pulse durationfor a wide range of noisy current magnitude. The measure-ment shows no obvious effects of noisy current on magneti-zation switching. The thin ﬁlm in the experiment has a di-mension of 160 nm in length, 80 nm in width, and 2 nm inthickness. The coercivity of the thin ﬁlm is around 250 Oeand the magnetic moment is about 2.56 /H1100310 −14emu. The maximum noisy current spectral in the experiment can reach10 −19C2/s. Although we are using a magnetic tunneling junction instead of a spin valve, the magnetic parameters andnoisy current magnitudes in the experiment are comparableto parameters in Ref. 6. While Ref. 6predicts a drastic ac- celeration of magnetization switching by noisy current, ourexperiment shows no obvious effect of noisy current on mag-netization switching for a wide range of noise magnitudesand measurement time scale. Analysis in Sec. III will showthat noisy current effect on accelerating magnetizationswitching is quite different for a magnetic thin-ﬁlm elementwithout rotational symmetry and a uniaxial anisotropy mag-netic element with rotational symmetry. III. THEORETICAL ANALYSIS The magnetization dynamics in the free layer of MTJ is described by the stochastic Landau-Lifshitz-Gilbert equationwith the spin-torque term at ﬁnite temperature, dm /H6023 dt=−/H9251m/H6109/H11003/H20851m/H6023/H11003/H20849h/H6023eff+h/H6023fluc/H20850/H20852 −m/H6023/H11003/H20851/H20849h/H6023eff+h/H6023fluc/H20850+/H9252m/H6023/H11003p/H6023/H20852, /H208491/H20850 where m/H6023is the normalized magnetization and tis the nor- malized time. h/H6023eff=H/H6023eff/Ms=/H11509/H9255 /H11509m/H6023is the normalized effective magnetic ﬁeld corresponding to a normalized energy density /H9255andh/H6023flucis the thermal ﬂuctuation ﬁeld at ﬁnite tempera- ture./H9251is the damping parameter, p/H6023is a unit vector pointing to the spin-polarization direction, and /H9252=/H9257hI/2eMs2Vis the normalized spin-torque polarization magnitude, where /H9257is the polarization efﬁciency, Vis the element volume, and Iis the applied current. The magnitude of thermal ﬂuctuationterm in Eq. /H208491/H20850is determined by ﬂuctuation-dissipation con- dition at room temperature as in Ref. 7. For a current with both dc and noisy components, I=I 0+I/H11032and/H9252=/H92520+/H9252/H11032, where the prime represents Gaussian white ﬂuctuations. Equation /H208491/H20850can be written in the spherical coordinate as a set of stochastic differential equations, d/H9258=/H20873−1 sin/H9258/H11509/H9255 /H11509/H9272−/H9251/H11509/H9255 /H11509/H9258+/H9254 tan/H9258+/H92520sin/H9258/H20874dt +/H208812/H9254Isin/H9258/H92643/H20881dt+/H208812/H9254T/H92641/H20881dt, sin/H9258d/H9272=/H20873/H11509/H9255 /H11509/H9258−/H9251 sin/H9258/H11509/H9255 /H11509/H9272/H20874dt+/H208812/H9254T/H92642/H20881dt, /H208492/H20850 where /H92641,/H92642,/H92643are Gaussian random variables with zero mean and variance one. /H9254T=/H9251/H9253kBT Ms2Vis the thermal ﬂuctuation magnitude and /H9254I=1 4Ms/H20849/H9257/H9253h eM sV/H208502PSD is the current ﬂuctuation magnitude with PSD representing power spectral density of current ﬂuctuations. Notice here that time is normalized byproduct of gyromagnetic ratio and magnetization saturation /H9253Ms. For the uniaxial anisotropy case, /H9255=E MsHcV=1 2sin2/H9258and the magnetization dynamics has rotational symmetry due to /H11509/H9255 /H11509/H9272=0. Dynamics of the magnetization is essentially one di- mensional, d/H9258=/H20873−/H9251/H11509/H9255 /H11509/H9258+/H9254 tan/H9258+/H92520sin/H9258/H20874dt +/H208812/H9254Isin/H9258/H92643/H20881dt+/H208812/H9254/H92641/H20881dt. /H208493/H20850 FIG. 1. Measured resistance versus current for magnetic tunnel- ing junction used in the experiment. FIG. 2. Measured MTJ critical switching current versus dc pulse duration for a wide range of noisy current power spectral density.The multiple curves represent data taken at different noise values,and there is no need to distinguish among them as they are alleffectively indistinguishable.WANG, ZHU, AND DIMITROV PHYSICAL REVIEW B 78, 024417 /H208492008 /H20850 024417-2Notice that even in the rotational symmetric case /H208493/H20850, noisy current effects on magnetization switching could notbe simply treated as the temperature increases because of the sin /H9258factor in the term /H208812/H9254Isin/H9258/H92643/H20881dt. Here sin /H9258factor ex- ists in the current ﬂuctuation term because only current mag-nitude ﬂuctuates /H20849polarization direction is ﬁxed by polariza- tion layer magnetization /H20850while both magnitude and direction of thermal magnetic ﬁeld ﬂuctuate. However, if we neglectsin /H9258factor in the current ﬂuctuation term, the total ﬂuctua- tions can be written as a summation of the thermal ﬂuctua-tion and the current ﬂuctuation, /H9254=/H9254T+/H9254I=/H9251/H9253kBT Ms2V+1 4Ms/H20873/H9257/H9253h eM sV/H208742 PSD. /H208494/H20850 Formula /H208494/H20850is the same as Eqs. /H208498/H20850and /H208499/H20850in Ref 6and it was used to analyze noisy current as an effective temperaturerising in Ref 6. We consider a uniaxial anisotropy magnetic element with coercivity H c=500 Oe and magnetic moment MsV=2.56 /H1100310−14emu at room temperature T=300 K. For damping parameter /H9251=0.02 and polarization efﬁciency /H9257=0.57, a noisy current spectral density PSD=10−19C2/s gives/H9254I /H9254/H110151. Figure 3shows critical switching current mag- nitude versus dc pulse width. The black dot curve is thesolution of Eq. /H208493/H20850at room temperature without noisy cur- rent. This solution can be ﬁtted well to the Néel-Arrheniusformula at long-time scale /H20849dash curve /H20850, /H9251/H9253Hct=/H20881/H9266/H9254 21 /H208731−/H9251/H9257hI 2eM sHcV/H208742/H208731+/H9251/H9257hI 2eM sHcV/H20874 /H11003e1/2/H9254/H208491−/H9251/H9257hI/2eMsHcV/H208502. /H208495/H20850 The square curve is the solution of Eq. /H208493/H20850at room tempera- ture with a noisy current magnitude/H9254I /H9254=1. The effect of noisy currents on accelerating magnetization switching is obvious.Figure 1also shows the solution of Eq. /H208493/H20850at 600 K tem- perature with/H9254I /H9254=0. This is the star curve and corresponds to neglecting sin /H9258factor in current ﬂuctuation term for a noisy current magnitude/H9254I /H9254=1 at room temperature T=300 K /H20849ef- fective temperature approach /H20850. It is clear that neglecting sin /H9258 factor or an effective temperature approach signiﬁcantly overestimates noisy current effect on accelerating magnetiza-tion switching. In the case of a thin-ﬁlm element in MTJ, spin polariza- tion points in the direction of the easy axis of the rectangular element p /H6023=e/H6023z. The energy of the magnetic system is /H9255 =E Ms2V=1 2Nxmx2+1 2Ny2my2+1 2Nzmz2, where Nx,Ny, and Nzare de- magnetization factors. For a thin-ﬁlm element, the perpen- dicular demagnetization factor is much stronger than the sur-face demagnetization factor /H20849N y/H11271Nx,Nz/H20850, rotational symmetry is broken and two-dimensional stochastic differen- tial Eq. /H208492/H20850with/H11509/H9255 /H11509/H9272/HS110050 needs to be solved for magnetization dynamics. In order to examine explicitly the interaction be-tween noisy current and unsymmetric magnetization dynam-ics, we simplify the above two-dimensional stochastic differ-ential Eq. /H208492/H20850to a one-dimensional system based on small damping approximation. For small damping parameter, a sto-chastic average technique 8allows Eq. /H208492/H20850to be integrated around constant energy levels to obtain the following one-dimensional stochastic differential equation: d/H9255= A/H20849/H9255/H20850dt+/H20881B/H20849/H9255/H20850dW /H20849t/H20850, /H208496/H20850 where A/H20849/H9255/H20850and B/H20849/H9255/H20850are the deterministic and stochastic terms, respectively. dW /H20849t/H20850is the increment of a standard Brownian process. A/H20849/H9255/H20850term can be explicitly written as FIG. 3. Critical switching current versus dc pulse duration for a uniaxial magnetic element. FIG. 4. Critical switching current versus dc pulse duration for a magnetic thin-ﬁlm element.CURRENT FLUCTUATIONS AND MAGNETIZATION … PHYSICAL REVIEW B 78, 024417 /H208492008 /H20850 024417-3A/H20849/H9255/H20850=/H20886d/H9272sin/H9258 /H11509/H9255 /H11509/H9258/H20902−/H9251/H20875/H20873/H11509/H9255 /H11509/H9258/H208742 +1 sin2/H9258/H20873/H11509/H9255 /H11509/H9272/H208742/H20876+/H92520sin/H9258/H11509/H9255 /H11509/H9258+/H9254/H20873/H115092/H9255 /H11509/H92582+1 sin2/H9258/H115092/H9255 /H11509/H92722/H20874+/H9254/H20873/H11509/H9255 /H11509/H9258/H20874 tan/H9258+/H9254I/H20873/H115092/H9255 /H11509/H92582/H20874sin2/H9258/H20903 /H20886d/H9272sin/H9258 /H11509/H9255 /H11509/H9258, /H208497/H20850 where /H9258and/H9272are magnetization angles in spherical coordinates. /H20859is the integration of gyromagnetic motion around a constant energy level /H9255/H20849/H9258,/H9272/H20850=/H9255.B/H20849/H9255/H20850term can be explicitly written as B/H20849/H9255/H20850=/H20886d/H9272sin/H9258 /H11509/H9255 /H11509/H9258/H208772/H9254/H20875/H20873/H11509/H9255 /H11509/H9258/H208742 +1 sin2/H9258/H20873/H11509/H9255 /H11509/H9272/H208742/H20876+2/H9254I/H20873/H11509/H9255 /H11509/H9258/H208742 sin2/H9258/H20878 /H20886d/H9272sin/H9258 /H11509/H9255 /H11509/H9258. /H208498/H20850 Notice that the stochastic term /H208498/H20850has a thermal term 2/H9254/H20851/H20849/H11509/H9255 /H11509/H9258/H208502+1 sin2/H9258/H20849/H11509/H9255 /H11509/H9272/H208502/H20852and a current ﬂuctuation term 2/H9254I/H20849/H11509/H9255 /H11509/H9258/H208502sin2/H9258. Only for rotational symmetric magnetization dynamic case,/H11509/H9255 /H11509/H9272is equal to zero and the current ﬂuctuation term could be written as an effective temperature rising for-mat /H20849of course sin /H9258term still needs to be neglected /H20850. How- ever, in the case of a MTJ thin ﬁlm, due to strong out-of-plane demagnetization factor, the magnetization dynamics isquite unsymmetric and the current ﬂuctuation effect is quite different from a temperature rising effect. Figure 4shows the critical switching current vs dc pulse duration for a magnetic thin-ﬁlm element. The thin ﬁlm is160 nm long, 80 nm wide, and 2 nm thick. Magnetizationsaturation is 1000 emu/cc /H20849corresponding to the magnetic moment of 2.56 /H1100310 −14emu /H20850. The damping parameter is 0.0057 and the polarization efﬁciency is 0.57. The black dotcurve is switching at room temperature T=300 K without noisy current. The star curve is magnetization switching atincreased temperature T=600 K. The square curve is mag- netization switching at room temperature with a noisy cur- rent magnitude /H9254I /H9254=1. It is clear that for a magnetic thin-ﬁlm, noisy current effects on magnetization switching are quitedifferent than that of the temperature rising. Figure 4shows no obvious effects of magnetization switching accelerationdue to noisy current for a magnetic thin ﬁlm, consistent withexperiment measurement. We have shown that noisy current effect on accelerating magnetization switching is quite different for a magnetic thinﬁlm without rotational symmetry and a uniaxial magneticelement with rotational symmetry. This is due to the interac-tion between magnetization dynamics symmetry and ﬂuctua-tions of the system. The model result of a magnetic system without rotational symmetry is consistent with experimentalmeasurement on MTJ thin ﬁlm. However, the prediction of amagnetic system with rotational symmetry could only betested on perpendicular anisotropy MTJ with magnetizationpointing in the perpendicular direction. Currently, MTJ withhigh perpendicular anisotropy and perpendicular magnetiza-tion are in active research for their potential beneﬁts in low-ering switching current and maintaining thermal stability atthe same time. 9 IV . CONCLUSION Noisy current effects on spin-torque-induced magnetiza- tion switching are studied using both models and experimentmeasurements. Although drastic magnetization switching ac-celeration due to noisy current has been predicted for auniaxial anisotropy element, both experiment measurementand model show no obvious effect of noisy current on mag-netization switching speed of a thin-ﬁlm element. This dif-ference between a uniaxial anisotropy element and a thin-ﬁlm element is due to the fact that the efﬁciency ofaccelerating magnetization reversal by noisy current stronglydepends on the symmetry of magnetization dynamics. Ourstudy shows that treating noisy current as an effective tem-perature rising in magnetization reversal in general signiﬁ-cantly overestimates current ﬂuctuation effects. Understand-ing the interaction between magnetization dynamicssymmetry and system ﬂuctuations are critical for predictingswitching behavior of spin-torque excited magnetic systemwith noise.WANG, ZHU, AND DIMITROV PHYSICAL REVIEW B 78, 024417 /H208492008 /H20850 024417-41J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 3See references cited in W. Wetzels, G. E. W. Bauer, and O. N. Jouravlev, Phys. Rev. Lett. 96, 127203 /H208492006 /H20850, and references cited in J. C. Sankey, Y.-T. Cui, J. Z. Sun, J. C. Slonczewski, R.A. Buhrman, and D. C. Ralph, Nat. Phys. 4,6 7 /H208492008 /H20850. 4M. Hosomi, H. Yamagishi, H. Ymamoto, T. Bessho, K. Higo, Y. Yamane, K. Yamada, H. Shoji, M. Hachino, H. Fukumoto, C.Nagao, and H. Kano, Tech. Dig. - Int. Electron Devices Meet. 2005, 479. 5J. Z. Sun, Phys. Rev. B 62, 570 /H208492000 /H20850.6W. Wetzels, G. E. Bauer, and O. N. Jouravlev, Phys. Rev. Lett. 96, 127203 /H208492006 /H20850. 7W. F. Brown, Phys. Rev. 130, 1677 /H208491963 /H20850. 8X. Wang, N. H. Bertram, and V. L. Safonov, J. Appl. Phys. 92, 2064 /H208492002 /H20850. 9H. Yoda, T. Kishi, T. Kai, T. Nagase, M. Yoshikawa, M. Na- kayama, E. Kitagawa, M. Amano, H. Aikawa, N. Shimomura, K.Nishiyama, T. Daibou, S. Takahashi, S. Ikegawa, K. Yakushiji,T. Nagahama, H. Kubota, A. Fukushima, S. Yuasa, Y. Nakatani,M. Oogane, Y. Ando, Y. Suzuki, K. Ando, and T. Miyazaki,IEEE International Magnetics Conference, 2008 /H20849unpublished /H20850.CURRENT FLUCTUATIONS AND MAGNETIZATION … PHYSICAL REVIEW B 78, 024417 /H208492008 /H20850 024417-5
PhysRevB.91.014438.pdf	PHYSICAL REVIEW B 91, 014438 (2015) Role of antisite disorder on intrinsic Gilbert damping in L10FePt ﬁlms X. Ma,1L. Ma,2P. He,2H. B. Zhao,3,S. M. Zhou,2and G. L ¨upke1,† 1Department of Applied Science, College of William and Mary, 251 Jamestown Road, Williamsburg, Virginia 23187, USA 2Shanghai Key Laboratory of Special Artiﬁcial Microstructure Materials and Technology and School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 3Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China (Received 13 June 2014; revised manuscript received 1 January 2015; published 30 January 2015) The impact of antisite disorder xon the intrinsic Gilbert damping α0in well-ordered L10FePt ﬁlms is investigated by time-resolved magneto-optical Kerr effect. The variation of xmainly affects the electron scattering rate 1/τe, while other leading parameters remain unchanged. The experimentally observed linear dependence of α0on 1/τeindicates that spin relaxation is through electron interband transitions, as predicted by the spin-orbit coupling torque correlation model. Measurements at low temperature show that α0remains unchanged with temperature even for FePt with very high chemical order, indicating that electron-phonon scattering is negligible.Moreover, as xdecreases, the perpendicular magnetic anisotropy increases, and the Landau gfactor exhibits a negative shift due to an increase in orbital momentum anisotropy. Our results will facilitate the design andexploration of magnetic alloys with large magnetic anisotropy and desirable damping properties. DOI: 10.1103/PhysRevB.91.014438 PACS number(s): 75 .78.Jp,75.30.Gw,75.50.Vv,75.70.Tj I. INTRODUCTION Ultrafast magnetization precessional switching in ferro- magnets utilizing magnetic ﬁeld pulses, spin polarized cur-rents, and ultrafast laser pulses [ 1–6] is currently a popular topic due to its importance in magnetic information storageand spintronic applications. The uniform magnetization pre-cession can be well modeled with the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation [ 7–9], where the Slonczewski torque term denotes the spin transfer torques (STTs), and theGilbert damping parameter αdetermines the spin relaxation time [ 10] and is crucial for device performance [ 11–14]. The extrinsic Gilbert damping is due to nonlocal spin relaxation,such as spin pumping and magnon-magnon scattering, whichcan be tuned by artiﬁcial substrates, specially designed bufferand coverage layers [ 15–21], while the intrinsic Gilbert damp- ing parameter α 0is thought to arise from spin-orbit interaction (SOI) [ 22–30], and recently its quadratic dependence on SOI is demonstrated experimentally in FePtPd alloys [ 31]. Theα0describes the energy ﬂow rate from spin to electronic orbital and phonon degrees of freedom throughelectron scattering and has been studied in various theoreticalmodels [ 22–30]. The breathing Fermi surface model [ 22] and torque-correlation model [ 23] based on ﬁrst-principle band structure calculations qualitatively match α 0in soft magnetic alloys such as Fe, Co, and Ni [ 32–37]. Moreover, contributions toα0can be categorized based on intraband and interband transitions [ 23,26]. The damping rate from intraband transi- tions scales linearly with the electron relaxation time τeand exhibits conductivitylike behavior. In contrast, the dampingrate from the interband transition is proportional to the electronscattering rate 1 /τ eand consequently exhibits resistivitylike behavior. Therefore, the transition from conductivitylike toresistivitylike behavior through modulation of 1 /τ eis used to hbzhao@fudan.edu.cn †gxluep@wm.eduqualitatively understand the temperature dependence of αin soft magnetic materials [ 23,26]. The underlying physics of α0in metallic magnets with large uniaxial magnetic anisotropy Ku, however, is not completely understood. The question is still open as to whether the well-developed theories can shed light on the damping behavior in such materials and therefore motivates extensive research [ 31,38–43]. For instance, S. Mizukami et al. demonstrated a low value of α0with considerable Kuin MnGa alloys due to the low density of states at the Fermi level,D(EF)[40]. One key investigation still missing is the relationship between α0and 1/τe. Theoretical studies [ 28–30] point out that 1 /τeinvolves various types of electron scattering events such as by phonon and impurity. It is difﬁcult to measure the electron-phonon scattering rate from experiments with sufﬁcient accuracy, while the electron-impurity scattering can be controlled by either doping or artiﬁcial disorder.However, no direct experiments have been reported to verify quantitatively the relationship between α 0and 1/τethrough impurity scattering despite many attempts [ 33,44–47]. The challenge lies in the fact that α0also depends on other leading parameters, such as magnetization MS[33], SOI [ 44], lattice distortion [ 47], andD(EF)[40], which may vary signiﬁcantly when the impurity concentration is modulated. It is difﬁcult to quantitatively investigate the impact of 1 /τeonα0in those material systems. In this paper, we investigate the effect of antisite disorder on α0in well-ordered L10FePt thin ﬁlm samples. Time-resolved magneto-optical Kerr effect (TRMOKE) measurements showthatα 0gradually increases by more than a factor of three when the antisite disorder xis varied from 3 to 16% by sample growth temperature ( Tg). The variation of xmainly affects the electron scattering rate 1 /τe, while other leading parameters remain almost unchanged. A linear correlation between α0and 1/τeis experimentally observed due to electron interband transitions.Moreover, α 0remains unchanged down to low temperature (20 K), indicating that the electron-phonon interaction andelectron intraband transitions are negligible. In addition, K u 1098-0121/2015/91(1)/014438(7) 014438-1 ©2015 American Physical SocietyMA, MA, HE, ZHAO, ZHOU, AND L ¨UPKE PHYSICAL REVIEW B 91, 014438 (2015) decreases, and the Landau gfactor increases with larger xdue to an increase in orbital momentum anisotropy. Our resultsprovide a pathway for designing magnetic alloys with desirableαandK u. II. EXPERIMENTS As e r i e so f L10ordered FePt thin ﬁlms are deposited on single crystal MgO (001) substrates by magnetron sputtering.The FePt composite target is fabricated by placing small Ptpieces on a Fe target. The base pressure of the depositionsystem is 1 .0∗10 −5Pa, and the Ar pressure is 0.35 Pa. During deposition, the rate of deposition was about 0.1 nm /s, and the substrates are kept at different temperatures Tg.A f t e r deposition, the samples are annealed in situ at the same temperature as their growth temperature for 2 hours. Two seriesof samples with different thickness are fabricated, and the ﬁlmthickness is determined by x-ray reﬂectivity to be 17 ±1n m and 22 ±1 nm. The microstructure analysis is performed by using x-ray diffraction (XRD), with Cu Kαradiation. Static magnetization hysteresis loops are measured by vibratingsample magnetometer (VSM) at room temperature. In order to measure α, TRMOKE measurements are performed at various temperatures Twith a pump-probe setup using pulsed Ti:sapphire laser with a pulse duration of 200 fsand a repetition rate of 250 kHz. The wavelength of pump(probe) pulses is 400 nm (800 nm). A modulated pump pulsebeam with a ﬂuence of 0 .16 mJ/cm 2is focused to a spot ∼1 mm in diameter on the sample to excite the magnetization precession, and the transient Kerr signal is detected by a probepulse beam that is time-delayed with respect to the pump. Thefocus area of the probe beam has a diameter of 0.7 mm, whichwas smaller than that of the pump beam so that the intensityratio of the pump to probe pulses is set to be about 6:1. Thegeometry of applied external magnetic ﬁeld and magnetizationprecession is depicted in Fig. 1(a). A variable magnetic ﬁeld H up to 6.5 T is applied at an angle of θ H=45◦with respect to the ﬁlm normal direction using a superconducting magnet [ 48]. III. RESULTS AND DISCUSSION A. Sample characterization Figure 1(b) displays the out-of-plane magnetization hys- teresis loops for 22-nm-thick ﬁlms with Tg=580◦C, 620 °C, and 680 °C and the in-plane hysteresis loop for the sample with Tg=620◦C. The out-of-plane hysteresis loops are almost square-shaped with coercivity Hc=0.3 T, but it is difﬁcult to reach the saturated magnetization with in-plane magnetic ﬁeld,indicating the establishment of high perpendicular magneticanisotropy K u. From the experiments, the saturation magneti- zation MSfor all samples is determined to be 1100 emu /cm3 and remains unchanged as a function of growth temperature and disorder x. Moreover, the magnetization is not fully saturated with H=2 T applied along the easy axis, indicating the existence of multiple magnetic domains at lower magneticﬁelds. Figure 1(c) displays the structure characterization of FePt samples with XRD measurements. Only face-centered-tetragonal (fct) (001) and (002) peaks of FePt are observedin the spectrum along with other peaks from MgO substrate, (arb. units) FIG. 1. (Color online) Schematic of TRMOKE measurement ge- ometry (a), static magnetic hysteresis loops measured by VSM (b), structure characterization results of FePt thin ﬁlms by XRD (c), and antisite disorder percentage xas a function of growth temperature Tg (d). The insets in (d) depict FePt alloy structure with low (left) and high (right) antisite disorder concentration. which indicates the L10ordering in the FePt alloys. The peak positions do not shift with different Tg, which indicates that the lattice constant varies by less than 1.0% for different Tg, and tetragonal distortion of lattice is not affected. The antisitedisorder percentage is derived from x=1−S 2=⎛ ⎝1−/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftbigI001 I002/parenrightbig meas/parenleftbigI001 I002/parenrightbig calc⎞ ⎠/2, (1) where Sis the degree of chemical order, I001andI002are integrated intensities of fct (001) and (002) peaks, and (1 −x) is the probability of the correct site occupation for eitherFe or Pt atoms in such an L1 0ordered alloy system [ 49]. (I001/I002)calcis calculated to be 2.0 for the perfect ordered ﬁlm with thickness ranging from 11 to 49 nm [ 49]. The x as a function of Tgfor ﬁlms with both thicknesses is shown in Fig. 1(d). Higher Tgleads to monotonous decrease of the antisite disorder xin FePt alloys, as depicted by the insets in Fig.1(d). Islands form throughout the ﬁlm with Tg=720◦Co r 740 °C, which prevents further sole reduction of bulk point defects. Figure 2displays the surface topography from samples prepared at Tg=620◦C and Tg=720◦C with scanning electron microscopy (SEM). With Tg/lessorequalslant680◦C, the FePt layer is homogeneously distributed throughout the thin ﬁlm, asindicated in Fig. 2(a). In contrast with T g=720◦C or 740◦C, the FePt layer is inhomogeneously distributed on the MgOsubstrate, and islands form. As shown in Fig. 2(b), the dark contrast in the image corresponds to areas without FePt grownon the MgO substrate. The formation of islands and particlesinL1 0ordered FePt thin ﬁlms leads to lower chemical order as the Fe-rich clusters are promoted at the surface regions [ 50]. Moreover, it will also introduce more surface contribution to 014438-2ROLE OF ANTISITE DISORDER ON INTRINSIC . . . PHYSICAL REVIEW B 91, 014438 (2015) FIG. 2. SEM results from FePt thin ﬁlms grown at Tg=620◦C (a) and Tg=720◦C( b ) . the magnetic properties, such as Ku,gfactor, and damping. For instance, the islands exhibit pronounced out-of-planeanisotropy K u, which is peaked when c=0.5i nF e cPt(1−c) alloys [ 50]. This is due to Dzyaloshinskii-Moriya interactions emerging at the surface as a result of the broken inversionsymmetry, which compete with exchange interactions causingthe complex magnetism at the FePt surface. With respect to theimpact on the gfactor, the orbital momentum is not entirely quenched due to the broken symmetry at the surface, while the formation of islands or enlarged surface areas results in anegative shift of gfactor [ 51]. B. Gilbert damping Figure 3(a) shows the TRMOKE results of FePt thin ﬁlms with various xatH=6.5 T. The uniform magnetization precession is demonstrated by the oscillatory Kerr signals θK, while the magnetic damping is indicated by the decaying pre-cession amplitude as the time delay increases. The measured(arb. units) (arb. units)f f FIG. 3. (Color online) TRMOKE data from FePt thin ﬁlms with different xand applied magnetic ﬁeld H=6.5 T (a) and with x=4% measured at different H(b). The dependence of spin precession frequency f(c) and effective Gilbert damping αeff(d) on Hobtained f r o mE q s .( 2)a n d( 4) for FePt thin ﬁlms with different x. The solid lines refer to ﬁtted results using Eqs. ( 2)a n d( 3).θKcan be well ﬁtted by the following equation: θK=a+b∗exp(−t/t0)+A∗exp(−t/τ)s i n ( 2 πf t+ϕ), (2) where parameters A,τ,f, andϕare the amplitude, magnetic relaxation time, spin precession frequency, and initial phaseof the magnetization precession, respectively [ 31]. Here, a, b, andt 0correspond to the background signal owing to the slow recovery process after fast demagnetization by laserpulse heating. It is well demonstrated in Fig. 3(a) that the spin precession frequency is larger and the magnetic dampingeffect becomes weaker for lower xwith the same H. In order to obtain αfor FePt samples with different x, magnetic ﬁeld ( H)-dependent TRMOKE measurements are performed. Figure 3(b) shows the measured results of the 22- nm-thick sample with T g=680◦C. It can be seen clearly that the precession period and relaxation time vary as Hincreases. The ﬁtted fas a function of Hfor different xare plotted in Fig.3(c). We note that fcan be tuned from 225 to 335 GHz by varying Handx. By solving the LLG equation, fcan be expressed as 2πf=γ(H1H2)1/2, (3) where H1=Hcos(θH−θ)+HKcos2θ,H2=Hcos(θH− θ)+HKcos 2θ,HK=2Ku/MS−4πMS, and γ=γeg/2, with γe=1.76×107Hz/Oe, Landau gfactor, satu- rated magnetization MS=1100 emu /cm3, and perpendicular anisotropy Ku. The equilibrium angular position θof the mag- netization satisﬁes the equation sin 2 θ=(2H/H K)s i n (θH− θ). The measured Hdependence of fcan be well ﬁtted by Eq. ( 3), as shown in Fig. 3(c), and we thus obtain Kuandg. Kudecreases from 5 .2t o3.2( 1 07erg/cm3), and the Landau g factor also displays a shift from 1.9 to 2.24 as xincreases, as shown in Fig. 4(a). Furthermore, using the ﬁtted values of τ, 1.82.02.2 0.080.16 0 6 12 180.040.08g factor 22 nm FePt 17 nm FePt 0 0 / g Anti-site percentage x FIG. 4. (Color online) Landau gfactor (a), intrinsic Gilbert damping α0(b), and α0/g(c) as a function of antisite defect concentration xfor FePt ﬁlms with thicknesses of 17 nm and 22 nm. 014438-3MA, MA, HE, ZHAO, ZHOU, AND L ¨UPKE PHYSICAL REVIEW B 91, 014438 (2015) we determine the effective Gilbert damping αeffwith αeff=α0+αex=2 τγ(H1+H2), (4) where αexis the extrinsic contribution to Gilbert damping [ 40]. As shown in Fig. 3(d),t h ev a l u eo f αeffgradually decreases with higher Hand saturates at high ﬁelds [ 52]. The decreasing trend of αeffwithHis attributed to the suppression of dephasing dynamics among magnetic domains [ 21,31] since multiple domains exist at low ﬁelds, as indicated in Fig. 1(b), and the magnon-magnon scattering is less effective for per-pendicularly magnetized samples [ 53]. The saturation values ofα effat higher ﬁelds are therefore used to approximate the intrinsic Gilbert damping α0[31]. Figure 4(b) shows α0as a function of xatT=200 K. The key ﬁnding here is that α0gradually increases with larger x. Since the scattering rate 1 /τeis enhanced with more impurity scattering sites xaccording to scattering theory [ 28– 30], the positive correlation between α0and 1/τequalitatively matches the resistivitylike behavior where α0is governed by interband transitions [ 23,26]. In the spin-orbit coupling torque correlation model for interband transitions [ 23,26,40], α0∝gμB2D(EF)ξ2 MSW2τe, (5) where Wis the dband width, ξthe spin-orbit coupling strength, and D(EF) is the density of states at Fermi level. It is demonstrated that antisite disorder in L10ordered alloys smoothens the density of states [ 54]. The calculated D(EF) increases within 5% as xincreases, and Wremains almost the same [ 55] when 0 <x< 15%, as in our case. In our previous paper [ 56], we investigated the anomalous Hall conductivity of L10ordered FePt ﬁlms with different ordering. The resistivity- independent term ( b0) of anomalous Hall conductivity remains almost unchanged with xwhen xis small, indicating that the variance of spin-orbit coupling strength ξis negligible in our samples. The gfactor increases by 19% with larger x,a s shown in Fig. 4(a), due to the modulation of orbital momentum anisotropy (discussed in the next subsection). The latticedistortion and M Sremain almost unchanged, as demonstrated by structure characterization and VSM measurements. Asimilar trend of α 0withxis observed for samples with two different thicknesses, which indicates that surface and straineffects on damping can be ruled out. Therefore, we attributethe signiﬁcant increase of α 0by more than three times, as revealed in Fig. 4(b), to the enhancement of 1 /τeas a result of the increasing x. To separate the effect of gon the damping, we further plot α0/gas a function of xin Fig. 4(c). We observe approximately a linear correlation between the two variables.The exchange of different types of atoms in L1 0ordered alloy ﬁlm leads to the scattering of itinerant electrons through localspin-dependent exchange potentials. Since the cross-sectionand strength of individual scattering events remain unchangedin weak scattering regime [ 28], the linear correlation indicates thatα 0is proportional to 1 /τe, where the damping process is dominated by interband contribution [ 57]. The damping process can be considered roughly as the decay of a uniformmode magnon into an electron spin-ﬂip excitation. Therefore,the antisite disorder works as spin-ﬂip scattering center foritinerant electrons transferring spin angular momentum to the01 0 0 2 0 00.10.20.3 0.10.20.3 Decay rate 1/ (THz) Frequency Decay rateFrequency f (THz) Temperature T (K) 34560.10.20.3 0.10.20.3 Decay rate 1/ (THz) Frequency Decay rateFrequency f (THz) Magnetic field H (T) FIG. 5. (Color online) The fand 1/τas functions of temperature Twith applied ﬁeld H=5 T (a) and as functions of Hat 20 K (b). The solid lines refer to ﬁtted results using Eq. ( 3). lattice via SOI. Moreover, complete suppression of the antisite defects might lead to a remnant α0, where the electron is mainly scattered by phonon instead of impurity. In order to check for intraband contribution to α0, temperature-dependent TRMOKE measurements are carriedout. The TRMOKE measurements are carried out for the17-nm-thick FePt ﬁlm with the lowest x=3% at low tem- peratures. Figure 5(a)shows that the frequency and decay rate of coherent spin precession at H=5 T varies slightly with temperature (from 20 to 200 K). The ﬁeld-dependent frequencyand decay rate at 20 K, as shown in Fig. 5(b), are analyzed to obtain α 0. It turns out that α0remains almost unchanged (from 0.053 ±0.013 to 0.054 ±0.013) when temperature decreases from 200 to 20 K, despite a signiﬁcant change inthe electron-phonon scattering rate [ 23,26]. The temperature- independent behavior of α 0indicates that electron-phonon scattering is negligible, and electron-impurity interactiondominates the scattering events. In our previous paper [ 31], we calculated the electron-phonon scattering rate from ﬁrstprinciples to be approximately 1 .33 ps −1for FePt at 200 K. The electron-impurity scattering rate must be considerablyhigher. Moreover, the weak temperature dependence of α 0 indicates that the 3% antisite disorder is still too high to observe the conductivitylike behavior for intraband contribution toα 0, which may become signiﬁcant with low 1 /τe. Further investigations at low temperature with fewer impurities arenecessary to get deeper insight on the relationship between α 0 andτegoverned by intraband transitions. 014438-4ROLE OF ANTISITE DISORDER ON INTRINSIC . . . PHYSICAL REVIEW B 91, 014438 (2015) 0 6 12 18345 22 nm FePt 17 nm FePtKu(107 erg/cm3) Anti-site percentage x FIG. 6. (Color online) Perpendicular magnetic anisotropy Kuas a function of antisite disorder percentage x. C. Perpendicular magnetic anisotropy ( Ku) and Landau gfactor Figure 6shows that the Kumaintains high values from 3.2t o5.2( 1 07erg/cm3) and gradually increases with smaller x, which is consistent with other experiments as well as the theory [ 49,58–61]. The Kuin FePt alloys results from the simultaneous occurrence of the spin polarization andlarge SOI [ 62]. The smaller xrepresents that more Pt atoms become the nearest neighbors of Fe, which results in strongerhybridization between Fe and Pt atoms. Consequently, Ptacquires larger spin polarization and orbital moment dueto the Fe-Pt hybridization in the higher chemical orderedsample and contributes signiﬁcantly to K u. As a result, the orbital momentum anisotropy is increased [ 60,61], and Kuis suggested to increase with decreasing x. Figure 4(a) shows that a gradual decrease of the Landau gfactor is observed with smaller x.T h e gfactor sets the proportionality of angular momentum and magnetic momentfor the individual spins, which also affects the dynamicresponse of a magnetic ﬁlm. In itinerant electron systems,thegfactor may be written as g=2m e e∗μS+μL /angbracketleftS/prime/angbracketright+/angbracketleftL/angbracketright, (6) where μS(uL) denotes the magnetic momentum from the spin (orbital) component, and /angbracketleftS/prime/angbracketright(/angbracketleftL/angbracketright) is the spin (orbital) con- tribution to the electron angular momentum. For a symmetriccrystal lattice, the orbital motion of the delectron is quenched by the crystal ﬁeld effect, i.e., /angbracketleftL/angbracketright=0. Nevertheless, the or- bital contribution to the magnetic moment is nonzero, thus thegfactor is equal to 2 ∗(1+uL/μS) and is typically greater than two in an itinerant electron system. However, as xdecreases, the enhanced hybridization between Fe and Pt restores theorbital momentum due to the large SOI strength of Pt [ 62] and raises the orbital momentum anisotropy [ 60,61]. This would lead to /angbracketleftL/angbracketright/negationslash=0 andg≈2 ∗(1−uL/μS), indicating a negative shift of the gfactor relative to the value of two. Such a negative shift of the gfactor is also observed at surface or interface, where the orbital momentum is not entirely quenched due tothe symmetry broken effect [ 51]. IV . CONCLUSION In conclusion, we demonstrate that in L10ordered FePt ﬁlms, control of the antisite disorder xwith proper growth temperature results in signiﬁcant variation of α0.A s x increases from 3 to 16%, α0increases by more than a factor of three from 0.05 to 0.19. The variation of xmainly affects the scattering rate 1 /τe, while other leading parameters remain unchanged. A linear correlation between α0and 1 /τeis observed experimentally due to electron interband transitions.Moreover, α 0remains unchanged with temperature, indicat- ing that electron-phonon scattering and electron intrabandtransitions are negligible. Moreover, as antisite occupationdecreases, the perpendicular magnetic anisotropy increases,and the Landau gfactor exhibits a negative shift due to an increase in orbital momentum anisotropy. Our resultswill facilitate the design and exploration of new magneticalloys with large magnetic anisotropy and desirable dampingproperties. ACKNOWLEDGMENTS The TRMOKE experiments, data analysis, and discussions performed at the College of William and Mary were sponsoredby the Department of Energy (DOE) through Grant No.DE-FG02–04ER46127. H. Z. acknowledges ﬁnancial supportfrom the National Natural Science Foundation of China(Grants No. 61222407 and No. 51371052) and the Ministry ofScience and Technology (MOST) of China through Grant No.2015CB921403. 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PhysRevB.87.174427.pdf	PHYSICAL REVIEW B 87, 174427 (2013) Topological chiral magnonic edge mode in a magnonic crystal Ryuichi Shindou,Ryo Matsumoto, and Shuichi Murakami Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, Japan Jun-ichiro Ohe Department of Physics, Toho University, 2-2-1 Miyama, Funabashi, Chiba, Japan (Received 6 April 2013; published 24 May 2013) Topological phases have been explored in various ﬁelds in physics such as spintronics, photonics, liquid helium, correlated electron system, and cold-atomic system. This leads to the recent foundation of emerging materials suchas topological band insulators, topological photonic crystals, and topological superconductors/superﬂuid. In thispaper, we propose a topological magnonic crystal which provides protected chiral edge modes for magnetostaticspin waves. Based on a linearized Landau-Lifshitz equation, we show that a magnonic crystal with the dipolarinteraction acquires a spin-wave volume-mode band with nonzero Chern integer. We argue that such magnonicsystems are accompanied by the same integer numbers of chiral spin-wave edge modes within a band gap for thevolume-mode bands. In these edge modes, the spin wave propagates in a unidirectional manner without beingscattered backward, which implements novel fault-tolerant spintronic devices. DOI: 10.1103/PhysRevB.87.174427 PACS number(s): 85 .75.−d, 73.43.−f, 03.65.Vf, 85 .70.Ec I. INTRODUCTION Topological phases in condensed matters have been at- tracting much attention because of their fascinating physicalproperties. Discoveries of topological band insulators 1–7open up an emerging research paradigm on spin-orbit interactionphysics. Relativistic spin-orbit interaction in the topologicalband insulator endows its Bloch electron bands with nontrivialglobal phase structures, which leads to novel surface metallicstates. 5,8Superconductor analogs of topological insulators have exotic edge modes,9,10which are composed only of Majorana fermion. Some aspects of these bound states areexperimentally conﬁrmed, 11,12fostering much prospect of the realization of quantum computers.13A photonics analog of quantum Hall phase with chiral edge modes is proposedtheoretically 14–16and is subsequently designed in actual photonics crystals.17Unidirectional propagations of an elec- tromagnetic wave along these edge modes were experimentallyobserved, which provides these metamaterials with uniquephotonic functionality. In this paper, we theoretically propose a spin-wave analog of topological phases, which has a topologically protected chiraledge mode for the spin-wave propagation. The spin wave is acollective propagation of precessional motions of magneticmoments in magnets. Depending on its wavelength, spinwaves are classiﬁed into two categories. One is the exchangespin wave with the shorter wavelength, whose motion isdriven by the quantum-mechanical exchange interactions(“exchange-dominated” region). The other is the magne-tostatic spin wave with the longer wavelength, 18,19whose propagation is caused by the long-range dipolar interaction(“dipolar” region). Magnonics research investigates how thesespin waves propagate in the submicrometer length scaleand subnanosecond time scale. 20–22Like in other solid-state technologies such as photonics and plasmonics, the mainapplication direction is to explore the ability of the spin waveto carry and process information. Especially, the propagationof spin waves in periodically modulated magnetic materialsdubbed as magnonic crystals 22–32is one of its central concerns.Owing to the periodic structuring, the spin-wave volume-mode spectrum in magnonic crystal acquires allowed frequencybands of spin-wave states and forbidden-frequency bands(band gap). 22–30 In the next section, we introduce a topological Chern number in these volume-mode bands with a band gap. InSec. III, we consider a two-dimensional magnonic crystal (MC), where the Chern number for the lowest spin-wavevolume mode takes nonzero integer values in the dipolarregion. In Sec. IV, we argue that the nonzero Chern integer for the lowest volume-mode band results in the same integernumbers of topological chiral edge modes (surface modes;Fig. 1), whose dispersions run across the band gap between the lowest volume-mode band and the second lowest volume-mode band. The existence of the topological chiral edge modein the MC is further justiﬁed by a micromagnetic simulationin Sec. V. The relevant length scale for the magnonic crystal turns out to be from sub- μm to sub-mm, well within the range of nanosize fabrication. Unidirectional propagations of spinwaves along the edge modes are experimentally measurableespecially in yttrium iron garnet (YIG), where the coherencelength of magnons is on the order of centimeters. 21Based on these observations, we argue in Sec. VIthat the topological chiral edge modes can be easily channelized, twisted, split, andmanipulated, which enables us to construct novel magnonicdevices such as spin-wave logic gate and spin-wave currentsplitter. II. CHERN NUMBER IN BOSON SYSTEMS A. Bosonic BdG Hamiltonian To introduce the topological Chern number for the spin- wave volume-mode band, we ﬁrst consider a quadratic formof generic boson Hamiltonian, ˆH=1 2/summationdisplay k[β† kβ−k]·Hk·/bracketleftBigg βk β† −k/bracketrightBigg , (1) 174427-1 1098-0121/2013/87(17)/174427(11) ©2013 American Physical SocietySHINDOU, MATSUMOTO, MURAKAMI, AND OHE PHYSICAL REVIEW B 87, 174427 (2013) FIG. 1. (Color online) Magnonic crystal with chiral edge modes. Periodic array of holes is introduced into YIG, where iron (Fe) is ﬁlledinside every hole. Chiral spin-wave edge modes are propagating along the boundary in a unidirectional way (light purple arrow). where β† k≡[β† 1,k,..., β† N,k] denote spin-wave (boson) cre- ation operators. Describing volume-type modes, the operatorsare already Fourier-transformed in a two-dimensional spacewith the periodic boundary conditions and the wave vector k≡(k x,ky).Nis a number of internal degrees of freedom considered within a unit cell. A 2 Nby 2NHermitian matrix (Hk) stands for a bosonic Bogoliubov–de Gennes (BdG) Hamiltonian, whose explicit form will be derived from alinearized Landau-Lifshitz equation later. With the magneticdipolar interaction, the Hamiltonian thus derived acquires notonlyNbyNnormal parts (particle-hole channel), a kanda∗ −k, but also NbyNanomalous parts (particle-particle channel), bkandb∗ −k, Hk≡/bracketleftbiggak bk b∗ −ka∗ −k/bracketrightbigg . Such a bosonic BdG Hamiltonian is diagonalized in terms of a paraunitary matrix Tkinstead of a unitary matrix,33 T† kHkTk=/bracketleftbiggEk E−k/bracketrightbigg , (2) with [ γ† k,γ−k]T† k=[β† k,β−k].Ekis a diagonal matrix, whose diagonal element gives a dispersion relation for respectivevolume-mode band. The orthogonality and completeness of anew basis ( γﬁeld) are derived as T † kσ3Tk=σ3,Tkσ3T† k=σ3, (3) respectively, where a diagonal matrix σ3takes ±1i nt h e particle/hole space, i.e., [ σ3]jm=δjmσjwithσj=+ 1f o rj= 1,..., N andσj=− 1f o rj=N+1,..., 2N. This additional structure comes from the fact that the magnon obeys the bosonstatistics. Each column vector encoded in the paraunitarymatrix T kstands for the (periodic part of) Bloch wave function for the respective volume-mode band. Provided that a Hermite matrix Hkis unitarily equivalent to a positive-deﬁnite diagonal matrix, a paraunitary matrix Tk which diagonalizes Hkcan be obtained by a method based on the Cholesky decomposition.33In the method, we ﬁrst decompose Hkinto a product between an upper triangle matrix Kkand its Hermite conjugate, Hk=K† kKk. The unitarily positive deﬁniteness of Hkalways allows this decomposition and also guarantees the existence of K−1 k. We next introduce a unitary matrix Ukwhich diagonalizes a Hermite matrixWk≡Kkσ3K† k, U† kWkUk=/bracketleftbiggEk −E−k/bracketrightbigg . Owing to Sylvester’s law of inertia, both EkandE−kcan be made positive-deﬁnite NbyNdiagonal matrices. One can see a posteriori that these two diagonal matrices are nothing but those in the right-hand side of Eq. (2). Namely, the following paraunitary matrix satisﬁes Eq. (3):33 Tk=K−1 kUk/bracketleftBigg E1/2 k E1/2 −k/bracketrightBigg (4) and it diagonalizes the Hamiltonian as HkTk=σ3Tk/bracketleftbiggEk −E−k/bracketrightbigg . (5) The upper NbyNdiagonal matrix in the right-hand side, Ek, is positive deﬁnite, so we will refer to them as (dispersionsfor) “particle bands,” while the lower NbyNdiagonal matrix, −E −k, is negative deﬁnite, whose diagonal elements are thus referred to as (dispersion for) “hole bands.” Due to the trivialredundancy, σ 1H∗ kσ1=H−kwith [ σ1]jm=δ\|j−m\|,N, either one of these two NbyNdiagonal matrices gives the full information of the dispersions for the volume-mode bands. B. Chern integers in bosonic BdG systems To introduce the Chern number for the jth volume-mode band, let us ﬁrst deﬁne a projection operator Pjin the 2 N- dimensional vector space, which ﬁlters out those bands otherthan the jth volume-mode band at each momentum point k, P j≡Tk/Gamma1jσ3T† kσ3. (6) Here /Gamma1jis a diagonal matrix taking +1f o rt h e jth diagonal component and zero otherwise. Equation (3)suggests that the operator obeys/summationtext jPj=1andPjPm=δjmPj. In terms of the projection operator, the Chern number for the jth band is given as follows:34 Cj≡i/epsilon1μν 2π/integraldisplay BZdkTr[(1−Pj)(∂kμPj)(∂kνPj)], (7) where the integral is over the ﬁrst Brillouin zone (BZ) in the two-dimensional kspace. Equation (7)is integer-valued and characterizes a certain global phase structure associated with a Bloch wave functionover the BZ. To see this, we follow the same argument as in thequantum Hall case, 35,36and introduce ﬁeld strength (Berry’s curvature) Bjand gauge connection (gauge ﬁeld) ( Aj,x,Aj,y) for each volume-mode band, Bj(k)≡∂kxAj,y(k)−∂kyAj,x(k), (8) Aj,ν(k)≡iTr[/Gamma1jσ3T† kσ3(∂kνTk)], (9) withj=1,..., 2N. The Chern number for a volume-mode band reduces to an integral of the respective Berry’s curvatureover the BZ, C j=1 2π/integraldisplay BZd2kBj(k). (10) 174427-2TOPOLOGICAL CHIRAL MAGNONIC EDGE MODE IN A ... PHYSICAL REVIEW B 87, 174427 (2013) Such a surface integral is zero, provided that the respective gauge ﬁeld can be deﬁned uniquely and smoothly over theﬁrst BZ. When [ T k]m,jhas a zero somewhere on the BZ for anym(m=1,..., 2N), however, the gauge ﬁeld for the jth band cannot be chosen uniquely over the BZ. In this case,it is necessary to decompose the BZ into two overlappedregions ( H 1andH2withH1∪H2=BZ and H1∩H2= ∂H 1=−∂H 2≡S), so that [ Tk]1,jdoes not have any zero within one region ( H1), while [ Tk]2,jhas no zero inside the other ( H2). In the former region, we then take the gauge, [T(1) k]m,j, such that [ T(1) k]1,jis always real positive, while we take another gauge in the other, making [ T(2) k]2,jto be always real positive. Provided that the jth band considered is isolated from the others ( Ek,j/negationslash=Ek,m/negationslash=jfor any k), these two gauge choices are related to each other by a U(1) transformation, T(2) k/Gamma1j=T(1) k/Gamma1jeiθk, (11) onk∈H1∩H2. Now that the gauge of T(1) k/Gamma1jand the gauge ofT(2) k/Gamma1jare uniquely deﬁned in H1andH2, respectively, A(m) j,ν≡iTr/bracketleftbig /Gamma1jσ3T(m) k†σ3/parenleftbig ∂kνT(m) k/parenrightbig/bracketrightbig (12) (m=1,2) are smooth functions in each of these two regions, respectively. The Stokes theorem is applied separately, so thatEq.(10) is calculated as C j=1 2π/contintegraldisplay Sdk·/parenleftbig A(1) j−A(2) j/parenrightbig =1 2π/contintegraldisplay Sdk·∇kθk,(13) with∇k≡(∂kx,∂ky). Two regions share a boundary ( S), which forms a closed loop. θkin Eq. (11) has a 2 πnphase winding along the loop. This leads to Cj=n(n=Z). C. Sum rule for Chern integer When all volume-mode bands in a system are physically stable, the sum of the Chern integer over all particle bands andthat over all hole bands are zero, respectively, N/summationdisplay j=1Cj=2N/summationdisplay j=N+1Cj=0. (14) To see this, let us linearly interpolate a 2 Nby 2Nspin-wave Hamiltonian and the 2 Nby 2Nunit matrix, Hk,λ=(1−λ)Hk+λ1. (15) We assume that Hkis paraunitarily equivalent to a diagonal matrix whose elements are all positive for any wave vector k; all volume-mode bands obtained from original spin-wave Hamiltonian ( λ=0) are physically stable. Thanks to the Sylvester’s law of inertia, such a Hermite matrix is unitarilyequivalent to a diagonal matrix whose elements are all positivedeﬁnite. Clearly, so is any H k,λduring 0 /lessorequalslantλ/lessorequalslant1. Given the unitarily positive deﬁniteness, we can then introduce fromEq.(4)a paraunitary matrix T k,λwhich transforms Hk,λinto a diagonal form as Hk,λTk,λ=σ3Tk,λ/bracketleftbiggEk,λ −E−k,λ/bracketrightbigg ,with positive-deﬁnite NbyNdiagonal matrices Ek,λand E−k,λ. With Tk,λ, the Chern integer can be explicitly deﬁned as a function of λfor 0/lessorequalslantλ/lessorequalslant1,Cj(λ). The sum of the Chern integer over a group of bands does not change, unless some band in the group forms aband touching (frequency degeneracy) with bands outside thegroup. The positive deﬁniteness of E k,λandE−k,λmeans that particle bands obtained from Hk,λare always in the positive frequency regime, while hole bands are in the negativefrequency regime; during the interpolation (0 /lessorequalslantλ/lessorequalslant1), they are always disconnected from each other in frequency. Thus,the sum of the Chern integer over all particle bands does notchange during the interpolation, N/summationdisplay j=1Cj(λ=0)=N/summationdisplay j=1Cj(λ=1). Since a paraunitary matrix at λ=1 is trivial, Tk,λ=1=1, the right-hand side reduces to zero and so does the case with theoriginal spin-wave Hamiltonian ( λ=0). In summary, Eq. (14) is derived only from the paraunitarily positive deﬁniteness of2Nby 2NHermite matrix H k. As a corollary of Eq. (14), one can argue that any topological chiral edge modes obtainedfrom proper spin-wave approximations appear only at a ﬁnite frequency region (see the Sec. IVfor the argument). In the following, we show that a two-dimensional bicomponentmagnonic crystal (MC) with the dipolar interaction supportsspin-wave bands with nonzero Chern integers. III. MAGNONIC CRYSTAL, DIPOLAR INTERACTION, AND CHERN INTEGER FOR VOLUME-MODE BANDS A. Plane-wave theory for magnonic crystals The MC considered is a ferromagnetic system with its magnetization and exchange interaction modulated periodi-cally in the two-dimensional ( x-y) direction. For simplicity, we assume that the system is translationally symmetric alongthezdirection, whereas the subsequent results are expected to be similar when a thickness of the system in the zdirection becomes ﬁnite (but large). The system is composed of twokinds of ferromagnets: iron and YIG. The unit cell of the MC is ana x×ayrectangle, inside which iron is embedded into circular regions, while the remaining region is ﬁlled with YIG(Fig. 1). A uniform magnetic ﬁeld H 0is applied along the zdirection, such that the static ferromagnetic moment Msin both regions is fully polarized in the longitudinal direction.Propagation of the transverse moments ( m x,my) is described by a linearized Landau-Lifshitz equation,27–30 1 \|γ\|μ0dm± dt=± 2iMs(∇·Q∇)m±∓2im±(∇·Q∇)Ms ∓iH0m±±ih±Ms (16) with ∇≡ (∂x,∂y),m±=mx±imy, and h±=hx±ihy. (hx,hy) stands for the transverse component of the long- ranged magnetic dipolar ﬁeld h, which is related to the ferromagnetic moment m≡(mx,my,Ms) via the Maxwell equation, i.e., ∇×h=c−1∂tezand∇·(h+m)=0. The former two terms in the right-hand side of Eq. (16) come from short-ranged exchange interaction, where Qdenotes a square of the exchange interaction length. MsandQtake 174427-3SHINDOU, MATSUMOTO, MURAKAMI, AND OHE PHYSICAL REVIEW B 87, 174427 (2013) values of iron inside the circular region and those of YIG otherwise. A ﬁlling fraction of the circular region with respectto the total area of the unit cell is represented by f. We further employ magnetostatic approximation, replacing the Maxwellequations by ∇×h=0 and∇·(h+m)=0; h ν=−∂ν/Psi1, /Delta1/Psi1 =∂xmx+∂ymy, (17) withν=x,y. This in combination with Eq. (16) gives a closed equation of motion (EOM) for the transverse moment. In order to obtain band dispersions and Chern integers for volume-mode bands, we need to reduce the EOM into ageneralized eigenvalue problem with a BdG Hamiltonian ( H k) deﬁned in the form of Eq. (1). To this end, we ﬁrst normalize the transverse moment, to introduce a Holstein-Primakov (HP)ﬁeld as 37 β(r)≡m+(r)√2Ms(r),β†(r)≡m−(r)√2Ms(r). (18) In terms of the HP ﬁeld, the Landau-Lifshitz equation is properly symmetrized as dβ dt=4iα(∇·Q∇)αβ−4iβ(∇·Q∇)α2 −iH0β−iα∂+/Psi1, (19) /Delta1/Psi1=∂+(αβ†)+∂−(αβ),∂ ±≡∂x±i∂y, (20) withα(r)≡√Ms(r)/2.\|γ\|μ0was omitted in Eq. (19) for clarity. The static magnetization and exchange interaction arespatially modulated with the lattice periodicity; α(r)=/summationdisplay Gα(G)eiG·r,Q (r)=/summationdisplay GQ(G)eiG·r, with the reciprocal vectors G,α∗(G)=α(−G) andQ∗(G)= Q(−G). It follows from the Bloch theorem that m(r) and/Psi1(r) take a form β(r)=/summationdisplay k/summationdisplay Gβk(G)ei(k+G)·r, β†(r)=/summationdisplay k/summationdisplay Gβ† k(G)e−i(k+G)·r, and /Psi1(r)=/summationdisplay k/summationdisplay G/Psi1k(G)ei(k+G)·r, where the ksummation is taken over the ﬁrst BZ. In terms of these Fourier modes, an equivalent generalized eigenvalue problem with a quadratic Hamiltonian for the HPﬁeld is derived as id dt/bracketleftBigg βk β† −k/bracketrightBigg =/bracketleftBigg/bracketleftBigg βk β† −k/bracketrightBigg ,ˆH/bracketrightBigg =σ3Hk/bracketleftBigg βk β† −k/bracketrightBigg ,(21) with ˆH=/summationdisplay ky>0[β† kβ−k]Hk/bracketleftBigg βk β† −k/bracketrightBigg ,and/bracketleftBigg βk β† −k/bracketrightBigg ≡[..., β k(G),..., β k(−G),...,\| ..., β† −k(−G),...β† −k(G),...]T, [β† kβ−k]≡[..., β† k(G),...β† k(−G),...,\| ..., β −k(−G),...β −k(G),...]. σ3in the right-hand side takes ±1 in the particle/hole space, which comes from the commutation relation of bosons,[β k,β† k]=1and [ β† −k,β−k]=−1. A comparison between Eqs. (19) and(20) and Eq. (21) dictates that Hkthus introduced is given by the following Hermitian matrix ( H† k=Hk): Hk≡/bracketleftbiggα·α+Bk+H01α·Ik·α α·I∗ k·αα ·α+Bk+H01/bracketrightbigg (22) with [α]G,G/prime≡α(G−G/prime),[Ik]G,G/prime≡δG,G/primee−2iθk(G), (23) eiθk(G)≡(k+G)x+i(k+G)y \|k+G\|, and [Bk]G,G/prime≡4/summationdisplay G1,G2α(G−G1)Q(G1−G2)α(G2−G/prime) ×(k+G1)·(k+G2) −4/summationdisplay G1,G2Q(G1)α(G2)α(G−G/prime−G1−G2) ×(G−G/prime)·(G−G/prime−G1). (24) After taking the summation over G1andG2in Eq. (24), one can further decompose [ Bk]G,G/primeinto three parts, [Bk]G,G/prime=4/summationdisplay μ=x,y[Qα2]G−G/prime(k+G)μ(k+G/prime)μ +4i/summationdisplay μ=x,y[Qα(∂μα)]G−G/prime(G−G/prime)μ +4/summationdisplay μ=x,y[Q(∂μα)(∂μα)]G−G/prime (25) with [Qα2]G≡1 S/integraldisplay Q(r)α2(r)e−iG·rd2r, [Qα(∂μα)]G≡1 S/integraldisplay Q(r)α(r)[∂μα(r)]e−iG·rd2r,(26) [Q(∂μα)(∂μα)]G≡1 S/integraldisplay Q(r)[∂μα(r)][∂μα(r)]e−iG·rd2r, where the two-dimensional (2D) integrals in the right-hand side are taken over the MC unit cell and Sdenotes an area of the cell ( S≡axay). In actual numerical calculation, the dimension of Hkis typically taken to be 512 ×512, where the reciprocal-vector Granges over [ −16π/ax,16π/ax]× [−16π/ay,16π/ay]. The MC considered is composed of two kinds of ferro- magnets, where the respective (square root of) magnetizationand (square of) exchange interaction length are speciﬁed by 174427-4TOPOLOGICAL CHIRAL MAGNONIC EDGE MODE IN A ... PHYSICAL REVIEW B 87, 174427 (2013) (αj,Qj)(j=1,2). Within the rectangular-shaped unit cell (ax×ay), one of these two ferromagnets ( α1,Q1) is embedded within the circular region, while the remaining region isﬁlled with the other ( α 2,Q2). Ifα(r) has a discontinuity at the boundary between these two regions, the last termin Eq. (25),[Q(∂ μα)(∂μα)], diverges, since it contains the second derivative with respect to a spatial coordinate alongthe radial direction. Physically, such an infrared divergence isremoved by a smooth variation of the saturation magnetizationat the boundary. For simplicity, we interpolate α(r) as a linear function of the radial coordinate measured from the center ofthe circular region, α(r)=⎧ ⎪⎨ ⎪⎩α 1 (\|r\|<R 0), α1−\|r\|−R0 R1−R0(α1−α2)(R0<\|r\|<R 1), α2 (R1<\|r\|).(27) A discontinuity in Q(r)i sa l s or e m o v e db yt h es a m e linear interpolation. Actual numerics are carried out with(r 0,r1)≡(R0/λ,R 1/λ)=(0.10,0.125), where λdenotes the linear dimension of the unit-cell size, λ≡√axay.A sf o rt h e material parameter, we used ( Ms,1,Ms,2)=(1.8,0.19)(A/μm) and (√Q1,√Q2)=(33,130) ( ˚A) with αj≡/radicalbigMs,j, while parameters of iron (Fe), cobalt (Co), and YIG are[M s(A/μm),√Q(˚A)]=(1.7,21),(1.4,29), and (0 .14,184), respectively. Under Tk, the EOM is paraunitarily equivalent to id dt/bracketleftBigg γk γ† −k/bracketrightBigg =/bracketleftbiggEk −E−k/bracketrightbigg/bracketleftBigg γk γ† −k/bracketrightBigg . [Ek]jgives a dispersion relation for the jth volume-mode band (j=1,...), while the Chern integer is calculated from Tkvia Eqs. (8)–(10). In its numerical evaluation, we employed an algorithm based on a “manifestly gauge-invariant” descriptionof the Chern integer. 38 B. Role of dipolar interaction When a system considered is either time-reversal sym- metric; H∗ −k=Hk, or mirror-symmetric with a mirror plane perpendicular to the xy plane, e.g., H(kx,ky)= H(kx,−ky), the Berry’s curvature satisﬁes −Bj(k)=Bj(−k) or−Bj(kx,ky)=Bj(kx,−ky), respectively, which reduces the Chern integer to zero. In the present situation, however,the magnetic dipolar ﬁeld brings about complex-valued phasefactors in the anomalous part, I k/negationslash=I∗ k, which removes from Eq. (22) both the time-reversal symmetry and the mirror symmetries. Without periodic modulation of the saturationmagnetization, α=α1, these phase factors can be erased by a proper gauge transformation, β† k→β† k/radicalbig I∗ kandβ−k→ β−k√Ik, so that both symmetries are recovered. In the presence of the periodic modulations, α/negationslash=α1, however, these two symmetries are generally absent in the 2D MCs and thebosonic Chern integer can take a nonzero integer value. This situation is quite analogous to what the relativistic spin-orbit interaction does in ferromagnetic metals. 39,40More- over, contrary to the spin-orbit interaction, a strength of thedipolar interaction is an experimentally tunable parameterin MCs. 22When characteristic length scale of MC (linear dimension of the unit-cell size λ≡√axay) becomes largerthan the typical exchange length√Q, the dipolar interaction is expected to prevail over the exchange interaction. C. Chern integer of volume-mode bands and role of band touchings In fact, we found that the Chern integer of the lowest magnonic band, C1, is always quantized to be 2 for the longer λ, while the integer reduces to zero for the shorter λ[Fig. 2(a)]. The respective quantization is protected by a ﬁnite direct band gap between the lowest band and the secondlowest band [Fig. 2(b)]. In the intermediate regime of λ, these two bands get closer to each other. With a fourfold rotationalsymmetry ( r≡a y/ax=1), the gap closes at the two Xpoints at a critical value of λ(∼0.28μm), where the two bands form gapless Dirac spectra [Fig. 2(c)]. Without the fourfold symmetry ( r/negationslash=1), the band touching at one of the two X points and that of the other occur at different values of λ. These band touchings are denoted as P1(π,0,λc,1) andP2(0,π,λ c,2)i n FIG. 2. (Color online) Chern-integer phase diagram and band dispersions with f=π×10−2. (a) Chern-integer phase diagram. The phases are distinguished by the Chern integer of the lowestmagnonic band, C 1.rstands for the aspect ratio of the unit-cell shape (r≡ay/ax). (b) Band dispersions of the lowest three volume-mode bands with r=1, and λ=0.35μm. A band gap appears between the ﬁrst and the second lowest band. (c) The band gap collapses at λ=0.28μm(r=1), where the lowest and second lowest volume mode form Dirac cones at the two inequivalent Xpoints. 174427-5SHINDOU, MATSUMOTO, MURAKAMI, AND OHE PHYSICAL REVIEW B 87, 174427 (2013) FIG. 3. (Color online) Band-touching points (dual magnetic charges) in a three-dimensional parameter space subtended by the wave vector ( kx,ky) and the unit-cell size ( λ≡√axay)f o rr> 1. Small green spheres denote the band-touching points, which emit the dual magnetic ﬁeld (blue arrows). a three-dimensional parameter space subtended by two wave vectors kx,kyand the unit-cell size λ(Fig. 3). The band touchings endow the lowest volume-mode band with nonzero Chern integers in the longer λregion. In generalized eigenvalue problems as well as usual eigenvalueproblems, 35,36,41,42a band-touching point in the 3D parameter space plays role of a dual magnetic monopole (charge).The corresponding dual magnetic ﬁeld is generalized fromEq. (8)as a rotation of the three-component gauge ﬁeld A j=(Aj,x,Aj,y,Aj,λ), Bj=∇×Aj (28) with∇≡(∂kx,∂ky,∂λ). Here the third component of the gauge ﬁeldAj,λis introduced in the same way as in (9): Aj,λ≡iTr[/Gamma1jσ3T† kσ3(∂λTk)]. (29) jspeciﬁes either one of the two magnonic bands which form the band touching. At the band-touching point, the dualmagnetic ﬁeld for the respective bands has a dual magneticcharge, whose strength is quantized to be 2 πtimes integer (see the Appendix). A numerical evaluation tells us that thedual magnetic charges for the lowest band at the band-touchingpoints at P 1andP2are both +2π(Fig. 3), ∇·B1=2πδ(λ−λc,1)δ(kx−π)δ(ky) +2πδ(λ−λc,2)δ(kx)δ(ky−π), where λc,1<λc,2foray>ax(r> 1) and λc,1>λc,2foray< ax(r< 1). Because the Chern integer can be regarded as the total dual magnetic ﬂux penetrating through the constant λplane [see Eq. (10)], the Gauss theorem suggests that, when λgoes across either the λ=λc,1plane or the λ=λc,2plane, the Chern integer for the lowest magnonic band always changesby unit, e.g., C 1\|λ>λ c,1−C1\|λc,1>λ=1. Without the fourfold rotational symmetry ( r/negationslash=1), the two critical values of λbound three phases for the lowest magnonicband, the phase with C1=0 (phase IV), that with C1=1 (phase III or I), and that with C1=2 (phase II). In the presence of the fourfold rotational symmetry ( r=1), two band touchings occur at the same critical value of λ, where C1 increases by 2 on increasing λ. This leads to a phase diagram shown in Fig. 2(a), which describes the Chern integer of the lowest magnonic band as a function of the unit-cell size λand the aspect ratio r. A dual magnetic charge is a quantized topological object, so that, upon any small change of parameters, it cannot disappearby itself. Instead, it only moves around in the 3D parameterspace. As a result, the global structure of the phase diagramdepicted in Fig. 2(a) widely holds true for other combinations of material parameters. For r=1 and f=π×10 −2,w e found λc=0.370μm for iron (circular region) and YIG (host), and λc=0.372μm for cobalt (circular region) and YIG (host). When varying the ﬁlling fraction for iron (circularregion) and YIG (host) with r=1, we found λ c=0.274μm forf=4π×10−2, andλc=0.348μmf o rf=9π×10−2. I V . C H I R A LS P I N - W A V EE D G EM O D EI NM C The chiral phases with nonzero Chern integers have chiral spin-wave edge modes, which are localized at a boundary withthe phase with zero Chern integer (phase IV) or the vacuum.The edge modes have chiral dispersions which go across theband gap between the lowest and the second lowest band. As an illustrative example, we consider a boundary ( yaxis) between the MC in phase III ( C 1=1) and the MC in the phase IV ( C1=0). The existence of a chiral spin-wave edge mode at the boundary is shown from a following 2 ×2D i r a c Hamiltonian derived near their phase boundary λ=λc,1(see the Appendix), Heff=ω0τ0+κ(x)τ3−ia∂xτ1−ib∂yτ2. (30) τjdenotes the Pauli matrices subtended by the twofold degenerate eigenstates at P1.aandbare positive material parameters. The difference of the Chern integers ( C1)f o rt h e two phases is represented as a change of sign of a Dirac masstermκ(x):κ(x)>0f o rx> 0 (phase III) and and κ(x)<0 forx< 0 (phase IV) [see Fig. 4(a) ]; lim x→±∞κ(x)=±κ∞. The Hamiltonian has a following eigenstate:43,44ψk(r)∝ eikye−(1/a)/integraltextxκ(x/prime)dx/prime[1,i]t, which is localized at the boundary (x=0). In terms of the surface wave vector kalong the yaxis, the corresponding eigenfrequency is given by E= ω0+bk. This connects the lowest magnonic band lying at E/lessorequalslantω0−κ∞and the second lowest band at E/greaterorequalslantω0+κ∞ [see Figs. 4(b) and4(c)]. The mode is chiral, since the group velocity is always positive, vk≡∂kE=b. Similarly, we can easily show that the phase with C1=1a tr> 1 (phase III) or r< 1 (phase I) has a chiral edge mode at its boundary with vacuum, whose dispersion crosses the direct band gap at the(π,0) or (0 ,π) point, respectively, while the phase with C 1=2 (phase II) has both at its boundary with vacuum. A number ofchiral modes localized at the interface between two MCs withdifferent Chern integers is equal to the difference of the twoChern integers. Generalizing the arguments so far into the linearly inter- polated Hamiltonian deﬁned in Eq. (15), we can argue that, in general, a number of those chiral spin-wave edge modes 174427-6TOPOLOGICAL CHIRAL MAGNONIC EDGE MODE IN A ... PHYSICAL REVIEW B 87, 174427 (2013) FIG. 4. (Color online) Chiral spin-wave edge modes. (a) Geometry of the system. (b),(c) Wave-vector frequency dispersions for Dirac Hamiltonian with the P ¨oschl-Teller potential (Ref. 15)κ(x)=κ∞tanh(x/d)f o rκ∞d=0.9a(b), and κ∞d=2.9a (c). The volume-mode magnonic bands have a band gap, inside which an edge mode has a chiral dispersion. In (c) there are somenonchiral edge states, whereas in (b) there is not. Nonetheless, the number of chiral edge modes is one, which is determined solely from the difference between the Chern integers for the two phases. whose dispersions run across a gap between the mth and the (m+1)th particle bands, Nm, is equal to the sum of the Chern integer over those particle bands below the gap, Nm≡m/summationdisplay j=1Cj. (31) Here clockwise (counterclockwise) chiral edge modes con- tribute by +1(−1) to the number of chiral edge modes, Nm [see Fig. 4(a)]. Namely, the sum rule, Eq. (14), suggests that the right-hand side of Eq. (31) counts the total number of the band touchings (including the sign of the respective dualmagnetic charges) which happen between the mth particle band and ( m+1)th particle band during the interpolation, λ=1→λ=0. On the one end, each band touching (dual magnetic monopole) is accompanied by the emergence of achiral edge mode between these two bands, whose sense ofrotation is either clockwise or counterclockwise, depending onthe sign of the respective dual magnetic charge. Since there areno chiral edge modes in the trivial limit ( λ=1), we can safely conclude Eq. (31) atλ=0. As a corollary of Eq. (31), one can also see that any topological chiral edge modes obtained fromlegitimate spin-wave approximation appear only at a ﬁnite frequency region, never a gapless mode. Chiral edge modes proposed in this paper share similar physical properties with the well-known Damon-Eshbach(DE) surface mode, 19instead of backward volume modes studied so far in actual magnonic cyrstals.45–48The topological modes as well as the DE surface mode are propagating in achiral way along boundaries (surfaces) of the systems, wherethe propagation directions are parallel (or antiparallel) tovector products between the polarized ferromagnetic momentand the normal vectors associated with surfaces. Experimentaltechniques for measuring the DE surface mode can be alsoutilized for detecting the proposed topological chiral edgemodes. Possible experiments include Brillouin light-scattering(BLS) measurements, 49–51time-resolved Kerr microscopy,52 infrared thermography,53and scanning local magnetic ﬁelds in terms of a wire loop or antenna.54 The topological edge mode always has a chiral dispersion within a band gap for volume-mode bands. When a radiofrequency (rf) of applied microwave is chosen inside theband gap, spin waves are excited only along these chiraledge modes, while other volume modes remain intact. Onecan test this situation by changing the position of an inputantenna from the boundaries to an interior far from theboundaries. Being protected by the topological Chern integersdeﬁned for volume modes, the proposed chiral edge modes areexpected to be robust against various perturbations introducednear the boundaries. The robustness can be also tested bychanging boundary shape or introducing boundary roughnessand obstacle. Contrary to the DE mode, Eq. (31) dictates that the number and the sense of rotation of the topological chiral edge modesare determined by the Chern integer for volume modes belowthe band gap. In fact, the chiral edge mode in the proposedmagnonic crystal rotates along the boundary in the clockwise manner with an up-headed magnetic ﬁeld (Fig. 1), while the DE surface mode with the same geometry rotates in thecounterclockwise way with the up-headed ﬁeld. 18,19Moreover, the Chern integer for a volume-mode band itself can bechanged by closing the band gap, as was shown in the previoussection. Thus, using band-gap manipulation, one can evencontrol the chiral direction 55or the number of the mode, which enable intriguing spintronic devices such as a spin-currentsplitter (see also Sec. VI). To our best knowledge, the DE mode in a uniform thin ﬁlm does not have such properties. V . MICROMAGNETIC SIMULATION FOR CHIRAL TOPOLOGICAL EDGE MODES A. Simulation procedure and material parameters To justify the existence of topological chiral spin-wave edge modes in the present MC model, we have numericallysimulated the Landau-Lifshitz-Gilbert equation, dm dt=−γ\|μ0\|m×Heff+α Msm×dm dt, (32) where αis set to the Gilbert damping coefﬁcient of YIG; α=6.7×10−5.56The magnetic ﬁeld Heffincludes a short- ranged exchange ﬁeld, a long-ranged dipolar ﬁeld h, the static 174427-7SHINDOU, MATSUMOTO, MURAKAMI, AND OHE PHYSICAL REVIEW B 87, 174427 (2013) longitudinal external ﬁeld H0=0.1 T, and a small temporally alternating transverse ﬁeld ( H1excosωt) with H1=1.0O e . A simulated magnonic crystal (MC) is as large as either 28 μm ×28μmo r7 μm×7μmi nt h e x-yplane, which is composed of 80 ×80 or 20 ×20 MC unit cells, respectively; each unit-cell size is 350 nm ×350 nm within the plane. A MC unit cell consists of an YIG region and a Fe region, wherethe size of the Fe region is 140 nm ×140 nm ( f=0.16). The MC studied in the previous sections is translationallysymmetric along the zdirection. To mimic this situation in a micromagnetic simulation, we take the thickness along the z direction L zto be sufﬁciently large ( Lz=1 mm). In actual simulation, each MC unit cell is further discretized into a bunch of smaller elements,57each of which is taken in this study as large as 70 nm ×70 nm ×Lzand each element is assigned with one ferromagnetic spin, which isuniformly distributed over the element. Namely, every MCunit cell has 25 spins consisting of 4 Fe spins and 21YIG spins. The short-ranged exchange stiffness between Feelements is taken to be A Fe=2.1×10−11J/m, between YIG elements AYIG=0.437×10−11J/m, and between Fe and YIG elements AFe−YIG=1.0×10−11J/m. These set material parameters to be those in the dipolar regime with C1=2 (H0=0.1T,λ=0.35μm,r=1, and f=0.16). The superelongated cell size (70 nm ×70 nm ×Lzwith Lz=1 mm) used in this simulation clearly ignores those spin-wave modes which have nodes along the zdirection. In the presence of very large thickness along the zdirection (e.g., 28 μm×28μm×1 mm), however, it is likely that the strong magnetic shape anisotropy pushes up such spin-wavemodes into higher frequency regimes; spin-wave excitationsin lower-frequency regime, which we focus on in the presentstudy, are mainly dominated by those modes having nonode along the zdirection. Even for spin-wave modes without nodes along the zdirection, their wave functions should be certainly modiﬁed near the boundaries along the direction. When thethickness along the direction is much larger than the lineardimension within the other two directions ( xyplane), however, the modiﬁcations of the wave functions are also expected tobe small. The time evolution is determined by Eq. (32), which is numerically integrated with a time interval of 1 ps by use of thefourth-order Runge-Kutta method. The demagnetization ﬁeld his calculated by the convolution of a kernel which describes the dipole-dipole interaction. With the Gilbert damping term,the system eventually reaches a certain steady state, in whichonly the spin-wave modes around the external frequency ωare excited permanently. In order to deduce spatial distribution ofspin-wave modes at a given frequency, we have studied steadystates ( ωt/greatermuch1) with changing the external frequency. To compare simulation results with dispersions for the volume-mode bands obtained from the plane-wave theory, wealso take a Fourier transformation of transverse moments overspace and time in a steady state. Speciﬁcally, we performa discrete Fourier transformation of m +≡mx+imywith respect to space and time coordinate as m+(kx,ky.ω)≡/summationdisplay X,Y/summationdisplay jm+(X,Y,j/Delta1T )eikxX+ikyY−iωj/Delta1T. FIG. 5. (Color online) A contour plot of \|m+(kx,ky,ω)\|as a function of kx,ky,a n d ω, which has stronger amplitude in blue- colored regions and smaller amplitude in white regions. The contour plot signiﬁes dispersion relations for volume-mode bands [comparewith Fig. 2(b)]. The wave vector ( k x,ky) is along (0 ,0) (/Gamma1point), ( π,0) (Xpoint), ( π,π)(Mpoint), and (0 ,0) (/Gamma1point). In this calculation, the system size is taken to be 7 μm×7μm. In this system size, we observe chiral edge modes similar to those in Figs. 6(c)–6(f) within 27 GHz <ω< 30 GHz. A contour plot of (absolute value of) the left-hand side as a function of kx,ky, andωis expected to give dispersion relations for spin-wave volume-mode bands. B. Result Throughout the micromagnetic simulation, we found that a lower frequency region can be roughly classiﬁed into threecharacteristic regimes: two volume-mode frequency regimes,(i)ω 0<ω<ω 1and (iii) ω2<ω, and a band-gap regime for volume modes, (ii) ω1<ω<ω 2.B e l o w ω0, the system remains intact against the small alternating transverse ﬁeld, in-dicating no spin-wave modes for ω<ω 0. Within the volume- mode frequency regimes (i) ω0<ω<ω 1[Fig. 6(a)] and (iii)ω2<ω [Fig. 6(b)], spin-wave excitations in steady states are always distributed over the entire system. A comparisonbetween Fig. 2(b) and the contour plot of the Fourier transform of the transverse moments (Fig. 5) indicates that these two frequency regimes correspond to the lowest volume-modeband and higher volume-mode bands, respectively. On the one hand, a steady state in the intermediate frequency regime, ω 1<ω<ω 2, has almost no weight for volume modes. Instead, spin-wave excitations in the intermediateregime are localized only around the boundaries of thesystem [see Fig. 6(c)], indicating the existence of spin-wave edge modes. Moreover, these edge modes propagate in achiral way [see Figs. 6(d)–6(f)], whose direction is consistent with a chiral direction determined from the Chern integerin the dipolar regime found in Fig. 2(a),C 1=+ 2>0. These observations indicate that the spin-wave edge modesfound in the intermediate frequency region is nothing butthe topological chiral spin-wave edge modes described inthe previous section. In fact, the three frequency regimesare qualitatively consistent with the plane-wave-theory 174427-8TOPOLOGICAL CHIRAL MAGNONIC EDGE MODE IN A ... PHYSICAL REVIEW B 87, 174427 (2013) FIG. 6. (Color online) Snapshots of spatial distribution of spin- wave excitations in steady states. In this calculation, the system sizeis taken to be 28 μm×28μm. A normalized transverse component of the ferromagnetic spin, m x≡nx/nmax, is plotted, where nmax denotes the maximum value of√ n2 x+n2 ywith (nx,ny,nz)≡m/M s. The static magnetic ﬁeld is taken along the +zdirection. (a) Snapshot ofmxatt=100 ns with the external frequency ω0<ω<ω 1(ω= 25.6 GHz). (b) Snapshot at t=100 ns with ω2<ω (ω=27.8 GHz). (c) Snapshot at t=100 ns with ω1<ω<ω 2(ω=26.8 GHz). (d)–(f) Spin-wave edge modes at ω=26.8 GHz are propagating in a chiral way. (d) t=100 ns, (e) t=100.02 ns, (f) t=100.04 ns. calculation [Fig. 2(b)]; (ω0,ω1,ω2)=(25.5,26.0,27.5 GHz) for 28 μm×28μm×1 mm, and (26.2, 27.1, 30.0 GHz) for 7μm×7μm×1 mm. Our micromagnetic simulation with a shorter sample thickness ( Lz=200μm) also justiﬁes the existence of topological chiral modes. VI. DISCUSSION By calculating a newly introduced bosonic Chern integer for spin-wave bands, we argue that a two-dimensional normallymagnetized magnonic crystal acquires chiral edge modes formagnetostatic wave in the dipolar regime. Each mode islocalized at the boundary of the system, carrying magneticenergies in a unidirectional way. Thanks to the topologicalprotection, spin-wave propagations along these edge modesare robust against imperfections of the lattice periodicityand boundary roughness; they are free from any types ofelastic backward scatterings with moderate strength. 58This robustness makes it possible to implement novel fault-tolerantmagnonic devices such as a spin-wave current splitter and amagnonic Fabry-Perot interferometer as discussed below. The chiral spin-wave edge modes studied in this paper can be easily twisted or split by changes of the size ( λ) and shape (r) of the unit cell, which we demonstrate in Figs. 7(a) and FIG. 7. (Color online) Examples of magnonic circuit made by chiral magnonic edge modes. (a),(b) Schematic pictures of spin-current splitter. (c) Magnonic analog of the Fabry-Perot interferometer. 7(b).I nF i g . 7(a), the MC in phase II ( r=1) is connected with the other MC in phase III ( r> 1), whose Chern integer for the lowest band differs by unit. A boundary between thesetwo MC systems supports the chiral spin-wave edge modewhich runs across the direct band gap at the (0 ,π) point. This means that the two chiral edge modes propagating along theboundary of the the MC in phase II ( r=1) are spatially divided into two, where one mode goes along the boundary of theMC in the phase III ( r> 1), while the other goes along the boundary between these two MCs. This conﬁguration realizesa spin-wave current splitter, an alternative to those proposedin other geometries. 59 The Fabry-Perot interferometer is made up of a couple of chiral spin-wave edge modes encompassing a singletopological MC [see Fig. 7(c)] .P a r t so ft h eM C sa r es p a t i a l l y constricted by the hole inside, so as to play a role of thepoint contact between these edge modes. 60A unidirectional spin-wave propagation is induced in a chiral mode via eitheran antenna attached to the boundary [“input” in Fig. 7(c)]o r a microwave-spin-wave transducer put near the boundary. 52,61 The wave is divided into two chiral edge modes at a point 174427-9SHINDOU, MATSUMOTO, MURAKAMI, AND OHE PHYSICAL REVIEW B 87, 174427 (2013) contact [PC1 in Fig. 7(c)]. Two chiral propagations merge into a single chiral propagation at the other point contact(PC2). Depending on a phase difference between these two, thesuperposed wave exhibits either a destructive or a constructiveinterference, which is detected as an electric signal fromthe other antenna (“output”). Note that local application ofmagnetic ﬁelds [PS1 or PS2 in Fig. 7(c)] changes the velocities of the two chiral edge modes locally. These modiﬁcationsresult in phase shifts of the two chiral waves, which thuschanges the interference pattern observed in the output signal.With the use of these local magnetic ﬁelds as an externalcontrol, 20,62the interferometer can serve as a solid-state based magnonic logic gate. A combination with a recently proposedresonant microwave-to-spin-wave transducer 61would also expand further prospects of spintronic applications of thesespin-wave devices. Though chiral edge modes are robust against static pertur- bations, magnetic energies excited in the edge mode decayinto either phonon states or other magnon states in the volumemodes via inelastic scatterings. The associated decay time orcoherence length depends on speciﬁc materials, and spin-wavepropagation along the chiral edge mode survives only withinthis coherence length. Due to the absence of conductionelectrons, however, spin waves in magnetic insulators havelong coherence lengths; the coherence length in YIG in themagnetostatic regime becomes on the order of centimeter. 21 In such magnetic insulators, the characteristic spin-wavepropagations depicted in Figs. 7(a)–7(c) are experimentally realizable especially in sizable MC systems. Experimentalmeasurement of these propagations in the space- and time-resolved manner is by itself remarkable, which surely leads tothe development of innovative spintronic devices in the future. Note added . After submission of an original version of the present paper into the preprint server, 63we found another submission,64which also theoretically explored the realization of a similar topological chiral magnonic edge mode in alocalized spin system. Their edge modes come from the short-ranged Dzyaloshinskii-Moriya exchange interaction instead ofthe (more classical) magnetic dipole-dipole interaction studiedhere. ACKNOWLEDGMENTS We would like to thank M. Hashisaka for discussions. R.S. also thanks T. Momoi for informing him of Ref. 33. This work was supported in part by Grant-in-Aids from the Ministry ofEducation, Culture, Sports, Science and Technology of Japan(Grants No. 21000004, No. 22540327, No. 23740284, andNo. 24740225) and by a Grant for Basic Scientiﬁc ResearchProjects from the Sumitomo Foundation. APPENDIX: MAGNETIC MONOPOLE AND DIRAC HAMILTONIAN When two bosonic (magnonic) bands form a band-touching point in the three-dimensional p-parameter space with p≡ (kx,ky,λ), the dual magnetic ﬁelds for these two bands [Eqs. (9),(28), and (29)] have a quantized source of their divergence at the point ( p=pc). Away from the band- touching point, the dual gauge ﬁelds [Eqs. (9)and(29)] can belocally determined, so that the dual magnetic ﬁelds [Eq. (28)] are clearly divergence free. At p=pcthe projection to each of these two bands cannot be deﬁned, which endows therespective dual magnetic ﬁeld with some singular structure.The singular structure can be studied by the degenerateperturbation theory for a generalized eigenvalue problem. Theeigenvalue problem takes a form H pTp=σ3Tp/bracketleftbiggEp −Ep/bracketrightbigg , with p≡(kx,ky,λ) and p≡(−kx,−ky,λ). The diagonal matrix σ3takes +1 for the particle space while it takes −1 in the hole space. Hpis a quadratic form of the boson Hamiltonian introduced in Sec. I.Epis a diagonal matrix, whose elements give dispersions for bosonic (magnonic) bandsand are physically all positive deﬁnite. We decompose this intothe zeroth-order part and the perturbation part, H p=H0+Vp (A1) with H0≡Hp=pcandVp≡Hp−H0. Suppose that H0 has twofold degenerate eigenstates tj(j=1,2) with its eigenfrequency ω0(>0), H0tj=σ3tjω0, where the states are normalized as t† jσ3tm=δjm.O ni n t r o - ducing the perturbation Vp, the degeneracy is split into two frequency levels. The eigenstate for the respective eigenfre-quency is determined on the zeroth order of p−p cas Tp=T0Up+O(\|p−pc\|), (A2) where T0diagonalizes H0withT† 0σ3T0=T0σ3T† 0=σ3and a unitary matrix Updiagonalizes a 2 by 2 Hamiltonian Veff formed by the twofold degenerate eigenstates, Veff≡/bracketleftBigg t† 1Vpt1t†1Vpt2 t2Vpt1t†2Vpt2/bracketrightBigg . (A3) By substituting Eq. (A2) into Eqs. (9),(28), and (29), one can easily see that near p=pc, the dual magnetic ﬁeld is given only by the unitary matrix; in the leading order of p−pc,i t is given as Bj=∇×Aj+O(\|p−pc\|−1),Aj=iTr[/Gamma1jU† p∇Up], with∇≡(∂kx,∂ky,∂λ). Now that Eq. (A3) reduces to a usual 2 by 2 Dirac-type Hamiltonian, we can show the quantizationof the dual magnetic charge at the band-touching point exactlyin the same way as in the 2 by 2 Dirac fermion system. 41,42 Thereby, the sign and the strength of the magnetic charge is determined only by the 2 by 2 effective Dirac Hamiltonian.With a proper gauge transformation and scale transformation,the effective Hamiltonian at the band-touching points P j(j= 1,2) takes a form Heff=ω0τ0+(λ−λc,j)τ3+apxτ1+bpyτ2, witha> 0,b> 0, (px,py)≡(kx−π,ky)f o r j=1 and (px,py)≡(kx,ky−π)f o rj=2. Equation (30) is derived by the replacement of pμ→−i∂μ. 174427-10TOPOLOGICAL CHIRAL MAGNONIC EDGE MODE IN A ... 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PhysRevB.95.024417.pdf	PHYSICAL REVIEW B 95, 024417 (2017) Energy repartition in the nonequilibrium steady state Peng Yan,1Gerrit E. W. Bauer,2,3and Huaiwu Zhang1 1School of Microelectronics and Solid-State Electronics and State Key Laboratory of Electronic Thin Film and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China 2Institute for Materials Research and WPI-AIMR, Tohoku University, Sendai 980-8577, Japan 3Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands (Received 6 December 2016; revised manuscript received 26 December 2016; published 17 January 2017) The concept of temperature in nonequilibrium thermodynamics is an outstanding theoretical issue. We propose an energy repartition principle that leads to a spectral (mode-dependent) temperature in steady-statenonequilibrium systems. The general concepts are illustrated by analytic solutions of the classical Heisenbergspin chain connected to Langevin heat reservoirs with arbitrary temperature proﬁles. Gradients of externalmagnetic ﬁelds are shown to localize spin waves in a Wannier-Zeemann fashion, while magnon interactionsrenormalize the spectral temperature. Our generic results are applicable to other thermodynamic systems such asNewtonian liquids, elastic solids, and Josephson junctions. DOI: 10.1103/PhysRevB.95.024417 I. INTRODUCTION Equilibrium thermodynamics provides a uniﬁed description of the macroscopic properties of matter and its response toweak stimuli without referring to microscopic mechanisms.Statistical mechanics, on the other hand, proceeds fromquantum mechanics and describes macroscopic observablesin terms of probabilities and averages. The combination ofboth ﬁelds leads to an understanding of many physical andchemical phenomena from ﬁrst principles. Temperature is aprincipal quantity in the study of equilibrium physics. Energyequipartition in classical equilibrium thermodynamics impliesthat every quadratic normal mode [ 1] carries on average an energy k BT/2 (quantum statistics can be disregarded when mode energies are small compared to kBT)[2]. Here, kBis the Boltzmann constant and Tis the absolute temperature. The system temperature of a given system can be obtainedby, e.g., the kinetic approach [ 1], the entropy method [ 3], and dynamical systems theory [ 4]. In recent years, the physics of nonequilibrium systems has attracted attention from widely different disciplines, such asstochastic thermodynamics [ 5], many-body localizations [ 6], and spin caloritronics [ 7]. One outstanding issue is the concept and proper deﬁnition of the temperature of a nonequilibriumsystem. Most common is the local thermal equilibriumapproximation, assuming that spatially separated componentsof a system thermalize with their immediate surroundings,while the global state of the system is out of equilibrium.The spatially distributed local temperature forms a spatialﬁeld that gives a good impression of the nonequilibriumdynamics of the full system. This approach, however, oftenleads to contradictions: the kinetic temperature has beenfound to differ from the entropic temperature [ 8]. This is no issue in equilibrium systems, in which the temperatureis constant and all modes in momentum space share the sametemperature. Recently, the (equilibrium) thermodynamic entropy has been identiﬁed as a Noether invariant associated with an in-ﬁnitesimal nonuniform time translation [ 9]. In nonequilibrium systems, however, the translational symmetry is broken, so theentropy appears to be not well deﬁned either.In this work, we propose the principle of energy repartition in nonequilibrium systems. It provides partial answers to thesefundamental questions by enabling us to deﬁne a spectral(mode-dependent) temperature [ 10]. We illustrate the principle for magnons in a classical Heisenberg spin chain connected toLangevin heat reservoirs with arbitrary temperature proﬁles.We analytically solve the non-Markovian Landau-Lifshitz-Miyazaki-Seki (LLMS) equation [ 11][ E q .( 1) below], and ﬁnd that the steady-state nonequilibrium properties are governedby a set of normal-mode temperatures that depend on thebath temperature proﬁle, the boundary conditions, and theratio between the ﬁeld gradient and the exchange couplingbetween spins. We show that gradients of external magneticﬁelds localize spin waves in the Wannier-Zeeman fashion,while weak many-body interactions (nonlinearities) lead to amode-temperature renormalization. The LLMS equation en-compasses allstandard equations for classical spin dynamics, reducing to the (stochastic) Landau-Lifshitz-Gilbert (LLG)equation [ 12–15] and the Bloch equation [ 16] in respective limits. Our generic results should be widely applicable todescribe the semiclassical dynamics of other thermodynamicsystems such as Newtonian liquids, elastic solids, and Joseph-son junctions. This paper is organized as follows. In Sec. II, the theoretical model is presented. Section IIIgives the results and discus- sions: we derive the the analytical solution of non-Markovianspin waves and propose the principle of energy repartitionin Sec. III A ; the temperature and chemical potential of nonequilibrium magnons are calculated in Sec. III B ;s p i n pumping and spin Seebeck effects are analyzed in Sec. III C ; Wannier-Zeeman localization due to inhomogeneous magneticﬁelds and its effect on magnon transport are predicted inSec. III D ; magnon-magnon interactions are perturbatively treated in Sec. III E . Section IVis the summary. II. MODEL We consider a classical monatomic spin chain along the x direction, consisting of N+1 local magnetic moments /vectorSn= S/vectorsn,where the unit vector /vectorsnis the local spin direction, S the total spin per site, and n=0,1,..., N . Each spin is in 2469-9950/2017/95(2)/024417(17) 024417-1 ©2017 American Physical SocietyPENG Y AN, GERRIT E. W. BAUER, AND HUAIWU ZHANG PHYSICAL REVIEW B 95, 024417 (2017) FIG. 1. Schematic of a monatomic spin chain consisting of N+1 local magnetic moments /vectorsncoupled with external Langevin bath at temperature Tn, respectively, with n=0,1,..., N. contact with a local Langevin bath at temperature Tn,a ss h o w n in Fig. 1. Long wave-length excitations of complex magnets such as yttrium iron garnet (YIG) can be treated by such amodel by coarse graining, i.e., letting each spin represents themagnetization of a unit cell. Artiﬁcially fabricated exchange-coupled atomic spins on a substrate [ 17] is another physical realization of this model. The magnetization dynamics canbe described by the so-called Landau-Lifshitz-Miyazaki-Seki(LLMS) equations [ 11]: d/vectors n dt=− /vectorsn×(/vectorHeff+/vectorhn),d/vectorhn dt=−1 τc(/vectorhn−χ/vectorsn)+/vectorRn, (1) where /vectorhnis the ﬂuctuating magnetic ﬁeld, /vectorHeff=/vectorHn+ Dsz n/vectorz+J(/vectorsn−1+/vectorsn+1)+/vectorHdis the effective ﬁeld consisting of the external magnetic ﬁeld /vectorHnand uniaxial anisotropy ﬁeld with constant Dalong the same (here z) direction, and the exchange constant Jinitially taken to be ferromagnetic, i.e.,J> 0./vectorHdis caused by long-range dipolar ﬁelds, but is disregarded in the following. /vectorRnis the random force with zero average and a time-correlation function that satisﬁes theﬂuctuation-dissipation theorem (FDT) [ 18]: /angbracketleftbig R i n(t)Rj n/prime(t/prime)/angbracketrightbig =(2χkBTn/τc)δnn/primeδijδ(t−t/prime), (2) where i,j=x,y,z , the parameter χdescribes the spin-bath coupling, and τcis the relaxation time. In the following, Hn, D,J,Hd,hn, andkBTnare all measured in hertz. Equation ( 1) has been very successful in atomistic simulations of ultrafastspin dynamics for constant bath temperatures [ 19] and can be derived from microscopic spin-lattice or spin-electron cou-plings [ 19,20]. Here we introduce a spatially inhomogeneous thermal bath with arbitrary temperature proﬁles. We assumestatistical independence of neigboring baths, i.e., a correlationlength between reservoirs is shorter than the (course-grained) lattice constant. By eliminating the ﬂuctuating ﬁeld /vectorh nin Eq. ( 1), we arrive at the following stochastic LLMS with non-Markovian damping: d/vectorsn dt=− /vectorsn×(/vectorHeff+/vectorηn)+χ/vectorsn ×/integraldisplayt −∞dt/primeκ(t−t/prime)d/vectorsn(t/prime) dt/prime, (3)and a new stochastic ﬁeld /vectorηn=/integraldisplayt −∞dt/primeκ(t−t/prime)/vectorRn(t/prime), (4) which is correlated as /angbracketleftbig ηi n(t)ηj n/prime(t/prime)/angbracketrightbig =χkBTnδnn/primeδijκ(\|t−t/prime\|), (5) with memory kernel κ(τ)=exp (−τ/τc).Equation ( 3) is gen- uinely non-Markovian and has been believed to be analyticallyintractable [ 11,19]. Nevertheless, here we present an analytical solution for non-Markovian spin waves, to the best of ourknowledge for the ﬁrst time. III. RESULTS AND DISCUSSIONS A. Linear spin-wave theory For small-angle dynamics /vectorsn.=/vectorz+(sx n/vectorx+sy n/vectory) with \|sx,y n\|/lessmuch 1 the stochastic LLMS equation reduces to idψn dt+χ/integraldisplayt −∞dt/primeκ(t−t/prime)dψn(t/prime) dt/prime =−N/summationdisplay m=0(JQnm+Hnδnm)ψm+ηn(t), (6) for the complex scalar-ﬁelds ψn(t)=sx n+isy nandηn(t)= ηx n+iηy n,which are correlated as /angbracketleftη∗ n(t)ηn/prime(t/prime)/angbracketright=2χkBTnδnn/primeδijκ(\|t−t/prime\|), (7) where∗is the complex conjugate. The extra factor 2 reﬂects energy equipartition since ηnincorporates two degrees of freedom. Qis a (N+1)×(N+1) symmetric quasiuniform tridiagonal canonical matrix that does not depend on materialparameters (see Appendix A). InH n=H+εn, ε models external or anisotropy ﬁeld gradients [ 21,22]. Since in general, matrices Qand diag {Hn}cannot be diagonalized simulta- neously, we introduce a new matrix ˇQ=Q+(ε/J)diag{n} that satisﬁes JQnm+Hnδnm=JˇQnm+Hδnm.We remove the integral in Eq. ( 6) by taking the time-derivative id2ψn dt2+N/summationdisplay m=0/bracketleftbig JˇQnm+/parenleftbig H+χ+iτ−1 c/parenrightbig δnm/bracketrightbigdψn dt =−τ−1 cN/summationdisplay m=0(JˇQnm+Hδnm)ψm+Rn(t), (8) where Rn(t)=Rx n+iRy nis correlated as /angbracketleftR∗ n(t)Rn/prime(t/prime)/angbracketright=(4χkBTn/τc)δnn/primeδ(t−t/prime). (9) In the limit of τc→0,the above equation reduces to the Markovian LLG: (i+α)dψn dt=−N/summationdisplay m=0(J˘Qnm+Hδnm)ψm+ξn(t), (10) with correlator /angbracketleftξ∗ n(t)ξn/prime(t/prime)/angbracketright=4αkBTnδnn/primeδ(t−t/prime) (11) expressed in terms of the Gilbert damping constat α=χτc. The mathematical structure is identical to that of ﬂuctuating 024417-2ENERGY REPARTITION IN THE NONEQUILIBRIUM . . . PHYSICAL REVIEW B 95, 024417 (2017) heat [ 23] and/or mass [ 24] transport and the widely studied macroscopic ﬂuctuation theory of ﬂuids [ 25], where the scalar ﬁeld ψrepresents temperature [ 23] or number density ﬂuctuations [ 24], while ξ(t) is the divergence of a heat or particle current. The symmetric tridiagonal matrix ˇQcan be diagonalized by a linear transformation P−1ˇQP with an orthogonal matrix Pwhich solely depends on the ratio ε/J. This is equivalent to an expansion of the ﬁeld into normal magnon modes φk=/summationtextN n=0P−1 knψnthat obey d2φk dt2+νkdφk dt−iωk τcφk=fk(t), (12) where ωk=H+Jλkis the eigenfrequency of the kth mode, λkis thekth eigenvalue of ˇQ,andνk=τ−1 c−i(χ+ωk). The structure of Eq. ( 12) is reminiscent of the thermal acoustic wave equations [ 1] and the dynamic equations of ﬂuctuating superconducting Josephson junctions [ 26]. The boundary conditions affect the dispersion relation ωk. The modes interact via the transformed stochastic variable fk=−i/summationtextN n=0P−1 knRn with nonlocal correlator /angbracketleftf∗ k(t)fk/prime(t/prime)/angbracketright=(4χkBTkk/prime/τc)δ(t−t/prime), (13) introducing the temperature matrix Tkk/prime=N/summationdisplay n=0PnkPnk/primeTn. (14) Tis diagonal in the absence of temperature gradients, i.e., whenTn=T∀n. We now show that the diagonal terms Tkkencode the energy distribution over the different magnon modes in thenonequilibrium steady state. The average energy of the k-th magnon mode is E k=ωk/angbracketleftφ∗ kφk/angbracketright/2,where the expectation value/angbracketleft ···/angbracketright is taken over different realizations of the thermal noiseRn(t) and/angbracketleftφ∗ kφk/angbracketright/2 is the magnon number. Equation ( 12) can be solved exactly by introducing the Green functioncorresponding to the left-hand side and integrating over thenoise source term: φ k(t)=/integraldisplayt −∞dt/prime1 c1−c2[e−c2(t−t/prime)−e−c1(t−t/prime)]fk(t/prime),(15) with two complex numbers c1,2=/parenleftbig νk±/radicalBig ν2 k+4iτ−1 cωk/parenrightbig /2. (16) We thus arrive at the central result of this work that the energy stored in mode kis nothing but the thermal energy as deﬁned by the diagonal elements of T: Ek=kBTkk. (17) The entropy of the nonequilibrium steady system then can be expressed as S=−kB/summationtext kpklnpk,with the probability distribution pk=/angbracketleft \|φk\|2/angbracketright//summationtext k/prime/angbracketleft\|φk/prime\|2/angbracketright.Interestingly, for ho- mogeneous external magnetic ﬁelds, Tkkis parameter-free, depending only on the bath temperature proﬁle Tnand the boundary conditions. A magnetic ﬁeld gradient modiﬁes themode temperature only via the ratio ε/J. The memory kernel with relaxation time τ cdoes not affect the repartition. Although we consider an exponential memory kernel here, we envisionthat the obtained energy repartition principle ( 17) should be robust to the speciﬁc form of the kernels. The generalizationto two spins in the unit cell leads to acoustic and opticalmagnon branches and can be used to study ferrimagnets andantiferromagnets [ 27]. In the following, we limit ourselves to the temperature distribution of nonequilibrium ferromag-netic magnons. Off-diagonal terms T kk/prime(k/negationslash=k/prime) encode the magnonic spin current, which can be obtained from the spincontinuity equation [ 28,29] /vectorj M,n=J/vectorsn−1×/vectorsn,(0<n≤N). (18) Its dc component can be expanded into normal modes as jz M,n=J/summationdisplay kk/primeP(n−1)kPnk/primeIm/angbracketleftφ∗ kφk/prime/angbracketright, (19) where Im denotes the imaginary part. The associated real space magnon density distribution [ 30]ρM,n=/angbracketleftψ∗ nψn/angbracketright/2 is conjugate to the magnon number in reciprocal space/angbracketleftφ ∗ kφk/angbracketright/2.These quantities are expressed in terms of spectral temperatures in Appendix A. B. Temperature and chemical potential of nonequilibrium magnons under uniform magnetic ﬁeld We ﬁrst consider a simple case with a vanishing ﬁeld gradient ( ε=0).Under free boundaries (no pinning), we derive (Appendix A) Tkk=/braceleftbigg¯T, k =0, ¯T+/summationtextN n=0Tn N+1cos(2n+1)kπ N+1,k/negationslash=0,(20) where ¯T=/summationtextN n=0Tn/(N+1) is the average bath temperature. The energy stored in mode kemerges as a correction to the average temperature ¯T,but never exceeds ±¯T.Tkk−¯Tis an average over the bath temperature proﬁle weighted by a cosinefunction. We study the spectrally resolved temperature T kkfor ﬁve different model baths, all with T0=300,TN=350,and N=99 [see Fig. 2(a)] (in arbitrary temperature units): (i) a linear temperature proﬁle, i.e., Tn=T0+(TN−T0)n/N , (ii) a quadratic proﬁle, i.e., Tn=T0+(TN−T0)(n/N )2, (iii) a “subduplicate” proﬁle, i.e., Tn=T0+(TN−T0)√n/N ,( i v ) a Sanders-Walton proﬁle, i.e., Tn=T0+TN−T0 N+2μsinh/parenleftbigN ν/parenrightbig ×/bracketleftbigg n+μ/parenleftbigg sinh2n−N ν+sinhN ν/parenrightbigg/bracketrightbigg (21) with adjustable parameters μandν[31–33] chosen to be μ=1 and ν=16, and (v) an asymmetric Heaviside step function [ 34]a t1 0 +(N+1)/2. While a linear and sinh proﬁles can make physical sense being solutions of a simpleheat diffusion equation, arbitrary temperature proﬁles can beengineered in terms of a string of heat sources such as Peltiercells placed along the spin chain. Figure 2(b) shows the resulting T kkfor free boundary conditions. The magnon temperature does not deviate fromthe average temperature ¯Tfor both the linear and the Sanders- Walton proﬁle. The correction terms in Eq. ( 14) vanish for all temperature proﬁles that are odd around ( N/2,¯T).For free boundary conditions the equipartition at equilibrium persists 024417-3PENG Y AN, GERRIT E. W. BAUER, AND HUAIWU ZHANG PHYSICAL REVIEW B 95, 024417 (2017) FIG. 2. (a) Thermal bath temperature proﬁles chosen to study the mode-resolved temperature of nonequilibrium magnons. (b)–(d) Dependence of the temperatures of normal magnon modes φkon boundary conditions: (b) both ends are free, (c) both ends are pinned,and (d) the left end is pinned, while the right one is free. (e) Temperature of kmagnons under a asymmetric Heaviside temperature distribution with free boundary conditions. The applied magnetic ﬁeldis uniform. for temperature proﬁles with odd symmetry. For quadratic (subduplicate) proﬁles, on the other hand, low- (high-) k magnons are heated and high- (low-) kmagnons cooled. In general, pinning can reduce the magnon amplitude at the sam-ple boundaries, which obviously affects transport. However,boundary conditions also modify the energy repartition ofnonequilibrium magnons, as demonstrated in Fig. 2(c) for ﬁxed (pinned) boundary conditions (the analytical expressionofT kkare given in Appendix A). Notably, long-wavelength magnons are strongly affected by the boundary conditions,which leads to the inverted temperature proﬁle when magnonsare pinned and thereby do not sense the temperature atthe edges. Figure 2(d) shows T kkas a function of kunder boundary conditions with a pinned left and a free rightterminals. Since the boundaries now break symmetry, even forthe antisymmetric proﬁles the magnon temperature becomesdistributed; the low- kmagnons are getting hotter. We ﬁnd that a higher asymmetry of either the bath temperature proﬁleor the boundary condition leads to a smaller decay length inthe reciprocal space ( kspace). Figure 2(e) shows oscillations of the mode-dependent temperatures for a nonsymmetric andnonadiabatic thermal bath proﬁle, i.e., with a Heaviside stepfunction displaced from the midpoint. Though calculated forfree boundary conditions this feature is robust with respect toother choices. For free boundary conditions and bath temperature proﬁles with odd symmetry with respect to ( N/2,¯T), all magnons share the same temperature ¯T,cf. Eq. ( 20). One might thereforeFIG. 3. (a) Spatial distribution of thermally induced magnon accumulations for different heat-bath proﬁles. Inset (upper-middle) γnas function of system size N. Inset (lower-left corner) Zoom of the accumulation for linear and Sanders-Walton bath proﬁlesat the sample center. (b) Magnon chemical potential distribution for different heat baths. In (a) and (b), we set damping parameter α=0.001.(c) Magnon accumulation as a function of the damping parameter for a linear heat-bath. In calculations, we consider free boundary conditions at the edges and set H/J=0.01. naively conclude that the magnon distribution is then not modiﬁed by the temperature gradient. However, the localtemperature differences between bath and magnon wouldmake the steady state unsustainable since we ﬁnd a heatcurrent-induced magnon accumulation /Delta1ρ M,n=ρM,n−γn¯T withγn=/summationtext k(Pnk)2kB/ωk(Appendix A). Figure 3(a) shows the calculated spatial distribution /Delta1ρM,n for different heat baths and free boundary conditions. For lattice temperatureswith odd symmetry, the magnon accumulation around thecenter N/2 increases linearly with site n[the lower-left-corner 024417-4ENERGY REPARTITION IN THE NONEQUILIBRIUM . . . PHYSICAL REVIEW B 95, 024417 (2017) inset of Fig. 3(a) zooms in on the details] with a slope that depends on the shape of the temperature proﬁle. Themagnon accumulation is distributed in space, in spite ofthe uniform magnon temperature T kk=¯T∀kat all sites n. Therefore the magnon distribution cannot be parameterizedby temperature alone. The solution is provided by introducinga distributed magnon chemical potential. A ﬁnite magnonchemical potential is the precursor of the magnon Bose-Einstein (or Rayleigh-Jeans) condensation that has beenobserved in magnetic insulators parametrically pumped bymicrowaves [ 20]. The semiclassical nonequilibrium distribution function of magnons can be described by Bose-Einstein statistics, f BE(k,n)=1 exp/parenleftbigωk−μM,n kBTkk/parenrightbig −1, (22) inphase space spanned by coordinate and momentum, which in the high-temperature limit approaches the Rayleigh-Jeansdistribution f BE(k,n)→kBTkk/(ωk−μM,n).The magnon chemical potential proﬁle μM,ncan therefore be determined by equating ρM,n=/summationdisplay k(Pnk)2kBTkk ωk−μM,n, (23) with/angbracketleftψ∗ nψn/angbracketright/2. The calculated μM,n for different heat baths under free boundary conditions are shown in Fig. 3(b). At equilibrium μM,n vanishes and the local magnon density is governed by the magnon temperature only. For quadratic, subdupli-cate, and Heaviside proﬁles, the magnon accumulation isnonmonotonic. In a subduplicate bath, it ﬁrst increases andthen decreases with n, opposite to the cases of quadratic and Heaviside proﬁles. We therefore conclude that heat-bath temperature proﬁles can strongly affect the magnonaccumulation. In Fig. 3(c), by tuning the damping parameter α,we ﬁnd that a larger dissipation causes a spatially steeper magnon accumulation (a smaller diffusion length) under freeboundary conditions. Using other boundary conditions doesnot change the results qualitatively. C. Spin pumping and spin Seebeck effects Thermal spin currents can be detected by heavy normal metal contacts that convert them into a transverse voltage bythe inverse spin Hall effect [ 35]. We can model this situation by contacting the spin chain either at the two ends or at someintermediate site. The former conﬁguration corresponds to the“longitudinal” spin Seebeck effect [ 36–41], while the latter one is referred to as “transverse” [ 35,42–46] or “nonlocal” [ 47]. The spin dynamics at the interface pumps a spin current intothe contact at site ngiven by /vectorj s,n=g↑↓ eff/planckover2pi1 4π/vectorsn×d/vectorsn dt, (24) where g↑↓ effis the effective spin-mixing conductance including a back-ﬂow correction [ 48] and/or spin-orbit coupling at the interface [ 49]. Its averaged dc component reads jz s,n=−g↑↓ eff/planckover2pi1 4π/summationdisplay k,k/primePn,kPn,k/primeIm/angbracketleft˙φ∗ kφk/prime/angbracketright. (25)In the small dissipation/Markovian limit, the pumped dc spin current can be expressed as jz s,n=2/planckover2pi1g↑↓ eff π(1+α2)/summationdisplay kk/primePnkPnk/primekB(Tkk/prime−Teδkk/prime)G(α,ωk,ωk/prime), (26) where G=α2ωkωk/prime α2(ωk+ωk/prime)2+(ωk−ωk/prime)2. (27) Experimentally, this spin current can be detected by the inverse spin Hall voltage in attached heavy metal contacts. Here weinclude the Johnson-Nyquist noise generated in the metalthat is proportional to the electron temperature T e, usually assumed to be in equilibrium with its phonon temperature.Disregarding the Kapitza interface heat resistance, the phonontemperature is continuous over the interface and T e=Tn. For small damping, α/similarequal10−5in YIG, the cross correlations between modes become unimportant and jz s,n/similarequalg↑↓ eff/planckover2pi1 2π/summationdisplay k(Pnk)2kB(Tkk−Te), (28) as found in conventional spin Seebeck theory [ 32] for uniform magnon temperature Tkk=Tm∀k. According to this theory, the spin Seebeck effect vanishes when magnon and electrontemperatures are the same. However, the full Eq. ( 26) reveals the limitations of this approximation: the off-diagonal termsgenerate an SSE even in the absence of a temperaturedifference between magnons and electrons. Figure 4shows the spatial distribution of the pumped spin current ( 26)f o r T e=¯T,i.e., the contribution to the SSE driven by the chemical potential alone, for different bath temperature proﬁles andmixed boundary conditions. The details of the bath proﬁlestrongly affect the distribution and magnitude of the spincurrent and spin Seebeck effect. FIG. 4. Spin Seebeck spin current (in units of /planckover2pi1g↑↓ effkB¯T/ π )i na metal contact attached to site nfor different heat-bath proﬁles and mixed boundary conditions. Parameters used in the calculations areα=0.001 and H/J=0.01. 024417-5PENG Y AN, GERRIT E. W. BAUER, AND HUAIWU ZHANG PHYSICAL REVIEW B 95, 024417 (2017) D. Wannier-Zeeman localization It follows from Eq. ( 6) that magnetic ﬁeld gradients act on magnons like electric ﬁelds act on electrons. Sufﬁciently strongelectric potential gradients in crystals can cause Wannier-Stark electron localization [ 50]. We may therefore expect an analogous Wannier-Zeeman magnon localization in stronglyinhomogeneous magnetic ﬁelds, which may modify the modetemperature of magnons. The matrix ˇQgenerally can in that limit not be diagonalized analytically anymore, but small ora large magnetic-ﬁeld gradient can be treated perturbatively.In the limit of large magnetic ﬁeld gradients \|ε/J\|/greatermuch 1 and free boundary conditions: ω 0=H+J, ω N=H+J+ εN, ω k=H+2J+εkfor 0<k<N , andPnk=δnk.The spectrum then becomes a Wannier-Zeeman ladder. The tem-perature matrix T kk/prime=δkk/primeTkis then diagonal even at nonequi- librium, i.e., the localization length is of the order of the latticeconstant. The magnon density becomes ρ M,n=kBTn/ωn, thereby recovering the classical Rayleigh-Jeans distributionwith zero chemical potential, i.e., local thermal equilibrium.Strong magnon localizations renders the spin chain insulatingsincej z M,n=0. In the limit of small damping, the pumped spin current becomes jz s,n=g↑↓ eff(/planckover2pi1/2π)kB(Tn−Te); the spin Seebeck effect becomes local and vanishes when electrons onthe metal side of the contact are at the same temperature as thethermal bath (phonons) on the magnetic side. Numerical calculations describe the transition from ex- tended Bloch states for small ﬁeld-gradients to localizedWannier-Zeeman ladder states under large magnetic ﬁeldgradients (referring to Appendix Afor details and ﬁgures). The localization length L=1//summationtext N n=0(Pnk)4(in units of the lattice constant) shrinks with increasing gradient, down to unity inthe limit of high ﬁeld-gradients. The localized magnon statesshift from the low- to the high-ﬁeld region with increasingenergy. For a long chain ( N→∞ ),we ﬁnd an asymptotic L∼− 1/[(ε/J)l n(ε/J)] forε/J→0.Magnon localization suppresses the transverse or nonlocal spin Seebeck effect.However, most experiments are carried out on YIG ﬁlmswith very small anisotropy, which makes observation difﬁcult.On the other hand, strong perpendicular anisotropies can beinduced by alloying and doping (but preserving high magneticquality) [ 51–53]. In (YBi) 3(FeGa) 5O12, this is reﬂected by domain wall widths of 8–11 lattice constants [ 54]. The material parameters at low temperatures are [ 54–56] an exchange coupling J=1.29 K and crystalline magnetic anisotropy D=0.3 K, and lattice constant a=1.24 nm. An upper bound for the ﬁeld gradient generated by a position dependentmagnetic anisotropy in a temperature gradient can be obtainedassuming its low temperature value on the cold side and avanishing one at the hot side, or ε=(D/l )a=4×10 −7K and ε/J=3×10−7.This leads to a magnon localization length L=− 1/[(ε/J)l n(ε/J)]×a=0.3 mm. When the magnons are localized on the scale of the metal contact widths (typically,0.1 mm, see, e.g., Ref. [ 46], and references therein) we predict a suppressed spin Seebeck signal. Magnon localization can alsobe induced by applying magnetic ﬁeld gradients, for example,by the stray ﬁelds of proximity ferromagnets or by the Oerstedﬁelds due to current-carrying wires close to the magnonconduits. Magnetic write heads generate local ﬁeld gradientsof up to 20 MT/m. Analogous to electronic Wannier-Stark localizations in semiconductor superlattices [ 57], magnonic crystals with tunable lattice periods can display magnon lo-calization at possibly much weaker inhomogeneous magneticﬁelds. E. Magnon-magnon interactions The results above assume the presence of magnon-phonon thermalization, but absence of magnon-magnon interactionsthat modify the equations of motion for higher magnondensities. Anisotropy-mediated magnon interactions dominatein the long-wave lengths regime considered here [ 58–60]. Adopting the Markov approximation and to leading order in themagnon density, we arrive at a dissipative discrete nonlinearSchr ¨odinger (DNLS) equation with stochastic sources and a local interaction, (i+α)dψ n dt=−N/summationdisplay m=0[J˘Qnm+(H−ν\|ψm\|2)δnm]ψm+ξn(t), (29) where νis the interaction strength governed by the anisotropy constant Dbut treated here as a free parameter. For ν= 0,eigenstates are affected by magnetic ﬁeld gradients ε, as discussed above. The mode frequency splitting /Delta1ω∼ min [J(λk+1−λk)],while for large ε,/Delta1ω∼ε.The non- linearity in Eq. ( 29) for the uniaxial anisotropy considered (D,ν > 0) corresponds to an attractive interaction and a frequency redshift δωn∼ν\|ψn\|2.The interaction is assumed short range, which is allowed when dipolar coupling betweenspins is small in our coarse grained model. We may thenexpect three qualitatively different regimes: (i) \|ν\|</Delta1 ω ; (ii)/Delta1ω < \|ν\|</Delta1 ; and (iii) /Delta1<\|ν\|,where the bandwidth /Delta1=ω N−ω0.In case (i), the local frequency shift is smaller than the spacing /Delta1ω. Therefore the long-time dynamics is not modiﬁed from the limit ν=0. For (ii), nonlinearities become important since the mode frequencies overlap. In the limit (iii),the interaction is stronger than the noninteracting bandwidth,drastically transforming the spectrum. Discrete bound statesmay develop at the band edges, leading to interaction inducedself-trapping [ 26]. We may expand ( 29) into normal modes as before to obtain (i+α)dφ k dt=−ωkφk+ν/summationdisplay k1,k2,k3Ik,k 1,k2,k3φ∗ k1φk2φk3+ζk(t), (30) where the matrix elements Ik,k 1,k2,k3=/summationdisplay nPnkPnk1Pnk2Pnk3 (31) describe four-magnon scattering events and the stochastic variables are correlated as /angbracketleftζ∗ k(t)ζk/prime(t/prime)/angbracketright=4αkBTkk/primeδ(t−t/prime). (32) For arbitrary ﬁeld gradients, we obtain the analytical formula of the nonlinearity correction to the energy repartition up to 024417-6ENERGY REPARTITION IN THE NONEQUILIBRIUM . . . PHYSICAL REVIEW B 95, 024417 (2017) the ﬁrst order of νas follows (Appendix B): kBT/prime kk=kBTkk+16ν/summationdisplay k1,k2,k3Ik,k 1,k2,k3α2(kBTkk3)(kBTk1k2)[(−3+α2)ωkωk1+(1+α2)(ωkωk2+ωk1ωk3+ωk2ωk3)] [(ωk1−ωk2)2+α2(ωk1+ωk2)2][(ωk−ωk3)2+α2(ωk+ωk3)2], (33) where we introduce the renormalized thermal energy kBT/prime kk= ωk/angbracketleftφ∗ kφk/angbracketright/2.It reduces to T/prime kk=(1+/Lambda1)Tkkin the strongly localized limit in leading order of the small parameter/Lambda1=4νk BTkk/ω2 k.The interaction generates a redshift of the spectrum and corresponding higher thermal occupation, asconﬁrmed by numerical simulations for few-spin systems(Appendixes C, D, and E) for both strong and relativelyweak localizations. The nonlinearity is therefore acting like anadditional heat source leading to mode-dependent correctionsto the temperature that are observable in the spin Seebeckeffect, e.g., by tuning the anisotropy while keeping othermaterial parameters approximately constant. IV . SUMMARY To conclude, we report here a principle of energy repartition for nonequilibrium system. We illustrate the general principleat the hand of analytical solutions of the non-MarkovianLandau-Lifshitz-Miyazaki-Seki equations. We ﬁnd that ﬂuc-tuations are governed by a set of normal-mode temperatureswithout strong effect of the non-Markovian memory kernel.The mode temperatures strongly depend on the temperatureproﬁle of the heat bath and the boundary conditions, while thenonequilibrium magnon density distribution can be describedonly by introducing a chemical potential. Gradients of mag-netic ﬁelds cause Wannier-Zeeman magnon localization thatshould be observable in the transverse or nonlocal spin Seebeckeffect on magnetic insulators with strong magnetocrystallineanisotropies such as (YBi) 3(FeGa) 5O12. Magnon-magnon interactions can to leading order be captured by increasedmode temperatures. Our generic results shed light on thefundamental concept of temperature and are applicable tomany disciplines beyond spintronics. ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11604041, theNational Key Research Development Program under ContractNo. 2016YFA0300801, the National Thousand-Young-TalentProgram of China, the DFG Priority Programme 1538 “Spin-Caloric Transport,” the NWO, EU FP7 ICT Grant No. 612759InSpin, and Grant-in-Aid for Scientiﬁc Research (Grant Nos.25247056, 25220910, and 26103006). APPENDIX A: SYMMETRIC TRIDIAGONAL MATRIX ˇQ Here we consider the effect of boundary conditions on the canonical ( N+1)×(N+1) matrix ˇQ=Q+(ε/J)diag{n} for the n=0,1,2,...,N spin chain with nearest-neighbor exchange coupling J.Qis diagonalized by a matrix P,i.e., P−1ˇQP=diag{λk},which must be orthogonal: P−1=PT. We ﬁrst consider the case of homogeneous magnetic ﬁelds(ε=0,soˇQ=Q) for different boundary conditions.Case I. For free boundaries at the ends, Q=⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1−10 ··· 0 −12 −10... 0−12 −1 ......... −12 −10 ... −12 −1 0··· 0−11⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(A1) has eigenvalues λ k=2/parenleftbigg 1−coskπ N+1/parenrightbigg , (A2) withk=0,1,2,...,N and eigenvectors vk=/bracketleftbigg coskπ 2(N+1),cos3kπ 2(N+1),..., cos(2N+1)kπ 2(N+1)/bracketrightbiggT (A3) that can be normalized as uk=/braceleftBigg1√ N+1vk,k =0/radicalBig 2 N+1vk,k/negationslash=0, (A4) leading to the orthogonal matrix P, Pnk=/braceleftBigg1√ N+1,k =0/radicalBig 2 N+1cos(2n+1)kπ 2(N+1),k/negationslash=0. (A5) The temperature matrix deﬁned as Tkk/prime=N/summationdisplay n=0PnkPnk/primeTn (A6) has diagonal elements Tkk=/braceleftBigg¯T, k =0 ¯T/bracketleftbig 1+/summationtextN n=0Tncos(2n+1)kπ N+1/summationtextN n=0Tn/bracketrightbig ,k/negationslash=0, (A7) with ¯T=/summationtextN n=0Tn/(N+1).At equilibrium, we recover Tkk=¯T∀k,since/summationtextN n=0cos(2n+1)kπ N+1=0 andTn=¯T∀n. In the limit of very small Gilbert damping, e.g., α/similarequal10−5 in YIG, the magnon density can be approximated as ρM,n/similarequal/summationtext k(Pnk)2kBTkk/ωk, which becomes exact for constant tem- peratures. f(ω,T )=kBT/(/planckover2pi1ω) is the Rayleigh-Jeans distri- bution function and ( Pnk)2the probability to ﬁnd a k-magnon at site n. At equilibrium, i.e., Tn≡T∀n,all magnons share the temperature of the heat bath ( Tkk/prime=Tδkk/prime) and ρM,n=γnTwithγn=/summationtext k(Pnk)2kB/ωk.This agrees with the low-temperature expansion of the Watson-Blume-Vineyardformula by introducing γ n≡βn/Tcwith the Curie temperature Tc.We thereby derive expressions for a site-dependent critical 024417-7PENG Y AN, GERRIT E. W. BAUER, AND HUAIWU ZHANG PHYSICAL REVIEW B 95, 024417 (2017) exponent βn.γnbecomes a constant in the thermodynamic limit (N→∞ ) as shown in the upper-middle inset of Fig. 3(a). In the present 1D model, we have γn kB/J=1 N+11 x+1 πN/summationdisplay k=01+cos(2n+1)kπ N+1 x+2/parenleftbig 1−coskπ N+1/parenrightbigπ N+1, (A8) where x=H/J. Its thermodynamic limit is lim N→∞γn kB/J=1 π/integraldisplayπ 01 x+2(1−cosy)dy =1√x(4+x). (A9) We therefore obtain lim N→∞γn=kB/√H(H+4J). Case II. For ﬁxed (pinned) boundaries at the two ends ,the number of spins is effectively reduced to N−1 and Q=⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝2−10 ··· 0 −12 −1... 0−12 −1 ......... −12 −1 ... −12 −1 0··· − 12⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(A10) has eigenvalues λ k=2/parenleftbigg 1−coskπ N/parenrightbigg , (A11) withk=1,2,..., N −1,and eigenvectors vk=/bracketleftbigg sinkπ N,sin2kπ N,..., sin(N−1)kπ N/bracketrightbiggT , (A12) normalized as uk=/radicalbigg 2 Nvk, (A13) and the matrix elements of P Pnk=/radicalbigg 2 Nsinnkπ N,n=1,2,..., N −1. (A14) Now Tkk=N−1 N¯T/bracketleftBigg 1−/summationtextN−1 n=1Tncos2nkπ N/summationtextN−1 n=1Tn/bracketrightBigg , k=1,2,..., N −1, (A15) with ¯T=/summationtextN−1 n=1Tn/(N−1).Since/summationtextN−1 n=1cos2nkπ N=− 1,we again recover Tkk=¯T∀kat equilibrium. Case III. For ﬁxed amplitude at site n=0 and free amplitude at site n=N,the number of spins is N.T h eN×Nmatrix Q=⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝2−10 ··· 0 −12 −10... 0−12 −1 ......... −12 −10 ... −12 −1 0··· 0−11⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(A16) has eigenvalues λ k=2/parenleftbigg 1−cos2k−1 2N+1π/parenrightbigg ,k=1,2,..., N, (A17) withk=1,2,..., N, and eigenvectors vk=/bracketleftbigg sin2k−1 2N+1π,sin2(2k−1) 2N+1π,..., sinN(2k−1) 2N+1π/bracketrightbiggT (A18) that can be normalized as uk=2vk/√ 2N+1 and matrix elements Pnk=2√ 2N+1sinn(2k−1) 2N+1π, n =1,2,..., N. (A19) Now Tkk=2N 2N+1¯T/bracketleftBigg 1−/summationtextN n=1Tncos2n(2k−1) 2N+1π /summationtextN n=1Tn/bracketrightBigg , k=1,2,..., N, (A20) with ¯T=/summationtextN n=1Tn/N. In this case,/summationtextN n=1cos2n(2k−1) 2N+1π= −1/2,and again we recover Tkk=¯T∀kat equilibrium. In the presence of ﬁnite ﬁeld gradients, the matrix ˇQ generally cannot be diagonalized analytically. Here, we areinterested in the limit of large magnetic ﬁeld gradients, i.e.,\|ε/J\|/greatermuch 1.With free boundary conditions, we obtain by perturbation theory λ 0=1,k =0, λk=2+ε Jk,1≤k≤N−1, (A21) λN=1+ε JN, k =N, and P=I(N+1)×(N+1)orPnk=δnk. (A22) Correspondingly, the eigenfrequency of the kth mode is ω0=H+J, k =0, ωk=H+2J+εk, 1≤k≤N−1, (A23) ωN=H+J+εN. k =N. The spectrum is no longer a trigonometric function of wave number but forms a Wannier-Zeeman ladder. The temperaturematrix T kk/prime=N/summationdisplay n=0PnkPnk/primeTn=N/summationdisplay n=0δnkδnk/primeTn=δkk/primeTk (A24) 024417-8ENERGY REPARTITION IN THE NONEQUILIBRIUM . . . PHYSICAL REVIEW B 95, 024417 (2017) FIG. 5. Magnon dispersion and wave functions without ﬁeld gradients. is now diagonal. The mangons are now Wannier-Zeeman localized to the unit cell rendering the spin chain insulatingfor spin and energy currents. This can be illustrated in smalldamping/Markovian limit with magnonic spin current j z M,n=J/summationdisplay k/negationslash=k/primePnkP(n−1)k/primekBTkk/primeF(α,ωk,ωk/prime), (A25) where F=4α(ωk−ωk/prime)/[α2(ωk+ωk/prime)2+(ωk−ωk/prime)2]i sa n antisymmetric Lorentzian that vanishes for a diagonal temper-ature matrix. The associated magnon density ρ M,n=1 2/angbracketleftψ∗ nψn/angbracketright=kBTn ωn(A26) indicates local equilibrium. In the following, we present numerical calculations for different ﬁeld gradients in order to illustrate the transitionfrom propagation Bloch to localized Wannier-Zeeman statesby increasing ε.Here, we adopt J=1,H=0 (its value only shifts the magnon band gap) and consider free boundaryconditions. Figure 5shows the results without ﬁeld gradients. The magnon dispersion is a cosine function. The magnon wavefunctions are spreading Bloch states.Figure 6shows the results at ε=0.1. The magnon dispersion is starting to deviate from the cosine function. Themagnon wave functions are localized. Figure 7shows the results at ε=1. The magnon disper- sion becomes linear. The magnon wave functions are morelocalized. Figure 8shows the results at ε=10. The magnon dis- persion is linear with strongly localized wave functions. Thelocalization length is close to a lattice constant. Figures 6–8 show that in the valleys of an inhomogeneous magneticﬁeld distribution only low-energy magnons contribute, sincehigh-energy magnons are localized to the hills. The case isopposite in the high-ﬁeld side that only high-energy magnonscontribute, since low-energy magnons are localized in thelow-ﬁeld side. The magnon localization length L(ε/J)=1 /summationtextN n=0(Pnk)4. (A27) is plotted in Fig. 9as a function of the ﬁeld gradient. 024417-9PENG Y AN, GERRIT E. W. BAUER, AND HUAIWU ZHANG PHYSICAL REVIEW B 95, 024417 (2017) FIG. 6. Magnon dispersion and wave functions with a ﬁeld gradient ε=0.1. APPENDIX B: PERTURBATION THEORY In this section, we present a perturbative solution of the stochastic nonlinear equation including the interaction term ν for arbitrary ﬁeld gradients. We expand the normal modes as φk(t)=φk,0(t)+νφk,1(t)+ν2φk,2(t)+··· , (B1) and ˙φk(t)=˙φk,0(t)+ν˙φk,1(t)+ν2˙φk,2(t)+··· . (B2) Keeping only ﬁrst-order terms, (i+α)(˙φk,0+ν˙φk,1) =−ωk(φk,0+νφk,1) +ν/summationdisplay k1,k2,k3Ik,k 1,k2,k3φ∗ k1,0φk2,0φk3,0+ζk(t). We therefore obtain zero order: (i+α)˙φk,0=−ωkφk,0+ζk(t), (B3)ﬁrst order: (i+α)˙φk,1=−ωkφk,1 +/summationdisplay k1,k2,k3Ik,k 1,k2,k3φ∗ k1,0φk2,0φk3,0. (B4) The stationary solution of the zero-order equation is φk,0(t)=1 i+α/integraldisplayt −∞dt/primeexp/bracketleftbigg −ωk i+α(t−t/prime)/bracketrightbigg ζk/parenleftbig t/prime/parenrightbig ,(B5) and that for the ﬁrst-order one is φk,1(t)=1 i+α/integraldisplayt −∞dt/primeexp/bracketleftbigg −ωk i+α(t−t/prime)/bracketrightbigg ×/summationdisplay k1,k2,k3Ik,k 1,k2,k3φ∗ k1,0(t/prime)φk2,0(t/prime)φk3,0(t/prime).(B6) The quantity we aim to evaluate is ωk 2/angbracketleftφ∗ k(t)φk(t)/angbracketright=ωk 2/angbracketleftφ∗ k,0(t)φk,0(t)/angbracketright +νωkRe/angbracketleftφ∗ k,0(t)φk,1(t)/angbracketright. (B7) 024417-10ENERGY REPARTITION IN THE NONEQUILIBRIUM . . . PHYSICAL REVIEW B 95, 024417 (2017) FIG. 7. Magnon dispersion and wave functions with a ﬁeld gradient ε=1. The ﬁrst term in the right-hand side of the above equation is simply kBTkk,while the second term is /angbracketleftφ∗ k,0(t)φk,1(t)/angbracketright=1 i+α/integraldisplayt −∞dt/primeexp/bracketleftbigg −ωk i+α(t−t/prime)/bracketrightbigg/summationdisplay k1,k2,k3Ik,k 1,k2,k3/angbracketleftφ∗ k1,0(t/prime)φk2,0(t/prime)φk3,0(t/prime)φ∗ k,0(t)/angbracketright, where the correlation is /angbracketleftφ∗ k1,0(t/prime)φk2,0(t/prime)φk3,0(t/prime)φ∗ k,0(t)/angbracketright=1 (1+α2)2/integraldisplayt/prime −∞dt/prime/prime/integraldisplayt/prime −∞dt/prime/prime/prime/integraldisplayt/prime −∞dt/prime/prime/prime/prime/integraldisplayt −∞dt/prime/prime/prime/prime/primeexp/bracketleftbigg −ωk1 −i+α(t/prime−t/prime/prime)−ωk2 i+α(t/prime−t/prime/prime/prime) −ωk3 i+α(t/prime−t/prime/prime/prime/prime)−ωk −i+α(t−t/prime/prime/prime/prime/prime)/bracketrightbigg /angbracketleftζ∗ k1(t/prime/prime)ζk2(t/prime/prime/prime)ζk3(t/prime/prime/prime/prime)ζ∗ k(t/prime/prime/prime/prime/prime)/angbracketright. By Isserlis’ (or Wick’s) theorem, we have /angbracketleftζ∗ k1(t/prime/prime)ζk2(t/prime/prime/prime)ζk3(t/prime/prime/prime/prime)ζ∗ k(t/prime/prime/prime/prime/prime)/angbracketright=/angbracketleftζ∗ k1(t/prime/prime)ζk2(t/prime/prime/prime)/angbracketright/angbracketleftζk3(t/prime/prime/prime/prime)ζ∗ k(t/prime/prime/prime/prime/prime)/angbracketright+/angbracketleftζ∗ k1(t/prime/prime)ζk3(t/prime/prime/prime/prime)/angbracketright/angbracketleftζk2(t/prime/prime/prime)ζ∗ k(t/prime/prime/prime/prime/prime)/angbracketright =(4αkB)2[Tkk3Tk1k2δ(t/prime/prime−t/prime/prime/prime)δ(t/prime/prime/prime/prime−t/prime/prime/prime/prime/prime)+Tkk2Tk1k3δ(t/prime/prime−t/prime/prime/prime/prime)δ(t/prime/prime/prime−t/prime/prime/prime/prime/prime)], where we only keep the nonzero terms. After straightforward substitutions, /angbracketleftφ∗ k,0(t)φk,1(t)/angbracketright=(4αkB)2(−i+α) αωk/summationdisplay k1,k2,k3Ik,k 1,k2,k3Tkk3Tk1k2 [ωk1(i+α)+ωk2(−i+α)][ωk(i+α)+ωk3(−i+α)]. 024417-11PENG Y AN, GERRIT E. W. BAUER, AND HUAIWU ZHANG PHYSICAL REVIEW B 95, 024417 (2017) FIG. 8. Magnon dispersion and wave functions with a ﬁeld gradient ε=10. The perturbative mode temperature ( B7) is thus given by ωk 2/angbracketleftφ∗ k(t)φk(t)/angbracketright=kBTkk+16ν/summationdisplay k1,k2,k3Ik,k 1,k2,k3α2(kBTkk3)(kBTk1k2)[(−3+α2)ωkωk1+(1+α2)(ωkωk2+ωk1ωk3+ωk2ωk3)] [(ωk1−ωk2)2+α2(ωk1+ωk2)2][(ωk−ωk3)2+α2(ωk+ωk3)2]. (B8) In the limit of a very strong Wannier-Zeeman localization, i.e., Pnk=δnk,Pnk1=δnk1,Pnk2=δnk2,andPnk3=δnk3, Ik,k 1,k2,k3=/summationdisplay nPnkPnk1Pnk2Pnk3=δkk1δkk2δkk3, (B9) which implies absence of mode coupling. The above mode temperature ( B8) is then modiﬁed to ωk 2/angbracketleftφ∗ k(t)φk(t)/angbracketright=kBTkk/parenleftbigg 1+4νkBTkk ω2 k/parenrightbigg . In the limit of a very weak Gilbert damping, only the trivial resonance terms, i.e., ωk=ωk3andωk1=ωk2,in Eq. ( B8)survive. We thus have ωk 2/angbracketleftφ∗ k(t)φk(t)/angbracketright=kBTkk+4ν/summationdisplay k1Ik,k 1,k1,k(kBTkk)/parenleftbig kBTk1k1/parenrightbig ωkωk1. Higher-order perturbation calculations are straightforward if necessary. APPENDIX C: SPIN MONOMER We implement numerical calculations for a single spin (spin monomer) in contact with a thermal bath correspondingto either an isolated classical atomic moment or a stronglylocalized normal mode in kspace. The equation of motion 024417-12ENERGY REPARTITION IN THE NONEQUILIBRIUM . . . PHYSICAL REVIEW B 95, 024417 (2017) FIG. 9. Localization length Las a function of the ﬁeld gradient ε/J. including the magnon interaction is simpliﬁed to (i+α)dφ dt=−ωφ+ν\|φ\|2φ+ζ(t), (C1) where we omitted subscripts. Here, the source term ζ(t)= ξ1(t)+iξ2(t) is the complex noise deﬁned in the main text, with two real-valued Gaussian white noise sources (Wienerprocess) ξ 1(t) andξ2(t). Figure 10shows the dynamics of the function ωφ∗(t)φ(t)/2. We simulate 2 ×106steps with a time step 0.01 for the time evolution. In numerical calculations, we use parametersω=k B=α=1,T=1,andν=− 0.5. The Ito interpretation is adopted when integrating the above stochastic differentialequation. The time average of ωφ ∗(t)φ(t)/2 represents the tempera- ture of the (single) normal mode. Numerical simulations foreveryνare repeated 20 times in order to suppress the statistical error (Fig. 10is just one of them at ν=− 0.5). Figure 11 shows the renormalized temperature of the normal mode as afunction of the nonlinearity strength ν. It demonstrates that an increasing nonlinearity increases the temperature of the mode.In the regime of weak nonlinearity ( \|ν\|≤0.02), the numerical results compare very well with the analytical formula. FIG. 10. Time evolution of function ωφ∗(t)φ(t)/2i nas p i n monomer driven by a stochastic white noise.FIG. 11. Renormalization of mode temperature in a spin monomer, tuned by the strength of nonlinearity ν. APPENDIX D: SPIN DIMER We implement numerical calculations on a spin dimer model contacting with two thermal baths with differenttemperatures. Under free boundary conditions, the 2 ×2 matrix ˇQis ˇQ=/parenleftbigg 1 −1 −11+ε/J/parenrightbigg . (D1) In the following, we set J=1.The corresponding diagonal matrix P=⎛ ⎝ ε+√ 4+ε2 2√ 1+1 4(ε+√ 4+ε2)2ε−√ 4+ε2 2√ 1+1 4(ε−√ 4+ε2)2 1√ 1+1 4(ε+√ 4+ε2)21√ 1+1 4(ε−√ 4+ε2)2⎞ ⎠ (D2) has the eigenvalues ω0=H+2+ε−√ 4+ε2 2, (D3) ω1=H+2+ε+√ 4+ε2 2. (D4) Forε=1, the equations of motions for the normal modes in the main text become (i+α)dφ0 dt=−ω0φ0+ν(−0.2\|φ0\|2φ0+0.8\|φ0\|2φ1 +0.2φ∗ 0φ2 1+0.4φ2 0φ∗ 1+0.4φ0\|φ1\|2 +0.6\|φ1\|2φ1)+ζ0(t), (D5) (i+α)dφ1 dt=−ω1φ1+ν(0.6\|φ0\|2φ0−0.4\|φ0\|2φ1 +0.4φ∗ 0φ2 1−0.2φ2 0φ∗ 1+0.8φ0\|φ1\|2 +0.2\|φ1\|2φ1)+ζ1(t), (D6) 024417-13PENG Y AN, GERRIT E. W. BAUER, AND HUAIWU ZHANG PHYSICAL REVIEW B 95, 024417 (2017) FIG. 12. Time evolution of function ωkφ∗ k(t)φk(t)/2f o rt h et w o normal modes ( k=0a n dk=1) in a spin dimmer. with ζ0(t)=− 0.850651 ξ0(t)−0.525731 ξ1(t), (D7) ζ1(t)=− 0.525731 ξ0(t)+0.850651 ξ1(t), (D8) in which source terms ξ0(t)=ξ01(t)+iξ02(t) and ξ1(t)= ξ11(t)+iξ12(t) with Gaussian white noises (Wiener process) ξ01(t),ξ02(t),ξ11(t),andξ12(t). Figure 12shows the dynamics of function ωkφ∗ k(t)φk(t)/2 fork=0 and 1. We simulate 2 ×106steps with a time step 0.01 for the time evolution. The parameters used in thenumerical calculations are H=ε=J=k B=α=1,T 1= 2T0=2,andν=− 0.6. Ito interpretation is adopted to integrate the above stochastic differential equations. The time-average of ωkφ∗ k(t)φk(t)/2 represents the temper- ature of the normal mode. Numerical simulations for everyνare repeated 20 times (Fig. 12is just one of them when ν=− 0.6). Figure 13shows the renormalized temperatures of normal modes as a function of the nonlinearity strength ν. It demonstrates that an increasing nonlinearity increases thetemperature of all modes. FIG. 13. Renormalization of mode temperatures in a spin dimer, tuned by the strength of nonlinearity ν.T00andT11represent the temperatures of normal modes for k=0a n d k=1, respectively, without nonlinearity.APPENDIX E: SPIN TRIMER Numerical calculations of a spin trimer model are presented here. Under free boundary conditions, the 3 ×3m a t r i x ˇQis ˇQ=⎛ ⎝1−10 −12+ε−1 0−11 +2ε⎞ ⎠, (E1) w h e r ew ea s s u m e J=1.Because the analytical form of the eigenvalues and eigenvector of the above matrix is toocomplicated, we assign a speciﬁc number to ε,e.g.,ε=0.5. The corresponding diagonal matrix then reads P=⎛ ⎝−0.313433 −0.516706 0 .796727 0.796727 0 .313433 0 .516706 −0.516706 0 .796727 0 .313433⎞ ⎠, and the eigenvalues of three normal modes are ω 0=H+0.351465 , (E2) ω1=H+1.6066, (E3) ω2=H+3.54194 . (E4) In the following numerical calculations, we use parameters H=kB=α=1,T 0=1,T 1=2,andT2=3.The three eigenfrequencies are then ω0=1.351465 ,ω 1=2.6066,and ω2=4.54194 .The equations of motions for normal modes become (i+α)dφ0 dt=−ω0φ0+ν/parenleftbig 0.193548 \|φ0\|2φ0+0.516129 \|φ0\|2φ2 −0.0645161 φ∗ 0φ2 2+0.258065 \|φ0\|2φ1 +0.258065 φ∗ 0φ1φ2−0.129032 φ∗ 0φ2 1 +0.258065 φ2 0φ∗ 2−0.129032 φ0\|φ2\|2 +0.483871 \|φ2\|2φ2+0.258065 φ0φ1φ∗ 2 −0.387097 φ1\|φ2\|2+0.258065 φ2 1φ∗ 2 +0.129032 φ2 0φ∗ 1+0.258065 φ0φ∗ 1φ2 −0.193548 φ∗ 1φ2 2−0.258065 φ0\|φ1\|2 +0.516129 \|φ1\|2φ2+0.0645161 \|φ1\|2φ1/parenrightbig +ζ0(t), (E5) (i+α)dφ1 dt=−ω1φ1+ν/parenleftbig 0.0645161 \|φ0\|2φ0+0.258065 \|φ0\|2φ2 +0.129032 φ∗ 0φ2 2+0.516129 \|φ0\|2φ1 −0.258065 φ∗ 0φ1φ2−0.193548 φ∗ 0φ2 1 +0.129032 φ2 0φ∗ 2+0.258065 φ0\|φ2\|2 −0.193548 \|φ2\|2φ2−0.258065 φ0φ1φ∗ 2 +0.516129 φ1\|φ2\|2+0.0645161 φ2 1φ∗ 2 +0.258065 φ2 0φ∗ 1−0.258065 φ0φ∗ 1φ2 +0.258065 φ∗ 1φ2 2−0.387097 φ0\|φ1\|2 +0.129032 \|φ1\|2φ2+0.483871 \|φ1\|2φ1/parenrightbig +ζ1(t), (E6) 024417-14ENERGY REPARTITION IN THE NONEQUILIBRIUM . . . PHYSICAL REVIEW B 95, 024417 (2017) FIG. 14. Time evolution of function ωkφ∗ k(t)φk(t)/2 for the three normal modes ( k=0,1, and 2) in a spin trimmer. The nonlinearity strength is ν=− 0.5. (i+α)dφ2 dt=−ω2φ2+ν/parenleftbig 0.483871 \|φ0\|2φ0 +0.387097 \|φ0\|2φ2 +0.258065 φ∗ 0φ2 2+0.129032 \|φ0\|2φ1 +0.258065 φ∗ 0φ1φ2+0.258065 φ∗ 0φ2 1 +0.193548 φ2 0φ∗ 2+0.516129 φ0\|φ2\|2 −0.0645161 \|φ2\|2φ2+0.258065 φ0φ1φ∗ 2 +0.258065 φ1\|φ2\|2−0.129032 φ2 1φ∗ 2 +0.0645161 φ2 0φ∗ 1+0.258065 φ0φ∗ 1φ2 +0.129032 φ∗ 1φ2 2+0.516129 φ0\|φ1\|2 −0.258065 \|φ1\|2φ2−0.193548 \|φ1\|2φ1/parenrightbig +ζ2(t), (E7) with ζ0(t)=0.796727 ξ0(t)+0.516706 ξ1(t)+0.313433 ξ2(t), (E8)FIG. 15. Renormalization of mode temperatures in a spin trimer, tuned by the nonlinearity parameter ν.T00,T11,a n dT22represent the temperatures of normal modes for k=0,1,andk=2, respectively, without nonlinearity. ζ1(t)=− 0.516706 ξ0(t)+0.313433 ξ1(t)+0.796727 ξ2(t), (E9) ζ2(t)=−0.313433 ξ0(t)+0.796727 ξ1(t)−0.516706 ξ2(t), (E10) in which source terms ξ0(t)=ξ01(t)+iξ02(t),ξ 1(t)= ξ11(t)+iξ12(t), and ξ2(t)=ξ21(t)+iξ22(t) with Gaus- sian white noises (Wiener process) ξ01(t),ξ 02(t),ξ 11(t), ξ12(t),ξ21(t),andξ22(t). Figure 14shows the dynamics of function ωkφ∗ k(t)φk(t)/2 withk=0,1, and 2. We simulate 2 ×106steps with a time step 0.01 for the time evolution. Ito interpretation is adoptedto integrate the above stochastic differential equations. The time average of ω kφ∗ k(t)φk(t)/2 represents the temper- ature of the normal mode. Numerical simulations for every ν are repeated 20 times in order to diminish the sample deviation(Fig. 14is one example of them at ν=− 0.5). 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PhysRevB.98.214410.pdf	PHYSICAL REVIEW B 98, 214410 (2018) Impact of Joule heating on the stability phase diagrams of perpendicular magnetic tunnel junctions N. Strelkov,1,2,A. Chavent,1A. Timopheev,1R. C. Sousa,1I. L. Prejbeanu,1L. D. Buda-Prejbeanu,1and B. Dieny1,† 1Université Grenoble Alpes, CNRS, CEA, Grenoble INP , INAC-SPINTEC, 38000 Grenoble, France 2Department of Physics, Moscow Lomonosov State University, Moscow 119991, Russia (Received 17 September 2018; revised manuscript received 21 November 2018; published 7 December 2018) Measured switching voltage-ﬁeld diagrams of perpendicular magnetic tunnel junctions exhibit unexpected behavior at high voltages associated with signiﬁcant heating of the storage layer. The boundaries deviate fromthe critical lines corresponding to the coercive ﬁeld, which contrasts with the theoretically predicted behaviorof a standard macrospin-based model. Combining recent experimental studies of the temperature dependence ofspin polarization and perpendicular magnetic anisotropy, we are proposing a modiﬁed model. Our approach takesinto account the Joule heating during the writing pulse, which reduces the spin polarization and the anisotropy,thereby reducing the spin torque efﬁciency and the coercive ﬁeld during the switching. The numerical macrospinsimulations based on this model are in agreement with our experimental measurements and consistent with theresults derived from the linearization of Landau-Lifshitz-Gilbert equation. DOI: 10.1103/PhysRevB.98.214410 I. INTRODUCTION A large number of studies related to spin transfer torque (STT) switching [ 1,2] in magnetic tunnel junctions (MTJ) were conducted over the past few years in support of thedevelopment of spin transfer torque magnetic random accessmemory (STT-MRAM). This memory uses as storage ele-ments MgO-based magnetic tunnel junctions with perpen-dicular (i.e., out-of-plane) magnetization (pMTJs) [ 3,4]. The common stacks involve MgO/FeCoB interfaces allowing toreach the technological requirements in terms of magnetore-sistance variation around 200% [ 5,6] and thermal stability factor at room temperature around 60 [ 7,8]. The main prop- erties of the memory MTJ cell can be extracted from theirwriting voltage-ﬁeld (V-H) diagram [ 9]. Experimental obser- vations and theoretical modeling of such diagrams in collinear[6,10–12] and noncollinear geometries [ 13]g i v ea na l m o s t complete description of STT-induced switching processes inMTJ. However, at high voltages (typically above 0.4–0.8 Vdepending on the junction resistance), most experimentaldiagrams show deviation from the theoretically predictedbehavior. A plausible origin of this effect is the heating dueto Joule dissipation around the tunnel barrier and the resultingvariation of the storage layer magnetic properties. It is wellknown that the coercive ﬁeld of magnetic layers decreaseswith temperature [ 14,15]. STT reduction due to the heating was also noticed earlier in experiments on current-induceddomain-wall motion [ 16,17] and it was observed both in spin valves [ 18] and MTJs [ 19]. Bandiera et al. proposed a concept of thermally assisted STT-MRAM in which the heating dueto the STT write current is also used to induce an anisotropyreorientation from out of plane to in plane [ 20]. Moreover, nik@magn.phys.msu.ru †bernard.dieny@cea.frrecent experiments have shown that the MgO/CoFeB interface with thin MgO tunnel barrier have a much lower (more thantwo orders of magnitude) thermal conductivity than whatcould be expected from the bulk value [ 21]. In double-barrier MTJs in which the storage layer is sandwiched between twoMgO barriers, this further enhances the Joule heating of thestorage layer. In this paper, we present a series of experimental studies and their theoretical interpretation using the macrospin modelthat takes into account the heating effects described above.These results point out the strong inﬂuence of the Joule heat-ing on switching abilities of MTJs at high voltages. Consider-ing the importance of these heating effects, it is mandatory toinclude these heating effects in MRAM design tools. II. EXPERIMENTS We performed a number of experiments using different pillars based on pMTJ stack with different compositions andsizes. Typical switching diagrams are summarized in Fig. 1for various kinds of samples varying the composition of the stackand the nanofabrication procedure (see Table I). The wafers in Figs. 1(a),1(b), and 1(d) were grown by dc and rf mag- netron sputtering on a thermally oxidized Si substrate. Thesample from Fig. 1(c) was grown on prepatterned conducting nonmagnetic pillar without post-deposition etching [ 22]. The storage layer from Figs. 1(a),1(b), and 1(d) consists of two ferromagnetic parts separated by a thin W ﬁlm andsandwiched between two MgO layers to increase the inter-facial anisotropy: the main MgO barrier which provides thetunneling magnetoresistance (TMR) and a thinner MgO layerwith lower resistance-area (RA) product. Next to the MgOlayer, there is a thin W capping layer is inserted to absorb theB away from the MgO interfaces upon annealing [ 6,12,23]. The magnetization of the bottom reference layer is pinned bya synthetic antiferromagnet (SAF). 2469-9950/2018/98(21)/214410(9) 214410-1 ©2018 American Physical SocietyN. STRELKOV et al. PHYSICAL REVIEW B 98, 214410 (2018) -400 -200 0 200 400 600 800-0.8-0.40.00.40.8 APPR(Ω)Voltage (V) 8451106136816291890 P/AP(a) S1 -500 0 500 1000 1500-0.8-0.40.00.40.8 S2(b) R(Ω) 13301791225327143175 -400 -200 0 200 400 600-0.6-0.30.00.30.6 S3(d) (c) H( O e )Voltage (V) 105107110112114R(Ω) -1600 -800 0 800 1600 2400-0.8-0.40.00.40.8 S4 H( O e )8601003114512881430R(Ω) FIG. 1. Experimental stability V-H diagrams for the different samples based on pMTJ stacks. The color scale corresponds to the resistance of pMTJ. Field shift with respect to the H=0 Oe corresponds to an uncompensated stray ﬁeld Hstrayfrom the SAF (a) 12 R-H loops, Hstray=192 Oe, TMR =124%; (b) 12 R-H loops, Hstray=352 Oe, TMR =139%; (c) 12 R-H loops, Hstray=72 Oe, TMR =9%; (d) 10 R-H loops, Hstray=379 Oe, TMR =66%. See sample details in Table I. The measurement process to derive the V-H diagrams is the same as described in Ref. [ 11]. A magnetic ﬁeld is applied out of plane, i.e., along the easy axis of magnetization ofthe storage layer and slowly varied (typically 50 kOe /s) after each write-read sequence. A 100 ns writing voltage pulse ofdeﬁned amplitude is applied at each magnetic ﬁeld point. Theresistance is measured under low constant (dc) current afterapplying the writing pulse. The resistance-ﬁeld (R-H) loop isrepeated several times, and the ﬁnal resistance value is deﬁnedby averaging all these loops. In the Fig. 1, one can see V-H diagrams with various shapes. Three main stability areas are identiﬁed: the high- TABLE I. Stack composition of the used samples. No. Figure Sample description S1 Fig. 1(a) SAF/MgO/FeCoB(1.3 nm)/W(0.3 nm)/ FeCoB(0.5 nm)/MgO/capping, D=110 nm, RA =10/Omega1×μm2 S2 Fig. 1(b) SAF/MgO/FeCoB(1.3 nm)/W(0.3 nm)/ S2b Fig. 2, FeCoB(0.5 nm)/MgO/capping, Fig. 7(b) D=80 nm, RA =10/Omega1×μm2 S3 Fig. 1(c), SAF/MgO/FeCoB(1.5 nm)/ Fig. 6 W(2 nm)/capping, Fig. 7(a) D=300 nm, RA =9/Omega1×μm2 S4 Fig. 1(d) SAF/MgO/FeCoB(0.9 nm)/W(0.1 nm)/ S4b Fig. 3 Co(0.5 nm)/W(0.1 nm)/FeCoB(0.8 nm)/ capping, D=70 nm, RA =9/Omega1×μm2resistance area with antiparallel (red, AP) conﬁguration of storage and reference layer, the low-resistance area with par-allel (blue, P) conﬁguration, and bistable (green, P/AP) areawhere both mentioned conﬁgurations are possible. The centerof the bistable area is shifted with respect to the zero ﬁeld.This shift corresponds to an uncompensated stray ﬁeld H stray from the SAF. The shape of the diagram in Fig. 1(a) is predictable by the theory developed in Ref. [ 11]: there are straight verti- cal boundaries corresponding to the coercive ﬁeld. But, thethree other diagrams show deviations from the theoreticallyexpected shape. This unpredictable behavior can be due toJoule heating effects inside the storage layer. A number ofrecent experiments support this interpretation. For example,it was observed earlier that the STT write current ﬂowingthrough pMTJs could induce a decrease of both PMA andcoercivity due to Joule heating around the tunnel barrier [ 20]. In Fig. 1(c), it can be clearly seen that the critical lines lean back towards the H strayvalue at high voltages and completely coincide with Hstray when the absolute value of the applied voltage exceeds ∼0.6 V. This could mean that the PMA and spin polarization in storage layer are reduced by Jouleheating to the point where the critical lines then correspondto the applied ﬁeld value which compensates the SAF strayﬁeldH stray. Further comparing Figs. 1(a) and1(b), a stronger thermal inﬂuence can be noted in the smaller junction. Thiscan be explained by considering that smaller samples shouldbe more sensitive to the temperature variation, as the stabilityfactor of MTJ in macrospin approximation /Delta1=K eff/Omega1/(kBT) depends on the volume /Omega1of the storage layer, Keffbeing the effective perpendicular anisotropy constant. 214410-2IMPACT OF JOULE HEATING ON THE STABILITY … PHYSICAL REVIEW B 98, 214410 (2018) -1200 -800 -400 0 400 800 1200-0.8-0.40.00.40.8S2R-Rmin Rmax-RminVoltage (V) H( O e )0.00.20.40.60.81.0 FIG. 2. Normalized experimental stability V-H diagram mea- sured under constantly applied (dc) voltage. Sample is the same asin Fig. 1(b) (see details in Table I). For a given voltage, one might expect that the longer the writing pulse, the higher the temperature reached in the stor-age layer and therefore stronger the impact of the heating. Tocheck this effect on our samples, the measurement procedurewas changed. A constant (dc) voltage was applied to thesample and the applied ﬁeld varied step by step from −1400 to 1400 Oe and vice versa. At each ﬁeld step, the current wasmeasured during 500 μs and the resulting resistance calcu- lated as R=V bias/I.I nF i g . 2, the normalized experimental V-H diagram measured using this procedure is shown. Ascan be seen, there are no more vertical straight boundariescorresponding to ﬁeld equal to coercive ﬁeld. Since here thecurrent is steadily ﬂowing through the sample, the magnetictunnel junction cannot cool down between each resistancemeasurement, so that a signiﬁcant decrease of the anisotropydue to Joule heating takes place as the bias voltage is graduallyincreased. To estimate the temperature variation and its voltage de- pendence in MTJs, we performed an experiment by themethod described in Ref. [ 24] based on the temperature de- pendence of Ruderman-Kittel-Kasuya-Yosida (RKKY) cou-pling, using the sample similar to that from Fig. 1(d) but with larger diameter D=170 nm. We measured the V-H diagram under dc voltage and extracted the voltage dependence ofspin-ﬂop ﬁeld H SF[white solid line in Fig. 3(a)], at which both SAF layers become magnetized in parallel conﬁgurationversus applied voltage. This dependence is parabolic, and itsexpression can be derived from Fig. 3(b). We then measured the variation of H SFﬁeld versus controlled temperature which is found to be almost linear [Fig. 3(c)]. By combining the data from Figs. 3(b) and3(c), the temperature elevation /Delta1T due to current ﬂow versus applied voltage can be derived. Aquadratic variation is obtained given by /Delta1T=308 V 2and shown in Fig. 3(d) (Tin K,Vin volt). On the V-H diagrams in Fig. 1one can note some asymme- try between the critical lines for positive and negative voltagepolarities. For example, the absolute value of switching volt-ageV swatH=Hstray(black vertical line) corresponding to zero applied ﬁeld is different for positive ( V+ sw) and negative (V− sw) voltages. One of the possible reasons for such asym- metry might be due to the different resistances for P-AP andAP-P transitions. If the heating power at ﬁxed voltage reads asV2/R, then lower resistance causes more heating and, as a result, a greater decrease of PMA and spin polarization.In Fig. 1(a), the TMR amplitude is 124%, V + sw=0.37 V (AP-P transition with high resistance and lower temperature),V − sw=0.33 V (P-AP transition with low resistance and higher temperature). The switching voltage difference for this sampleis 12%. In Fig. 1(b), the TMR amplitude is 139%, V + sw= 0.37 V, V− sw=0.3 V, and the difference is 23%. For the sample with TMR amplitude 66% in Fig. 1(d),V+ sw=0.54 V, V− sw=0.52 V, and the difference is 4%. So, we can state that the sample with larger TMR amplitude has more pronouncedasymmetry in its V-H diagram due to larger Joule heating in Pstate (low resistance) than in AP state (high resistance). In Fig. 1(c), the sample has a very low TMR amplitude, so the mechanism of asymmetric heating described aboveis negligible. But, one can still see a small difference inswitching voltage: V + sw=0.32 V,V− sw=0.34 V. This polarity dependence in the shape of V-H diagram might originate fromthe heating asymmetry induced by tunneling current [ 25]. Most of Joule energy is released after the electrons’ tunnelingin the receiving magnetic electrode adjacent to the MgObarrier. Therefore, considering that the thermal conductivityof thin MgO barrier is much lower than that of bulk MgO[21], the storage layer reaches up to 10% higher temperature if the current polarity is such that the storage layer receives thetunneling electrons. This effect becomes more signiﬁcant inthe case of MgO-based capping layer since the storage layerappears to be thermally insulated between two MgO layerswith low thermal conductivity. Another source of asymmetry in the V-H diagram can result from the voltage dependence of the torque amplitudeassociated with elastic tunneling in MTJ [ 26]. Indeed, the parallel component of the torque T /bardblvaries as T/bardbl∼a1V+ a2V2. The transport parameter a2is known to vanish for MTJs with ferromagnetic electrodes having large spin splitting orbeing half-metallic. Such MTJs are expected to exhibit highTMR amplitude. In such MTJs, if heating effects are nottaken into account, the switching voltages for P-AP and AP-Ptransitions are therefore close to opposite ( V + sw=V− sw). On the other hand, for low TMR MTJs, the parallel torque exhibitsa nonsymmetric quadratic voltage behavior so that switchingvoltages do not coincide ( V + sw/negationslash=V− sw). To get deeper insight on the above-mentioned features of the experimental V-H diagrams, we developed a model ac-counting for the heating effects in the storage layer. The stan-dard phenomenological macrospin-based model was modiﬁedby including a dependence of the storage layer temperatureon voltage and a resulting dependence of magnetization, spinpolarization, and anisotropy constant on the applied voltagepulse. III. MODEL The ﬁrst adjustment of the model concerns the change in anisotropy energy while changing the bias voltage. There areseveral possible mechanisms affecting the uniaxial anisotropyconstant. A ﬁrst one is a direct voltage induced variation ofanisotropy due to interfacial charge modulation [ 27–30], but this effect is relatively weak compared to the inﬂuence of 214410-3N. STRELKOV et al. PHYSICAL REVIEW B 98, 214410 (2018) 0.0 0.1 0.2 0.3 0.4 0.50.50.60.70.80.91.0 0 1 02 03 04 05 06 07 00.850.900.951.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8020406080100120140160-3000 -2000 -1000 0 1000 2000 3000-0.6-0.4-0.20.00.20.40.6(b) H( O e )Voltage (V) 374436497559620S4bR(Ω)(a) HSF=1 . 0-0 . 5 9 1 ⋅V2Normalized spin-flop field Voltage squared (V2)(c)Normalized spin-flop field ΔT( K )HSF= 1.0 - 0.00192 ⋅ΔT(d) ΔT = 308 ⋅V2ΔT( K ) Voltage (V) FIG. 3. (a) Experimental stability V-H diagram measured under dc voltage for the sample similar to that from Fig. 1(d) withD=170 nm and extracted spin-ﬂop ﬁeld-voltage dependence (white solid line). (b) Linear ﬁt of normalized spin-ﬂop ﬁeld-voltage squared dependence. (c) Linear ﬁt of normalized spin-ﬂop ﬁeld-temperature variation dependence. (d) Temperature elevation due to current ﬂow versus voltage derived by combining the ﬁts from Figs. 3(b) and3(c). the temperature variation associated with Joule heating. A second mechanism is a temperature-induced variation of theanisotropy due to Joule heating, which is the most signiﬁcantfor our samples. Concerning other thermoelectric effects, asit was shown by Flipse et al. [31] and later by Avery and Zink [ 32] the Peltier coefﬁcient is of order /Pi1∼10 -3V and thus the Peltier power at R∼103/Omega1andV∼1Vi s Q/Pi1∼ /Pi1×V/R∼10-6W, while the Joule power QJ=V2/R∼ 10-3W is larger by three orders of magnitude. Seebeck effect is also negligible as its coefﬁcient Sis expressed through /Pi1 by Thomson-Onsager relation /Pi1=ST. We ﬁrst estimated the temperature dependence on the bias voltage in MTJ starting from the ﬁrst law of thermodynamicsand describing the MTJ with a simple one-dimensional modelaccording to the following differential equation: V 2 R−Q(T−T0)=CdT dt, (1) where V2/Ris a Joule power, Ris resistance, Qis the heat transfer coefﬁcient between the tunnel barrier and theconductive bottom and top electrodes assumed to be at a ﬁxedtemperature T 0,T0is the room temperature, and Cis the MTJ heat capacity. Equation ( 1) gives a solution for the temperature Tas a function of pulse duration tp: T=T0+V2 RQ(1−e−(Q/C )tp). (2) We assume that the temperature of the storage layer follows this square dependence on voltage pulse amplitude ( 2) and all other parameters are kept constant for a given sample. Then, asecond step in the model development consists in assuming that the temperature dependence of the uniaxial anisotropyconstant follows the Callen-Callen law [ 33]: K(T)=K 0/parenleftbiggMs(T) Ms0/parenrightbiggξ , (3) where K0is the uniaxial anisotropy constant at zero tempera- ture,Ms0is the spontaneous magnetization of the storage layer at zero temperature. From previous studies, the value of theexponent ξwas experimentally estimated to lie in the range between 2 and 3 [ 20,34–37]. The temperature dependence of the storage layer magnetization is supposed to follow a Blochlaw: M s(T)=Ms0(1−(T/T c)3/2), (4) where Tcis the Curie temperature of the storage layer. The exponent “3 /2” is slightly different for thin deposited ﬁlms, and its experimentally determined value equals to 1.73 [ 34]. Taking into account ( 3) and ( 4) and assuming that pulse durations are long enough so that the temperature varies withthe voltage as T=T 0+kVV2according to ( 2), the voltage dependencies of the magnetization and anisotropy constantread as M s(V)=Ms0/bracketleftBigg 1−/parenleftbiggT0+kVV2 Tc/parenrightbigg1.73/bracketrightBigg , K(V)=K0/bracketleftBigg 1−/parenleftbiggT0+kVV2 Tc/parenrightbigg1.73/bracketrightBiggξ ,(5) 214410-4IMPACT OF JOULE HEATING ON THE STABILITY … PHYSICAL REVIEW B 98, 214410 (2018) where kVis a parameter dependent on the writing pulse duration, the thermal capacity of the storage layer, electricalresistance, and the thermal resistance of the stack around thestorage layer. The parameter k Vdepends also on the polarity of the applied voltage as mentioned earlier [ 25]. Actually in the following of our model, the heating asymmetry was nottaken into account and k Vwas kept independent on current polarity. We want to underline that in case of pulsed V-H diagram, the observed back switching anomalies are not only causedby the voltage-induced anisotropy reduction but also by thereduction of spin polarization. As the anisotropy amplitudeis restored at the end of the voltage pulse, the fact that thestorage layer had time to switch or not during the pulse has agreat importance. If STT efﬁciency becomes too weak due tothe heating, the storage layer may not be able to switch anymore. This is the reason why we can cross the critical linestwice on V-H diagram at a given applied ﬁeld [e.g., Fig. 1(c) increases voltage at 150 Oe from 0 to 0.6 V]. The phenomenological macrospin model used here is the same as described in Refs. [ 11,13] but it takes into account the voltage dependence of several parameters. The Landau-Lifshitz-Gilbert (LLG) equation for the magnetization dynam-ics is written as dm dt=−γ[m×μ0Heff(V)]+α/parenleftbigg m×dm dt/parenrightbigg −γa/bardbl(V)V[m×(m×p)]+γa⊥(V)V2(m×p), (6) where mis the unit vector of the magnetization of the storage layer, αis the Gilbert damping factor, γis the gyromagnetic ratio of free electrons, pis the unit vector along the spin current polarization (magnetization of the reference layer),a /bardbl(V) anda⊥(V) are phenomenological transport parameters depending on the applied voltage. The effective ﬁeld Heff(V) in our model depends on applied voltage and derives from theGibbs free-energy E(V) density functional: E(V)=−K(V)(u K·m)2−μ0Ms(V)Hext(V)·m +1 2μ0M2 s(V)/parenleftbig Nxm2 x+Nym2 y+Nzm2 z/parenrightbig , (7) where Hextis the external applied ﬁeld, uKis the unit vector along the easy axis (here Oz), Nx,y,z are diagonal terms of the demagnetizing tensor, Ms(V) andK(V) are deﬁned from ( 5). The ﬁrst term in ( 7) represents the anisotropy energy EK(V), so the corresponding effective anisotropy ﬁeld reads as HK(V)=2K(V) μ0Ms(V) =HK0/bracketleftBigg 1−/parenleftbiggT0+kVV2 Tc/parenrightbigg1.73/bracketrightBiggξ , (8) where HK0=2K0/(μ0Ms0). The dependence of normalized effective anisotropy ﬁeld ( 8) on applied voltage for different values of exponent ξis presented in Fig. 4. In the inset, the same dependence is plotted versus the temperature reachedby the storage layer during the voltage pulse T=T 0+kVV2. This dependence is almost linear as it was shown in theexperiment [ 34,38]. Assuming the ground state of storage0 . 00 . 20 . 40 . 60 . 80.00.20.40.60.81.0Hk(V) / Hk0 Voltage (V) ξ 2.0 2.2 2.4 2.6 2.8 3.0400 600 800 1000 12000.00.20.40.60.81.0Hk(V) / Hk0 Temperature (K) FIG. 4. Normalized effective anisotropy ﬁeld versus applied voltage according to ( 8) for different values of the exponent ξusing the parameters T0=300 K, kV=1900 K /V2,Tc=1200 K. The curve for ξ=2 (solid black line) repeats the normalized magneti- zation Ms(V)/Ms0. In the inset, the same function versus resulting temperature T=T0+kVV2. layer to be out of plane, the ﬁeldlike torque term ∼a⊥(V)i n (6) is neglected here since the Slonczewski term ∼a/bardbl(V)i s known to play a dominant role in the switching process. Thephenomenological transport parameter a /bardbl(V) is proportional to the spin polarization which follows the magnetization vari-ation given by ( 5) (see solid black line in Fig. 4) as reported in Ref. [ 19]. Consequently, due to the Joule heating, the STT prefactor has the following dependence on voltage: a /bardbl(V)=a/bardbl0/bracketleftBigg 1−/parenleftbiggT0+kVV2 Tc/parenrightbigg1.73/bracketrightBigg , (9) where a/bardbl0represents the transport parameter value at zero temperature. Having set the model as described above, the numerical switching V-H diagrams are computed following the experi-mental procedure. The magnetization of the storage layer isrelaxed under the given applied ﬁeld. After that, a writingpulse is applied for 100 ns. The total integration time is 1 μs. The amplitude of applied ﬁeld is changed from its minimum tomaximum value and vice versa to reproduce a hysteresis loop.The initial room temperature value was set at T 0=300 K. The thermal ﬂuctuations are described by random ﬁeld addedto the total effective ﬁeld as a white-noise ﬁeld with thefollowing properties: /angbracketleftH TR(t)/angbracketright=0, /angbracketleftHTR(t)HTR(t/prime)/angbracketright=2αkBT γμ 0Ms/Omega1δ(t−t/prime), (10) where kBis the Boltzmann constant and /Omega1is the sample volume. Each hysteresis loop at ﬁxed voltage was repeated10 times, resulting in an averaged V-H loop similar to theexperimental diagram. The numerical V-H diagrams computed for a disk of 100 nm diameter and 1.6 nm thick are presented in Fig. 5for several values of the exponent ξ. The shape of the diagram 214410-5N. STRELKOV et al. PHYSICAL REVIEW B 98, 214410 (2018) -0.8-0.40.00.40.8mzVoltage (V)ξ=2.0-1.0 -0.5 0.0 0.5 1.0 ξ=2.2 -0.8-0.40.00.40.8 ξ=2.4Voltage (V)ξ=2.6 -4000 -2000 0 2000 4000-0.8-0.40.00.40.8ξ=2.8 H( O e )Voltage (V) -4000 -2000 0 2000 4000ξ=3.0 H( O e ) FIG. 5. Numerical stability V-H diagrams for different values of exponent ξat room temperature T0=300 K, kV=1900 K /V2, Tc=1200 K, Ms0=106A/m,K0=778319 J /m3,α=0.01, a⊥0=0,a/bardbl0=8×10-3T/V,p=(0,0,−1),Nx=Ny=0.025, Nz=0.95 using 10 averaged loops for each voltage with resolution 64×64 points. Critical lines (white solid curves) calculated from Eq. ( 18) and superposed with the numerical V-H diagrams. Vertical dashed lines correspond to the coercive ﬁeld calculated from the solution of the Eq. ( 17) at zero applied voltage. For ξ=3.0t h e intersection points of two solutions are circled. evolves progressively with the value of the exponent ξ.T w o regimes are identiﬁed. At low values of ξ(around 2), the numerical diagram is very similar to the experimental diagramplotted in Fig. 1(d). In contrast, when ξapproaches the upper limit of 3, a strong similarity with the experimental diagram,Figs. 1(b) and 1(c), is observed. In this second regime, two speciﬁc critical lines are identiﬁed: a ﬁrst one, STT driven, atlow voltages and a second one, heat driven, at high voltages.The parameter ξaccounts for the strength of the dependence of the material parameters on the voltage. The impact ofits value on the shape of the V-H diagrams is quite drastic,conﬁrming the important role of Joule heating on switchingabilities of MTJs. To better understand the underlying param-eters behind these critical lines and eventually ﬁnd how tocontrol them, we developed an analytical model based on thestability analysis of the linearized LLG equation.IV . ANALYTICAL MODEL The critical lines of the numerically obtained V-H di- agrams presented in Fig. 4(white lines) can be extracted analytically at zero temperature using the generic model of anonlinear auto-oscillator proposed by Slavin and Tiberkevich[39]. Following the Holstein-Primakoff transformation, the components of the unitary magnetization vector min (6)a r e replaced by the canonical variables candc ∗as proposed in Ref. [ 40]: c=mx−ımy√2(1+mz). (11) It is convenient to express the modiﬁed LLG equation ( 6) in complex variables with four terms: precession, damping,ﬁeldlike, and dampinglike: dc dt=dc dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle prec+dc dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle damp+dc dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle FL+dc dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle DL. (12) Assuming that the polarization vector p=(0,0,pz) and the applied ﬁeld Hext=(0,0,Hz) are both perfectly out of plane, the full expression of each term is explicitly given below: dc dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle prec=−ıc[ωH+(ωM−ωA)(2\|c\|2−1)], dc dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle damp=−αc(1−\|c\|2) ×[ωH+(ωM−ωA)(2\|c\|2−1)], dc dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle FL=ı(αω/bardbl+ω⊥)pzc, dc dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle DL=− (ω/bardbl−αω⊥)(1−\|c\|2)pzc,(13) where ωH=γ 1+α2Hz,ω A=γ 1+α22Ku μ0Ms, ω/bardbl=γ 1+α2a/bardblV μ0,ω ⊥=γ 1+α2a⊥V2 μ0, (14) ωM=γ 1+α2(Nz−Nx)Ms. Since the complex variable chas an amplitude pand a phase φassociated by the relation c(t)=√p(t)eφ(t), one can then derive two equations: dp dt=− 2α[ωH+(ω/bardbl/α−ω⊥)pz +(ωM−ωA)(2p−1)](p−1)p, (15) dφ dt=−[ωH−(αω/bardbl+ω⊥)pz+(ωM−ωA)(2p−1)]. Looking for the stationary solutions dp/dt =0, two static solutions are found: (i) p0=0 meaning magnetization “up” mz=+ 1 and (ii) p0=1 meaning magnetization “down” mz=− 1. By linearizing the equation dp/dt in (15) around the solution p0and keeping only the ﬁrst-order terms, one can 214410-6IMPACT OF JOULE HEATING ON THE STABILITY … PHYSICAL REVIEW B 98, 214410 (2018) derive an equation for the power deviation δp: d(δp) dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle p0=⎧ ⎪⎨ ⎪⎩−2α[ωH+(ω/bardbl/α−ω⊥)pz +(ωM−ωA)]δp, p 0=1 +2α[ωH+(ω/bardbl/α−ω⊥)pz −(ωM−ωA)]δp, p 0=0.(16) If the prefactor in front of δpbecomes positive, the deviation δpwill diverge and the static equilibrium becomes unstable. From this condition, by setting the prefactors to zero, onecan derive the boundaries of stability under static ﬁeld and/orvoltage. Two critical lines for the applied ﬁeld without anyapplied voltage are given by ω H+(ωM−ωA)=0, ωH−(ωM−ωA)=0.(17) The two critical lines for dc ﬁeld and voltage are ωH+(ω/bardbl/α−ω⊥)pz+(ωM−ωA)=0, ωH+(ω/bardbl/α−ω⊥)pz−(ωM−ωA)=0.(18) Substituting Ms,Ku, anda/bardblin (14) by the corresponding functions of the bias voltage given by expressions ( 5) and ( 9), the solution of Eqs. ( 17) and ( 18) gives critical the curves H(V) which coincide well with the boundaries of the numer- ical diagrams. In Fig. 5, the critical lines are superposed to the numerical V-H diagrams. Vertical dashed lines correspondto the coercive ﬁeld calculated from the solution of Eq. ( 17) at zero applied voltage, which corresponds to a ﬁnite pulseduration. If the voltage pulse is inﬁnite (dc), the solution(17) reproduces the boundaries like in Fig. 2(a): coercive ﬁeld decreases with applied bias voltage. The small differencebetween the numerical and analytical coercive ﬁeld is dueto the ﬂuctuating term ( 10). It increases the probability of magnetization switching before the ﬁeld reaches its coercivevalue and yields this 10% deviation. The critical lines under dcﬁeld and dc voltage are rather complex and partially recoverthe numerical boundaries at low ξand low voltage pulses. At ξ value close to 3, the critical lines at high voltages are closer tothat in the dc diagram from Fig. 2(a). The analytical dc curves reproduce well the two intersection points through which thenumerical critical lines pass. In the case where ξequals 2 or 3 and the magnetization follows the Bloch law ( 4), one can ﬁnd the solution for intersection points of the critical lines deﬁned by Eq. ( 18) (see yellow circles in Fig. 5forξ=3.0). Considering that the magnetization depends on voltage due to Joule heating andintroducing the function f B(V)i n( 5) deﬁned by fB(V)= Ms(V)/Ms0, neglecting the ﬁeldlike torque term ω⊥=0 and assuming the polarization is “up” pz=1, one gets the follow- ing equations from ( 14) and ( 18): Hz+a/bardbl0V μ0αfB(V)+/bracketleftbig Ms0(Nz−Nx)fB(V) −HK0fξ−1 B(V)/bracketrightbig =0, Hz+a/bardbl0V μ0αfB(V)−/bracketleftbig Ms0(Nz−Nx)fB(V) −HK0fξ−1 B(V)/bracketrightbig =0, (19) where HK0=2K0/(μ0Ms0). Now, it is clear that to satisfy Eq. ( 19) at the same value of applied ﬁeld Hz, the last termin brackets should be equal to zero. If the exponent ξ=2, then the solution is trivial: fB(V)=0. The intersection points correspond to heating the storage layer up to the Curietemperature: V c0=±√(Tc−T0)/kV. If the exponent ξ=3, then aside from the trivial solution, there is another one whichreads as V c1=±/radicalBigg Tc kV/radicalBigg (1−η)2/3−T0 Tc, (20) where η=Ms0(Nz−Nx)/HK0. As one can see, the ratio Vc0/Vc1depends on the factor ηand Curie temperature Tc. The thermally induced slope that passes through these intersection points also depends on these parameters but doesnot depend on the heating coefﬁcient k V: dV dH/vextendsingle/vextendsingle/vextendsingle/vextendsingle ξ=3=Vc1−Vc0 Hc1 =−1 ημ0α a/bardbl0/parenleftBigg 1−√1−T0/Tc/radicalbig (1−η)2/3−T0/Tc/parenrightBigg .(21) Therefore, this slope should not depend on the voltage pulse duration as it does not depend on kV. On the other hand, if the exponent ξ=2, the thermally induced slope can be calculated as an inverted derivative of the function Hz(V) extracted from ( 19) taken at the point V=Vc0: dV dH/vextendsingle/vextendsingle/vextendsingle/vextendsingle ξ=2=/parenleftBigg dH dV/vextendsingle/vextendsingle/vextendsingle/vextendsingle V=Vc0/parenrightBigg−1 =−1 3/bracketleftbigga/bardbl0 μ0α/parenleftbigg 1−T0 Tc/parenrightbigg +Ms0(Nz−Nx) ×1−η η/radicalBigg kV Tc/radicalbig 1−T0/Tc/bracketrightbigg−1 . (22) The slope slightly increases with decreasing writing pulse duration. In Fig. 6, the critical lines extracted from the ex- perimental diagrams for different values of the writing pulse -200 -100 0 100 200 300 400-0.50.00.5Voltage (V) H( O e )t( n s ) 15 20 30 50 100 150 200 FIG. 6. Critical lines extracted from the experimental diagrams of the sample from Fig. 1(c) for different lengths of applied voltage pulse. Thermally induced boundaries remain almost constant at pulses longer than 30 ns and move upwards for shorter pulses with slight increase of the slope. See sample details in Table I. 214410-7N. STRELKOV et al. PHYSICAL REVIEW B 98, 214410 (2018) -400 -200 0 200 400 600-0.6-0.30.00.30.6 H( O e )Voltage (V) 105107110112114R(Ω)(a) S3 -400 0 400 800 1200-0.8-0.40.00.40.8S2b(b) R(Ω) H( O e )186023902920345039804510 FIG. 7. Experimental diagrams ﬁt with the analytical solution. (a) Sample from Fig. 1(c) with low TMR, ξ=2.16,Ms0=900 kA /m, K0=570 kJ /m3,kV=1200 K /V2,Tc=700 K, a/bardbl0=1.4m T/V,Nx=Ny=0.01,Nz=0.98. Other parameters the same as in Fig. 5.( b ) Sample from the same wafer as the sample in Fig. 1(b) with high TMR =142%, ξ=2.3,Ms0=900 kA /m,K0=580 kJ /m3,kV=340 K /V2, Tc=700 K, a/bardbl0=2m T/V. Dashed line corresponds to the critical curve at kV=500 K /V2. Other parameters the same as in Fig. 5. duration are presented. The switching voltage is almost con- stant at pulse durations more than 100 ns and increases sharplyfor shorter pulses as it was observed earlier [ 13]. The ther- mally induced boundaries remain almost constant at pulseslonger than 30 ns and move upward for shorter pulses withslight increase of the slope. The macrospin model used in this study gives an overes- timated value for the coercive ﬁeld which can be found fromEq. ( 17) at zero applied voltage: H c=±Ms0(Nz−Nx)/bracketleftbigg 1−/parenleftbiggT0 Tc/parenrightbiggx/bracketrightbigg ×/braceleftBigg 1 η/bracketleftbigg 1−/parenleftbiggT0 Tc/parenrightbiggx/bracketrightbiggξ−2 −1/bracerightBigg . (23) For the FeCoB-based storage layer with a diameter of 100 nm, theoretical Hcin macrospin approximation should be of the order of ∼1T( 1 04Oe). But, in the experiment, its value is usually 10 times smaller ∼0.1T( 1 03Oe) due to the nonuniform switching mode (not macrospin). Similarly, theSTT switching is not macrospin for the used MTJ diameters.Therefore, to ﬁt the experimental diagrams with this model,one has to use an effective value for the uniaxial anisotropyconstant to account for the fact that the barrier height forswitching is different from that of macrospin uniform switch-ing. The result of the ﬁtting is presented in Fig. 7. The ﬁrst sample in Fig. 7(a) has a low TMR value and the critical lines are almost symmetric. Figure 7(b) shows a diagram of the sample from the same wafer as the sample in Fig. 1(b) measured under ﬁnite voltage pulse. As the TMR amplitudeis larger than for the sample of Fig. 7(a), the asymmetry of STT boundaries is more pronounced; the STT bound-ary corresponding to the P-AP transition is shifted towardsthe zero voltage because more heat is produced in the lowresistance state (P) in the storage layer which reduces theswitching voltage. To take into account the heating asymmetryin our model, one can just take different k Vvalues for AP-P and P-AP critical lines. To roughly estimate the dependenceof the asymmetry factor ζ=(V AP→P c0−VP→AP c0 )/VP→AP c0 on TMR value, assuming that kVdepends only on resistance ofMTJ ( 2), one can use the expression ζ=√RAP/RP−1=√1+TMR−1. V . SUMMARY To conclude, we measured the stability V-H diagrams in a variety of samples of different composition and diametersin which signiﬁcant Joule heating takes place during STTwriting. We explained the high-voltage anomaly of the V-Hdiagrams as due to heating effects in the storage layer. Twomechanisms of the asymmetry of the critical lines for positiveand negative voltages were suggested: heating asymmetryinduced by the asymmetric inelastic relaxation of tunnelingelectrons and heating asymmetry due to the difference of theresistance in P and AP states. The second mechanism makesthe largest contribution to the heating asymmetry versus biasvoltage when the TMR is large. We also developed a modiﬁedmacrospin-based model by including a dependence of thematerial parameters on the applied voltage, such as uniaxialanisotropy constant ( K u), spontaneous magnetization ( Ms), and STT prefactor a/bardbl. The model was used to numerically simulate the V-H diagrams. Good agreement with the experi-mental results was obtained. Furthermore, we derived explicitanalytical expressions for the critical lines that exactly followthe contours of the numerical diagram. The obtained expres-sions can be applied to ﬁt also the experimental diagrams, butsome effective values for K uandMsshould be used, as the switching of the real samples occurs in a nonuniform regime(for example, by domain-wall propagation) which reduces thecoercive ﬁeld. 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PhysRevLett.92.126601.pdf	Dynamic Ferromagnetic Proximity Effect in Photoexcited Semiconductors Gerrit E.W . Bauer,1Arne Brataas,2Y aroslav Tserkovnyak,3Bertrand I. Halperin,3 Maciej Zwierzycki,4,and Paul J. Kelly4 1Kavli Institute of NanoScience, Delft University of T echnology, 2628 CJ Delft, The Netherlands 2Department of Physics, Norwegian University of Science and T echnology, N-7491 Trondheim, Norway 3Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 4Faculty of Science and T echnology and MESA /.0135Research Institute, University of Twente, 7500 AE Enschede, The Netherlands (Received 1 October 2003; published 22 March 2004) The spin dynamics of photoexcited carriers in semiconductors in contact with a ferromagnet is treated theoretically and compared with time-dependent Faraday rotation experiments. The long-time response of the system is found to be governed by the ﬁrst tens of picoseconds in which the excited plasma interacts strongly with the intrinsic interface between the semiconductor and the ferromagnet inspite of the existence of a Schottky barrier in equilibrium. DOI: 10.1103/PhysRevLett.92.126601 P ACS numbers: 72.25.Mk, 75.70.–i, 76.70.Fz, 78.47.+p Magnetoelectronics, the science and technology of us- ing ferromagnets in electronic circuits, is divided intotwo subﬁelds, metal-based [1] and semiconductor-basedmagnetoelectronics [2], with little common ground. Metal researchers focus mainly on topics derived from giant magnetoresistance, the large difference in the dcconductance for parallel and antiparallel magnetizationconﬁgurations of magnetic multilayers, where theoreticalunderstanding has progressed to the stage of materials-speciﬁc predictions [3]. The dynamics of the magnetiza-tion vectors in the presence of charge and spin currentshas recently received a lot of attention [4–7]. Semiconductor-based magnetoelectronics is motivated by the prospect of integrating new functionalities withconventional semiconductor electronics. The emphasishas been on the basic problem of spin injection intosemiconductors; theoretical understanding is less ad-vanced, and detailed electronic structure calculationsare just starting [8,9]. Unlike metals, semiconductorscan be studied by optical spectroscopies such as the pow- erful time-resolved Faraday or Kerr rotation techniques, in which a selected component of a spin-polarized ex-citation cloud in the semiconductor can be monitored onps time scales [10]. In n-doped GaAs, these experiments revealed long spin coherence of the order of /.0022s[10]. When the semiconductor is in contact with a ferromagnet,an initially unpolarized electron distribution preparedusing linearly polarized light was found to very quickly acquire a spin polarization—the dynamic ferromag- netic proximity (DFP) effect [11–13]. In turn, DFP efﬁ-ciently imprints spin information from a ferromagnetonto nuclear spins by dynamic nuclear polarization,opening new options for quantum information storage. In this Letter we show that metal and semiconductor- based magnetoelectronics can both be understood interms of coherent spin accumulations. Speciﬁcally, the DFP can be treated by the same formalism that success- fully describes the dynamics of the magnetization vectorin metallic hybrids [6,14]. The experiments can be under-stood in terms of a time-dependent interaction between the conduction-band electrons and the ferromagnet in a‘‘ﬁreball-afterglow’’ scenario. Photoexcited holes are in-strumental in helping the electrons to overcome the Schottky barrier between metal and semiconductor in the ﬁrst &50 ps (‘‘ﬁreball’’ regime) and induce the prox- imity effect. The interaction weakens with vanishing holedensity, thus preventing fast decay of the created spinaccumulation in the ‘‘afterglow.’’ Two groups have already contributed important in- sights into this problem. Ciuti et al. [15] interpreted the DFP in terms of a spin-dependent reﬂection of electrons at a ferromagnetic interface through a Schottky barrier in equilibrium but did not address the time dependence ofthe problem. Gridnev [16] did investigate the dynamics ofthe photoexcited carriers but postulated a phenomeno-logical relaxation tensor with a speciﬁc anisotropy thatwe ﬁnd difﬁcult to justify. W e show how both approachescan be uniﬁed and extended by ab initio magnetoelec- tronic circuit theory [17–20]. W e ﬁrst summarize the experimental evidence [12,13]. Initially, a /.0024100 fs pulse with frequency close to the band gap is absorbed by the semiconductor (100 nm of GaAs).The polarization state is then monitored by time-dependent Faraday rotation measurements of the coherentspin precession in an applied magnetic ﬁeld. The homo-geneously excited carriers (ﬁreball) thermalize within aps, in which the holes also lose any initial spin polariza- tion. The interaction time scale with the ferromagnet (Fe or MnAs) can be deduced from the rise time of thepolarization after excitation with a linearly polarized(LP) light pulse to be /.002050 ps [12]. For long delay times (afterglow), the spin relaxation is very slow ( >2n s), comparable to GaAs reference samples in the absence ofa ferromagnet. The sample can be also excited by circu-larly polarized (CP) light, in which case the ﬁreball is polarized from the outset. Dynamic nuclear polarization (DNP) by the hyperﬁne interaction can be detected bydeviations of the precession frequency from the barePHYSICAL REVIEW LETTERSweek ending 26 MARCH 2004 VOLUME 92, N UMBER 12 126601-1 0031-9007 =04=92(12) =126601(4)$22.50 2004 The American Physical Society 126601-1Larmor frequency, i.e., a modiﬁed g-factor [11]. DNP should vanish when the external magnetic ﬁeld is nor-mal to the spin accumulation. It therefore remains to be explained that Epstein et al. [12] observe a modiﬁed g-factor for this conﬁguration that differs for LP andCP excitation. Interesting additional information relatesto the material dependence, indicating that the polariza-tion induced by Fe is of the opposite sign to that inducedby MnAs [12], and to the modulation of the afterglowLarmor frequency by an applied bias [13]. Let us consider a semiconductor (SC) ﬁlm in which a nonequilibrium electron chemical potential h/.0022j/.0136 h/.0022 c;~/.0022/.0022sjis excited with charge and spin components /.0022c and~/.0022/.0022s/.0017h/.0022x;/.0022y;/.0022zjrespectively (in energy units). The bilayer parallel to the yzplane (see Fig. 1) consists of a semiconductor in contact with a metallic ferromagneticﬁlm (F) with ﬁxed single-domain magnetization in thedirection of the unit vector ~mm. Because of its relatively huge density of states, a metallic ferromagnet may be treated as a reservoir in equilibrium. A current hIj/.0136 hI c;~IIsj(in units of reciprocal time), with charge and spin components Icand~IIs, respectively, ﬂows through the ferromagnet/semiconductor ( FjSC) interface, which is governed by the spin-dependent (dimensionless) conduc-tances g ""andg##as well as the complex spin-mixing conductance g"#[17]. Physically, the real part of the mixing conductance expresses the angular momentum transfer to and from the ferromagnet, such as the strength of the spin-current induced magnetization torque [4,18]or nonlocal Gilbert damping [14], whereas the imaginarypart is an effective magnetic ﬁeld [21–23]. The micro-scopic expression for the conductances is Landauer like g /.0027/.00270/.0136X nm/.0137/.0014nm/.0255r/.0027nm/.0133r/.00270 nm/.0134/.0003/.0138; (1)where the reﬂection coefﬁcient r/.0027nmof an electron in the SC with spin /.0027at theSCjFcontact between nth and mth transverse modes is accessible to ab initio calculations [8,9,19,20]. The time dependence of the system is gov- erned by charge and spin conservation [17]: /.02552hD/.0018d/.0022c dt/.0019 bias/.0136/.0133g""/.0135g##/.0134/.0133/.0022c/.0255e’/.0134 /.0135/.0133g""/.0255g##/.0134/.0133~mm/.0001~/.0022/.0022s/.0134; (2) /.02552hD/.0018d~/.0022/.0022s dt/.0019 bias/.01362Reg"#~/.0022/.0022s/.02552Img"#/.0133~mm/.0002~/.0022/.0022s/.0134 /.0135/.0137 /.0133g""/.0255g##/.0134/.0133/.0022c/.0255e’/.0134 /.0135/.0133g""/.0135g##/.02552Reg"#/.0134/.0133~mm/.0001~/.0022/.0022s/.0134/.0138~mm; (3) whereDis the SC single-spin energy density of states. The electrostatic potential ’due to an applied bias and/or a charge imbalance between electrons and holes will bedisregarded in the following (see below). In the presenceof a magnetic ﬁeld ~BB, the sum of externally applied and hyperﬁne (Overhauser) ﬁelds with ordered nuclear spins, we have to add /.0018 d~/.0022/.0022s dt/.0019 field/.0136ge/.0022B /.0022h~BB/.0002~/.0022/.0022s; (4) where geis the electron g-factor ( /.0025/.02550:4in GaAs) and /.0022Bis the Bohr magneton. These equations can be sum- marized in terms of a 4/.00024matrix equation: /.0255TIdj/.0022i dt/.0136/.0255j/.0022i; (5) where TI/.01362hD=gis an interface-mediated relaxation time in terms of the total conductance g/.0136g""/.0135g##. Choosing ~mmparallel to the zaxis, /.0255/.01360 BBB@10 0 p 0 /.0017 r /.0017i/.0135/.0010z/.0255/.0010y 0/.0255/.0017i/.0255/.0010z /.0017r/.0010x p/.0010y /.0255/.0010x11 CCCA: (6) j/.0010 /.0011j=TI/.0136jgej/.0022BB/.0011=/.0022his the Larmor frequency, the mixing conductance has been normalized as /.0017/.01362g"#=g with subscripts iandrdenoting its real and imaginary part, and the polarization is deﬁned as p/.0136/.0133g""/.0255g##/.0134=g. Equation (5) can be solved easily [16] for the boundaryconditions corresponding to LP excitation h/.0022 LP/.01330/.0134j /.0136 h1;0;0;0jor CP excitation with wave vector in the xdirection h/.0022CP/.01330/.0134j /.0136 h1;1;0;0j. When the conductance is expressed in terms of an interface transparency parameter /.0020times the intrinsic SC single-spin Sharvin conductance of a ballistic con-striction of area A,g/.0136/.0020/.01332/.0025m /.0003"A=h2/.0134, we can write FIG. 1. Spin dynamics in the ﬁreball-afterglow scenario of an excited semiconductor in proximity with a ferromagnet polar- ized in the zdirection. A (DNP enhanced) magnetic ﬁeld 0.24 T is applied in the ydirection. Plotted is /.0022x/.0133t/.0134in arbitrary units for CP (shifted upwards by 0.6) and LP excitation, with the wave vector in the xdirection. Time is measured in units of TI. The transition from ﬁreball to afterglow with /.0021TTI/.013610TIis taken to be abrupt at t/.0136TI=2.PHYSICAL REVIEW LETTERSweek ending 26 MARCH 2004 VOLUME 92, N UMBER 12 126601-2 126601-2TI/.00253:5 /.0020LSC 100 nm/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 m/.0003 0:067m010 meV "s ps: (7) For an Ohmic interface with /.0020’0:1, an SC layer thick- ness of LSC/.0136100 nm and at a characteristic electron kinetic energy (depending on doping and excitation den- sity) of "/.013610 meV , the time constant for GaAs (effec- tive mass m/.0003/.01360:067m0)i sTI/.002435 ps , of the order of the experimental rise time of the proximity effect [12]. Atlong-time scales, experiments ﬁnd /.0021TT I>2n s, which cor- responds to a strongly reduced transparency of /.0021/.0020/.0020& 0:002, introducing the embellishment /.0021to indicate pa- rameters in the long-time regime. With a few exceptions (notably InAs), Schottky bar- riers are formed at metal-semiconductor interfaces wheninterface states in the gap of the semiconductor becomeﬁlled giving rise to space charges. Photoexcited holes arestrongly attracted by the barrier, thereby dragging theelectrons with them [24] and/or screen the barrier. Theobserved large /.0020in the ﬁreball regime reﬂects the facil- itation of electron transport to the SCjFinterface by the holes. In the limit of predominant ambipolar electron- hole transport (and in the absence of an applied bias),which should apply for the low doping densities in theexperiments, we may neglect the electrostatic potential ’, but the electron conductance gmight be affected by the scattering of the holes. At long time scales, the holes disappear into the ferro- magnet or recombine with electrons in the semiconductor, but the electron spin accumulation persists [15]. In this afterglow, remaining space charges vanish when thesample is earthed and net charge transport is suppressed.Equation (2) thus vanishes and /.0255/.0021TT Id~/.0022/.0022s dt/.0136/.0021/.0255/.0255s~/.0022/.0022s (8) with /.0021/.0255/.0255s/.01360 B@/.0021/.0017/.0017r/.0021/.0017/.0017i/.0135/.0021/.0010/.0010z/.0255/.0021/.0010/.0010y /.0255/.0021/.0017/.0017i/.0255/.0021/.0010/.0010z/.0021/.0017/.0017r/.0021/.0010/.0010x /.0021/.0010/.0010y /.0255/.0021/.0010/.0010x1/.0255/.0021pp21 CA; (9) where the parameters are now governed by the full Schottky barrier. The kinetic equations are valid when the system is dif- fuse or chaotic (as a result of interface roughness or bulkdisorder). The ferromagnetic elements should have an ex-change splitting /.0001which is large enough that the mag- netic coherence length /.0021 c/.0136/.0022h=/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 /.0129 2m/.0001p <min/.0133‘F;LF/.0134, where LFis the thickness of the ferromagnetic layer (typically 50 nm) and ‘Fis the mean free path. These conditions are usually fulﬁlled in hybrid systems exceptfor very thin layers with nearly perfect interfaces. The spin relaxation time in GaAs is taken to be very long. W e also require that L SC</.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129/.0129 2/.0022hvF‘SC=3ge/.0022BBextp for diffuse systems [21] or LSC</.0022hvSC;f=ge/.0022BBextfor ballistic sys-tems, where ‘SCis the mean free path, vSC;fis the Fermi velocity, and Bextis the externally applied ﬁeld. For samples with LSC/.0136100 nm the applied magnetic ﬁelds should therefore not much exceed 5 T. Thus a microscopic justiﬁcation can be given for Gridnev’s phenomenological relaxation tensor [16]. Butwhereas we can explain how the longitudinal relaxation[/.0024gor/.0024/.0021gg/.01331/.0255/.0021pp 2/.0134] can differ from the transverse com- ponents [ /.0024Reg"#or/.0024Re/.0021gg"#], Gridnev postulated a large difference between the two components normal to~mm. Such large magnetic anisotropies can be excluded for Fe and the scenario sketched by Gridnev cannot be a generic explanation for all experiments. To explain the proximitypolarization, Gridnev’s 3/.00023Bloch equation must be extended to the 4/.00024kinetic Eq. (5) that includes a charge current component. First-principles calculations of the bare interface con- ductance (with the Schottky barrier assumed quenched)provide a ﬁrst indication of the transport properties in the ﬁrst tens of ps. W e choose here the FejInAs system, which apart from the Schottky barrier, is very similar toFejGaAs , and as such, of great interest in itself. Table I summarizes results obtained from ﬁrst-principles scatter-ing matrix calculations [9]. W e ﬁnd the reversal of polar-ization sign with disorder as noted before [9], which mayexplain the negative polarization found for FejGaAs [12]. The real part of the mixing conductance /.0017 ris close to unity similar to metallic interfaces, but in contrast to these, the imaginary part /.0017iis strongly enhanced. The latter can be explained by the focus on a small number ofstates with wave vectors close to the origin, which pre-vents the averaging to zero found in metals [20]. W e now model the experiments of Epstein et al. on GaAs jMnAs (concentrating on Fig. 1 in [12]) with an applied magnetic ﬁeld of 0.12 T in the ydirection (in our coordinate system). Analytic solutions of Eq. (5) can be obtained for /.0017 r/.01361/.0133/.0136/.0021/.0017/.0017r/.0134(see Table I and [20]) such that all modes are exponentially damped by a singleinterface relaxation time T I. Time is measured in units TABLE I. Ab initio interface transport parameters for a cleanInAsjFe(001) interface with In (or As) termina- tion.Gis the electric conductance e2g=h,pis the polarization, /.0020is the ratio between Gand the intrinsic SC Sharvin con- ductance [ 5:2/.000210/.02555=/.0133f/.0010m2/.0134],/.0017is the relative mixing con- ductance, all at a kinetic energy of 20 meV in the InAs conduction band. The dirty interfaces are modeled as a mono- layer of random alloy with 1=4of the interface In (or As) replaced by Fe in a 7/.00027lateral supercell. G/.01371=/.0133f/.0010m2/.0134/.0138 p/.0020 Re/.0017Im/.0017 CleanFejInAs 1 :5/.000210/.025550.98 0.14 1.3 /.02551:3 FejAsIn 3 :6/.000210/.025550.88 0.35 1.6 /.02551:05 DirtyFejInAs 5 :7/.000210/.02555/.02550:29 0.56 1.1 /.02550:18 FejAsIn 7 :4/.000210/.02555/.02550:22 0.71 1.3 /.02550:30PHYSICAL REVIEW LETTERSweek ending 26 MARCH 2004 VOLUME 92, N UMBER 12 126601-3 126601-3ofTIand the polarization is chosen to be p/.01361/.0133/.0136/.0021pp/.0134.I n the ﬁreball regime the quality factor /.0010y/.00281, and we adopt /.0017i/.0136/.02551/.0133/.0136/.0021/.0017/.0017i/.0134. For LP excitation the charge com- ponent relaxes in favor of the zcomponent polarized along the magnetization direction 2/.0022z/.0133t/.0134/.0136e/.0255/.01331/.0255p/.0134t=TI/.0255 e/.0255/.01331/.0135p/.0134t=TIfor/.0010y/.01360, which is the essence of the DFP effect. The time scale on which the Schottky barrierrecovers determines (together with p) the modulus of the spin accumulation in the afterglow. It is of the sameorder, but smaller, than T Ibecause of competing electron- hole recombination in the semiconductor. Our Fig. 1 is similar to Fig. 1 in [12], but additional experimental data on a short time scale are required to guide the develop-ment of a more reﬁned model. As mentioned above, a modiﬁed Larmor frequency has been observed [12] even when the photon wave vec-tor is normal to the ﬁeld. This is at odds with the notionthat a CP excited spin accumulation should rotate aroundthe ﬁeld without net angular momentum transfer to the nuclei. This could be evidence for a DFP effect for the CP conﬁguration. In the ﬁreball, a signiﬁcant /.0017 iacts like a magnetic ﬁeld in the zdirection, causing the initial spin ensemble to precess into the direction of the externalmagnetic ﬁeld, which is then able to polarize the nuclearspins. This effect is weaker for LP excitation since, in thebrief ﬁreball interval, any spin accumulation has to begenerated before it can precess. For LP excitation the Larmor frequency depends [13] on an applied bias, proving that the electrostatic potential’, in general, cannot be neglected in Eqs. (2) and (3). This does not invalidate our qualitative arguments since, com-pared to the bare interface exchange the modiﬁcationsneeded to explain the shifts in the effective Larmorfrequencies are small, beyond the accuracy of our model.The decreasing spin lifetime in the afterglow with in- creasing bias has been explained by inhomogeneous nuclear polarization [13], but lowering the Schottky bar-rier by a forward bias also reduces /.0021TT I. In conclusion, we propose a physical picture for the spin dynamics of photoexcited carriers in semiconductor/ferromagnet bilayers. The experiments can be understoodin terms of at least two time scales. In the ﬁrst 50 ps or so,the photoexcited carriers screen the Schottky barrier efﬁciently and the interaction of the electrons with the ferromagnet is described by nearly intrinsic interfaceconductances that can be calculated from ﬁrst principles.After delay times of >100 ps , the Schottky barrier pro- tects the semiconductor carriers from fast decay and anyresidual exchange interaction is very weak. More insightinto the interaction of carriers in semiconductors withferromagnets could be gained by a faster (ps) time reso- lution and higher applied magnetic ﬁelds. A quantitative explanation of the experiments requires self-consistentmodeling of the combined electron and hole carrier dy-namics as well as ab initio calculations of the interface scattering matrices for electrons and holes.W e thank D. D. A wschalom, Y . Kato, and R. J. Epstein for helpful discussions on the Faraday rotation ex-periments. This work has been supported by the FOM, by the NEDO joint research program ‘‘Nano Magneto- electronics,’’ by the Norwegian Research Council‘‘NanoMat,’’ DARP A Grant No. MDA 972-01-1-0024,by the European Commission’s RT Network Computa- tional Magnetoelectronics (Contract No. HPRN-CT- 2000-00143), and by KBN Grant No. PBZ-KBN-044/P03-2001. Permanent address: Institute of Molecular Physics, P .A.S., Smoluchowskiego 17, 60-179 Poznan ´, Poland. [1]Applications of Magnetic Nanostructures ,e d i t e db y S. Maekawa and T. Shinjo (Taylor and Francis,New Y ork, 2002). [2]Semiconductor Spintronics and Quantum Computing , edited by D. D. A wschalom, D. Loss, and N. Samarth (Springer, Berlin, 2002). [3] E.Y . Tsymbal and D. G. Pettifor, Solid State Phys. 56,1 1 3 (2001). [4] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); L. Berger, Phys. Rev. B 54, 9353 (1996). [5] M. Tsoi et al. , Phys. Rev. Lett. 80, 4281 (1998); J.-E. We g r ow e et al. , Europhys. Lett. 45, 626 (1999); J. Z. Sun, J. Magn. Magn. Mater. 202, 157 (1999); E. B. Myers et al. , Science 285, 867 (1999), and several more recent others. [6] B. Heinrich et al. , Phys. Rev. Lett. 90, 187601 (2003). [7] S. I. Kiselev et al. , Nature (London) 425, 380 (2003). [8] O. Wunnicke et al. , Phys. Rev. B 65, 241306 (2002). [9] M. Zwierzycki et al. , Phys. Rev. B 67, 092401 (2003). [10] D. D. A wschalom and N. Samarth, in Ref. [2]. [11] R. K. Kawakami et al. , Science 294, 131 (2001). [12] R. J. Epstein et al. , Phys. Rev. B 65, 121202 (2002). [13] R. J. Epstein et al. , Phys. Rev. B 68, 041305 (2003). [14] Y . Tserkovnyak, A. Brataas, and G. E.W . Bauer, Phys. Rev. Lett. 88, 117601 (2002). [15] C. Ciuti, J. P . McGuire, and L. J. Sham, Phys. Rev. Lett. 89, 156601 (2002); J. P . McGuire, C. Ciuti, and L. J. Sham, cond-mat/0302088. [16] V . N. Gridnev, JETP Lett. 77, 187 (2003) [Pis’ma Zh. Eksp. Teor. Fiz. 77, 217 (2003)]. [17] A. Brataas, Y u. V . Nazarov, and G. E.W . Bauer, Phys. Rev. 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PhysRevLett.113.215501.pdf	Classical Mobility of Highly Mobile Crystal Defects T. D. Swinburne,1,2,S. L. Dudarev,2and A. P. Sutton1 1Department of Physics, Imperial College London, Exhibition Road, London SW7 2AZ, United Kingdom 2CCFE, Culham Science Centre, Abingdon, Oxon OX14 3DB, United Kingdom (Received 19 June 2014; published 20 November 2014) Highly mobile crystal defects such as crowdions and prismatic dislocation loops exhibit an anomalous temperature independent mobility unexplained by phonon scattering analysis. Using a projection operator, without recourse to elasticity, we derive analytic expressions for the mobility of highly mobile defects anddislocations which may be efficiently evaluated in molecular dynamics simulation. The theory explains how a temperature-independent mobility arises because defect motion is not an eigenmode of the Hessian, an implicit assumption in all previous treatments. DOI: 10.1103/PhysRevLett.113.215501 PACS numbers: 61.72.Bb, 05.10.Gg, 63.20.kp, 66.30.Lw Plastic deformation of crystals is effected by the motion of dislocations and point defects [1]. Away from shock loading and the melting temperature this motion is usuallymodeled with the viscous damping law _x¼γ −1·f, employing a matrix of friction or drag coefficients γ, which set the time scales of defect dynamics [2]. To reproduce the stochastic trajectories of highly mobile defects seen inexperiment [3,4] this mobility law has been supplemented with a stochastic force to give the Langevin equation [5] _x¼γ −1·½fþηðtÞ/C138, where hηðtÞ⊗ηðt0Þi ¼ 2kBTγδðt−t0Þ [6,7]. The stochastic force is usually more significant for small dislocation loops and point defects because theconfigurational force f λis determined only by gradients in the stress field. For larger extended defects the configu-rational force usually dominates over the stochastic force.In both cases γcontrols the rate of important micro- structural processes such as swelling and post-irradiationannealing [8], but no universal theory for γexists. Phonon scattering calculations [9–11]and soliton mod- els[12]predict γshould increase linearly with temperature in the classical regime ( γ¼k BTγw, where γwis a constant). While this “phonon wind ”relationship is seen to hold with varying degrees of quantitative agreement in moleculardynamics (MD) simulations of extended highly mobiledislocation lines [13,14] , no theory has explained the widely observed [6,7,15 –17] temperature independent mobility ( γ¼γ 0) of highly mobile defects such as crow- dions [17], kinks on screw dislocations [7], and small prismatic dislocation loops [16], which all exhibit highly stochastic trajectories particularly sensitive to γ. In this Letter we use the Zwanzig projection technique [18] to show that γ¼γ0þkBTγw, in quantitative agree- ment with MD simulations of defects and dislocations. γ0arises because the defect displacement vector is not an eigenvector of the Hessian, so that thermal vibrations caninduce a force on defects to linear order. This is missed in previous treatments [11,19] as by perturbing a quadratic integrable Hamiltonian one implicitly assumes that defectmotion is an eigenmode, an assumption that we explicitlyshow to be false. Violation of this assumption is the origin of the anomalous mobility. Defect coordinates. —We describe a crystal using a 3N- dimensional vector of atomic positions X∈R 3Nand velocities _X∈R3N. In this treatment crystal defects are not elastic singularities but localized deformations, which may be assigned a set of M≪N“position ”labels xλ∈R3M and“velocity ”labels _xλ∈R3Mto characterize the state of a defective crystal. Common methods for determining xλ,_xλ include analysis of the atomic disregistry [20]or an energy filter [7], though in the following the only requirement is a repeatable protocol. By definition, the zero temperature configurations X¼UðxλÞof the crystal potential energy VðXÞmay be entirely characterized by the parameters xλ, while variation of UðxλÞwithxλcan be determined through nudged elastic band calculations [21] or simply a finite difference derivative in the case of a defect with a wide core.To complete the discrete representation of a crystal at finitetemperature, we must include displacements due to thermal vibrations Φ∈R 3N. The crystal configuration Xat any given instant can now be expressed as X¼ΦþUðxλÞ; _X¼_Φþ_xλ·∂λUðxλÞ; ð1Þ where ∂λU¼∂=∂xλ⊗UðxλÞ∈R3M×3N.B yi n t r o d u c i n g a defect position and velocity the coordinate set Φ⊕_Φ⊕ xλ⊕_xλhas6Mmore dimensions than X⊕_X. To rectify this we require the vibrational displacements Φto be independent to the displacements caused by defect motion∂ λU,g i v i n gt h e 6Mconstraints [12] ∂λU·Φ¼0; ∂λU·_Φ¼0: ð2ÞPublished by the American Physical Society under the terms of theCreative Commons Attribution 3.0 License . Further distri- bution of this work must maintain attribution to the author(s) and the published article ’s title, journal citation, and DOI.PRL 113, 215501 (2014) PHYSICAL REVIEW LETTERSweek ending 21 NOVEMBER 2014 0031-9007 =14=113(21) =215501(5) 215501-1 Published by the American Physical SocietyTo obtain a dynamical equation for xλ, it now suffices [22]to project the exact equation of motion m ̈X¼−∇VðXÞonto the direction ∂λUorthogonal to the crystal vibrations. Defining an effective mass tensor ~m¼m∂λU·ð∂λUÞT, we exploit the time invariance of (2) to obtain ~m· ̈xλ−_xλ·∂2 λU·_Φ¼−∂λðVþ_xλ·~m·_xλ=2Þ.S i m i l a r equations of motion are standard in dynamical quasiparticletheories [12,22] and, in common with other authors, we will neglect the “hydrodynamic ”term−_x λ·∂2 λU·_Φand the effective kinetic energy gradient −_xλ·∂λ~m·_xλ=2.T h i si s justified as we consider the motion of only subsonic defects, and it can be shown that these terms are of order j_xλj=c≪1, where cis the speed of sound. As a result, the defect coordinates evolve according to m· ̈xλ¼fλ≡−∂λU·∇VðXÞjX¼UðxλÞþΦ; ð3Þ where we have defined the instantaneous defect force fλas the projection of the total force −∇Vin the direction of defect motion ∂λU. The vibrational coordinates evolve in the subspace orthogonal to ∂λU, implying that m ̈Φ¼−½I−mð∂λUÞT·~m−1·∂λU/C138·∇V≡−∇ΦV. Removing the vibrational coordinates. —From the form of the potential energy V½UðxλÞþΦ/C138, it is clear that the evolution of the defect and vibrational coordinates are coupled, as they must be for a frictional force to exist.However, for highly mobile subsonic defects, which necessarily possess a wide defect core [23], the defect coordinates may be considered as slowly varying comparedto the vibrational coordinates, a conclusion which will be explicitly demonstrated in molecular dynamics simulation below. Over a Debye period τ D∼a=c∼0.1ps, where ais the lattice parameter, the displacements of any atom due to thermal vibrations will approximately average to zero, with an oscillation amplitude of ∼τDﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kBT=mp . Since the defect speed will be approximately _xλ∼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kBT=~mp ≪c, the dis- placement of any one atom due to defect motion in a time interval τDwill be at most τDjj∂λUjj∞ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kBT=~mp , where jj∂λUjj∞is the largest component of ∂λU. These calcu- lations imply that if jj∂λUjj∞≪j∂λUj, then the displace- ment due to defect motion will be much less than the magnitude of displacements due to thermal motions, which implies that the Φare effectively ergodic [24] over a time scale∼τD, where the defect coordinates are essentially stationary. But the condition ∥∂λU∥∞≪j∂λUjamounts to a requirement that the deformation associated with thedefect is spread over many atomic sites, which is always satisfied by highly mobile defects with a wide core. We therefore assume that vibrational displacements average tozero over periods of ∼0.1ps while the defect remains effectively stationary, an assumption that we will test explicitly when calculating the defect force autocorrelation. We can exploit this separation of time scales to remove thermal vibrations from the defect equation of motion using the formalism of dimensional reduction by Zwanzig[18,25] . In this formalism the solution of the “fast” equation of motion for Φis substituted into the “slow” equation of motion for x λ. It may be shown, to order τ3 D, that Φ,_Φare adiabatic with respect to xλ,_xλand ergodic over the partial Gibbs distribution h…i≡Z−1ðxλÞZ Φ;_Φ…e−β½V(UðxλÞþΦ)þm_Φ·_Φ=2/C138; ð4Þ where ZðxλÞ¼exp½−βFðxλÞ/C138is the partial partition func- tion and we integrate on the hypersurface defined by (2). The defect coordinates now evolve on a coarse time scale τDand follow the stochastic equation of motion ~m· ̈xλðtÞ¼−γ·_xλðtÞþhfλiþηðtÞ: ð5Þ It is usual in dislocation dynamics to neglect the inertial term ~m· ̈xλðtÞ, which is valid when the potential energy landscape is slowly varying over the thermal lengthﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kBT=~mjγjp [5].I n(5)we have introduced the expected force hfλi¼−h∂λVi¼−∂λF, the stochastic force ηðtÞ, where hηðtÞ⊗ηðt0Þi ¼ 2kBTγδðt−t0Þ, and our central quantity, the friction matrix γ. In this timescale separated regime it is a standard result that γis proportional to the time integral of the force autocorrelation CðτÞ, namely, γ≡ðkBTÞ−1Z∞ 0CðτÞdτ; ð6Þ where when xλ∈R,CðτÞ≡hfλðτÞfλð0Þi−hfλð0Þi2and may be expressed by ergodicity as CðτÞ¼lim t→∞/C20Zt 0fλðt0þτÞfλðt0Þ tdt0−/C18Zt 0fλðt0Þ tdt0/C192/C21 : ð7Þ We evaluate CðτÞ, and hence γ, in two ways: first by deriving in closed form the thermal averages (4)and second by numerical calculation of fλðtÞin MD simulation. Analytic derivation. —To derive an expression for γwe expand the potential energy Vand the defect force fλin powers of Φ. For the evaluation of the partition function the constraints (2)and the requirement that the UðxλÞdescribes the zero temperature configurations results in an expansion V¼VðxλÞþ1 2Φ·∇2 ΦV·Φþ1 3!Φ·∇3 ΦV∶Φ⊗Φþ/C1/C1/C1 ; ð8Þ where all inner products are with respect to Φand all partial derivatives are evaluated at X¼UðxλÞ. Although ∇ΦV¼ 0[so that m ̈Φ¼−∇2 ΦV·ΦþOðΦ2Þ] there is no restriction on the existence of mixed derivatives ∂λ∇n ΦV≠0. This is important as these mixed derivatives couple the defect and vibrational coordinates, as can be seen in the defect force expansionPRL 113, 215501 (2014) PHYSICAL REVIEW LETTERSweek ending 21 NOVEMBER 2014 215501-2fλ¼∂λVðxλÞþ∂λ∇ΦV·Φþ1 2∂λ∇2 ΦV∶Φ⊗Φþ/C1/C1/C1⋅ ð9Þ While we retain anharmonicity in the defect force, in order to perform analytical evaluation of expectation values we truncate Vto quadratic order in Φin the Gibbs distribution (4), allowing us to explicitly evaluate the expectation values in terms of the 3ðN−MÞdimensional vibrational eigenset fωl;vlg, where ∇2 Φ·vl¼mω2 lvl. This truncation neglects any thermal expansion arising from the purely vibrational anharmonicities ∇3 ΦVand∇4 ΦV. In the Supplemental Material [26] we systematically include these terms to produce an expression for γup to linear order in temper- ature. It is shown that the anomalous temperature inde- pendent mobility γ0is unaffected by these additional terms. Using a quadratic Gibbs distribution, the expected force isfound to be hf λi¼−∂λðV−TSÞ, where Sis the harmonic entropy kBP llogωl[27]; to evaluate CðτÞwe evolve the vibrational coordinates Φfrom a given xλ. This is justified by the time scale separation and achieved by evaluating propagator terms of the form hΦðtÞ⊗Φð0Þi ¼X lkBT mω2 lvl⊗vlcosðωltÞ: ð10Þ As appropriate for nonconservative dynamics, the propa- gator is evaluated using only the initial conditions hΦð0Þ⊗ Φð0Þi ¼P lkBT=m ω2 lvl⊗vland, consequently, is closely related to the retarded Green ’s function GðtÞ¼ ΘðtÞðkBTÞ−1hΦðtÞ⊗Φð0Þi[28]. All that now remains is to perform elementary Gaussian integrations to obtain our main result γ¼Z∞ 0∂λ∇ΦV·GðtÞ·∂λ∇ΦVdt þkBT 2Z∞ 0Trð∂λ∇2 ΦV·GðtÞ·∂λ∇2 ΦV·GðtÞÞdt þkBT 2Z∞ 0∂λ∇ΦV·GðtÞ·∂λ∇3 ΦV∶GðtÞdt: ð11Þ We see that the friction coefficient takes the form γ¼γ0þkBTγw, with the new temperature independent γ0a function of the mixed quadratic derivative ∂λ∇ΦV, and the temperature dependent kBTγwa function of the mixed cubic and quartic derivatives ∂λ∇2 ΦVand∂λ∇3 ΦV. These terms may, in principle, be evaluated after diagonalizing∇ 2 ΦVto obtain fωl;vlgand computing the tensorial derivatives ∂λ∇n ΦV. However, in common with modern methods to evaluate dispersion relations [29],w eh a v e found dynamical measurement of the thermal averages to be much more efficient than complete diagonalization of the vibrational Hessian ∇2ΦV.Numerical evaluation. —We have developed a method to calculate fλðtÞby MD simulation, which yields CðτÞand hence γ, yielding a numerical evaluation of the analytic expressions (11). In an ensemble of MD runs, with no stress applied, we time average the output for each run XðtÞusing a coarse-grained time step between τD=4andτDto give hXi. To eliminate any errors, we find the zero temperature configuration Uλwhich minimizes j∂λhXi−∂λUj2. The calculated ∂λUis then used to project out the defect force fλðtÞ¼−∂λU·∇V½XðtÞ/C138over the same averaging time interval that produced hXi. We have found this method to be robust to variation in the averaging period and especiallyefficient for short line segments or nanoscale defects, where the zero temperature structures are typically related by rigid translation [30]. An example of such calculations is shown in Fig. 1for a 7 atom SIA cluster in tungsten, which exhibits the anomalous temperature independent mobility γ¼γ 0[17], and in Fig. 2for a highly mobile edge dislocation in iron, which exhibits a mixed temperaturedependence γ¼γ 0þkBTγw[15]. In both cases we see that CðτÞloses all coherence after the first zero at ∼τD=2, over which time the defect is observed to be essentially sta-tionary. This validates our assumption of time scale separation between thermal vibrations and defect motion. We identify the subsequent force autocorrelation (FAC)signal as noise because it flattens with the system and ensemble size, limiting the integration CðτÞonly to the first zero. As shown in the figures, this method gives values inexcellent agreement with conventional trajectory analysis. We also calculated the FAC for the 7-atom SIA [∂ λ∇ΦV·GðtÞ·∂λ∇ΦV] via full diagonalization of FIG. 1 (color online). Evaluation of the defect FAC in unbiased molecular dynamics simulation at three temperaturesand the first analytic term in (11) for a 7 atom SIA cluster in tungsten using LAMMPS [31] and an interatomic potential by Marinica et al. [32]. We see a very similar peak in all methods which loses coherence after a time period ∼τD=2, and we approximate the time integral in (11) by the area under this first peak. Inset: Comparison of the predicted diffusivity D¼ kBT=γand the direct measurement D¼hx2i=2t.PRL 113, 215501 (2014) PHYSICAL REVIEW LETTERSweek ending 21 NOVEMBER 2014 215501-3∇2 ΦV. We find excellent agreement with the dynamical method, as shown in Fig. 1. Discussion. —Terms similar to (11) appear in phonon scattering predictions of γ, where they may be interpreted diagrammatically, with ∂λ∇n ΦVapproximately representing a vertex of one defect with nphonons [11,34] . In this continuum picture, defects and phonons are separable to harmonic order, conserving energy and momentum in collisions. As a result, each term in (11)becomes dependent on the phase space available for the scattering process it represents. The anomalous term γ0is forbidden in such models as it represents the pure absorption or emission of a phonon, a process which has a vanishing phase space for subsonic defect speeds due to the linear phonon dispersion relation [34,35] . It turns out that the second term in (11) dominates, describing a two-phonon elastic scattering proc-ess known as the phonon wind. With a cubic anharmonicity parameter A[36] this term has an approximate magnitude ∼k BTðA=μÞ2τD, where μis the shear modulus, in agreement with more detailed continuum treatments [11]. However, the prediction γ0¼0from continuum analysis does not explain the observed simulation results. To see how the present treatment allows an anomalous temperature independent mobility, we express γ0in the eigenbasis fvkgof the vibrational Hessian ∇2 ΦV. Using (10) and the expansion ∂λ∇ΦV¼P kvk∂λ∇kV, where ∇k¼ vk·∇Φ, the temperature independent component γ0readsR∞ 0P kð∂λ∇kVÞ2=ðmωkÞ2cosðωktÞdt. For this term to van- ish, as in all continuum theories, we require ∂λ∇kV¼0.B u t this implies that the defect displacement operator ∂λUis an eigenvector of the total Hessian ∇2Vas the “off-diagonal ” terms ∂λ∇kVthat mix ∂λUand the vibrational modes must vanish. We have explicitly demonstrated that this is not thecase; it is precisely this effect, which relies on the weaker identification of a defect as a localized deformation that is not an eigenvector of the Hessian, in contrast to a canonicalquantity separable from vibrations, that gives rise to γ 0.O f course, anharmonic vibrations still affect the dynamics in a manner which becomes analogous to typical scatteringtheories in a continuum picture, giving the phonon wind term k BTγwin(11). These terms are appreciable for only extended line dislocations, which significantly deform thehost lattice, while the anomalous γ 0is the leading term for nanoscale defects which are typically elastically neutral in the far field. For some extended dislocations in close-packedcrystals the defect translational operator is very nearly an eigenvector of the Hessian, implying that the anomalous mobility vanishes and γ∼k BTγw[13]. But, in general, we have found this not to be the case, with the mixed depend- ence γ¼γ0þkBTγwoccurring across a wide range of crystal defects. Concluding remarks. —Our main result is an explicit form (11) for the friction tensor γof highly mobile crystal defects. We believe this is a new result. It may be used to parametrize accurately deterministic ( _x¼γ−1·f) or sto- chastic f_x¼γ−1·½fþηðtÞ/C138gdefect mobility laws. The result was obtained by identifying defects through a pro- jection operator with no recourse to elasticity. An anomaloustemperature independent mobility γ∼γ 0arises because the displacement vector corresponding to defect motion is not an eigenvector of the Hessian, inviolation of elasticity theory orsolitonlike models, where vibrations are canonical. This finding highlights the importance of intrinsically discrete (i.e., atomistic) analysis to understand nanoscale crystalplasticity. We note that the form of γ 0in(11) is closely analogous to the famous Kac-Zwanzig heat bath formula [18]. But rather than a random array of harmonic oscillators with a constant coupling strength, we have here the vibra- tional modes of the entire crystal coupling to a localized deformation through ∂λ∇ΦV. It is hoped that our explicit expression for γand the method of evaluation may be used to provide further connections between analytic heat bath models and the thermal dynamics of real systems. T. D. S. was supported through a studentship in the Centre for Doctoral Training on Theory and Simulation of Materials at Imperial College London, funded byEPSRC under Grant No. EP/G036888/1. This work was partially funded by the RCUK Energy Programme (Grant No. EP/I501045) and by the European Unions Horizon2020 research and innovation programme under Grant Agreement No. 633053. To obtain further information on the data and models underlying this Letter pleasecontact PublicationsManager@ccfe.ac.uk. The views and opinions expressed herein do not necessarily reflect those of the European Commission. This work was also partiallyfunded by the United Kingdom Engineering and Physical Sciences Research Council via a programme Grant No. EP/ G050031. FIG. 2 (color online). Evaluation of CðτÞfor a 1=2h111ið10¯1Þ edge dislocation in Fe, using an interatomic potential by Gordonet al. [33], normalized to the unit length aj½1¯21/C138j∼7Å. The FAC increases with temperature such that γ¼γ 0þkBTγw, exhibiting both anomalous and phonon wind drag. Inset: Comparison withdirect measurement of the diffusivity. The values are in quanti-tative agreement with finite stress simulations [15].PRL 113, 215501 (2014) PHYSICAL REVIEW LETTERSweek ending 21 NOVEMBER 2014 215501-4tds110@ic.ac.uk [1] J. P. Hirth and J. Lothe, Theory Of Dislocations (Krieger, Malabar, FL, 1991). [2] V. V. 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PhysRevB.102.214436.pdf	PHYSICAL REVIEW B 102, 214436 (2020) L10-ordered (Fe 100−xCrx)Pt thin ﬁlms: Phase formation, morphology, and spin structure Nataliia Y . Schmidt ,1Ritwik Mondal ,2Andreas Donges,2Julian Hintermayr ,1Chen Luo ,3,4Hanjo Ryll,4Florin Radu,4 László Szunyogh ,5,6Ulrich Nowak ,2and Manfred Albrecht1 1Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86159, Augsburg, Germany 2Department of Physics, University of Konstanz, D-78457 Konstanz, Germany 3Experimental Physics of Functional Spin Systems, Technische Universität München, James-Franck-Str. 1, D-85748 Garching b. München, Germany 4Helmholtz-Zentrum Berlin, D-14109 Berlin, Germany 5Department of Theoretical Physics, Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary 6MTA-BME Condensed Matter Research Group, Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary (Received 6 August 2020; accepted 4 December 2020; published 28 December 2020) Chemically ordered L10(Fe 100−xCrx)Pt thin ﬁlms were expitaxially grown on MgO(001) substrates by magnetron sputter-deposition at 770◦C. In this sample series, Fe was continuously substituted by Cr over the full composition range. The lattice parameter in the [001] growth direction steadily increases from L10-FePt toward L10-CrPt, conﬁrming the incorporation of Cr in the lattice occupying Fe sites. With the observed high degree of chemical ordering and (001) orientation, strong perpendicular magnetic anisotropy is associated, which persistsup to a Cr content of x=20 at. %. Similarly, the coercive ﬁeld in the easy-axis direction is strongly reduced, which is, however, further attributed to a strong alteration of the ﬁlm morphology with Cr substitution. The latterchanges from a well-separated island microstructure to a more continuous ﬁlm morphology. In the dilute alloywith low Cr content, isolated Cr magnetic moments couple antiferromagnetically to the ferromagnetic Fe matrix.In this case, all Cr moments are aligned parallel, thus forming a ferrimagnetic FeCrPt system. With increasing Crconcentration, nearest-neighbor Cr-Cr pairs start to appear, thereby increasing magnetic frustration and disorder,which lead to canting of neighboring magnetic moments, as revealed by atomistic spin-model simulations withmodel parameters based on ﬁrst principles. At higher Cr concentrations, a frustrated ferrimagnetic order isestablished. With Cr substitution of up to 20 at. %, no pronounced change in Curie temperature, which is inthe range of 700 K, was noticed. But with further addition the Curie temperature drops down substantially evendown to room temperature at 47 at. % Cr. Furthermore, x-ray magnetic circular dichroism studies on dilute alloyscontaining up to 20 at. % of Cr revealed similar spin moments for Fe and Cr in the range between 2.1–2.5 μ B but rather large orbital moments of up to 0.50 ±0.10μBfor Cr. These results were also compared to ab initio calculations. DOI: 10.1103/PhysRevB.102.214436 I. INTRODUCTION FePt alloy thin ﬁlms with chemically ordered L10structure have been investigated intensively for high-density mag-netic recording media [ 1–5] and magnetic random access memory applications [ 6,7] due to their large magnetocrys- talline anisotropy energies, providing high thermal stabilityof magnetization against thermal ﬂuctuations. The orderedstructure consists of alternate stacking of Fe and Pt atomsalong the caxis of the face-centered tetragonal structure. As the easy magnetization axis of the L1 0crystal is par- allel to the caxis, a (001) orientation is mandatory for practical device applications requiring perpendicular mag- netic anistropy. Despite the large lattice misﬁt of about9%, MgO(001) substrates [ 8,9] or seed layers [ 10–12]a r e commonly employed for epitaxial growth of L1 0FePt(001). Typically, for ﬁlm deposition, a substrate temperature higherthan 500 ◦C is required to promote L10order, while lowertemperatures result in the chemically disordered fcc (A1) structure. As key magnetic properties such as uniaxial magnetic anisotropy ( Ku), saturation magnetization ( MS), and Curie temperature ( TC) are not fully independent, adjustment and optimization of these parameters are of supreme importance for applications. One promising route is the substitution withthird elements into the L1 0FePt lattice. For instance, partial substitution of Fe with Cu occupying Fe sites induce loweringof the ordering temperature and reduction of T C[13–19], which is attractive for heat-assisted magnetic recording [ 2,3]. However, Kuand MSare lowered as well, thus a delicate tuning of the Cu concentration is necessary for applications. A recent theoretical work [ 20] studied the effect of substituting Fe by various 3 dtransition metal (TM) elements (i.e., Cr, Mn, Co, Ni, Cu), whose introduction changes the effective numberof valence electrons and thus allows tuning of the magneticanisotropy energy (MAE). However, these studies need to be 2469-9950/2020/102(21)/214436(10) 214436-1 ©2020 American Physical SocietyNATALIIA Y . SCHMIDT et al. PHYSICAL REVIEW B 102, 214436 (2020) FIG. 1. Schematic picture of the ferromagnetic L10- FePt struc- ture, intermediate ferrimagnetic L10unit cell with substitution of Fe by Cr, where frustration leads to spin canting, and the antiferromag- netic L10CrPt structure. supported by experiments that also consider the chemical and structural changes induced by third element doping. Another important characteristic is the difference in the magnetic exchange interaction. While Ni couples ferromag-netically to the Fe matrix [ 21–24], antiferromagetic coupling is expected for Mn [ 25–27] and Cr [ 28,29]. For the latter case, complex spin structures are expected. In this regard,frustration and spin canting should arise from the competitionof ferromagnetic Fe-Fe, antiferromagnetic Cr-Cr (Mn-Mn),as well as antiferromagnetic Cr(Mn)-Fe nearest-neighbor ex-change interactions, depending on the concentration [ 28,30]. This magnetic frustration is expected to reduce the saturationmagnetization and lower the effective Curie temperature [ 30]. Schematics of various spin structures are illustrated in Fig. 1, showing the expected transition from ferromagnetic L1 0FePt to antiferrimagnetic L10CrPt magnetic order [ 31] via a mag- netically disordered ferrimagnetic state upon Cr addition. Experimentally, the magnetic and structural properties of epitaxial (Fe 100−xMn x)Pt (0/lessorequalslantx/lessorequalslant68) thin ﬁlms were in- vestigated in detail by Meyer and Thiele [ 25]. Maximum magnetic anisotropy and magnetization were reported for thepure FePt alloy, and substitution with Mn resulted in a steadyreduction of magnetocrystalline anisotropy, Curie tempera-ture, and saturation magnetization, the latter being a result ofantiparallel alignment of Fe and Mn moments. Further, it wasreported by Kuo et al. for nonstoichiometric (FePt) 100−xCrx thin ﬁlms that the addition of Cr also leads to a reduction of magnetization and coercivity, but also inhibits grain growthduring annealing [ 32]. These ﬁlms were sputter deposited at room temperature on natural-oxidized Si(111) substrates andpostannealed up to 750 ◦C for 15 min showing strong (111) texture and the presence of further phases, e.g., A1 FePt andFeCr. Here, a systematic study on the evolution of magnetic and structural properties of epitaxial (Fe 100−xCrx)Pt thin ﬁlms covering the full composition range is presented. Differentexchange interactions between Cr-Cr, Cr-Fe, and Fe-Fe pairsstimulates magnetic frustration and spin canting, which wasinvestigated in more detail by ab initio and atomistic spin- model calculations. II. EXPERIMENTAL AND THEORETICAL METHODS A. Preparation As e r i e so f( F e 100−xCrx)Pt thin alloy ﬁlms was deposited by magnetron cosputtering from individual Fe, Pt, and Crtargets on MgO(001) substrates at 770◦C using an Ar working pressure of 5 μbar. The Cr content xwas varied between 0 –100 at. % by altering the deposition rates while keeping the(Fe+Cr) to Pt ratio roughly equal. An overview of the sample names and corresponding composition and ﬁlm thickness, asevaluated by Rutherford backscattering spectrometry, is givenin Table I. All samples in the series are protected from surface oxidation by a 3-nm-thick Al capping layer. The surface mor-phology of the samples was investigated by scanning electronmicroscopy (SEM). The evolution of the L1 0phase formation was stud- i e db yx - r a yd i f f r a c t i o n( X R D )u s i n gC u - K αradiation. The chemical order parameter Swas calculated as S=/radicalBig (Iexp (001)Itheo (002))/(Itheo (001)Iexp (002)), using the ratio of theoretical ( Itheo) and experimental ( Iexp) integrated intensities [ 33].Iexp (001)and Iexp (002)are the integrated intensities of the respective reﬂections, corrected for absorption, Lorentz, Debye-Waller, and angular-dependent atomic scattering factors. The structure factor wascalculated assuming a pseudobinary TM:Pt alloy, where Feand Cr atoms are randomly distributed among TM sites. Anexperimental Debye-Waller factor of 0.0136 nm, reported forL1 0FePt [ 34], was used for all samples. The linear attenu- ation factor, required to compute the absorption factor, wasestimated from weighted elemental mass absorption factors incombination with the assessed unit cell volume a 2c. For this, a value of ain the range of 0 .97−0.99cwas assumed for varying Cr content. The order parameter is normalized to thecalculated maximum value, S Max=1−2/Delta1, where /Delta1is the deviation from the elemental 50:50 ratio of TM:Pt content.Please note that in addition to conventional error calculations,an additional error on the experimental ratio of intensities ofup to±0.1 was added due to overlap of the L1 0-(002) peak with the main substrate reﬂection and possibly present (020)reﬂections, especially on the Cr-rich side. B. Magnetic measurements Static magnetic properties were characterized using a su- perconducting quantum interference device-vibrating samplemagnetometer (SQUID-VSM) in the temperature range of25–800 K. For selected samples, x-ray magnetic circulardichroism (XMCD) absorption experiments were performedat the high-ﬁeld end station VEKMAG [ 35] installed at the PM2 beamline of the Helmholtz-Zentrum Berlin (HZB). XASspectra were collected at the Fe and Cr L 2,3edges using the total electron yield (TEY) detection mode. The XMCD signalwas calculated as the difference between XAS spectra mea-sured with two opposite directions of the external magneticﬁeld of up to ±80 kOe. Besides evaluating spin and orbital moments of Cr and Fe, element-speciﬁc magnetic hysteresisloops were measured at room temperature for both Fe and Crat the energy of maximum XMCD signal. C.Ab initio calculations The magnetic moments, exchange interactions, and magnetic anisotropy energies were calculated by densityfunctional theory (DFT) within the local spin densityapproximation (LSDA) [ 36]. First, self-consistent calcula- tions were performed using the fully relativistic screened 214436-2L10-ORDERED (Fe 100−XCrX)Pt THIN … PHYSICAL REVIEW B 102, 214436 (2020) TABLE I. Overview of the structural and magnetic properties of the (Fe 100−xCrx)Pt sample series at room temperature. Film composition (uncertainty ±2 at. %) and thickness (uncertainty 10%) was determined by Rutherford backscattering spectrometry. Sample name Fe Pt Cr tS c/a MS Ku HC TC (at. %) (at. %) (at. %) (nm) (emu /cm3)( M e r g /cm3) (kOe) (K) Fe52Pt48 52 48 – 10 0.93 0.964 1092 22.8 42.9 707 (Fe 89Cr11)Pt 47 47 6 9 0.91 0.965 1037 19.9 22.8 719 (Fe 85Cr15)Pt 45 47 8 9 0.81 0.965 1016 20.3 16.7 692 (Fe 80Cr20)Pt 43 46 11 10 0.91 0.971 890 12.9 10.8 653 (Fe 65Cr35)Pt 33 49 18 10 0.54 0.976 680 1.6 0.2 504 (Fe 53Cr47)Pt 27 49 24 10 0.51 0.980 226 – – 300 (Fe 42Cr58)Pt 22 48 30 10 0.34 0.982 <70 – – – (Fe 31Cr69)Pt 16 49 35 10 0.34 0.988 <50 – – – (Fe 20Cr80)Pt 10 49 41 10 0.39 0.991 <50 – – – Cr52Pt48 – 48 52 9 0.41 0.987 <50 – – – Korringa-Kohn-Rostoker method [ 37–39] employing the atomic sphere approximation by expanding the partial wavesup to l max=2(spd basis) inside the atomic spheres. The MAE and the exchange interactions were then determined bymeans of the magnetic force theorem [ 39,40]. The (Fe 100−xCrx)Pt ﬁlms were modeled in the bulk geom- etry in the ab initio calculations. We considered the systems with low Cr content, 0 /lessorequalslantx/lessorequalslant40 at. %, and for each concen- tration the lattice parameters a=3.863 Å and c=3.784 Å, resulting in a c/aratio of 0.979 of FePt were used, as in previous theoretical works [ 41,42]. Random distribution of the Fe and Cr atoms was supposed in the nominal Fe layersof FePt, which was treated in terms of the coherent potentialapproximation [ 43]. Chemical disorder with respect to inter- mixing of the Fe 100−xCrxand the Pt layers was not considered in the calculations. D. Spin dynamics simulation Furthermore, to extract the magnetic properties of the Cr doped FePt system in the ground state as well as at elevatedtemperatures, the stochastic Landau-Lifshitz-Gilbert (LLG)equation of motion via atomistic spin simulations was solved, ∂s i ∂t=−γ (1+α2)μs isi×/bracketleftbig Heff i+αsi×Heff i/bracketrightbig , (1) using a Heun’s algorithm [ 44,45]. The LLG equation contains a gyromagnetic ratio γ=1.76×1011T−1sec−1, magnetic moment μs i, and a scalar damping parameter αthat accounts for the energy dissipation to a heat bath. The effective ﬁeldH effcan be calculated by using the total energy ( H)o ft h e system as Heff i=ζi(t)−∂H ∂si. (2) To account for the inﬂuence of the temperature T, a stochastic white noise term ζi(t) was introduced, which obeys the fol- lowing properties: /angbracketleftζi(t)/angbracketright=0, (3) /angbracketleftbig ζη i(t)ζθ j(t/prime)/angbracketrightbig =2kBTαμs i γδijδηθδ(t−t/prime), (4)with kBas Boltzmann constant and the indices {i,j}and {η,θ}represent the atomic lattice sites and components of the stochastic noise ﬁeld. To account for the total energy, theHamiltonian H=−/summationdisplay i/negationslash=jJijsi·sj−/summationdisplay iKus2 iz−/summationdisplay iμs iBi·si (5) was considered, where the three energy terms describe the Heisenberg exchange parameters ( Jij), the uniaxial magnetic anisotropy constant ( Ku), and the Zeeman energy, respectively. III. STRUCTURAL PROPERTIES AND MICROSTRUCTURE XRD ( θ−2θ) scans of the (Fe 100−xCrx)Pt sample series are displayed in Fig. 2. The reference FePt sample shows pronounced (001) superstructure and (002) reﬂections, char-acteristic for the chemically ordered L1 0-FePt structure. With addition of Cr, no apparent changes of the diffractrogramsare observed, maintaining the L1 0structure over the whole composition range. The a- and c- lattice parameters were ex- tracted from the (111) and L10-(001) superstructure reﬂection positions, respectively. While the a- lattice parameter is close to bulk L10-FePt and remains almost constant, the c- lattice parameter starting from pure bulk L10-FePt ( c=0.371 nm) is continuously increasing with Cr addition, indicating the incor-poration of Cr into the L1 0- FePt lattice [Fig. 3(a)]. For higher Cr content, the lattice parameter approaches the value of L10- CrPt (bulk c=0.381 nm) [ 46]. Initially, a linear increase with Cr content up to 20 at. %, following Vegard’s law for a solidsolution, is observed. However, with further Cr addition somedeviation occurs. The higher Pt concentration in these alloys ismost likely the reason for that, but it should be noticed that thislaw is seldom perfectly obeyed, as discussed in Refs. [ 47,48]. The evaluation of the L1 0order parameter for the FePt ref- erence layer revealed a high degree of ordering with a valueof 0.93, which is close to the theoretical maximum S maxof 1 [Fig. 3(b)]. This high degree of order remains for ﬁlm samples up to 20 at. % Cr. With further increase of Cr content, theorder parameter gets reduced down to about 0.3–0.4. In thisregard, in an earlier study on L1 0CrPt phase formation, it was reported that L10ordering is quite limited when grown at elevated temperatures up to 770◦C compared to postannealed 214436-3NATALIIA Y . SCHMIDT et al. PHYSICAL REVIEW B 102, 214436 (2020) FIG. 2. XRD ( θ−2θ)-scans of the (Fe 100−xCrx)Pt sample series. The additional weak peaks between 2 θ=28.8◦−29.2◦are substrate related. samples [ 49]. Thus, in contrast to FePt, for CrPt even higher deposition temperatures are required for sufﬁcient L10phase formation. To investigate the microstructure in detail, SEM imaging was performed, as displayed in Fig. 4. For reference, L10- FePt ﬁlm pronounced agglomeration with well-separatedislands is found. Most of the individual islands cover an aver-aged area of about 2 ×10 −2μm2but with a broad distribution in sizes. The lattice misﬁt with respect to the MgO crystalis about −9% (tensile stress) and both materials have very similar thermal expansion coefﬁcients [ 50]. Together with a larger surface energy of FePt [ 51] compared to MgO [ 52], the formation of an isolated island morphology when grownat high temperatures is favored. Adding Cr up to 20 at. %shows almost no changes in average island size [Figs. 4(b)– 4(d)]. However, with substitution of Fe by 35 at. % Cr, the morphology changes drastically: Instead of island growth,the ﬁlm appears more continuous with hole structures–largeelongated ones extended over an area of up to 4 .5×10 −2μm2 and small ones with a diameter of about 10 nm [Fig. 4(e)]. With further increase in Cr content, the elongated holes shrinkin size while the density of the small holes increases steadily[Figs. 4(f)–4(h)] until they mostly vanish for pure L1 0-CrPt. It is well known for thin ﬁlms that surface and interfacetension effects play a major role in determining the overallstability of a ﬁlm to agglomeration [ 53]. Further, it is generally observed that tensile stresses within a ﬁlm can enhance holeformation /agglomeration, while compressive stresses oftenlead to hillock formation. Both provide stress relaxation in the ﬁlm. It is speculated that reduced tensile stress with additionof Cr leads to the observed change in morphology. However,due to the lack of literature data, it is hard to relate themicrostructure to all these varying contributions. IV . STATIC MAGNETIC PROPERTIES Room-temperature M-H hysteresis loops of the (Fe 100−xCrx)Pt sample series obtained in in-plane (ip) and out-of-plane (oop) geometry are shown in Fig. 5. Samples with up to 20 at. % Cr reveal strong uniaxialmagnetic anisotropy with an easy axis of magnetization inthe out-of-plane direction. The effective magnetic anisotropyK effwas extracted from the differences of the areas enclosed by the averaged hard axis (ip) and easy axis (oop) loopsand the magnetization axis of the M-Hloops. The uniaxial magnetic anisotropy K uwas calculated as the sum of Keffand the magnetic shape anisotropy ( Ksh=2π·M2 S). Values of KeffandKuobtained at room temperature are summarized in Figs. 6(a) and6(b), respectively. Films with up to 20 at. % Cr show strong perpendicular magnetic anisotropy whichgets gradually reduced from 23 down to 13 Merg /cm 3with increasing Cr content. However, at 35 at. % of Cr, the uniaxialmagnetic anisotropy drops almost to zero. A similar behavioris observed for the coercivity, which continuously decreasesfrom 43 down to 11 kOe for up to 20 at. % Cr and almostvanishes at 35 at. % Cr [Fig. 6(d)]. 214436-4L10-ORDERED (Fe 100−XCrX)Pt THIN … PHYSICAL REVIEW B 102, 214436 (2020) FIG. 3. (a) a-a n d c- lattice parameters and (b) order parameter Sof the (Fe 100−xCrx)Pt sample series in dependence of the Cr content x. Solid lines mark the bulk values of the c-lattice parameter for L10- FePt and L10-CrPt while the dotted line follows Vegard’s law. Of particular interest is the evolution of the saturation magnetization MS. At low Cr concentration, a ferrimagnetic ground state is expected, as the Fe moments couple anti-ferromagnetically to the Cr moments. In this case, as theCr is heavily diluted in the FePt matrix, an overall paral-lel alignment of the Cr moments is supposed. At higherconcentrations, nearest-neighbor Cr-Cr pairs start to appear,thereby increasing magnetic frustration and disorder. Frustra-tion arises from the competition of ferromagnetic Fe-Fe andantiferromagnetic Cr-Cr(Fe) nearest-neighbor (NN) exchangeinteractions, resulting in canting of neighboring magneticmoments, as will be discussed later in more detail by atom-istic spin-model calculations. Nevertheless, in both cases astrong reduction in magnetization can be expected. Indeed,the saturation magnetization gets strongly reduced by almost40% down to 700 emu /cm 3with doping by 35 at. % of Cr [Fig. 6(c)]. At an almost equiatomic Fe to Cr ratio, the MSvalue becomes negligibly small due to the formation of antiferromagnetic order [ 31]. Additional magnetization versus temperature measure- ments were performed at temperatures up to 800 K to studythe inﬂuence of Fe substitution by Cr on the Curie temper-ature [Fig. 7(a)]. For Cr concentrations below 20 at. %, no signiﬁcantly impact on T Cwas observed revealing a value of about 700 K. However, further substitution of Fe with 35 and47 at. % Cr lowers the magnetic ordering temperature substan-tially down to 500 and 300 K, respectively, as summarized inFig.7(b). FIG. 4. (a)–(j) SEM images of the (Fe 100−xCrx)Pt sample series. V . X-RAY MAGNETIC CIRCULAR DICHROISM STUDY To investigate the relative orientation of the magnetic mo- ments between Fe and Cr and to evaluate their spin andorbital moments, XMCD measurements were performed atroom temperature. The XMCD spectra at the Fe and Cr L 2,3 edges are shown in Figs. 8(a) and 8(b), respectively. The resulting dichroism signals have opposite signs, indicating anantiparallel alignment of the Fe and Cr magnetic moments.Element-speciﬁc hysteresis loops, recorded in out-of-planegeometry for samples with perpendicular magnetic anisotropy(up to 20 at. % Cr), show exactly the same shape of mag-netic reversal behavior but mirrored relatively to each other,which is due to strong antiferromagnetic exchange coupling[Figs. 8(c)–8(h)]. Furthermore, using sum rule considerations, the spin ( μ s) and orbital moments ( μl) of Fe and Cr were cal- culated. The results are summarized in Table II. It was found that the Fe spin magnetic moment per atom is in the rangeof 2.1–2.5 μ Bwith a small orbital contribution of about 0.1 μB. For the Cr magnetic moments, values in a similar range 214436-5NATALIIA Y . SCHMIDT et al. PHYSICAL REVIEW B 102, 214436 (2020) FIG. 5. (a)–(j) M-H hysteresis loops measured in out-of-plane and in-plane geometry at 300 K for the (Fe 100−xCrx)Pt sample series. Insets show the enlarged central part of the corresponding loops. TABLE II. Experimental spin ( μs), orbital ( μl), and total mag- netic moment ( μtotal=μs+μl) of Fe and Cr (per atom) of the (Fe 100−xCrx)Pt sample series. Sample Absorption μs μl μtotal edge [ μB][ μB][ μB] Fe52Pt48 Fe 2 .29±0.23 0.14±0.01 2.43±0.24 (Fe 89Cr11)Pt Fe 2 .54±0.25 0.05±0.01 2.59±0.26 Cr 2 .43±0.35 0.30±0.05 2.73±0.40 (Fe 85Cr15)Pt Fe 2 .42±0.24 0.05±0.01 2.47±0.25 Cr 2 .50±0.38 0.34±0.05 2.84±0.43 (Fe 80Cr20)Pt Fe 2 .11±0.21 0.12±0.01 2.23±0.22 Cr 2 .36±0.47 0.50±0.10 2.86±0.57 FIG. 6. Dependence of (a) effective and (b) uniaxial magnetic anisotropy, (c) saturation magnetization, and (d) coercive ﬁeld on the Cr content. All values are obtained at 300 K. Results from ab initio calculations are included in (b) and (c). between 2.3–2.5 μBwere obtained. The magnetic moments of Cr are comparable to values reported for bulk L10-CrPt alloys of 2.24 ±0.15μB[31] but substantially larger than for epitaxial Fe 1−xCrxﬁlms of up to 1.0 ±0.1μB[28]. Interestingly, a rather high orbital moment of up to 0.5 μB is observed for Cr. FIG. 7. (a) Magnetization versus temperature curves measured along the easy axis direction of the respective (Fe 100−xCrx)Pt sample. A guiding ﬁeld of 100 Oe was applied during measurement. (b) De- pendence of the Curie temperature on the Cr content including results from atomistic spin model calculations. 214436-6L10-ORDERED (Fe 100−XCrX)Pt THIN … PHYSICAL REVIEW B 102, 214436 (2020) FIG. 8. XMCD spectra of (a) Fe and (b) Cr for the (Fe 100−xCrx)Pt sample series. (c)–(h) XMCD element-speciﬁc hysteresis loops forselected samples obtained at the Fe- (c), (e), (g) and Cr-edge (d), (f), (h). The dips in the hysteresis loops at zero magnetic ﬁeld are x-ray intensity artifacts which are typical for the TEY detection mode [ 54]. VI. THEORETICAL RESULTS The spin, orbital, and total magnetic moments of the Fe, Pt, and Cr atoms in the L10(Fe 100−xCrx)Pt alloys, as obtained from the ab initio calculations ( T=0 K) up to a concentration of 40 at. %, are summarized in Table III. As can be seen from this table, the magnetic moments of Fe, Cr, and Pt decreaseslightly with increasing Cr content. Comparing these valueswith the experimental ones (see Table II), reasonable agree- ment is obtained considering that the experimental results FIG. 9. Ab initio calculated exchange energies for Fe-Fe, Fe-Cr, Cr-Cr (left scale) and Fe-Pt (right scale) for (Fe 90Cr10)Pt. were obtained at room temperature and, also, represent spatial averages. Furthermore, it should be noted that the ab initio calculated spin moments of 3d elements using DFT-LSDAare in general overestimated compared to experimental val-ues, as spin ﬂuctuations are neglected [ 42]. The calculated uniaxial magnetic anisotropy constants displayed in Fig. 6(b) are systematically larger in magnitude than the experimentalvalues. One reason for this deviation is that the calculationsare performed for the ground state ( T=0 K) while the static magnetic properties were measured at room temperature. Itis well known both from experiment [ 55] and from theory [56,57] that the MAE of FePt decreases almost linearly with increasing temperature. The theoretical MAE monotonouslydecreases with increasing Cr concentration, however, therapid breakdown of the experimental K uabove x=20 at. % is not recovered by the ab initio calculations. This can be attributed to the fact that the decrease of the order parameterSabove 20 at. % [see Fig. 3(b)], was not included in the calculations, which is known to strongly reduce the MAE ofFePt [ 55,56,58,59]. Next, the isotropic exchange interactions for the Fe-Fe, Cr-Cr, and Cr-Fe, as well as for the Fe-Pt pairsof atoms were calculated for the L1 0(Fe 100−xCrx)Pt alloys, as exemplarily presented for x=10 at. % in Fig. 9.M o s t importantly, the NN Fe-Fe interaction is ferromagnetic, whilethe NN Fe-Cr and Cr-Cr interactions are antiferromagnetic,the latter being the largest one in magnitude. Furthermore,we found that the second-NN and also the ﬁfth-NN Fe-Feinteractions that are interlayer couplings are antiferromagetic,as also reported earlier [ 60]. These interactions would sta- bilize antiferromagnetic order of neighboring Fe layers. Theferromagnetic NN and NNN Fe-Pt couplings, see Fig. 9,h o w - ever, stabilize ferromagnetic coupling between adjacent Felayers. Since the induced moments of Pt are not consideredin the spin-model [Eq. ( 5)], we employed the renormalization of the Fe-Fe interactions via the induced moments of Pt, asintroduced by Mryasov et al. [57,61], J ij=Jb ij+1 SFP/summationdisplay νJiνJjν, (6) 214436-7NATALIIA Y . SCHMIDT et al. PHYSICAL REVIEW B 102, 214436 (2020) TABLE III. Ab initio calculated spin moment ( μs), orbital moment ( μl), total magnetic moment ( μtotal=μs+μl)o fF e ,C r ,a n dP t (per atom), saturation magnetization ( MS), uniaxial magnetic anisotropy ( Ku), and the spin model simulated Curie temperatures ( TC)o ft h e (Fe 100−xCrx)Pt series. Sample Element μs μl μtotal MS Ku TC [μB][ μB][ μB][ e m u /cm3] [Merg /cm3][ K ] (Fe 95Cr5)Pt Fe 2.875 0.082 2.957 Cr 2.530 0.018 2.548 993 26.5 742 Pt 0.287 0.046 0.332 (Fe 89Cr11)Pt Fe 2.868 0.083 2.951 Cr 2.539 0.021 2.56 873 25.2 719 Pt 0.263 0.044 0.307 (Fe 85Cr15)Pt Fe 2.863 0.084 2.947 Cr 2.543 0.023 2.566 794 24.4 696 Pt 0.248 0.043 0.291 (Fe 80Cr20)Pt Fe 2.853 0.085 2.938 Cr 2.540 0.026 2.566 693 23.2 672 Pt 0.229 0.041 0.270 (Fe 70Cr30)Pt Fe 2.827 0.087 2.914 Cr 2.504 0.030 2.534 495 19.8 556 Pt 0.188 0.037 0.225 (Fe 60Cr40)Pt Fe 2.799 0.089 2.888 Cr 2.346 0.029 2.375 317 14.9 417 Pt 0.149 0.032 0.181 where Jb ijdenotes the bare exchange interaction between the Fe atoms, Jiνstands for the exchange interaction between t h eF ea t o m iand the Pt atom ν, and SFP=/summationtext iJiν. These renormalized Fe-Fe interactions are then displayed in Fig. 9. Though the ﬁfth-NN Fe-Fe interaction is not affected bythe renormalization, the ﬁrst three Fe-Fe interactions are en-hanced and, in particular, the second-NN Fe-Fe interactionbecame positive (ferromagnetic), as expected. Based on the ab initio calculated J ijand Kuvalues, the magnetic properties of the (Fe 100−xCrx)Pt system in the ground state was investigated. Our results suggest that at lowerconcentrations of Cr, the ground state forms a ferrimagneticsystem due to the antiferromagnetic coupling between theFe and Cr sublattices. However, as the concentration of Cr increases, the probability of ﬁnding nearest Cr-Cr pairs in- creases as well. Such spin arrangement makes the system afrustrated ferrimagnet due to the strong NN Cr-Cr antifer-romagnetic coupling. In Fig. 10, the average angle of the ground-state normalized magnetic moments as a function ofCr concentration is plotted. Due to the exchange frustrationin the system, the Cr spins are not exactly aligned antiparallelto Fe but show a canting behavior. While the canting angleof Fe changes only slightly with Cr content, a strong cantingeffect is observed for the Cr moment, resulting in a cantingangle of up to about 70 ◦with respect to the caxis at 40 at. %. Taking into account this canting effect, the saturationmagnetization was determined which shows good agreementwith the experimental results, as displayed in Fig. 6(c). Starting from the ground-state spin conﬁgurations at zero temperature, the Curie temperatures of the (Fe 100−xCrx)Pt al- loys were calculated. An example is presented in Fig. 11also showing the variation of the magnetization with temperaturefor the two individual sublattices of Fe and Cr. For Cr con-centration below 20%, the Curie temperature decreases ratherslowly and stays within the temperature range of 650–750 K.However, further increase of Cr concentration brings the Curie temperature to decrease rapidly, which is in excellent agree-ment with the experimental results [Fig. 7(b)]. VII. CONCLUSIONS Epitaxial pseudobinary (Fe 100−xCrx)Pt thin ﬁlms were grown on MgO(001) covering the full composition range(x=0−100). As expected from the phase diagrams of the FePt and CrPt bulk alloys near equiatomic composition, Fesubstitution by Cr in the L1 0structure with (001) orientation is obtained. However, beyond 20 at. % the chemical order pa-rameter decreases continuously, as the deposition temperatureof 770 ◦C appears not to be sufﬁciently high for develop- ment of the L10CrPt phase. High perpendicular magnetic anisotropy and coercivity were observed with addition of up FIG. 10. The average angle of the ground-state spin moments with respect to the caxis shown as a function of Cr content for (Fe 100−xCrx)Pt. 214436-8L10-ORDERED (Fe 100−XCrX)Pt THIN … PHYSICAL REVIEW B 102, 214436 (2020) FIG. 11. Magnetization as a function of temperature for (Fe 89Cr11)Pt. Green dots and red line denote theoretical and ex- perimental values, respectively. The inset shows the temperature dependent magnetization of the two sublattices Fe and Cr. to 20 at. % Cr. The change in coercivity is further attributed to a strong alteration of the ﬁlm morphology changing fromislandlike to a more continuous ﬁlm structure. Comparing our measurements with multiscale simulations, combining ab initio theory with spin-model simulations, we ﬁnd that in dilute alloys with low Cr concentration, isolatedCr magnetic moments couple antiferromagnetically to theferromagnetic Fe matrix, reducing the saturation magneti-zation. With increasing Cr concentration, nearest-neighborCr-Cr pairs start to appear, thereby increasing magneticfrustration and disorder. Frustration arises from the competi-tion of ferromagnetic Fe-Fe, and antiferromagnetic Cr-Cr(Fe)nearest-neighbor exchange interactions, which lead to cantingof neighboring magnetic moments, as disclosed by our cal- culations. Furthermore, this effective exchange coupling willalso impact the Curie temperature of the (Fe 100−xCrx)Pt alloy ﬁlms. With Cr substitution of up to 20 at. %, no pronouncedchange in Curie temperature, which is in the range of 700 K,was noticed. But with addition of more than 35 at. % Cr, theCurie temperature drops down substantially. These results arein excellent agreement with our calculations. Furthermore, XMCD studies on alloys containing up to 20 at. % of Cr conﬁrmed strong antiferromagnetic exchangecoupling between Fe and Cr and revealed spin magneticmoments in the range between 2.1– 2.5 μ B, in reasonable agreement with ab initio calculations. ACKNOWLEDGMENTS Financial support provided by the German Research Founda- tion (DFG) under Grants No. AL 618 /31-1 and No. 290 /5-1 are gratefully acknowledged. The authors are thankful forthe allocation of synchrotron radiation beam time and ac-knowledge the ﬁnancial support from HZB (Project No.17105004-ST). We would like to thank O. Ciubutariu, M.Heigl, and R. Wendler for assistance at the HZB beamlineand S. Rudorff for technical support, as well as F. Jung forassistance in SEM measurements. We are thankful for addi-tional SEM measurement time at the CEITEC Nano ResearchInfrastructure with technical support from O. Man. 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PhysRevB.98.174115.pdf	PHYSICAL REVIEW B 98, 174115 (2018) Thermal ﬂuctuations of dislocations reveal the interplay between their core energy and long-range elasticity Pierre-Antoine Geslin* ELyTMaX UMI 3757, CNRS - Université de Lyon - Tohoku University, International Joint Unit, Tohoku University, Sendai, Japan; Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aobaku, Sendai 980-8577, Japan; and Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Aramaki, Aoba 6-3, Aobaku, Sendai 980-8578, Japan David Rodney† Univ. Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, LYON, F-69622 France (Received 21 May 2018; revised manuscript received 6 November 2018; published 29 November 2018) Equilibrium ﬂuctuations of dislocations are central to the plastic response of metals and alloys because they control the attempt frequency of thermally activated events. We analyze here atomic-scale simulationsof thermally vibrating dislocations with the help of an analytical model and show that thermal ﬂuctuationsintimately involve both long-range elasticity and short-range core effects. In addition, equilibrium ﬂuctuationsof edge and screw dislocations in aluminum are used to derive quantitative parameters that characterizetheir energetics and dynamics and we discuss how large-scale models such as dislocation dynamics can beparametrized based on these results. In particular, we show that the core parameters found here throughﬂuctuations are transferable and can be used to predict dislocation bow-out under an applied stress. DOI: 10.1103/PhysRevB.98.174115 I. INTRODUCTION Dislocations are linear defects responsible for the plastic ﬂow of metals and alloys. While an important body of workhas focused on their athermal properties (core energy, latticefriction, elastic interactions, stress-induced curvature, etc.)[1–4], their behavior at ﬁnite temperatures remains only par- tially explored. While recent progress was obtained on the freeenergy barrier against dislocation motion [ 5,6], dislocation thermal vibrations remain poorly understood despite theirimportance in setting the attempt frequency of thermally acti-vated dislocation processes such as dislocation movement insolution-strengthened alloys, kink-pair nucleation, and crossslip [ 7,8]. Dislocation vibrations were the subject of the seminal works of Friedel [ 9] and Granato and Lücke [ 10], who modeled a dislocation based on the simplifying assumptionthat it behaves like a vibrating string characterized by a linetension /Gamma1and an energy E=/Gamma1/lscript, with /lscriptthe dislocation length. At thermal equilibrium, the mean square amplitude ofa vibrational mode is then inversely proportional to the squareof its wave vector k(see, e.g., Ref. [ 11]), a consequence of the equipartition theorem. Such dependence was measured usingmolecular dynamics (MD) on ﬂuctuating solid/liquid inter-faces [ 12]. Using this so-called capillary ﬂuctuation method, the analysis of solid/liquid interfacial ﬂuctuations has beenused to assess the solid/liquid anisotropy [ 12,13] and the *pierre-antoine.geslin@insa-lyon.fr †david.rodney@univ-lyon1.frkinetics coefﬁcient [ 14], necessary to parametrize quantita- tively phase-ﬁeld models of solidiﬁcation [ 15]. In contrast, it was shown that thermal ﬂuctuations of grain boundaries follow a different scaling because of the long-range elastic interactions induced by their ﬂuctuations [ 16,17]. As for dislocations, a line tension model accounts only for short-range effects and ignores their long-range elasticinteractions. The latter may affect the vibrational spectrumand hence the attempt frequency of thermally activated events[18]. In this paper, we use face centered cubic (FCC) alu- minum as a benchmark and study from the atomic scale thevibrational spectrum of edge and screw dislocations. We ﬁnda1/k 2dependence but only at short wavelengths where core effects dominate, while a different scaling in 1 /[k2log(k0/k)] is obtained at long wavelengths due to long-range elasticity, inaccordance with previous works [ 19,20]. The spectra can be fully explained by modeling core effects with a line tensionand elasticity with the nonsingular theory of dislocations ofCaiet al. [21]. This theory avoids the core singularity by spreading the Burgers vector over a ﬁnite distance calledthe core parameter. Fitting the vibrational spectra againstour analytical model allows us to extract character-dependentcore parameters and line tension coefﬁcients. The reliabil-ity and transferability of these parameters are demonstratedby reproducing dislocation bow-out under an applied shearstress. Moreover, analyzing the time-correlations of the vi-brations, we obtain wavelength-dependent mass and damp-ing coefﬁcient for both dislocation characters. Use of theseparameters in mesoscale dislocation dynamics (DD) mod-els [ 22–24] as well as in recent attempts to model thermal ﬂuctuations of crystalline defects as Langevin forces [ 25]i s discussed. 2469-9950/2018/98(17)/174115(7) 174115-1 ©2018 American Physical SocietyPIERRE-ANTOINE GESLIN AND DA VID RODNEY PHYSICAL REVIEW B 98, 174115 (2018) FIG. 1. Normalized elastic energy of a sinusoidal perturbation on screw and edge dislocations compared to numerical calculations (thin lines with symbols). The total energy Eel+Ecois shown as dashed lines for values of the line tension reported in Table II. II. ENERGY OF A PERTURBED DISLOCATION We consider an inﬁnite dislocation line of Burgers vector [bs,be,0] (see Fig. 1) and a core parameter athat represents the core spreading distance in the nonsingular dislocationtheory [ 21]. We assume a perturbation h(x),Lperiodic in the xdirection, expressed as a Fourier series h(x)=/summationtext Nmax n=−NmaxCneiknxwhere kn=2πn/L andC−n=C∗ n[h(x) is real]. We do not consider ﬂuctuations out of the glideplane because at large wavelengths, they would involve climb[26] and at small wavelengths, they would correspond to line tension terms independent of the in-plane ﬂuctuations.Within isotropic elasticity, and assuming that the perturbationvaries slowly [i.e., ∀x,h /prime(x)/lessmuch1], the excess self-energy due to the perturbation can be expressed analytically within thenonsingular theory of dislocations as E el(h)=L 2Nmax/summationdisplay n=1Kel(kn)\|Cn\|2(1) with Kelan elastic stiffness expressed as: Kel(k)=μ π(1−ν)a2/braceleftbigg −/parenleftbig 2b2 s(1−ν)+b2 e/parenrightbig +a2k2K0(ak)/parenleftbig 2b2 s−νb2 e/parenrightbig +ak ×K1(ak)/bracketleftbigg (3−ν)b2 s−b2 e/parenleftbigg 1−a2k2(1−ν) 2/parenrightbigg/bracketrightbigg −a2k2K2(ak)/parenleftbiggb2 s(1+ν) 2−b2 e/parenrightbigg/bracerightbigg , (2) where μandνdenote the shear modulus and Poisson ratio, and Ki(i=0,1,2) are modiﬁed Bessel functions of the second kind (see Sec. 1 of Ref. [ 27] for details of thecalculations). We note that this analytical solution assumes isotropic elasticity. Generalizing our approach to anisotropicelasticity would require the use of a numerical approach (see,e.g., Refs. [ 24,28]). In addition to the elastic contribution, we account for the core energy of the perturbation using a line tension model.As shown previously [ 2,29,30], this contribution allows us to account for the additional energy due to the nonlinearitiesin the core region that cannot be accounted for with linearelasticity. The energy E co(h) has the same form as Eq. ( 1) and a stiffness (see Sec. 2 of Ref. [ 27]) Kco(k)=2/Gamma1k2. (3) To assess the domain of validity of Eqs. ( 1)–(2), we con- sider a sinusoidal perturbation h(x)=Asin(kx). We compute the elastic energy in Eq. ( 1) and compare with a numerical estimate obtained from the interaction energy between straightsegments of the discretized dislocation line [ 21]. We consider here a dislocation length L=400 Å with elastic parameters of aluminum ( ν=0.34,μ=29.8 GPa). As shown in Fig. 1, the numerical solutions converge towards the prediction ofEqs. ( 1)–(2) in the limit of small amplitude Aor small wave vector k. It is worth mentioning that the elastic energy of a screw dislocation grows faster with kthan for an edge dislocation due the different nature of elastic interactions.We also note that for large wave vectors, the elastic energybecomes negative. This occurs for ka/greaterorsimilar1, which is when the wavelength of the ﬂuctuations becomes of the order of thecore parameter, i.e. in a region where spreading the Burgersvector may lead to artifacts [ 31]. The potential instability associated with a negative elastic energy is, however, coun-terbalanced by the core energy in Eq. ( 3) (see dashed lines in Fig. 1), which increases as k 2and far exceeds the elastic energy at large k. Vibrational spectra are predicted from the dislocation en- ergy using the equipartition theorem. Adding both the coreand elastic contributions yields the total energy of a perturbeddislocation, expressed in the same form as in Eq. ( 1) but with the total stiffness K=K el+Kco. Since the energy is the sum of quadratic contributions of independent degrees of freedom(two terms per Fourier mode because C nis complex), the equipartition theorem ensures that at thermal equilibrium: /angbracketleft\|Cn\|2/angbracketright=2kBT LK(kn). (4) III. MOLECULAR DYNAMICS SIMULATIONS AND POWER SPECTRA In this section, we compare the prediction of Eq. ( 4) with MD simulations. We chose FCC aluminum because ofits near isotropic elasticity and the availability of a reliableinteratomic potential [ 32]. For both screw and edge characters, a dislocation dipole is initially introduced in a large simulationcell through its displacement ﬁeld [ 2,29,33,34]. The position of the dislocations in the cell was chosen such that the interac-tions between the dislocations of the dipole and their periodicimages is reduced to a minimum (see Fig. 2). The dimensions of the periodic cells reported in Table Iwere selected to keep comparable dislocation lengths and interdislocation distances 174115-2THERMAL FLUCTUATIONS OF DISLOCATIONS REVEAL … PHYSICAL REVIEW B 98, 174115 (2018) FIG. 2. Visualization of the simulation cell used for the (a) screw and (b) edge dislocations (only the atoms having a HCP stacking that belong to the stacking fault are shown). for both characters. We then performed long MD simulations with LAMMPS [35] in the microcanonical (NVE) ensemble to generate a large number of dislocation conﬁgurations. The dislocation position was estimated using the common neighbor analysis (CNA) implemented in LAMMPS : the atoms belonging to the stacking fault between the two partials aredetected as having a hexagonal (HCP) stacking. To improvethe estimate of the dislocation position, we also include theatoms of the partial dislocation core deﬁned as defective FCCatoms, nearest neighbors of the stacking fault atoms. Theposition of the dislocation is then obtained by averaging thepositions of the stacking fault and core atoms in equispacedbins along the dislocation line. We discuss here results obtained at 300 K, but we checked that our methodology also applies at other temperatures (seeSec. 5 of Ref. [ 27]). From ten independent 0 .2 ns long simula- tions, we extract about 20000 dislocation conﬁgurations, andcompute the vibrational spectra from their Fourier transform.Figure 3displays the resulting spectra for edge and screw characters. We ﬁrst note that the power spectrum of the edge dis- location is signiﬁcantly higher than that of the screw, asexpected from the lower energy of a perturbation on an edgethan on a screw dislocation seen in Fig. 1. Two regimes can be distinguished with a transition at around k∼0.5Å −1, (i.e., a wavelength ∼13 Å): large kﬂuctuations are domi- nated by the core contribution and scale approximately as1/k 2, while small kﬂuctuations show a different scaling and are dominated by the elastic energy. As expected fromprevious works [ 19,20], an expansion of Eq. ( 2)i nt h e limitak/lessmuch1s h o w sa1 /[k 2log(k0/k)] scaling displayed in Fig. 3. Furthermore, ﬁtting the power spectra with Eq. ( 4) allows us to determine the best choice of core parameter aand line tension /Gamma1for both orientations. The ﬁts were performed in the central part of the spectra (full lines in Fig. 3) because the long TABLE I. Dimensions of the molecular dynamics simulation cells. character N[1¯10]N[11¯2]N[111] Natoms L/bardbl[Å] Ldip[Å] screw 173 50 70 3 633 000 495.4 245.5 edge 120 100 48 3 441 600 496.0 240.6FIG. 3. Vibrational spectra obtained at 300 K for screw and edge dislocations (symbols) and corresponding ﬁts (lines) from Eq. ( 4). Fluctuation amplitudes are averaged over a large number of conﬁgu- rations such that uncertainties are smaller than the symbol size. Thedashed lines are guides for the eye representing asymptotic behaviors in the limit of small and large wave vectors. and short wavelength ﬂuctuations are affected, respectively, by long-range interactions with periodic images and the dis-crete nature of the crystalline structure. The ﬁts are of highquality, showing that dislocation ﬂuctuations can be faithfullyreproduced using the analytical model of Eq. ( 4). Table IIlists the parameters aand/Gamma1obtained from ﬁtting the power spectra obtained at 300 K. We also report here theline tension computed at 0 K from static energy calculationson straight dislocations (see Ref. [ 24] and Sec. 4 of Ref. [ 27]). For straight dislocations, the choice of ais arbitrary [ 4,24] and the 0 K line tensions in Table IIwere computed for the same avalue as found at 300 K. By way of contrast, the present calculations yield a unique pair ( a,/Gamma1) for each orientation because for ﬂuctuating curved dislocations, adeﬁnes uniquely the wavelength where long-range elasticity ceases to dominateover core effects. We added in Table IIthe dissociation distance d 0of edge and screw dislocations to highlight that aincreases with d0. The values of aobtained here therefore capture the larger physical spreading of the edge dislocationcore compared to the screw. In addition, the line tensionsdeduced from the ﬂuctuation spectra are in good agreementwith the ones obtained at 0 K. TABLE II. Fitted values of the core parameters and line tension obtained at 300 K and comparison with dissociation distances andline tensions obtained from molecular static simulations. character screw edge a(from ﬁt) [Å] 3.34 5.60 d0(0 K) [Å] (from [ 24]) 6.34 12.19 /Gamma1(from ﬁt) [eV /Å] 0.114 0.053 /Gamma1a(0 K) [eV /Å] 0.134 0.085 174115-3PIERRE-ANTOINE GESLIN AND DA VID RODNEY PHYSICAL REVIEW B 98, 174115 (2018) FIG. 4. Comparison between bow-out obtained from atomistic calculations (atoms belonging to the stacking fault are shown) andfrom the elastic model (continuous red line). IV . TRANSFERABILITY TO DISLOCATION BOW-OUT To check the reliability of our results and in analogy with the ﬂuctuation-dissipation theorem, we investigated thebow-out of periodically pinned dislocations under a small external stress, as considered in the past by Szajewski et al. [4]. We considered as above dipoles of either edge or screw dislocations but with a smaller length L≈130 Å. A resolved shear stress was applied by straining the simulation cells. Thedislocations are pinned by freezing the atoms located within adistance ¯d=bof the sides of simulation cell and belonging to the dislocation core. Reasonable stresses below 110 MPa areconsidered to remain in the limit of a small bow-out. Figure 4displays the averaged atomistic conﬁgurations obtained for both screw and edge dislocations at 300 K. To compare these atomistic results with our analytical approach, we use the energy of a deformed dislocation dis-cussed in Sec. II. We use the fact that the dislocation shape is symmetric with respect to L/2( s e eF i g . 4) to express the dislocation shape as a sum of cosine functions: h bo(x)=A0+Nmax/summationdisplay n=1Ancos(knx), (5) where A0is the average height of the dislocation. To en- force the fact that the dislocation is pinned, we impose h(x=0)=0 by the condition A0=−/summationtextNmax n=1An. The energy related to the bow-out is given by Eqs. ( 1)–(3) with C−n= Cn=An/2 and an extra term to account for the work of the Peach-Koehler force: Epk=−A0L[(σab∧l/bardbl)·l⊥] =−A0Lfpk =LfpkNmax/summationdisplay n=1An, (6) where l/bardbland l⊥are unit vectors lying in the slip plane of the dislocation, respectively, parallel and perpendicular to thestraight dislocation line (see Fig. 4). In addition, to enforce thefact that the dislocation position remains ﬁxed on a length ¯d close to the sides of the cell, we add a Lagrangian multiplierL({A n}) to the total energy to enforce h(¯d)=h(L−¯d)=0. Therefore, the energy of the dislocation is expressed as afunction of the coefﬁcients A n: Etot({An})=Eel({An})+Eco({An}) +Epk({An})+L({An}). (7) The equilibrium conﬁguration of the dislocation is obtained by minimizing this total energy with respect to the amplitudesA nusing a conjugate gradient algorithm. Figure 4shows the comparison between atomistic results and elastic calculations for both screw and edge characterswith parameters taken from Table II. We ﬁrst notice the very good agreement between the molecular dynamics results andelastic calculations for both characters. Slight differences canbe seen near the obstacles, due to the difﬁculty to imposeperfectly equivalent boundary conditions in atomistic and theelastic models. On the other hand, the curvature of the dis-locations is practically identical far from the blocked atoms.Therefore, the parameters deduced from the analysis of ﬂuc-tuations at ﬁnite temperature can be directly used to predictquantitatively bow-out conﬁgurations under ﬁnite stresses. V . TIME-DEPENDENT BEHA VIOR The dynamical behavior of the dislocations can be fur- ther investigated by computing the time correlation of thevibrational modes, /angbracketleftC n(0)C∗ n(t)/angbracketrightin connection with Langevin models of defect dynamics [ 25]. In this case, the time correla- tions should decrease following underdamped dynamics: /angbracketleftCn(0)C∗ n(t)/angbracketright=/angbracketleft \|Cn\|2/angbracketrightcos(ωnt)e−t/τn, (8) where the coefﬁcients ωnandτnare related to the effective massMnand drag coefﬁcient Bnof the Langevin equation [36] (see Sec. 3 of Ref. [ 27] for details). Figures 5(a) and 5(b) conﬁrm that the time correlations exhibit the expected underdamped behavior for the screw and edge characters.Using nonlinear ﬁts, we obtained the drag coefﬁcient andeffective mass displayed in Figs. 5(c) and5(d). We note that the ﬁts are of better quality for the edge than for the screwdislocation. We attribute this to the larger Peierls stress of thescrew (not accounted for in the present approach), that slowsdown its dynamics and affects the ﬂuctuation kinetics. As expected from dislocation theory [ 1], the drag coef- ﬁcient of the screw dislocation is larger than the edge forany wave vector. Figures 5(c) and5(d) also demonstrate that the drag coefﬁcient and effective mass depend signiﬁcantlyon the wave vector, showing that perturbations are dumpedmore quickly for small wavelengths due to the increase of B and decrease of Mwithk. In particular, the drag coefﬁcient of the edge dislocation depends linearly on the wave vectorand extrapolates to B 0=13.4μPa/s in the limit of inﬁnite wavelengths, close to the value obtained by Bitzek [ 37,38]f o r straight dislocations. By way of contrast, the effective massdoes not depend signiﬁcantly on the dislocation character anddecreases roughly exponentially with the wave vector [seedashed line in Fig. 5(d)]. Using this crude ﬁt, the extrapolation to a straight line yields M 0=48 fg/m, again close to the 174115-4THERMAL FLUCTUATIONS OF DISLOCATIONS REVEAL … PHYSICAL REVIEW B 98, 174115 (2018) FIG. 5. Examples of time correlation functions for the (a) screw and (b) edge dislocation. (c) Drag coefﬁcient and (d) effective mass are shown as a function of wave vector and compared with dataobtained for a straight edge dislocation [ 37]. value reported by Bitzek (with a different method and at 30 K) [37,38]. VI. EXTRACTING PARAMETERS FOR HIGHER-SCALE MODELS In Sec. III, we have shown that reproducing accurately the power spectra obtained from atomistic simulations requiresdifferent core parameters for the edge and screw characters(see Table II). This character dependence reﬂects the change of core structure with dislocation orientation and the particulardifﬁculty to represent dissociated dislocations with perfect,albeit spread, dislocations. However, the nonsingular theoryand its numerical implementation in DD codes require a singlecharacter-independent value of a[21,22,39,40]. To obtain parameters compatible with the DD formalism, the power spectra can also be ﬁtted at long wavelengths byenforcing a single value of aas shown in Fig. 6(a).W e then obtain a core parameter a=3.50 Å and line tensions FIG. 6. (a) Fit of the power spectra obtained at 300 K by enforc- ing the same core parameter for the screw and edge characters. The ﬁt is only performed for intermediate wave vectors in between thevertical dashed lines. (b)–(c) Comparison of bowing-out dislocation shape between atomistic simulations at 300 K and two sets of parameters: listed in Table II(red) obtained from the ﬁt shown in (a) (dashed blue). coefﬁcients /Gamma1s=0.119 and /Gamma1e=0.0297 eV /Å for the screw and edge characters, respectively. While this ﬁt does notreproduce accurately the details of the ﬂuctuations at smallwavelengths ( <10 Å), it reproduces satisfactorily the large wavelength regime. In addition, as shown in Figs. 6(b) and 6(c), the bow-out obtained with this set of parameters (shown with dashed blue lines) is essentially identical to the bow-outobtained with character-dependent core parameters (shown inred) because of the long wavelength involved in this bow-out. Moreover, the kdependence of the mass and drag coef- ﬁcient pointed out in Sec. Vcannot be accounted for in real space implementations of DD. Because DD is often used to in-vestigate mesoscale dislocation microstructures, much largerthan the dislocation lengths investigated here, this kdepen- dence could in principle be neglected. Then, limiting valuesat large wavelengths can be used ( M 0andB0mentioned previously) to investigate these large dislocation assemblies.However, it is worth keeping in mind that these M 0and B0parameters would not reproduce faithfully the dislocation 174115-5PIERRE-ANTOINE GESLIN AND DA VID RODNEY PHYSICAL REVIEW B 98, 174115 (2018) dynamics when using DD to investigate mechanisms occur- ring on nanometric scales [ 4,24,41–43]. VII. CONCLUSION In conclusion, we propose here a method to analyze dislo- cation ﬂuctuations by means of an analytical approach. Ouranalysis reveals the interplay between long-range elasticityand short-range core effects and shows the necessity to ac-count for both to reproduce faithfully the dislocation behaviorover a wide range of wavelengths. The reliability of the ob-tained parameters is tested by showing that they can reproducevery faithfully atomistic bow-out conﬁgurations. In addition,analysis of the time correlations of the ﬂuctuations reveals amarked wavelength dependence of the dislocation dynamics.Finally, we discuss how to extract reliable parameters that canbe used as input of higher-scale models such as dislocationdynamics. We note that we have considered here the thermal ﬂuc- tuations of isolated dislocations. They are not representativeof the temperature-dependent plastic behavior at the mi-crostructural scale, which is often controlled by collectiveeffects [ 44–46]. Our work can, however, be used to parametrized quantitatively DD simulations that are the ap-propriate tool to study collective effects. Also, the analyticalexpression derived here [Eqs. ( 1)–(3)] can be used to investi- gate quantitatively other dislocation processes: natural exten-sions include thermally-activated processes such as kink-pairnucleation in friction-limited dynamics, the cross slip of screwdislocations or the thermally activated dislocation glide indilute or high-entropy alloys. ACKNOWLEDGMENTS We would like to thank Emmanuel Clouet for fruitful dis-cussions concerning the calculation of dislocation core en-ergies from molecular static simulations. We acknowledgesupport of the Agence Nationale de La Recherche throughGrant No. ANR-13-MERA-0001-02. P.A.G. gratefully ac-knowledges SR16000 supercomputing resources from theCenter for Computational Materials Science of the Institutefor Materials Research, Tohoku University. D.R. acknowl-edges support from LABEX iMUST (ANR-10-LABX-0064)of Université de Lyon (program “Investissements d’Avenir”,ANR-11-IDEX-0007). [1] J. Hirth and J. Lothe, Theory of Dislocations (McGraw-Hill, New York, 1968). [2] E. Clouet, L. Ventelon, and F. Willaime, P h y s .R e v .L e t t . 102, 055502 (2009 ). [3] D. Rodney, L. Ventelon, E. Clouet, L. Pizzagalli, and F. Willaime, Acta Mater. 124,633(2017 ). [4] B. Szajewski, F. Pavia, and W. Curtin, Modell. Simul. Mater. Sci. Eng. 23,085008 (2015 ). [5] M. R. Gilbert, P. Schuck, B. Sadigh, and J. Marian, Phys. Rev. Lett. 111,095502 (2013 ). [6] T. D. Swinburne and M.-C. Marinica, Phys. Rev. Lett. 120, 135503 (2018 ). [7] T. Vegge, T. Rasmussen, T. Leffers, O. B. Pedersen, and K. W. Jacobsen, Phys. Rev. Lett. 85,3866 (2000 ). [8] D. Caillard and J.-L. Martin, Thermally Activated Mechanisms in Crystal Plasticity (Elsevier, Amsterdam, 2003), V ol. 8. [9] J. Friedel, Dislocations (Pergamon Press, Oxford, 1964). [10] A. Granato, K. Lücke, J. Schlipf, and L. Teutonico, J. Appl. Phys. 35,2732 (1964 ). [11] C. Misbah, O. Pierre-Louis, and Y . Saito, Rev. Mod. Phys. 82, 981(2010 ). [12] J. J. Hoyt, M. Asta, and A. Karma, Phys. Rev. Lett. 86,5530 (2001 ). [13] M. Asta, J. J. 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PhysRevB.82.144404.pdf	Spin torque dynamics with noise in magnetic nanosystems J. Swiebodzinski,1,2A. Chudnovskiy,1T. Dunn,3and A. Kamenev3,4 1I. Institute of Theoretical Physics, University of Hamburg, Jungiusstr. 9, 20355 Hamburg, Germany 2Theoretische Physik, Universität Duisburg–Essen and CeNIDE, 47048 Duisburg, Germany 3Department of Physics, University of Minnesota, Minneapolis, Minnesota 55455, USA 4Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA /H20849Received 12 May 2010; revised manuscript received 30 August 2010; published 4 October 2010 /H20850 We investigate the role of equilibrium and nonequilibrium noise in the magnetization dynamics on mono- domain ferromagnets. Starting from a microscopic model, we present a detailed derivation of the spin shot-noise correlator. We investigate the ramiﬁcations of the nonequilibrium noise on the spin torque dynamics, bothin the steady-state precessional regime and the spin switching regime. In the latter case, we apply a generalizedFokker-Planck approach to spin switching, which models the switching by an Arrhenius law with an effectiveelevated temperature. We calculate the renormalization of the effective temperature due to spin shot noise andshow that the nonequilibrium noise leads to the creation of cold and hot spot with respect to the noise intensity. DOI: 10.1103/PhysRevB.82.144404 PACS number /H20849s/H20850: 75.70. /H11002i, 75.75.Jn, 85.75. /H11002d I. INTRODUCTION The manipulation of magnetization of a ferromagnet by means of spin-polarized currents is a key issue of the state-of-the-art spintronics concepts /H20849for a review see Ref. 1/H20850.I n this respect the most important phenomenon is the so-calledspin-transfer-torque /H20849STT /H20850effect, which was predicted by Slonczewski and Berger. 2,3A spin-polarized current may transfer angular momentum to a free ferromagnetic layer re-sulting in a macroscopic torque on the latter’s magnetization.In very small ferromagnets, in which the magnetization canbe assumed spatially uniform, the STT results in the rotationof the magnetization as a whole rather than in the excitationof spin waves. The STT can, in particular, lead to two dy-namical regimes: the reversal of the free layer’s magnetiza-tion or a steady-state precession of the magnetization. Due tothe giant magnetoresistance effect, the dynamics of magne-tization is reﬂected in the change in resistance of the circuitcontaining magnetic junctions. Current-induced resistancevariations, identiﬁed with STT, were reported in Ref. 4.I n subsequent years, both spin torque induced magnetizationreversal were observed 5,6as well as steady-state precession.7–10Both dynamical regimes are interesting for applications: as a tempting alternative to Oersted ﬁelds inswitching the magnetization in ferromagnetic memory ele-ments on one hand, and as clock devices used to synchronizethe CPU with other logic units, on the other. Hence a reliabledescription of these phenomena is very important. On a semiclassical level, magnetization dynamics of a monodomain ferromagnet can be well described by theLandau-Lifshitz-Gilbert /H20849LLG /H20850equation, dm dt=−/H92530m/H11003Heff+/H92510m/H11003dm dt+/H92530 MsVm/H11003/H20849m/H11003Is/H20850. /H208491/H20850 Here mis a unit vector in the free layer’s magnetization direction, Heffthe effective magnetic ﬁeld, Vthe volume of the switching element, Msthe absolute value of the free lay- er’s magnetization, /H92530the gyromagnetic ratio, /H92510the Gilbert damping parameter, and Isthe spin-polarized current. How-ever, as the extension of associated devices are very small, effects of noise may play a signiﬁcant role and should beincluded into the dynamical description. A ﬁrst inclusion ofnoise into the LLG was given in the seminal paper ofBrown, 11who considered the effect of thermal ﬂuctuations on the dynamics of a monodomain particle by a randomcomponent h Rof the effective magnetic ﬁeld entering the LLG Eq. /H208491/H20850. As a consequence of the ﬂuctuation-dissipation theorem one ﬁnds for the equilibrium correlator of the ran-dom ﬁeld, 11 /H20855hi/H20849t/H20850hj/H20849t/H11032/H20850/H20856/H11008/H92510kBT/H9254ij/H9254/H20849t−t/H11032/H20850, /H208492/H20850 where hi/H20849t/H20850denotes the ith Cartesian component of the ran- dom ﬁeld at time t,kBis the Boltzmann constant, and Tthe temperature. Since then, temperature effects on the LLGequation have been considered, both with 12–16and without the spin torque term.17,18The emphasis of these approaches has been on the inﬂuence of noise on switching rates, oftenby performing explicit numerical calculations. 12,13,17,18Since the spin torque experiments8,9,19–21are performed under clear nonequilibrium conditions, it is natural to address, besidestemperature, other sources of noise. One possible source ofnonequilibrium noise is the spin shot noise. By analogy withthe charge shot noise, the quantization of the angular mo-mentum transfer leads to spin shot noise, which manifestsitself in a random Langevin force entering the equations ofmotion for the free magnetic layer. It was shown by Foros et al. 16in the context of normal-metal/ferromagnet/normal- metal structures that the spin shot noise is the dominant con-tribution to magnetization noise at low temperatures. In therealistic experiments on spin torque and spin switching, thenonequilibrium noise starts to dominate at temperatures be-low several kelvins. In Ref. 22, it was shown that inclusion of the nonequilibrium noise into the dynamical descriptioncan explain the experimentally observed nonmonotonic de-pendence of the microwave power spectrum on the voltage,as well as the saturation of the spectral linewidth at lowtemperatures. In this paper, we concentrate on the effect of the nonequi- librium noise on spin switching. Without noise, the spinPHYSICAL REVIEW B 82, 144404 /H208492010 /H20850 1098-0121/2010/82 /H2084914/H20850/144404 /H208499/H20850 ©2010 The American Physical Society 144404-1switching takes place when the spin current /H20849and with it the resulting spin torque /H20850exceeds a critical value. The critical current is determined by the strength of the magnetic aniso-tropy, external magnetic ﬁeld, and Gilbert damping, and itcan be obtained from the solution of the deterministic LLGEq. /H208491/H20850. Magnetization noise opens a possibility of activated switching at currents much less than critical. Moreover, inthe presence of noise the switching becomes a random pro-cess that requires a probabilistic description. The latter isbased on the solution of the Fokker-Planck /H20849FP/H20850equation for the probability distribution of magnetization as derived inRef. 22. A crucial step toward this description has been made in Ref. 14, where the authors reduced the FP equation for magnetization to the effectively one-dimensional FP equa-tion for the probability distribution of energies and applied itto the description of activated switching processes. A number of experiments on current-induced switching have been carried out previously. 23–27From them it was found that the noise reduces the typical switching time, asone might expect. Myers et al. 23observed a broad distribu- tions of switching currents strongly depending on tempera- ture, indicating a thermally activated switching process al-tered by the STT. To ﬁt the measured data, they used a Neel-Brown model 11,28with a ﬁeld- and current-dependent potential barrier height U/H20849H,I/H20850. In this model, the probability for the magnetization to switch decays exponentially withtime over a characteristic relaxation time /H9270that obeys the Arrhenius law /H9270/H11011eU/kBT. An implicit assumption in the Neel- Brown theory is that magnetization dynamics is governed bya torque from an effective magnetic ﬁeld H eff, which is de- rivable from the free energy E/H20849M/H20850of the system via Heff= −1 /H92620/H11612ME/H20849M/H20850. The spin torque however is nonconservative and the concept of a corresponding potential barrier is ill deﬁned, which makes the situation signiﬁcantly more com-plicated. For thermally activated switching in presence ofSTT Urazhdin et al. 24,25found that the activation energy strongly depends on the magnitude as well as the direction ofthe current. To capture the observed features, they introducedan effective temperature distinct from the real temperature inthe Neel-Brown formula. Its current directional dependenceindicated that the heating is not the ordinary Joule heating.Based on a stationary solution of the Fokker-Planck equationApalkov and Visscher, 14,15and Li and Zhang,12linked this effective temperature to the spin torque. In their model, thealteration of switching rates is due to the change in the el-evated effective temperature in the Arrhenius factor, which ingeneral yields a non-Boltzmann probability distribution. Weextend the approach of Ref. 14considering the inﬂuence of the noise on the switching in the whole range of currents,from the noise-induced activated switching at small currentsup to the almost deterministic switching by large critical cur-rents. We also take into account speciﬁc angular dependenceof the nonequilibrium noise and analyze the applicability ofthe effective temperature description to the nonequilibriumnoise in detail. With the present paper, we hope to give a contribution toward a better understanding of noise in magnetic systems.In Sec. II, we start our discussion with a detailed derivation of the spin shot-noise correlator by means of the Keldyshtechnique. It is our goal to depict the underlying mechanismsthat lead to the occurrence of spin shot noise in magnetic nanodevices and to provide a general mathematical frame-work for their description. We then turn our attention to theramiﬁcations of the noise on the magnetization dynamics.We address the question of switching rates estimation inSecs. IIIandIVby applying a generalized Fokker-Planck approach. Within this approach, the alteration of switchingrates due to spin torque is determined by an effective tem-perature T eff. The latter differs from the real temperature T, as it incorporates the effects of the damping, the spin torque,and noise. We calculate the renormalization of the effectivetemperature due to the nonequilibrium noise. A conclusion ofour ﬁndings is given in Sec. V II. SPIN SHOT-NOISE CORRELATOR In this section, starting from a microscopic model of a magnetic tunnel junction /H20849MTJ /H20850we will derive a stochastic version of the LLG equation. Fluctuations will naturallycome about due to the nonequilibrium situation, and willcomprise the random part of the stochastic LLG. In particu-lar, performing a perturbative expansion of the Keldysh ac-tion in terms of the spin-ﬂip processes and the tunnelingamplitude we will be able to derive the spin shot-noise cor-relator. The model MTJ consists of two itinerant ferromagnets separated by a tunneling barrier. Let us introduce the corre-sponding model Hamiltonian allowing for an external mag-netic ﬁeld H, tunneling of itinerant electrons through the barrier and exchange coupling between the itinerant elec-trons and the free layer’s magnetization. It reads H 0=/H20858 k,/H9268/H9280k/H9268ck/H9268†ck/H9268+/H20858 l/H9268/H9280ldl/H9268†dl/H9268−/H9253S·H−2JS·s +/H20875/H20858 kl/H9268Wklck/H9268†dl/H9268+ H.c./H20876. /H208493/H20850 The notation is as follows: the creation /H20849annihilation /H20850opera- tors ck/H9268†/H20849ck/H9268/H20850anddl/H9268†/H20849dl/H9268/H20850describe the itinerant electrons of the ﬁxed and the free magnetic layer, respectively. /H9268=+ cor- responds to the respective majority- and /H9268=− to the minority-spin band, and the indices kandllabel momentum. The operator Sdescribes the total spin of the free layer. It is connected to the free layer’s magnetization via S=MV//H9253.s =1 2/H20858l/H9268/H9268/H11032dl/H9268†/H9268/H6023/H9268/H9268/H11032dl/H9268/H11032is the quantum operator associated with the spin of itinerant electrons, where /H9268/H6023denotes the vector of Pauli matrices. Jis the exchange coupling constant and Wkl are tunneling matrix elements. For the subsequent discussion, we assume that the time between two tunneling processes is much larger than the re-laxation time in the free ferromagnet, which is equivalent toassuming a complete spin relaxation in the free magneticlayer. This allows us to introduce an instantaneous reference frame with the spin quantization axis directed along the freelayer’s magnetization direction. To render the free layer’smagnetization, a dynamical variable, we make use of theHolstein-Primakoff /H20849HP/H20850parametrization, 29SWIEBODZINSKI et al. PHYSICAL REVIEW B 82, 144404 /H208492010 /H20850 144404-2Sz=S−b†b;S−=b†/H208812S−b†b;S+=/H208812S−b†bb, /H208494/H20850 where b†,bare usual bosonic operators and S/H11006=Sx/H11006iSy.A t low temperatures, we can assume that the expectation valueofb †bis much smaller than 2 Sallowing to treat the square root to zeroth order in b†b. Taking all of the above mentioned into account, Hamiltonian /H208493/H20850can be written in the instanta- neous reference frame as H0=/H20858 k,/H9268/H9280k/H9268ck/H9268†ck/H9268+/H20858 l/H9268/H20849/H9280l−JS/H9268/H20850dl/H9268†dl/H9268−/H9253SHz+/H9253b†bHz +Jb†b/H20858 l/H9268/H9268dl/H9268†dl/H9268+/H20875/H20858 kl,/H9268/H9268/H11032Wkl/H9268/H9268/H11032ck/H9268†dl/H9268/H11032 −b/H208812S/H20873J/H20858 ldl↓†dl↑+/H9253 2H−/H20874+ H.c./H20876, /H208495/H20850 where we used the notation H/H11006=Hx/H11006iHy.Wkl/H9268/H9268/H11032are spin- dependent tunneling matrix elements given by Wkl/H9268/H9268/H11032=/H20855/H9268/H20841/H9268/H11032/H20856Wkl, /H208496/H20850 /H20855/H9268/H20841/H9268/H20856= cos/H9258 2e−i/2/H9268/H9278,/H20855/H9268/H20841/H9268/H11032/H20856=/H9268/H11032sin/H9258 2ei/2/H9268/H9278. /H208497/H20850 Hamiltonian /H208495/H20850can be now readily translated into a Keldysh action using the general scheme of the Keldyshtechnique. 30To this end we switch to symmetric /H20849cl/H20850and antisymmetric /H20849q/H20850linear combinations of the ﬁeld operators. In accordance with parametrization /H20851Eq. /H208494/H20850/H20852the former are connected to the m/H11006components of the free layer’s magne- tization in the instantaneous reference frame via bcl/H20849t/H20850=/H20881MsV 2/H9253m+/H20849t/H20850,b¯cl/H20849t/H20850=/H20881MsV 2/H9253m−/H20849t/H20850. /H208498/H20850 For the retarded and advanced components of the fermionic Green’s functions for the itinerant electrons of the free andﬁxed layer, we obtain in the energy domain G l/H9268R/A=1 /H9280−/H9280l/H9268/H11006i0,Gk/H9268R/A=1 /H9280−/H9280k/H9268/H11006i0, /H208499/H20850 where /H9280l/H9268=/H9280l−/H9268JSare the energies of the itinerant electrons with momentum land spin /H9268in the free ferromagnet, and /H9280k/H9268 the corresponding energies for the ﬁxed layer. The Keldysh components are Gl/H9268K=/H208511−2 nFd/H20849/H9255/H20850/H20852/H9254/H20849/H9280−/H9280l/H9268/H20850, /H2084910/H20850 Gk/H9268K=/H208511−2 nFc/H20849/H9255/H20850/H20852/H9254/H20849/H9280−/H9280k/H9268/H20850, /H2084911/H20850 where chemical potentials /H9262d/cfor the free and ﬁxed layer are included in the fermionic distribution functions nFc/d. For future use we also deﬁne the matrices in Keldysh space, /H9253cl=/H2087310 01/H20874,/H9253q=/H208730110/H20874. /H2084912/H20850The Keldysh action can be now solved perturbatively in terms of the tunneling amplitude and the spin-ﬂip processes.In second order in both quantities, this leads to the diagramsof Fig. 1. The corresponding equations of motion are ob- tained when varying the action Awith respect to the quan- tum component /H9254A /H9254bq=0 ,/H9254A /H9254b¯q=0 . /H2084913/H20850 Finally we note that, in the instantaneous reference frame, we have for m/H11006=mx/H11006imy /H20855m/H11006/H20856=0 , /H20855/H11509tm/H11006/H20856/HS110050, /H2084914/H20850 where, on the other hand /H20855mz/H20856=1 , /H20855/H11509tmz/H20856=0 . /H2084915/H20850 We may now translate diagrams 1 /H20849a/H20850and 1 /H20849b/H20850into the analytical expressions. However, let us start with the contri-bution of zeroth order /H20849in spin ﬂips and in tunneling /H20850.I t reads A 0=/H20885dtb¯q/H20849t/H20850/H20851i/H11509tbcl/H20849t/H20850+/H9253/H20881S/2H+/H20852+ c.c. /H2084916/H20850 The resulting equations of motion are i/H11509tbcl+/H9253/H20881S/2H+=0 /H2084917/H20850 and a corresponding complex conjugate equation for b¯cl. Equation /H2084917/H20850describes the precession of the magnetization around the magnetic ﬁeld Hand forms the ﬁrst term of the LLG Eq. /H208491/H20850. Let us come to the diagram of Fig. 1/H20849a/H20850. To extract its contribution to the action we have to calculate −J/H20881S/H20858 kl/H9268/H9268/H11032Wkl/H9268/H11032−/H9268W¯ kl/H9268/H11032/H9268b/H9268Tr/H20853Gl/H9268d/H9253qGl−/H9268dGk/H9268/H11032c/H20854, /H2084918/H20850 where for brevity the symbolic notation b/H9268with b↑=bqand b↓=b¯qwas introduced. The resulting action reads A1=i 2/H20881SIs/H20885dt/H20853b¯q/H20849t/H20850sin/H9258e−i/H9278−bq/H20849t/H20850sin/H9258ei/H9278/H20854./H2084919/H20850 Variation in Eq. /H2084919/H20850with respect to bqand b¯qgives the following contribution to the equations of motion:FIG. 1. Diagrams for spin ﬂip processes: /H20849a/H20850First order. /H20849b/H20850 Second order. Solid /H20849dashed /H20850lines denote electronic propagators in the free /H20849ﬁxed /H20850layer. Bold dashed lines are propagators of HP bosons. Tunneling vertices are denoted by circles with crosses.SPIN TORQUE DYNAMICS WITH NOISE IN MAGNETIC … PHYSICAL REVIEW B 82, 144404 /H208492010 /H20850 144404-3/H9254A1 /H9254bq/H20849t/H20850=−iIs /H208812Ssin/H9258ei/H9278,/H9254A1 /H9254b¯q/H20849t/H20850=iIs /H208812Ssin/H9258e−i/H9278. /H2084920/H20850 Again, using the HP parametrization /H20851Eq. /H208494/H20850/H20852and the rela- tion between Sandm, Eq. /H2084920/H20850can be readily translated into the corresponding equation of motion for the magnetization.The result is the spin torque term of Eq. /H208491/H20850, /H11509tm=/H9253 MsVm/H11003/H20849Is/H11003m/H20850. /H2084921/H20850 As far as the remaining diagram /H20851Fig.1/H20849b/H20850/H20852is concerned we have to distinguish two contributions: one with two quan-tum components and one with a quantum and a classicalcomponent, respectively. 31In the ﬁrst case we obtain J2Sbqb¯q/H20858 kl/H9268/H9268/H11032/H20841Wkl/H9268/H11032−/H9268/H208412Tr/H20853Gl−/H9268d/H20849/H9255/H20850/H9253qGl/H9268d/H20849/H9255−/H9275/H20850 /H11003/H9253qGl−/H9268d/H20849/H9255/H20850Gk/H9268/H11032c/H20849/H9255/H20850/H20854. /H2084922/H20850 In the second case we have J2Sbclb¯q/H20858 kl/H9268/H9268/H11032/H20841Wkl/H9268/H11032−/H9268/H208412Tr/H20853Gl−/H9268d/H20849/H9255/H20850/H9253qGl/H9268d/H20849/H9255−/H9275/H20850 /H11003/H9253clGl−/H9268d/H20849/H9255/H20850Gk/H9268/H11032c/H20849/H9255/H20850/H20854, /H2084923/H20850 and the corresponding contribution with q↔cl. The resulting action is A2=/H20885dt/H20875/H9251¯/H20849/H9258/H20850/H20849b¯q/H11509tbcl−b¯cl/H11509tbq/H20850+2i SD/H20849/H9258/H20850b¯qbq/H20876, /H2084924/H20850 where /H9251¯/H20849/H9258/H20850=/H6036/H9253 eMV/H20873dIsf/H20849/H9258/H20850 dV/H20874, /H2084925/H20850 D/H20849/H9258/H20850=MsV /H9253/H92510kBT+/H6036 2Isf/H20849/H9258/H20850coth/H20873eV 2kBT/H20874. /H2084926/H20850 The spin-ﬂip current Isfcan be calculated from the electric conductances GP/H20849AP/H20850in the parallel /H20849antiparallel /H20850conﬁgura- tion as follows: dIsf/H20849/H9258/H20850 dV=/H6036 4e/H20875GPsin2/H20873/H9258 2/H20874+GAPcos2/H20873/H9258 2/H20874/H20876. /H2084927/H20850 Action /H2084924/H20850consists of two parts. The ﬁrst term is a damping term. In the LLG equation, it will result in a renormalizationof the Gilbert damping parameter. The renormalization is dueto the coupling to the reservoirs. The enhancement of thedamping, Eq. /H2084925/H20850, is closely related to the spin-pumping enhanced damping as discussed in Refs. 32and33in the framework of the Landauer-Büttiker formalism. As far as thesecond term of Eq. /H2084924/H20850is concerned we introduce a Hubbard-Stratonovich auxiliary ﬁeld which decouples the action. Let us denote this /H20849complex /H20850ﬁeld by I +R=Is,xR+iIs,yR. We can write/H20885dI+RdI¯ +Re−1 /4DI+RI¯ +ReiA21 =/H20885dI+RdI¯ +Re−1 /4DI+RI¯ +Rexp/H20877−i1 /H208812S/H20849I+Rb¯q+I¯ +Rbq/H20850/H20878, /H2084928/H20850 where we abbreviated the second term of Eq. /H2084924/H20850byA22.A s one can see the result is a noise-averaged term which islinear in the quantum component. The linear action consti-tutes a resolution of functional /H9254functions of the Langevin equations on bcl/H20849t/H20850and its complex conjugate. The stochastic properties are encoded in the auxiliary ﬁeld I+R, precisely in the correlator /H2084926/H20850. For bclthe Langevin equations read i/H11509tbcl=1 /H208812SI+R. This corresponds to i/H11509tm+=/H9253 MVI+Rleading to the random term of the stochastic LLG equation. Adopting the notation IsR/H11013/H9254Is, in conclusion we have found /H11509tm=/H9253 MsVm/H11003/H20849/H9254Is/H11003m/H20850, /H2084929/H20850 where the stochastic ﬁeld is characterized by /H20855/H9254Is,i/H20849t/H20850/H9254Is,j/H20849t/H20850/H20856=2D/H20849/H9258/H20850/H9254ij/H9254/H20849t−t/H11032/H20850/H20849 30/H20850 with the correlator D/H20849/H9258/H20850given by Eq. /H2084926/H20850. To complete our discussion, we add some comments con- cerning the correlator /H2084926/H20850. To start with, we note that D contains two parts, an equilibrium part /H20849which is phenom- enological, and in compliance with the FDT proportional to /H92510taking into account intrinsic damping processes /H20850and a nonequilibrium part. The nonequilibrium part exhibits a de-pendence on the mutual orientation of the ﬁxed and freelayer’s magnetizations. This angle dependence enters the cor-relator through the spin-ﬂip current I sf. The physical meaning behind this quantity is the following: Isfcounts the total number of spin-ﬂip events, irrespective of their direction.Hence, even if there is no contribution to the spin current I s, the spin-ﬂip current Isfmay acquire a nonzero value. The discreteness of angular momentum transfer in each spin-ﬂipevent leads to the occurrence of the nonequilibrium noise. Inthis sense, the nonequilibrium part of Eq. /H2084926/H20850can be iden- tiﬁed with the spin shot noise. In conclusion we have derived the following stochastic LLG equation: dm dt=−/H92530m/H11003Heff+/H92510m/H11003dm dt+/H92530 MsVm /H11003/H20851m/H11003/H20849Is+IsR/H20850/H20852, /H2084931/H20850 where the random ﬁeld correlator is given by Eq. /H2084930/H20850along with Eqs. /H2084926/H20850and /H2084927/H20850. III. FOKKER-PLANCK APPROACH TO SPIN TORQUE SWITCHING Spin torque switching is observable in two different re- gimes. On the one hand, the spin torque can switch the mag-netization of a free ferromagnet when the current exceeds acritical value I c. On the other hand, switching is also ob- served for currents below Ic. In the second case the actualSWIEBODZINSKI et al. PHYSICAL REVIEW B 82, 144404 /H208492010 /H20850 144404-4switching procedure is mainly noise induced. A suitable de- scription of switching times in this regime can be obtainedfrom the Fokker-Planck approach which was recently intro-duced by Apalkov and Visscher in the context of thermalﬂuctuations. 14,15Within this approach switching rates are speciﬁed by an Arrhenius-type law with an effective tempera- ture Teff. The latter differs from the real temperature T,a si t is inﬂuenced by the damping and the spin torque. In thesequel we present a generalization of the method to nonequi-librium noise, and show that the spin shot noise alters theeffective temperature. Let us start our consideration with the Fokker-Planck equation as introduced by Brown. 11We denote the probabil- ity density for the magnetization of a monodomain particleby /H9267/H20849m,t/H20850. The corresponding Fokker-Planck equation can be written in the form of a continuity equation /H11509/H9267/H20849m,t/H20850 /H11509t=−/H11612·j/H20849m,t/H20850/H20849 32/H20850 with probability current11 j/H20849m,t/H20850=/H9267/H20849m,t/H20850m˙det/H20849m/H20850−D/H11612/H9267/H20849m,t/H20850. /H2084933/H20850 Here m˙detdenotes the deterministic part of the stochastic LLG Eq. /H208491/H20850andDis the random ﬁeld correlator. We recall that the dynamics governed by Eq. /H208491/H20850conserves the absolute value of m. As a consequence the movement of the tip of m is restricted to the surface of a sphere, which we will call themsphere. The gradient and the divergence in Eqs. /H2084932/H20850and /H2084933/H20850are two-dimensional objects, both living on the m sphere. We now observe that in presence of anisotropy the phase space will be in general separated. The potential landscapewill exhibit different minima referring to stable and meta-stable states of the magnetization. Precession of the magne-tization takes place around one /H20849or more /H20850of these equilib- rium positions. We refer to orbits of constant energy asStoner-Wohlfarth /H20849SW /H20850orbits. Now, considering the dynam- ics of the magnetization vector one can distinguish two dif-ferent time scales. The time scale for the angular movement,on the one hand, is characterized by the precession fre-quency. On the other hand, there is also a time scale for apossible change in energy. In the following, we will requirethat the time scale for the change in energy is much longerthan the time scale for constant energy precession. In otherwords: we assume that the magnetization vector stays ratherlong on a SW orbit before changing to higher/lower energies.In this low damping and small current limit, we can intro-duce an energy-dependent probability density by identifying /H9267i/H11032/H20849E/H20849m/H20850,t/H20850/H11013/H9267/H20849m,t/H20850, where the index itakes into account that the energy dependence may be different in different re-gions of the msphere. The above mentioned time-scale sepa- ration allows us to average out the movement along the SWorbit and to be concerned with only the long-time dynamics. The idea of the FP approach is to translate Eq. /H2084932/H20850into a corresponding equation for /H9267i/H11032/H20849E/H20850. For thermal noise, this has been done in Ref. 14. We now give a generalization of the method to the angle-dependent spin shot noise of Sec. II.T o this end we write the correlator /H2084926/H20850in the formD/H20849/H9258/H20850=Dth+D0/H208511−Pcos/H9258/H20852, /H2084934/H20850 where Dth=/H9253/H9251kBT MsVis the thermal part and D0/H208511−Pcos/H9258/H20852the nonequilibrium part of the correlator.34 We abbreviated the angle-independent part of the spin shot noise by D0. We also used P=GP−GAP GP+GAP. /H2084935/H20850 In general we can write the Fokker-Planck equation for the distribution /H9267i/H11032/H20851E/H20849m/H20850,t/H20852in the form14 /H9253Pi/H20849E/H20850 Ms/H92620/H11509/H9267i/H11032/H20849E,t/H20850 /H11509t=−/H11509 /H11509EjiE/H20849E,t/H20850, /H2084936/H20850 where Pi/H20849E/H20850is the period of the orbit with energy E.jiEis the probability current in energy. It is given by jiE/H20849E,t/H20850=/H20886/H20851j/H20849m,t/H20850/H11003dm/H20852·m =−/H9253/H9251/H9267i/H11032/H20849E,t/H20850IiE/H20849E/H20850+/H9253J/H9267i/H11032/H20849E,t/H20850mp·IiM −/H11509/H9267/H11032/H20849E/H20850 /H11509EMsDthI/H9258,iE. /H2084937/H20850 The constant Jis deﬁned in such a way that Jmp =/H9253//H20849MV/H20850Isifmpis a unit vector in direction of Is. Further- more we have introduced the following integrals along theSW orbit: I /H9258,iE=IiE+D0 Dth/H20849IiE−P/H20886cos/H9258Heffdm/H20850, /H2084938/H20850 IiE/H20849E/H20850=/H20886Heffdm, /H2084939/H20850 IiM/H20849E,t/H20850=/H20886dm/H11003m. /H2084940/H20850 A steady-state solution of the FP equation is obtained by setting jiE=0. From Eq. /H2084937/H20850we get the following differential equation for the probability density /H9267i/H11032: /H11509ln/H9267i/H11032/H20849E/H20850 /H11509E=/H9253 DthMs/H9261i/H20849E/H20850/H20851−/H9251+/H9257i/H20849E/H20850J/H20852/H11013−V/H9252i/H11032/H20849E/H20850, /H2084941/H20850 where the right-hand side serves as a deﬁnition of an inverse effective temperature /H9252i/H11032/H20849E/H20850. From Eq. /H2084941/H20850one can see that, depending on the sign of the spin current, the spin torquemay either enhance or diminish the damping, leading to alower or higher effective temperature, respectively. In Eq./H2084941/H20850we have deﬁned /H9257i/H20849E/H20850=mp·IiM/H20849E/H20850 IiE/H20849E/H20850/H2084942/H20850 andSPIN TORQUE DYNAMICS WITH NOISE IN MAGNETIC … PHYSICAL REVIEW B 82, 144404 /H208492010 /H20850 144404-5/H9261i/H20849E/H20850=IiE I/H9258,iE. /H2084943/H20850 /H9257ican be viewed of as the ratio of the work of the Sloncze- wski torque to that of the damping.14The quantity /H9261gives the renormalization of the effective temperature as comparedto the pure thermal case. We can write for /H9261 /H9261/H20849E/H20850=T eff Teff/H11032, /H2084944/H20850 where Teffis the effective temperature when only equilibrium noise is present, and Teff/H11032the effective temperature when both, equilibrium and nonequilibrium noise are included. Itshould be observed from Eq. /H2084941/H20850that the effective tempera- ture is in general energy dependent. The corresponding prob-ability distribution will thus, in general, differ from theBoltzmann distribution. However, when we turn off the non-equilibrium, /H9261/H20849E/H20850/H110131 and J=0. In this case the solution of Eq. /H2084941/H20850is exactly a Boltzmann distribution. In the remainder of this section, we evaluate /H9261for an exemplary system with easy-axis and easy-plane anisotropy.The easy axis is chosen to be the zaxis and the easy plane is they-zplane. The magnetization direction of the ﬁxed layer, m p, is taken to be antiparallel to the zaxis. Let us use the following convention for the spherical coordinates: mx =cos/H9277,my=sin/H9277sin/H9272, and mz=sin/H9277cos/H9272. The SW con- dition deﬁnes the orbits of constant energy. For our system itreads E/H20849M/H20850 /H92620=−1 2HKMS/H20849mez/H208502+1 2MS2/H20849mex/H208502. /H2084945/H20850 We abbreviate /H92601=HKMS/H20849characterizing the strength of easy-axis anisotropy /H20850,/H92602=MS2/H20849characterizing the strength of easy-plane anisotropy /H20850andd=/H92601 /H92602, being the ratio of easy-axis to easy-plane anisotropy so that /H20849taking the magnetic con- stant/H92620=1/H20850we can obtain from Eq. /H2084945/H20850the dimensionless energy c c/H110132E /H92602=−dmz2+mx2. /H2084946/H20850 This relation deﬁnes the “potential landscape” of our system. We can distinguish three regions: two potential wells, onearound /H9272=0 /H20849well 1 /H20850and one around /H9272=/H9266/H20849well 2 /H20850, and a third region /H20849region 3 /H20850with energies above the saddle-point energy, separating the two wells. Switching takes place if themagnetization vector changes from some orbit in the onewell to an orbit in the other well. Equation /H2084946/H20850deﬁnes the orbits of integration for the evaluation of Eq. /H2084943/H20850. Let us concentrate on orbits lying in the potential well around /H9272 =0 with energies c/H113490.35In addition we assume a strong easy-plane anisotropy, allowing to consider small deviations of/H9277around/H9266 2. We ﬁx the Gilbert damping to /H9251=0.01, the ratio of anisotropies to d=0.028, the polarization to P=0.81, and MsV//H9253=10/H6036. These values deﬁne the ratio D0/Dthas a func- tion of eV /kBT. The results for /H9261=Teff/Teff/H11032are plotted in Fig. 2. As can be seen from the plot taking into account the non-equilibrium noise results in a renormalization of the effective temperature. This renormalization is proportional to the ap-plied voltage Vand can be very strong for sufﬁciently large values of V. The deviation from the purely thermal case /H20849/H9261 =1/H20850approaches 15% for eV=5k BTand is thus experimen- tally not negligible. For eV=10kBTthe deviation is even in the order of 25% and grows further with the voltage. Thevariation in /H9261with energy is on the other hand very weak. This indicates that the inﬂuence of the angle dependence israther small or in other words: the angle dependence of thecorrelator does not lead to a signiﬁcant variation in T effwith precession orbit. Let us continue our discussion of the renormalized effec- tive temperature by considering the limit where the equilib-rium part of the correlator is much smaller than its nonequi-librium part and thus may be neglected. In this case wedeﬁne the following quantity of interest: /H9261 /H11032/H20849E/H20850=D0 Dth/H9261/H20849E/H20850. /H2084947/H20850 One should note the difference between /H9261and/H9261/H11032. From Eq. /H2084944/H20850we see that /H9261is the ratio of the effective temperatures Teffand Teff/H11032for systems without and with nonequilibrium noise, respectively. On the other, from the deﬁnition Eq. /H2084947/H20850 it is clear that /H9261/H11032is a measure for the inﬂuence of the angle dependence of the correlator. The stronger /H9261/H11032deviates from /H9261/H11032=1 the stronger is the inﬂuence of the angle dependence. In Fig. 3we plot /H9261/H11032for our model system for d=0.028 and different values of P. As one can see from Fig. 3the largest deviation from /H9261/H11032/H20849E/H20850=1 /H20849corresponding to the stron- gest inﬂuence of the angle dependence /H20850is present at the minimum of the well /H20849c=−d=−0.028 /H20850. The smallest devia- tion from /H9261/H11032/H20849E/H20850=1 is observed for orbits which lie near the separatrice. The overall change in /H9261/H11032/H20849E/H20850forP=1 is on the order of 10%. These results provide a good insight into the inﬂuence of the angle dependence. As /H9261/H11032/H110111/Teff/H11032, a small value of /H9261/H11032 indicates a “hot” spot whereas large values of /H9261/H11032correspond to “cold” spots on the msphere. For the particular system under consideration, cf. Eq. /H2084945/H20850, the equilibrium position of the magnetization is roughly along the zaxis. SW orbits of precession are symmetric with respect to this axis. At the0.9 0.8 0.7 0.6 0.5λ= -0.025 -0.020 -0.015 -0.010 -0.005 0.00 0 c(E)Teff Teff' FIG. 2. /H20849Color online /H20850/H9261=Teff/Teff/H11032as a function of cin the case mp↑↓ezforeV=kBT/H20849red/H20850,eV=5kBT/H20849magenta /H20850,eV=10kBT/H20849blue /H20850, andeV=20kBT/H20849green /H20850.SWIEBODZINSKI et al. PHYSICAL REVIEW B 82, 144404 /H208492010 /H20850 144404-6bottom of the well /H9258=/H9266and the spin shot noise has its maxi- mal value. We hence expect a hot spot at the minimum of thewell. With increasing energy the orbits will become larger.The angle /H9258will vary along these orbits. However as the orbit energy grows the trajectories increasingly go throughregions of smaller /H9258so that the average value of /H9258will di- minish with orbit energy. As a consequence the nonequilib-rium noise will become smaller as well. Cold orbits shouldbe hence those that are in the vicinity of the separatrice. Thisis exactly what can be read off from Fig. 3. Our ﬁndings are thus in agreement with the geometrical situation. Cold spotsand hot spots on the msphere are shown in Fig. 4. IV. SWITCHING TIME OF SPIN TORQUE STRUCTURES The switching process can be analyzed by performing nu- merical simulations of the Langevin equations of motionwith the inclusion of temperature and shot noise via the ran-dom ﬁeld term. In this section we present such simulationsfor Gilbert damping of /H9251=0.01, an anisotropy ratio of d =0.028, and with a spin torque current characterized by J and polarized in the mp=−ezdirection. Before going further though, it would be useful to con- sider how the system acts in the absence of the noise. In sucha case the switching occurs when the energy current /H208491.34 /H20850is positive for all values of energy between the starting position/H20849say positive zdirection /H20850and the saddle point. Since the probability function /H9267iE/H20849E,t/H20850is always positive it stand to reason that a switch will only happen if 0 /H11349−/H9251IiE/H20849E/H20850 +Jmp·IiM/H20849E/H20850. We plot this quantity as a function of energy for various values of the spin current Jin Fig. 5. From this we also gain a useful reference value for the critical current which is Jc=/H9251IiE/H20849Esad/H20850 m·IiM/H20849Esad/H20850=0.00645 Ms. The positive value signi- ﬁes the tendency toward the switching. In the ﬁrst example with J=0.77 Jcthe noiseless system, being driven by the dis- sipation toward the stable position, does not switch. It isworth noticing that in the presence of the noise the switchingnevertheless does occur but it takes exponentially long time.In the three other examples J/H11350J cand the magnetization cur- rent is always directed toward the saddle. Therefore, even thenoiseless system does switch and the noise serves to intro-duce an uncertainty in the switching time. Putting the thermal noise back into the system, we set the noise strength parameter to D th=0.00001 /H9253Ms. Simulations are then run by starting each particle at /H9258=0, allowing it to come into thermal equilibrium with the system, turning thecurrent on and calculating how long it takes for it to go pastthe saddle point into the second well. This is done for manyparticles for a given current value and over several differentcurrent values. A typical trajectory of the system is repre-sented by the graph /H9258as a function of time in Fig. 6forJ =1.08 Jc. It may be seen that it takes many revolutions before the system ﬁnally switches to the basin of attraction of thetrue stationary points at t/H11015700. Let us estimate the contribution of the nonequilibrium noise to the switching process under realistic experimentalconditions. From Eq. /H2084926/H20850we obtain the relationship between equilibrium noise and nonequilibrium noise,0.70 0.65 0.60 0.55 0.50λ' -0.025 -0.020 -0.015 -0.010 -0.005 0.00 0 c(E) FIG. 3. /H20849Color online /H20850/H9261/H11032as a function of cin the case mp↑↓ez forP=1 /H20849red/H20850,P=0.9 /H20849green /H20850,P=0.7 /H20849blue /H20850, and P=0.5 /H20849magenta /H20850. d=0.028. FIG. 4. /H20849Color online /H20850Hot spots /H20849red/H20850and cold spots /H20849blue /H20850on theM-half-sphere in case of mp↑↓ez. The noise intensity is highest at the bottom of the well. FIG. 5. /H20849Color online /H20850Plots A/H20849E/H20850=−/H9251 MsIiE/H20849E/H20850+J Msmp·IiM/H20849E/H20850as a function of energy c=2E//H92602over a range of spin torque current. From bottom to top: blue=0.77 Jc, red=1.08 Jc, yellow=1.55 Jc, and green=3.10 Jc.SPIN TORQUE DYNAMICS WITH NOISE IN MAGNETIC … PHYSICAL REVIEW B 82, 144404 /H208492010 /H20850 144404-7Dneq Deq=/H60362/H9253 8ekBMsV/H9251I T. /H2084948/H20850 Replacing Iwith the critical current Ic=Jc4MseV /H6036, /H2084949/H20850 where Jcis the minimum spin current needed to cause a switch in the absence of noise, we obtain Dneq Deq=/H6036/H9253 2kB/H9251Jc T. /H2084950/H20850 Using the material parameters in Ref. 6,Ms=1440 emu, nanopillar volume=1.97 /H1100310−17cm3, and switching current Ic/H11015109A/cm2, the shot noise, D0, at the critical current is equivalent to the thermal noise, Dth, of temperature Tc /H1101515 K. Even though the theoretical switching current and the experimentally determined switching current of Ref. 6 differ by more than an order of magnitude /H20849such a discrep- ancy is mentioned in many experimental studies, includingRef. 6/H20850the theoretical approach can be used to gain insight into how the shot noise at the switching current scales withthe parameters of a nanopillar. If we set D 0=Dthagain and include our calculation of the switching current we ﬁnd Tc/H11008Jc//H20849/H9251P/H20850where Jcis the criti- cal spin torque current needed to cause a spin-ﬂipping event/H20849in the absence of noise /H20850andPis the degree of polarization of the current. J cis determined by the relative values of Hk andMsand has the units of magnetization. For the simple case with only easy-axis anisotropy /H20849E=−1 2/H92620HkMscos2/H9258/H20850, Jc=/H9251Hk, and therefore Tc/H11008Hk P. Therefore for materials and/or geometry with a bigger anisotropy ﬁeld the shot noise maybe a dominant source of noise at temperatures well above 15K. Moreover, in the simplest case where we only have uniaxial anisotropy and an applied external ﬁeld along theeasy axis it can be shown that J c=/H9251/H20849Hk+Hext/H20850. /H2084951/H20850 This means the relative strength of the nonequilibrium noise scales with the potential well depth of our system. By apply-ing an external ﬁeld we can increase the maximum allowedcurrent before a switch takes place and thus increase theimportance of nonequilibrium noise. Since the initial condition is taken out of a stationary dis- tribution /H20849without the spin current /H20850and subsequent evolution is subject to the Langevin noise, the time of the switching isa random quantity. The percentage of trial systems that haveswitched as a function of time is shown in the left panel ofFig.7for four different values of the spin current. The time derivatives of these graphs provide probability distributionfunctions of the switching time. One may then evaluate theﬁrst moment of these distributions which gives the meanswitching time for a given value of the spin current. Theright panel of Fig. 7shows such a mean switching time as a function of J/J c. One may notice that for J/H11349Jcthe switching time grows exponentially while for J/H11350Jcthe switching time becomes relatively short /H20849although it is still substantially longer than the inverse precession frequency /H20850. V. CONCLUSION We would like to conclude with the following remarks. The study of noise in dynamical magnetic systems is a broadand fascinating ﬁeld. In particular, in view of potential ap-plications of magnetic nanodevices, both equilibrium andnonequilibrium noise may play an important role. For ex-ample, the stability of magnetic storage devices is stronglyinﬂuenced by thermal ﬂuctuations. The functionality of newgeneration technologies /H20849such as the magnetic random access memory 36with STT writing or the racetrack memory37/H20850is largely based on the spin torque phenomenon. The latter is anonequilibrium effect and thus besides the temperature alsononequilibrium sources of noise may play an important role. A way to introduce ﬂuctuations into magnetization dy- namics is to add a random component to the effective ﬁeld orto the current in the phenomenological LLG equation. Thenoise is then deﬁned by the value of its correlator /H20849and its higher-order cumulants /H20850. The determination of the noise cor- relator is of great importance as it deﬁnes the noise proper-ties. In addition it may give insight into the physical context. A powerful and very ﬂexible tool is the Langevin ap- proach based on the Keldysh path-integral formalism. Start-0 100 200 300 400 500 600 700t 0.00.51.01.52.0Θ FIG. 6. A typical realization of /H9258as a function of time /H20851in units of/H20849/H9253Ms/H20850−1/H20852forJ=1.08 Jc. 0 500 1000 1500 2000t 0.00.20.40.60.81.0Psw 0246050010001500200025003000TAve 8J JcFIG. 7. /H20849Color online /H20850/H20849Left /H20850Shows the switching probability as a function of time /H20851in units of /H20849/H9253Ms/H20850−1/H20852for various current values. /H20849From left to right: green=3.10 Jc, yellow =1.55 Jc, red=1.08 Jc, and blue=0.77 Jc./H20850/H20849Right /H20850 Shows the average switching time /H20851in units of /H20849/H9253Ms/H20850−1/H20852as a function ofJ Jc.SWIEBODZINSKI et al. PHYSICAL REVIEW B 82, 144404 /H208492010 /H20850 144404-8ing from a microscopic model, one derives the equations of motion for the magnetic system. Fluctuations naturally ariseas a generic feature of the Keldysh approach. We have dem-onstrated the applicability of this method to magnetic sys-tems on the example of spin shot noise in magnetic tunneljunctions. The spin shot-noise correlator arose naturally, as aconsequence of the sequential tunneling approximation, insecond order in spin-ﬂip processes. The Keldysh formalism is however not restricted to the system described above. In particular, it may be used in thecontext of nonuniform magnetic textures /H20849as domain walls, for instance /H20850. Promising advances in this direction have al- ready been reported 38and demonstrate the versatility of the method as well as inspire us with curiosity about future de-velopments.Finally, to investigate the inﬂuence of the spin shot noise on spin torque switching rates we have generalized theFokker-Planck approach of Ref. 14. We have shown that the nonequilibrium noise manifests itself in a renormalized ef-fective temperature. In particular, at low temperatures wecould observe a signiﬁcant variation in the noise with orbitenergy, reﬂecting cold and hot trajectories of the magnetiza-tion vector with respect to the noise intensity. ACKNOWLEDGMENTS J.S. and A.C. acknowledge ﬁnancial support from DFG through Sonderforschungsbereich 508 and T.D. and A.K. ac-knowledge support from NSF under Grant No. DMR-0804266. 1D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 /H208492008 /H20850. 2J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 3L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 4M. Tsoi, A. G. M. Jansen, J. Bass, W. C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 /H208491998 /H20850. 5E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285, 867 /H208491999 /H20850. 6J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 /H208492000 /H20850. 7M. Tsoi, A. G. M. Jansen, J. Bass, W. C. Chiang, V. Tsoi, and P. Wyder, Nature /H20849London /H20850406,4 6 /H208492000 /H20850. 8S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. 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PhysRevB.103.214433.pdf	PHYSICAL REVIEW B 103, 214433 (2021) Impact of intragrain spin wave reﬂections on nanocontact spin torque oscillators Anders J. Eklund ,1,Mykola Dvornik ,2Fatjon Qejvanaj,2Sheng Jiang,3Sunjae Chung,4,5Johan Åkerman,2,3,4 and B. Gunnar Malm6 1Department of Physics, University of Oslo, Box 1048 Blindern, 0316 Oslo, Norway 2NanOsc AB, Electrum 205, 164 40 Kista, Sweden 3Department of Applied Physics, School of Engineering Sciences, KTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden 4Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden 5Department of Physics Education, Korea National University of Education, Cheongju 28173, Korea 6Division of Electronics and Embedded Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden (Received 21 August 2020; revised 13 April 2021; accepted 25 May 2021; published 21 June 2021) We investigate the origin of the experimentally observed varying current-frequency nonlinearity of the propagating spin wave mode in nanocontact spin torque oscillators. Nominally identical devices with 100 nmdiameter are characterized by electrical microwave measurements and show large variation in the generatedfrequency as a function of drive current. This quantitative and qualitative device-to-device variation is describedin terms of continuous and discontinuous nonlinear transitions between linear current intervals. The thin-ﬁlmgrain microstructure in our samples is determined using atomic force and scanning electron microscopy to beon the scale of 30 nm. Micromagnetic simulations show that the reﬂection of spin waves against the grainboundaries results in standing wave resonance conﬁgurations. For a simulated device with a single artiﬁcial grain,the frequency increases linearly with the drive current until the decreased wavelength eventually forces anotherspin wave antinode to be formed. This transition results in a discontinuous step in the frequency versus currentrelation. Simulations of complete, randomly generated grain microstructures additionally shows continuousnonlinearity and a resulting device-to-device variation in frequency that is similar to the experimental levels.The impact of temperature from 4 to 300 K on the resonance mode-transition nonlinearity and frequency noiseis investigated using simulations and it is found that the peak levels of the spectral linewidth as a function ofdrive current agree quantitatively with typical levels found in experiments at room temperature. The impact ofthe grain microstructure on the localized oscillation modes is also investigated. DOI: 10.1103/PhysRevB.103.214433 I. INTRODUCTION The nanocontact (NC) spin torque oscillator (STO) [ 1,2]i s a spintronic microwave oscillator in which the spin transfertorque (STT) [ 3–5], induced by an electrical direct current, counteracts the Gilbert damping and enables a persistent pre-cession of the magnetization in the free layer. Through thegiant magnetoresistance effect [ 6,7], this precessing magne- tization direction results in a correspondingly time-varyingdevice resistance, which together with the DC drive currentproduces an oscillating voltage signal. The excitation of themagnetic free layer can take place in the form of a circularlytrajecting magnetic vortex [ 8,9] at lower frequencies (from hundreds of megahertz to a couple of gigahertz) and, on theorder of tens of gigahertz [ 10], as the localized “bullet” mode a.j.eklund@fys.uio.no Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.[11], the propagating spin wave mode [ 12–14], and the droplet soliton [ 15,16]. Out of these high-frequency modes, the propagating spin wave mode possesses several features that make it more at-tractive for applications. First, it can be excited exclusivelywhile the bullet mode can only be excited in conditionswhere also the propagating mode exists [ 17]. Simultaneous excitability with the possibility of mode hopping betweenthese modes accounts for the comparatively high frequency-domain linewidths observed during these conditions. Second,its propagating nature increases the size of the oscillatingsystem and thereby increases the frequency stability. Third,it blueshifts with the application of increased drive currentmagnitude and thereby provides higher frequencies. Apartfrom the high frequency range and comparatively high fre-quency stability most useful in microwave rf applications, thelarge-amplitude propagating spin waves are also attractive foruse in magnonic circuits [ 18–20]. One general property of the propagating spin wave mode that has so far not been explained or modeled is the richvariety of features in the frequency versus current behavior f(I DC), which generally shows regions of linear depen- dence joined by nonlinear transitions [ 21,22]. These nonlinear transitions can be either continuous or discontinuous, i.e., 2469-9950/2021/103(21)/214433(13) 214433-1 Published by the American Physical SocietyANDERS J. EKLUND et al. PHYSICAL REVIEW B 103, 214433 (2021) appearing as f(IDC) bending or discrete steps in frequency. It may be considered natural to describe the linear regions assubmodes of the propagating spin wave mode, but a physi-cal mechanism behind such a degeneracy has not yet beenreported. The frequency steps are of the order of 1 GHzand cannot be explained as the much larger ∼10 GHz steps between the different higher-order Slonczewski modes [ 23]. We would like to point out that the phenomenon under inves-tigation is a higher-order nonlinearity not to be confused withthe general auto-oscillator nonlinearity property [ 24], which concerns the presence of coupling between the amplitudeand frequency of the magnetization precession. While theamplitude-frequency coupling is the mechanism that makesit possible to tune the STO frequency by changing the magni-tude of the drive current I DC, we see no physical reason why this coupling by itself should give rise to the type of complex-ity found in the experimentally measured f(I DC) behavior. The f(IDC) nonlinearity is of direct interest for any tech- nological application of the NC STO for two main reasons:frequency stability (phase noise) and device-to-device vari-ability. Within the nonlinear transition intervals, the frequencystability is decreased which is commonly observed as anincrease in the spectral linewidth [ 21,25]. More detailed measurements have shown that the f(I DC) nonlinearity is associated with increased levels of both white and 1 /ffre- quency noise [ 25]. Nominally identical devices also differ signiﬁcantly in the position and type of the f(IDC) nonlin- earity [ 22], which translates to device-to-device variation in f(IDC). The same type of qualitative and quantitative fre- quency variability that is found between devices can alsobe seen when changing the angle of the in-plane compo-nent [ 22,26] or polarity [ 26] of the applied magnetic ﬁeld. These studies have concluded that the measured behavior isconsistent with an oscillation that takes place in magnetic“hotspots” [ 22] or “subregions” [ 26] deﬁned by an inhomo- geneous effective magnetic ﬁeld and /or spin polarization ratio originating from microstructural inhomogeneity. The linearityand nonlinearity of f(I DC) would in this context arise due to a complex interplay between these different subregions.Although this is certainly a possible scenario, we considerit less likely that the selection of the dominating subregionwould be so sensitive to the drive current I DC.W ed on o ts e ea clear reason why this selection would change and would ratherexpect that the frequency would be set by the same subregionthroughout most of the operating current range. When setting up our simulations in an attempt to recre- ate the nonlinearity, we initially noted the sensitivity to theboundary conditions of the simulation space. In particular, wefound that periodic boundary conditions in combination witha simulation space being about one order of magnitude largerthan the nanocontact resulted in discontinuous steps in f(I DC). In this conﬁguration, the spin wave propagated from the NCto any of the simulation space borders, reentered from the op-posing border, and still had a notable amplitude as it reenteredthe NC area. This led us into the conceptually simpler hypoth-esis that the f(I DC) nonlinearity originates from spin wave propagation and STO self-interaction. Microstructural inho-mogeneity and spin wave reﬂection in combination with axialasymmetry in the oscillation mode would be consistent withboth the device-to-device variability and in-plane magneticﬁeld dependence. Micromagnetic simulations have shown that for applied magnetic ﬁelds having an in-plane component,the inclusion of the current-induced Oersted magnetic ﬁeldindeed breaks the symmetry of the propagating mode andpropagation instead takes the form of a directed spin wavebeam [ 27,28]. In this work, we investigate the microstructure of the thin ﬁlm in terms of the size of the metal grains and perform mi-cromagnetic simulations with included grain boundaries. Withreduced magnetic exchange coupling at the grain boundaries,the propagating spin wave becomes reﬂected and travels backto the active region. By self-locking, the spin wave reﬂectionsresult in resonating spin wave paths that each depend on thedistance to the reﬂecting grain boundary and the wavelength.This leads to multiple sets of resonance frequencies for thedifferent reﬂecting grain boundaries and provides a directphysical model for the f(I DC) submodes and their associated variability. As will be shown, the model is able to recreate bothcontinuous and discontinuous f(I DC) nonlinearity, the device- to-device variability (with reasonable quantitative agreement),and the correct variation of the spectral linewidth. For com-pleteness, the grain model is tested also for the localizedoscillation modes that exist when the magnetic ﬁeld is directedmore into the ﬁlm plane, i.e., further away from the normaldirection of the ﬁlm stack. II. EXPERIMENTAL METHODS The samples were fabricated by sputter deposition to form the stack Si /SiO x/Pd8/Cu30/Co8/Cu8/NiFe4.5 /Cu3/Pd3 (thicknesses in nanometers). The ﬁlm was then patternedinto 16 ×8μm 2mesas by optical lithography and lift-off, followed by sputter deposition of a 30-nm SiO 2insulat- ing layer. Circular nanocontacts with diameter dNC=100 nm were patterned using electron beam lithography and etchedusing reactive ion etching. The nanocontact vias were thenmetallized by Cu during the deposition of the Cu1000 /Au400 top contact, which was also deﬁned by optical lithogra-phy and lift-off. The top contact has a coplanar waveguideground-signal-ground (GSG) conﬁguration, where the S padis connected to the nanocontact and the G pads are connectedto the outer regions of the mesa through two 2 ×4μm 2vias. The electrical microwave measurements were performed using a 40-GHz-rated GSG microwave probe followed by abias-T and a 20–40 GHz low-noise ampliﬁer (gain 28 dB,noise ﬁgure of 3.0 dB) before recorded on a spectrum an-alyzer. The bias current was supplied by a Keithley 6221precision current source, with the positive current directiondeﬁned as electrons ﬂowing from the free NiFe layer tothe ﬁxed Co layer. The sample and microwave circuit weremounted on an electrically controlled rotating holder withthe sample positioned inside the pole gap of an electro-magnet. The current driving the electromagnet was feedbackcontrolled using a proportional-integral controller with a cal-ibrated Hall sensor positioned at the center of one of thepoles. In all measurements presented in this work (except forSec. IV E ), the strength and angle of the applied ﬁeld (away from the ﬁlm plane) are μ 0Hext=1.00 T and θext=70◦.T h i s ﬁeld was selected based on Ref. [ 29] to optimize the trade- off balance between oscillation power and frequency stability 214433-2IMPACT OF INTRAGRAIN SPIN WA VE REFLECTIONS ON … PHYSICAL REVIEW B 103, 214433 (2021) (spectral linewidth), well above the critical angle of θext,c= 58◦under which also the localized bullet mode is coexistingly excited. Atomic force microscopy (AFM) and scanning electron microscopy (SEM) were performed on a separately prepared,unpatterned Si /SiO x/Pd8/Cu30/Co8/Cu8/NiFe4.5 /Ta3 ﬁlm to capture the structure of the NiFe free layer. AFM was con-ducted with a JPK NanoWizard 3 NanoScience microscopein the AC tapping mode using an AppNano ACTA tip with a(nominal) radius of curvature of 6 nm. The SEM measurementwas performed using the in-lens detector of a Zeiss Ultra55 microscope at ∼3 mm working distance, with a speciﬁed resolution of 1.6 nm at 1 kV accelerating voltage. Ferromagnetic resonance (FMR) measurements of the same unpatterned ﬁlms, using a NanOsc InstrumentsPhaseFMR-40, gave the saturation magnetization and Gilbertdamping for the NiFe free layer of μ 0Ms,NiFe=1.01 T, αG,NiFe=0.0135 and for the Co ﬁxed layer of μ0Ms,Co= 1.98 T,αG,Co=0.0088. III. SIMULATION METHODS Simulations were performed using the open-source, GPU- accelerated micromagnetic simulation package MUMAX 3[30]. For the homogeneous free layer simulations, a 512 ×512×1 quadratic grid was used with a cell size of 2 .5×2.5×4n m3 for the 1280 ×1280×4n m3free layer representation. The ﬁxed layer and spacer layer were modeled as 1280 ×1280× 8n m3. An initial settling step was used to let the full stack relax into its static conﬁguration, taking into account theexternally applied ﬁeld, the dipolar ﬁeld, the exchange ﬁeld,and the Oersted ﬁeld. After settling, the ﬁxed layer cells werekept static in order to reduce the computation time. By thisapproach, the dipolar ﬁeld of the (static) ﬁxed layer is au-tomatically included in the simulation. Absorbing boundaryconditions were implemented similar to those in Ref. [ 31] with three encapsulated frames, each with a width of 5% ofthe simulation space, with the damping parameter succes-sively increasing to α G=0.05, 0.15, and 0.45. Using these settings and sweeping the current resulted in a frequencyversus current behavior free from continuous or discontinuousnonlinearities for the oscillation regime above the threshold,while a linear 50% reduction in the simulation space resultedin slight frequency stepping due to wave reﬂection against thesimulation space borders. The majority of the material parameters for the fully processed samples were taken from our previous work inRef. [ 28] with slight adaptation to ﬁt the threshold current and frequency of our experimental sample batch. Selected valueswere as follows: saturation magnetization μ 0Ms,NiFe=0.85 T, μ0Ms,Co=1.70 T; exchange stiffness Aex,NiFe =1.1×10−11 J/m,Aex,Co=2.1×10−11J/m. For the spin torque, the po- larization was 0.3 and the Slonczewski parameter /Lambda1=1.0. The Gilbert damping parameter was taken from our FMRmeasurements: α G,NiFe=0.0135 and αG,Co=0.0088. The Oersted ﬁeld was calculated as that from an inﬁnitely long conductor running down the nanocontact. The magnitudeof the ﬁeld increases linearly from the center of the nanocon-tact out to the edge, outside which it decays with the inversedistance from the center.For the simulations including the grain structure, the grains were randomly generated using the V oronoi tessellation ex-tension to MUMAX 3. Using another extension, the exchange coupling across the grain boundaries was reduced by scaling.Tests using a constant scaling factor [ 32] between all grains showed well-deﬁned oscillation for 30–100% coupling, while20% showed oscillation only for currents below 30 mA and0–10% resulted in broadband noise. Since the exchange cou-pling is highly sensitive to the interatomic distance (it has beencalculated [ 33] to drop to zero already at a distance of 1.5 times the crystalline distance), we consider it more realistic tohave a random inter-grain exchange distribution. Knowing lit-tle about the grain-to-grain interface structure, we here makea ﬁrst approximation with a uniform distribution of 0–100%coupling. Both the grain tessellation and exchange scalingare set using the same speciﬁed seed number for the randomnumber generator, ensuring reproducibility. When simulatingthe grain structure, no signiﬁcant difference was observedbetween the full 1280 ×1280 nm 2and halved 640 ×640 nm2 simulation spaces, indicating that only an insigniﬁcant amount of energy is propagated to the simulation border and back tothe active region in this case. This is consistent with spin wavereﬂection occurring at the grain boundaries having a higherinﬂuence than the simulation space border effects. Because ofthis, the simulations of the grainy free layer were performedusing the smaller simulation space in order to reduce thesimulation time. The majority of the simulations were performed at temperature T=300 K using an adaptive [ 34] time step which usually settled at around 50 fs. The durations of thesimulations were 1 μs except for the homogeneous ﬁlm pre- sented in Fig. 3, where we used 100 ns. For the calculation of the spectral linewidth plotted in Fig. 7the 1-μs time trace was split into two and the spectra averaged, resulting in anapproximate spectral resolution of 2 MHz. IV . RESULTS AND DISCUSSION A. Experiment We begin by exemplifying the diversity in the device- speciﬁc frequency versus current behavior. We do this bymeasuring nine devices located in a line next to each other,on the same chip. Any wafer-scale manufacturing variabilityis hence kept at its minimum possible inﬂuence. The spectraldensity as a function of the drive current is shown in Fig. 1for the nine devices. The frequency as a function of current showsa blueshifting trend for all the devices, consistent with thepropagating spin wave mode. At low currents, before the onsetof the blueshifting propagating mode, all the devices show aweaker prethreshold mode with low, zero, or even negativetunability. The prethreshold mode does in some cases, but notall, connect to the propagating mode. Apart from those generalfeatures, it can be said that the behavior differs qualitativelybetween the devices in terms of the number of simultaneouslyexcited frequencies, the position and height of the discontin-uous frequency steps, and the linearity or curvature. We referto both the discontinuous frequency steps and continuouslynonlinear current dependence as nonlinearities, since the sim-ulated behavior for an ideal, homogeneous thin ﬁlm was found 214433-3ANDERS J. EKLUND et al. PHYSICAL REVIEW B 103, 214433 (2021) FIG. 1. [(a)–(i)] Experimental power spectral density in decibels over noise as a function of drive current for nine devices on the same chip, adjacent to each other. to be highly linear above the threshold current. The devices in Figs. 1(d),1(h), and 1(i)show a particular instability in the low-current section of the propagating mode, with worsedeﬁned frequencies. The devices in Fig. 1show resemblance to previously characterized NC STOs in similar magnetic ﬁelds, in partic-ular in terms of the presence of linear regions connected bynonlinearities that are either discontinuous or continuous. Thediversity among our devices is large but we would like topoint out that we here present completely nonselected data in its unprocessed form, without reducing it by extractingand displaying only the dominant peak frequencies. The largesample-to-sample variation in the frequency versus currentbehavior implies that the magnetization dynamics is highlydifferent between the devices. The differences cannot easilybe explained by device-to-device variation in the nominal pa-rameters such as the ﬁlm thicknesses since this is not expectedto result in qualitatively different device characteristics. Nordoes such reasoning explain the emergence of the discon-tinuous and continuous nonlinearities. The device-to-devicevariation is complex and can be described in terms of nonlin-ear transitions between more linear segments. Multiple linearsegments can even coexist over limited current intervals asshown in Figs. 1(a),1(d), and 1(e). The observation of qualitative differences in the device characteristics indicates qualitative differences in the devicestructure. Figure 2(a) shows the AFM measurement of the structure of the free layer ﬁlm and Fig. 2(b) shows the SEMmeasurement of the same ﬁlm. Both the microscopes reveal a microstructure with grain size on the order of 30 nm. This iscomparable to the size found in previous works for magneticﬁlm stacks with Co bottom layer: 15–40 nm [ 35], 9 nm [ 36], and 23.6 nm [ 37]. Two of these studies [ 36,37] performed transmission electron microscopy measurements that showedcolumnar grain growth throughout the stack. Due to the4.5-nm small thickness of our NiFe free layer, it is reasonableto assume that all the grains are columnar also in our samples. The grain structure constitutes a possible complex source for the similarly complex device-to-device variation andbrings up the question of how reasonable the homogeneous-ﬁlm approximation is for modeling exchange-dominatedpropagating spin waves. It can be assumed that at the grainboundaries, the exchange interaction may be signiﬁcantly re-duced [ 33] with the magnitude of the reduction depending on the individual grain-to-grain interfaces. The random geometryof the grain structure together with the likely random natureof the intergrain exchange coupling reduction constitute a vastvariability space. We next turn to micromagnetic simulationsin order to gain insights of the potential effects that the grainstructure has on the propagation as well as generation of thespin waves. B. Simulation: Homogeneous thin ﬁlm As a basis for the investigation of the impact of spin wave barriers in the free layer thin ﬁlm, we ﬁrst perform simulationsof the nominal case of a perfect, homogeneous ﬁlm. 214433-4IMPACT OF INTRAGRAIN SPIN WA VE REFLECTIONS ON … PHYSICAL REVIEW B 103, 214433 (2021) FIG. 2. (a) Atomic force microscopy (phase contrast mode) and (b) scanning electron microscopy of the surface of the free layer (Si/SiO x/Pd8/Cu30/Co8/Cu8/NiFe4.5 /Ta3 stack). Figures 3(a)and3(b) show the simulated frequency versus current behavior for the nominal case of a perfectly homoge-neous free layer ﬁlm in the cases without and with the Oerstedﬁeld. In both cases and for sufﬁciently high drive current, the f(I DC) relation is blueshifting and perfectly linear. We identify this as the propagating spin wave mode. The threshold currentis lower in Fig. 3(b) but the linear dependence in both cases starts at 20.4 GHz. The inclusion of the Oersted ﬁeld doesnot result in any qualitative difference from the linear f(I DC) behavior. For a given current, the only effect of the Oerstedﬁeld is to shift the frequency up; this shift is ∼0.6 GHz close to the threshold around 20 mA and increases to ∼0.9 GHz at 40 mA. The linear f(I DC) dependence is an important result, since it shows that there is no inherent mechanism for the magne-tization precession or the propagating spin wave mode thatintroduces nonlinear f(I DC) behavior. In other words, the amplitude-frequency coupling has a constant nonlinearity co-efﬁcient within the operating frequency range. This shows thata more complex model of the system is required. Before the onset of the propagating mode, the so-called prethreshold behavior shows a more dramatic difference.When the Oersted ﬁeld is included [Fig. 3(b)], there is a frequency jump of 0.6 GHz from the prethreshold modeto the propagating mode whereas there is a continuous transition without the Oersted ﬁeld [Fig. 3(a)]. This can be understood by considering the local FMR landscapes.Without the Oersted ﬁeld, the local FMR frequency is ho-mogeneous in the entire NC region and the small-amplitude,prethreshold oscillation also has the same frequency ev-erywhere. This symmetry remains when the drive currentis increased and the increased spin torque eventually be-comes strong enough to launch the propagating wave. Inthe case of the Oersted ﬁeld, there is an effective magneticﬁeld asymmetry between the center and the edges of theNC. This results in a spatially inhomogeneous FMR fre-quency, which facilitates simultaneous excitation of multiplelocal prethreshold modes (with different FMR frequencies)at different locations within the NC region [ 28]. This allows the propagating mode to be excited on the high-frequencyedge of the NC region, independently of the lower-frequencyprethreshold mode that remains on the opposite side. Asthe drive current is increased and the mode volumesgrow and start to overlap, the propagating mode eventu-ally extinguishes the prethreshold mode. The intermodulationproducts visible in the threshold current range 15–17 mAin Fig. 3(b) are a result of this coexistence [ 28,38] in time and space of the two modes. FIG. 3. Fourier amplitude of my(in decibels) from simulations with a homogeneous free layer ﬁlm at T=4 K, (a) without and (b) with the Oersted ﬁeld. 214433-5ANDERS J. EKLUND et al. PHYSICAL REVIEW B 103, 214433 (2021) Our result of a high degree of linearity in the frequency as a function of IDCfor the propagating mode is opposite to the result in Ref. [ 39], where nonlinearity was found despite the effort of having implemented spin wave absorbing boundaryconditions. However, when we employed identical boundaryconditions (in a circle outside the nanocontact) and performeda spatial analysis we did actually observe a certain amountof reﬂection from the “absorbing” boundary that resulted inan artiﬁcially introduced standing spin wave pattern. We willsee in the following sections that reﬂection and standing spinwaves in the physical system are highly dominant mechanismsaffecting the frequency selection in a way that introduces the f(I DC) nonlinearity. C. Simulation: Impact of a single barrier As a ﬁrst step towards investigating the possible effects of the grain structure, we simulate the STO behavior in the casewhere a single barrier is placed in the spin wave path. Beinga wave phenomenon, we expect that part of the incident waveis reﬂected back towards the source, i.e., the active, current-driven region directly below the NC. We set up a barrier inthe free layer in the form of an artiﬁcial rectangular grainwith the exchange coupling between it and the surroundingﬁlm set to zero. The rectangular grain has a width of 100 nmfacing the NC and is 50 nm deep. The NC is at the origin ofthexyplane of the sample ﬁlm; the externally applied ﬁeld is aligned in the ﬁrst quadrant of the xzplane and the barrier is positioned along the positive ydirection. This is the direction into which the spin wave beam propagates in the case whenthe Oersted ﬁeld is included [ 28]; this is the side where the in-plane component of the external ﬁeld and the Oersted ﬁeldoppose each other, resulting in a decrease of the local FMRfrequency. On the opposite side, the in-plane components addup and bring the local FMR frequency up to a level above thespin wave generation, hence blocking spin wave propagationin that direction. The result from the simulations with the single artiﬁcial “wall” grain is shown in Fig. 4. Figure 4(a)displays the power spectral density as a function of the edge-to-edge separationdistance d sepbetween the NC and the wall, for a ﬁxed current ofIDC=27.75 mA. At this level of drive current, the nominal, ideal ﬁlm oscillation is stable at 23.1 GHz [see Fig. 3(b)]. Figure 4(a) shows a periodic pattern with a period of 45 nm. This corresponds very well to half of the spin wave lengthof 85–90 nm which we read off from the simulation spacein the unperturbed situation. At its peaks, the wall shiftsthe oscillation frequency away from the homogeneous-ﬁlmcase frequency up to 1.0 GHz positive and 0.3 GHz negative,where the magnitude of the frequency shift decreases withd sep. The effect of the wall is visible up to dsep=210 nm, where the STO once again attains its ideal, homogeneous-ﬁlmfrequency. The downward frequency slope can be understoodas the consequence of a forced and gradually enlarged spinwave length as the STO strives for spin wave resonance at agradually longer distance. The upward jumps in frequency oc-cur when resonance eventually occurs for one additional nodeand antinode, which rapidly forces a shorter wavelength. Wenote that the effect is asymmetric towards higher frequency,corresponding to a preferred situation of shorter wavelength FIG. 4. Simulations of a barrier with no exchange coupling to the remaining free layer ﬁlm. Fourier amplitude of my(in decibels) for (a) varied NC edge to obstacle separation distance dsepforIDC= 27.75 mA and (b) current sweep for dsep=125 nm. (more nodes). This is a consequence of the general coupling in STOs between the oscillation amplitude and frequency: thefrequency increases with the amplitude for the propagatingspin wave mode in this magnetic ﬁeld. The strongest and dom-inating resonance occurs for the standing wave conﬁgurationthat has the highest amplitude and, as a consequence, alsohas the highest frequency, shortest wavelength, and highestnumber of nodes. Larger amplitude and higher frequency isobtained for resonance at small d sepdue to the lower amount of spin wave damping along the shorter propagation path. Figure 4(b) shows a current sweep for the case of ﬁxed dsep=125 nm. The artiﬁcially introduced spin wave barrier introduces nonlinearity in f(IDC) similar to the type that is characteristic for the experimental devices; we notice the ap-pearance of a discontinuous frequency step of 0.65 GHz at27 mA and a small degree of continuous nonlinear behaviorboth below and above the step. Compared to the homogeneousﬁlm case in Fig. 3(b), the introduction of the barrier generally pushes the frequency higher. This is consistent with the gen-eral preference of selecting a higher frequency (i.e., squeezingin an additional standing wave node), as found in Fig. 4(a). This exempliﬁes how a reﬂected spin wave can alter the generation frequency, and can be considered as STO self-interaction. This is made possible by the ability of the STOto be pulled towards the frequency of an injected signal andphase-lock to it, which has previously been shown to occur 214433-6IMPACT OF INTRAGRAIN SPIN WA VE REFLECTIONS ON … PHYSICAL REVIEW B 103, 214433 (2021) FIG. 5. Simulations with grain microstructure. Fourier amplitude of my(in decibels) for (a) seed 1 and (b) seed 5. (c) Extracted frequency versus current for nine simulated devices and a homogeneous ﬁlm. Mean values for simulated and experimental device series. Inset: The standard deviation of the frequency σfof the simulated and experimental device series. both for injected electrical rf signals [ 40–45] and incoming spin waves [ 46–51]. We ﬁnd that in the case of spin wave reﬂection and self-interaction, the wavelength becomes tunedto form a standing wave between the NC and the barrier in apositive feedback loop. The STO will stabilize its oscillationby tuning its frequency such that an antiphase reﬂected waveis avoided. The degree to which the STO adapts the frequency to the available standing wave frequencies depends on the amplitudeof the incoming reﬂected spin wave in relation to the STOamplitude. If an incoming antiphase wave is weak enough, theSTO frequency will be unaffected. In such a case, the reﬂectedwave merely constitutes a perturbation that can be expected tointroduce phase noise but not shift the frequency. In Fig. 4(a) we ﬁnd that the distance where frequency shifting becomesnegligible for our simulated devices is around d sep=210 nm away from the NC edge. Similar standing spin wave effects were obtained when the same barrier was left with full exchange coupling across itsboundaries and instead the NiFe Gilbert damping parameterα Gwas increased to 1.0. D. Simulation: Impact of grain structure Figures 5(a) and5(b) show the simulated behavior of two grainy free layers with random intergrain exchange reduc-tion within 0–100%. These two “devices” exemplify both continuous and discontinuous nonlinearities: the sample inFig. 5(a) shows a highly continuous behavior while in Fig. 5(b) there is a discontinuous frequency transition at 29 mA. The inclusion of grains in the model thus provides adirect explanation of the origin of the f(I DC) nonlinearity in NC STOs. The grain boundaries form spin wave barriers withvarying degree of reﬂectance and orientation with respect tothe propagating spin waves and thus have varying degrees ofimpact on the STO. We next investigate the simulated device statistics of f(I DC) for randomly generated grainy ﬁlms. Figure 5(c)shows the extracted frequency versus current for nine simulated de-vices together with the mean value for both the simulated andexperimental (Fig. 1) cases. The homogeneous-ﬁlm case is included for reference. The simulated grainy ﬁlms all havetheir frequencies shifted upwards compared to the nominalhomogeneous-ﬁlm case. The shift of the mean frequency in-creases gradually with the drive current from 500 MHz at20.5 mA to 1400 MHz at 32.5 mA. In fact, we do not observea single frequency below the homogeneous-ﬁlm case for anysimulated device at any current level. This shows that theSTO also for the grain microstructure case always selects astanding spin wave pattern that results in an upward ratherthan downward frequency shift and is in line with the asym-metry towards higher frequency (and amplitude) found in the 214433-7ANDERS J. EKLUND et al. PHYSICAL REVIEW B 103, 214433 (2021) single-barrier case, Fig. 4(a). In the grain case, the standing spin wave pattern can be altered both by changing the numberof nodes towards a given grain boundary, or by changing toanother dominant grain. The large number of possible modeconﬁgurations gives a high probability of always ﬁnding amode with the preferred positive frequency shift. We note that the experimental mean f(I DC) actually evolves with a factor of 2 lower slope df/dIDCthan the simulated grain and homogeneous cases. Our free simula-tion parameters were initially tuned to give largely correctthreshold current and frequency for the propagating spin wavemode, but were not adjusted to ﬁt the experimental f(I DC) relation for higher drive currents. We believe that this discrep-ancy in the simulations might be due to the true spin torqueefﬁciency being lower than the value used for the simula-tions. It can also be related to the real Oersted ﬁeld, which isprobably lower than what is calculated using the inﬁnite-wireapproximation. There is also a lateral spread in current due tothe device design [ 52], where the current is intended to ﬂow at the bottom of the spin valve mesa from the NC region outto the ground contacts. The lateral current spread decreasesthe current density in the NC region and further modiﬁesthe Oersted ﬁeld. We have also not taken into account anypossible temperature dependence for the magnetization (i.e.,the FMR frequency), which would decrease at higher I DC due to the higher electrical power dissipation. Exploring this parameter space while ﬁne-tuning the distribution functionfor the intergrain exchange coupling to achieve even bettercorrespondence is beyond the scope of the present work. Exactcorrespondence is also not necessary for discussing the mech-anisms of spin wave generation, reﬂection, and interaction andtheir consequences. The inset of Fig. 5(c) shows the standard deviation (be- tween devices) of the frequency σ ffor the simulated and experimental devices. The standard deviation of the simulateddevices increases linearly over the simulated current intervalwith a factor of 3, from 0.2 to 0.6 GHz. This factor of 3coincides with the increase of the grain-induced frequencyshift as a function of drive current in Fig. 5(c). It is a natural behavior that the device-to-device variation that is due tograin-induced frequency shifting is directly proportional to themean magnitude of the frequency shift. In the upper half of the simulated current range the stan- dard deviation of the frequency reaches the levels found inthe experiment, i.e., 400–600 MHz. This quantitative corre-spondence strengthens the hypothesis of grain-induced spinwave reﬂection as a main source of the device-to-devicevariation. Since the trends are not the same (constant ver-sus linearly increasing), there remain modeling aspects inparticular for lower drive currents. The most straightforwardapproach would be to further decrease the intergrain exchangecoupling to force the grain effect down to lower oscilla-tion amplitudes. For the experimental devices at lower drivecurrent, there is also a more prominent appearance of theprethreshold mode which perturbs the propagating mode inthe devices in Figs. 1(d),1(e),1(h), and 1(i). It is beyond the scope of this paper to fully reproduce the prethresholdbehavior. Figure 6(a) shows the mode structure for seed 5 at I DC=23.5 mA. The structure is shown as the time-averaged FIG. 6. Simulations of the grain structure with seed 5. (a) Fourier amplitude of myfor the propagating mode at its peak frequency 22.7 GHz for IDC=23.5 mA. (b) The underlying grain structure, showing the normalized strength of the intergrain exchange coupling.The nanocontact region is indicated by the circle. oscillation amplitude, as opposed to an instantaneous snapshot of the propagating waves. The oscillation amplitude falls offnonmonotonously outside the nanocontact and forms nodesand anti-nodes in a complex interference pattern. Points ofspin wave reﬂection can be identiﬁed where there is a discon-tinuous drop in the oscillation amplitude. Four clear reﬂectorsare indicated at points A, B, C, and D in Fig. 6(a). Looking at the exchange coupling at the same points in Fig. 6(b) re- veals that the reﬂection occurs at grain boundaries where theexchange coupling has been strongly reduced. Grain bound-aries A and B are oriented so that their normal direction ispointing approximately towards the NC. At point C thereare two possible strong reﬂectors oriented at approximatelya4 5 ◦angle relative to the NC direction and it is not clear that they reﬂect spin waves back directly to the NC. However,the proximity to the NC still results in interference effects atstrong amplitudes close to the main oscillation. Around pointD there are multiple grain boundaries that are too close to theNC to form a node and antinode, but that nonetheless act toconﬁne the oscillation. The STO stabilizes at the stationaryoscillation state that results in the least amount of con-ﬂict between the reﬂected spin waves returning from pointsA, B, C, and D. 214433-8IMPACT OF INTRAGRAIN SPIN WA VE REFLECTIONS ON … PHYSICAL REVIEW B 103, 214433 (2021) FIG. 7. Simulation of seed 5 grain microstructure and the homogeneous-ﬁlm case at sample temperatures T=300, 150, 77, and 4 K. (a) Frequency versus drive current. (b) Spectral linewidth (FWHM) versus drive current. Linewidth values below our effective resolution of 0.5 MHz have been plotted at this value. The homoge- neous ﬁlm at T=4 K is below the linewidth resolution for the entire current range and has been omitted. We ﬁnally investigate the impact of the grain structure on the frequency stability of the oscillation. For this study weselect seed 5 since it contains the cases of both continuous anddiscontinuous nonlinearity in f(I DC) and compare its spectral linewidth to the homogeneous-ﬁlm case. Simulations werecarried out for sample temperatures Tof 300, 150, 77, and 4 K. Figure 7shows the frequency and spectral linewidth [full width at half maximum (FWHM)] as functions of the drivecurrent I DC.Figure 7(a)shows that as the temperature is decreased, the frequency for a given current also decreases. This simulationresult agrees with our previous experimental results [ 53]f o r similar devices and magnetic ﬁeld. More interestingly, thetemperature decrease induces a shift of the nonlinear op-erating points to higher drive currents. The nonlinearity isprimarily a function of the frequency, which can best be seenat the operating point around 30 mA, where there is a discon-tinuous step which goes from 25 GHz up to 25.5 GHz for alltemperatures. This is a natural consequence of the standingspin wave landscape, where the operating frequency is setby the optimum spin wave length. The nonlinearity around26 mA is continuous for all temperatures. The nonlinearity at 23 mA changes character as the tem- perature changes: at 300 K it is continuous but breaks up intoa discontinuous transition at 150, 77, and 4 K. This illustratesthat there is no fundamental difference in the origin of thecontinuous and discontinuous nonlinearity. In the discontinu-ous case there are two resonance states where the oscillatorselects one at a time. Around the transition, thermal energymay be able to kick the trajectory back and forth betweenthe states which can be observed as mode jumping [ 54,55]. In the continuous case, the oscillator enters a trajectory thatis intermediate to the two underlying resonances. In this con-tinuous transition case, the nanocontact spin torque oscillatorcan readily be analyzed within the framework of the generalnonlinear auto-oscillator theory. Figure 7(b) shows that the nonlinear operating points are associated with a destabilization of the frequency. Forhomogenous ﬁlms, the linewidth is largely independent ofthe drive current and stays within a narrow interval of3.6–6.7 MHz for T=300 K. With the grain microstructure, the linewidth varies from the homogenous-ﬁlm values of sin-gle megahertz (minimum 4.6 MHz is observed) inside thelinear regions up to 85–90 MHz at the nonlinear operatingpoints. The total range for the spectral linewidth of singlemegahertz up to around 100 MHz at T=300 K agrees well with the experimentally observed ranges [ 21,25]. As the tem- perature is decreased, the maximum linewidth points shiftto higher currents. This occurs since the entire f(I DC) rela- tion is moved to higher currents, as previously discussed forFig.7(a). The two nonlinearities at higher current (26 and 30 mA) both show a decreasing value for the maximum linewidthas the temperature is decreased. This is the generally expectedbehavior for single modes described by the nonlinear auto-oscillator theory [ 24], where the nonlinear ampliﬁcation factor νalong with the temperature determines the linewidth. The same nonlinear ampliﬁcation factor has also been shown to beapplicable in the multimode case [ 56] with thermally activated mode jumping. The situation of mode hopping between multi-ple excitable modes creates an increased sensitivity to thermalﬂuctuations of the oscillation power through a decrease inthe power restoration rate /Gamma1 p. Since ν∼1//Gamma1p, this theory can be used to at least qualitatively explain the substantiallyincreased linewidth at the nonlinear operating points. Con-versely, our work explains the origin of the different modesand gives a physical justiﬁcation of the applicability of themultimode theory for the analysis of the nanocontact STO ascarried out in Ref. [ 56]. 214433-9ANDERS J. EKLUND et al. PHYSICAL REVIEW B 103, 214433 (2021) FIG. 8. Mode localization for in-plane magnetic ﬁelds (low θext). Experimental: power spectral density in decibels over noise at (a) θext= 70◦and (b) θext=30◦. Simulated: Fourier amplitude of mxin decibels for seed 5 at (c) θext=70◦and (d) θext=30◦. The low-current nonlinearity at 23 mA again shows a different behavior. Here the maximum linewidth instead in-creases when the temperature is decreased from 300 to 77 K.At 4 K the linewidth has decreased to single-megahertzvalues, similar to the case of a well-deﬁned discontinuousnonlinearity. Since this nonlinearity changes its nature fromcontinuous to discontinuous when changing Tfrom 300 to 77 K we cannot expect either single- or multimode theoryto accurately describe the temperature dependence of thefrequency stability across the transition. There may also beadditional instability induced by the simultaneous availabilityof both single- and multimode solutions. More detailed studyof the transition from continuous into discontinuous nonlin-earity is beyond the scope of this work. E. Mode localization at low ﬁeld angles When the applied magnetic ﬁeld is directed more in plane, the intrinsically localized bullet mode is excited and the prop-agating mode also becomes localized [ 28]. The localization of the propagating mode occurs when the generated frequencybecomes lower than the FMR frequency of the surrounding ﬁlm, thereby breaking the condition for propagation. The ab-sence of spin wave propagation implies the elimination of theeffects of wave reﬂection, i.e., the elimination of nonlinearityinf(I DC). Figure 8shows the device behavior at θext=70◦and 30◦ for an additional experimental device and the simulated device with the seed 5 grain structure. The simulated device wasselected based on its signiﬁcant nonlinearity of the propa-gating mode characteristics at θ ext=70◦.I nF i g s . 8(a) and 8(c), where only the propagated mode is excited, the degree of nonlinearity is similar for the experimental and simulateddevices. The experimental device shows multiple propagat-ing modes at the highest current levels, similar to our workin Ref. [ 54] where the oscillator was jumping between the multiple frequencies. We interpret this behavior as the randomselection between different standing wave patterns. When the applied magnetic ﬁeld is directed more into the plane, θ ext=30◦,F i g s . 8(b) and8(d) both show the red- shifting localized bullet mode. Compared to the experimentalpropagating mode in Fig. 8(a), the bullet mode in Fig. 8(b) 214433-10IMPACT OF INTRAGRAIN SPIN WA VE REFLECTIONS ON … PHYSICAL REVIEW B 103, 214433 (2021) has a signiﬁcantly more linear f(IDC) relation. This is also seen in the simulated device in Fig. 8(d) and shows that the grain structure does not have a signiﬁcant inﬂuence on thebullet mode. For the blueshifting, propagating mode at θ ext=30◦the simulated device in Fig. 8(d) now shows two simultane- ous and highly linear modes. The experimental device inFig. 8(b) shows a very weak electrical signal, just above the noise ﬂoor of our measurement apparatus, but we areable to note two features of its behavior. First, the propa-gating mode is much less stable than the bullet mode andspreads its oscillation energy over a wider spectrum range—itbetter resembles broadband noise rather than a spectral peak.Second, at high currents (above 37 mA) we again see anadditional propagating mode. A spatial frequency analysisof the propagating mode in the simulated device shows thatthe blueshifting mode is localized, with its two differentfrequencies in Fig. 8(d) being dominant in different grain clusters. We believe that this occurs also for the experimentaldevice in Fig. 8(b). Since the grain conﬁgurations in both experimental and simulated samples appear to give rise tomultiple propagating modes with weak mutual coupling wethink that the agreement is still convincing. In simulationswe could observe as many as three individual propagatingmodes and the actual number is determined by the chosenaverage grain size and exchange coupling values at the grainboundaries. V . CONCLUSIONS Experimental spectra of the propagating spin wave mode from nine nominally identical devices have been presentedand their qualitative behavior has been described. The sample-to-sample variation in terms of the frequency as a functionof current is signiﬁcant quantitatively as well as qualita-tively, with a common feature being linear regions that areconnected by nonlinearities that can be either continuousor discontinuous (in the form of a frequency step). Thisqualitative behavior has been reproduced in simulations in-corporating the ∼30-nm grain structure measured using AFM and SEM, with randomly reduced intergrain exchange cou-pling. The reduction of the intergrain exchange couplingresults in spin wave reﬂection, which in turn facilitates self-locking of the STO to geometry-deﬁned resonant frequencies.The spin wave resonance preferably acts to increase the os-cillation frequency compared to the homogeneous ﬁlm case.Each of the strongly reﬂecting grain boundaries constitutesone resonance condition and the ﬁnal frequency selectionfor a given current is determined by the inherent STO fre-quency and the relative strengths of the different reﬂections.The different standing spin wave modes act to increase thespectral linewidth by more than one order of magnitude atoperating points where several of them are simultaneouslyexcitable. Spin wave reﬂection and resonance due to grain boundaries constitute a physically reasonable model that is able to explainthe origin of the continuous and discontinuous nonlinearitiesin the frequency versus current. This model also explains a large part of the device-to-device variation as stemming fromthe random grain structure, with partially quantitative agree-ment with the experimental device variation. For improvedagreement, we suggest future modeling work to further re-duce the intergrain exchange coupling in order to increasethe effects of spin wave reﬂection. Other possible sourcesof variability are inhomogeneity in the magnetic parameterssuch as the saturation magnetization (or ﬁlm thickness) andspin polarization ratio. Given the grain microstructure andassuming columnar growth throughout the thin-ﬁlm stack, itwould be natural to assign these varying properties at thegrain level. The different grain-induced resonance conditions can be viewed as separate spin wave submodes which are simul-taneously excitable at the nonlinear operating points. Thisinstability explains the elevated level of the spectral linewidthat the nonlinearities, provides a physical motivation for multi-mode oscillator theory, and explains the origin of the apparentnonlinearity as it is treated in single-mode theory. For the dynamics at low ﬁeld angles, experiment and simulations show that the localized bullet mode is largelyunaffected by the grain conﬁguration. The propagating modebecomes localized [ 28], and our results indicate that this localization is conﬁned to individual grain clusters. Sincethe FMR frequency varies within the NC region due to theOersted ﬁeld and there are no longer any propagating spinwaves to strongly couple and synchronize the grain clustersinto a common frequency, the localization of the propagatingmode results in a multitude of separate but weakly interactingoscillators. Future work for improvement of the frequency stability and spin wave propagation, as well as reduction of the device-to-device variation, should aim at reducing the effect of thegrain boundaries. Possible actions include optimizing the de-position process for larger grain size, and using annealingto improve the intergrain exchange coupling. Further workon controlling and improving the device performance is sug-gested to focus also on the machining of artiﬁcial spin wavereﬂection boundaries that can dominate over the random grainboundaries. We also suggest future work to investigate thepotential impact of grain boundaries on the other magneto-dynamical excitations in nanocontact spin torque oscillators:vortex gyration and the droplet soliton. ACKNOWLEDGMENTS The authors gratefully acknowledge Ahmad Abedin for as- sistance with the SEM measurements and Federico Peveré andFaraz Khavari for assistance with the AFM measurements.The computations were enabled by resources provided bythe Swedish National Infrastructure for Computing (SNIC)partially funded by the Swedish Research Council throughGrant Agreement No. 2018-05973. The computational resultspresented have in part been achieved using the Vienna Sci-entiﬁc Cluster (VSC). This work was ﬁnancially supportedby the Swedish Research Council (VR) through the projects2017-04196 and 2012-05372. 214433-11ANDERS J. EKLUND et al. PHYSICAL REVIEW B 103, 214433 (2021) [1] T. J. Silva and W. H. 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PhysRevApplied.10.054013.pdf	PHYSICAL REVIEW APPLIED 10,054013 (2018) Magnetic Conﬁgurations and State Diagram of Nanoring Magnetic Tunnel Junctions Houfang Liu,1Hongxiang Wei,1Xiufeng Han,1,Guoqiang Yu,1Wenshan Zhan,1Sylvain Le Gall,2 Yuan Lu,2Michel Hehn,2Stephane Mangin,2,†Mingjuan Sun,3Yaowen Liu,3,‡and Cheng Horng4 1Beijing National Laboratory of Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2Institut Jean Lamour, UMR 7198, CNRS-Université de Lorraine, BP 70239, 54506 Nancy, France 3School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 4TDK-Headway Technologies Inc., Milpitas, California 95035, USA (Received 16 November 2016; revised manuscript received 24 July 2018; published 6 November 2018) Nonvolatile magnetic random-access memory is one of the most promising memory candidates to meet the requirements of high-density and low-power data storage. Downsize scalability and energy eﬃciencyof conventional memory-unit cells with elliptic shape, however, remain matters of great concern. The development of alternative memory-unit-cell architecture that would potentially enhance the performance of practical devices is thus particularly interesting for applications. Here the magnetic conﬁgurations fornanoring magnetic tunnel junctions with an in-plane magnetic storage layer are studied by micromagnetic simulation, revealing that, for an appropriate ring width, the outer diameter can be scaled down to less than 20 nm, where the magnetic onion conﬁguration becomes energetically favored. We also study thespin-transfer-torque switching process and dynamic resistance state diagram with respect to the applied magnetic ﬁeld and spin-polarized current. A low switching-current density is demonstrated. The results indicate the advantage of using nanoring magnetic tunnel junctions as memory units, which may providean alternative solution for high-storage-density and low-power-consumption nonvolatile memory. DOI: 10.1103/PhysRevApplied.10.054013 I. INTRODUCTION Since 1975, microelectronics has been following Moore’s law, which states that the density and speed of integrated circuits will double every 18 months. However, this trend is presently coming to an end, due to insur- mountable physical limits and high power dissipation of CMOS. Besides, the next generation of portable, ﬂexi- ble, and wearable electronics devices, especially in the growing Internet-of-things ﬁeld, are gradually demand- ing lower-power-consumption and higher-storage-density devices [ 1,2]. The use of devices based on nonvolatile magnetic random-access memory (MRAM), which can signiﬁcantly reduce the total energy and be integrated into a system on chip without alteration of baseline logic plat- forms both in process and in design [ 3–8], has been consid- ered as a promising solution to reduce power consumption, improve reliability, and oﬀer new functionalities. Spin- transfer-torque (STT) MRAM with elliptic unit cells with xfhan@iphy.ac.cn †stephane.mangin@univ-lorraine.fr ‡yaowen@tongji.edu.cnan in-plane magnetic anisotropy or circular unit cells with a perpendicular magnetic anisotropy (PMA) has been exten- sively explored [ 9–12]. However, low thermal stability and high switching-current density at reduced dimensions stilllimit the reliability and high density of MRAM. Alter- natively, nanoring-shaped magnetic-tunnel-junction (MTJ) cells with closure magnetic ﬂux and circular Oersted ﬁeld can enable more reliable and lower power consumption [13–16]. Here the magnetic conﬁgurations for the free layer of a nanoring MTJ are studied by micromagnetic simulation, revealing an energetically stable onion conﬁg- uration even with the size scaled down to less than 20 nm for the outer diameter and 0.5 nm for the ring width. The switching properties and dynamic resistance state diagram for nanoring MTJs are also been explored. A low switching current of 1.3 ×10 6A/cm2is experimentally demonstrated in MTJs with tunnel magnetoresistance ratio of 100%. The nanoring MTJ exhibits the attractive capability of achiev- ing ultralow writing energy (less than 0.4 pJ per bit with a pulse width of 10 ns) and high storage density (up to 10 Gb/in.2). These results provide insights for the further understanding of the magnetization-switching mechanism in nanoring MTJs and oﬀer a design rule for MRAM unit cells with high density and low power consumption. 2331-7019/18/10(5)/054013(10) 054013-1 © 2018 American Physical SocietyHOUFANG LIU et al. PHYS. REV. APPLIED 10,054013 (2018) II. METHODS Magnetic multilayers with a core structure of buﬀer/layer/IrMn (10 nm)/Co 75Fe25(2.5 nm )/Ru(0.8 nm )/ Co40Fe40B20(3n m)/Mg(0.4 nm )/MgO(0.6 nm )/Co40Fe40 B20(2n m)/Ta(5n m)/Ru(6 nm) are deposited onto ther- mally oxidized Si wafers by dc and rf magnetron sput- tering, and then annealed at 330 °C in a vacuum under an in-plane magnetic ﬁeld for 1 h. The bottom, 3-nm- thick(Co, Fe )B layer is a reference layer, and the upper, 2-nm-thick (Co, Fe )B layer is a free layer. The fabrica- tion processes for MTJs with an outer diameter of 100 nm and ring width ranging from 25 to 30 nm are as fol- lows. First, the bottom electrode is fabricated by ultraviolet lithography combined with Ar-ion beam milling. Sec- ond, the nanoring-shaped junction area is patterned by electron-beam lithography using a poly(methyl methacry- late) and hydrogen silsesquioxane bilayer resist technique. The junctions are etched by reactive-ion etching and an Ar-ion beam. Third, the nanoring-shaped MTJs with the top resists are then buried by SiO 2deposition. The resists and SiO 2are removed by a lift-oﬀ process before the top electrode is deposited and patterned. Finally, a Cu(50 nm)/Au(100 nm )bilayer is deposited as the top electrode [ 17]. Theoretically, the magnetization reversal in nanoring elements can be well described by micromagnetic sim- ulations [ 18]. In this study, the magnetic conﬁguration and current-induced dynamics in the nanoring MTJs are modeled with the ﬁnite-element micromagnetics package TETRAMAG developed by Hertel et al .[19]. The code is based on the Landau-Lifshitz-Gilbert (LLG) equation extended to include the in-plane Slonczewski and ﬁeldlike spin torques [ 3,20]: dM dt=−γ0(M×Heﬀ)+α Ms/parenleftbigg M×dM dt/parenrightbigg +aJ MsM×(M×ˆmp)+bJ(M×ˆmp),( 1 ) where Mis the local magnetization of the free layer, ˆmpis the unit magnetization vector of the reference layer, and Heﬀis the eﬀective ﬁeld that contains the exchange, anisotropy, demagnetizing, stray ﬁeld (caused by the ﬁxed reference layer) and external magnetic ﬁelds. The last two terms in Eq. (1)are the in-plane Slon- czewski torque and the ﬁeldlike spin torque, respectively; aJ=/planckover2pi1γPJ/[2eM sd(1+ξ2)]a n d bJ=ξaJ, where /planckover2pi1is the Planck constant, γis the gyromagnetic ratio, Pis the spin polarization, Jis the electric current density, eis the electron charge, Msis the saturation magne- tization, dis the thickness of the free layer, and ξ= lex/lsfis the ratio of the spin decoherence length and spin-ﬂip relaxation length [ 20]. In this study, the follow- ing parameters are used for the free layer [ 21,22]: thethickness of the free layer is 2 nm, Ms=1000 emu/cm3, the exchange constant A=2×10−6erg/cm, the uniaxial anisotropy Ku=3×104erg/cm3,P=0.56, the damping constant is 0.01, and ξ=0.01. The average size of the tetrahedron mesh cell is around 2 nm for a nanoring with a diameter of 100 nm. For samples with a very small ring width of 0.5 nm, the mesh size is decreased to 0.2 nm. In principle, for such small samples, the LLG simulation at the atomistic level may be principally more eﬀective [ 23]. All the simulations are performed at zero temperature. III. RESULTS AND DISCUSSION A. Magnetic conﬁgurations and extended stability for nanoring MTJs First, we study the energetically stable magnetic con- ﬁgurations in the nanoring MTJ using the LLG micromag- netic simulations. The results show that the free-layer mag- netization in the nanoring MTJ can exhibit three diﬀerent magnetic textures—the onion, vortex, and antivortex con- ﬁgurations—as shown in Fig. 1(a). Although each of the three magnetic states is energetically stable (corresponding to one of the local minima of the Gibbs free energy), only the onion or vortex magnetic conﬁguration has the global smallest energy, depending on the structure parameters. This can be recognized from the averaged energy density of the magnetic conﬁgurations, which include exchange energy, anisotropy energy, demagnetization energy, and stray ﬁeld energy caused by the magnetic reference layer of the nanoring MTJ (if any). Figure 1(b) shows the depen- dence of the energy density on the width of the nanoring for the three magnetic conﬁgurations, where the thickness of the free layer is 2 nm and the outer diameter is ﬁxed to be 100 nm. In this simulation, the stray ﬁeld generated by the reference layer is assumed to be 150 Oe to reproduce the experimental observations. The vortex conﬁguration (with lower energy) is more energetically favorable when the nanoring width is smaller than a critical value wc,a n d the onion conﬁguration is more favorable for width larger than wc. In all cases, the antivortex conﬁguration has much higher system energy. The simulations show the magnetic conﬁguration of the nanoring sample with 2 R=100 nm and w=30 nm is an onion state, which is consistent with the experimental observations as discussed below. Figure 1(c) displays the magnetization phase diagram for the energetically favored conﬁgurations of magnetiza- tion for diﬀerent parameters. The black “nanopillar” line indicates the extreme condition (i.e., the nanoring becomes a pillar without a hole). The blue diamonds and red circles are critical boundaries between the magnetic vortex and onion conﬁgurations obtained from the simulations with astray ﬁeld and without a stray ﬁeld, respectively. The ener- getically favorable conﬁguration of either the onion or the vortex conﬁguration is mainly determined by the compar- ison of the exchange energy and demagnetization energy, 054013-2MAGNETIC CONFIGURATIONS AND STATE DIAGRAM . . . PHYS. REV. APPLIED 10,054013 (2018) Width of nanoring (nm) Width of nanoring (nm)(a) (b) (c) FIG. 1. (a) The appearance of the stable onion, vortex, and antivortex conﬁgurations in the free layer of the nanoring MTJs. The magnetization of the reference layer is ﬁxed to the – xdirection (i.e., antiparallel alignment for the onion state). (b) Comparison of the energy density as a function of nanoring width simulated at a ﬁxed outer diameter of 100 nm. wcis the critical value of the ring width after which the energy amplitude of the onion conﬁguration is lower than that of the vortex conﬁguration. (c) Calculated magnetization phase diagram in the nanoring MTJs. Here we set the free-layer thickness of the nanorings to be 2 nm. The yellow region is where the vortex (onion) conﬁguration is stable for simulations without (with) the stray ﬁeld. The inset shows an enlargement for ring width scaling down to 0.5 nm. as the other energy terms are comparably negligible. The onion conﬁguration is stabler for a ring width greater than the critical value wc[see Fig. 1(b)], at which the inner cur- vature of the ring leads to increased exchange energy by the misaligned moments of the neighbor cells. The vor- tex conﬁguration is the favorable texture for a very narrow ring width. Our simulation results indicate that the stable vortex conﬁguration remains for a ring width scaled down to 0.5 nm while the outer diameter ranges between 20 and 100 nm. Such an ultrathin nanoring structure looks like a spin benzene ring, in which the circular vortex conﬁgu- ration is obviously a stable favorite texture. On the other hand, the stable onion conﬁguration can also be continu- ously scaled down to smaller size with outer diameter less than 20 nm and ring width about 0.5 nm [see the inset in Fig. 1(c)]. Such a phase diagram of magnetic texturesshould be treated with a lot of caution, especially for practical memory applications at very small lateral sizes. Although the spin vortex is very stable against external perturbations due to its soliton structure with topological protection, the stability factor should still decrease with the downscaling of the in-plane-magnetized free layer due to the relatively low magnetic anisotropy. Another interesting feature in the inset in Fig. 1(c) is that the nanoring favors the out-of-plane (OOP) mag- netization when the ring width is further decreased. This OOP magnetization state is attributed to the shape anisotropy for these geometric parameters, at which the thickness (2 nm) is larger than the ring width (0.5 nm). According to Refs. [ 9,24,25], the t(Co,Fe)Bdependence of the eﬀective anisotropy energy density is given by K=Kb+Ki/t(Co,Fe)B−μ0M2 s/2, where Kbis the bulk 054013-3HOUFANG LIU et al. PHYS. REV. APPLIED 10,054013 (2018) Nanoring MRAM (Co, Fe)B (Co, Fe)B(a) (b)(c) FIG. 2. (a) The architecture for nanoring MRAM containing MTJ cells. Access transistors of MRAM are not shown here. (b)Scanning electron micrograph of nanoring MTJ arrays with an outer diameter of 100 nm and ring width of 30 nm. (c) Cross- section HRTEM image of the magnetic multilayer ﬁlm afterbeing annealed at 330 °C in a vacuum in a magnetic ﬁeld of 0.8 T. The bottom, 3-nm-thick (Co, Fe )B layer is a reference layer. The upper, 2-nm-thick (Co, Fe )B layer is the free layer, whose mag- netization can be switched by application of a magnetic ﬁeld and injection of polarized current. crystalline anisotropy, Kiis the interfacial PMA between MgO/(Co, Fe )B interfaces, and the third term is the demagnetization energy. The interfacial PMA is enhanced with the decrease of the free-layer thickness [ 24]. This will lead to the OOP magnetic texture becoming favorable. Our simulations indicate that the region of the OOP mag- netic conﬁguration is signiﬁcantly enlarged when the freelayer thickness, t(Co,Fe)B, decreases to 0.5–1.5 nm and the interfacial PMA is considered (not shown). B. Dynamic resistance state diagram in nanoring MTJs As a magnetic onion state is preferred for nanoring- shaped MTJs at the dimensions given, we next systemati- cally study the current-driven magnetic dynamics in MTJs hosting an onion state. To make the onion state favored, we fabricate nanoring-shaped MTJs with outer diameter of 100 nm and ring width ranging from 25 to 30 nm. From Fig.1(c), the onion conﬁguration appears in the case of the stray-ﬁeld eﬀect. The architecture for the nanoring MRAM is shown in Fig. 2(a), with nanoring MTJs as the mem- ory cells. Figure 2(b) shows an array of nanoring MTJs. The thickness of the MgO barrier is 0.6 nm to obtain a low resistance-area ( RA) product. An ultrathin Mg layer of 0.4 nm is inserted before deposition of the MgO barrier to improve the (001) texture of the MgO barrier and the interface between the (Co, Fe )B layers and the MgO layer [26]. Figure 2(c)shows a HRTEM image of the multilayer cross section after annealing. The interfaces between the (Co, Fe )B electrodes and the MgO barrier are sharp and the roughness of the barrier is low [ 27]. Figure 3shows the magnetic properties of the multi- layer thin ﬁlm, and R-Hmajor and minor loops of the nanoring MTJ. In Fig. 3(a), the M-H loop of the multilayer ﬁlm measured at room temperature (RT) shows two clear steps, indicating that the magnetization switching of the (a) (b) Bo/g425om(Co,Fe)BTop(Co,Fe)BCo, Fe)BTopHysteresis loop at RT Co, Fe)B CoFeBottom FIG. 3. Magnetic and magnetoelectric transport properties after annealing at 330 °C with a magnetic ﬁeld of 0.8 T. (a) M-Hloop of the multilayer ﬁlm measured with a VSM at RT. To show the magnetic-switching processes, three kinds of arrows represent the magnetization direction of the ferromagnetic ﬁlms: dashed arrows, top (Co, Fe )B layer; solid arrows, bottom (Co, Fe )B layer; dotted arrows, CoFe layer. (b) Major (black line) and minor (red line) R-Hloops of the pattered nanoring MgO-based MTJ. The arrows are only guides for the eye. The black arrows indicate the applied magnetic ﬁeld sweeps from positive to negative, while the blue arrows indicate the applied magnetic ﬁeld sweeps from negative to positive. The magnetic ﬁeld is applied in the plane of the multilayer ﬁlm and the ring. 054013-4MAGNETIC CONFIGURATIONS AND STATE DIAGRAM . . . PHYS. REV. APPLIED 10,054013 (2018) Free (Co, Fe)B Reference (Co, Fe)BMgO barrier(a) (b) (c)FIG. 4. (a) A nanoring memory cell with positive current from the free layer to the reference layer and magnetizationalignment of the two layers: parallel onion state with parallel resistance and antiparallel onion state with antipar-allel resistance. (b) Measurements of magnetic-ﬁeld-driven magnetization switching in a nanoring MTJ at RT. (c)Polarized-current-driven magnetization switching in a nanoring MTJ at RT from the measurements. Positive biascorresponds to electrons ﬂowing from the reference layer to the free layer and thus favors parallel alignment of the twolayers. TMR, tunnel magnetoresistance. free and reference (Co, Fe )B layers driven by the magnetic ﬁeld is clearly observed. The coercivity of the free layer is about 10 Oe, which is smaller than that obtained from the R-Hloop as shown in Fig. 3(b), probably due to the shape anisotropy or the suppression of domain-structure formation after nanostructure fabrication [ 24]. The switching properties of nanoring MTJs are shown in Figs. 4(b) and4(c). Each MTJ is measured by the four- probe method via leads connected to the top and bottom electrodes. Two types of methods are applied to measurethe magnetoelectric transport of the nanoring MTJ: the injection of direct current for spin-transfer torque and small alternating current for the I-Hphase diagram using the lock-in technique [ 28]. The positive bias corresponds to electrons ﬂowing from the reference layer to the free layer and thus favors parallel magnetization alignment of the two layers in Fig. 4(a). Figure 4(b) shows magnetic-ﬁeld- driven magnetization switching for the nanoring MTJ. A tunnel magnetoresistance ratio of about 100% with an RA product lower than 8 /Omega1µm 2is observed at RT. For the nanoring with no current injected, the reversal of the free layer occurs for a coercive ﬁeld of about 100 Oe, which is signiﬁcantly larger than the coercive ﬁeld (10 Oe) of the full ﬁlm. Figure 4(c) shows current-driven magnetiza- tion switching for a nanoring MTJ. The average switching current is deﬁned as Jc=(I+−I−)/2A, where I+and I− are the critical current for the antiparallel-to-parallel mag- netization switching and parallel-to-antiparallel switching, respectively, where Ais the MTJ cross-section area. In our samples, Jcof 1.3 ×106A/cm2is obtained. Owing to spin excitations at higher bias voltages, the antiparal- lel resistance ( RAP) decreases with increasing current I− [29]. The center of the R-Iloop is shifted about 0.5 mA, which likely results from the diﬀerent spin-accumulation eﬃciency with the diﬀerent current-ﬂowing directions aswell as the ferromagnetic coupling between the reference layer and the free layer. In contrast, the asymmetry of the R-Hloop is mainly caused by the ferromagnetic coupling. The impact of the injected spin-polarized currents on magnetic ﬁeld switching is also studied. The R-Hloops are measured for diﬀerent values of injected currents, as shown in Fig. 5(a). The orange curve (about zero current) shows a central hysteresis loop. The hystere- sis loops move to the left for relatively large positive currents, since positive currents favor the parallel con- ﬁguration. In contrast, for negative currents, which favor the antiparallel conﬁguration, the hysteresis loops move to the right. With an injected current of I=−0.15 mA, an abnormal switching-back phenomenon (green curve) is observed with parallel-to-antiparallel and antiparallel-to- parallel switching coexisting, as in nanopillar MgO-based MTJs [ 30]. For a large current I=−0.25 mA, the purple curve is even farther to the right, with its entire hystere- sis outside both the orange curve and the green curve. On the other hand, the R-Hcurves shift to the left for positive injected current. Similarly, the magnetic ﬁelds also have an impact on the current-driven magnetization switching, shown in Fig. 5(b). With ﬁelds decreasing from 200 to 50 Oe, the R-Iloop (blue curve) shifts to the right with a wider hysteresis. A negative ﬁeld makes the conﬁgu- ration favor the antiparallel state and prevents a positive current reversing the magnetization. When the ﬁelds are equal to −50 and −200 Oe, the R-Iloops (orange and purple curves) are further shifted to the right. A series of R-Hloops with diﬀerent injected currents (or a series of R-Iloops at various applied magnetic ﬁelds) allow us to plot the experimental state diagram. The dif- ference in resistance [ Rdiﬀ(H)=Rinc(H)-Rdec(H)] between the increasing and decreasing parts of the hysteresis loop can be extracted from the R-Hloops [ 28], from which a 054013-5HOUFANG LIU et al. PHYS. REV. APPLIED 10,054013 (2018) (a) (b) (c) (d) (e) (f) FIG. 5. (a) R-Hloops with a series of bias injected currents I=0.0025 mA, ±0.15 mA, and ±0.25 mA, respectively. (b) R-Iloops with a series of magnetic ﬁelds H=±50 Oe and ±200 Oe, respectively. Corresponding micromagnetic simulation curves based on the LLG equation for normalized magnetization ( mx) varied with the magnetic ﬁeld and spin-polarized current. (c) mx-Hloops with diﬀerent bias injected currents. For clarity, diﬀerently colored arrows denote the corresponding switching curves. (d) mx-Iloops with various bias magnetic ﬁelds. (e) A series of transient snapshots of free-layer magnetization conﬁgurations driven at H=−200 Oe and I=0.6 mA. (f) The I-Hstate diagram of the diﬀerence in resistance Rdiﬀ(H)=Rinc(H)-Rdec(H). The colored scale bar corresponds to the value of the diﬀerence in resistance Rdiﬀ. AP, antiparallel; P, parallel. 054013-6MAGNETIC CONFIGURATIONS AND STATE DIAGRAM . . . PHYS. REV. APPLIED 10,054013 (2018) (a) (c)(b) FIG. 6. (a) Micromagnetic simulation performed in the switching-parameter region, J=−6×106A/cm2(i.e., I=−0.39 mA) and H=0. Current-induced magnetization switching simulated with the in-plane Slonczewski STT, ﬁeldlike STT, and their combination. The inset shows the orientations of the in-plane Slonczewski STT (green arrow), the ﬁeld-like STT (blue arrow), and the dampingtorque (pink arrow). The in-plane Slonczewski STT drives the free layer to switch into the opposite direction, while the ﬁeldlike torque changes the frequency of magnetization precession. (b) Current-driven magnetization oscillations. Normalized magnetization vector (m:m x,my,a n d mz) as a function of time for magnetization precession with J=4×106A/cm2and H=−200 Oe. (c) The snapshots of the magnetic conﬁguration during the magnetization switching of the free layer in nanoring MTJs. region of dynamic magnetization states driven by the ﬁeld or current can be highlighted. Therefore, the state diagram is built as a two-dimensional colored ( I,H) map, where each point corresponds to a speciﬁc pair of the current and ﬁeld values. The color of the ( I,H) map represents the value of Rdiﬀat these coordinates. Figure 5(f)shows the state diagram of Rdiﬀas a function of the magnetic ﬁeld Hand current I, which can be mainly divided into three regions: the parallel, antiparallel, and bistable states. The bistable-state region corresponds to the area where the MTJ can be in either a parallel state or an antiparallel state depending on the history of the device. More interesting, in the zone with H∼−250 Oe and 0.1 mA <I<0.25 mA, an area can be observed with Rdiﬀ/negationslash=0, which has been identiﬁed as current-induced steady-state magnetizationprecession and spin-transfer nano-oscillators [ 28]. This feature is discussed further in Fig. 6(b). C. Micromagnetic simulation for the state diagram of nanoring MTJs To gain insight into the observed state diagram, micro- magnetic simulations based on the LLG equation includ- ing the current-induced STT eﬀect are performed for the nanoring MTJs. We show the spatiotemporal dynamics of the free layer driven by the combined eﬀect of theSTT and an applied magnetic ﬁeld. In our simulations, both the current-induced Slonczewski STT and the ﬁeld- like STT are considered. The simulated curves indicate the xcomponent of magnetization ( m x) varies with the 054013-7HOUFANG LIU et al. PHYS. REV. APPLIED 10,054013 (2018) magnetic ﬁelds and injected currents as shown in Figs. 5(c) and5(d), respectively, where the simulations are at each ﬁeld or current step run for 10 ns. Similarly to the exper- imental results shown in Figs. 5(a) and5(b), shifts of the mx-Hand mx-Iloops are also observed with a series of bias currents and magnetic ﬁelds. A central hysteresis loop at zero current is observed (orange curve) in Fig. 5(c). As the negative bias current increases, the switching loop starts to move right. This is caused by the negative-current- induced STT eﬀect, which drives the free layer to be the antiparallel state. For a large current (e.g., −1 mA), the magnetization-switching curve exhibits strong telegraph- noise-like ﬂuctuation behavior. This telegraphlike noise is obtained at zero temperature, rather than excited by eﬀec- tive temperature. This cannot be explained by the coherent spin-torque model, because the dwell time in one state at a certain current Iand magnetic ﬁeld His proportional to exp[ E(I,H)/kBT] [31], where E(I,H) represents the energy barrier corresponding to available magnetic states. According to the results of Lee et al.[32], this phenomenon can be ascribed to the spatial inhomogeneous magne- tization dynamics, which could generate the incoherent local precession frequency and spin-wave excitation. A series of transient snapshots taken from the magnetization- evolution process for I=0 . 6m Aa n d H=−200 Oe [cor- responding to the purple curve in Fig. 5(d)] are given in Fig. 5(e), showing the typical spatially inhomogeneous magnetic conﬁguration dynamics. The simulation starts from an antiparallel state ( mx=+1) at H=−200 Oe. When a current of 0.6 mA is applied, the current-induced STT eﬀect tries to drive the free layer to the parallel state ( mx=−1). The competition between the magnetic ﬁeld and the STT eﬀect, together with the nonuniform local demagnetizing ﬁeld, results in chaotic inhomoge- neous magnetic conﬁgurations. Consequently, ﬂuctuation magnetization of mxis observed in Fig. 5(d)(purple curve). In contrast, the telegraphlike noise is also observed in the case of large negative currents and positive magnetic ﬁeld bias [e.g., 200-Oe curve in Fig. 5(d)]. In this case, the current-induced STT tries to drive the free layer to the antiparallel state ( mx=+1) and the bias ﬁeld tries to drive it to the parallel state ( mx=−1). Similarly, the M-H curves in Fig. 5(c)shift to the left for positive bias current and to the right with negative bias current. Figure 6(a) shows the simulated current-driven mag- netization switching in a 100-nm nanoring, in which we compare the combined STT eﬀect (black curve) with the individual eﬀect of the Slonczewski STT (red curve) and the ﬁeldlike STT (blue curve). The simulations start with an initial parallel state (i.e. mx=−1) and are performed at a current density of J=−6×106A/cm2and H=0O e . The results clearly indicate that the ﬁeldlike STT alone cannot cause magnetization switching. In contrast, the Slonczewski STT alone as well as the combined STT can reproduce the magnetization switching to the antiparallelstate ( mx=+1). Additionally, the latter switches a little faster. These simulation results can be easily explained by a torque model as illustrated in the inset in Fig. 6(a).T h e Slonczewski STT can overcome the damping torque and drive the free layer in the opposite direction, while the ﬁeldlike STT speeds up the magnetization rotating around the eﬀective ﬁeld. Obviously, the ﬁeldlike STT alone can- not switch the magnetization, as conﬁrmed by the simula- tion results. The current-induced magnetization switching is mainly determined by the competition between the Slonczewski and Gilbert damping torque. In contrast to metallic or giant-magnetoresistance structures [ 33,34], an enhanced ﬁeldlike STT with typically 10%–30% of the amplitude of the in-plane Slonczewski spin torque has been reported in MTJs [ 35,36]. As observed in Fig. 6(a) the ﬁeldlike STT can change the frequency of the mag- netization’s precession motion. Figure 6(c) shows a series of transient snapshots taken during the magnetization evo- lution in the case of both the in-plane Slonczewski STT and the ﬁeldlike STT. The free layer starts from an onion conﬁguration, which is a parallel magnetization-alignment state with respective to the reference layer. After the cur- rent is applied, the current-induced STT eﬀect breaks the equilibrium, leading to the local moment departure from the −xdirection. The magnetization misalignment between the reference layer and the free layer will in turn locally enhance the spin-transfer torque [ 3]. Thus, two magnetic domains with opposite direction (along the +xaxis) nucleate at t=6.65 ns. After that, the reversed domain expands toward the neighboring parts through the exchange coupling and the STT eﬀect, resulting in more magnetic moments switching to the +xdirection. Finally, the free layer is fully switched to the antiparallel state at t=7.10 ns. To obtain a better understanding of the magnetization- oscillation state experimentally observed in the top-left and bottom-right precession regions of the state diagram in Fig. 5(f), we further perform a series of simulations. Figure 6(b) shows a typical magnetization precession of a free layer excited at H=−200 Oe and J=4×106A/cm2 (i.e., 0.26 mA). In this simulation, the initial magnetization starts from an antiparallel alignment (onion conﬁguration) with<mx>≈1. The STT generated by the positive current tries to drive the free-layer magnetization to the parallel state, while the applied negative magnetic ﬁeld prevents this rotation. The competition between those terms leads to a state of dynamical equilibrium, where the magneti- zation undergoes steady-state precession along an approx- imately elliptic trajectory. The calculated FFT spectrum of the magnetization precession exhibits a microwave fre- quency of 3.7 GHz. With the magnetic ﬁeld amplitudeincreasing from −300 to −400 Oe, the oscillation fre- quency can be tuned from 4.2 to 5.7 GHz. Similarly, the oscillation frequency can be tuned by the variation of the current. 054013-8MAGNETIC CONFIGURATIONS AND STATE DIAGRAM . . . PHYS. REV. APPLIED 10,054013 (2018) IV . CONCLUSION In summary, we theoretically predict the scalability of the static phase diagram of ring-shaped magnetic tunnel junctions with outer diameter of less than 20 nm. Our results show that with appropriate adjustment of the nanor- ing shape parameters, the onion conﬁguration is the ener- getically stablest state. The properties of current-driven magnetization switching in a (Co, Fe )B/MgO/(Co, Fe )B nanoring MTJ are experimentally reported. A switching- current density as low as 1.3 ×106A/cm2in a nanoring MTJ with an outer diameter of 100 nm and narrow ring width between 25 and 30 nm is experimentally demon- strated. The dynamic resistance state diagram as a function ofI-Hfor the nanoring MTJ is explored, and the ﬁnd- ings are supported by micromagnetic simulations. The results provide insights for the further understanding of the magnetization-switching mechanism in nanoring MTJs, oﬀering a design for MRAM with high density and low- power consumption. ACKNOWLEDGMENTS The project was supported by the National Key R&D Program Project of the Ministry of Science and Technol- ogy (MOST; Grant No. 2017YFA0206200), the MOST National Key Scientiﬁc Instrument and Equipment Devel- opment Projects (Grant No. 2011YQ120053), the NationalNatural Science Foundation of China (NSFC; Grants No. 11434014, No. 51229101, No. 11674373, No. 51501098, and No. 51620105004), the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB07030200), and an international collaborative research program between NSFC and the French Agence Nationale de la Recherche (ANR; Grant No. F040803). Y.W.L. thanks NSFC for support (Grants No. 11774260 and No. 51471118). Y.L. acknowledges the joint ANR- NSFC SISTER project (Grants No. ANR-11-IS10-0001 and NNSFC 61161130527). We thank Professor Zhong- ming Zeng for discussions. [1] M. Stoppa and A. Chiolerio, Wearable electronics and smart textiles: A critical review, Sensors 14, 11957 (2014). 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PhysRevApplied.12.064053.pdf	PHYSICAL REVIEW APPLIED 12,064053 (2019) Editors’ Suggestion Skyrmion Logic System for Large-Scale Reversible Computation Maverick Chauwin ,1,2,†Xuan Hu ,1,†Felipe Garcia-Sanchez ,3,4Neilesh Betrabet,1 Alexandru Paler,5Christoforos Moutaﬁs ,6and Joseph S. Friedman1,* 1Department of Electrical & Computer Engineering, The University of Texas at Dallas, Richardson, Texas 75080, USA 2Department of Physics, École Polytechnique, 91128 Palaiseau, France 3Istituto Nazionale di Ricerca Metrologica, 10135 Torino, Italy 4Departamento de Física Aplicada, Universidad de Salamanca, 37008 Salamanca, Spain 5University of Transilvania, Brasov 500091, Romania 6Department of Computer Science, The University of Manchester, Manchester M13 9PL, United Kingdom (Received 21 May 2019; revised manuscript received 8 October 2019; published 24 December 2019) Computational reversibility is necessary for quantum computation and inspires the development of computing systems, in which information carriers are conserved as they ﬂow through a circuit. While conservative logic provides an exciting vision for reversible computing with no energy dissipation, the large dimensions of information carriers in previous realizations detract from the system eﬃciency, andnanoscale conservative logic remains elusive. We therefore propose a nonvolatile reversible computing system in which the information carriers are magnetic skyrmions, topologically-stable magnetic whirls. These nanoscale quasiparticles interact with one another via the spin Hall and skyrmion Hall eﬀects asthey propagate through ferromagnetic nanowires structured to form cascaded conservative logic gates. These logic gates can be directly cascaded in large-scale systems that perform complex logic functions, with signal integrity provided by clocked synchronization structures. The feasibility of the proposed sys-tem is demonstrated through micromagnetic simulations of Boolean logic gates, a Fredkin gate, and a cascaded full adder. As skyrmions can be transported in a pipelined and nonvolatile manner at room tem- perature without the motion of any physical particles, this skyrmion logic system has the potential todeliver scalable high-speed low-power reversible Boolean and quantum computing. DOI: 10.1103/PhysRevApplied.12.064053 I. INTRODUCTION There is a fundamental minimum quantity of energy dissipated by a logic gate in which information-carrying signals are continuously created and destroyed, as deter- mined by Landauer [ 1]. Reversible computing aims to circumvent this limitation by conserving information - and therefore energy - as signals propagate through a logic cir- cuit [ 2]. In this scheme, conservative logical operations are executed through dissipation-free elastic interactions among these information carriers that conserve momen- tum and energy [ 3]. While Fredkin and Toﬀoli’s original thought experiment considered billiard balls as the infor- mation carriers, Prakash et al. recently demonstrated con- servative logic experimentally with micron-sized droplets driven through planar computing structures by pressure [ 4] and magnetism [ 5]. However, the large dimensions of the information carriers in these demonstrations detract *joseph.friedman@utdallas.edu †These authors contributed equally to this work.from the system eﬃciency and limit potential utility; a nanoscale information carrier for reversible computing remains elusive. Magnetic skyrmions [ 6] are intriguing information car- riers for reversible computing due to their small diameter (∼50 nm at room temperature, below 20 nm at low tem- perature) [ 7–10] and the small current required to induce skyrmion motion [ 11]. These quasiparticles are topolog- ically stable regions of magnetization that comprise a central core oriented antiparallel to the bulk of a mag- netic structure [ 12–14]. Skyrmion motion involves the propagation of magnetization rather than the transport of physical particles, and can be induced by the spin Hall eﬀect through the application of an electrical current [ 15]. These skyrmion quasiparticles move not along the axis of an electrical current, but rather deviate from this axis due to the skyrmion Hall eﬀect, which is equivalent to the Mag- nus eﬀect [ 16,17]. The skyrmion Hall eﬀect is generally deleterious to device functionality, so a track structure, as shown in Fig. 1, is often used to suppress the skyrmion Hall eﬀect and restrict motion to a single dimension 2331-7019/19/12(6)/064053(10) 064053-1 © 2019 American Physical SocietyMAVERICK CHAUWIN et al. PHYS. REV. APPLIED 12,064053 (2019) FIG. 1. Skyrmion track structure. Skyrmions propagate along a track comprised of Pt heavy metal (blue) and Co ferromag- net (gray, with polarization P), where an interfacial spin-orbit coupling induces a Dzyaloshinskii-Moriya interaction. This spin-orbit coupling also causes the externally-applied electrical cur- rent ( J) ﬂowing through the heavy metal in the +ydirection to create a spin current ( J S) polarized in the +zdirection via the spin Hall eﬀect. The skyrmion (multicolor circle) lies in the fer- romagnetic layer at the interface with the heavy metal, and the surrounding ferromagnet walls prevent the skyrmion from leav-ing the track. Spin current in the +zdirection produces a force F SHon the skyrmions in the +ydirection, which is the direction of the electrical current. The track constriction prevents the – x- directed skyrmion Hall force FSkHfrom inﬂuencing the skyrmion trajectory. Note that the axes are inverted for visual clarity; the Pt heavy metal is below the Co ferromagnet. [18–21]. Magnetic skyrmions propagating along ferromag- netic nanowire tracks [ 22,23] have been proposed for memory storage and individual logic gates [ 24–29], but the development of a scalable skyrmion computing sys- tem has been impeded by the need to directly cascade skyrmion logic gates without control and ampliﬁcation cir- cuitry that signiﬁcantly reduces the system eﬃciency. Fur- thermore, previous skyrmion logic proposals require the continual creation and annihilation of skyrmions, which is an energetically expensive process that requires an external control system. We therefore propose a reversible skyrmion logic sys- tem in which skyrmions are conserved as they ﬂow through nanowire tracks. Logical operations are performed by thor- oughly leveraging the rich physics of magnetic skyrmions: the spin Hall eﬀect [ 15], the skyrmion Hall eﬀect [ 16,17, 30], skyrmion-skyrmion repulsion [ 31], repulsion between skyrmions and the track boundaries [ 31], and electrical current control of notch depinning [ 11]. Binary informa- tion is encoded by the presence (“1”) or absence (“0”) of magnetic skyrmions, with the skyrmions ﬂowing directlyfrom the output nanowire track of one logic gate to the input track of another logic gate without an external con- trol or ampliﬁcation circuit. These reversible skyrmion logic gates can provide FAN-OUT and be integrated intoa large-scale system, with signal integrity provided by simple electronic clock pulses applied to the entirety of the system. This logic-in-memory computing system is nonvolatile due to the topological stability and ferromag- netic nature of skyrmions, providing eﬃcient pipelining that enhances the potential for high speed and low power. Furthermore, the availability of a Fredkin gate inspires the consideration of quantum computing with magnetic skyrmions. II. MICROMAGNETIC SIMULATION METHODOLOGY A. Micromagnetic simulation technique The Landau-Lifshitz-Gilbert (LLG) equation of motion describes magnetization dynamics in ferromagnetic mate- rials: ∂M ∂t=−γ(M×Heﬀ)+α Ms/parenleftbigg M×∂M ∂t/parenrightbigg +τCPP, (1) where Mis the magnetization vector, γis the gyro- magnetic ratio, Msis the saturation magnetization, and αis the Gilbert damping parameter. Heﬀis the eﬀective ﬁeld, which includes Heisenberg exchange, magneto- crystalline anisotropy, magnetostatic, Dzyaloshinskii- Moriya exchange, and external magnetic ﬁelds. τCPP implements the injection of spin Hall current perpendic- ularly to the sample and is described by τCPP=−β/epsilon1/prime(M×mP)−β Ms[M×(mP×M)], (2) with mPas the spin Hall polarization direction and β= (θSH/planckover2pi1J/2MsetCo,track), where θSHis the spin Hall angle, eis the electronic charge, Jis the electrical current density, and tCo,track is the thickness of the Co track. /epsilon1/primeis the ﬁeldlike torque, which is here considered to be zero; modiﬁcations to the skyrmion proﬁle by that homogeneous in-plane ﬁeld are not considered. Simulations are therefore performed using MUMAX3 ,a n open-source GPU-accelerated micromagnetic simulation software [ 32] that integrates the LLG equation of motion with a ﬁnite diﬀerence approach. We discretize the sam- ple into cuboid cells, the dimensions of which are set to 1×1×0.4 nm3, and we neglect thermal ﬂuctuations by setting the temperature to 0 K. The zero-temperature sim- ulation results are predictive of room-temperature exper- imental behavior of the proposed computing system, in which a randomly ﬂuctuating thermal ﬁeld will impact theskyrmion motion; as described in Sec. VA, notch synchro- nizers are used to prevent this random motion from causing logical errors. Furthermore, these phenomena have been suppressed experimentally in multilayer structures [ 33], 064053-2SKYRMION LOGIC SYSTEM FOR. . . PHYS. REV. APPLIED 12,064053 (2019) and the eﬀect of this thermal ﬁeld will be small relative to the applied electrical current. B. Magnetic parameter selection The following numerical values, which are adopted from Purnama et al. [18], model a multilayer of Pt and Co: saturation magnetization Ms=5.80×105A/m, exchange stiﬀness Aex=1.5×10−11J/m, Gilbert damp- ing coeﬃcient α=0.1, Dzyaloshinskii-Moriya interac- tion (DMI) constant Dint=3.0×10−3J/m2, magnetocrys- talline anisotropy constants Ku1=6×105J/m3and Ku2= 1.5×105J/m3, and spin polarization in the transverse direction mP=(1, 0, 0). The anisotropy direction points upwards. The spin Hall angle θSHis considered to be equal to one. The thickness of the Pt layer is tPt=0.4 nm and the thickness of the Co layer varies between tCo,track = 0.4 nm (Fig. 1)a n d tCo,sample =0.8 nm elsewhere. As the skyrmions travel in lithographically-feasible 20-nm- wide nanowire tracks [ 34], they are conﬁned to slightly smaller dimensions in a manner similar to the conﬁnement shown in Ref. [ 9], where geometric conﬁnement results in skyrmions with diameters of roughly 50 nm. III. REVERSIBLE SKYRMION LOGIC GATES A. Reversible skyrmion AND/OR gate The reversible AND/ORlogic function described in Table Iis the primary workhorse of the proposed skyrmion computing system. As this function is reversible, the total number of skyrmions, N, provided to inputs Aand Bis always equal to the total number of skyrmions emitted by the AND and ORoutputs. This reversible logic func- tion is performed by the structure shown in Fig. 2, with micromagnetic simulations [ 32] depicting the skyrmion trajectories for each input combination (see the Supple- mental Material [ 35]). The spin Hall eﬀect pushes the skyrmions in the +ydirection through the vertical tracks of Fig. 1, while the skyrmion Hall eﬀect introduces a –x-directed force that is mediated by repulsion from the track boundaries. The skyrmions are therefore free to move laterally within the central junction, where the skyrmion Hall eﬀect causes leftward skyrmion propagation unless repulsed by a second skyrmion. Whenever a skyrmion enters one of the input ports of the AND/ORgate, this logic gate geometry forces a skyrmion TABLE I. Truth table for the AND/ORgate. Inputs Outputs ABN AND OR 000 0 0 011 0 1 101 0 1112 1 1 (a) (b) (c) A = 0 B = 1 A = 1 B = 0 A = 1 B = 1 FIG. 2. Micromagnetic simulation results for the AND/ORgate are shown for input combinations (a) A=0,B=1; (b) A=1, B=0; and (c) A=B=1. In this and other ﬁgures, the path of each skyrmion is shown as a grayscale line that is black at t=0 and gradually lightens as the simulation advances, with skyrmion snapshots provided at the times noted in the ﬁgure. The spin current JSresulting from constant electrical current J=5×1010A/m2pushes the skyrmions in the +ydirection, with a skyrmion Hall force directed in the – xdirection. There- fore, when conﬁned laterally by their tracks, the input skyrmions travel directly in the +ydirection until they reach the central junction. In the lateral opening of the constrictive tracks at the central junction, the skyrmion Hall force induces a – x-directed component to the skyrmion trajectory that is counteracted by skyrmion-skyrmion repulsion. to propagate to the ORoutput port to represent binary “1.” If two skyrmions are input to this logic gate, one skyrmion is emitted by the ORoutput port and the other skyrmion is emitted by the AND output port, such that both produce binary “1.” Finally, if no skyrmions enter either input port, then no skyrmions are emitted by either output port, representing binary “0” outputs. The opera- tion of the AND/ORgate is shown in Video 1for these VIDEO 1. Micromagnetic simulation of AND/ORlogic gate with input combinations (a) A=0,B=1; (b) A=1,B=0; and (c)A=B=1. This video is analogous to Fig. 2. 064053-3MAVERICK CHAUWIN et al. PHYS. REV. APPLIED 12,064053 (2019) TABLE II. Truth table for the INV/COPY gate. Inputs Outputs CTRL IN N COPY1 NOT COPY2 10 1 0 1 0 11 2 1 0 1 three input combinations. The combined forces result- ing from the spin Hall eﬀect, the skyrmion Hall eﬀect, skyrmion-skyrmion repulsion, and repulsion between the skyrmions and the boundaries thus cause this structure to simultaneously calculate the logical functions A∨Band A∧Bwhile conserving the skyrmions. B. Reversible skyrmion INV/COPY gate The inclusion of an inversion operation enables the generation of all possible Boolean logic functions, which cannot be achieved by the AND and ORoperations alone. The proposed INV/COPY gate shown in Table II, Fig. 3, and Video 2functions similarly to the AND/ORgate shown above, here with an additional output port and the require- ment that a skyrmion always be provided to the control (CTRL ) input. (see Fig. S1, Table S1, and Video S1 within the Supplemental Material for the behavior when CTRL is “0” [ 35].) This reversible INV/COPY gate simultaneously duplicates and inverts the skyrmion input signal. As shown in Table II, the NOT output is therefore “1” whenever the INinput is “0,” and “0” whenever the INinput is “1.” This reversible logic gate also performs the FAN-OUT function, through which skyrmions are conserved such that the IN signal is duplicated to the two COPY outputs. This sig- nal duplication is an essential component of a large-scale (a) (b) FIG. 3. Micromagnetic simulation results for the INV/COPY gate are shown for input combinations (a) IN=1,CTRL=1; and (b) IN=0;CTRL=1. VIDEO 2. Micromagnetic simulation of INV/COPY logic gate forIN=1a n d IN=0. In order to provide the inversion and dupli- cation behaviors, CTRL=1 in both simulations. This video is analogous to Fig. 3. computing system, and can be performed repeatedly by cascaded INV/COPY gates to generate numerous copies of a signal. IV. POTENTIAL FOR SKYRMION QUANTUM COMPUTATION A magnetic skyrmion quantum computing system can be envisioned in which quantum non-binary operators operate in concert with the binary operations of a skyrmion Fredkin gate. In particular, the availability of atomic- scale skyrmions [ 7,36] and the fact that skyrmions cou- pled to conventional superconductors support Majorana fermions [ 37] open a pathway to topological quantum computation [ 38,39]. Building on our proposed skyrmion Fredkin gate and various potential physical solutions for qubit encoding and manipulation, we propose two quan- tum gate sets for a universal skyrmion quantum computing system: one based on the conventional combination of Tof- foli and Hadamard gates, and the other based on Cliﬀord and T gates. A. Skyrmion Fredkin gate In addition to reversible Boolean computation, this skyrmion logic system has potential applications in quan- tum computation. As single-spin states are promising can- didates for qubits, the recent experimental demonstration of atomic-scale magnetic skyrmions provides a potential pathway to quantum computing with skyrmion qubits [ 7, 36]. We therefore propose in Fig. 4, Video 3, and Table III the reversible Fredkin gate implemented in this skyrmion logic paradigm. Here, skyrmions provided to the Cinput propagate to the Coutput and determine whether or not the I1and I2input signals are swapped as they travel to the O1 and O2outputs (see also Fig. S2 within the Supplemen- tal Material [ 35]). A controlled- NOT (CNOT ) gate can also 064053-4SKYRMION LOGIC SYSTEM FOR. . . PHYS. REV. APPLIED 12,064053 (2019) (a) (b) FIG. 4. Micromagnetic simulation results for the Fredkin gate. (a) When C=0, the input signals travel directly to the output ports without swapping; that is, O1=I1and O2=I2.( b )W h e n C=1, the input skyrmions swap paths such that O1=I2and O2=I1.T h e Csignal is duplicated, with both skyrmions at the Cinput propagating directly to the Coutput. be achieved, and is implicitly included in the full adder described in Sec. VB [Fig. 6]. B. Toﬀoli +Hadamard universal quantum gate set The Fredkin gate is an intermediary between classi- cal Boolean logic and full/universal quantum logic, andthree Fredkin gates can be cascaded to realize a Toﬀoli gate. As a Toﬀoli gate in concert with a Hadamard gate are quantum computationally universal, the only missing component required for quantum computing within this skyrmion logic system is the Hadamard gate. The avail- ability of multiple types of skyrmions (antiferromagnetic, etc.) opens a pathway to achieve a skyrmion Hadamard VIDEO 3. Micromagnetic simulation of Fredkin gate for all nontrivial input combinations. This video is analogous to Fig. 4 and Fig. S2.TABLE III. Truth table for the Fredkin gate. Inputs Outputs CI 1 I2 NC O 1 O2 000000 0 001100 1010101 0 011201 1 100210 0101311 0 110310 1 111411 1 gate analogous to an optical beam splitter with half prob- ability. Here, when a “gate skyrmion” interacts with a “qubit skyrmion,” the “gate skyrmion” changes the state of the “qubit skyrmion.” The “gate skyrmion” repre- sents the unitary transformation of the quantum Hadamard gate, creating an equal superposition between two possi- ble positions of a particle (the “qubit skyrmion”). “Qubit skyrmions” may be left initialized in “0” or “1” states, which are represented by the absence or presence of a skyrmion, respectively, as described throughout the text. This Hadamard gate thus provides the necessary quantumstate superposition. C. Cliﬀord +T universal quantum gate set Given their high number of spins and intrinsic topologi- cal properties, skyrmions are intrinsically quantum-error- corrected logical qubits. It may therefore be feasible to consider the quantum universal Cliﬀord +T gate-set [ 40], in which all skyrmions are qubit carriers. The single qubit T gate can act on individual skyrmions (qubits), while the Cliﬀord gates (eﬃcient to simulate on a classical com- puter) are implemented by interacting the skyrmions [ 37]. The most observed and investigated skyrmions are two- dimensional, which suggests that these Cliﬀord gates may be implemented by braiding the movement trajectories of the skyrmions in a manner similar to that of conventional surface quantum-error-correcting code [ 41]. V. LARGE-SCALE INTEGRATED SKYRMION COMPUTING SYSTEM While these reversible logic gates are interesting in their own right, a mechanism for cascading logic gates is neces- sary for practical computing applications. In the proposed reversible computing system, the output skyrmions emit- ted by one logic gate are used as input skyrmions for another gate. As the logic gate functionality is based onskyrmion interactions at the central junctions, a synchro- nization mechanism must also be provided to ensure that skyrmions arriving from diﬀerent input paths reach the central junction simultaneously, despite thermal eﬀects. 064053-5MAVERICK CHAUWIN et al. PHYS. REV. APPLIED 12,064053 (2019) A. Notch synchronization with global clock This synchronization is achieved with the notch struc- ture [ 11] of Fig. 5(b) by applying a large spin Hall current pulse that enables the skyrmions to traverse the notch only when this large pulse is applied. This large current pulse causes a decrease in skyrmion diameter, while also increasing the skyrmion velocity, as shown in Fig. 5(a).A s depicted in Fig. 5(c)and Video 4, a small spin Hall current is continuously applied to the entire system to propagate the skyrmions through the tracks and logic gate junctions; this current magnitude is below the threshold required for skyrmions to traverse the notches. At regular intervals, a large spin Hall current pulse is provided to the entire sys- tem to drive the skyrmions past the notches; this regular pulse represents the global system clock that synchronizes the computing system. These notch synchronizers can be placed following the output of a logic gate, with the INport of the notch synchronizer connected to an output port of a logic gate; the OUT port of the notch synchronizer is con- nected to an input port of a cascaded logic gate. Notches are inserted between every logic gate input and output where synchronization is required, with each notch syn- chronizer handling zero or one skyrmion during each clock cycle. Alternatively, this synchronization can be similarly (a) (b) (c) OUT IN FIG. 5. Signal synchronization. (a) Skyrmion radius as a func- tion of applied electrical current density. (b) A 7-nm-wide notch is formed in the 20-nm-wide nanowire track to create a constric- tion that permits skyrmion passage only when a large currentis applied. (c) The electrical current applied to the entirety of the computing system maintains a constant low magnitude of J=5×10 10A/m2, which is periodically ampliﬁed to J=2× 1011A/m2for 150 ps to enable skyrmions to traverse notches throughout the system. The skyrmion traverses the notch when this large clock pulse is applied at t=1n s . OUT IN 2 VIDEO 4. Micromagnetic simulation of notch structure that synchronizes the skyrmions ﬂowing through various portions of the circuit. This video is analogous to Fig. 5. achieved through clocked electrical control of the magnetic anisotropy [ 42]. B. Directly-cascaded skyrmion logic gates Integrating the basis logic gates with the cascading and synchronization mechanisms enables the scaling of this reversible computing paradigm to large systems that eﬃciently perform complex functions. An example is pro- vided in Fig. 6and Video 5, where the input A,B, and carry-in skyrmion signals interact as they propagate (a) (b) FIG. 6. Cascaded one-bit full adder. (a) Cascaded logic gates are synchronized by large 150-ps-wide electrical pulses applied to the entire structure with a clock period of 5 ns. (b) A one-bitfull adder computes the binary SUM and carry-out of two one- bit binary numbers, Aand B, and a carry-in bit. The notches ensure that the skyrmions are synchronized with one another asthey enter each logic gate, thereby providing proper skyrmion- skyrmion repulsion and logical functionality. After two (three) clock cycles, the skyrmion signals reach the C out(SUM) output. 064053-6SKYRMION LOGIC SYSTEM FOR. . . PHYS. REV. APPLIED 12,064053 (2019) VIDEO 5. Micromagnetic simulation of one-bit full adder for all nontrivial input combinations. This video is analogous to Fig.6and Fig. S4. through the circuit to produce the sum and carry-out skyrmion signals, thus executing the one-bit full-addition function with two half adders (half adder simulations are found in Fig. S3 and Video S2, the full adder truth table is given in Table S2, and simulation results for all full adder input combinations are found in Fig. S4 within the Supplemental Material [ 35]). A 150-ps-wide clock pulse is provided every 5 ns to synchronize the skyrmions, ensuring proper conservative logic interactions within each component logic gate. The SUM output is produced within three clock cycles, while the carry-out output is produced within two clock cycles; the carry-in to carry-out delay is only one clock cycle. These clocked skyrmion signals pro- vide a natural means for pipelining, enabling the execution ofn-bit addition within n+2 clock cycles; this pipelining procedure can be observed in Fig. 7and Video 6.F u r - thermore, although the 200 MHz clock frequency and the electrical current magnitudes used in simulation provide inferior eﬃciency to that of conventional computing sys- tems, the nonvolatility and pipelining inspire a vision for highly-eﬃcient computing with alternative materials [ 43] and improvements in the Rashba coeﬃcient and spin Hall angle [ 44]. VI. EXPERIMENTAL APPROACHES The skyrmion logic system proposed here is experimen- tally feasible with industrially relevant magnetic multilay- ers of the required material systems at lithographically- accessible length scales. Furthermore, room-temperature ﬁeld-free operation is achievable, as nanoscale chiral skyrmions were recently stabilized at room temperature with zero magnetic ﬁeld [ 33,45,46]. Similar to the mate- rial parameters used in the above micromagnetic simula- tions, a Co-based multilayer (Ir/Pt/Co)stack, grown by sputtering, has been proposed that should enable room-temperature sub-100-nm skyrmions stabilized by a large additive DMI [ 33]. By further tuning material parame- ters, such as the perpendicular anisotropy, the thickness of the magnetic layers, or the DMI, even smaller skyrmion (a) (b) (c) FIG. 7. Pipelined full adder. The clocked full adder structure can be used for pipelined operations such that each logic gate junction is used within each clock cycle. (a) The clocking scheme used in the full adder micromagnetic simulation is extendedfor ﬁve cycles to perform three separate full adder operations simultaneously. (b) The initial state ( t=0 ns) of a three-stage pipeline is shown here for the following inputs: in stage one,A=B=C in=1; in stage two, A=1a n d B=Cin=0; and in stage three, A=0a n d B=Cin=1. (c) The ﬁnal state ( t=24.8 ns) is seen after ﬁve clock cycles, with the following results: for stageone, SUM=Cout=1; for stage two, SUM=1a n d Cout=0; and for stage three, SUM=0a n d Cout=1. The entirety of the simulation is shown in Video 6. dimensions can be achieved [ 11]. Full-ﬁeld x-ray magnetic imaging techniques can be used to demonstrate the device functionality and logic operations, including full-ﬁeld magnetic transmission soft-x-ray microscopy (MTXM) or photoemission electron microscopy (PEEM), which is a VIDEO 6. Micromagnetic simulation of pipelined full adder demonstrating the ability to provide a throughput of one full addi-tion per clock cycle despite the two- and three-cycle latencies to produce the carry-out and SUM outputs, respectively. This video is analogous to Fig. 7. 064053-7MAVERICK CHAUWIN et al. PHYS. REV. APPLIED 12,064053 (2019) full-ﬁeld surface-sensitive x-ray imaging technique. X-ray imaging is magnetization-sensitive and nondisruptive to the sample. For the proposed system in particular, imaging of the magnetic states of the devices must utilize the x-ray magnetic circular dichroism (XMCD) eﬀect and tune the incoming photon energy to the Co L3edge (∼779 eV). A. Skyrmion generation In a fully reversible skyrmion logic system, skyrmions must only be generated once: at the beginning of the system operation. After these initial skyrmions are gen- erated, they are continually propagated through the logic gates such that the output skyrmions of each conservative skyrmion logic gate are used as the input skyrmions of other conservative skyrmion logic gates (see Sec. VI C ). These skyrmions generated at system initialization are therefore suﬃcient for long-term use of this system, and their nonvolatility enables the skyrmions to maintain their states even when the power supply is removed. As skyrmion generation is signiﬁcantly more challeng- ing than the generation of many other types of informa- tion carriers, this ability to conserve skyrmions marks a major advance in comparison to previous proposals for skyrmion logic [ 24–29]. As this reversible computing sys- tem requires only a single skyrmion initialization process, the precise technique used has limited impact on the over- all system eﬃciency. Several techniques are available, with the application of nanosecond electrical current pulses [ 8] appearing particularly promising. Alternatively, ease-of-demonstration as well as opti- mization of the conventional metrics of speed, power, and delay may inspire a compromise regarding the con- servation of skyrmions. Another approach to skyrmion generation is to continually generate skyrmions at spe- ciﬁc points within the system with a dedicated skyrmion generation structure. This can be achieved with homoge- neous currents [ 8] or with nanosecond electrical current pulses, as achieved in Ref. [ 47] for a device-compatible stripline geometry. This skyrmion generator will generate a skyrmion in each clock cycle, which then can be pro- vided as an input to a particular logic gate where an input skyrmion is always required (e.g., for the CTRL input of the INV/COPY gate). These skyrmions will then travel through the circuit and eventually be annihilated after they have performed the desired operations. While this alternative system will no longer be fully reversible, the simplicity of this approach may be advantageous for proof-of-concept experiments and eventual high-performance computing systems. B. Skyrmion detection To read the binary outputs of this conservative logic system, it is necessary to detect the presence (binary “1”) or absence (binary “0”) of skyrmions at variouspoints throughout the circuit. While standard magnetic force microscopy imaging can be used to detect skyrmions in a laboratory setting, computing applications require transformation of the skyrmion information into electri- cal signals. Determination of the presence or absence of a skyrmion at a particular location can be achieved by plac- ing a tunneling barrier and hard ferromagnet above the Co ferromagnet to form a magnetic tunnel junction (MTJ), as described in previous logic and memory proposals with domain walls and skyrmions [ 25,27,29]. The magnetore- sistance of this MTJ indicates the presence or absence of a skyrmion within the free layer, as the skyrmion modiﬁes the local magnetization within the free layer and therefore the current through the MTJ tunneling barrier. C. Skyrmion conservation While each of the skyrmion logic gates conserves skyrmions by propagating each input skyrmion to an out- put port, the operation of a complete conservative logic system requires the conservation of skyrmions through- out the system. It is therefore critical that every skyrmion transmitted to the output port of a skyrmion logic gate is then used as an input to another skyrmion logic gate. As can be observed in relation to the full adder of Fig. 6, skyrmions propagate to several superﬂuous output ports that contain logical byproducts of the computation of the SUM and carry-out signals. In particular, the full adder also produces the signals A,B,A∧B,Cin,a n d Cin∧(A⊕B); these signals are byproducts of the full adder computation that are not the primary objectives of the full adder circuit. These signals are therefore available for use in other logic gates; it may be noted that one signal is available at the output for each of the three input signals. To enable a reversible system, these skyrmion signals must be able to propagate to other logic gates. As the lateral ﬂow of information through this two-dimensional structure is unidirectional (left to right), it is necessary to provide a technique by which skyrmions are provided to the left side of the circuit. Many potential techniques may be available, such as the use of an additional circuit layer, which would also enable the interaction-free crossover of skyrmion tracks. Furthermore, the direction of the spin cur- rent can be modiﬁed through static or dynamic modulation of the electrical current direction or the ferromagnetic Co magnetization. Finally, it should be noted that it may be worthwhile to shed the requirement of complete skyrmion conservation in order to maximize the primary metrics of a computing system (energy consumption, processing speed, and area footprint). VII. CONCLUSIONS The proposed skyrmion structure enables the realization of a nanoscale reversible computing system with nearly 064053-8SKYRMION LOGIC SYSTEM FOR. . . PHYS. REV. APPLIED 12,064053 (2019) ideal characteristics. Small spin Hall currents are sup- plied and the energy required for motion approaches the reversible computing ideal of frictionless informa- tion propagation. Rather than avoiding the skyrmion Hall eﬀect, the proposed system leverages the rich physics of magnetic skyrmions to provide cascaded logic oper- ations that are pipelined and synchronized to maintain signal integrity when scaled to large systems. In addi- tion to Boolean logic, the availability of a Fredkin gate inspires a vision for quantum computing with magnetic skyrmions. Furthermore, the stability of the ferromag- netic materials provides nonvolatility that can be exploited in non–von Neumann architectures that overcome the limitations of conventional computing systems. 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PhysRevA.99.053415.pdf	PHYSICAL REVIEW A 99, 053415 (2019) Optimal periodic control of spin systems: Application to the maximization of the signal-to-noise ratio per unit time N. Jbili,7,8K. Hamraoui,2S. J. Glaser,1J. Salomon,4,5,6and D. Sugny2,3,* 1Department of Chemistry, Technische Universität München, Lichtenbergstrasse 4, D-85747 Garching, Germany 2Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 6303 CNRS-Université de Bourgogne–Franche Comté, 9 Avenue Alain Savary, BP 47 870, F-21078 DIJON Cedex, France 3Institute for Advanced Study, Technische Universität München, Lichtenbergstrasse 2 a, D-85748 Garching, Germany 4INRIA Paris, ANGE Project Team, 75589 Paris Cedex 12, France 5Sorbonne Universités, UPMC Université Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France 6CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris 7Paris Dauphine University, PSL Research University, CNRS, CEREMADE, 75016 Paris, France 8University of Sousse, High School of Sciences and Technology, LAMMDA, 4011 Hammam Sousse, Tunisia (Received 6 January 2019; published 15 May 2019) We propose an optimal control algorithm for periodic spin dynamics. This nontrivial optimization problem involves the design of a control ﬁeld maximizing a ﬁgure of merit, while ﬁnding the initial and ﬁnal states of thedynamics, which are not known but are subjected to speciﬁc periodic conditions. As an illustrative example, weconsider the maximization of the signal-to-noise ratio per unit time of spin systems. In the case of a homogeneousspin ensemble and a very short control duration, we show numerically that the optimal ﬁeld corresponds to theErnst angle solution. We investigate the optimal control process for longer control durations and their sensitivityto offset inhomogeneities. DOI: 10.1103/PhysRevA.99.053415 I. INTRODUCTION Optimality with respect to a given criterion is vital in many applications, but it presents a complexity that requires a lotof ingenuity to provide a solution [ 1–4]. Quantum control [5–11] is no exception to this rule and many efforts have been made recently in this domain to develop tools and methods inorder to conduct a systematic analysis of optimal control prob-lems (see, e.g., some recent reviews [ 12–15] and references therein). In quantum control, there is a genuine desire to solveconcrete questions and contribute beyond the purely theoret-ical analysis [ 12,13]. Different numerical iterative algorithms have been proposed to solve the optimal equations [ 16–21] in a variety of domains [ 12,13], extending from photochem- istry [ 5,6], nuclear magnetic resonance (NMR) [ 22–24] and magnetic resonance imaging (MRI) [ 25–34], and quantum information science [ 12]. Several modiﬁcations of standard algorithms have been proposed to account for experimentallimitations and uncertainties [ 17,22–24,35–41], showing their ﬂexibility and the possibility to adapt them to new classes ofcontrol problems. We propose in this work to investigate an issue in quantum control, namely the optimal control of periodic processes.The difﬁculty of this optimization problem comes from the fact that the initial and ﬁnal states of the dynamical system are not known but have to be determined together with the *dominique.sugny@u-bourgogne.frcontrol ﬁeld to maximize a ﬁgure of merit. The optimalcontrol of periodic processes is well established in mathe-matics [ 42,43] and have been applied in different domains, such as robotics or biology [ 44], to mention a few. To the best of our knowledge, this issue has not been explored inquantum control. As an illustrative example, we consider inthis paper a question of fundamental and practical interestin NMR and MRI [ 45–48], namely the maximization of the signal-to-noise ratio per unit time (SNR) of spin-1 /2 particles. The SNR is practically enhanced in spin systems by usinga multitude of identical cycles. In this periodic regime, theSNR increases as the square root of the number of scans.Each elementary block is composed of a detection time andof a control period where the spin is subjected to a radio-frequency magnetic pulse, this latter being used to guaranteethe periodic character of the overall process. A ﬁrst solutionto this problem was established in the sixties by Ernst and hiscoworkers [ 48]. In this protocol, the control law is made of a δpulse, characterized by a speciﬁc rotation angle, called the Ernst angle solution . This pulse sequence is currently used in magnetic resonance spectroscopy and imaging. Related con-trol procedures, known as steady-state free precession (SSFP),have been also intensively investigated in the literature formedical applications (see Refs. [ 49–56], to mention a few). Some of us have revisited recently the question of the Ernstangle procedure by applying the tools of geometric controltheory [ 3,4]. In Ref. [ 57], it is shown in the general case of unbounded controls which also includes ﬁnite-amplitudeshaped pulses (only δpulses were considered in Ref. [ 48]) 2469-9926/2019/99(5)/053415(8) 053415-1 ©2019 American Physical SocietyN. JBILI et al. PHYSICAL REVIEW A 99, 053415 (2019) that the Ernst angle solution is the optimal solution of the control problem aiming at maximizing the SNR of a spin. Thisanalysis was generalized in Ref. [ 58] to spin dynamics in the presence of radiation damping effects and crusher gradients. The maximization of the SNR is used in this work as a motivation to extend the scope of quantum optimal algorithmsto periodic dynamics. The periodicity constraint is enforcedthrough the introduction of a Lagrange multiplier. We outlinethe principles of a gradient optimization procedure in thisgeneralized framework. We investigate numerically the opti-mization of the SNR. For short control times, we show that thealgorithm converges to the Ernst angle solution in the case ofa homogeneous spin ensemble. For longer control durations,we observe that the optimal solution heavily depends onthe guess ﬁeld used to initialize the algorithm. We describegeometrically the different solutions. Finally, we consider thecase of an inhomogeneous ensemble of spins with severaloffsets. For a ﬁxed control time, we analyze the differentperiodic trajectories and the maximum achievable SNR. The paper is organized as follows. Section IIintroduces the model system and gives a complete description of theoptimal control problem. Section IIIoutlines the principles of our optimization procedure. Special attention is paid to thedifference with a standard iterative algorithm. Section IVis dedicated to the numerical implementation of the algorithm.The numerical results are presented and discussed in Sec. V for homogeneous and inhomogeneous spin ensembles. Ourconclusion and prospective views are given in Sec. VI. Some analytical computations about the Ernst angle solution arereported in the Appendix. II. DESCRIPTION OF THE CONTROL PROCESS FOR THE MAXIMIZATION OF THE SNR We consider an inhomogeneous ensemble of uncoupled spin-1 /2 particles with different offset terms [ 47]. In a given rotating frame, the equation of motion for the spin ensemblereads⎛ ⎝˙x (ω) ˙y(ω) ˙z(ω)⎞ ⎠=⎛ ⎝−2πx(ω)/T2 −2πy(ω)/T2 2π(1−z(ω))/T1⎞ ⎠ +⎛ ⎝−ωy(ω)+ωy(t)z(ω) ωx(ω)−ωx(t)z(ω) ωx(t)y(ω)−ωy(t)x(ω)⎞ ⎠, where the Bloch vector X:=(x(ω),y(ω),z(ω))/intercalrepresents the state of an element of the ensemble, T1andT2are the two relaxation parameters, ωis the offset term, and ωx(t),ωy(t)a r e the two control ﬁelds. We use normalized coordinates so thatthe Bloch ball is deﬁned by x 2+y2+z2/lessorequalslant1 for each spin. Normalizing the time by the detection time Td(see below for the deﬁnition) and setting γ=2πTd/T1and/Gamma1=2πTd/T2, we arrive at ˙X=A(/vectorω)X+D, (1) where D:=(0,0,γ)/intercalandA(/vectorω)i sa3 ×3 matrix: A(/vectorω):=⎛ ⎝−/Gamma1 −ωω y(t) ω −/Gamma1−ωx(t) −ωy(t)ωx(t)−γ⎞ ⎠, FIG. 1. Schematic representation of the cyclic process used in the maximization of the SNR. with/vectorω(t)=[ωx(t),ωy(t)]. Note that the matrix Aleads to a decrease of the modulus of the Bloch vector, while thevector Dcan enhance this modulus. The optimization of the SNR per unit time is described by a simple scenario (see theschematic description in Fig. 1and Refs. [ 57,58] for details). The point M (ω)reached at the end of the control process is the measurement point for the spin of offset ω. The corresponding spin has then a free evolution from this point to the steadystate S (ω)where the pulse sequence starts. The times Tdand Tcdenote the detection time (ﬁxed by the experimental setup) and the control time, respectively. The total time during whicha series of identical experiments are made is ﬁxed. The totalnumber Nof experiments is then given by T=N(T c+Td). The optimization problem is deﬁned through the introductionof a ﬁgure of merit, R=N √ N/radicaltp/radicalvertex/radicalvertex/radicalbt/bracketleftBigg/summationdisplay ωx(ω)(Tc)/bracketrightBigg2 +/bracketleftBigg/summationdisplay ωy(ω)(Tc)/bracketrightBigg2 , (2) where [/summationdisplay ωx(ω)(Tc)]2+[/summationdisplay ωy(ω)(Tc)]2is the square modulus of the strength of the signal (transverse magnetization) at time Tc. We consider a white noise, which leads to the√ Nfactor inR. Using the relation N=T/(Tc+Td) and setting Td=1, we deﬁne the normalized ﬁgure of merit: Fopt=1√1+Tc/radicaltp/radicalvertex/radicalvertex/radicalbt/bracketleftBigg/summationdisplay ωx(ω)(Tc)/bracketrightBigg2 +/bracketleftBigg/summationdisplay ωy(ω)(Tc)/bracketrightBigg2 .(3) This ﬁgure of merit, Foptwill be used in the numerical simu- lations of Sec. V. III. OPTIMAL PERIODIC CONTROL OF SPIN SYSTEMS A. Optimal control algorithm In this paragraph, we generalize the optimal control al- gorithm GRAPE to periodic dynamics [ 16]. To simplify the presentation of the optimization procedure, we speciﬁcallyconsider the case of the maximization of the SNR of asingle spin, but the algorithm can be applied to any periodiccontrol of quantum systems. We focus here on the generalcharacteristics of the algorithm; a numerical implementationis described in Sec. IV. 053415-2OPTIMAL PERIODIC CONTROL OF SPIN SYSTEMS: … PHYSICAL REVIEW A 99, 053415 (2019) We describe the dynamics of the model system on bounded intervals ([0 Tc], [TcTc+Td],...), keeping in mind that all the variables are ( Tc+Td) periodic. Since the dynamics are governed by two different regimes over a period, we split thepropagation into the intervals [0 ,T c] and [ Tc,Tc+Td]: /braceleftbigg X(Tc)=UcX(0)+Ec X(0) =UdX(Tc)+Ed,(4) where the indices canddstand respectively for the controlled and detection periods, UcandUd(respectively, EcandEd) denote the linear (respectively, afﬁne) parts of the propa-gators over the intervals [0 ,T c] and [ Tc,Tc+Td]. Setting W:=UcUdandL:=UcEd+Ec, we ﬁnd that the periodicity constraint on Xreads X(Tc)=WX(Tc)+L, (5) that is, X(Tc)=(1−W)−1L, (6) in the case where 1−Wis invertible, with 1being the 3 ×3 identity matrix. The invertibility of 1−W, which is con- nected to the existence of periodic trajectories, is examinedin Sec. III B . Here, we stress that, for a given control ﬁeld /vectorω, Eq. ( 6) gives the point of the trajectory X(T c). The optimal control problem is deﬁned through the ﬁgure of merit F(X(Tc))=(1+Tc)F2 opt. We ﬁx here the control time Tc. To take into account the different constraints on the dynamics, we use the method of Lagrange multipliers byintroducing the Lagrangian of the problem [ 2]: ˜F=F(X(T c)) +/integraldisplay2Tc+Td TcY(t)/intercal(A(/vectorω)X(t)+D−˙X(t))dt,(7) where Y(t)i st h e( Tc+Td) periodic adjoint state of the system at time t. Differentiating ˜Fwith respect to its variables, we obtain the necessary conditions for /vectorωandX(Tc) to be optimal, namely, ⎧ ⎨ ⎩F/prime(X(Tc))−(Y(T− c)−Y(T+ c))=0 ˙Y(t)=A(/vectorω)Y(t) Y/intercal∂/vectorωA(/vectorω)X=0,(8) completed by Eq. ( 1), where F/prime(X(Tc))=∇X(Tc)F. We recall that X(Tc) is determined from Eq. ( 6). Since Fexplicitly depends on X(Tc), note that Yhas a jump at time Tc, i.e., Y(T− c)/negationslash=Y(T+ c). Using Y(T+ c)=U/intercal dY(Tc+Td) and Y(Tc+ Td)=U/intercal cY(T− c), we obtain Y(T+ c)=U/intercal dU/intercal cY(T− c)=W/intercalY(T− c). Combining the latter with Eq. ( 8), we get (1−W/intercal)Y(T− c)=F/prime(X(Tc)). If1−W/intercalis invertible, we deduce that the condition at time t=T− cfor the adjoint state is Y(T− c)=(1−W/intercal)−1F/prime(X(Tc)). (9) For a given control ﬁeld /vectorω,E q .( 9) leads to the adjoint state at time T− c. The optimality system ( 1) and ( 8) can be solved by using Eqs. ( 6) and ( 9) and the following iterative approach similarin spirit to a standard gradient algorithm. The iteration is initialized by a guess ﬁeld /vectorω0(t). At each step, X(Tc)i s computed using Eq. ( 6) which then enables to ﬁnd a solution of Eq. ( 1). Then, we deduce the state Y(T− c)f r o mE q .( 9), which allows us to compute the time evolution of the adjointstate through Eq. ( 8). The correction to the control ﬁeld is Y /intercal∂/vectorωA(/vectorω)X. The detailed numerical procedure is described in Sec. IVin a time-discretized setting. B. Proof of the existence of periodic trajectories We present a proof of the existence of periodic trajectories for any control ﬁeld /vectorω(t) and any offset ω. We start by writing the Bloch equation ( 1) as follows: ˙X(t)=(B+C(t))X(t)+D, (10) where B:=⎛ ⎝−/Gamma1−ω 0 ω−/Gamma1 0 00 −γ⎞ ⎠ and C(t):=⎛ ⎝00 ωy(t) 00 −ωx(t) −ωy(t)ωx(t)0⎞ ⎠. We denote by P(t) the solution of Eq. ( 10) when D=0: ˙P(t)=(B+C(t))P(t). Since the matrix Bin Eq. ( 1) is skew symmetric, the norm of P(t) decreases when tincreases. Indeed, we have d/bardblP(t)/bardbl2 dt=− 2[/Gamma1x2(t)+/Gamma1y2(t)+γz2(t)]<0. As a consequence, for t∈[0,T], we have d/bardblP(t)/bardbl2 dt/lessorequalslant−2m i n ( /Gamma1,γ )/bardblP(t)/bardbl2. Deﬁning g(t):=d/bardblP(t)/bardbl2 dt+2m i n ( /Gamma1,γ )/bardblP(t)/bardbl2and multi- plying both sides by e2m i n ( /Gamma1,γ)t, we obtain d(e2m i n ( /Gamma1,γ)t/bardblP(t)/bardbl2) dt=e2m i n ( /Gamma1,γ)tg(t). (11) The solution of Eq. ( 11) can be expressed as /bardblP(t)/bardbl2=e−2m i n ( /Gamma1,γ)t/bardblP(0)/bardbl2+/integraldisplayt 0e2m i n ( /Gamma1,γ)(s−t)g(s)ds. Since g(t)/lessorequalslant0, it follows that /bardblP(t)/bardbl/lessorequalslante−min(/Gamma1,γ)t/bardblP(0)/bardbl. (12) The inequality ( 12) is used in the proof of the following lemma. Lemma 1. Let the function fbe deﬁned by f(X(0)):=eTBX(0)+/integraldisplayTc 0e(T−s)B(C(s)X(s)+D)ds +/integraldisplayT Tce(T−s)BDds, 053415-3N. JBILI et al. PHYSICAL REVIEW A 99, 053415 (2019) where Xis the solution of Eq. ( 10). The function fhas a unique ﬁxed point. Proof. LetXandX/primebe the solutions of Eq. ( 10) corre- sponding respectively to initial conditions X(0) and X/prime(0). By integrating Eq. ( 10) over a period T=Tc+Td, we obtain that Xsatisﬁes X(T)=eTBX(0)+/integraldisplayTc 0e(T−s)B(C(s)X(s)+D)ds +/integraldisplayT Tce(T−s)BDds, and the same for X/prime. Subtracting the two identities, we get X(T)−X/prime(T)=eTB(X(0)−X/prime(0)) +/integraldisplayTc 0e(T−s)BC(s)(X(s)−X/prime(s))ds. Setting P(t)=X(t)−X/prime(t), we obtain P(T)=eTBP(0)+/integraldisplayTc 0e(T−s)BC(s)P(s)ds, and it can be easily checked that Psatisﬁes inequality ( 12) on one hand, and P(T)=f(X(0))−f(X/prime(0)), on the other hand. It follows that /bardblf(X(0))−f(X/prime(0))/bardbl/lessorequalslante−min(/Gamma1,γ)t/bardblP(0)/bardbl, which leads to /bardblf(X(0))−f(X/prime(0))/bardbl/lessorequalslante−min(/Gamma1,γ)t/bardblX(0)−X/prime(0)/bardbl. Since e−min(/Gamma1,γ)t<1, we obtain that fis a contraction map- ping, which implies that fadmits a unique ﬁxed point. /squaresolid As a consequence of this result, we observe that the peri- odic dynamical model introduced in this study is well posed ina very general mathematical setting, which includes the caseof an ensemble of inhomogeneous spins. IV . DISCRETE COMPUTATION We now repeat the previous computation in a time- discretized setting. This corresponds to a standard experimen-tal framework in NMR and MRI where the used magneticﬁelds are piecewise constant ﬁelds [ 46,47]. We introduce the time discretization parameters KanddTsatisfying Kd T= T cand a subdivision of the time interval [0 ,Tc] given by the sequence ( tk)k=1,···,K+1,tk=kdT. Equation ( 1) is discretized using a Crank-Nicholson scheme, corresponding to the itera-tion X k+1−Xk dT=A(/vectorωk)Xk+1+Xk 2+D,k=1,. . . , K, (13) where Xk,A(/vectorωk), and /vectorωkstand respectively for X(kdT ), A(/vectorω(kdT )), and/vectorω(kdT ). Introducing the matrices Bk:=1− dT 2A(/vectorωk),˜Bk:=1+dT 2A(/vectorωk),Uk:=B−1 k˜Bkand the vector Ek:=dTB−1 kD,E q .( 13) becomes Xk+1=UkXk+Ek. (14)For a given control ﬁeld /vectorω(t)a c t i n go n[ 0 ,Tc] and the initial state X1, we obtain Xk+1=UkUk−1...U1X1 +k−1/summationdisplay n=1⎛ ⎝k/productdisplay j=n+1Uj⎞ ⎠En+Ek. (15) Note that, up to a diagonalization, Udand Edcan be ex- actly computed. The diagonalization of A(0) gives A(0)= PD 1P/intercal, which leads to Ud=Pexp(TdD1)P/intercaland Ec= Pexp [( Tc+Tds)D1P/intercal]Dds. We therefore keep them at the discrete level, and the discrete expressions of Eq. ( 4)a r eg i v e n by /braceleftbigg XK+1=UdT cX1+EdT c X1=UdXK+1+Ed,(16) where UdT candEdT ccan be expressed as /braceleftBigg UdT c:=UKUK−1...U1 EdT c:=/summationtextK−1 k=1/parenleftBig/producttextK j=k+1Uj/parenrightBig Ek+EK.(17) Introducing WdT:=UdT cUdand LdT:=UdT cEd+EdT c,w e deduce that Eq. ( 16) leads to XK+1=WdTXK+1+LdT. (18) It follows that a solution of Eq. ( 18) can be written as XK+1=(1−WdT)−1LdT, (19) which is a time-discretized version of Eq. ( 5). In this setting, the Lagrangian can be expressed as ˜FdT=F(XK+1)−2K−1/summationdisplay k=KY/intercal k+1(Xk+1−UkXk−Ek). (20) The necessary conditions for /vectorωto be optimal are given by Eq. ( 14), as well as by F/prime(XK+1)−YK+1+U/intercal K+1YK+2=0, (21) and, for k=1,. . ., K, /braceleftbigg Yk−U/intercal kYk+1=0 ∇/vectorωkFdT=0,(22) with ∇/vectorωkFdT=Y/intercal k+1∂/vectorωkUkXk. (23) Note that YK+1and U/intercal K+1YK+2correspond respectively to Y(T− c) and Y(T+ c) in the continuous setting. As Y1= (UdT c)/intercalYK+1andU/intercal K+1YK+2=U/intercal dY1, we deduce U/intercal K+1YK+2=U/intercal d/parenleftbig UdT c/parenrightbig/intercalYK+1=(WdT)/intercalYK+1. Combining the latter with Eq. ( 21) leads to [1−(WdT)/intercal]YK+1=F/prime(XK+1). Assuming once again that 1−(WdT)/intercalis invertible (the in- vertibility can be shown by adapting the reasoning presentedin Sec. III B to the discrete setting), we obtain Y K+1from YK+1=(1−(WdT)/intercal)−1F/prime(XK+1). (24) 053415-4OPTIMAL PERIODIC CONTROL OF SPIN SYSTEMS: … PHYSICAL REVIEW A 99, 053415 (2019) which corresponds to a discrete version of Eq. ( 9). Using Eq. ( 23), we propose the following gradient-type algorithm that solves iteratively Eq. ( 22). Algorithm. Given /vectorω(0)the initial control ﬁeld, we calculate the operator WdT=UKUK−1···U1Ud. Let us assume that /vectorω(/lscript) is known at iteration /lscript; then the control /vectorω(/lscript+1)is computed by the following procedure: (1) Compute X(/lscript) K+1by Eq. ( 19). (2) Compute Y(/lscript) K+1according to Eq. ( 24). (3) Propagate forward X(/lscript)using Eq. ( 14) and X(/lscript) K+1. (4) Propagate backward Y(/lscript)from Y(/lscript) K+1using Eq. ( 22). (5) Compute the gradient ∇/vectorω(/lscript)FdTaccording to Eq. ( 23), and the optimal ascent step ρ. Then, update the control ﬁeld as follows: /vectorω(/lscript+1)=/vectorω(/lscript)+ρ∇/vectorω(/lscript)FdT. Numerical results are presented in Sec. IVbased on the implementation of the Crank-Nicholson approach ( 14). In these numerical tests, different values of Tcare used. In order to control the accuracy of the numerical evolution of thetrajectory of the state /vectorX, we consider an adaptive time step in which the parameters dTandKare ﬁxed with respect to the maximum amplitude of the control ﬁeld /vectorω ∗(t). In this way, we deﬁne dT=/Delta1T max t\|/vectorω∗(t)\| and K=Tc+dT dT. Step 1 corresponds to the implementation of this discretization technique. The control is updated in step 5 where an optimal step gradient iteration is used. One could alternatively considerother optimization methods (optimized gradient method orpseudo-Newton methods, conjugate gradient method, ...). In the examples discussed below, the choice of a gradientmethod with optimal step proved to be the most efﬁcient froma numerical point of view, in terms of number of iterations andcomputational time. V . NUMERICAL RESULTS A. Control of a homogeneous ensemble This section is dedicated to the numerical maximization of the SNR in the case of a homogeneous spin ensemble. The Ernst angle solution is the time-optimal solution max- imizing the SNR [ 48,57]. In this paragraph, we use this control problem as a benchmark to evaluate the efﬁciencyof the optimization algorithm. Without loss of generality, wecan assume that the offset term is zero and that ω y(t)=0, the spin trajectory belonging to the ( y−z) plane. We ﬁrst recall the deﬁnition of the Ernst angle solution. For sake ofcompleteness, the derivation of this control protocol is givenin the Appendix. In this approach, the pulse sequence is onlymade of a δpulse characterized by the Ernst angle θ E: cosθE=e−γ+e−/Gamma1 1+e−/Gamma1−γ. (25)-8 -6 -4 -2 0-10-50 FIG. 2. Evolution of the ﬁgure of merit Fopt(Tc) for different control times Tc(open circles). The relaxation parameters are set to/Gamma1=4a n d γ=2. The maximum is taken over 20 realizations of the algorithm for each time Tcwith different random trial ﬁelds. Dimensionless units are used. The logarithm is taken in base 10. The dissipation effect is neglected during the control time. The coordinates of the corresponding measurement point Mare z(E) m=1 1+eγ;y(E) m=e/Gamma1 1+eγ/radicalBigg e2γ−1 e2/Gamma1−1. (26) The ﬁgure of merit FEis then given by FE=y(E) msince the detection time Tdis set to 1 and the control time Tcis zero for an ideal δpulse. We start the analysis of the optimal control algorithm by a general study of the maximum SNR that can be achievedfor short control durations. As can be seen in Fig. 2,w e ﬁrst verify that the algorithm converges to the Ernst anglesolution when the control duration goes to 0. We observethat the convergence is almost linear as a function of T c. Figure 3represents the different positions of the steady state and the measurement point during the optimization process.Figure 3clearly shows that the two points converge very quickly toward the points of the Ernst angle solution. In a second series of numerical tests, we consider a range of times T cin the interval [10−4,7.3×10−4]. Several opti- mizations are performed for increasing control times, in whichthe previous optimal ﬁeld is used as a guess ﬁeld to initializethe next optimization. Coarse and ﬁne discretizations of thetime interval [10 −4,7.3×10−4] are used, with 4 and 13 points, respectively. The results of the optimization processare given in Fig. 4. A smooth evolution is observed in the case of a ﬁne discretization, while an abrupt change occursfor the coarse one. The different control mechanisms canbe described from the analysis of the uniform norm of theoptimal ﬁeld, \|\|ω\|\| ∞=max t∈[0,Tc]\|ωx(t)\|. Figure 4shows that this norm is almost constant for the coarse case, which leadsto different trajectories as can be seen in Fig. 5. (See also the movies in the Supplemental Material [ 59].) For times longer than 3 ×10 −4, the steady state and the measurement points change and the system follows more complex trajectories. Forthe ﬁne case, the ﬁgure of merit changes very little from onecontrol time to the next. For all the possible values of T c,w e observe that the optimal solution is very similar to the one forT c=10−4. The uniform norm of the control ﬁeld decreases 053415-5N. JBILI et al. PHYSICAL REVIEW A 99, 053415 (2019) FIG. 3. Evolution of the position of the steady state S(blue or dark gray) and the measurement point M(red or light gray) during the optimization process (the points of the ﬁrst 25 iterations of the algorithm are plotted). Since the guess ﬁeld is a zero control, the initial position of SandMcorresponds to the north pole of Bloch sphere of coordinates (0,1). The Sand the Mpoints for the Ernst angle solution are depicted respectively in black and in green. The control time Tcis set to 10−7and the relaxation parameters to /Gamma1=4 andγ=2. Dimensionless units are used. FIG. 4. Evolution of the ﬁgure of merit Fopt(a) and of the corre- sponding uniform norm of ω(t) (b) as a function of Tc(open circles). The coarse and the ﬁne time discretizations are plotted respectively inblue (dark gray) and red (light gray) lines. The relaxation parameters /Gamma1andγare set respectively to 4 and 2. Dimensionless units are used. -1 0 1-101 -1 0 1 -1 0 1 FIG. 5. Optimal trajectories for Tc=10−4(a), 4.16×10−4(b), and 7.33×10−4(c) corresponding to the coarse discretization of Fig. 4. The open blue (dark gray) and red (light gray) circles represent respectively the steady state and measurement points. The controlled trajectory is plotted in solid red (light gray) line and thefree relaxation in solid blue (dark gray) line. The measurement point of the Ernst angle solution corresponds to the green open circle. Dimensionless units are used. FIG. 6. Plot of the controlled trajectories for different offsets in the ( x,y,z) space. The parameters are set to Tc=5.22×10−4, /Gamma1=1.8,γ=1 and the four offsets are ω1=3.3333, ω2=5.5555, ω3=7.7778, and ω4=10. The dashed lines indicate the positions of the steady states and measurement points in the interval [0,10]. Dimensionless parameters are used. with Tcso that the area of the ﬁeld remains approximately constant. B. Extension to a spin ensemble with different offsets We investigate in this section the efﬁciency of the nu- merical algorithm for optimizing an inhomogeneous spinensemble. As a ﬁrst example, we consider the control offour spins with different offsets. The optimization algorithmhas been used for a speciﬁc control time T c=5.22×10−4. The optimal trajectories are plotted in Fig. 6for the four offsets. The set of steady state and measurement points arealso represented for ω∈[0,10]. The two sets of points are distributed along two circles of the Bloch ball. The ﬁgure ofmerit is given in Fig. 7where the result for a homogeneous ensemble is also indicated. Note the robustness of F optagainst variation of the offset ωin the interval [0,10]. This observation is not valid for larger offsets. 05 1 0456710-1 FIG. 7. Evolution of the ﬁgure of merit Foptas a function of the offsetω. The red dots represent the four offsets used in the numerical optimization. The same dimensionless parameters as in Fig. 6are used. The horizontal dashed line displays the ﬁgure of merit for a homogeneous spin ensemble with the same relaxation parameters /Gamma1 andγ. Dimensionless parameters are used. 053415-6OPTIMAL PERIODIC CONTROL OF SPIN SYSTEMS: … PHYSICAL REVIEW A 99, 053415 (2019) VI. CONCLUSION We have proposed in this study a numerical optimization algorithm for quantum systems with a periodic time evolution.The difﬁculty and the originality of the procedure rely on thefact that the initial and target states of the dynamics are notknown but have to be optimized together with the control ﬁeld.The algorithm is built on a standard framework, except for thecomputation of the initial state and adjoint state of the system.A time discretization scheme of the algorithm is presented. Ithas the advantage of simplicity and general applicability. Asan illustrative example, we have considered the maximizationof the SNR for an ensemble of spin-1 /2 particles. We have shown that the algorithm converges to the Ernst angle solutionwith a very high efﬁciency in the limit of a control durationgoing to 0. This analysis leads also to important insights intothe design of optimal pulses. According to the used guessﬁeld, we have observed that the algorithm converges towarddifferent ﬁelds associated with different steady states andmeasurement points. The different trajectories in the Blochball can be geometrically characterized. Some preliminarycomputations have also be done in the case of an inhomo-geneous spin ensemble with different offsets. The results ofthis paper can be viewed as an important step forward inthe development of numerical optimal algorithms in quantumdynamics. In the case of an inhomogeneous spin ensemble,only four spins have been considered in this paper and animprovement would be speeding up the algorithm in order toreduce the computational time, e.g., by using parallelizationtechniques [ 60]. Another open question is the mathematical and numerical description of the transient regime; only thepermanent periodic dynamics have been investigated in thiswork. Some results have been established in this direction forthe Ernst angle solution, but it will be interesting to generalizethis analysis to optimal ﬁelds of nonzero duration [ 49–51]. ACKNOWLEDGMENTS D. Sugny and S. J. Glaser acknowledge support from the ANR-DFG research program Explosys (No. ANR-14-CE35-0013-01 and No. DFG GL 203 /9-1). The work of D. Sugny has been done with the support of the Technische UniversitätMünchen–Institute for Advanced Study, funded by the Ger-man Excellence Initiative and the European Union SeventhFramework Programme under Grant Agreement No. 291763.J. Salomon is funded by ANR Ciné-Para (No. ANR-15-CE23-0019). This project has received funding from the EuropeanUnion’s Horizon 2020 research and innovation programmeunder the Marie-Sklodowska-Curie Grant Agreement No.765267. APPENDIX: THE ERNST ANGLE SOLUTION The goal of this Appendix is to recall the main features of the Ernst angle solution. We consider the case of a spin-1 /2 particle subjected to a control ﬁeld along the xdirection. We denote respectively by S(ys,zs) and M(ym,zm) the steady state and the measurement point of the control process. The SandtheMpoints are connected by a ﬁeld-free evolution: /braceleftbigg ys=yme−/Gamma1 zs=1−e−γ+zme−γ. (A1) In this limiting case, the pulse sequence is reduced to an ideal δpulse, and the radial coordinates rsandrmare therefore the same. We deduce that y2 me−2/Gamma1+(1−e−γ+zme−γ)2=y2 m+z2 m. The ﬁgure of merit Fis here given by ym. A necessary condition to maximize Fisdym dzm=0. We obtain the following relation: zm−(1−e−γ+zme−γ)e−γ=0, leading to zm=1−e−γ eγ−e−γ, which can be transformed into zm=1 1+eγ. It is then straightforward to show that ym=e/Gamma1 1+eγ/radicalBigg e2γ−1 e2/Gamma1−1. We then deduce that the Ernst angle which characterizes the δ pulse can be expressed as cos(θ)=cos(θm−θs)=zmzs+ymys y2m+z2m. 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PhysRevB.103.064501.pdf	PHYSICAL REVIEW B 103, 064501 (2021) Role of compensating current in the weak Josephson coupling regime: An extended study on excitonic Josephson junctions Ya-Fen Hsu1,2,and Jung-Jung Su2,† 1Physics Division, National Center for Theoretical Science, Hsinchu, 30013, Taiwan 2Department of Electrophysics, National Chiao Tung University, Hsinchu 300, Taiwan (Received 29 June 2020; revised 7 December 2020; accepted 19 January 2021; published 1 February 2021) Huang’s experiment [ Phys. Rev. Lett. 109, 156802 (2012) ] found, in the quantum Hall bilayer of the Corbino geometry, the interlayer tunneling currents at two edges are coupled to each other and one of twotunneling currents is referred to as the compensating current of the other. The recent theoretical work [Hsuet al. ,arXiv:2006.15329 ] has explained this exotic coupling phenomenon as a result of excitonic Josephson effect induced by interlayer tunneling current. In this paper, we study the same setup—excitonic Josephsonjunction—but in the weak Josephson coupling regime, which occurs for large junction length. Interestingly,we ﬁnd the compensating current drives the other edge to undergo a nonequilibrium phase transition from asuperﬂuid to resistive state, which is signaled by an abrupt jump of the critical tunneling current. We also identifythe critical exponent and furthermore offer more experimental prediction. DOI: 10.1103/PhysRevB.103.064501 I. INTRODUCTION Josephson effect is particularly attractive to condensed matter researchers because it serves as the striking manifes-tation of coherent condensation and the promising candidatefor quantum technology. Unrelenting and strong attention hasbeen received recently in optically-excited exciton or exciton-polariton cold gases [ 1–7] and graphene electron-hole bilayer exciton [ 8,9]. However, being the best platform to achieve the exciton condensation, the quantum Hall bilayer [ 10–33] remains not studied extensively in the land of Josephson ef-fect. Actually, the search for Josephson effect in quantumHall bilayer ever arouses intense interest since the observationof Josephson-like tunneling [ 34,35], in which the interlayer voltage abruptly increases once exceeding a critical tunnel-ing current [ 12,15,19,21,23,26]. However, to the end, the Josephson-like tunneling is attributed to a mixture of coherentand incoherent interlayer tunneling [ 36–38] instead of the “real” Josephson effect. Once exceeding a critical current, theincoherent tunneling dominates over the coherent one. The scattering approach by solving the Bogoliubov-de Gennes Hamiltonian [ 39–41] is the standard one to explore the Josephson effect, but it is difﬁcult to access in the contextof quantum Hall bilayer. In our previous works [ 42,43], we therefore turn to a new method within the frame of pseu-dospin dynamics based on the idea that the layers can betreated as pseudospin quantum degrees of freedom [ 38,44,45]. We ﬁrst employ this new method to study the exciton-condensate /exciton-condensate (EC /EC) [ 42] and exciton- condensate /normal-barrier /exciton-condensate (EC /N/EC) junctions [ 43] with a constant relative phase between two yafen.hsu.jane@gmail.com †jungjsu@nctu.edu.twECs that is generated by perpendicular electric ﬁeld [ 46]. We found that excitonic Josephson effect occurs only whend J/lessorequalslantξ, where dJandξare barrier length and correlation length [ 42,43]. When dJ>ξ, a new transport mechanism, namely tunneling-assisted Andreev reﬂection, occurs at a sin-gle N/EC interface [ 43]. While the excitonic Josephson effect gives rise to novel fractional solitons [ 42], the new mechanism leads to a half portion of fractional solitons [ 43]. Notably, these new types of solitons have potential to improve thestability and efﬁciency of quantum logic circuits [ 47]. We next study another setup suggested to have a relative phaseby externally applying interlayer tunneling current [ 48]. Inspired by Huang’s experiment [ 49], we consider the setup of interlayer tunneling currents exerted on two edgesof quantum Hall bilayer as shown in Fig. 1(a). The tunneling currents ( J tL,JtR) twist the condensate phases of two edges so as to create the relative phases between three condensates:EC1, EC2, and EC3. Such structure is regarded as two con-densates (EC1 and EC3) sandwiched by a superﬂuid barrier(EC2), which is a type of excitonic Josephson junctions [ 50]. Reference [ 51] has explored this setup but focuses on the short junction whose junction length Lis smaller than Josephson length λ[52]. Its results demonstrated that the exotic coupling phenomenon of edge tunneling currents observed by Huanget al [49] is originated from excitonic Josephson effect, and Huang’s experiment is very robust evidence for quantum Hallbilayer exciton condensation. In this paper, we turn our attention to the opposite case— the long junction of L∼10λ, which corresponds to the typical quantum Hall bilayer [ 12,53]. Our calculation of the conden- sate phase [see Fig. 1(b)] reﬂects that the Josephson current is essentially negligible in the bulk since the phase goes to zeroand becomes ﬂat there [ 54]. Because the two edges are weakly Josephson coupled, the long junction can be approximated astwo independent EC /EC junctions with the boundary between 2469-9950/2021/103(6)/064501(9) 064501-1 ©2021 American Physical SocietyYA-FEN HSU AND JUNG-JUNG SU PHYSICAL REVIEW B 103, 064501 (2021) FIG. 1. (a) Schematic layout of an excitonic Josephson junction induced by interlayer tunneling current. The relative phases betweenthree condensate regions: EC1,EC2 and EC3, are generated by ex- ternally applying tunneling currents J tLandJtR./lscriptBandLdenote the magnetic and junction length. (b) The calculated phase distribu-tions for parallel polarity ⊕(J tL=JtR) and anti-parallel polarity /circleminus (JtL=−JtR) with L=12λ. The green (black) and pink (grey) lines correspond to the parallel and anti-parallel polarity, respectively. Theemployed values of J tRare 5,10,15,20,25 Jt0and with increasing JtR, the phase φdeparts from the xaxis. The length unit λand the current unit Jt0are given later in Sec. II C. Such a long junction is sim- ilar to two weakly coupled exciton-condensate /exciton-condensate (EC/EC) junctions. The cross is the breakpoint between two EC /EC junctions and it is located where the Josephson current Jsapproaches zero. The left (right) part of the bulk combines with the left (right) edge forming an EC /EC junction. Leffdenotes the effective junc- tion length of the right EC /EC junction. (c) Schematic layout of a Corbino-geometry excitonic Josephson junction. The two tunneling currents JtLandJtRare exerted on the orange (lower) and blue (upper) shadow zones. RminandRmaxare the minimum and maximum radius. them occurring where Josephson current Jsgoes to zero [see the inset of Fig. 1(b)]. It is therefore highly desirable that the long junction can display entirely different properties fromthe short junction in which two edges are strongly Josephson coupled [ 51]. It turns out that the long junction indeed exhibits a unique property: One edge undergoes a nonequilibrium phase transi-tion [ 55,56] with increasing the tunneling current at the other edge (i.e., the compensating current). During this phase transi-tion, the critical tunneling current of the edge sharply falls andthe corresponding critical exponent is identiﬁed as γ∼0.5. Since the Josephson coupling is weak, we wonder why thecompensating current can inﬂuence the other edge so largely.According to our analysis, this is because the compensatingcurrent reduces the effective junction length of the constituentEC/EC junction on the opposite side. We furthermore calcu- late the magnetic ﬁeld induced by Josephson current (denotedbyB J) for the Corbino-geometry excitonic Josephson junction shown in Fig. 1(c). We ﬁnd the length reduction effect is revealed by the crossover of the BJversus /Delta1Jtcurve into the short junction regime [ 51] (a linear one) with increasing the compensating current, where /Delta1Jt=JtR−JtL. The induced magnetic ﬁeld is estimated at ∼100 pT that is large enough to be detected by the scanning superconducting interferencedevice (SQUID). In the main body of this paper, we show theresults of the rectangle-shaped junction in Figs. 3–7and those of the Corbino-geometry junction in Figs. 8and9. II. MODEL AND METHOD Burkov and MacDonald treated two layers of the quantum Hall bilayer as pseudospin quantum degrees of freedom andaccordingly deduced a lattice model Hamiltonian [ 45]: H=1 2/summationdisplay ij/parenleftbig 2Hij−Fintra i,j/parenrightbig Sz iSz j−Finter i,j/parenleftbig Sx iSx j+Sy iSy j/parenrightbig , /vectorSi=1 2/summationdisplay σ,σ/primea† i,σ/vectorτσ,σ/primeai,σ/prime. (1) Here a† i,σ(ai,σ) is the Schwinger boson creation (annihilation) operators [ 57] where iandσlabel the site and layer indexes and/vectorτis the Pauli matrix vector. The Hartree term Hijde- scribes the direct Coulomb interaction while the Fock termF intra i,j(Finter i,j) serves the intralayer (interlayer) exchange in- teraction. This lattice Hamiltonian possesses eigenstate wavefunction which can be generally expressed as \|/Psi1/angbracketright=/productdisplay i/bracketleftbigg cosθ(/vectorXi) 2c† i↑+sinθ(/vectorXi) 2eiφ(/vectorXi)c† i↓/bracketrightbigg \|0/angbracketright. (2) The operator c† i↑(c† i↓) creates an electron at the lattice site location /vectorXiin the top (bottom) layer. It is difﬁcult to study the present issue through quantum scattering approach whichis based on this wave function since we cannot simply writedown the explicit forms of θ(/vectorX i) andφ(/vectorXi). We therefore request a SU(2) to O(3) mapping, and the wave function is transformed into a classical pseudospin [ 44] /vectorm(/vectorXi)=(m⊥cosφ,m⊥sinφ,mz), m⊥=sinθ,mz=cosθ. (3) Accordingly, the dynamics of the quantum Hall bilayer can be described by the Landau-Lifshitz-Gilbert (LLG) 064501-2ROLE OF COMPENSATING CURRENT IN THE WEAK … PHYSICAL REVIEW B 103, 064501 (2021) FIG. 2. (a) Illustration of the effect of external tunneling current Jt. Here the top and bottom layers are selected as up pseudospin (mz=1) and down pseudospin ( mz=−1). The notation Adenotes the area that tunneling current passes through. Over the time durationdt, the electrons number that ﬂows out of the top layer or ﬂows into the bottom layer is counted by J tAdt/e. (b), (c) and (d), (e) depict the ﬂows of electrons and excitons when applying the externaltunneling current to the right and left edges, respectively. The solid red and dashed pink arrows indicate the direction of exciton ﬂow and single-particle tunneling, respectively. The insets at their upperright corner are the individual corresponding counterﬂow currents. For convenience in discussion, we choose +ea st h ec h a r g eo fa n electron and eis actually a negative amount. The current therefore goes along the ﬂow direction of electrons. equation [ 38,42,43] d/vectorm dt=/vectorm×(2/n¯h)(δE[/vectorm]/δ/vectorm)−α/parenleftbigg /vectorm×d/vectorm dt/parenrightbigg , E[/vectorm]=Aunit/summationdisplay i/bracketleftbigg βm2 z+ρsm2 ⊥ 2\|∇/vectorXiφ\|2 −n/Delta1tm⊥ 2cosφ/bracketrightbigg , (4) where Aunitis the area of the unit cell for the pseudospin lattice andnis the pseudospin density. The excitonic superﬂuid loses its coherence after traveling over one correlation length ξso the size of the unit cell is equal to ξ, which is estimated at ∼200 nm [ 58]. In unit of the magnetic length lB,ξ∼10lB (lBhas the typical value of ∼20 nm). On the other hand, the energy functional E[/vectorm] is composed of the capacitive penalty, the exchange correlation, and the interlayer tunneling energy,which are characterized by the parameters: anisotropic energyβ, pseudospin stiffness ρ s, and single-particle tunneling /Delta1t,FIG. 3. (a) and (b) depict the junction geometry: (a) the standard Hall bar geometry and (b) the Corbino geometry. (c)–(e) summa-rize the key results of Huang’s experiment (a realization of short Josephson junction): (c) Josephson-like I-Vcharacteristic without the compensating current applied. The inset shows the measurementconﬁguration. (d) Josephson-like I-Vcharacteristic for different val- ues of the compensating current. The numbers below the traces label the corresponding value of the compensating current I tL/It0. TheI-Vcurves are offset by ( ItL/2It0)mV. (e) The critical currents as a function of the compensating current. The measured current ItL(R)=JtL(R)Aand the unit It0=1nA, where Alabels the effective cross area of external tunneling currents. Ais difﬁcult to determine through the existing experimental information. The data of (c)–(e) are reproduced from Ref. [ 49]. respectively. These model parameters are up to which kind of samples we are discussing. The second term for the LLGequation is the Gilbert damping which relaxes the energytoward the minimum. A. Modeling excitonic Josephson junctions The key breakthrough of the present work is to introduce the effect of external tunneling currents. When exerting the+ˆz-direction tunneling current J ton an area of Aover a short duration of dt, there are electrons as many as JtAdt/epouring out of the top layer and trickling into the bottom layer simul-taneously (see Fig. 2), giving rise to the change of −2J tAdt/e in the total pseudospin nAm z. Under the effect of tunneling current, the z-component LLG equation thus can be modiﬁed as dmz dt=−2ρs n¯hm2 ⊥∇2φ+/Delta1t ¯hm⊥sinφ−2Jt ne+αm2 ⊥dφ dt.(5) In the rectangle-shaped excitonic Josephson junction as shown in Fig. 1(a), two tunneling current JtLand JtRare 064501-3YA-FEN HSU AND JUNG-JUNG SU PHYSICAL REVIEW B 103, 064501 (2021) applied to two edges over a length as large as one lattice size 10/lscriptB. We can therefore model the junction through setting Jtto Jt=JtL/Theta1(x+L/2)/Theta1(L/2−10lB−x) +JtR/Theta1(L/2−x)/Theta1(x−L/2+10lB). (6) Notice that from here on we use the continuous varying x instead of the discrete X ifor convenience in presentation and/Theta1(x) is the Heaviside step function. The origin x=0 is deﬁned to be located at the center of the junction. Afterevolving with time, we ultimately acquire the static solutionsforφ,m ⊥, and mzthat specify the pseudospin orientation. The counterﬂow Josephson current is furthermore calculatedby J s=eρs∇φ/¯h. (7) The physical picture for the effect of external tunneling currents can be depicted through Figs. 2(b)–2(e). When ap- plying the +ˆz-direction tunneling current to the left edge [see Fig. 2(b)], holes and electrons are injected into the top and bottom layer from the left side, respectively. The electronscan ﬂow into the top layer to annihilate holes via single-particle tunneling /Delta1 tor combine with holes to form excitons \|h↑;e↓/angbracketrightand then transmit right into the junction bulk, where \|h↑;e↓/angbracketrightindicates a state composed of a hole in the top layer bound to an electron in the bottom layer. However,single-particle tunneling destroys the excitons everywhereand leads to the attenuation of counterﬂow Josephson cur-rent in the bulk. When reversing the direction of externaltunneling current [see Fig. 2(c)], the roles of electrons and holes are exchanged and right-going but opposite polarizedexcitons \|e↑;h↓/angbracketrightoccur, where \|e↑;h↓/angbracketrightindicates a state composed of an electron in the top layer bound to a holein the bottom layer. Similarly, applying the +ˆz(−ˆz)-direction tunneling current to the right edge will generate “left”-goingexcitons \|h↑;e↓/angbracketright(\|e↑;h↓/angbracketright) [see Figs. 2(d) and2(e)]. It turns out that the external tunneling currents with parallel(antiparallel) polarity will inject counterﬂow Josephson cur-rent in the opposite (same) direction as shown in the insets ofFigs. 2(b)–2(e). B. Calculation of induced magnetic ﬁeld due to excitonic Josephson effect We next consider a Corbino-geometry excitonic Joseph- son junction that can generate circular Josephson current[see Fig. 1(c)]. The Corbino can be divided into a set of rings with radius which ranges from R mintoRmax.As i n - gle ring of the speciﬁc radius rcan be viewed as a bent Josephson junction with L=2πr. We ﬁrst calculate the phase distribution for the junction of L=2πRminby the LLG equa- tion and then acquire the phase distribution for other valuesofrby taking the azimuthal symmetry into account. The Josephson current is similarly calculated by Eq. ( 7). By using the Biot-Savart law, we ﬁnally obtain the induced magneticﬁeld: B J(z)=μ0/angbracketleftJs(Rmin,θ)/angbracketrightθzdR min 2 ×/bracketleftBigg 1 /parenleftbig R2 min+z2/parenrightbig3/2−1 /parenleftbig R2max+z2/parenrightbig3/2/bracketrightBigg ,(8) where dis the interlayer separation, zis the distance above the center of the bilayer, and /angbracketleft ··· /angbracketright θis the average over the angular axis of polar coordinate. C. Choice of units, identiﬁcation of critical current, and determination of parameters Both geometries we consider are discussed based on a length scale, namely, Josephson length: λ=/radicalbig 2ρs/n/Delta1t. (9) Two units for Josephson current and tunneling current read Js0=eρs/¯hλandJt0=en/Delta1t/2¯hthroughout this paper. We identify the critical interlayer tunneling current by ﬁndingthe upper and lower boundaries at which the junction de-partures from the coherent state, i.e., m zbegins to become nonzero. The main focus of the present work is the typi-cal quantum Hall bilayer of λ∼45μm(/Delta1 t=10−8E0)[53], which corresponds to the samples fabricated by Eisen-stein’s group [ 12]. Here the Coulomb interaction E 0= e2//epsilon1lBserves as the energy scale and E0∼7 meV. The other parameters we use are listed as follows: β=0.02E0 andρs=0.005E0, which were derived from the mean-ﬁeld calculation [ 53]. III. ANALYSIS OF ROLE OF THE COMPENSATING CURRENT Figures 3(a) and 3(b) show that edge-state currents in- evitably contribute to the coupling of the left and right edgesfor the Hall-bar geometry while two edge-state currents sep-arately ﬂow along the inner and outer boundaries so as not toconnect the left and right edges for the Corbino geometry [ 24]. To avoid the contribution of edge-state currents, Fig. 3(b) is the main setup we consider here and its corresponding junc-tion length roughly approximates the difference of the innerand outer radius. The realistic Corbino geometry possesses thejunction length L∼0.54 mm [ 49] and in the context of the typical quantum Hall bilayer [ 12](λ=45μm), the junction length reads L∼12λ. The large part of this paper therefore focuses on the case of L=12λlater. A. Nonequilibrium phase transition The realization of the short junction with L=0.6λ[51]— Huang’s experiment [ 49]—is devoted to analyzing Josephson-like behavior, in which the interlayer voltagesuddenly emerges when applying tunneling current up tocritical values: the upper and lower I c[see Fig. 3(c)]. They found the upper and lower critical currents are correlated withits compensating current—the tunneling current exerted onthe other edge and such coupling of the tunneling currentsat two edges would disappear when \|I tL\|>16 nA [see Fig. 3(d)]. The disappearance phenomenon will be discussed 064501-4ROLE OF COMPENSATING CURRENT IN THE WEAK … PHYSICAL REVIEW B 103, 064501 (2021) -40 -20 0 20 40 JtL / Jt0-40-2002040 JtR / Jt0 -40 -20 0 20 40 JtL / Jt00100200300400\|ΔJtR / ΔJtL\| 0.001 0.01 0.1 1 jtL-20-1001020ΔJtR / Jt0upper Jc lower Jc30.692 -30.692-30.692 30.692 (a) (b)upper Jc lower Jc++ --parallel anti-parallel FIG. 4. (a) The calculated upper and lower critical values of the external tunneling current JtRas a function of its compensat- ing current JtL. The inset: the corresponding slopes /Delta1JtR//Delta1JtLas a function of JtL. (b) The identiﬁcation of critical exponents near two phase transition points occurring at JtL=±30.692Jt0. Here /Delta1JtR=Jc(JtL)−Jc(±30.692Jt0)a n d jtL=\|(JtL−±30.692Jt0)/± 30.692Jt0\|. The choice of ±is up to which phase transition point we are discussing. By ﬁtting to the numerical results presented in this ﬁgure, we extract the exponent γ, which is deﬁned as /Delta1JtR∝jγ tL, and ﬁnd γ∼0.5 for any phase transition point. later in Sec. IV A , and we focus on how the tunneling currents at two edges correlate with each other here. Huang’sexperiment quantiﬁes this coupling through the plot of thecritical currents as a function of the compensating current[see Fig. 3(e)]. Therefore, we also display the similar plot for the long junction in Fig. 4to analyze the role of the compensating current. Over a wide range of J tL, the upper and lower critical currents nearly keep constant and aresymmetric with respective to J tR=0[ s e eF i g . 4(a)]. Near JtL=±30.692Jt0, however, the critical currents rapidly fall to zero. The sharp jump of critical currents Jcindicates the right edge is switched from a superﬂuid to resistive state. Theright edge undergoes a phase transition under the conditionof compensating-current-driven nonequilibrium [ 55,56]. With slowly adjusting J tL, it is identiﬁed as a ﬁrst-order phase transition since \|Jc(JtL=±30.692J0)\|=15.999J0 and \|Jc(JtL=±30.6925 J0)\|=0 (the giant change in critical currents hints possible incontinuity). We20 22 24 26 28 30 JtL / Jt055.25.45.65.86 Leff / λ -30 -28 -26 -24 -22 -20 JtL / Jt055.25.45.65.86 048 1 2 1 6 L / λ0102030Jc / Jt0JtR = 6J0JtR = 12 J0JtR = 18 J0 (a) (b)-+ JtR = 6J0JtR = 12 J0JtR = 18 J0 (c) FIG. 5. (a) and (b) are the effective length of the right EC /EC junction as a function of the corresponding compensating current JtLfor the parallel polarity ⊕and antiparallel polarity /circleminuswith the right tunneling current JtR=6,12,18Jt0. (c) The junction-length dependence of critical current Jcwithout the compensating current applied ( JtL=0). furthermore deﬁne new critical exponents: /Delta1JtR∝/braceleftBigg (30.692Jt0−JtL)γ+forJtL/lessorsimilar30.692Jt0, (JtL+30.692Jt0)γ−forJtL/greaterorsimilar−30.692Jt0,(10) where /Delta1JtR=Jc(JtL)−Jc(±30.692Jt0). The ﬁts to our nu- merical results extract the values of exponents [see Fig. 4(b)]: γ+=0.4939, γ−=0.4999 for the upper Jccurve. For the lower Jccurve, the values of γ+andγ−are exactly exchanged because of electron-hole symmetry. B. Junction-length reduction effect Why can the compensating current largely reduce the critical currents as JtL≈±30.692Jt0even if the Josephson coupling is so weak? As has been illustrated in Fig. 1(b),t h e long junction can be decomposed into two nearly independentEC/EC junctions. We here identify the breakpoint occurring atJ s=0 or where Jsreaches its minimum and determine the effective length of the right EC /EC junction as shown in Figs. 5(a) and 5(b). We ﬁnd, regardless of the polarity, the compensating current JtLdecreases the effective length of the right junction and hence leads to the jump of the criticalcurrents. It is quite intuitive or shown in Fig. 5(c) that the critical current would decrease with decreasing the junctionlength. 064501-5YA-FEN HSU AND JUNG-JUNG SU PHYSICAL REVIEW B 103, 064501 (2021) IV . OTHER INTERESTING PREDICTION A. Discussion on Josephson breakdown effect Now let us turn our attention to the disappearance phe- nomenon of the coupling of the two edge tunneling currentsshown in Fig. 3(b) occurring as \|I tL\|>16 nA. For this disappearance phenomenon, the main body of Ref. [ 49] furthermore demonstrates that it is accompanied by theoccurrence of the interedge voltage. Reference [ 51] has at- tributed this phenomenon to the breakdown of Josephsoneffect—when Josephson current attains some critical value,the Josephson effect would collapse and the external tunnelingcurrents will prefer to convert into edge-state currents. Wehere comment on whether this breakdown effect occurs alsoin the long junction or not. Differing from the short junction,the upper and lower J ccurves are always symmetric with re- spect to JtR=0 as if the Josephson breakdown effect already happens and the applied compensating current is limited to arange of J tL=−30.692Jt0∼30.692Jt0beyond which coher- ent interlayer tunneling disappears [see Fig. 4(a)]. We have performed a numerical calculation demonstrating that overthe range of J tL=−30.692Jt0∼30.692Jt0, static solutions can exist and there was not found any critical variation. Wetherefore believe that the breakdown effect does not occur inthe long junction. We furthermore give more detail analysis through Fig. 6. The difference of external tunneling currents /Delta1J tplays the similar role as the relative phase in the conventional Joseph-son junction [ 50] while it is easier to compare with the experiment directly based on the compensating current J tL. In Fig. 6, we therefore plot the spatial extrema of Joseph- son current Jextre s as a function of not only /Delta1Jtbut also JtL. We ﬁnd that Jextre s rises or drops to saturation over the range of /Delta1Jt=20Jt0∼40Jt0or/Delta1Jt=−20Jt0∼−40Jt0 [see Fig. 6(a)], which corresponds to JtL=−20Jt0∼20Jt0 [see Fig. 6(b)]. With increasing the compensating current, if the Josephson-breakdown regime is achieved, it necessarilyoccurs at J tL=−20Jt0∼20Jt0where the Jccurves hold hor- izontal [see Fig. 4(a)]. Measuring the interedge voltage will help us clarify the junction being in the weak Josephson cou-pling regime or Josephson-breakdown regime. Alternatively,after increasing the compensating current beyond ±20J to,\|Jc\| begins to fall [see Fig. 4(a)], providing a unique signature for the weakly Josephson coupling, namely, Josephson fall. B. The crossover behavior with varying junction length Since the dependence of the critical currents on the com- pensating current is so distinct for the short and long junctions,we next want to understand the crossover behavior with in-creasing junction length through Fig. 7. Because the lower J c curve can be produced through doing the electron-hole trans- formation: JtR→− JtR,JtL→− JtLon the upper Jccurve, in Fig. 7, we display only the upper Jccurve for conciseness. Figure 7shows that, with increasing the junction length, the curve is gradually skewed and no abrupt change occurs. More-over, the Josephson fall already can be found as L=4λwhile the weakly “symmetric” Josephson regime can be achieved asL∼5λ.T h ev a l u eo f5 λhappens to meet the junction length for the typical quantum Hall bilayer [ 12] of Hall-bar geometry-60 -40 -20 0 20 4 0 ΔJt / Jt0-2-1012 Jsextre / Js0 upper Jc lower Jc -40 -20 0 206 04 0 JtL / Jt0-2-1012 Jsextre / Js0upper Jc lower Jc(a) (b) upper Jc lower Jc FIG. 6. The spatial extrema of Josephson current Jextre sas a func- tion of (a) the difference of two tunneling currents /Delta1Jt=JtR−JtL and (b) the compensating current JtLfor the upper and lower critical points of JtR. (L∼225μm) but the Hall-bar geometry may be difﬁcult to coincide with our calculation due to the inﬂuence of edge-statecurrent. Replacing the usually-used side electrodes with thetop and back electrodes would be a method to avoid edge-statecurrents although it is a big technological challenge. -40 -20 0 20 40 JtL / Jt0-10010203040 JtR / Jt0L = 0.6 λ L = 1λ L = 2λ L = 4λ L = 6λ L = 12 λ FIG. 7. The critical value of the external tunneling current JtR versus the compensating current JtLfor different junction length L. 064501-6ROLE OF COMPENSATING CURRENT IN THE WEAK … PHYSICAL REVIEW B 103, 064501 (2021) -40 -20 0 20 40 ΔJt / Jt0-3-2-101210000BJ / μ0Js0 JtL = 30 Jt0 JtL = 17 Jt0 JtL = 0 JtL = -17 Jt0 JtL = -30 Jt0Rmin = 1.9 λ, Rmax = 9.56 λ FIG. 8. The induced magnetic ﬁeld BJdue to circular Joseph- son current of a Corbino-geometry excitonic Josephson junction atz=2.22λas a function of the difference of two external tunnel- ing currents /Delta1J t=JtR−JtLforRmin=1.9λandRmax=9.56λ.T h e curves are offset by the corresponding JtL. The interlayer separation d=1.6/lscriptB,w h e r e /lscriptBis the magnetic length. C. The induced magnetic ﬁeld due to Josephson current in a Corbino geometry Next Fig. 8shows the results for the Corbino-geometry excitonic Josephson junction, which is depicted in Fig. 1(c) (the curves are offset by the corresponding compensatingcurrent for clarity and a without-offset version is given inAppendix A). In Fig. 8, except for the minimum radius R min, the other parameters are determined according to the real-istic situation of experiments. The minimum radius for thetypical Corbino is roughly 0.16mm or equivalently R min∼ 3.56λinstead of Rmin=1.9λthat we choose for increasing the numerical efﬁciency. But, the investigated Corbino of λ< 2πRmin<2πRmaxcan already capture the physics of the long junction to a qualitative level and such a Corbino with smallerR minis easily realized by etching. We ﬁnd, differing from the short junction [ 51], the dependence of the induced magnetic ﬁeld BJon the difference of two tunneling currents /Delta1Jtcan have apparent curvature. The curve however becomes linearwhen J tLreaches ±30Jt0. This is because JtLdecreases the effective length of the EC /EC junction on the opposite side and drives the investigated Corbino into the short-junction regime of a linear dependence [ 51]. Moreover, the extremely subtle magnetic ﬁeld must be measured by the scanning su-perconducting quantum interference device (SQUID). To ourbest knowledge, the resolution of the typical scanning SQUIDis up to ∼10 pT at a sensor-to-sample distance of ∼100 nm and the current technology even improves the resolution to∼1p T[ 59]. We estimate B Jon the scale ∼100 pt and it is measurable without doubt. V . CONCLUSION In conclusion, we predict a nonequilibrium phase transition occurring in the long junction of weak Josephson coupling andﬁnd the effective length reduction effect of the compensating-60 -40 -20 0 20 40 60 ΔJt / Jt0-1.6-0.800.81.610000BJ / μ0Js0 JtL = 30 Jt0 JtL = 17 Jt0 JtL = 0 JtL = -17 Jt0 JtL = -30 Jt0Rmin = 1.9 λ, Rmax = 9.56 λ FIG. 9. The without-offset version for Fig. 8,w h e r e BJand/Delta1Jt denote the induced magnetic ﬁeld and the difference of two external tunneling currents, respectively. current. The sample size is not highly tunable in experimental measurement and therefore this length reduction effect will belargely helpful in observing the interesting crossover behaviorpredicted in Ref. [ 43]. We furthermore discuss the possibility of the breakdown of Josephson effect and suggest measuringthe interedge voltage and Josephson fall [ 60] to distinguish the Josephson breakdown effect from weak Josephson cou-pling. We also calculate the induced magnetic ﬁeld in theCorbino-geometry Josephson junction to suggest the detectionof Josephson current. It should be noted that there is stillvery much theoretical effort called for, such as developingBogolubov-deGennes description, exactly identifying phasetransition (especially for it being ﬁrst order or second order),systematically exploring the Josephson breakdown effect andetc. We believe the present work together with Ref. [ 51]— excitonic Josephson effect induced by interlayer tunnelingcurrent—will bring new attention to the condensed matterphysics community. ACKNOWLEDGMENTS We thank W. Dietsche, A. H. MacDonald, B. Rosenstein, Jheng-Cyuan Lin, Sing-Lin Wu, and Chien-Ming Tu for valu-able discussion. We also thank the support from Ministry ofScience and Technology and National Center for TheoreticalSciences in Taiwan. This work was ﬁnancially supported byGrant No. MOST 106-2112-M-009-011-MY3. 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PhysRevLett.120.097203.pdf	Strong Enhancement of the Spin Hall Effect by Spin Fluctuations near the Curie Point of FexPt1−xAlloys Yongxi Ou,1,D. C. Ralph,1,2and R. A. Buhrman1,† 1Cornell University, Ithaca, New York 14853, USA 2Kavli Institute at Cornell, Ithaca, New York 14853, USA (Received 1 October 2017; revised manuscript received 19 November 2017; published 1 March 2018) Robust spin Hall effects (SHE) have recently been observed in nonmagnetic heavy metal systems with strong spin-orbit interactions. These SHE are either attributed to an intrinsic band-structure effect or toextrinsic spin-dependent scattering from impurities, namely, side jump or skew scattering. Here we report on an extraordinarily strong spin Hall effect, attributable to spin fluctuations, in ferromagnetic Fe xPt1−x alloys near their Curie point, tunable with x. This results in a dampinglike spin-orbit torque being exerted on an adjacent ferromagnetic layer that is strongly temperature dependent in this transition region, with a peak value that indicates a lower bound 0.34/C60.02for the peak spin Hall ratio within the FePt. We also observe a pronounced peak in the effective spin-mixing conductance of the FM/FePt interface, anddetermine the spin diffusion length in these Fe xPt1−xalloys. These results establish new opportunities for fundamental studies of spin dynamics and transport in ferromagnetic systems with strong spin fluctuations, and a new pathway for efficiently generating strong spin currents for applications. DOI: 10.1103/PhysRevLett.120.097203 The manipulation of the magnetization in ferromagnetic (FM) nanostructures with pure spin current densities Jshas become a primary tool in spintronics since the demon- strations of nanomagnet switching driven by large spinorbit torques (SOT) originating from the spin Hall effect(SHE) [1–4]in an adjacent heavy metal carrying a longitudinal electrical current density J e. Most SOT efforts so far have focused on the utilization of large intrinsic spin Hall ratios, θSH≡ð2e/ℏÞJs/Je, for certain heavy metals that are compatible with the requirements of a successfulspintronics technology [5–9]. An alternative approach to enhance θ SHis to introduce dopants into a metallic system whereby strong spin-orbit interactions can strengthen the spin Hall effect, either by enhanced extrinsic spin-dependent skew or side-jump scattering, or by theintrinsic effect through a beneficial modification of theelectronic band structure at the Fermi level [10]. Typically the dopant has been a heavy element without a strong magnetic moment, e.g., Ir, Bi, Au [11–14], and the resulting enhancement, usually attributed to skew scatter-ing, has been measured by the inverse spin Hall effect or bya nonlocal spin accumulation technique. Recently Wei et al. [15] have reported a moderate, but notable, temperature- dependent enhancement of the inverse spin Hall effect in dilute NiPd alloys, attributed to spin-fluctuation-enhancedskew scattering by the Ni ions in the vicinity of theferromagnetic transition. We note that NiPd is one of aclass of ferromagnetic alloys that have long been known to exhibit giant magnetic moments per ferromagnetic solute, particularly in the dilute limit [16]. Reference [15]provides strong motivation for examining the direct SHE in otherferromagnetic alloys in which there can be a stronger spin- orbit interaction between the conduction electrons and the ferromagnetic component. Here we demonstrate that for Fe- doped Pt alloys, Fe xPt1−x, the effective spin Hall angle as measured directly from the dampinglike torque exerted on anadjacent ferromagnetic layer is increased by more than afactor of 3 in the vicinity of its Curie temperature T cin comparison to the already-substantial value it has well above Tc, and at its peak has a value at least comparable to that of beta-W[5], and with a much lower electrical resistivity. FexPt1−xalloys are well known for their unusually robust magnetic anisotropy properties arising from the strong conduction electron spin-orbit interaction with the Fe orbital moment [17,18] , and also for the dependence of the magnetic state on the chemical order. For example,well-ordered Fe 0.25Pt0.75exhibits antiferromagnetism, while chemically disordered Fe 0.25Pt0.75is ferromagnetic [19,20] . Ferromagnetic Fe xPt1−xfilms also exhibit quite large anomalous Hall effects (AHE) [21–23], which sug- gests that Fe xPt1−xcan be a promising material for the generation of spin currents by the extrinsic SHE. To investigate this possibility we prepared multilayers containing two different sets of Fe xPt1−xthin films made by codeposition at room temperature via dc magnetron sput-tering; in one case the nominal composition was Fe 0.15Pt0.85 and in the other Fe 0.25Pt0.75. Multilayer stacks consisting of substrate/Ta/IrMn/Fe xPt1−x/MgO/Ta were used for thin film characterization and substrate/Ta/IrMn/Fe xPt1−x/Hf/ FeCoB/Hf/MgO/Ta stacks were used for the SOT mea- surements. These samples were prepared by direct current(dc) sputtering (with rf magnetron sputtering for the MgOPHYSICAL REVIEW LETTERS 120, 097203 (2018) 0031-9007 =18=120(9) =097203(6) 097203-1 © 2018 American Physical Societylayer) in a deposition chamber with a base pressure <5×10−8Torr. The dc sputtering condition was 2 mTorr Ar pressure. The Fe xPt1−xalloy was grown by cosputtering from two pure sputtering targets (i.e., Fe andPt targets). All samples in this work had a Ta(1 nm) seed layer to provide a smooth base layer and a Ta(1 nm) top layer to provide an oxidized protection layer for the stack.All samples were annealed in an in-plane magnetic field (2000 Oe) in a vacuum furnace at 300°C for 1 h to enhance the PMA. For measurements of the AHE and SOT, Hall bar devices with lateral dimensions of 5×60μm 2were pat- terned via photolithography and ion milling [see the sample schematic in Fig. 1(d)]. We performed x-ray diffraction (XRD) measurements on two multilayer samples with the layer structures IrMn 3ð10Þ/Fe0.15Pt0.85ð10Þ/MgO and IrMn 3ð10Þ/ Fe0.25Pt0.75ð10Þ/MgO (the number in parentheses is the thickness in nanometers). The IrMn 3layer was included to provide antiferromagnetic pinning of the Fe xPt1−xlayers when cooled to well below their Curie points for researchthat will be presented elsewhere; the Fe xPt1−xlayers are thick enough that in the experiments to be considered below the IrMn 3does not contribute any significant SOT on the free magnetic layer. We show in Fig. 1(a)the (111) XRD peaks for the Fe 0.15Pt0.85and the Fe 0.25Pt0.75samples, and also for separate 10 nm Pt, Fe 0.50Pt0.50, and IrMn filmsfor comparison. The (111) peak is reasonably narrow, shifting to higher 2θangle as the Fe content increases, indicating a decrease in the unit cell size with increased Fecontent. As expected from the use of room temperaturedeposition there was no evidence of a (110) peak in theXRD of these samples that would indicate significantchemical order [24] [The small peak at 2θ≈41°i n Fig. 1(a) is due to the IrMn 3base layer]. Finally, as expected for a disordered metal the resistivity of the filmswas only weakly temperature dependent, decreasing by lessthan 10% from room temperature to 160 K, indicating thedominance of impurity scattering. The resistivity of thefilms did vary with Fe content, from ρ Ptð10Þ≈15μΩ·c m toρFe0.15Pt0.85ð10Þ≈55μΩ·cmto ρFe0.25Pt0.75ð10Þ≈75μΩ·cm, indicating an increased electron scattering rate with increased Fe content. To further confirm the chemical disorder and the ferromagnetic character of these alloys we made temper-ature-dependent vibrating sample magnetometry (VSM) measurements of the samples [Fig. 1(b)]. Both Fe 0.15Pt0.85 and Fe 0.25Pt0.75were found to be ferromagnetic at suffi- ciently low temperature, with fits of the spontaneous magnetization MsðTÞto the empirical function MsðTÞ¼ Msð0Þ½1−ðT/TcÞα/C138β[25] yielding a Curie temperature of Tc≈174K for the Fe 0.15Pt0.85sample and Tc≈288K for the Fe 0.25Pt0.75sample. In Fig. 1(c)we show the temperature dependence of the “anomalous Hall angle ”¼ρxy/ρxxof the Fe 0.15Pt0.85ð10Þ and Fe 0.25Pt0.75ð10Þsamples as measured in a magnetic fieldHz¼2kOe applied perpendicular to the plane of the film. (Here ρxxis the resistivity in the direction of current flow and ρxyis the transverse Hall resistivity). As can be seen in the Fe 0.15Pt0.85ð10Þsample, there is a significant AHE at high temperature that increases gradually as thetemperature is decreased toward 200 K and then increasesmore rapidly as the magnetization in the film develops as T decreases below T c, qualitatively as would be expected for the case of strong skew scattering from the Fe ions. In theinset of Fig. 1(c), there is a similar temperature dependence trend for the Fe 0.25Pt0.75ð10Þsample below its Curie temper- ature. In the AHE what is detected is the charge flow in thedirection perpendicular to the plane defined by the biascurrent direction yand the direction of the internal magnetic field z. This transverse charge flow is accompanied by a diffusive spin current arising from the spin-dependentscattering. The resulting V AHE, or, equivalently, ρxy,s c a l e s for the extrinsic case with the rate of skew scattering, but it also depends on the strength of the internal magnetization ofthe material and its spin dependent charge transport proper-ties. This makes it challenging to quantify the underlyingspin flow based only on AHE measurements. To achieve better quantitative measurements of the spin currents produced by an electrical current in the FePtalloys we employed the harmonic response SOT technique[26,27] , whereby we measured the magnetic deflection of (a) (c) (d)(b) FIG. 1. (a) XRD measurements on the samples IrMn 3ð10Þ/ F0.50Pt0.50ð10Þ/MgO, IrMn 3ð10Þ/Fe0.25Pt0.75ð10Þ/MgO, and IrMn 3ð10Þ/Fe0.15Pt0.85ð10Þ/MgO, and two control samples IrMn 3ð10Þ/Ptð1Þ/MgO and IrMn 3ð10Þ/Ptð8Þ/MgO. (b) Tempera- ture dependent VSM measurements on the samplesIrMn 3ð10Þ/Fe0.25Pt0.75ð10Þ/MgO and IrMn 3ð10Þ/Fe0.15Pt0.85ð10Þ/ MgO. The dashed lines are fits to the empirical equationM sðTÞ¼Msð0Þ½1−ðT/TcÞα/C138β. (c) Temperature dependence of the anomalous Hall angle of the samples IrMn 3ð10Þ/ Fe0.15Pt0.85ð10Þ/MgO (main) and IrMn 3ð10Þ/Fe0.25Pt0.75ð10Þ/ MgO (inset). (d) Schematic of the Hall bar device.PHYSICAL REVIEW LETTERS 120, 097203 (2018) 097203-2an adjacent, perpendicularly magnetized ferromagnetic thin film that occurs as the result of the spin torque arising fromthe absorption of the transverse polarized component of the spin current emanating from the spin source, the FePt alloys in this case. Such measurements of SOT effective fieldsusually only set a lower bound on θ SHdue to the expected less than perfect spin transparency of the interface between the spin source and spin sink [28,29] . For the harmonic response SOT measurements we fabricated two sets of FePt-based multilayer samples: IrMn 3ð10Þ/Fe0.15Pt0.85ð10Þ/Hfð1Þ/FeCoB ð1Þ/Hfð0.35Þ/MgO (A), and IrMn 3ð10Þ/Fe0.25Pt0.75ð10Þ/Hfð0.8Þ/FeCoB ð1Þ/ Hfð0.35Þ/MgO ( B), where FeCoB represents Fe 60Co20B20. The amorphous Hf insertion layer [1 and 0.8 nm for ( A) and (B), respectively] between the FePt and the FeCoB was employed to counter the detrimental effect of the fcc crystal structure of the FePt on obtaining perpendicular magnetic anisotropy (PMA) in the thin FeCoB layer, while the verythin (0.35 nm) Hf insertion layer between the FeCoB and the MgO enhanced the interfacial magnetic anisotropy energy density, strengthening the PMA [30]. In Fig. 2(a), we show the response of the anomalous Hall resistance of one of the Fe 0.15Pt0.85heterostructure Hall bars (sample A) to an applied out-of-plane field Hzat different temperatures between 300 and 140 K. The sharpfield-induced switching events, with an increasing coer- civity upon decreasing temperature, are from the PMA FeCoB layer. When the temperature is lower than the Curietemperature of Fe 0.15Pt0.85there is also a quasilinear background evident for Hzgreater than the coercive fields that is much larger than can be expected from the ordinaryHall effect, and instead is due to the AHE of the in-plane magnetized FePt layer. The AHE resistance for sample ( B) is similar [31]. We determined the dampinglike (DL) and fieldlike (FL) effective fields ( ΔH DLand,ΔHFLrespectively) arising from the SOT by measuring the first and second harmonic transverse Hall signals VωandV2ω[27], from which we obtain ΔHLðTÞ¼−2ð∂V2ω/∂HLðTÞÞ/ð∂2Vω/∂H2 LðTÞÞ and, hence, ΔHDL¼ðΔHLþ2δΔHTÞ/ð1−4δ2ÞandΔHFL¼ðΔHTþ2δΔHLÞ/ð1−4δ2Þ. Here, HLðTÞis the external bias field applied parallel to (transverse to) the current direction and δis the ratio of the planar Hall resistance to the anomalous Hall resistance [26,27] . In Fig. 2(b)we show the temperature dependences of the DL effective fields for both sample ( A) and sample ( B), plotted as a function of T/Tc, with Tcdetermined from the fits to the magnetization of the samples [Fig. 1(b)] (See Supplemental Material [31] for discussion of the FL SOT behavior of these samples). For sample ( A) (Fe 0.15Pt0.85) the measurement is from room temperature 293 to T¼160K, and for sample ( B) (Fe 0.25Pt0.75) from 330 to 275 K. For sample ( A) for which we have measurements starting around 100° K above Tc, we see that the DL effective field per current density ΔHDL/ΔJeis more or less constant untilT/Tc≈1.44 (250 K), below which it increases, at first gradually, then very rapidly reaching a peak near Tc (172 K) more than 3 times its 293 K value. This behavior is dramatically different from that of the DL torque foundwith conventional heavy metal systems [32,33] . Below this peakΔH DL/ΔJethen drops off even more quickly until below T/Tc¼0.92 (160 K) the observable development of spin torque from the PMA FeCoB on the emerging strongferromagnetism of the in-plane polarized Fe 0.15Pt0.85makes further quantitative harmonic response measurementsuntenable (see Supplemental Material [31] for more information). As also shown in Fig. 2(b), the behavior of sample ( B) over the same scaled temperature range above and belowT c, is quite similar, with the peak value of ΔHDL/ΔJebeing less than 20% different than that of sample ( A), and even less if we take into account the spin attenuation effect of thedifferent Hf spacer thickness (1 nm for sample Aand 0.8 nm for sample B) where Hf has a spin diffusion length of approximately 1 nm [34]. This close similarity in the values of the peak antidamping spin torque is observeddespite the 35% difference in resistivity, and 67% differ-ence in Fe concentration. This is consistent with skewscattering being the dominant spin Hall effect in thesematerials, at least in the ferromagnetic transition region, butmore study will be needed to confirm that attribution. Some years ago Kondo [35] developed a theory for the scattering of conduction electrons by localized orbitalmoments to explain an anomaly in the magnetoresistanceand AHE of ferromagnetic Ni and Fe near their Curie points[36], with Kondo attributing the anomaly to increasingly stronger spin fluctuations as T→T cfrom below. Recently Guet al. [37]extended this theory to explain results by Wei et al. [15]on inverse spin Hall effect (ISHE) measurements of NiPd alloys near their Tc, including the effect of correlations between neighboring localized moments. We surmise that spin fluctuations are also the origin of the strongpeak in the SOT torque (spin current) that we observe withthe FePt alloys, although our results, in addition to being adirect measure of the J sgenerated by the spin Hall effect,(a) (b) FIG. 2. (a) Temperature dependent AHE resistance of sample (A)F e 0.15Pt0.85. (b) Dampinglike effective fields of sample ( A) Fe0.15Pt0.85and ( B)F e 0.25Pt0.75as a function of normalized temperature T/Tc. Their temperature dependent magnetizations are also plotted here for comparison.PHYSICAL REVIEW LETTERS 120, 097203 (2018) 097203-3differ from the earlier work by the strength of the effect, which we attribute to the exceptionally strong spin-orbit interaction in the Fe-Pt system. Our results are also distinc- tive in that the peak in the SOT effective field (emitted spincurrent) is followed by a sharp decline with decreasing temperature that we attribute to the effect of the internal exchange field in the FePt that develops as Tis lowered below T c, which, once well established, acts to quickly dephase a spin current that is polarized in a direction not collinear with the internal magnetization [23]. If spin fluctuations are indeed the origin of the enhanced SHE in the FePt alloys near Tc,t h e ni ti sp r e d i c t e d [38]that there should also be an enhancement of the effective interfacial spin-mixing conductance g↑↓ effbetween the FeCoB and FePt alloy in the vicinity of the latter ’sC u r i e point, as has been recently observed with antiferromagnetic spin sinks near their N´ eel point by inverse spin Hall measurements [39] a n ds p i np u m p i n g [40]. In the latter case the interfacial enhancement of damping Δα≡ αðtFMÞ−α0,w h e r e α0is the Gilbert damping parameter for the bulk FM material, can be related to g↑↓ effbyg↑↓ eff¼ 4πMstFeCoBΔα/ðγℏÞ[41]. We do indeed observe a pro- nounced peak in magnetic damping of the FeCoB layer in our samples as they are cooled through the Tcof the FePt. In Fig.3(a)we show the effective spin mixing conductance, g↑↓ eff as determined by resonant linewidth measurements made by flip-chip field-modulated FMR, on a Fe 0.25Pt0.75ð10Þ/ Hfð0.25Þ/FeCoB ð7.3Þ/MgO sample (See SupplementalMaterial [31]for details of measurements). As can be seen g↑↓ effincreases rapidly as Tmoves below Tc, and then drops abruptly by more than a factor of 3 to a value ( ≈30nm−2) much closer to that expected for a typical FM/Pt interface[29]. The temperature-dependent behavior observed here is distinctly different from the temperature-insensitive g↑↓ effin Pt/ferromagnet bilayer systems [42]. Note that the peak ing↑↓ effdoes not occur simultaneously with the peak in ΔHDL/ΔJewhich indicates that the latter ’sp e a ki sn o tj u s t due to an enhanced g↑↓ eff. To better quantify the peak strength of SHE in the FePt alloys and to account for the spin current attenuation in the Hf spacer layer, we prepared another Fe 0.25Pt0.75hetero- structure, IrMn 3ð10Þ/Fe0.25Pt0.75ð10Þ/Hfð0.5Þ/FeCoB ð0.9Þ/ Hfð0.35Þ/MgO, i.e., with a thinner (0.5 nm) Hf spacer layer. In Table Iwe compare the peak ΔHDL/ΔJevalue that we obtained with a Hall bar measurement of this sampleto that previously measured [33] for a Pt ð4Þ/Hfð0.5Þ/ FeCoB ð1Þ/MgO sample at 293 K (room temperature). Assuming the same spin current attenuation in both casesfrom the 0.5 nm Hf/FeCoB interface, this indicates that the peak spin Hall effect in the Fe 0.25Pt0.75is approximately 5.5x larger than in Pt(4). More quantitatively, with thetorque-field relation ξ DL¼ð2e/ℏÞMstFeCoB ðΔHDL/ΔJeÞ [29], we can calculate the DL spin torque efficiency ξDL¼0.34/C60.02for the sample Fe 0.25Pt0.75ð10Þ/Hfð0.5Þ/ FeCoB ð1Þ. Considering the attenuation of the spin current as it passes through the 0.5 nm of Hf, and the likely less than ideal spin transparency of the Hf/FeCoB interface, this only sets the lower bound for the peak spin Hall ratio of the Fe0.25Pt0.75material as ≥0.34. Another key parameter for understanding and optimizing the effectiveness of SOTs arising from the SHE is the spin diffusion length λswithin the material. We obtained a measure of λsby producing a series of PMA samples without the IrMn layer Fe 0.25Pt0.75ðtFePtÞ/Hfð0.8Þ/ FeCoB ð1Þ/Hfð0.35Þ/MgO/Ta ð1Þ, where the thickness tFePtof the FePt alloy was varied from 2 to 10 nm. The measured dampinglike effective fields for these samples are plotted in Fig. 3(b) as a function of tFePtfor two different temperatures 293 and 330 K, i.e., in the near vicinity of Tcand somewhat above it. The solid lines are a fit of the functionΔH DLðtFePtÞ/ΔJe¼½ΔHDLð∞Þ/ΔJe/C138½1−sechðtFePt/λsÞ/C138[43] to the data. The results at the two temperatures are quite similar, with λs≈1.5nm.(a) (b) FIG. 3. (a) The effective spin mixing conductance of an in- plane magnetized Fe 0.25Pt0.75ð10Þ/Hfð0.25Þ/FeCoB ð7.3Þsample as determined by a flip-chip FMR measurement of the dampingparameter for the FeCoB resonance. The temperature dependenceof the DL effective field of sample ( B) is also plotted here for comparison. (b) Spin diffusion length measurement of thesamples Fe 0.25Pt0.75ðtÞ/Hfð0.8Þ/FeCoB ð1Þ. TABLE I. Comparison of the dampinglike effective fields as measured at 293 K for two Fe 0.25Pt0.75/Hf/FeCoB samples and a Pt/Hf/FeCoB sample. Sample Fe 0.25Pt0.75ð10Þ/Hfð0.8Þ/FeCoB ð1Þ Fe0.25Pt0.75ð10Þ/Hfð0.5Þ/FeCoB ð1Þ Ptð4Þ/Hfð0.5Þ/FeCoB ð1Þ DL effective field× 10−6Oe/ðA/cm2Þ5.6 12.2 2.3 Reference This work This work Ou et al. [33]PHYSICAL REVIEW LETTERS 120, 097203 (2018) 097203-4We further confirm the strength of the SHE in the FePt alloys with current-induced switching measurements. In Fig. 4we show the switching behavior of sample ( B)a s measured at 293 K, in close vicinity to Tc. The direction of current-induced switching is reversed upon changing the sign of a small in-plane applied magnetic field, in a way that is characteristic of antidamping torque SHE switching [44]. The switching current density for this case was≈6×106A/cm2. In summary, we have studied the spin-orbit torques resulting from the SHE in chemically disordered FePt alloys, above and through their ferromagnetic transition points Tc. The SOTs exerted by these materials on an adjacent FeCoB thin film exhibit a striking temperature dependence in which the DL SOT displays a strong maximum in the vicinity of Tc. We attribute this pro- nounced SOT behavior to spin fluctuation enhancement of the spin Hall effect arising from the strong spin-orbit interaction between the conduction electrons and the localized Fe moments. There is also a strong T-dependent enhancement of the effective spin-mixing conductance of the FeCoB/FePt interface that we similarly attribute to spin fluctuations in the FePt ferromagnetic transition region. The peak strength of the DL SOT indicates an exceptionally large spin Hall angle, >0.34near the Curie point of the FePt alloys. We also realized current-induced magnetiza- tion switching by the DL SOT in close vicinity to the Curie point of these ferromagnetic FePt alloys and measured the spin diffusion length to be quite similar, ≈1.5nm, both above and in close vicinity to Tc. This fluctuation enhanced spin Hall effect, which is tunable through the composition of the FePt alloy, provides new opportunities for the study of spin-dependent scattering and transport in systems with very strong spin-orbit interactions, and for applications where a very strong spin current from a relatively low resistivity material can be particularly beneficial. Y . 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PhysRevB.94.184415.pdf	PHYSICAL REVIEW B 94, 184415 (2016) Multiscale model approach for magnetization dynamics simulations Andrea De Lucia,1,2Benjamin Kr ¨uger,1Oleg A. Tretiakov,3,4and Mathias Kl ¨aui1,2 1Institute of Physics, Johannes Gutenberg University, Staudingerweg 7, 55128 Mainz, Germany 2Graduate School of Excellence—Materials Science in Mainz, Staudingerweg 9, 55128 Mainz, Germany 3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 4School of Natural Sciences, Far Eastern Federal University, Vladivostok 690950, Russia (Received 8 July 2016; revised manuscript received 27 September 2016; published 14 November 2016) Simulations of magnetization dynamics in a multiscale environment enable the rapid evaluation of the Landau- Lifshitz-Gilbert equation in a mesoscopic sample with nanoscopic accuracy in areas where such accuracy isrequired. We have developed a multiscale magnetization dynamics simulation approach that can be applied tolarge systems with spin structures that vary locally on small length scales. To implement this, the conventionalmicromagnetic simulation framework has been expanded to include a multiscale solving routine. The softwareselectively simulates different regions of a ferromagnetic sample according to the spin structures located withinin order to employ a suitable discretization and use either a micromagnetic or an atomistic model. To demonstratethe validity of the multiscale approach, we simulate the spin wave transmission across the regions simulatedwith the two different models and different discretizations. We ﬁnd that the interface between the regions is fullytransparent for spin waves with frequency lower than a certain threshold set by the coarse scale micromagneticmodel with no noticeable attenuation due to the interface between the models. As a comparison to exact analyticaltheory, we show that in a system with a Dzyaloshinskii-Moriya interaction leading to spin spirals, the simulatedmultiscale result is in good quantitative agreement with the analytical calculation. DOI: 10.1103/PhysRevB.94.184415 I. INTRODUCTION To model magnetization dynamics, currently two paradigms are commonly used in the ﬁeld: the micromagneticmodel and the Heisenberg spin model. The micromagneticmodel [ 1] is ideal when simulating systems with linear dimensions of the order of a few nanometers or larger; sinceit is a continuous model that is discretized for computationalapplication, its reliability decreases dramatically when sim-ulating magnetic structures exhibiting a large gradient thatcannot be resolved by the ﬁnite size cells. A textbook examplefor this scenario is offered by Bloch points [ 2]( s e eF i g . 1); domain walls and spin waves also belong to this category forparticular values of the material parameters. The Heisenberg model [ 3–5] is a discrete description, where with every atom in the lattice of the ferromagnet,a magnetic moment is associated. Since this is a discretemodel, its capability to simulate any magnetic structure isnot limited by computational artifacts originating from thediscretization of a continuum model, which makes it distinctfrom micromagnetism. On the other hand, the Heisenbergmodel cannot be efﬁciently used to simulate systems largerthan a few nanometers due to the computational time in- creasing faster than linearly with the number of atoms [ 6,7]. In the presented approach (Fig. 2), the entire system is simulated using the micromagnetic model, while one or moreregions of it containing large gradient structures (e.g., Blochpoints) are simulated using the discrete Heisenberg model. Themain obstacle for the development of a combined multiscaletechnique consists of devising accurate conditions to make theinterface between regions on two different scales magneticallysmooth, in order to prevent any interface related artifacts. While in magnetization dynamics, adaptive mesh reﬁne- ment techniques [ 8,9] have been used, none of these employed different models for different scales. One related approach hasbeen proposed, addressing the problem of interfaces between layers of different magnetic materials [ 10–12]. However, the lack of proper interface conditions, in particular, thechoice of applying a coarse scaled exchange ﬁeld on themagnetic moments along the interface in the ﬁne scale region,restricts the validity of this approach to systems with uniformmagnetization across the interface. While this shortcominghas been later resolved in Refs. [ 13,14], these approaches were devised to evaluate equilibrium conﬁgurations rather thansimulating dynamical systems. One further related approach [ 15] employed the ﬁnite element method. It should be noted, however, that while in this case the atomic lattice in the Heisenberg model can berendered more accurately, the computational times cannotbe dramatically reduced, as shown for our ﬁnite differenceapproach in Ref. [ 6], making this approach considerably slower. One further multiscale approach [ 16], devised for a different scale combination than the presented one, proposed to use the micromagnetic model as the ﬁne scale model and the Maxwell equations as the coarse scale model; this is,however, restricted to systems with slowly varying magne-tization. Another work [ 17] uses special relativity to evaluate a corrective term to the Landau-Lifshitz-Gilbert equation in thecase of domain wall motion. In continuum mechanics [ 18,19], multiscale approaches are commonly applied to the investi- gation of mechanical properties of materials, such as their response to deformations and fractures. However, so far it isunclear whether one can develop such a multiscale modelfor magnetization dynamics that allows one to carry outvalid simulations of systems that cannot be modeled with thecurrently available approaches. In this paper we show the details of the multiscale approach, with a particular focus on the interface conditions that wedeveloped to obtain a smooth interaction between regions 2469-9950/2016/94(18)/184415(7) 184415-1 ©2016 American Physical SocietyDE LUCIA, KR ¨UGER, TRETIAKOV , AND KL ¨AUI PHYSICAL REVIEW B 94, 184415 (2016) FIG. 1. (a) Schematic of a magnetization structure with a mi- cromagnetic singularity (Bloch point). The two gray domains are separated by two Bloch walls (black). The Bloch walls have anopposite sense of rotation and are separated by two N ´eel/Bloch lines (blue). Between the two N ´eel/Bloch lines with opposite orientations, a micromagnetic singularity (red) is formed. A magniﬁcation of thered square is shown in (b). (c) A micromagnetic singularity also occurs during the reversal of a magnetic vortex core [ 20,21]. These diagrams were adapted from Ref. [ 22]. on different scales. Finally, demonstrations of the validity for the approach are shown, revealing the transmission ofspin waves across the scale interface without attenuation, andcomparing the simulated ground state for structures exhibitingDzyaloshinskii-Moriya interaction to the analytical theory. II. METHOD The multiscale approach solves the Landau-Lifshitz-Gilbert equation numerically for two different models: the coarsegrained micromagnetic model, which simulates the wholesample, and the ﬁne scale model, which is used for mag-netic structures that cannot be accurately described by themicromagnetic model, discretizing the magnetization ﬁeld atatomic resolution and simulating it in the intrinsically discreteHeisenberg spin model. Our software executes in parallel twoindependent solving routines, one for each model (it is inprinciple possible to execute any number of ﬁne scale solvingroutines), performing one full computational step on the coarsescale one, and then a short series of steps on the ﬁne scale onecentered around the time coordinate of the coarse one (seeFig.3). The main task towards the development of this technique consisted in modeling the interaction between different re-gions. This was achieved by applying, after each coarse scalestep, a set of magnetic ﬁelds designed to approximate theeffect of the nonlocal terms of the effective magnetic ﬁeldfrom one region on the other (see Fig. 4), namely, exchange and stray ﬁeld. These magnetic ﬁelds are designed as follows:The exchange ﬁeld, generated by the ﬁne scale magneticmoments closest to the interface (“interfacial moments”),on their “neighboring” cells in the coarse scale (“interfacialcells”) is evaluated by averaging all the interfacial momentsinside each coarse scale cell. The average vector is rescaledby the volume V aof a cell in the atomic lattice, in order FIG. 2. Schematic diagram showing the basis of the multiscale model. (a) In this example each cell in the vortex core region is simulated in the coarse scale. (b) depicts the multiscale simulation,where a small region (central nine cells) is simulated using the atomistic model, while the rest of the sample is simulated using the micromagnetic model. The color code shows the out of plane component of the magnetization in units of M s. to obtain the magnetization (A /m), rather than the magnetic moment (A m2). A new ﬁnite difference mesh, with coarse scale discretization, is created and the cells corresponding tothe internal surface of the ﬁne scale region are ﬁlled withthe difference between the magnetization of the same cell in the original coarse mesh and the new vectors. In this way, the linearity of the exchange ﬁeld with respect to themagnetization is exploited to evaluate a correction to the ﬁeld,calculated in the micromagnetic formulation, generated by theoriginal coarse scale cells alone. The corrected exchange ﬁeld,exerted by the multiscale cell jon the micromagnetic cell i,i s calculated as H ex(Mj)=Hex(Mint,j−Mj)+Hex(Mj). (1) Here, Mjdenotes the magnetization in the cell jin the purely micromagnetic simulation, while Mint,jis deﬁned as Mint,j=Ms \|μ\|NintNint/summationdisplay kμk=/summationtextNint kμk VaNint, (2) where the sum runs over all the magnetic moments μlocated along the interface on the side of cell jthat is neighboring cell i. This formula is only fully valid in the case of complete cells, without additional or missing atoms. This effective ﬁeld termis evaluated in the micromagnetic model. Likewise, to evaluatethe exchange ﬁeld generated by interfacial cells on interfacial FIG. 3. Diagram showing the multiscale model in the time domain: After each coarse computational time step, the corrections to the effective ﬁeld in the ﬁne scale region, generated by the coarseone, are evaluated, a short series of ﬁne steps centered around the latest coarse step, of length h, is executed, and then corrections to the coarse scale effective ﬁeld (generated by the Heisenberg ﬁne scaleone) are evaluated. 184415-2MULTISCALE MODEL APPROACH FOR MAGNETIZATION . . . PHYSICAL REVIEW B 94, 184415 (2016) FIG. 4. The key players in the evaluation of the cross-scale effective ﬁeld terms: magnetic moments (red), micromagnetic cells (black), interfacial moments and cells (highlighted in blue), ghost moments, which are not part of the LLG solving routine (white). The dashed lines show how the ghost moments, and, in particular, the onemarked in green, are evaluated as the bilinear interpolation of ﬁne scale moments and coarse scale magnetization. moments, interpolation is employed in order to deﬁne a set of new magnetic moments (“ghost moments” [ 8]) to act as ﬁrst neighbors to the interfacial ones. The exchange ﬁeldgenerated by the ghost moments is evaluated in the Heisenbergspin model. A combination of ﬁne scale moments and coarsescale magnetization is used in the interpolation in order toensure a smooth transition in the magnetic pattern across theinterface. This means that each ghost moment results fromthe interpolation of atomistic and aptly renormalized micro-magnetic vectors. The interpolation can be linear, bilinear, orquadrilinear according to the dimensionality of the coarse scalemesh. The same techniques, based on the average of interfacialmagnetic moments, and the calculation of ghost momentsthrough interpolation across the interface, are employed whenevaluating antisymmetric exchange (Dzyaloshinskii-Moriyainteraction) across the scale interface. The stray ﬁeld contains all the long range contributions to the effective ﬁelds. The implementation of this ﬁeldconstitutes one of the main differences between the twomodels. In both scales the demagnetization tensor formulationwas employed [ 23], as well as the calculation method based on fast Fourier transform (FFT) for efﬁcient calculation [ 6]. While for the coarse cells the demagnetization tensor describesthe interaction between two uniformly magnetized solidrectangles, according to the calculations carried on by Newellet al. [23], the demagnetization tensor used for two full magnetic moments in the ﬁne scale is deﬁned as 1 4π/bracketleftbigg1 \|ri−rj\|3−3(ri−rj)⊗(ri−rj) \|ri−rj\|5/bracketrightbigg , (3) where riandrjare the positions of two magnetic moments, 1is the 3 ×3 identity matrix, and the symbol ⊗denotes the tensor product. Similarly to the exchange ﬁeld, the stray ﬁeld is linear in the magnetization vector and this property is exploited likewise.The correction to the stray ﬁeld generated in the micromagneticsystem by ﬁne scale regions is evaluated using the averagedvalue of magnetic moments in each cell.In order to evaluate the complete demagnetization ﬁeld acting on the ﬁne scale system, the coarse scale magnetizationstructure is copied into a new mesh and the cells correspondingto the ﬁne scale region are ﬁlled with zero vectors. The strayﬁeld generated by this system is evaluated. This technique isemployed in order for the ﬁeld generated by the ﬁne scaleregion on itself not to be evaluated twice. Since the ﬁeld hasthe same discretization as the structure generating it, the resultis then interpolated, in order for it to have the discretizationof the ﬁne scale mesh. The type of linear interpolationdepends, as for the ghost moments, on the dimensionalityof the mesh. This is the only case for an effective ﬁeldterm evaluated micromagnetically to be applied on the ﬁnescale region. This approximation is made necessary by thecomputational complexity of the algorithm calculating theﬁeld, increasing with Nlog(N), where Nis the number of cells. This dependence is due to the method employed forcalculating the demagnetization tensor based on FFT [ 6,7]. III. SIMULATIONS Having implemented the approach, we run a series of tests as a demonstration of the validity of our model. The simulatedsystem was a one-dimensional nanowire, 1 .8μm long with a square 0 .3×0.3n m 2section. The ﬁne scale domain was 90 nm long (Fig. 5), and the material parameters for this system are those commonly used for permalloy, namely, Ms= 8×105A/m, exchange constant A=1.3×10−11J/m, and Gilbert damping constant α=0.01 [24]. For the purpose of efﬁciency and due to the constraints of the ﬁnite differencemethod, upon which the original software is based, the crystalin the atomistic region is considered to be ordered in a simplecubic lattice with a lattice constant l=0.3 nm, comparable to the ones of iron and nickel. Spin waves of different frequencies ω 0were excited by applying an alternating transversal magnetic ﬁeld Haltto a short (3 nm long) section of the wire. The magnetization asa function of time was measured on the atomistic momentfurthest from the region where H altis applied [ μj(t)], and on the neighboring micromagnetic cell [ Mi(t)], the transversal component of the two arrays was normalized, and thenanalyzed using FFT in order to ﬁnd μ j(ω) andMi(ω). Peaks with frequency corresponding to the frequency of Haltwere easily identiﬁable. The height of such peaks increased linearlywith the amplitude of H alt. The peaks μj(ω0) andMi(ω0)w e r e squared and the transmission coefﬁcient Tacross the interface has been evaluated by calculating the ratio between the two: T(ω0)=\|μj(ω0)\|2 \|Mi(ω0)\|2. (4) FIG. 5. Diagram showing the ﬁne scale region of the nanowire and its immediate surroundings. An oscillating magnetic ﬁeld Haltis applied to a section of the ﬁne scale region to excite spin waves. The amplitude of the spin wave is evaluated in the atomistic ﬁne scale celljand the coarse scale cell iwhich is described micromagnetically. 184415-3DE LUCIA, KR ¨UGER, TRETIAKOV , AND KL ¨AUI PHYSICAL REVIEW B 94, 184415 (2016) FIG. 6. Linear regression used to measure the relation between the excited wave vector kand the excitation frequency ω. For some values of the frequency, a purely atomistic simulation was performed for comparison and with the aim ofobtaining the relation between the frequency and wavelength.Using FFT in the space domain, the corresponding wavenumber kwas measured for each value of the excitation fre- quency. In particular, such Fourier transforms were evaluatedat different time instants and then averaged. Once again peakswere easily identiﬁable. By means of linear regression (seeFig.6) the dependence k 2(ω) was measured and the wavelength corresponding to each value of the excitation frequency wascalculated as λ(ω)=2π/k(ω). IV . RESULTS Three sets of simulations were performed, with different lengths of the micromagnetic cells, corresponding to 10, 20,and 30 times l(0.3 nm). The data show ideal transmission for frequency values smaller than a sharply deﬁned cutofffrequency. The same data, as a function of the wavelength,show consistently that the transmission drops to zero at a cutoffwavelength corresponding to a speciﬁc value of the coarse cellsize. This universal behavior can be considered as a limitationof computational micromagnetism, which does not allow oneto simulate very short wavelength spin waves without reﬁningthe mesh, introducing therefore a dramatic increase in thecomputation time (Fig. 7). Since we assume that the frequency cutoff is a consequence of the coarse scale not being able to resolve waves with such ahigh frequency, we simulated a similar system, this time withthe excitation being applied on the coarse scale region only.Here, the waves propagate into and then out of the ﬁne scaleregion and the transmission is measured for waves leaving theﬁne scale region (Fig. 8). The test was repeated using periodic boundary conditions to make sure that the sharp cutoff was notcaused by the waves being reﬂected at the end of the wire. Bothtests were then repeated for different values of the exchangeconstant. In order to measure the cutoff frequencies, a linear regres- sion was executed on all the transmission values between 0.1and 0.9, and the intersection of this line with the transmissionvalue of 0.5 was deﬁned as the cutoff frequency. We assume the FIG. 7. The measured transmission for waves excited in the ﬁne scale region with open boundary conditions as a function of their (a) frequency ωand (b) wavelength λ. A transmission of 1 (100%) for a wide range of wavelengths demonstrates the numerical validity of the model. FIG. 8. The measured average transmission for waves of all possible frequencies excited in the coarse scale region, before entering the ﬁne scale one, with closed and periodic boundary conditions (BC) for different values of the exchange constant A. The data shown are the result of an average over all the frequencies. Peaks with frequency higher than 3 .5 THz were not visible in the Fourier transform, underlining the fact that the cutoff is a consequence ofthe waves not being resolved for the chosen cell size. The observed transmission of approximately 1 shows the validity of the method with no artiﬁcial attenuation at the interface between the regionswhere different models are used. 184415-4MULTISCALE MODEL APPROACH FOR MAGNETIZATION . . . PHYSICAL REVIEW B 94, 184415 (2016) FIG. 9. The measured cutoff frequency ωcutfor waves excited in the ﬁne scale region with closed and periodic boundary conditions(BC) for different values of the exchange constant A.Ac u t o f f frequency depending on the exchange constant demonstrates that this phenomenon is strictly micromagnetic and is not introduced bythe multiscale approach. cutoff to be a direct consequence of the exchange interaction not being accurately evaluated in the micromagnetic modelwhen the angle in the magnetization between two neighboringcells is too large. The dependence of the cutoff frequency onthe exchange constant supports this hypothesis (see Fig. 9). FIG. 10. (a) The two components of the magnetization for a multiscale DMI helix, showing continuity and consistency of theperiod in the coarse and ﬁne scale region. Below the direction of the spin helix is schematically shown. (b) The wave number of the helix increases linearly with the DMI constant and is consistent with theexpected value [ 26]. FIG. 11. Domain wall displacement after the application of a unidirectional Gaussian-shaped magnetic ﬁeld pulse with differentvalues of height and width as a function of the tracking distance. This is the distance traveled by the domain wall before the ﬁne scale region is centered around it. We expect this parameter not to inﬂuence thedynamics of the system and the data conﬁrm this assumption. V . DZYALOSHINSKII-MORIYA INTERACTION To demonstrate the reliability of the method used to evaluate effective ﬁelds across the interface by direct comparisonto analytical theory, a system exhibiting antisymmetric ex-change [ 25,26] was simulated. The nanowire was similar in shape to the one used to test spin wave transmission, withthe parameters M s=1.05×106A/m and exchange constant A=11×1011J/m. Different values of D=\|Dij\|were used. The vector Dijscales the energy density of the Dzyaloshinskii- Moriya interaction (DMI) as calculated in Ref. [ 26]: eDMI=Dij·(μi×μj)/\|μ\|2. The system was relaxed in a coarse scale simulation, then a ﬁne scale region was applied on a section of the wire,and the system was relaxed again. The relaxed state (seeFig. 10) showed continuity in the helix structure, typical of systems exhibiting DMI, with a pitch in agreement with thepredicted [ 26] value of D/(4πA). The pitch was evaluated from the Fourier transform in the space domain for the twocomponents of the helix, using the data points from both scalesand taking the peak value from the Fourier transform. Thecomponents of Mevidently have a perfectly sinusoidal shape [see Fig. 10(a) ]. VI. TRACKING A tracking algorithm was devised in order to keep the ﬁne scale region as small as possible; it scans the ﬁne scale regionfor the position of the structure of interest (SOI), usually thespin structure with large magnetization gradients, and shifts 184415-5DE LUCIA, KR ¨UGER, TRETIAKOV , AND KL ¨AUI PHYSICAL REVIEW B 94, 184415 (2016) the ﬁne scale region by an integer number of coarse scale cells units, in order to always have the SOI close to its center.When micromagnetic cells previously not part of the ﬁne scaleregion become included, interpolation is applied in order to ﬁllin the ﬁne scale mesh with magnetic moments that accuratelyreproduce the coarse scale magnetization and are continuouswithin and across the scale interface. To show that the ﬁne scale area can be reliably moved, a test was performed. This test simulated domain wallmotion in a nanostrip (3 μm×33 nm ×0.3 nm) induced by a unidirectional magnetic ﬁeld. The material parameters of thestrip are the same as the nanowire from the previous test, withthe only exception being the Gilbert damping α=0.1. The domain wall is initially in the center of the ﬁne scale region,and when the distance from the starting position becomeslarger than a certain threshold (tracking distance), the wholeﬁne scale region is shifted, in order to keep it centered. Thetest was repeated for different tracking distances to show thatthis process does not inﬂuence the dynamics of the system(Fig. 11). VII. CONCLUSIONS We have presented an innovative methodology to perform magnetization dynamics simulations in the systems whichcannot be accurately simulated otherwise. Since some ofthese systems describe phenomena, including vortex coreswitching [ 20,27–30] and skyrmion nucleation [ 31,32], that are considered to be important problems in spintronics, wedeem this methodology a key step to advance this ﬁeld. In orderto improve the technique and establish multiscale simulationsas a valuable tool, its basic features have been described andits limits have been tested. The transmission data for the spin waves show that information about magnetic structures in the ﬁne region cancross perfectly the scale interface, thus demonstrating thereliability and numerical validity of our model. A thoroughanalysis of the cutoff phenomenon found for spin wavetransmission shows that, in the presence of spin waves with ashort wavelength, the multiscale approach can be reliably usedunder the condition that the waves do not leave the ﬁne scaleregion. Meanwhile, the traditional approach—a reﬁnement of the whole mesh—would increase the computational timedramatically. The simulations including the DMI furthershow that the method employed for evaluating cross-scaleinteractions ensures continuity between the regions of differentscales and yields quantitative agreement with the analyticaltheory. Moreover, the domain wall data indicate the reliabilityof the tracking algorithm and its effectiveness as a method tokeep the size of the ﬁne scale regions at a minimum and doesnot introduce artifacts to the simulated results. As a future direction, we propose to analyze the dynamics of magnetic vortex core reversal, a phenomenon that requiresa similar approach in order to be accurately simulated [ 20]. Further research will include magnetic structures such asskyrmions which are stabilized by DMI and where the nu-cleation involves Bloch points. In the long term, there is roomfor further improvements: Generalizing the approach beyondsimple cubic lattices in the ﬁne scale region, optimization ofthe computational routines, and extension of this approach toantiferromagnets and nonzero temperatures are some of theexamples that will broaden the applicability even further. ACKNOWLEDGMENTS A.D.L. acknowledges support as a recipient of a schol- arship through the Excellence Initiative by the GraduateSchool Materials Science in Mainz (GSC 266), B.K. ac-knowledges support as the recipient of the Carl ZeissPostdoc Scholarship–Multiskalensimulationen f ¨ur energies- parende Magnetisierungsmanipulation. The authors acknowl-edge the support of SpinNet (DAAD Spintronics network,Project No. 56268455), the EU (ERC-2014-PoC 665672,MultiRev), and the DFG (SFB TRR 173 SPIN +X). O.A.T. acknowledges support by the Grants-in-Aid for ScientiﬁcResearch (Grants No. 25800184, No. 25247056, and No.15H01009) from MEXT, Japan. M.K. thanks ICC-IMR at To-hoku University for the hospitality during a visiting researcherstay at the Institute for Materials Research. We are thankful toU. Nowak and D. Hinzke from the University of Konstanz fortheir help with testing the ﬁne scale model. [1] A. Aharoni, Domain walls and micromagnetics, J. Phys. Colloq. 32,C1-966 (1971 ). [2] A. 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PhysRevB.101.174431.pdf	PHYSICAL REVIEW B 101, 174431 (2020) Unusual anomalous Hall effect in perpendicularly magnetized YIG ﬁlms with a small Gilbert damping constant Q. B. Liu,1,2K. K. Meng ,1,Z. D. Xu,3Tao Zhu,4X. G. Xu,1J. Miao,1and Y . Jiang1,† 1Beijing Advanced Innovation Center for Materials Genome Engineering, University of Science and Technology Beijing, Beijing 100083, China 2School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA 3Department of Physics, South University of Science and Technology of China, Shenzhen 518055, China 4Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China (Received 6 March 2020; revised manuscript received 19 April 2020; accepted 29 April 2020; published 20 May 2020) The Y 3Fe5O12(YIG) ﬁlms with perpendicular magnetic anisotropy (PMA) have recently attracted a great deal of attention for spintronics applications. Here, we report the induced PMA in the YIG ﬁlms grown on (Gd 2.6Ca0.4)(Ga 4.1Mg 0.25Zr0.65)O12(SGGG) substrates by epitaxial strain without preprocessing. Reciprocal space mapping shows that the ﬁlms are lattice matched to the substrates without strain relaxation. Throughferromagnetic resonance and polarized neutron reﬂectometry measurements, we ﬁnd that these YIG ﬁlms havean ultralow Gilbert damping constant ( α< 1×10 −5) and a magnetic dead layer, which is negligible at the YIG/SGGG interfaces. Moreover, the transport behavior of Pt/YIG/SGGG ﬁlms reveals an enhancement of spinmixing conductance and a large unusual anomalous Hall effect (UAHE) as compared with Pt /YIG/Gd 3Ga5O12 (GGG) ﬁlms. Although the UAHE in Pt/YIG/SGGG ﬁlms show different characteristics with varying YIG thickness, they are all ascribed to the possible noncollinear magnetic order at the Pt/YIG interfaces inducedby epitaxial strain. DOI: 10.1103/PhysRevB.101.174431 I. INTRODUCTION Spin transport in ferrimagnetic insulator (FMI) based de- vices has received considerable interest due to its free ofcurrent-induced Joule heating and beneﬁcial for low-powerspintronics applications [ 1,2]. Especially, the high-quality Y 3Fe5O12(YIG) ﬁlm as a widely studied FMI has low damp- ing constant, low magnetostriction, and small magnetocrys-talline anisotropy, making it a key material for magnonics andspin caloritronics. Though the magnons can carry informationover distances as long as millimeters in YIG ﬁlm, thereremains a challenge to control its magnetic anisotropy whilemaintaining the low damping constant [ 3], especially for the thin ﬁlm with perpendicular magnetic anisotropy (PMA),which is very useful for spin polarizers, spin-torque oscilla-tors, magneto-optical devices, and magnon valves [ 4–7]. In addition, the spin-orbit torques (SOTs) induced magnetizationswitching with low current densities has been realized innonmagnetic heavy metal (HM)/FMI heterostructures, pavingthe road towards ultralow-dissipation SOT devices based onFMIs [ 8–10]. Furthermore, previous theoretical studies have pointed that the current density will become much smallerif the domain structures were topologically protected (chiral)[11]. However, most FMI ﬁlms favor an in-plane magnetic easy axis dominated by shape anisotropy, and the investigation kkmeng@ustb.edu.cn †yjiang@ustb.edu.cnis eclipsed as compared with ferromagnetic materials, whichshow abundant and interesting domain structures such aschiral domain walls and magnetic skyrmions, et al. [12–17]. Recently, the interface-induced chiral domain walls have beenobserved in centrosymmetric oxides Tm 3Fe5O12(TmIG) thin ﬁlms, and the domain walls can be propelled by the spin cur-rent from an adjacent platinum layer [ 18]. Similar to the TmIG ﬁlms, the possible chiral magnetic structures are also expectedin the YIG ﬁlms with lower damping constant, which wouldfurther improve the chiral domain walls’ motion speed [ 19]. Recently, several ways have been reported to attain per- pendicularly magnetized YIG ﬁlms, one of which is utilizingthe lattice distortion and magnetoelastic effect induced byepitaxial strain [ 20–23]. It is noted that strain control can- not only enable ﬁeld-free magnetization switching but alsoassist stabilization of the noncollinear magnetic textures ina broad range of magnetic ﬁeld and temperature. Therefore,abundant and interesting physical phenomena could emerge inepitaxial grown YIG ﬁlms with PMA. However, either varyingbuffer layer or doping could increase the Gilbert dampingconstant of YIG, which will affect the efﬁciency of SOT-induced magnetization switching [ 21,22]. On the other hand, this preprocessing would lead to more complicated magneticstructures and impede further discussion of spin transportproperties such as the possible topological Hall effect (THE). In this work, we realized the PMA YIG ﬁlms deposited on (Gd 2.6Ca0.4)(Ga 4.1Mg 0.25Zr0.65)O12(SGGG) substrates due to epitaxial strain. Through ferromagnetic resonance (FMR)and polarized neutron reﬂectometry (PNR) measurements, 2469-9950/2020/101(17)/174431(9) 174431-1 ©2020 American Physical SocietyLIU, MENG, XU, ZHU, XU, MIAO, AND JIANG PHYSICAL REVIEW B 101, 174431 (2020) FIG. 1. (a) AFM images of the YIG ﬁlms grown on two substrates (scale bar, 1 μm). (b) XRD ω-2θscans of the two different YIG ﬁlms grown on two substrates. (c) High-resolution XRD reciprocal space map of the 40-nm-thick YIG ﬁlm grown on SGGG substrate. (d) Fielddependence of the normalized magnetization of the 40-nm-thick YIG ﬁlms grown on two different substrates. we have found that the YIG ﬁlms had small Gilbert damp- ing constant and magnetic dead layer is negligible at theYIG/SGGG interfaces. Moreover, we have carried out theYIG thickness dependence of transport measurements inPt/YIG/SGGG ﬁlms and observed large unusual anomalousHall effect (UAHE), which did not exist in the comparedPt/YIG/GGG ﬁlms. The UAHE in both Pt/YIG(3 nm)/SGGGand Pt/YIG(40 nm)/SGGG ﬁlms are all ascribed to the pos-sible noncollinear magnetic order at the Pt/YIG interfacesinduced by epitaxial strain. II. METHODS The epitaxial YIG ﬁlms with varying thickness from 3 to 90 nm were grown on [111]-oriented GGG substrates (lat-tice parameter a=1.237 nm) and SGGG substrates (lattice parameter a=1.248 nm), respectively by pulsed laser depo- sition technique. The growth temperature was T S=780◦C and the oxygen pressure was varied from 10 to 50 Pa. Then,the ﬁlms were annealed at 780 °C for 30 min at the oxygen pressure of 200 Pa. The Pt (5 nm) layer was deposited on topof YIG ﬁlms at room temperature by magnetron sputtering.After position, electron beam lithography and Ar ion millingwere used to pattern Hall bars, and a lift-off process was usedto form contact electrodes. The size of all the Hall bars is20μm×120μm. After the deposition, we have investigated the surface morphology of the two kinds of ﬁlms using atomicforce microscopy (AFM) as shown in Fig. 1(a), and the two ﬁlms have similar and small surface roughness ∼0.1 nm. Figure 1(b) shows enlarged x-ray diffraction (XRD) ω-2θ scan spectra of 40-nm-thick YIG ﬁlms grown on two differentsubstrates, and more details are shown in the SupplementalMaterial Note 1 [ 24], and they all show predominant (444) diffraction peaks without any other diffraction peaks, ex-cluding impurity phases or other crystallographic orientationsand indicating single-phase nature. According to the (444)diffraction peak position and the reciprocal space map (RSM)of (642) reﬂection of a 40-nm-thick YIG ﬁlm grown on 174431-2UNUSUAL ANOMALOUS HALL EFFECT IN … PHYSICAL REVIEW B 101, 174431 (2020) FIG. 2. Room temperature XPS spectra of (a) Fe 2pand (b) Y 3dfor YIG ﬁlms grown on two substrates. (c) PNR signals (with a 900-mT in-plane ﬁeld) for the spin-polarized R++andR−−channels. Inset: The experimental and simulated SA as a function of scattering vector Q. (d) SLD proﬁles of the YIG/SGGG ﬁlms. The nuclear SLD and magnetic SLD is directly proportional to the nuclear scattering potential and the magnetization, respectively. SGGG as shown in Fig. 1(c), we have found that the lattice constant of SGGG ( ∼1.248 nm) substrate was larger than YIG layer ( ∼1.236 nm). We quantify this biaxial strain as ξ=(aOP−aIP)/aIP, where aOPandaIPrepresent pseudo cubic lattice constant calculated from out-of-plane lattice constantd(4 4 4) OPand in-plane lattice constant d( 110) IP, respec- tively, following the equation of d=a√ h2+k2+l2, with h,k, and lstanding for Miller indices of the crystal planes. It indicates that the SGGG substrate provides tensile stress ( ξ∼0.84%) [22]. At the same time, the magnetic properties of the YIG ﬁlms grown on two different substrates were measured viavibrating sample magnetometry at room temperature. Accord-ing to the magnetic ﬁeld ( μ 0H) dependence of magnetization (M) as shown in Fig. 1(d) and Fig. S2 [ 24], the magnetic anisotropy of YIG ﬁlms grown on SGGG substrates havebeen modulated by strain, while they show similar in-planebehaviors with normal YIG/GGG ﬁlms. III. RESULTS AND DISCUSSION To further investigate the quality of YIG ﬁlms grown on SGGG substrates and exclude the possibility of the straininduced large stoichiometry and lattice mismatch, composi-tional analyses were carried out using x-ray photoelectronspectroscopy (XPS) and PNR. As shown in Fig. 2(a),t h edifference of binding energy between the 2 p 3/2peak and the satellite peak is about 8.0 eV , and the Fe ions are determinedto be in the 3 +valence state. It is found that there is no obvious difference for Fe elements in YIG ﬁlms grown onGGG and SGGG substrates. The Y 3 dspectrums show small energy shift as shown in Fig. 2(b) and the binding energy shift may be related to lattice strain and variation of bond length[22]. Therefore, the stoichiometry of YIG surface has not been dramatically modiﬁed with strain control. Furthermore, wehave performed PNR measurement to probe depth dependentstructure and magnetic information of YIG ﬁlms grown onSGGG substrates. The PNR signals and scattering length den-sity (SLD) proﬁles for YIG (12.8 nm)/SGGG ﬁlms by apply-ing an in-plane magnetic ﬁeld of 900 mT at room temperatureare shown in Figs. 2(c) and2(d), respectively. In Fig. 2(c), R ++andR−−are the non-spin-ﬂip reﬂectivities, where the spin polarizations are the same for the incoming and reﬂectedneutrons. The inset of Fig. 2(c) shows the experimental and simulated spin asymmetry (SA), deﬁned as SA =(R ++− R−−)/(R+++R−−), as a function of scattering vector Q.A reasonable ﬁt was obtained with a three-layer model for thesingle YIG ﬁlm, containing the interface layer, main YIGlayer, and surface layer. The nuclear SLD and magnetic SLDare directly proportional to the nuclear scattering potentialand the magnetization, respectively. Then, the depth-resolved 174431-3LIU, MENG, XU, ZHU, XU, MIAO, AND JIANG PHYSICAL REVIEW B 101, 174431 (2020) structural and magnetic SLD proﬁles delivered by the ﬁtting are shown in Fig. 2(d).T h eZaxis represents the distance for the vertical direction of the ﬁlm, where Z=0 indicates the position at the YIG/SGGG interfaces. It is obvious that thereis little Gd diffusion into the YIG ﬁlm, and the dead layeris much thinner than the reported values (5–10 nm) betweenYIG (or TmIG) and substrates [ 33–35]. The net magnetization of YIG is 3 .36μ B(∼140 KA /m), which is similar to that of bulk YIG [ 36]. The PNR results also showed that besides the YIG/SGGG interfaces region, there is also 1.51-nm-thicknonmagnetic surface layer, which may be Y 2O3and is likely to be extremely important in the magnetic proximity effect[33]. To quantitatively determine the magnetic anisotropy and dynamic properties of YIG ﬁlms, the FMR spectra were mea-sured at room temperature using an electron paramagnetic res-onance spectrometer with rotating the ﬁlms. Figure 3(a)shows the geometric conﬁguration of angle-resolved FMR measure-ments. For FMR measurements, the DC magnetic ﬁeld wasmodulated with AC ﬁeld. The transmitted signal was detectedby a lock-in ampliﬁer. We carried out the FMR spectrum bysweeping external magnetic ﬁeld. The data obtained were thenﬁtted to a sum of symmetric and antisymmetric Lorentzianfunction to extract the linewidth. We use the FMR absorptionline shape to extract resonance ﬁeld ( μ 0Hres) and peak-to-peak linewidth ( μ0/Delta1Hpp) at different θfor 40-nm-thick YIG ﬁlms grown on GGG and SGGG substrates, respectively. The re-sults for 3-nm-thick YIG ﬁlms are shown in the SupplementalMaterial Note 3 [ 24]. According to the angle dependence of μ 0Hresas shown in Fig. 3(b), one can ﬁnd that as compared with YIG ﬁlms grown on GGG substrates, the minimumμ 0Hresof 40-nm-thick YIG ﬁlm grown on SGGG substrate increases with varying θfrom 0° to 90°. We have also quanti- ﬁed through the FMR results the anisotropy ﬁeld μ0Heff kand the effective magnetization μ0Meff=μ0Ms−μ0Heff kof YIG (40 nm)/SGGG ﬁlms, and the values are 23.82 and 149.58 mT,respectively. More detailed information about the procedure isshown in the Supplemental Material Note 3 [ 24]. On the other hand, according to the frequency dependence of μ 0Hresfor 40-nm-thick YIG ﬁlms with applying μ0Hin the XYplane as shown in Fig. 3(c), in contrast to YIG/GGG ﬁlms, the μ0Hres in YIG/SGGG ﬁlms could not be ﬁtted by the in-plane mag- netic anisotropy Kittel formula f=(γ/2π)[μ0Hres(μ0Hres+ 4πMeff)]1/2and more detailed discussion is shown in Sup- plemental Material Note 3 [ 24]. All the results indicate that the easy axis of YIG (40 nm)/SGGG ﬁlms lies out-of-plane.The angle dependent μ 0/Delta1Hppare also compared as shown in Fig. 3(d), the 40-nm-thick YIG ﬁlm grown on SGGG substrate has an optimal value of μ0/Delta1Hppas low as 0.4 mT atθ=64◦, and the corresponding FMR absorption line and Lorentz ﬁtting curve are shown in Fig. 3(e). Generally, the μ0/Delta1Hppis expected to be minimum (maximum) along the magnetic easy (hard) axis, which is basically coincident withangle dependent μ 0/Delta1Hppfor the YIG ﬁlm grown on GGG substrates. However, as shown in Fig. 3(d),t h eμ0/Delta1Hppfor YIG/SGGG ﬁlms shows an anomalous variation. The lowestμ 0/Delta1Hppatθ=64◦could be ascribed to high YIG ﬁlm quality and ultrathin magnetic dead layer at the YIG/SGGG inter-faces. It should be noted that, as compared with YIG/GGGﬁlms, the μ 0/Delta1Hppis independent on the frequency from 5 to14 GHz as shown in Fig. 3(f). Then, we have calculated the Gilbert damping constant αof YIG (40 nm)/SGGG ﬁlms by extracting μ0/Delta1Hppat each frequency as shown in Fig. 3(f). The obtained αis smaller than 1 ×10−5, which is one order of magnitude lower than the results in Ref. [ 21] and would open perspectives for magnetization dynamics. According tothe theoretical theme, the μ 0/Delta1Hppconsists of three parts: Gilbert damping, two magnons scattering relaxation processand inhomogeneities, in which both Gilbert damping and two-magnons-scattering relaxation process depend on frequency.Therefore, the large μ 0/Delta1Hppin YIG/SGGG ﬁlms mainly stems from inhomogeneities, which will be discussed nextwith the help of transport measurements. Notably, although the 3-nm-thick YIG ﬁlm grown on SGGG has not shown PMA as compared with YIG(40 nm)/SGGG ﬁlms, the FMR results of YIG (3 nm)/SGGGﬁlms reveal different magnetic anisotropy with YIG (3nm)/GGG ﬁlms as discussed in Supplemental Material Note3[24], indicating the strain will also modify the magnetic anisotropy of thinner YIG ﬁlms. To further explore thestrain-induced magnetic order, we have investigated the YIGthickness dependence of spin transport properties in Pt/YIGSGGG ﬁlms, which are basically sensitive to magnetic de-tails of YIG ﬁlms. The magnetoresistance (MR) has beenproved as a powerful tool to effectively explore magneticinformation originating from the interfaces [ 37]. The temper- ature dependent spin Hall magnetoresistance (SMR) of Pt (5nm)/YIG (3 nm) ﬁlms grown on two different substrates weremeasured using a small and nonperturbative current density(∼1×10 6A/cm2), and the sketches of the measurement are shown in Fig. 4(a). It should be noted that the inﬂuence of thermoelectric/thermomagnetic effect can be negligible inour samples as discussed in Supplemental Material Note 4[24]. The βscan of longitudinal MR, which is deﬁned as MR=/Delta1ρ XX/ρXX(0)=[ρXX(b)−ρXX(0)]/ρXX(0)in the YZ plane for the two ﬁlms under a 3 T ﬁeld (enough to saturatethe magnetization of YIG), shows cos 2βbehaviors with vary- ing temperature for Pt/YIG/GGG and Pt/YIG/SGGG ﬁlmsas shown in Figs. 4(b) and 4(c), respectively. The SMR of Pt/YIG/SGGG ﬁlms is larger than that of Pt/YIG/GGGﬁlms with the same thickness of YIG at room temperature,indicating an enhancement of spin mixing conductance ( G ↑↓) in the Pt/YIG/SGGG ﬁlms. Here, the spin transport propertiesof Pt layers are expected to be the same because of the similarresistivity and ﬁlms quality. Therefore, the SGGG substratenot only induces PMA but also enhances G ↑↓at the Pt/YIG interfaces. We have also investigated the ﬁeld dependent Hallresistivities in Pt/YIG (3 nm)/SGGG ﬁlms in the temperaturerange from 260 to 350 K as shown in Fig. 4(d). Though the conduction electrons cannot penetrate into the FMI layer,the possible anomalous Hall effect (AHE) at the HM/FMIinterfaces is proposed to emerge, and the total Hall resistivitycan usually be expressed as the sum of various contributions[38,39]: ρ H=R0μ0H+ρS+ρS-A, (1) where R0is the normal Hall coefﬁcient, ρSis the transverse manifestation of SMR, and ρS-Ais the spin Hall anomalous Hall effect (SHAHE) resistivity. Notably, the external ﬁeld isapplied out-of-plane, and ρ s(∼/Delta1ρ 1mxmy) can be neglected 174431-4UNUSUAL ANOMALOUS HALL EFFECT IN … PHYSICAL REVIEW B 101, 174431 (2020) FIG. 3. (a) The geometric conﬁguration of the angle dependent FMR measurement. (b) The angle dependence of the μ0Hresfor the 40-nm- thick YIG ﬁlms grown on GGG and SGGG substrates. (c) The frequency dependence of the μ0Hresfor 40-nm-thick YIG ﬁlms grown on GGG and SGGG substrates. (d) The angle dependence of μ0/Delta1Hppfor the 40-nm-thick YIG ﬁlms grown on GGG and SGGG substrates. (e) FMR spectrum of the 40-nm-thick YIG ﬁlm grown on SGGG substrate with 9.46 GHz at θ=64◦. (f) The frequency dependence of μ0/Delta1Hppfor the 40-nm-thick YIG ﬁlms grown on GGG and SGGG substrates. [32]. Interestingly, as compared with the total Hall resistivities of Pt/YIG(3 nm)/GGG ﬁlms as discussed in SupplementalMaterial Note 5 [ 24], the YIG ﬁlms grown on SGGG substrate show bump and dip features during the hysteretic measure-ments in the temperature range from 260 to 350 K. In thefollowing discussion, we term the part of extra Hall signals asρ U-S-A.T h e( ρS-A+ρU-S-A) clearly coexist with the large background of a normal Hall effect. Notably, the broken(space) inversion symmetry with strong spin-orbit couplingwill induce the Dzyaloshinskii-Moriya interaction (DMI). If 174431-5LIU, MENG, XU, ZHU, XU, MIAO, AND JIANG PHYSICAL REVIEW B 101, 174431 (2020) FIG. 4. (a) The deﬁnition of the angle, the axes and the measurement conﬁgurations. (b) and (c) Longitudinal MR at different temperatures in Pt/YIG/GGG and Pt/YIG/SGGG ﬁlms, respectively (The applied magnetic ﬁeld is 3 T). (d) Total Hall resistivities vsμ0Hfor Pt/YIG(3 nm)/SGGG ﬁlms in the temperature range from 260 to 300 K. (e) ( ρS-A+ρU-S-A)vsμ0Hfor two ﬁlms in the temperature range from 260 to 300 K. (f) ρU-S-Avsμ0Hfor Pt/YIG(3 nm)/SGGG ﬁlms at 300K. Inset: ρS-AandρS-A+ρU-S-Avsμ0Hfor Pt/YIG(3 nm)/SGGG ﬁlms at 300K. (g) Temperature dependence of the ρMax U-S-A. We have carried out the transport properties of many samples for several times, and the data have been label −ed with error bars. the DMI could be compared with the Heisenberg exchange interaction and the magnetic anisotropy that were controlledby strain, it could stabilize noncollinear magnetic texturessuch as skyrmions, producing a ﬁctitious magnetic ﬁeld andTHE. The ρ U-S-Aindicates that a chiral spin texture may exist, which is similar to B20-type compounds MnSi andMnGe [ 40,41]. To more clearly demonstrate the origin of ρ U-S-A, we have extracted it by subtracting the normal Hall term as discussed in Supplemental Material Note 6 [ 24], and the temperature dependence of ( ρS-A+ρU-S-A) has been shown in Fig. 4(e). Then, we can further discern the peakand hump structures in the temperature range from 260 to 350 K. The SHAHE contribution ρS-Acan be expressed as ρS-A= −/Delta1ρ 2mZ[38,42,43], where /Delta1ρ 2is the coefﬁcient depending on the imaginary part of G↑↓, and mzis the magnetization alongZdirection. The further extracted ρU-S-Ahas been shown in Fig. 4(f), and the temperature dependence of the largest ρU-S-A(ρMax U-S-A) in all the ﬁlms have been shown in Fig. 4(g). Finite values of ρMax U-S-Aexist in the temperature range from 150 to 350 K, which is much different from that in B20-type bulk chiral magnets, which are subjected to low temperature andlarge magnetic ﬁeld [ 44]. The large nonmonotonic magnetic 174431-6UNUSUAL ANOMALOUS HALL EFFECT IN … PHYSICAL REVIEW B 101, 174431 (2020) FIG. 5. (a) and (b) The Hall resistances vsμ0Hfor the Pt/YIG(40 nm)/SGGG ﬁlms in the temperature range from 50 to 150 K in small and large magnetic ﬁeld range, respectively. (c) The Hall resistances vsμ0Hat small magnetic ﬁeld range after sweeping a large out-of-plane magnetic ﬁeld +0.8 T (black line) and −0.8 T (red line) to zero. (d) An illustration of the orientations of the magnetizations Fe ( a)a n dF e( d) in YIG ﬁlms with the normal in-plane magnetic anisotropy (IMA), the ideal strain induced PMA and the actual magnetic anisotropy grown on SGGG substrates in our work. ﬁeld dependence of anomalous Hall resistivity could not stem from the Weyl points, and a more detailed discussion is foundin Supplemental Material Note 7 [ 24]. Furthermore, robust UAHE with different characteristics as compared with Pt (5 nm)/YIG (3 nm)/SGGG ﬁlms has alsobeen found in Pt (5 nm)/YIG (40 nm)/SGGG ﬁlms. First,we have investigated the small ﬁeld dependence of the Hallresistances for Pt (5 nm)/YIG (40 nm)/SGGG ﬁlms as shownin Fig. 5(a), and more details are shown in Supplemental Material Note 8 [ 24]. The out-of-plane hysteresis loops of Pt/YIG/SGGG ﬁlms are not central symmetry, indicating theexistence of an internal ﬁeld leading to opposite velocities ofup to down and down to up domain walls in the presence ofcurrent along the +Xdirection. The large ﬁeld dependencesof the Hall resistances are shown in Fig. 5(b), which could not be described by Eq. ( 1). There are large variations for the Hall signals when external magnetic ﬁeld is lower thansaturation ﬁeld ( μ 0Ms)o fY I Gﬁ l m s( ∼50 mT at 300 K and ∼150 mT at 50 K). More interestingly, we have applied a large out-of-plane external magnetic ﬁeld of +0.8 T ( −0.8 T) above μ0Msto saturate the out-of-plane magnetization component MZ>0(MZ<0), then decreased the ﬁeld to zero, ﬁnally the Hall resistances were measured in the small ﬁeld range(±40 mT), from which we could ﬁnd that the shape was reversed as shown in Fig. 5(c). Here, we infer that the mag- netic structures at the Pt/YIG interfaces could not be a simplelinear magnetic order. Theoretically, an additional chirality-driven Hall effect might be present in the ferromagnetic 174431-7LIU, MENG, XU, ZHU, XU, MIAO, AND JIANG PHYSICAL REVIEW B 101, 174431 (2020) regime due to spin canting [ 45–48]. It has been found that the strain from an insulating substrate could produce tetragonaldistortion, which would drive orbital selection, modifyingelectronic properties and magnetic ordering of manganites.ForA 1−xBxMnO 3perovskites, a compressive strain makes the ferromagnetic conﬁguration relatively more stable than theantiferromagnetic state [ 49]. On the other hand, the strain could induce spin canting [ 50]. A variety of experiments and theories have reported that ion substitute, defect, and mag-netoelastic interaction would cant the magnetization of YIG[51–53]. Therefore, if we could modify the magnetic order by epitaxial strain, the noncollinear magnetic structure is ex-pected to emerge in YIG ﬁlms. For YIG crystalline structure,the two Fe sites are located on octahedrally coordinated 16( a) site and tetrahedrally coordinated 24( d) site, aligning antipar- allel with each other [ 28]. According to the XRD and RSM results, the tensile strain due to SGGG substrate would resultin a distortion angle of the facets of the YIG unit cell smallerthan 90° [ 54]. Therefore, the magnetization of Fe atoms at two sublattices should be discussed separately rather than as awhole. Then, the unusual signals of Pt/YIG/SGGG ﬁlms couldbe ascribed to the emergence of four different Fe 3+magnetic orientations in strained Pt/YIG ﬁlms as shown in Fig. 5(d). To be more clearly, we assume that, in analogy with ρS,t h e ρU-S-Ais larger than ρS-Aand scales linearly with mymzand mxmz. With applying a large external ﬁeld μ0Halong the Z axis, the uncompensated magnetic moment at tetrahedrallycoordinated 24( d) is along with external ﬁelds μ 0Hdirection for\|μ0H\|>μ 0Ms, and the magnetic moment tends to be alongA(−A) axis when external ﬁeld is swept from 0.8 T (−0.8 T) to 0 T. Then, if the Hall resistance was measured at a small out-of-plane ﬁeld, the uncompensated magneticmoment would switch from the A(−A) axis to the B(−B) axis. In this case, the ρ U-S-Athat scales with /Delta1ρ 3(mymz+mxmz) would change sign because mzis switched from the Zaxis to the −Zaxis as shown in Fig. 5(c). However, there is still some problem that needs to be further clariﬁed. There are nounusual signals in Pt/YIG/GGG ﬁlms that could be ascribed tothe weak strength of /Delta1ρ 3or strong magnetic anisotropy. It is still valued for further discussion of the origin of /Delta1ρ 3whether it could stem from skrymions et al. , but until now we have not observed any chiral domain structures in Pt/YIG/SGGGﬁlms through Lorentz transmission electron microscopy.Finally, we want to mention that, as shown in Figs. 4(d) and 5(b), the Hall measurements in Pt/YIG (3 nm)/SGGG and Pt/YIG (40 nm)/SGGG ﬁlms are different, which could berelated to the thickness dependence of magnetic anisotropyin YIG/SGGG ﬁlms. Notably, according to the FMR re-sults, although the magnetic anisotropy is different in YIG(3 nm)/SGGG and YIG (40 nm)/SGGG ﬁlms, a similar mag-netic order will exist for both ﬁlms and we should discriminateit from the normal YIG/GGG ﬁlms. The mechanisms thatdetermine the UAHE are very complex, but the dominantrole could be the modiﬁed magnetic order induced by strain.We hope that future work would involve more detailed mag-netic microscopy imaging and microstructure analysis, whichcan further elucidate the real microscopic origin of largenonmonotonic magnetic ﬁeld dependence of anomalous Hallresistivity. IV . CONCLUSION In conclusion, PMA YIG ﬁlms could be realized using epitaxial strain. These YIG ﬁlms grown on SGGG substrateshad low Gilbert damping constants ( <1×10 −5) and the magnetic dead layer is negligible at the YIG/SGGG inter-faces. Moreover, we observed large UAHE in Pt/YIG/SGGGﬁlms with varying YIG thickness, which did not exist in thecompared Pt/YIG/GGG ﬁlms. The UAHE in Pt/YIG/SGGGﬁlms are all ascribed to the possible noncollinear magneticorder at the Pt/YIG interfaces induced by epitaxial strain. 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RevModPhys.87.1213.pdf	Spin Hall effects Jairo Sinova Institut für Physik, Johannes Gutenberg Universität Mainz, 55128 Mainz, Germany and Institute of Physics, Academy of Science of the Czech Republic, Cukrovarnická 10, 162 00 Praha 6, Czech Republic Sergio O. Valenzuela ICN2—Catalan Institute of Nanoscience and Nanotechnology, the Barcelona Institute of Science and Technology and CSIC, Campus UAB, Bellaterra, 08193 Barcelona, Spain, and ICREA—Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain J. Wunderlich Institute of Physics, Academy of Science of the Czech Republic, Cukrovarnická 10, 162 00 Praha 6, Czech Republic and Hitachi Cambridge Laboratory, Cambridge CB3 0HE, United Kingdom C. H. Back Universität Regensburg, Universitätstraße 31, 93040 Regensburg, Germany T. Jungwirth Institute of Physics, Academy of Science of the Czech Republic, Cukrovarnická 10, 162 00 Praha 6, Czech Republic and School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom (published 27 October 2015) Spin Hall effects are a collection of relativistic spin-orbit coupling phenomena in which electrical currents can generate transverse spin currents and vice versa. Despite being observed only a decade ago,these effects are already ubiquitous within spintronics, as standard spin-current generators anddetectors. Here the theoretical and experimental results that have established this subfield of spintronicsare reviewed. The focus is on the results that have converged to give us the current understanding of thephenomena, which has evolved from a qualitative to a more quantitative measurement of spin currentsand their associated spin accumulation. Within the experimental framework, optical-, transport-, andmagnetization-dynamics-based measurements are reviewed and linked to both phenomenological andmicroscopic theories of the effect. Within the theoretical framework, the basic mechanisms in both theextrinsic and intrinsic regimes are reviewed, which are linked to the mechanisms present in their closelyrelated phenomenon in ferromagnets, the anomalous Hall effect. Also reviewed is the connection to the phenomenological treatment based on spin-diffusion equations applicable to certain regimes, as well as the spin-pumping theory of spin generation used in many measurements of the spin Hall angle. Afurther connection to the spin-current-generating spin Hall effect to the inverse spin galvanic effect isgiven, in which an electrical current induces a nonequilibrium spin polarization. This effect oftenaccompanies the spin Hall effect since they share common microscopic origins. Both can exhibit thesame symmetries when present in structures comprising ferromagnetic and nonmagnetic layers throughtheir induced current-driven spin torques or induced voltages. Although a short chronological overviewof the evolution of the spin Hall effect field and the resolution of some early controversies is given, themain body of this review is structured from a pedagogical point of view, focusing on well-establishedand accepted physics. In such a young field, there remains much to be understood and explored, hencesome of the future challenges and opportunities of this rapidly evolving area of spintronics are outlined. DOI: 10.1103/RevModPhys.87.1213 PACS numbers: 72.25.Ba, 75.76.+j, 72.25.Dc, 75.78. −n CONTENTS I. Introduction 1214 II. Overview 1215 A. Spin Hall, anomalous Hall, and Mott polarimetry 1215B. Intrinsic spin Hall and quantum Hall effects 1216 C. Spin Hall effect and magnetic multilayers 1217D. Spin Hall effect, spin galvanics, and spin torques 1218 III. Theory of the Spin Hall Effect 1220 A. Mechanisms of the spin Hall effect 1221REVIEWS OF MODERN PHYSICS, VOLUME 87, OCTOBER –DECEMBER 2015 0034-6861 =2015=87(4) =1213(47) 1213 © 2015 American Physical Society1. Intrinsic mechanism 1222 2. Skew-scattering mechanism 12233. Side-jump mechanism 12244. Cancellation of mechanisms in model systems 1225 B. Phenomenological drift-diffusion theory 1226C. Crépieux-Bruno model of extrinsic side jump and skew scattering 1226 D. Theory of the inverse spin Hall effect induced by spin pumping 1228 E. Kubo formalism 1230 IV. Experimental Studies of Spin Hall Effect 1232 A. Early experiments of anomalous Hall effect in paramagnets 1232 B. Optical tools in spin Hall experiments 1233 1. Optical detection of the spin Hall effect 12332. Optical generation of the inverse spin Hall effect 12353. All-optical generation and detection 12364. Electrical manipulation 1237 C. Transport experiments 1237 1. Concepts of nonlocal spin transport: Electrical injection and detection 1238 2. Nonlocal detection of inverse spin Hall effect with lateral spin current 1238 3. Nonlocal detection of spin Hall effects with vertical spin current 1240 4. Direct detection of the spin Hall induced spin accumulation 1242 5. Spin Hall injection and detection without ferromagnets 1243 6. Spin Hall magnetoresistance 1244 D. Spin Hall effect coupled to magnetization dynamics 1244 1. Ferromagnetic resonance spin pumping 12452. Spin Hall effect modulation of magnetization damping 1248 3. Spin Hall effect: Spin-transfer torque 12494. Spin Hall effect induced switching of the magnetization 1250 E. Spin Hall angles 1254 V. Future Directions and Remaining Challenges 1254List of Symbols and Abbreviations 1255Acknowledgments 1255References 1256 I. INTRODUCTION Spintronics is a field that jointly utilizes the spin and charge degrees of freedom to control equilibrium and nonequilibriumproperties of materials and devices ( Wolf et al. , 2001 ;Zutic, Fabian, and Sarma, 2004 ;Bader and Parkin, 2010 ). The generation, manipulation, and detection of spin currents is oneof the key aspects of the field of spintronics. Among the several possibilities to create and control spin currents, the spin Hall effect (SHE) has gained a distinct place since its first observation a decade ago ( Kato et al. , 2004a ;Wunderlich et al. , 2004 ;Day, 2005 ;Wunderlich et al. , 2005 ). In the direct SHE, an electrical current passing through a material can generate a transverse pure spin current polarized perpendicular to the plane defined by the charge and spin current. In its reciprocal effect, the inverse SHE (ISHE), a pure spin current through the material generates a transverse chargecurrent. In both cases, the material must possess spin-orbit coupling.The SHE borrows its concept from the well-established anomalous Hall effect (AHE), where relativistic spin-orbit coupling generates an asymmetric deflection of the charge carriers depending on their spin direction ( Nagaosa et al. , 2010 ). The AHE can be detected electrically in a ferromagnet (FM) via a transverse voltage because of the difference inpopulation of majority and minority carriers. The generali-zation of this effect to a pure spin current generated by the SHE in a nonmagnetic material (NM) was proposed over four decades ago ( Dyakonov and Perel, 1971b ) based on the idea of asymmetric Mott scattering ( Mott, 1929 ). This so-called extrinsic SHE remained unexplored until recent proposals thatput forward a similar prediction ( Hirsch, 1999 ;Zhang, 2000 ) as well as the possibility of a strong intrinsic SHE ( Murakami, Nagaosa, and Zhang, 2003 ;Sinova et al. , 2004 ). The initial challenge for SHE detection was primarily the lack of direct electrical signals; therefore initial experi- ments detected it by optical means, in both the extrinsicregime ( Kato et al. , 2004a ) and the intrinsic regime (Wunderlich et al. , 2004 ,2005 ). The ISHE was detected soon thereafter by Saitoh et al. (2006) ,Valenzuela and Tinkham (2006) , and Zhao et al. (2006) . Early measurements were mostly qualitative. However, more accurate quantitativemeasurements of spin Hall angles have been established in later experiments through the aid of FM detectors in static or dynamic magnetization regimes, and a much firmer situationhas arisen in the field. Adding to this flurry of activity and increased understanding, recent experiments in magnetic structures have aimed to use spin currents injected from an adjacent spin Hall NM for spin- transfer torque (STT) switching of a FM ( Miron, Garello et al. , 2011 ;Liu, Pai, Li, et al., 2012 ). In addition to this SHE-induced torque, there is also a spin-orbit torque (SOT) ( Bernevig and Vafek, 2005 ;Chernyshov et al. , 2009 ), which is generated via the inverse spin galvanic effect (ISGE) ( Belkov and Ganichev, 2008 ). In the ISGE, a charge current can generate a non- equilibrium uniform spin polarization via spin-orbit couplingand it is often a companion effect to the spin current generatingSHE ( Kato et al. , 2004a ,2004b ;Wunderlich et al. , 2004 , 2005 ). These results underscore the relevance of the SHE for applications. As mentioned, the SHE borrows directly from the physics and mechanisms of the AHE and correspondingly much of their descriptions are parallel. The family of the AHE, SHE,and ISHE is illustrated in Fig. 1. The important caveat is that, unlike the AHE which correlates charge degrees of freedom via relativistic spin-orbit interaction, the SHE and ISHEcorrelate the charge degree of freedom, a conserved quantity,and the spin degree of freedom, a nonconserved quantitysubject to decay and dephasing. The aim of this review is to survey the rapid developments of the SHE field, to give an overview of its current exper- imental understanding, the basic theoretical tools that arebeing applied to describe it and their current level of successand limitations, the connection to important related phenom-ena, as well as the potential of the SHE for applications, particularly in the area of magnetization dynamics. Given the enormous volume of work that has been published in just a decade, we can only highlight a selection1214 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015of reports that contributed significantly to the field. Our review covers most of the major aspects of the field. It surveys bothmetals and semiconductors, as well as optical, transport, and magnetization dynamics experiments. The theory survey covers most of the relevant microscopic and phenomenologi-cal modeling, as well as resolutions of earlier controversies. The reader interested in this field should complement this reading with other recent reviews. Hoffmann (2013b) reviewed extensively the transport measurements in metallic systems. Other focused reviews published recently are Gradhand et al. (2012) ,Jungwirth, Wunderlich, and Olejník (2012) ,Maekawa and Takahashi (2012) ,Raimondi et al. (2012) ,Valenzuela and Kimura (2012) . II. OVERVIEW In this section, we provide an overview that starts from the original seeds of the SHE field and connects afterwards tothe broader context of the phenomenon within spintronics. The overview is organized as follows: First, we look back to how the Mott scattering of electron beams in vacuum and theskew scattering of electrons in FMs germinated into the prediction of the extrinsic SHE in NMs. Second, we discuss that in a solid-state system there is in addition an intrinsicspin-deflection, arising from the internal spin-orbit coupling forces in a perfect crystal. This key distinction from electrons in a vacuum makes the spin-dependent Hall physics incondensed matter systems much richer. We also note here the connection of this intrinsic mechanism to the quantum Hall effects. Third, we summarize studies of spin injection anddetection in hybrid FM-NM structures, which were particu- larly impactful on the research of the SHE. Here, we highlight dc transport as well as ac ferromagnetic resonance (FMR)experiments. Finally, we connect the physics of the SHE, which considers pure spin currents and nonuniform spin accumulations, to the physics of the spin galvanic effects.The latter effects represent a seemingly distinct family ofrelativistic phenomena relating to the generation or detection of uniform nonequilibrium spin polarizations. However, as wepoint out, the spin Hall and spin galvanic effects can have common features in their microscopic physical origins and both can contribute to spin-charge conversion phenomena. These two relativistic effects are now at the forefront ofcurrent-induced magnetization dynamics research and botheffects also contribute to the reciprocal conversion of mag-netization dynamics into electrical signals. A. Spin Hall, anomalous Hall, and Mott polarimetry In their original work, Dyakonov and Perel (1971b) referred to the phenomena of Mott scattering ( Mott, 1929 ) and of the AHE ( Hall, 1881 ) to theoretically predict the extrinsic SHE. In particular, they pointed out the following: (i) spin-dependentasymmetric deflection is observed in electron beams in vacuum due to Mott scattering ( Mott, 1929 ,1932 ;Shull, Chase, and Myers, 1943 ;Gay and Dunning, 1992 ). (ii) Mott ’s skew scattering is regarded among the origins of the AHE ofelectron carriers in FMs ( Karplus and Luttinger, 1954 ;Smit, 1955 ,1958 ;Berger, 1970 ;Nagaosa et al. , 2010 ). The two points imply that under an applied electrical current, asym- metric spin-dependent deflection should occur in NMs.Unlike in FMs, NMs in equilibrium have the same numberof spin-up and spin-down electrons and no transverse chargeimbalance will occur. Instead, the SHE generates an edge spinaccumulation that has opposite polarization at opposite edges. We now explore the Mott scattering seed of the SHE in more detail. In 1925, the spin of the electron was inferredindirectly by atomic spectra ( Uhlenbeck and Goudsmit, 1925 ). Based on the then recently derived Dirac equation ( Dirac, 1928 ), Mott proposed his scattering experiment ( Mott, 1929 , 1932 ) to provide direct evidence that spin is an intrinsic property of a free electron. The ensuing quest for theexperimental verification of Mott scattering ( Shull, Chase, and Myers, 1943 ) was among the founding pillars of the entire relativistic quantum mechanics concept. Since Mott scatteringof electron beams from heavy nuclei in a vacuum chamber canbe regarded as the SHE in a non-solid-state environment, theseeds of the SHE date back to the very foundations of theelectron spin and relativistic quantum mechanics. Figure 2(a)shows the Mott (1929) double-scattering experi- ment proposal. First, an unpolarized beam of electrons isscattered from heavy nuclei in a target. Because of the relativistic spin-orbit coupling, large angle ( ∼90°) scattering from the first target produces a polarized beam with the spin polarizationtransverse to the scattering plane. Scattering of these polarizedelectrons from the second target results, again due to the spin-orbit coupling, in a left-right scattering asymmetry that is proportional to the polarization induced by the first scattering. In a complete analogy to the Mott double-scattering effect, but instead of vacuum now considering a solid-state system, Hankiewicz et al. (2004) proposed an H-bar microdevice schematically shown in Fig. 2(b). In the SHE part of the device, an unpolarized electrical current generates a transverse spincurrent due to an effective spin-orbit force F sothat acts on the carriers. The spin current injected into the second leg generates,via the ISHE, an electrical current, or in an open circuit geometry a voltage across the second leg. The first attempt to implement this H-bar SHE-ISHE experiment was carried outbyMihajlovic et al. (2009) in gold, but no signature of the spin magneticAHE SHE non-magnetic non-magneticISHE FIG. 1 (color online). An illustration of the connected family of the spin-dependent Hall effects. In the AHE, a charge currentgenerates a polarized transverse charge current. In the SHE, anunpolarized charge current generates a transverse pure spincurrent. In the ISHE, a pure spin current generates a transversecharge current.Jairo Sinova et al. : Spin Hall effects 1215 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015Hall effects was observed. The first successful experiment was realized in a NM semiconductor by Brüne et al. (2010) . In Fig. 2(c), we show an earlier variant of the double-scattering experiment proposed by Hirsch (1999) for observing the SHE- ISHE in a solid-state device. Instead of considering the spincurrent produced directly by the charge current via the SHE,Hirsch focused on the edges of the SHE part of the sample. Here,the transverse spin current accumulates, forming a nonequili-brium spin polarization of opposite sign at the two opposite edges. In NMs, the nonequilibrium spin polarization corre- sponds to a splitting of the spin-up and spin-down chemicalpotentials. When connecting the two edges, the gradient of thespin-dependent chemical potentials will generate a circulatingspin current which is then detected by the ISHE spin currentmeter inserted into the closed spin-current circuit. The idea for the experiment was borrowed from the ordinary Hall effect (HE) in which opposite charge accumulates at opposite edges due tothe Lorentz force, and the resulting electrochemical potentialgradient generates a circulating charge current when the twoedges are connected in the closed circuit geometry.Realizing the Hirsch (1999) SHE-ISHE device remains a challenge. Similarly to the Hankiewicz et al. (2004) design directly copying the Mott double-scattering experiment, the wires connecting the SHE and ISHE parts of Hirsch ’s device have to be shorter than the characteristic spin-conserving lengthscale. The spin-orbit coupling required for the SHE-ISHE in thefirst place, however, tends to make the spin lifetime short. The additional complication is that the spin-orbit coupling also limits, again via the finite spin lifetime, the width of the sampleedge with nonzero spin accumulation from which the spincurrent is extracted in Hirsch ’s device proposal. While difficult to realize experimentally, Hirsch ’s concept is stimulating for comprehending the general key distinctionsbetween charge and spin current. Electron charge is a con-served quantity but its spin direction is not conserved. In the charge HE, the difference between electrochemical potentials at the edges determines the uniform charge current which insteady-state flows through the closed circuit. In the SHE, on theother hand, the spin current in the connecting wire of Hirsch ’s device is not uniform and is not determined by the difference between the spin-dependent chemical potentials at the left and right edges. It is determined by the local gradient of the spin-dependent chemical potentials which vanishes (i.e., the spincurrent also vanishes) on the length scale given by the spinlifetime. As long as the connecting wire is longer than the characteristic spin-conserving length scale, there is no differ- ence between a closed and an open spin-current circuit. Hirsch ’s concept also points to the general applicability of the ISHE as an electrical spin detector. Even in electrically open circuits, the nonconserving, nonuniform spin current canstill flow. It is then readily separated from the charge currentand can be detected by the ISHE. The Mott polarimetry ofelectron beams in vacuum chambers and AHE polarimetry ofcharge currents in itinerant magnets is, therefore, comple- mented by the ISHE polarimetry of pure spin currents. A spin current in a NM of any origin (not only of the SHE origin) can be detected by the ISHE. Indeed, ISHE detectors of pure spin currents became a standard measurement tool. They led to, e.g., the discovery of the spin Seebeck effect(Uchida et al. , 2008 ,2010 ;Jaworski et al. , 2010 ) and helped establish the emerging field of spin caloritronics ( Bauer, Saitoh, and van Wees, 2012 ). Given the inherent challenges in realizing Hirsch ’s device, it is not surprising that experimentalists initially avoidedattempts to perform the SHE-ISHE double-scattering experi- ments and that the first observations of the SHE ( Kato et al. , 2004a ;Wunderlich et al. , 2004 ,2005 ) and ISHE ( Saitoh et al. , 2006 ;Valenzuela and Tinkham, 2006 ;Zhao et al. , 2006 ) were made separately. When the Hankiewicz et al. (2004) H-bar microdevice was eventually realized in experiment by Brüne et al. (2010) , both the SHE and ISHE had already been established independently. B. Intrinsic spin Hall and quantum Hall effects Remarkably, the H-bar experiment ( Brüne et al. , 2010 ) discussed in the previous section was performed in a ballistic transport regime where the picture of Mott scattering, single ordouble, did not apply. A fundamental physics principle makesthe SHE in solid-state systems richer than in the Mott electron beam (a) (b) (c) FIG. 2 (color online). (a) Schematics of the Mott (1929) original double-scattering proposal, (b) SHE-ISHE analog of Mott doublescattering in Hankiewicz et al. (2004) H-bar device, (c) SHE (left) and ISHE (right) wired as proposed by Hirsch (1999) . Instead of directly injecting a spin current generated in the SHE part of theexperiment, as suggested by Mott and by Hankiewicz et al., Hirsch considered that the pure spin current is generated from the oppositespin accumulations at the edges of the SHE part of the “double- scattering ”device. The effective spin-orbit force that deflects the spins in the SHE-ISHE is represented by straight black arrowsacting on the carriers. (a) Adapted from Gay and Dunning, 1992 .1216 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015beams scattered from spin-orbit-coupled targets in vacuum chambers. For electrons moving in a crystal, a transverse spin- dependent velocity can be generated by the relativistic spin- orbit field of a perfect crystal even in the absence of scattering. The roots of this intrinsic SHE are clearly distinct from theMott (skew) scattering AHE and from the Mott scattering of free electron beams. The reactive term responsible for the intrinsic SHE is akin to the ordinary HE in which the transverse deflection of electrons is a reaction to the Lorentz force of the applied magnetic field acting on the moving carriers [see Fig. 3(a)]. In strong magnetic fields, the quantum Hall effect (QHE) provides a disorder-independent measure of the quantum conductance e 2=h. The integer multiples of e2=hobserved in the QHE correspond to the number of occupied dissipa- tionless chiral edge states in the conductor [see Fig. 3(b)]. Besides the externally applied Lorentz force, electrons moving in a crystal can experience an internal spin-orbit force. The effect was first recognized in FMs where it generates the intrinsic AHE [see Fig. 3(a)](Karplus and Luttinger, 1954 ;Jungwirth, Niu, and MacDonald, 2002 ; Onoda and Nagaosa, 2002 ).Murakami, Nagaosa, and Zhang (2003) andSinova et al. (2004) predicted that the same spin-orbit force derived directly from the relativistic band structure of a NM can induce the SHE without involving Mott scattering [see Fig. 3(c)]. The first experimental obser- vations confirmed that the SHE can indeed have the two distinct origins. While Wunderlich et al. (2004 ,2005) ascribed the circularly polarized luminescence signal from the edge of thep-GaAs sample to the intrinsic SHE, Kato et al. (2004a) detected an edge Kerr rotation signal in n-GaAs due to the extrinsic, skew-scattering SHE. Following the discovery of the phenomenon, SHE experi- ments in semiconductors using optical spin detection haveexplored the basic phenomenologies of the extrinsic and intrinsic SHEs ( Kato et al. , 2004a ;Wunderlich et al. , 2004 , 2005 ;Nomura et al. , 2005 ;Sihet al. , 2005 ,2006 ;Stern et al. ,2006 ,2007 ;Chang et al. , 2007 ;Matsuzaka, Ohno, and Ohno, 2009 ). They also demonstrated the potential of the SHE as a spin-current source ( Sih et al. , 2006 ). The experimental observation of the ISHE in a semiconductor was performedby a two-color optical excitation technique with perpendicularlinear polarizations ( Zhao et al. , 2006 ). The spin current produced by the laser excitation is transferred due to the ISHEinto a transverse electrical current, resulting in a spatially dependent charge accumulation which was detected by the optical transmission signal of a probe laser beam. These all-optical measurements were eventually performed on timescales shorter than the scattering time and provided a directdemonstration of the intrinsic SHE signal ( Werake, Ruzicka, and Zhao, 2011 ). The intrinsic SHE proposal triggered an intense theoretical debate which is summarized in several review articles(Murakami, 2006 ;Engel, Rasbha, and Halperin, 2006 ; Schliemann, 2006 ;Sinova et al. , 2006 ;Sinova and MacDonald, 2008 ;Culcer, 2009 ;Hankiewicz and Vignale, 2009 ;Vignale, 2010 ;Raimondi et al. , 2012 ). Combined with the established physics of the dissipationless QHE, theintrinsic SHE led to the prediction and subsequent exper-imental verification of the quantum spin Hall effect (QSHE) in a HgTe 2D system ( Murakami, Nagaosa, and Zhang, 2004 ; Kane and Mele, 2005 ;Bernevig, Hughes, and Zhang, 2006 ; König et al. , 2007 ;Hasan and Kane, 2010 ). In the time- reversal symmetric QSHE, the chiral edge states of the QHEare replaced by pairs of helical spin-edge states [seeFig.3(d))]. This leads to a 2e 2=hquantization of the observed transport signal ( König et al. , 2007 ) and resistance values in nonlocal experiments that can be expressed as specific integerfractions of the inverse conductance quanta ( Büttiker, 2009 ; Roth et al. , 2009 ). The QSHE initiated the new research field of topological insulators ( Hasan and Kane, 2010 ;Moore, 2010 ). In this context, we also point out the connection of the spin Hall phenomena to the research of 2D systems with spin and pseudospin degrees of freedom, including graphene andlayered transition-metal dichalcogenides ( Avsar et al. , 2014 ; Qian et al. , 2014 ;Xuet al. , 2014 ). C. Spin Hall effect and magnetic multilayers Among the early SHE device proposals, Zhang (2000) suggested to electrically detect the edge spin accumulationproduced by the SHE using an attached FM probe ( Silsbee, 1980 ;Johnson and Silsbee, 1985 ). In a broader context, the idea of connecting the SHE with the more mature field which utilizedFMs for injection and detection of spins in NMs fueled numerous studies of fundamental importance for the SHE field. Electrical spin injection from a FM contact and electricalobservation of the ISHE on a Hall cross patterned in the NM was demonstrated by Valenzuela and Tinkham (2006) . The original proposal by Zhang (2000) was first demon- strated in metal spin Hall devices by Kimura et al. (2007) . They showed that the same NM electrode attached to the FM can generate the SHE or the ISHE, i.e., can be used as an electrical spin injector or detector ( Valenzuela and Tinkham, 2006 ;Kimura et al. , 2007 ;Vila, Kimura, and Otani, 2007 ; Seki et al. , 2008 ;Mihajlovic et al. , 2009 ). FIG. 3 (color online). Schematics of the HE and the AHE (a), the QHE (b), the SHE (c), and the QSHE (d). In the HE and QHE, thecarrier deflection is a reaction to the Lorentz force. In the cases ofthe intrinsic AHE, SHE, and QSHE, the carriers experience aninternal spin-orbit force.Jairo Sinova et al. : Spin Hall effects 1217 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015Compared to metals, semiconductor spin transport devices with FM metal electrodes can suffer from the problem of the resistance mismatch, which hinders efficient spin transport across the interface ( Schmidt et al. , 2000 ). The introduction of a highly resistive tunnel barrier between theFM metal electrode and the semiconductor channel solves this problem ( Rasbha, 2000 ;Lou et al. , 2007 ) and FM tunnel contacts were successfully used to detect the SHE-induced spin-accumulation in a semiconductor ( Garlid et al. , 2010 ). Similarly, an electrical spin injection from a FM- semiconductor tunnel contact was used to demonstrate, sideby side, the electrical spin detection by the ISHE and by theFM electrode ( Olejník et al. , 2012 ). Using FMs contributed significantly to the basic under- standing of the SHE. Apart from the transport measurements,NM-FM hybrid structures also allow one to combine the SHE physics with the field of magnetization dynamics. The ISHE and SHE can be investigated using spin pumping (SP) andother related dynamic methods in structures comprising FMsand NMs, as illustrated in Fig. 4(Saitoh et al. , 2006 ;Ando et al. , 2008 ,2009 ;Mosendz, Pearson et al. , 2010 ;Mosendz, Vlaminck et al. , 2010 ;Czeschka et al. , 2011 ;Liuet al. , 2011 ; Miron, Garello et al. , 2011 ;Liu, Pai, Li et al. , 2012 ;Saitoh and Ando, 2012 ;Baiet al. , 2013 ;Wei et al. , 2014 ). In return, the SHE was found to provide efficient means for injecting spin currents into the FM, generating the STT ( Ralph and Stiles, 2008 ), and by this electrically controlling magnetiza- tion in FMs with potential applications in spintronic infor- mation technologies ( Miron, Garello et al. , 2011 ;Miron et al. , 2011 ;Liu, Pai, Li et al. , 2012 ;Emori et al. , 2013 ;Ryu et al. , 2013 ). Moreover, the ISHE detection of pure spin currents did not remain limited to NMs but is now used also in FMs ( Miao et al. , 2013 ;Azevedo et al. , 2014 ) and antiferromagnets (Freimuth, Blügel, and Mokrousov, 2010 ;Mendes et al. , 2014 ;Zhang et al. , 2014 ). In general, when SHE-induced torques in the adjacent FM are considered in the description of the dynamic magnetiza-tion (the Landau-Lifshitz-Gilbert equation), two types of torques can occur. An (anti-)damping-like torque which has the same functional shape as the Gilbert damping term (andthus can manifest itself in an increased or decreased Gilbert damping) and a field-like term which alters the magnetic energy landscape and can be observed as a shift of the resonance line in a FMR experiment. FMR allows thedetermination of the total internal magnetic field in a sampleas well as investigation of dissipation. Thus, in principleFMR-like techniques enable determination of field-like and (anti-)damping-like contributions of SHE-induced torques. If these current-induced torques arise only from the absorption of the spin current generated by the SHE in the NM, the analysis of field-symmetric and field-antisymmetric contributions of the detected dc output voltage at FMR allowsfor a quantitative determination of the strength and symmetryof the SHE-induced torques, as well as the spin Hall angle ofthe NM. Note that the torque can induce not only the small-angle FMR precession but the lateral current along a NM-FMinterface can also drive domain walls ( Miron et al. , 2011 ; Emori et al. , 2013 ;Haazen et al. , 2013 ;Ryu et al. , 2013 )o r switch the magnetization in the FM ( Miron, Garello et al. , 2011 ;Liu, Pai, Li et al. , 2012 ). This may have practical implications for designing domain-wall based memories or forthree-terminal magnetic tunnel junction bits with the lateral writing current decoupled from the perpendicular readout current. D. Spin Hall effect, spin galvanics, and spin torques From the early experiments with the relativistic torques, it was realized that the SHE is not the only possible mechanism responsible for torques induced by the lateral current in theNM-FM bilayers ( Manchon et al. , 2008 ). The interface breaks the structural inversion symmetry which implies that the SHE-STT can be accompanied by another microscopic mechanism. Its origin is in the so-called spin galvanic phenomena that were explored earlier in inversion-asymmetric NMs ( Ivchenko and Ganichev, 2008 ). In the picture discussed in the previous section, the spin current generated in the NM via therelativistic SHE is absorbed in the FM and induces theSTT. In the competing scenario, a nonequilibrium spin density of carriers is generated in inversion-asymmetric systems via the relativistic ISGE ( Ganichev et al. , 2004 ;Kato et al. , 2004b ;Silov et al. , 2004 ;Wunderlich et al. , 2004 ,2005 ; Belkov and Ganichev, 2008 ;Ivchenko and Ganichev, 2008 ). A SOT is then directly induced if the carrier spins are exchange coupled to magnetic moments ( Bernevig and Vafek, 2005 ;Manchon et al. , 2008 ;Chernyshov et al. , 2009 ;Miron et al. , 2010 ). From the early observations in nonmagnetic semiconduc- tors, SHE and ISGE are known as companion phenomena,both allowing for electrical alignment of spins in the samestructure ( Kato et al. , 2004a ,2004b ;Wunderlich et al. , 2004 , 2005 ). Hand in hand, SHE and ISGE evolved from subtle academic phenomena to efficient means for electrically reorienting magnets. Understanding the relation betweenthe spin Hall and spin galvanic phenomena is, therefore,important not only from the basic physics perspective but hasalso practical implications for spintronic devices. The term spin galvanic effect (SGE) is derived from the analogy to the galvanic (voltaic) cell. Instead of a chemical FIG. 4 (color online). Illustration of the SP spin-current gen- eration by magnetization dynamics from a FM into a NM. FromAndo et al. , 2011b .1218 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015reaction, however, it is the spin polarization that generates an electrical current (voltage) in the SGE. Inversely, an electrical current generates the spin polarization in the ISGE. Following theoretical predictions of the phenomena (Ivchenko and Pikus, 1978 ;Aronov and Lyanda-Geller, 1989 ;Ivchenko, Lyanda-Geller, and Pikus, 1989 ;Edelstein, 1990 ;Malâshukov and Chao, 2002 ;Inoue, Bauer, and Molenkamp, 2003 ), it was the SGE that was initially observed in an asymmetrically confined two-dimensional electron gas (2DEG) in a GaAs quantum well ( Ganichev et al. , 2002 ). The key signature of the SGE is the electrical current-induced by a nonequilibrium, but uniform, polarization of electron spins. The microscopic origin of the effect is illustrated in Fig. 5.I n the nonequilibrium steady state, the spin-up and spin-down subbands have different populations, induced in the Ganichev et al. (2002) experiment by a circularly polarized light excitation. Simultaneously, the two subbands for spin-up and spin-down electrons are shifted in momentum space due to the inversion asymmetry of the semiconductor struc- ture, which leads to an inherent asymmetry in the spin-flipscattering events between the two subbands. This results in the flow of the electrical current. A microscopic picture of the ISGE is illustrated in Fig. 6. The uniform nonequilibrium spin density occurs as a conse- quence of an electric-field and scattering induced redistribution of carriers on the Fermi surface whose texture of spin expect- ation values has a broken inversion symmetry. For the Rashbaspin-orbit coupling, illustrated in Fig. 6for one chirality, the uniform in-plane spin polarization is perpendicular to the applied electrical current.Initial observations of the ISGE were made in parallel with the initial SHE experiments, in both cases employing the Kerr-Faraday magneto-optical detection methods or circularly polarized luminescence ( Ganichev et al. , 2004 ;Kato et al. , 2004a ,2004b ;Silov et al. , 2004 ;Wunderlich et al. , 2004 , 2005 ;Belkov and Ganichev, 2008 ;Ivchenko and Ganichev, 2008 ).Kato et al. (2004a ,2004b) observed the SHE and ISGE in the same strained bulk n-InGaAs sample and Wunderlich et al. (2004 ,2005) detected the two effects in the same asymmetrically confined 2D hole gas (2DHG) in a AlGaAs/GaAs heterostructure. Subsequently, it was predicted ( Bernevig and Vafek, 2005 ) and experimentally verified ( Chernyshov et al. , 2009 ) that the ISGE can generate relativistic SOTs in a ferromagnetic semi-conductor (Ga,Mn)As with broken inversion symmetry in the strained crystal structure of a thin film sample. The reciprocal relativistic effect converting magnetization dynamics into acharge signal has also been observed in this inversion- asymmetric (Ga,Mn)As material ( Ciccarelli et al. , 2014 ). In the NM-FM bilayers with broken structural inversion symmetry, both the SHE- and ISGE-based mechanisms have been found to contribute to the relativistic spin torques (Manchon et al. , 2008 ;Miron et al. , 2010 ;Piet al. , 2010 ; Miron, Garello et al. , 2011 ;Suzuki et al. , 2011 ;Garello et al. , 2013 ;Kim et al. , 2013 ;Paiet al. , 2014 ). Similarly to the bulk inversion-asymmetric materials, in the structurally asymmet-ric NM-FM bilayers the reciprocal effects converting mag- netization dynamics into charge signals have been observed and attributed to the ISHE and SGE ( Saitoh et al. , 2006 ; Rojas-Sánchez, Vila et al. , 2013 ). As mentioned, the SHE and the Mott scattering of free electron beams can have the same extrinsic skew-scattering origin (captured by the second-order Born approximation). Moreover, in condensed matter systems, the SHE can arise from the spin-dependent transverse deflection induced by theintrinsic spin-orbit coupling in a perfect crystal with no impurities. We also mentioned that this intrinsic SHE has its direct counterpart in systems with broken time-reversalsymmetry in the intrinsic AHE.E k 0kkx xx- ++1/2 y-1/2 y FIG. 5. Microscopic origin of the spin galvanic current in the presence of k-linear terms in the electron Hamiltonian. The σykx term in the Hamiltonian splits the conduction band into two parabolas with the spin /C61=2in the ydirection. If one spin subband is preferentially occupied, for example, by spin injection(the j−1=2i ystates shown in the figure) asymmetric spin-flip scattering results in a current in the xdirection. The rate of spin- flip scattering depends on the value of the initial and final k vectors. There are four distinct spin-flip scattering events pos-sible, indicated by the arrows. The transitions sketched by dashedarrows yield an asymmetric occupation of both subbands andhence a current flow. If, instead of the spin-down subband, thespin-up subband is preferentially occupied the current direction isreversed. From Ganichev et al. , 2002 . ky kx Jkxky FIG. 6 (color online). Left panel: Rashba spin texture for one of the chiral states in equilibrium with zero net spin density. Rightpanel: nonequilibrium redistribution of eigenstates in an appliedelectric field resulting in a nonzero spin density due to brokeninversion symmetry of the spin texture. The opposite chiralityspin texture with lower Fermi wave vector is not drawn for clarity.This reversed chirality will give and opposite but lower con-tribution to the one shown, hence not changing the basic physicsillustrated here.Jairo Sinova et al. : Spin Hall effects 1219 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015The spin galvanic phenomena, on the other hand, are tradi- tionally considered to originate in NMs only from extrinsicorigins (seen already in the first-order Born approximation scattering). Nevertheless, the physics of the SHE, AHE, spin galvanics, and relativistic spin torques can be entangled evenwhen considering the intrinsic effects. In Fig. 7we illustrate that the same current-induced reactive mechanism that generates the transverse spin current in the intrinsic SHE can induce a uniformspin polarization, i.e., a signature characteristic of the ISGE, in systems with broken space and time-reversal symmetry. Relativistic SOTs generated by the nonequilibrium uniformspin polarization of this intrinsic origin were identified in the FM semiconductor (Ga,Mn)As ( Kurebayashi et al. , 2014 ). Current-induced torques generated by the companion spin Hall and spin galvanic phenomena are not limited to magnets with FM order. In antiferromagnets, the SHE- or ISGE- induced effective fields can have a microscopically staggerednature, i.e., can alternate in sign between the antiferromag- netic spin sublattices, and by this couple strongly to the Néel magnetic order ( Železný et al. , 2014 ). Since external magnetic fields couple weakly to antiferromagnetic moments, the electrically generated staggered fields are rather unique in providing efficient means for the manipulation of antiferro-magnets ( Wadley et al. , 2015 ). III. THEORY OF THE SPIN HALL EFFECT The SHE is a prime example of a field germinated directly from several key theoretical predictions and one which neededthe correct timing to come to its full life. It all began with the seminal prediction of the extrinsic SHE by Dyakonov and Perel (1971b) based on a phenomenological theory that considered the consequences of chiral Mott scattering in asolid-state system. This prediction laid dormant for almost three decades until Hirsch (1999) andZhang (2000) made a similar prediction, but at a time that the nascent field of spintronics could fully exploit the notion of the SHE. Shortly after this, Murakami, Nagaosa, and Zhang (2003) andSinova et al. (2004) predicted the intrinsic SHE based on linear response microscopic theories of strong spin-orbit- coupled materials. It is perhaps at this point that the fieldof SHE surged forward in a flurry of enormous theoretical activity, later on culminating in the parallel discoveries of the extrinsic ( Kato et al. , 2004a ) and intrinsic ( Wunderlich et al. , 2004 ,2005 ) SHE. The theories of the SHE have naturally emerged from the theory of the AHE. However, the ever-present key difference between the SHE and the AHE is that spin, unlike charge, is not a conserved quantity in most cases. This makes the examination of experiments and predictions more involved in the case of the SHE. In the initial predictions of the extrinsic SHE, this was dealt with by writing down phenomenological theories based on coupled spin-charge drift-diffusion equations derived from symmetry considerations. The approach is well justified in the weak spin-orbit-coupling regime ( Dyakonov and Perel, 1971b ;Hirsch, 1999 ;Zhang, 2000 ;Dyakonov and Khaetskii, 2008 ). However, within the strong spin-orbit-coupling regime, the dominant coherent effects of the intrinsic SHE are moredifficult to couple to such phenomenological theories. This is particularly relevant for heavy transition metals. It is within this strong spin-orbit-coupling regime that the AHE has made its furthest progress within the last decade. The intrinsic, skew-scattering, and side-jump mechanisms thatgive rise to the AHE were initially introduced and defined only through the nonsystematic semiclassical formalism. This did not allow for a direct application of fully microscopic computational approaches to calculate the effect in real materials. Over the last decade, a more systematic approachhas been followed that aimed at reaching agreement in nontrivial models using different linear response formalisms. This has led to a better established, microscopic-theory description of the mechanisms that is applicable to ab initio computational techniques of complex materials ( Sinitsyn et al. , 2007 ;Kovalev, Sinova, and Tserkovnyak, 2010 ; Nagaosa et al. , 2010 ). We spend the first part of Sec. III.A defining and explaining each of the contributions and their origins in the more modern parsing of the spin-dependent Hall transport theory. We will try to clarify, in particular, the typical misconceptions that sometimes linger in the literature regarding which aspect of the spin-orbit coupling —within the crystal itself or within the disorder potential —contributes to each mechanism. We bor- row in this part extensively from Nagaosa et al. (2010) and direct the interested reader to this previous review for detailed explanations of the different linear response theories and theresolution of some of the historical controversies. We follow in Sec. III.B with a description of the phenom- enological spin-charge drift-diffusion equations that are often ky ky ky kykxkx kxkxJ J(a) (b) (c) (d) FIG. 7 (color online). (a) A model equilibrium spin texture in a 2D Rashba spin-orbit-coupled system with spins (thick arrows)pointing perpendicular to the momentum. (b) In the presence ofan electrical current along the xdirection the Fermi surface (circle) is displaced along the same direction. When moving inmomentum space, electrons experience an additional spin-orbitfield (thin arrows). In reaction to this nonequilibrium current-induced field, spins tilt up for k y>0and down for ky<0, creating a spin current in the ydirection. (c) A model equilibrium spin texture in a 2D Rashba spin-orbit-coupled system with anadditional time-reversal symmetry breaking exchange field of astrength much larger than the spin-orbit field. In equilibrium, allspins in this case align approximately with the direction of theexchange field. (d) The same reactive mechanism as in (b)generates a uniform, nonequilibrium out-of-plane spin polariza-tion. Adapted from Sinova et al. , 2004 and Kurebayashi et al. , 2014 .1220 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015used to fit experiments. These equations are symmetry based and their phenomenological parameters are often extracted from fits to experiments. Because of the challenge of merging the strong spin-orbit- coupled microscopic theories and the phenomenological weak spin-orbit-coupling theories, one of the more popular models that is used to describe the SHE is based on a simpleHamiltonian in which the spin-orbit coupling is only presentin the disorder potential. We discuss such a model inSec. III.C. This theory has the benefit of having a single parameter —the strength of the spin-orbit coupling of the disorder potential. The parameter can be fitted to the spin-diffusion length and from this the value of the SHE can beestimated ( Zhang, 2000 ;Crépieux and Bruno, 2001 ;Engel, Halperin, and Rashba, 2005 ;Maekawa and Takahashi, 2012 ). However, as seen by comparing to experiment, this model gives sensible results in the weak spin-orbit-coupling regimebut misses the coherent effects of the band structure instrongly spin-orbit-coupled materials. In Sec. III.D , we discuss in detail the theory of SP and how it is utilized to measure the ISHE and the spin Hall angle. Thetheory of SP introduces the concept of the spin-mixingconductance ( Tserkovnyak, Brataas, and Bauer, 2002a ), another parameter borrowed from the weak spin-orbit-coupled systems, which is at present often used in analyzing mag-netization-dynamics experiments in connection to the physicsof the SHE ( Saitoh et al. , 2006 ). Besides introducing the basic concepts of SP and its connection to the measurements of the ISHE, we discuss the range of assumptions and limits which are often used when analyzing experiments. In Sec. III.E, we present the formalism primarily used in the strong spin-orbit-coupled systems. The formalism is based in the Kubo formula and exploited successfully in transitionmetals ( Tanaka et al. , 2008 ;Freimuth, Blügel, and Mokrousov, 2010 ). Calculations seem to indicate that for these metals the principal contribution, as in the AHE, arises from the intrinsic deflection mechanism. A. Mechanisms of the spin Hall effect The spin-dependent Hall effects (AHE, SHE, and ISHE) originate from three distinct microscopic mechanisms that they all share: the skew, the side jump, and the intrinsicmechanisms. The mechanisms are caused by coherent bandmixing effects induced by the external electric field and thedisorder potential. This makes them more complex than the simpler single-band diagonal transport. As with other coherent interference transport phenomena, they cannot be directly explained using traditional semi- classical Boltzmann theory. It is then not surprising that the original proposals based on semiclassical theory for theintrinsic, skew-scattering, and side-jump mechanisms broughtinsightful new concepts, as well as seeds for ensuing con-troversies in the debate over the quantum-mechanical micro- scopic origins of the AHE and SHE. There exists now a more modern, stricter definition of the mechanisms within microscopic theories. However, to keep continuity and not create further confusion, this more modernapproach has inherited the already established lexicon [seeNagaosa et al. (2010) , Sec. IV].The new parsing of the microscopic mechanisms is based on both experimental and microscopic-transport theory con- siderations, rather than on the identification of one particulareffect within semiclassical theory. The justification here is, of course, primarily on the AHE, not the SHE. For the SHE, the spin Hall conductivity and its consequences have to beultimately coupled to the spin accumulation that it induces and can therefore depend on the method of measurement. In other words, depending on the measurement, the spin accu- mulation induced by the SHE may vary, e.g., in nonlocal transport measurements versus FMR-based measurements. As mentioned in the introduction of Sec. III, the key recent development that led to a better understanding of the mech-anisms was linking directly the semiclassical and microscopic theory of spin-dependent Hall transport. This link between the semiclassically defined processes and their fully equivalentmutliband microscopic theories was established by fully generalizing the Boltzmann transport theory to take interband coherence effects into account ( Sinitsyn et al. , 2007 ;Nagaosa et al. , 2010 ). Based on what we have learned from the AHE, a very natural classification of contributions is to separate them according to their dependence on the transport lifetime τ. This classification is directly guided by experiment and by themicroscopic theory of metals. Within the metallic regime,disorder is treated perturbatively and higher order terms varywith a higher power of the quasiparticle scattering rate τ −1.A s we discuss, it is relatively easy to identify contributions to theanomalous or spin Hall conductivity σ Hxy, which vary as τ1and asτ0. In experiments of the AHE, a similar separation can sometimes be achieved by plotting σxyvs the longitudinal conductivity σxx∝τ,w h e n τis varied by altering disorder or varying temperature. However, it is important to note that several microscopically distinct contributions can share the same τdependence (Sinitsyn et al. , 2007 ;Sinitsyn, 2008 ). The contribution proportional to τ1we define as the skew-scattering contribu- tion, σH−skewxy . The second contribution proportional to τ0(or independent of σxx) we further separate into two terms: intrinsic andside jump . The first term arises from the evolution of spin-orbit-coupled quasiparticles as they are accelerated by an external electric fieldin the absence of disorder. The second term arises from scatteringevents from impurities that do not include the skew-scatteringcontribution. This then leaves a unique definition for the side-jump term, as σ H−sj xy≡σHxy−σH−skewxy−σH−intxy. We further describe these contributions below. We note that the above definitions have not relied on linking the terms tosemiclassical processes such as side-jump scattering ( Berger, 1970 ) or skew scattering from asymmetric contributions to the semiclassical scattering rates ( Smit, 1958 ), as was done in earlier theories. The ideas explained briefly in this section are substantiated in the recent review by Nagaosa et al. (2010) , which analyzes the tendencies in the AHE data of several material classes and extensively discusses the AHE theory. The extensions to theother spin-dependent Hall effects, such as SHE and ISHE, require the coupling of these spin-current generating mech- anisms to spin-charge drift-diffusion transport equations thatJairo Sinova et al. : Spin Hall effects 1221 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015are appropriate to describe the particular experiment, be it optical or electrical. 1. Intrinsic mechanism Among the three contributions, the easiest to evaluate accurately and the one that has dominated most theoretical studies is the intrinsic contribution. There is a direct link of theintrinsic effect to the semiclassical theory in which the induced interband coherence is captured by an anomalous velocity arising from a momentum-space Berry phase. In the context of the AHE, this contribution was first derived by Karplus and Luttinger (1954) but its topological nature was not fully appreciated until recently ( Jungwirth, Niu, and MacDonald, 2002 ;Onoda and Nagaosa, 2002 ). The work of Jungwirth, Niu, and MacDonald (2002) was moti- vated by the experimental importance of the AHE in FM semiconductors and also by the analysis of the relationship between momentum-space Berry phases and anomalous trans-verse velocities in semiclassical transport theory ( Sundaram and Niu, 1999 ;Xiao et al. , 2010 ). Its connection to the SHE was described by Murakami, Nagaosa, and Zhang (2003) and Sinova et al. (2004) . The intrinsic contribution to the spin Hall conductivity is dependent only on the band structure of the perfect crystal, hence its name. Pictorially, it can be seen to arise from the nonequilibrium electron dynamics of the Bloch electrons asthey are accelerated in an electric field and undergo spin precession due to the induced momentum-dependent mag- netic field, as illustrated in Sec. II.D, Fig. 7. Here the system is described by a 2D Rashba Hamiltonian, H¼ p2 2m−λ ℏσ·ðˆz×pÞ; ð3:1Þ where p¼ℏkis the 2D electron momentum, λis the Rashba coupling constant, σthe Pauli matrices, mthe electron effective mass, and ˆzthe unit vector perpendicular to the 2DEG plane. For this example, the dynamics of an electron spin in the presence of time-dependent spin-orbit coupling is described by the Bloch equation ( Sinova et al. , 2004 ), ℏdˆn dt¼ˆn×ΔðtÞþαℏdˆn dt×ˆn; ð3:2Þ where ˆnis the direction of the spin and αis a damping parameter that we assume is small. In Eq. (3.2), the p-dependent effective Zeeman coupling induced by the spin-orbit-coupling term is given by −s·Δ=ℏ, where Δ¼ 2λ=ℏðˆz×pÞ. For a Rashba effective magnetic field with magnitude Δ1that initially points in the ˆx1direction, the effective field then tilts (arbitrarily slowly) slightly toward ˆx2, where ˆx1and ˆx2are orthogonal in-plane directions. It follows from the linear response limit of Eq. (3.2) that ℏdn2 dt¼nzΔ1þαdnz=dt; ℏdnz dt¼−Δ1n2−αdn2=dtþΔ2;ð3:3Þwhere Δ2¼Δ·ˆx2. By solving these inhomogeneous coupled equations, it follows that to leading order in the slow-time dependences n2ðtÞ¼Δ2ðtÞ=Δ1, i.e., the ˆx2component of the spin rotates to follow the direction of the spin-orbit field, andthat n zðtÞ¼1 Δ2 1ℏdΔ2 dt: ð3:4Þ The dynamics give rise to the spin current in the ˆydirection, js;y¼Z annulusd2p ð2πℏÞ2ℏnz;p 2py m¼−eEx 16πλmðpFþ−pF−Þ; ð3:5Þ where pFþandpF−are the Fermi momenta of the majority and minority spin Rashba bands ( Sinova et al. , 2004 ). We choose the example based on the Rashba system because it is simple to see pictorially the intrinsic contribution. However,for this particular simple example, in a large range of Fermienergies the result for the intrinsic spin Hall conductivity turns out to be σ H−intxy ¼−ðe=ℏÞjs;y=Ex¼e2=8πℏ. This contribution is eventually canceled by short-range disorder scatteringbecause the induced spin current is proportional to the spindynamics, which should vanish in the steady state ( Inoue, Bauer, and Molenkamp, 2004 ;Dimitrova, 2005 ;Raimondi et al., 2012 ). The issue of the cancellation between the intrinsic and side-jump contribution, the so-called vertex corrections, has been debated extensively and we discuss it briefly below. Here, we point out that the exact cancellation is only present in theparabolic 2D linear Rashba Hamiltonian and is not present inother spin-orbit-coupled Hamiltonians corresponding to real- istic material systems ( Shytov et al. ,2 0 0 6 ;Sinova et al. , 2006 ; Raimondi et al. , 2012 ). The above result, illustrated in a simple semiclassical form, is usually best evaluated directly from the Kubo formula for the spin Hall conductivity for an ideal lattice ( Sinova et al. , 2004 ), σ H−intxy ¼e2 VX k;n≠n0ðfn0;k−fn;kÞ ×Im½hn0kjˆjz spinxjnkihnkjvyjn0ki/C138 ðEnk−En0kÞðEnk−En0k−ℏω−iηÞ; ð3:6Þ where n; n0are band indices, jz spin¼ℏ 4fσz;vgis the spin- current operator, ωandηare set to zero in the dc clean limit, and the velocity operators at each pare given by ℏvi¼ ℏ∂HðpÞ=∂pi. It is important to emphasize that the semi- classical derivation describing the time-dependent polariza- tion of the Bloch states as they are accelerated and the Kubo formalism are entirely equivalent, as they should be from acorrect treatment of linear response of this contribution. Theintrinsic contribution to the AHE and SHE conductivity can also be obtained from the semiclassical theory of wave packet dynamics ( Sundaram and Niu, 1999 ;Jungwirth, Niu, and MacDonald, 2002 ;Culcer et al. , 2004 ;Xiao et al. , 2010 ). What makes the intrinsic contribution quite unique, par- ticularly in the AHE, is that it is directly linked to thetopological properties of the Bloch states. Specifically, it is1222 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015proportional to the integration over the Fermi sea of the Berry ’s curvature of each occupied band. One of the motivations for identifying the intrinsic con- tribution σH−intxy is that it can be evaluated accurately even for materials with relatively complex electronic band structure using microscopic ab initio theory techniques. In many materials which have strongly spin-orbit-coupled bands, theintrinsic contribution seems to dominate the SHE and AHE. The calculations have given semiquantitative predictions of the expected spin Hall angles, particularly in metals. This isillustrated in the density-functional calculation for Pt ( Guo et al. , 2008 ), shown in Fig. 8, and in the microscopic tight- binding calculations for other 4dand5dmetals ( Tanaka et al. , 2008 ), shown in Fig. 9. As is clear from Fig. 8, the largest contributions to the spin Hall conductivity arise, similar toAHE, whenever bands connected via spin-orbit coupling are near each other at the Fermi energy. The calculated spin Hall conductivities are predicted to be large in these transitionmetals, and, in particular, a sign change is predicted going from Pt to Ta which has been observed in experiments. More recent density-functional calculations on a range of hcp metalsand antiferromagnetic Cr, show a strong anisotropy of the spinHall conductivity ( Freimuth, Blügel, and Mokrousov, 2010 ), as illustrated in Fig. 10. 2. Skew-scattering mechanism The skew-scattering contribution to the SHE and the AHE is the mechanism proportional to the Bloch state transportlifetime τ. It will therefore tend to dominate in nearly perfect crystals. It is the only contribution to the SHE and AHE whichappears in traditional Boltzmann transport theory where interband coherence effects are usually neglected. Skew scattering is due to chiral features which appear in the disorderscattering in the presence of spin-orbit coupling. This mecha-nism was first identified in FMs by Smit (1958) and has its origins in the Mott scattering in relativistic physics ( Mott, 1929 ,1932 ). Typical treatments of semiclassical Boltzmann transport theory found in textbooks often appeal to the principle ofdetailed balance. This states that the transition probabilityW n→mfrom state ntomis identical to the transition probability in the opposite direction ( Wm→n). In Fermi ’s golden-rule approximation, where Wn→n0¼ð2π=ℏÞjhnjVjn0ij2δðEn−En0Þ, with Vbeing the perturbation inducing the transition, the detailed balance indeed holds. However, detailed balance inthe microscopic sense is not generic. In the presence of spin-orbit coupling, either in the Hamiltonian of the perfect crystal orin the disorder Hamiltonian, a transition which is right handedwith respect to the magnetization direction has a differentprobability than the corresponding left-handed transition. Whenthe transition rates are evaluated perturbatively, asymmetricchiral contributions appear at third order. In simple models, the -11-10-9-8-7-6-5-4-3-2-101234Energy (eV)SOC noSOC -20 0 20 σxyz (102Ω-1cm-1)-11-10-9-8-7-6-5-4-3-2-101234 W L X W Γ Γ(b) (a)fcc Pt (c) (d) FIG. 8 (color online). Band structure (a) for Pt calculated with (solid lines) and without (dotted lines) spin-orbit coupling. Thespin Hall conductivity (b) is shown calculated at each energy. Inthe lower figure, the Berry curvature is calculated (total) (c) aswell as the one corresponding for each subband (d). From Guo et al. , 2008 . SHC ( ) −0.50.00.51.0γ = 0.02 4d 5dSHC ( ) −0.1−0.050.00.05 5d4d 4d 5dγ = 0.2 n7 91 1 AuAg6 5 81 0 Tc Ru Os ReMo Nb Rh Pd Ta W Ir Pt10 cm3 −1 Ω−1 10 cm3 −1 Ω−1 FIG. 9 (color online). Intrinsic spin Hall conductivity for 4dand 5dtransition metals. A key feature is the change in sign from Pt to Ta. From Tanaka et al. , 2008 .Jairo Sinova et al. : Spin Hall effects 1223 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015asymmetric chiral contribution to the transition probability of momenta k;k0is often assumed to have the form WA kk0∼ðk×k0Þ·M: ð3:7Þ Inserting this asymmetry into the Boltzmann equation leads to a current proportional to the longitudinal current driven by theelectric field Eand perpendicular to both EandM, where Mis the magnetization direction in case of the AHE and the direction of the polarization of the spin current in case of the SHE. The corresponding contribution to the Hall conductivity σ H−skewxy and the conductivity σxxare proportional to the transport lifetime τ. Therefore, the spin or anomalous Hall resistivity is proportional to the longitudinal resistivity ρxx, whenever this contribution dominates, since ρH−skewxy ≈σH−skewxy ρ2xx∼ρxx. There are several specific mechanisms for skew scattering [see Sec. IV.B and Sec. V.A. in Nagaosa et al. (2010) ]. To evaluate the skew-scattering contribution to the Hall conduc-tivity, one needs an accurate solution of the chiral part to thecollision integral of Boltzmann equation. In practice, our ability to accurately estimate the skew-scattering contribution to the SHE and AHE of a real material is limited only by theaccuracy of the characterization of its disorder. In simple models, the skew-scattering contributions to the SHE or AHE are considered to arise only from the spin-orbitcoupling in the disorder potential. This is only valid when the typical disorder broadening is larger than the splitting of the bands due to the spin-orbit coupling. In systems with strongspin-orbit coupling in the bands, such as heavy transitionmetals, considering the spin-orbit coupling only in the disorder potential would be incorrect. The reason is because, in this case, a strong contribution to the skew scattering also arises from thescattering of the spin-orbit-coupled quasiparticle from thescalar potential. In fact, the spin-orbit coupling of the disorder potential is typically strongly renormalized by the other nearby subbands as well, and therefore the effect of the multibandcharacter can never be ignored in these materials. Studies focused on the skew scattering from an ab initio perspective were started by Gradhand et al. (2010) . Further recent studies of skew scattering based on ab initio electronic structure and the Boltzmann equation in systems with impu- rities of Cr, Mn, Fe, Co, and Ni in Pt, Au, and Pd hosts haveyielded contributions to the spin Hall angle of a fraction of apercent ( Long et al. , 2014 ;Zimmermann et al. , 2014 ). Resultsrelated to 1% doping of impurities to a Pt host are shown in Fig. 11. An additional contribution, not considered often, can arise from the variation of the spin-orbit coupling in real space. This possibility exists in certain semiconductor devices with 2D-Rashba spin-orbit coupling. Such variations have beenshown to yield a contribution to the spin Hall effect in these2D systems ( Dugaev et al. , 2010 ). We end this section with a note directed to the interested reader who is more versed in the latest development of thelinks between the full semiclassical and the microscopictheory of the SHE and AHE. We have been careful abovenot to define the skew-scattering contribution as the sum of allthe contributions arising from the asymmetric scattering rate present in the collision term of the Boltzmann transport equation. We know from microscopic theory that this asym-metry also makes an AHE contribution of order τ 0(Sinitsyn et al. , 2007 ). There exists a contribution from this asymmetry which is present in the microscopic theory treatment asso-ciated with the so-called ladder-diagram corrections to theconductivity, and therefore of order τ 0. In the more modern parsing of the contributions to the SHE and AHE, we do notassociate this contribution with skew scattering but place itunder the umbrella of side-jump scattering even thoughit does not physically originate from any side-step type ofscattering. 3. Side-jump mechanism Given the sharp definition we have provided for the intrinsic and skew-scattering contributions to the SHE andAHE conductivity, the equation FIG. 10 (color online). Intrinsic spin Hall conductivity for hcp metals Sc, Ti, Zn, Y, Zr, Tc, Ru, Cd, La, Hf, Re, and Os and forantiferromagnetic Cr. From Freimuth, Blügel, and Mokrousov, 2010 . FIG. 11 (color online). Skew-scattering spin Hall angle in a Pt host with 1% level of impurities. From Zimmermann et al. , 2014 .1224 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015σHxy¼σH−intxy þσH−skewxy þσH−sj xy ð3:8Þ defines unambiguously the side-jump contribution as the difference between the full SHE-AHE conductivity and the skew and intrinsic contributions. In using the term side jump for the remaining contribution, we are appealing to thehistorically established taxonomy outlined in the previoussection. Establishing this connection rigorously has been the most controversial aspect of the AHE theory and, not surprisingly, some confusion has spilled over to the discussionof the SHE. The basic semiclassical argument for a side-jump contribu- tion can be stated straightforwardly: when considering thescattering of a Gaussian wave packet from a spherical impurity with spin-orbit interaction [ H so¼ð1=2m2c2Þðr−1∂V=∂rÞSzLz], a wave packet with incident wave vector kwill undergo a displacement transverse to kequal to1 6kℏ2=m2c2. This type of contribution was first noticed, but discarded, by Smit (1955 , 1958) and reintroduced by Berger (1964) who argued that it was the key contribution to the AHE. Most of the earlierdevelopments were based on physical arguments of how toincorporate this physics in a semiclassical Boltzmann formal- ism. Because this cannot be done systematically, errors ensued (Nagaosa et al. , 2010 ). A common misconception is that the side jump can be generally computed by taking only into account the spin- orbit-coupling interaction of the disorder scattering potential.This can only be justified in a weak spin-orbit-coupled system,e.g.,n-doped GaAs, where indeed this is likely to be the case. This is the consideration in the Crépieux-Bruno model (Crépieux and Bruno, 2001 ), where the spin-orbit coupling is only present in the disorder potential and which has beensubsequently used by others to model the extrinsic SHE (Engel, Halperin, and Rashba, 2005 ;Maekawa and Takahashi, 2012 ). However, when addressing materials with strong spin- orbit coupling, it is important to remember that there arealways two sources of side-jump scattering: (1) Extrinsic side jump: the contribution arising from the non-spin-orbit-coupled part of the wave-packet scat- tering off the spin-orbit-coupled disorder. (2) Intrinsic side jump: the contribution arising from the spin-orbit-coupled part of the wave packet formed by the Bloch electrons scattering off the scalar potential alone without spin-orbit coupling. Both can be important and independent of each other, depending on the crystalline environment and the type of scattering impurity. In heavy-element materials, such as Pt andTa, the dominant contribution is likely to be the second type ofcontribution. In FMs, it has been demonstrated that the second type of contribution, termed here intrinsic side jump to distinguish them clearly, can be very large. Both of theseside-jump contributions add to the scattering-independentmechanisms, i.e., they are independent of τ(Weischenberg et al. , 2011 ). The intrinsic SHE-AHE contribution and the side-jump contribution, which we further separated into the extrinsic side jump and the intrinsic side jump, have all quite differentdependences on specific material parameters, particularly insystems with complex band structures. [For a detailed reviewon these delicate issues, see Sinitsyn (2008) ]. Most of the prior mistakes surrounding the theory of side jump can be tracedback to the physical meaning ascribed to quantities whichwere gauge dependent, like Berry ’s connection, which is typically identified as the definition of the side step uponscattering. Studies of simplified models, e.g., semiconductorconduction bands, also gave results in which the intrinsic- side-jump contribution seemed to be of the same magnitude but opposite in sign compared to the intrinsic contribution(Nozieres and Lewiner, 1973 ). It is well understood now that these cancellations are unlikely in more complex models(Sinitsyn et al. , 2007 ;Weischenberg et al. , 2011 ). The prior cancellations can be traced back to the fact that, in these verysimple band structures, Berry ’s curvature of the Bloch electrons is a constant independent of momentum. One isreminded in this case of the famous quote attributed to AlbertEinstein, “Things should be made as simple as possible but not simpler. ” It is only through a careful comparison between different fully microscopic linear response theory calculations, basedon equivalently valid microscopic formalisms such as Keldysh(nonequilibrium Green ’s function), the Kubo formalisms, and the systematically developed semiclassical theory, that thespecific contributions due to the side-jump mechanism can beidentified with confidence ( Sinitsyn et al. , 2007 ;Nagaosa et al. , 2010 ). Recently, there have been major steps forward in the theory of the AHE in developing full theories with predictive powerto calculate all the AHE contributions in FM materials with a complex band structure ( Freimuth, Blügel, and Mokrousov, 2010 ;Kovalev, Sinova, and Tserkovnyak, 2010 ;Lowitzer, Ködderitzsch, and Ebert, 2010 ;Weischenberg et al. , 2011 ; Czaja et al. , 2014 ). In the theory of the SHE, on the other hand, such progress still remains to be undertaken fully. Thereason is perhaps because of the complexity of the measure-ments, the dephasing of spin, and the lack of practical generaltheories that can bring one from a weak to a strong spin-orbit-coupled regime. 4. Cancellation of mechanisms in model systems We mentioned earlier that in certain simplified models there exist relative cancellations, either total or partial, between thecontributions that depend to zeroth order on the scatteringlifetime. This is a topic that has entertained the researchcommunity of the AHE and the SHE for quite some time. The reader familiar with the early history of the AHE and with the early history of the intrinsic SHE will have seen manyworks debating these issues. Some simplified models thatallow for an analytical treatment show such cancellations. Thetwo key ones are the linear wave vector 2D Rashba model(with parabolic dispersion) and the direct gap conduction bandIII-V semiconductor model (Kane model). In the 2D Rashba model, the intrinsic contribution and the intrinsic-side-jump contribution cancel each other directly inthe presence of short-range disorder scattering. The easiestway to see this is by noticing that the spin dynamics is directlyproportional in this model to the spin current generated.Therefore, in the steady state, the spin current must vanish(Inoue, Bauer, and Molenkamp, 2004 ;Dimitrova, 2005 ;Jairo Sinova et al. : Spin Hall effects 1225 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015Shytov et al. , 2006 ;Sinova et al. , 2006 ). For an extensive discussion of the issues of this particular model, see Raimondi et al. (2012) . When one incorporates to the model higher dimensions or a nonparabolic dispersion, such cancellations do not occur. For example, a graphene-like model (lineardispersion) does not exhibit such cancellations and provides a useful model to study the relative dependence of the mech- anisms ( Sinitsyn et al. , 2007 ). Also, as mentioned, another extensively studied model is the Kane model ( Nozieres and Lewiner, 1973 ). In this model, the relative cancellation is a 2 to 1 ratio. The fact that theintrinsic contribution and the intrinsic-side-jump contribution have the same dependence on the parameters (up to the factor of−2) can be traced back to Berry ’s curvature being constant (independent of momentum) in such a model. For most other spin-orbit-coupled Hamiltonians, corre- sponding to a realistic materials system, these exact cancella-tions do not seem to arise. This has been verified primarily by comparison to experiments. Nevertheless, this remains an important topic in the SHE field which will continue to berefined as better approximations are created to treat the effects of disorder. B. Phenomenological drift-diffusion theory Dyakonov and Perel (1971a ,1971b) considered the phe- nomenological theory of the SHE by coupling the usual drift- diffusion equation for charge transport to the spin-currentdrift-diffusion equations. Hence, the spin-charge drift-diffusion equations applicable to electrical transport measure- ments can be written from symmetry considerations as (Dyakonov and Khaetskii, 2008 ) j c¼eμnEþD∇nþeαSHμðE×PÞþeαSHDð∇×PÞ;ð3:9Þ js ij¼−ℏμnEiPjþD∂Pj ∂xi−ℏαSHϵijk/C18 μnEkþD∂n ∂xk/C19 ;ð3:10Þ where the first two terms of Eqs. (3.9) and(3.10) correspond to the definition of the uncoupled charge and spin currents. HerePcorresponds to the spin polarization, Dis the electron diffusion constant, μis the spin-independent electron mobility, nis the electron density, Eis the electric field, and αSHis the spin Hall angle defined by the ratio of the spin Hallconductivity to the diagonal charge conductivity. In Eq. (3.9), the third term corresponds to the AHE. The fourth term describes the ISHE if a charged diffusive current isabsent, i.e., in the case of the pure spin current in the system. We distinguish this from the situation in which a polarized charge diffusive current, e.g., generated by optical excitation ( Bakun et al. , 1984 ), leads to a charge transverse current which we associate here with a regime closer to the AHE. This distinction is made more clear by the fact that the SHE has a precisedefinition of a pure spin current being generated by a charge current, and therefore its inverse is associated with a pure spin current generating a transverse charge current. In Eq. (3.10) , the third term represents the SHE from an electric field, while the fourth term represents its diffusion-driven counterpart. The equations are written to first order in the spin Hall angle.Equations (3.9) and (3.10) can be directly extended to include junctions ( Johnson and Silsbee, 1987 ). Recently, there has been also an extension of these equations to incorporatethermal SHEs within the emerging field of spin caloritronics (Bauer, MacDonald, and Maekawa, 2010 ;Bauer, Saitoh, and van Wees, 2012 ). The treatment, for the most part, remains phenomenological with connections, in particular, to theOnsager relations between the thermodynamic forces andtheir corresponding entropy fluxes. Within this emergingsubfield of spintronics, many theoretical challenges remain,not least a better treatment of scattering coherent effects drivenby statistical forces and the ability of going beyond the simpleadiabatic frozen phonon approximations. A connection has also been made between the diffusion regime treated by the above equations and the microscopictreatment considering the specific boundary conditions in 2DRashba systems ( Adagideli and Bauer, 2005 ). In this case, the spin accumulation at the edges remains in a diffusive systemeven though no spin current exists in the bulk for this model.The results are also relevant when connecting the drift-diffusion equations to the indirect detection of the SHE byferromagnetic contacts ( Adagideli, Bauer, and Halperin, 2006 ). C. Crépieux-Bruno model of extrinsic side jump and skew scattering We discuss here the Crépieux and Bruno (2001) model that incorporates spin-orbit coupling only through the disorderpotential, i.e., there is no spin-orbit coupling present directlyin the Bloch electron bands at the Fermi surface. This modelhas been applied to weak spin-orbit-coupled materials, such asn-GaAs, to explain the extrinsic origin of their SHE ( Engel, Halperin, and Rashba, 2005 ), schematically illustrated in Fig. 12, and has also been applied to weakly spin-orbit- coupled metals ( Maekawa and Takahashi, 2012 ). The model builds on the influential work of Nozieres and Lewiner (1973) , where they studied the AHE in semiconductors with a simpleband structure. In particular, they focused on how to accountfor the effects of the spin-orbit coupling by projecting multi-band systems to an effective two-band model. There are manysubtle issues in such treatment already at the level of thissimple model. However, extrapolating some of its results togeneralities, e.g., specific cancellations, is dangerous since k (a) k2 (b) FIG. 12. Schematic of the (a) skew-scattering and the (b) extrinsic-side-jump mechanisms from a quantum point ofview (⊙corresponds to spin up and ⊗to spin down). The bold curves represent the anisotropic enhancement of the amplitude ofthe wave packet due to spin-orbit coupling. Here the electronsthemselves contain no spin-orbit coupling. From Crépieux and Bruno, 2001 .1226 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015“the side-jump is no longer given by the simple expression ” derived in these works, as they themselves state ( Nozieres and Lewiner, 1973 ). The two-band model Hamiltonian is given by H¼ℏ2k2 2m/C3þVðrÞþλe−soσ·ðk×∇VÞ¼H0þW: ð3:11Þ Here m/C3is the effective mass of the Bloch electron, VðrÞis the disorder potential, σare the Pauli matrices, and λe−sois the effective spin-orbit coupling parameter. For a freeelectron, λ 2e−so¼ℏ2=2m2c2is an extremely small parameter (∼10−12Å2), but in a solid-state environment it is strongly renormalized by nearby bands. For the effective two-bandmodel of conduction electrons, obtained from the 8×8Kane description of the semiconductor band structure, λ e−so¼ ðP2=3Þ½1=E2g−1=ðEgþΔsoÞ/C138,w i t h Egbeing the gap, Pthe s-pdipole matrix element, and Δsothe spin-orbit splitting of the valence band. For n-GaAs, for example, this value is 5.3Å2. The total scattering potential is W. In this model, the velocity operator is modified by the spin- orbit-coupled term to read ˆv¼ˆp m/C3þλe−so ℏ½σ×∇V/C138; ð3:12Þ and the scattering amplitude due to the disorder potential from state jk;sitojk0;s0iis given by hk0;s0jWjk; si¼ ~Vkk0½δss0þiλe−soðσs0s×k0Þ·k/C138;ð3:13Þ where ~Vkk0is the Fourier transform of V. The disorder potential is considered to be short ranged for simplicity[for ionic scattering, see Engel, Halperin, and Rashba (2005) ] such that VðrÞ¼u iP jδðr−rjÞand ~Vkk0¼uiδk;k0. One can then connect this procedure with the Boltzmannequation, which will also yield the spin-diffusion equation (Zhang, 2000 ). Microscopically, the scattering from this disorder potential induces a collision integral in theBoltzmann formalism of the form /C18 ∂fk;s ∂t/C19 coll¼X k0;s0½Pks;k0s0fk0s0−Pk0s0;ksfks/C138; ð3:14Þ with the transition scattering probabilities available from the T matrix, which yields a symmetric and antisymmetric contri-bution, P sym k0s0;ks¼2π ℏni Vu2 i½δs;s0þλ2e−sojðk0×kÞ·σs;s0/C138δðϵk0−ϵkÞ;ð3:15Þ Pant k0s0;ks¼−ð2πÞ2 ℏλe−soni Vu3 i½Nð0Þδs;s0ðk0×kÞ·σs;s0/C138 ×δðϵk0−ϵkÞ: ð3:16Þ Here, niis the density of impurities and Nð0Þis the density of states at the Fermi level. Within the framework of thesemiclassical Boltzmann equation, one then writes ( Zhang, 2000 ;Maekawa and Takahashi, 2012 )v k·∇fksþeE ℏ·∇kfks¼−δfks τtr−f0 ks−f0 k0s τsfðθÞ;ð3:17Þ where τ−1 tr¼X k0;s0Psym ks;k0s0¼1 τ0 trð1þ2k4 Fλ2e−so=3Þð 3:18Þ and τ−1 sf¼X k0Psym k1;k0−1¼k4 Fλ2e−so 3τ0 tr½1þcos2ðθÞ/C138: ð3:19Þ Here τtris the transport lifetime, τ0 tris the transport lifetime when neglecting the spin-orbit coupling part of the disorderpotential, and τ sfis the spin-flip time. Further expanding the equilibrium and nonequilibrium distribution functions yieldsthe spin-diffusion equation, ∇ 2ðμ1−μ−1Þ¼1 λ2 sdðμ1−μ−1Þ; ð3:20Þ with λ2 sd¼Dτsf=2,D¼ð1=3Þτtrv2 F, and μsrepresenting the spin-dependent chemical potentials ( s¼/C61is the spin index). Averaging over the scattering angle, one obtains thatthe ratio of spin-flip time and transport time for this particularmodel is τtr τsf≈1 2k4 Fλ2e−so: ð3:21Þ This is one of the key aspects that has made this model appealing. The model provides a means to obtain its effectivespin-orbit-coupling parameter λ e−soby measuring the spin- diffusion length, independently of the spin Hall angle. At thispoint, it should be emphasized that the model is applicable tothe weak spin-orbit-coupling regime, i.e., when τ tr=τsf≪1. From either a microscopic or Boltzmann-like analysis, the result for this model for the extrinsic-side-jump contribution tothe spin Hall angle is ( Crépieux and Bruno, 2001 ;Engel, Halperin, and Rashba, 2005 ) α sj SH≡σH−sj xy σxx¼−2λe−som/C3 τtr¼−2k2 Fλe−so kFl:ð3:22Þ Here, l¼τtrkF=m/C3is the mean free path. The skew-scattering contribution is given by αsk SH¼4π 3k2 Fλe−soNð0Þui: ð3:23Þ In the SHE experiments in n-GaAs, the spin Hall angle is dominated by the skew-scattering contribution versusthe extrinsic-side-jump contribution by a factor of 2, σ H−skewxy = σH−sj xy∼−1.7=0.8(Engel, Halperin, and Rashba, 2005 ). When this simplified model is used for metals, it yields a mixture of results and inconsistencies. This can be seen inTable I, where we show for a series of metals the experimental SHE angles and the independently experimentally inferredJairo Sinova et al. : Spin Hall effects 1227 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015effective spin-orbit-coupling parameters k2 Fλe−so. It is not possible to estimate the skew scattering from the model expression (3.23) without knowing the specific value for ui. However, we know from the AHE that skew scattering has only been seen to dominate for extremely conductive metals, so it is neglected in the discussion of Table I. When comparing the extrinsic-side-jump contribution estimates to the measured values of αSH, the results vary extensively. In some cases, like Ta, the theoretical extrinsic-side-jump contribution is 24 times larger than the measured αSH. The failure of Eq. (3.22) is not surprising here, since it was derived assuming the weak spin- orbit-coupling regime and Ta is a 5dheavy transition metal. In others, like in Pt, Eq. (3.22) gives a value that is smaller than the measured αSH, in some cases approaching the experi- mental value. However, even in this case, ascribing the measured spin Hall angle to the extrinsic-side-jump contri- bution of Eq. (3.22) is questionable because the independently inferred parameter k2 Fλe−sois close to 1. This is inconsistent with τtr=τsf≪1, i.e., with the weak spin-orbit-coupling regime assumed in the model. In the heavy-element materials, the intrinsic SHE estimates have had much more success. Hence, the simple expression arising from the model [see Eq.(3.22) ], although illustrative and appealing, should only be considered as such, not as a quantitative predictive theory of the SHE. Beyond this simplified model based on the spin-orbit coupling scattering which ignores interband coherent effectsin strongly spin-orbit-coupled metals, an even more simplified model was put forward based a classical treatment (Drude model) of the equations of motions of the Bloch electrons (Chudnovsky, 2007 ,2009 ). For a cubic crystal it predicts a spin Hall angle of hσ xx=2mc2which disagrees withexperimental observations ( Hoffmann, 2013a ). Although the theory has also been questioned ( Kravchenko, 2008 ; Chudnovsky, 2009 ), it is nonetheless popular, particularly in the treatment of the spin-motive force in ferromagnetic systems. The symmetry being built directly into the theory, itwill always give a parameter to which experiment can be fitted, similar to the model presented. D. Theory of the inverse spin Hall effect induced by spin pumping To measure the ISHE, it is necessary to generate a spin current that flows into the NM whose spin Hall angle is beingmeasured. In the nonlocal transport schemes, this is achieved indirectly by spin diffusion into the NM. A key alternative to generating spin currents is to exploit SP in a FM-NM bilayer system. This phenomenon was observed experimentally in early 2000s ( Mizukami, Ando, and Miyazaki, 2001 ,2002 ;Urban, Woltersdorf, and Heinrich, 2001 ). The experiments showed an enhanced Gilbert damping in FMR measurements associated with the loss of angular momentum by a spin current flowing from the FM to the NM.In this setup, the NM served as a spin sink. The associated SP theory based on the scattering formalism was developed by Tserkovnyak, Brataas, and Bauer (2002a , 2002b) andTserkovnyak, Brataas, and Halperin (2005) .I t extends the theory of adiabatic quantum pumping ( Büttiker, Thomas, and Prêtre, 1993 ;Brouwer, 1998 ) by incorporating the spin degrees of freedom. It can be shown that the precessingmagnetization in the FM generates a time-dependent spincurrent at the FM-NM interface that flows into the NM given byTABLE I. Experimental spin Hall angles and effective spin-orbit-coupling parameters, k2 Fλe−so. The values marked by an asterisk are not measured but taken from the literature. The Fermi momenta are taken to be kF¼1.75×108cm−1(Al), 1.21×108cm−1(Au), 1.18× 108cm−1(Nb), and 1.0×108cm−1(Mo, Pd, Ta, Pt). Here, kFl¼ð3π=2Þσ=kFðh=e2Þ. References: (1) Valenzuela and Tinkham (2006 ,2007) ; (2)Seki et al. (2008) ; (3)Mosendz et al. (2010b) ; (4)Niimi et al. (2011) ; (5)Morota et al. (2009) ; (6)Morota et al. (2011) ; (7)Ando and Saitoh (2010) ; (8) Kimura et al. (2007) ; (9) Vila, Kimura, and Otani (2007) ; (10) Ando et al. (2008) ; and (11) Liuet al. (2011) . λsd(nm) kFlk2 Fλe−so αSH(%) jαsj SH=αSHj Refs. (see caption) Al (4.2 K) 455/C615 73 0.0079 0.032/C60.006 0.67 1 NL Al (4.2 K) 705/C630 118 0.0083 0.016/C60.004 0.88 1 NL Au (295 K) 86/C610 371 0.3 11.3 0.014 2 NL Au (295 K) 35/C63/C3253 0.52 0.35/C60.03 1.17 3 SP CuIr (10 K) 5 –30 2.1/C60.6 4N L Mo (10 K) 10 36.8 0.32 −0.20 8.7 5 NL Mo (10 K) 10 8.11 0.07 −0.075 23 5 NL Mo (10 K) 8.6/C61.3 34.1 0.34 −ð0.8/C60.18Þ 2.5 6 NL Mo (295 K) 35/C63/C356.7 0.14 −ð0.05/C60.01Þ 9.9 3 SP Nb (10 K) 5.9/C60.3 11.3 0.14 −ð0.87/C60.20Þ 2.9 6 NL Pd (295 K) 9/C324.0 0.23 1.0 1.9 7 SP Pd (10 K) 13/C62 26.8 0.18 1.2/C60.4 1.1 6 NL Pd (295 K) 15/C64/C348.6 0.28 0.64/C60.10 1.8 3 SP Pt (295 K) 77.9 0.74 0.37 5.1 8 NL Pt (5 K) 14 97.3 0.61 0.44 2.9 9 NL Pt (295 K) 10 67.6 0.58 0.9 1.9 9 NLPt (10 K) 11/C62 98.5 0.77 2.1/C60.5 0.74 5 NL Pt (295 K) 7 /C377.8 0.97 8.0 0.31 10 SP Pt (295 K) 3 –6 60.8 0.88 –1.75 7.6þ8.4 −2.0 0.57 11 SP Pt (295 K) 10/C62/C329.2 0.25 1.3/C60.2 1.31 3 SP Ta (10 K) 2.7/C60.4 3.90 0.17 −ð0.37/C60.11Þ 24 6 NL1228 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015js;pumpσðtÞ¼ℏ 4πArˆm×dˆm dt: ð3:24Þ Here ˆmðtÞis the unit vector of the magnetization, σis the unit vector of the spin-current polarization, js;pump its magnitude, andAris defined as the SP conductance of the particular sample. The spin current generated at the interface whichpropagates into the NM decays on a length scale connected tothe effective spin-diffusion length λ sdof the NM. A sketch of the physics is shown in Fig. 13. Note that in systems with strong spin-orbit coupling, this length scale can be difficult to definesince it can be as short as several atomic layers. Also, proximityeffects as well as roughness at the interface with the NM can blur the sharpness of such an interface. The scattering-matrix theory introduces the concept of a complex spin-mixing conductance at the interface based on spin-conserving channels and no spin losses at the interface.Theoretical ab initio calculations and phase randomization at the scattering interface seem to indicate that only the real part of the mixing conductance dominates the physics. In the diffusive regime, this will be approximately the Sharvinconductance given by the number of conducting channels.In this approximation ( Tserkovnyak, Brataas, and Bauer, 2002b ), A r≈Re½g↑↓/C138¼k2 F 4π≈1 4πð3π2nÞ2=3; ð3:25Þ where kFandnare the Fermi wave vector and electron density in the NM, respectively. Direct ab initio calculations of the mixing conductance ( Xiaet al. , 2002 ;Zwierzycki et al. , 2005 ; Carva and Turek, 2007 ) have verified that, for a FM-NM interface with moderate spin-orbit coupling, the spin-mixingconductance is of this order of magnitude. As the magnetization rotates, the spin current injected from the FM into the NM is time dependent, but the ac spin current when averaged over time has a nonzero dc component, which is given by j s;dc¼ℏω 4πArsin2Θ: ð3:26Þ Here ωis the driving radio frequency (rf) and Θis the cone angle of precession (see Fig. 13). Under the assumption of the NM being a perfect spin sink, the SP conductance will be thespin-mixing conductance. However, in systems where the NM has a finite thickness of the order of the spin-diffusion length, the induced spin accumulation in the NM due to the pumped spin current from the FM will create a spin accumulation,which in turn will generate a spin-current backflow(Tserkovnyak, Brataas, and Bauer, 2002a ;Costache, Sladkov, van der Wal, van Wees, 2006 ;Wang et al. , 2006 ). The spin accumulation in the NM within the spin-diffusiveregime μ s≡μ↑−μ↓is governed by dμs dt¼D∂2zμs−μs τsf; ð3:27Þ with the boundary conditions z¼0∶∂zμs¼−4e2ρ ℏjs;0; z¼tNM∶∂zμs¼0;ð3:28Þ where ρis the NM resistance and tNMis the thickness of the NM. In the NM, the spin current decays away from the FM-NM interface due to the combination of spin-diffusion and spin-flip scattering. The z-dependent spin-current density jsðzÞin the NM ( Mosendz, Pearson et al. , 2010 ; Azevedo et al. , 2011 ) with the above boundary conditions reads jsðzÞ¼−ℏ 4e2ρ∂zμsðzÞ¼js;0sinh ½ðtNM−zÞ=λsd/C138 sinh ðtNM=λsdÞ: ð3:29Þ The backflow current density js;backat the interface can be taken into account with js;backð0Þ≈2Re½g↑↓/C138μsð0Þ. This allows the following expression to be solved for the total spin current crossing the interface: js;0σðtÞ¼ðjs;pump−js;backÞσðtÞ¼ℏ 4π~Arˆm×dˆm dt: ð3:30Þ The result is that the effective spin-mixing conductance gets reduced due to a backflow factor given by ( Tserkovnyak, Brataas, and Bauer, 2002b ) β≡τsfδsd=h tanh ðtNM=λsdÞ; ð3:31Þ where δsdis an effective spin-flip scattering energy obtained by the inverse of the product of the volume defined by thescattering cross section and spin-diffusion length and thedensity of states. This then gives the result ( Tserkovnyak, Brataas, and Bauer, 2002b ) ~A r≈g↑↓1 1þβg↑↓; ð3:32Þ ~Ar≈g↑↓1 1þ1=4ﬃﬃϵ 3ptanh ðtNM=λsdÞ≈g↑↓ eff; ð3:33Þ where g↑↓is now the real part of the spin-mixing conductance. The last approximation assumes a weak spin-orbit-coupling FIG. 13 (color online). Schematic of SP comprising a spin-pump current flowing from the FM to the NM and a backflow currentthat depends on the thickness of the NM.Jairo Sinova et al. : Spin Hall effects 1229 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015limit. More explicitly, it assumes ϵ¼τtr=τsf≪1(see Sec. III.C). Hence, the larger ϵ, the more efficiently the injected spin current is relaxed in the NM and the smaller is the amount of backflow ( Tserkovnyak, Brataas, and Bauer, 2002b ). However, one has to be aware of the limitations of the approximation since in the strongly spin-orbit-coupled systems many of these assumptions fail. The detection of the net spin current flowing into the NM can be done electrically via the ISHE, as was demonstrated by Saitoh et al. (2006) . Earlier experiments by Azevedo et al. (2005) showed similar results in trilayer FM-NM-FM thin films, although they were not identified with the ISHE. By measuring the Hall voltage induced by the spin current, one can infer the spin Hall angle of the material, jc¼αSH2e ℏjs×σðtÞ: ð3:34Þ Here the vector of the spin-current density jspoints perpendicular to the NM-FM interface. Note that the vector of the spin current polarization σðtÞis a time varying quantity. In the geometry sketched in Fig. 13the propagation direction of the spin current is along zand its polarization is along thexaxis. For the detection of a dc voltage along the y direction, one has to consider the charge current jcˆy¼αSH2e ℏð1=tNMÞZtNM 0jsðzÞˆz×ˆx with magnitude ( Azevedo et al. , 2011 ) jc¼αSH2e ℏjs;0λsd tNMtanh/C18tNM 2λsd/C19 : ð3:35Þ To convert this charge current density into the actual measured voltage, one has to take into account the details of themeasurement geometry and the resistivity of the bilayer. This will be discussed in Sec. IV .D. In addition, as described as well in Sec. IV .D, the ac component can be directly measured. An extension of the above theory to incorporate the ac componenthas been done by Jiao and Bauer (2013) , with the result that backflow is important to distinguish between the measured voltages for both the ac and dc configurations. We conclude this section with a discussion regarding the assumptions of the SP theory. In the derivations, whenever ϵ∼1, the approximations do not hold anymore since for the given boundary conditions and for the use of the spin-diffusion equation (and the spin-resolved spin-mixing con- ductance) one assumes ϵ≪1(Tserkovnyak, Brataas, and Bauer, 2002b ). However, for ϵ>0.1, most of the spin scattering occurs right at the interface and consequently thefilms are almost perfect sinks. Hence, in this case there is no dependence on the thickness of the film. Since in such films the interface plays the prominent role and scattering occurs ator near the interface, many issues regarding proximity effects, the induced spin-accumulation, and the spin Hall angle inferred from the measurements should be taken as phenom-enological parameters rather than direct connections to aquantitative value of the bulk spin Hall angles.E. Kubo formalism The Kubo formalism provides a fully quantum-mechanical formally exact expression for the spin and anomalous Hall conductivity in linear response theory ( Mahan, 2000 ). In this section, we review how it is employed in the calculations ofthe intrinsic SHE. The key approximation within the formal-ism is how disorder is treated. For most studies, it is incorporated through a simple finite quasiparticle lifetime, but can also have more sophisticated treatments, such ascoherent potential approximations when treating metal alloys.Here we emphasize the key issues in studying the SHE within this formalism and how it relates to the semiclassical formal- ism described in the previous sections. For the purpose of studying the SHE and AHE, it is best to reformulate the current-current Kubo formula for the conduc- tivity in the form of the Bastin formula [see Appendix A inCrépieux and Bruno (2001) ], which can be manipulated into the more familiar form for the conductivity of the Kubo-Streda formula for the zero-temperature Hall conductivity, σ Hxy¼ σIðaÞ xyþσIðbÞ xyþσIIxy, where σIðaÞ xy¼e2 2πVTrhfˆsz;ˆvgxGRðϵFÞˆvyGAðϵFÞic; ð3:36Þ σIðbÞ xy¼−e2 4πVTrhfˆsz;ˆvgxGRðϵFÞˆvyGRðϵFÞþc:c:ic;ð3:37Þ σIIxy¼e2 4πVZþ∞ −∞dϵfðϵÞTr/C20 fˆsz;ˆvgxGRðϵÞvyGRðϵÞ dϵ −fˆsz;ˆvgxGRðϵÞ dϵvyGRðϵÞþc:c:/C21 : ð3:38Þ Here the subscript cindicates a disorder configuration average. The last contribution σIIxywas originally derived by Streda in the context of the QHE ( Streda, 1982 ). In these equations GR=AðϵFÞ¼ð ϵF−H/C6iδÞ−1are the retarded and advanced Green ’s functions evaluated at the Fermi energy of the total Hamiltonian. Looking more closely at σIIxywe notice that every term depends on products of retarded Green ’s functions only, or on products of advanced Green ’s functions only. It can be shown that only the disorder free part of σIIxyis important in the weak disorder limit, i.e., this contribution is zeroth order in theparameter 1=k Fl. The only effect of disorder on this contri- bution (for metals) is to broaden the Green ’s functions (see below) through the introduction of a finite lifetime ( Sinitsyn et al. , 2007 ). By a similar argument, σIbxyis of order 1=kFland can be neglected in the weak scattering limit ( Mahan, 2000 ). Thus, important disorder effects beyond a simple quasiparticle lifetime broadening are contained only in σIaxy. For these reasons, it is standard within the Kubo formalism to neglectσ Ibxyand evaluate the σIIxycontribution with a simple lifetime broadening approximation to the Green ’s function. In this formalism, the effect of the disorder-configuration averaged Green ’s function is often captured by the use of the T matrix, defined by the integral equation T¼WþWG 0T, where W¼P iV0δðr−riÞandG0are the Green ’s function of the pure lattice. From this, one obtains1230 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015¯G¼G0þG0TG 0¼G0þG0Σ¯G: ð3:39Þ Upon disorder averaging, we obtain Σ¼hWicþhWG 0WicþhWG 0WG 0Wicþ/C1/C1/C1 .ð3:40Þ To linear order in the impurity concentration nithis translates to Σðz;kÞ¼niVk;kþni VX kVk;k0G0ðk0;zÞVk0;kþ/C1/C1/C1 ;ð3:41Þ with Vk;k0¼Vðk−k0Þbeing the Fourier transform of the single impurity potential, which in the case of delta scatterers is simply V0. Note that ¯GandG0are diagonal in momentum but, due to the presence of spin-orbit coupling, nondiagonal inspin index in the Pauli spin basis. One effect of disorder on the spin and anomalous Hall conductivity is taken into account by inserting the disorderaveraged Green ’s function ¯G R=Adirectly into Eqs. (3.36) and (3.38) forσIaxyand σIIxy, respectively. This step captures the effect of disorder on the intrinsic contribution to the SHE and AHE, which is generally weak in metallic systems. The so-called ladder diagram vertex corrections contribute to the AHE and SHE at the same order in 1=kFlas the intrinsic contribution. To capture their effect, we define a ladder-diagram corrected velocity vertex ~v αðϵFÞ≡vαþδ~vαðϵFÞ, where δ~vαðϵFÞ¼niV2 0 VX k¯GRðϵFÞ½vαþδ~vαðϵFÞ/C138¯GAðϵFÞ: ð3:42Þ Note again that ~vαðϵFÞandvα¼∂ˆH0=∂ℏkαare matrices in the spin-orbit-coupled band basis. The skew-scattering con-tributions are obtained by evaluating third order processes inthe disorder scattering, without doing an infinite partial sum as in the case of the ladder diagrams. As may seem obvious from the above machinery, calculat- ing the intrinsic contribution is not very difficult. However, calculating the full effects of the disorder in a systematic way (beyond calculating a few diagrams) is challenging for anydisorder model beyond the simple delta-scattering model. An important recent development has taken place within the theory of the AHE, which we expect will have a direct analogyto the spin Hall conductivity. Assuming uncorrelated Gaussian noise disorder, i.e., ignoring any skew-scattering contribution, it has been shown that all the scattering-independent con-tributions (side-jump and intrinsic) can be formulated in termsof the band structure of the crystal alone ( Kovalev, Sinova, and Tserkovnyak, 2010 ;Weischenberg et al. , 2011 ). Note that spin-orbit coupling is not included in the scattering potentialWin the microscopic theories we discuss in this section and, therefore, the side-jump contribution is given by the intrinsic side jump only. For the short-range scattering disorder model, the starting point to calculate the scattering-independent intrinsic-side- jump contribution is the retarded Green ’s function in equi- librium and the Hamiltonian Hof a general multiband noninteracting system. The first step in the calculation is toexpand the self-energy of the system Σ eqin powers of the potential V0, which describes scattering off impurities. One then inserts the expression for the self-energy into the equations for the current densities (spin or charge) derived following the Kubo-St ředa formalism mentioned earlier. The next step is to rotate into the chiral eigenstate represen-tation (eigenstates with spin-orbit coupling) and keeping only the leading order terms in the limit of vanishing disorder parameter V 0. Having done this, the scattering-independent part of the AHE conductivity may be written as σH−ð0Þ xy ¼ σH−intxy þσH−sj xy, where σH−int ij ¼2e2 ℏZd3k ð2πÞ3ImX n≠mðfn−fmÞvnm;iðkÞvmn;jðkÞ ðωn−ωmÞ2 ð3:43Þ is the intrinsic contribution. In this expression, the band indices nandmrun from 1 to N,vnm;iare the matrix elements of the velocity operator ˆvi¼∂ℏkiˆH, and ωnðkÞ¼εnðkÞ=ℏ. The scattering-independent intrinsic-side-jump contribution to the AHE conductivity for inversion-symmetric systems reads σH−sj ij¼e2 ℏXN n¼1Zd3k ð2πÞ3ReTr/C26 δðεF−εnÞγc ½γc/C138nn ×/C20 SnAkið1−SnÞ∂εn ∂kj−SηAkjð1−SnÞ∂εn ∂ki/C21/C27 :ð3:44Þ Here the imaginary part of the self-energy Im ½Σeq/C138¼−ℏV0γis taken to be in the eigenstate representation, i.e., γc¼U†γU, with γ¼1 2XN n¼1Zd3k ð2πÞ2USnU†δðωF−ωnÞ; ð3:45Þ Uas the unitary matrix that diagonalizes the Hamiltonian at point k, ½U†HðkÞU/C138nm¼εnðkÞδnm: ð3:46Þ Here Snis aN×Nmatrix that is diagonal in the band indices, ½Sn/C138ij¼δijδin, and the so-called Berry connection matrix is given by Ak¼iU†∂kU. Not included in Eq. (3.44) are the TABLE II. AHE conductivities for bcc Fe and hcp Co in S/cm for selected high-symmetry orientations of the magnetization. σH−intxy, σH−sj xy and σH−intþsj xy stand for intrinsic contribution, intrinsic- side-jump contribution, and their sum, respectively. The experimentalvalues are for the scattering-independent conductivity. From Weischenberg et al. , 2011 . Fe [001] [111] [110] Co [001] [100] σ H−intxy 767 842 810 σH−intxy 477 100 σH−sj xy 111 178 141 σH−sj xy 217 −45 σH−intþsj xy 878 1020 951 σH−intþsj xy 694 55 Exp. 1032 Exp. 813 150Jairo Sinova et al. : Spin Hall effects 1231 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015vertex corrections, which vanish for an inversion-symmetric system in the Gaussian disorder model. For an inversion-asymmetric system, the vertex corrected velocity operator would have to be explicitly calculated. Because the intrinsic- side-jump contribution in the short-range disorder model issolely determined by the electronic structure of the pristinecrystal, it is thus directly accessible by ab initio methods. Table IIshows a comparison of the improvement in predictive power of the AHE theory when including theintrinsic-side-jump term. Figure 14shows the nontrivial angular dependence, within the Fermi surface, of the intrinsic-side-jump contribution and the intrinsic contribution. This is reminiscent of the spin-hot spots observed previouslyin the theory of spin dephasing, and emphasizes the impor-tance of anisotropies induced by the band structure itself. Finally we note that a study that incorporates the intrinsic contribution as well as the skew-scattering contribution withinab initio calculations has shown a good semiquantitative agreement within certain simple alloys ( Lowitzer et al. , 2011 ). IV. EXPERIMENTAL STUDIES OF SPIN HALL EFFECT Several experimental schemes to detect the SHE were outlined by Dyakonov and Perel (1971b) in their seminal theory work. They proposed to use paramagnetic resonance for detecting the edge spin polarization, to measure the nuclear magnetization resulting from the Overhauser effect, to exploitthe gyrotropy, i.e., the difference in the propagation ofelectromagnetic waves with opposite helicities through thespin-polarized edges, or, in semiconductors, to detect circular polarization of the luminescence excited by an unpolar- ized light. Variants of the two latter schemes, namely, the Kerr magneto-optical microscopy and circularly polarized electro- luminescence from the sample edges, were indeed employed in the pioneering SHE experiments ( Kato et al. , 2004a ; Wunderlich et al. , 2004 ,2005 ). These were, however, per- formed more than 30 years after the original proposal byDyakonov and Perel (1971b) . Within these three decades, the interest in the phenomenon was scarce. The experimental SHE research only picked up momentum after the theoretical workbyHirsch (1999) who rediscovered the phenomenology of the extrinsic SHE, and after the prediction of the intrinsicSHE ( Murakami, Nagaosa, and Zhang, 2003 ;Sinova et al. , 2004 ). The renewed theoretical interest occurred in the midst of an extraordinary growth of the nascent field of spintronics ( Zutic, Fabian, and Sarma, 2004 ), which had already found important applications in the hard-disk-drive industry and promisedrevolutionary concepts for memory and logic devices. In thissetting, the theoretical SHE proposals not only ignited anextensive theoretical debate for their inherent fundamental interest but also attracted significant attention due to the potential of spin Hall phenomena as new spin injection anddetection tools. The proposals started to materialize shortlythereafter with the observations of the SHE in n-doped semiconductors ( Kato et al. , 2004a ) and in the 2DHG (Wunderlich et al. , 2004 ,2005 ), and of the ISHE in metallic systems ( Saitoh et al. , 2006 ;Valenzuela and Tinkham, 2006 ,2007 ). In this section, we review the experimental studies of the spin Hall phenomena. In Sec. IV.A, we summarize AHE experiments in nonferromagnetic materials that were per-formed within the three decades separating the first theoreticalproposal and the experimental observations of the SHE. Therest of the section is devoted to modern experiments divided according to the techniques used to generate, detect, and manipulate the SHE and ISHE in experimental samples(Secs. IV.B–IV.D). The overall understanding of the experi- ments is still incomplete regarding some materials andstructures, in particular, when trying to quantify the magnitude of the SHE. Therefore, in those cases, we attempt to provide an overview of the current status of the field while stressingthe strengths and weaknesses of the different techniques andmethods employed. Apart from the basic research interest in this relativistic quantum-mechanical phenomenon, Sec. IV.D provides an illustration of the application potential of the SHE in spin-tronic devices. This prompted detailed studies of the SHE efficiency for the charge-spin conversion in a variety of materials. Measurements of the corresponding spin Hallangles are summarized in Sec. IV.E. A. Early experiments of anomalous Hall effect in paramagnets Chazalviel and Solomon (1972) reported a pioneering work on a spin-dependent Hall effect in nonmagnetic semiconduc-tors. They detected the AHE in InSb and n-doped Ge at low temperatures ( <25K), where the spin polarization was FIG. 14 (color online). Angular distribution of the (a) intrinsic- side-jump contribution for Ni [001], (b) intrinsic-side-jumpcontribution for Ni [110], (c) intrinsic contribution for Ni[001], and (d) intrinsic-side-jump contribution for Fe [001] ona sphere in the Brillouin zone. The dark regions correspond tolarge contributions. The color code of each surface point corresponds to the sum of all contributions along the path from the origin to the particular surface point. From Weischenberg et al. , 2011 .1232 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015created by the application of a magnetic field and the spin-dependent Hall effect was separated from the largerordinary HE by magnetic resonance of the conductionelectrons ( Chazalviel and Solomon, 1972 ;Chazalviel, 1975 ). The magnitude of the measured anomalous Hall angles was of the order of 10 −4for InSb, and of 10−5for Ge, while its sign was observed to change depending on the degree ofcarrier compensation (InSb) and temperature (Ge). Thechange in sign was associated with competing contributionsfrom the side-jump and skew-scattering mechanisms. The former was expected to be favored in low mobility samples, which was confirmed in the experiment. In the early 1980s, Fert and collaborators studied diluted magnetic alloys based on nonmagnetic hosts, such as Au andCu, and magnetic impurities such as Mn, Fe, or Cr ( Fert, Friederich, and Hamzic, 1981 ). They found that CuMn showed negligible skew-scattering effects, but that theexchange scattering by polarized Mn impurities created aspin-polarized current. They also noted that the addition ofnonmagnetic impurities to CuMn gave rise to skew scatteringof the polarized current by the unpolarized impurities. By analyzing variations of the Hall coefficient, they were able to extract the Hall angle for the nonmagnetic impurities.They found that they varied from −1.4×10 −2for Lu to −2.6×10−2for Ir. In another type of AHE measurement, a circularly polarized beam at the normal incidence to the surface of a bulksemiconductor was used to excite spin-polarized photoelec-trons ( Bakun et al. , 1984 ;Miah, 2007 ). These electrons diffused in the vertical direction from the surface and afteraligning their spins along an axis parallel to the surface by an applied magnetic field (via Hanle precession), an electrical voltage was detected in the transverse in-plane direction(Bakun et al. , 1984 ). Alternatively, the vertically spin- polarized electrons can be accelerated in the in-planedirection by an applied electrical bias yielding also a trans- verse in-plane voltage ( Miah, 2007 ). Since in these experi- ments the source spin current is accompanied by a diffusive ordrift charge current, the geometry corresponds to the AHE(see Sec. III.B). B. Optical tools in spin Hall experiments 1. Optical detection of the spin Hall effect The experimental discovery of the SHE was prompted by the intrinsic SHE proposals ( Murakami, Nagaosa, and Zhang, 2003 ;Sinova et al. , 2004 ) which focused on semiconductors and suggested to utilize the optical activity of these materialsfor detecting the SHE. Similar to the original work byDyakonov and Perel (1971b) ,Murakami, Nagaosa, and Zhang (2003) , and Sinova et al. (2004) proposed a circularly polarized electroluminescence or a spatially resolved magneto-optical Faraday and Kerr effects. These methodswere indeed used in the first measurements of the phenome-non. Kato et al. (2004a) employed a magneto-optical Kerr microscope to scan the spin polarization across the channel while Wunderlich et al. (2004 ,2005) used coplanar p-n diodes to detect circularly polarized electroluminescence atopposite edges of the spin Hall channel. Wunderlich et al.(2004 ,2005) ascribed the observed signal to the intrinsic SHE while Kato et al. (2004a) to the extrinsic SHE. Kato et al. (2004a) performed the experiments in n-GaAs andn-In 0.07Ga0.93As films grown by molecular-beam epitaxy on (001) semi-insulating GaAs substrates. The films were doped with Si with n¼3×1016cm−3in order to obtain long spin relaxation lifetimes of τs∼10ns, which result in spin-diffusion lengths λsd¼ﬃﬃﬃﬃﬃﬃﬃﬃDτsp∼10μm. The unstrained GaAs sample consisted of 2μmo f n-GaAs grown on 2μmo fu n d o p e d Al0.4Ga0.6As, whereas the strained InGaAs sample had 0.5μm ofn-In0.07Ga0.93As and 0.1μm of undoped GaAs. Static Kerr rotation measurements were performed at 30 K with a pulsed Ti: sapphire laser tuned to the absorption edge of the semiconductor with normal incidence to the sample. In this technique, the laserbeam is linearly polarized and the polarization axis of thereflected beam is determined. The rotation angle is proportional to the net magnetization along the beam direction. Figure 15(a) shows a schematic of the experimental geometry. The epilayers were patterned into 300×77μm 2 (GaAs) and n-InGaAs 300×33μm2(InGaAs) channels. An electric field was applied along the channel while a magneticfieldBcould be applied perpendicular to it in the film plane. Figure 15(b) shows a two-dimensional scan of the GaAs sample, which demonstrates the existence of spin accumu-lation close to the edges. The amplitude of the measured edgespin polarizations reaches ∼0.1%. The polarization has opposite sign at the two edges and decreases rapidly with the distance from the edge as expected for the SHE. This isclearly seen in the one-dimensional profile in Fig. 15(c) . Further experiments demonstrated the effect of spin (Hanle) FIG. 15 (color online). Observation of the SHE by the magneto- optical Kerr microscope. (a) Schematics of the GaAs sample.(b) Two-dimensional images of the spin density n s(left) and reflectivity R(right) for an unstrained GaAs sample measured at temperature 30 K and applied driving electric fieldE¼10mVμm −1. (c) Kerr rotation as a function of xand external magnetic field BextforE¼10mVμm−1. (d) Spatial dependence of the peak Kerr rotation A0across the GaAs channel. From Kato et al. , 2004a .Jairo Sinova et al. : Spin Hall effects 1233 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015precession, and associated suppression of the observed signal to the applied magnetic field, as predicted by Dyakonov and Perel (1971a) andHirsch (1999) . Zhang (2000) showed, by solving the spin-dependent drift-diffusion equations for a finite width channel, that the spin-diffusion length λsddefines the length scale of the edge spin accumulation. By fitting to the spin-drift-diffusion equation, Kato et al. (2004a) extracted the transverse spin current and the spin Hall resistivity ρH. This analysis, which assumes well-resolved spin-up and spin-down transport channels ( Hirsch, 1999 ;Zhang, 2000 ), is valid in the weak spin-orbit limit, which is verified by noting that Δsoτ=ℏ∼ 10−3≪1, where Δsois the spin-orbit coupling energy and τis the momentum scattering time. The measured value of ρH∼ 2Ωm is consistent with that obtained from modeling based on scattering by screened and short-range impurities ( Engel, Rasbha, and Halperin, 2006 ;Tse and Das Sarma, 2006 ). Noting that the charge resistivity ρ∼4×10−6Ωm, this corresponds to a spin Hall angle of ∼10−4. In the weak spin-orbit coupling regime, the spin-orbit splitting of the quasiparticle bands is smeared out by disorder which favors the extrinsic SHE interpretation of the measured signal. Theabsence of the intrinsic SHE was confirmed by measurements in the strained InGaAs sample which showed no dependence of the SHE signal on the strain induced anisotropies of the spin-orbit-coupled band structure. Experiments in 2DHG devices (Fig. 16) were carried out in the strong spin-orbit coupling limit, Δ soτ=ℏ∼4, which favors the intrinsic mechanism ( Wunderlich et al. , 2004 ,2005 ; Nomura et al. , 2005 ). The device comprised coplanar p-njunction light emitting diodes (LEDs) that were patterned inðAl;GaÞAs=GaAs heterostructures grown by molecular- beam epitaxy and using modulation donor (Si) and acceptor (Be) doping in (Al,Ga)As barrier materials. The heterostruc- ture consisted of an n-doped AlGaAs/GaAs heterojunction, followed by the growth of 90 nm of intrinsic GaAs and a p-doped AlGaAs/GaAs heterojunction. The coplanar p-n junctions were created by removing the p-doped layer of the wafer and thus creating a hole channel, with a carrier density 2×1012cm−2. The 2DEG at the bottom heterojunc- tion was almost depleted. The removal of p-doped surface layer populated the 2DEG, forming the n-side of the coplanar p-njunction. A current Ipwas applied to drive the electroluminescence at the edge of the channel due to recombination near p-n junctions. The detection of spin polarization in the 2DHG was done by measuring the circular polarization of the emitted light, shown in Figs. 16(d) and 16(e) . The magnitude of the signal reached ∼1% at 4 K. Consistent with the SHE phenomenology, the experiments demonstrated that the spin accumulation was opposite at opposite sides of the channel and that it reversed sign following current reversal. Calculations of the SHE conductivity showed that the SHE originating from the spin-orbit-coupled quasiparticle bands of the 2DHG is only weakly affected by disorder for the parameters of the studied system ( Wunderlich et al. , 2005 ). A quantitative microscopic description of the measured edge spin-accumulation signal was developed and further exper-imentally tested by Nomura et al. (2005) . The theory analysis pointed out that the length scale of the edge spin accumulationis defined in the strong spin-orbit coupling regime by the spin- orbit precession length L so¼vFτso, where τso¼ℏπ=Δsois the precession time of the spin in the internal spin-orbit fieldandv Fis the Fermi velocity. With increasing strength of the spin-orbit coupling, the edge spin-accumulation region nar- rows down and, simultaneously, the amplitude of the spinpolarization increases. For the experimental parameters of thestudied 2DHG, L so∼10nm and the calculated amplitude of the edge spin polarization was 8%, in good agreement with the1% polarization of the measured electroluminescence signal which was averaged over a ∼100nm sensitivity range of the coplanar light emitting diode. A comparison between mea-surements in devices with 1.5 and 10μm wide channels confirmed the expectation that the SHE signal is independentof the channel width. Subsequent magneto-optical measurements of the SHE in then-GaAs 3D epilayers have experimentally demonstrated that the SHE-induced spin accumulation is due to a transverse spin current which can drive spin polarization tens of microns into a region in which there is minimal electric field ( Sihet al. , 2006 ). The work proved experimentally that the SHE can be used as a source of spin current generated in a NM. FIG. 16 (color online). Observation of the SHE by the circularly polarized electroluminescence of coplanar p-ndiodes. (a) Sche- matic configuration of the lateral p-njunction to detect spin accumulation. (b) Light emission from the p-njunction recorded by a charged-couple device camera. (c) Electron microscopeimage of the microdevice with symmetrically placed p-ndiodes at both edges of the 2DHG channel. (d),(e) Emitted lightpolarization of recombined light in the p-njunction for the current flow indicated in (c) at 4 K. From Nomura et al. , 2005 .1234 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015A systematic doping dependence of the SHE angle was studied in n-GaAs 3D epilayers with electron densities n¼1.8×1016–3.3×1017cm−3and the results were found consistent with theory predictions for the extrinsic SHE(Matsuzaka, Ohno, and Ohno, 2009 ). The measured SHE angles of ∼5×10 −4–5×10−3increase with increasing dop- ing with a tendency to saturate at the high doping end of thestudied set of samples at a value corresponding to ∼1%edge spin polarization. It was concluded from this systematicanalysis that the spin accumulation is reduced by an enhancedspin relaxation due to the Dyakonov-Perel mechanism, whilethe spin current induced by the SHE is enhanced withincreasing n(Matsuzaka, Ohno, and Ohno, 2009 ). The SHE was observed also in other semiconductor systemsincluding n-ZnSe 3D epilayers ( Stern et al. , 2006 ) and InGaN/GaN superlattices ( Chang et al. , 2007 ). 2. Optical generation of the inverse spin Hall effect A traditional way of generating spin-polarized photocar- riers in semiconductors is by absorption of circularly polarizedlight ( Meier and Zakharchenya, 1984 ). Because of the optical selection rules, the out-of-plane spin polarization of photo- carriers is determined in this technique by the sense and degree of the circular polarization of vertically incident light.This technique was used to observe the AHE in semicon-ductors ( Bakun et al. , 1984 ;Miah, 2007 ), which was discussed in Sec. IV.A, and eventually led also to the detection of the ISHE generated by the pure spin current. Ando et al. (2010) reported an experiment in a NM- semiconductor hybrid structure in which they demonstratedthe conversion of circularly polarized light absorbed in asemiconductor to an electrical signal in the attached NM ISHEsensor (Fig. 17). The photoinduced ISHE was observed in a Pt/GaAs hybrid structure. In the GaAs layer, circularlypolarized light generates spin-polarized carriers, inducing apure spin current into the Pt layer through the interface. Thispure spin current is converted into an electrical voltage due tothe ISHE in Pt. Systematic changes of the ISHE signal wereobserved upon changing the direction and ellipticity ofthe circularly polarized light, consistent with the expectedphenomenology of the photoinduced ISHE. The observedphenomenon allows the direct conversion of circular-polarization information into the electrical voltage and canbe used as a spin photodetector. Using a similar detector configuration, Kampfrath et al. (2013) demonstrated the control of the transmission of terahertz spin current pulses. The samples consisted ofFe/Au and Fe/Ru heterostructures. The absorption of afemtosecond laser pulse in the Fe layer generates a non-equilibrium electron distribution and associated spin current,dominated by the majority-spin sp-like electrons, that flow into the Au(Ru) nonmagnetic layer. The transport dynamics isdifferent in the Fe/Au and Fe/Ru heterostructures because ofthe much larger electron mobility of Au; the flow of thenonequilibrium electrons occur much more slowly in Ru thanin Au, and are accompanied by significantly more spinaccumulation. The nonmagnetic layer can thus be used toeither trap or transmit electrons, and thus engineer ultrafastspin pulses, which change in temporal shape and delay.The detection of the spin-current pulses used by Kampfrath et al. (2013) relied on the ISHE. While in the static experiments by Ando et al. (2010) the resulting charge current is measured as a voltage, Kampfrath et al. (2013) detected the electromagnetic pulse emitted by the charge current burst by electro-optical sampling using a GaPcrystal. The feasibility of the experiment demonstrated theoperation of the ISHE as a spin-current detector up to frequencies as high as 20 THz. Wunderlich et al. (2009 ,2010) , using the same type of lateral p-ndiodes as in Nomura et al. (2005) andWunderlich et al. (2005) , exploited optical spin injection by a circularly polarized laser beam to observe the ISHE and to fabricate experimental optospintronic and spin-transistor devices. In theSHE measurements in Nomura et al. (2005) andWunderlich et al. (2005) , the p-njunctions were fabricated along the edges of the 2DHG channel and under forward bias could sense the spin state of recombining electrons and holesthrough polarized electroluminescence. In Wunderlich et al. (2009 ,2010) , on the other hand, the spin Hall channel was fabricated in the etched part of the epilayer with the 2DEG, the channel was oriented perpendicular to the p-njunction, and (a) (c)(b) FIG. 17 (color online). (a) Schematic illustration of the band structure of GaAs and spin-polarized electrons generated by theabsorption of circularly polarized light. (b) Schematic illustrationof the ISHE induced by photoexcited pure spin currents in thePt/GaAs system. (c) The illumination angle θdependence of V R−VLmeasured for the Pt/GaAs hybrid structure. θis the in- plane angle between the incident direction of the illumination andthe direction across the electrodes attached to the edges of the Ptlayer as shown in the inset. V R−VLis the difference in the electromotive force for illumination with right and left circularlypolarized light. The filled circles are the experimental data. Thesolid curve shows a fitting result using a function proportional tosinθ. From Ref Ando et al. , 2010 .Jairo Sinova et al. : Spin Hall effects 1235 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015the diode was under zero or reverse bias, operating as a photocell as shown in Fig. 18. The optical activity of the lateral diode confined to a submicron depletion region, combined with a focused ( ∼1μm) laser beam, allowed for a well localized injection of spin-polarized photoelectrons intothe planar 2DEG channel. The Hall signals were detected electrically on multiple Hall crosses patterned along the channel. Two regimes of operation of the device are distinguished: One corresponds to an AHE regime, in which the reverse-bias charge current is drainedbehind the Hall crosses at the opposite end of the channel fromthep-njunction injection point [Fig. 18(a) ]. The other regime corresponds to the ISHE measurement since in this case the charge current is drained before the Hall crosses, allowingonly the pure spin current to diffuse further in the channel[Fig. 18(b) ]. In both cases, the measured transverse electrical signals were consistent with the phenomenology of the spin- dependent Hall phenomena ( Wunderlich et al. , 2009 ,2010 ). The sign of the voltage was opposite for opposite helicities ofthe incident light, i.e., opposite spin polarizations of injected photoelectrons. Moreover, the amplitude of the electrical signals was found to depend linearly on the degree of circularpolarization of the light, rendering the device an electrical polarimeter ( Wunderlich et al. , 2009 ). The electrical signals were observable over a wide temperature range with spin Hall angles of 10 −3–10−2. The measured 2DEG was in the weak spin-orbit coupling regime, Δsoτ=ℏ∼10−1, and the measured data were consistent with the extrinsic mechanism(Wunderlich et al. , 2009 ). 3. All-optical generation and detection The SHE and ISHE were also observed using two-color optical coherence control techniques in intrinsic GaAs at 80 Kwith polarized 70 fs, 715 and 1430 nm pulses ( Zhao et al. , 2006 ). When the pulses were orthogonally polarized, a pure spin source current was generated that yielded a transverse Hall pure charge current via the ISHE. When the pulses wereparallel polarized, a pure charge source current was generatedthat yielded a pure spin current via the SHE. By varying therelative phase or polarization of the incident pulses, the type,magnitude, and direction of both the source and transverse currents were tuned without applying electric or magnetic fields. In contrast to the previous steady-state experiments,where drift currents are generated by electric fields, theinjected currents are ballistic with electrons traveling initiallyat∼1000 km=s. The generation of spin and charge currents results from the quantum interference between absorption pathways for one-and two-photon absorption connecting the same initial andfinal states as illustrated in Fig. 19(a) . For a spin current, a coherent pulse centered at frequency ωwith phase φ ωis normally incident along ˆzand linearly polarized along the ˆx direction which can be arbitrary with respect to crystal axessince the effects are not strongly sensitive to crystal orienta-tion. A copropagating 2ωpulse with phase φ 2ωis linearly polarized along the orthogonal ˆydirection. Excited spin-up(a) (b) (c) FIG. 18 (color online). ISHE based transistor. (a) Schematics of the spin-injection Hall effect measurement setup with opticallyinjected spin-polarized electrical current propagating through theHall bar and corresponding experimental Hall effect signals atcrosses H1 and H2. The Hall resistances R H¼VH=IPHfor the two opposite helicities of the incident light are plotted as afunction of the focused light spot position, i.e., of the position ofthe injection point. The optical current I PHis independent of the helicity of the incident light and varies only weakly with the lightspot position. (b) Same as (a) for the ISHE measurementgeometry in which electrical current is closed before the firstdetecting Hall cross H1. (c) Schematics of the setup of the spinHall transistor and experimental Hall signals as a function of thegate voltage at a Hall cross placed behind the gate electrodefor two light spot positions with a relative shift of 1μm. From Wunderlich et al. , 2010 . 2 z [100]^x^ y^(a) FIG. 19 (color online). Observation of the ISHE using the two- color optical pump-and-probe technique. (a) Illustration oforthogonally polarized ωand2ωpulses producing a pure spin current (double headed straight arrow) along the ωbeam polarization direction ( ˆx). The charge current due to the ISHE (curved arrows) along ˆyleads to electron accumulation near one edge of the illuminated region. (b) Measured charge accumu-lation due to the ISHE. From Zhao et al. , 2006 .1236 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015electrons are polarized along ˆzand move preferentially in one direction along ˆx, while spin-down electrons move in the opposite direction. Together they generate a spin currentproportional to cos ðΔφÞwhere Δφ¼2φ ω−φ2ω. The spin current is dominated by electrons as holes lose their spin in<100fs. Because of the ISHE, a charge current is generated [Fig. 19(a) ] that has the same cosine dependence as the spin- current source. Consistent with the ISHE phenomenology, theexcess charge on one side and the deficit on the other side ofthe sample, shown in Fig. 19(b) , was observed along the direction perpendicular to the driving spin current. The ballistic nature of transport in these experiments was fully exploited in fs time-resolved measurements ( Werake, Ruzicka, and Zhao, 2011 ). They allowed one to infer the momentum scattering time τ≈0.45ps and with a much shorter time delay of the probe pulses to observe in real timethe transverse charge current. The measurements showed thatthe charge current was generated well before the firstscattering event, providing a direct demonstration of theintrinsic ISHE. 4. Electrical manipulation A distinct feature of the ISHE experiments in the 2DEG is the observed spin precession due to internal Rashba andDresselhaus spin-orbit fields ( Wunderlich et al. , 2009 ,2010 ). Since the spin-diffusion length scales approximately as∼L 2so=w(Wunderlich et al. , 2010 ), it was possible to observe a few spin precessions in channels of a width w¼1μm for Lso∼1μm of the studied 2DEG. The corresponding oscil- lations of the spin Hall voltages were consistently observed bymeasuring at different Hall crosses along the channel or byshifting the laser spot, i.e., the spin-injection point (Fig. 18). The lateral ISHE channels also allow one to place top gateelectrodes in between the Hall crosses as shown in Fig. 18(c) . (The gates are formed by unetched regions of the wafer.) Thestrength of the Rashba and Dresselhaus spin-orbit fields and,therefore, also the spin precession can be manipulated electri-cally in the device shown in Fig. 18(c) . To demonstrate an AND logic functionality, two gates were fabricated on top of the channel and the Hall electrical signal was measured at a crossplaced behind both gates. Intermediate gate voltages on bothgates represented the input value 1 and gave the largestelectrical ISHE signal, representing the output value 1. Whena large reverse gate voltage was applied to any of the twogates, representing input 0, the electrical ISHE signal dis-appeared, i.e., the output was 0. A different approach to achieve the control of spin currents is by directly modifying the spin-orbit coupling strength on agiven material, which in turn determines the spin Hall angle.The electronic band structure and impurity states are weaklydependent on an external electric field and therefore cannot beused to change the spin-orbit strength. However, Okamoto et al. (2014) noted that the electric field can induce a carrier redistribution within a band or multiple bands. Therefore, ifthe electrons generating the SHE can be controlled bypopulating different areas (valleys) of the electronic structure,the spin-orbit interaction (and the spin Hall angle) can betuned directly within a single sample. Okamoto et al. (2014) reported such a tuning in bulk GaAs at room temperature bymeans of an electrical intervalley transition induced in the conduction band. The spin Hall angle was determined by measuring an electromotive force driven by photoexcited spin- polarized electrons drifting through n-GaAs Hall bars. By controlling electron populations in the ΓandLvalleys with an applied electric field (part of the pcharacter in the Lvalley provides a larger effective spin-orbit interaction), the spin Hall angle was changed by a factor of 40, from 0.0005 to 0.02 formoderate electric fields beyond 100kV m −1. Thus the highest spin Hall angle achieved is comparable to that of Pt. C. Transport experiments Hirsch (1999) andZhang (2000) discussed specific con- cepts for the experimental detection of the SHE and ISHEusing dc transport techniques. Hirsch (1999) proposed a device that consists of a metallic slab in which spin accu- mulation is generated by an electrical current via the SHE, asdescribed in Sec. II(see Fig. 2). A transverse strip connects the edges of the slab, allowing the spin current to flow through it. Because of the ISHE, a voltage is generated that can be measured with a voltmeter. In an alternative approach, Zhang (2000) proposed to detect the spin accumulation electrically using a FM probe. The concept borrows from techniques for spin injection and detection in NM implemented in nonlocal spin devices ( Silsbee, 1980 ;Johnson and Silsbee, 1985 ). Shortly after the optical SHE detection in semiconductors (Kato et al. , 2004a ;Wunderlich et al. , 2004 ,2005 ), Valenzuela and Tinkham (2006) reported an observation of the voltage generated by the ISHE. Instead of generating thespin current by the SHE, which would render a second-order voltage in the spin Hall angle, they used electrical spin injection from a FM in combination with a Hall crosspatterned in the ISHE paramagnet. Independently, Saitoh et al. (2006) observed the voltage generated by the ISHE in a setup where the spin injection from the FM to the NM was achieved using the SP techniques. Kimura et al. (2007) combined the concept of the spin Hall cross and the proposalbyZhang (2000) to detect both the SHE and ISHE in the same device. It took a few more years to demonstrate the idea of Hirsch of simultaneously exploiting both the SHE and theISHE in an electrical device. It was first attempted by Mihajlovic et al. (2009) using a gold H-bar device (Hankiewicz et al. , 2004 ), but they did not succeed in observing a spin Hall related signal. Eventually, this wasachieved by Brüne et al. (2010) , who performed the experi- ment in a ballistic semiconductor H-bar device. The transport SHE and ISHE experiments are described in detail inSecs. IV.C.1 –IV.C.5 . More recently, the SHE was also detected via the manipu- lation of magnetization in FMs ( Liu et al. , 2011 ;Miron, Garello et al. , 2011 ). Spin currents generated by the SHE were shown to be sufficiently large to induce magnetization dynamics, drive domain walls, or switch magnetization in the FM, demonstrating the potential of the SHE for applica-tions ( Miron, Garello et al. , 2011 ;Miron et al. , 2011 ;Liu, Pai, Li et al. , 2012 ;Emori et al. , 2013 ;Ryu et al. , 2013 ). These SHE experiments together with the ISHE measurements via SP are discussed in Sec. IV.D.Jairo Sinova et al. : Spin Hall effects 1237 Rev. Mod. Phys., Vol. 87, No. 4, October –December 20151. Concepts of nonlocal spin transport: Electrical injection and detection Johnson and Silsbee (1985) reported the injection and detection of nonequilibrium spins using a device that con-sisted of a NM (N), with two attached FM electrodes (F1, F2), illusrated in Fig. 20. In this device, spin-polarized electrons are injected from F1 into N by applying a current Ifrom F1 that results in spin accumulation in N. The population of, say,spin-up electrons in N increases by shifting the electrochemi- cal potential by δμ N, while the population of spin-down electrons decreases by a similar shift of −δμN. Overall, this corresponds to a spin-accumulation splitting of 2δμN. The spin accumulation diffuses away from the injection point and reaches the F2 detector, which measures its local magnitude. As first suggested by Silsbee (1980) , the spin accumu- lation in N can be probed by the voltage VNL, which is induced at F2. Silsbee noted that the polarization density inN, or equivalently the nonequilibrium magnetization, acts as the source of the spin electromotive force that produces V NL. The magnitude of VNLis associated with δμN, while its sign is determined by the relative magnetization ori-entation of F1 and F2. Because the current is drained to the left of N, there is no charge current toward the right, where the detector F2 lies[Fig. 20(a) ]. For this reason, the spin detection is said to be implemented nonlocally, where no charge current circulates by the detection point, and thus V NLis sensitive to the spin degree of freedom only. Accordingly, nonlocal measurements elimi-nate the presence of spurious effects associated to charge transport, such as anisotropic magnetoresistance (AMR) or the ordinary HE that could mask subtle signals related to spininjection. Typically, nonlocal devices exhibit a small output background allowing sensitive spin-detection experiments. This approach has been widely used in recent years tocharacterize the spin transport in metals, semimetals, semi-conductors, superconductors, carbon nanotubes, and graphene. It has also been used to study the spin-transfer properties of FM-NM material interfaces.2. Nonlocal detection of inverse spin Hall effect with lateral spin current Valenzuela and Tinkham (2006 ,2007) adapted the nonlocal detection techniques to study the ISHE. Their device is schematically shown in Fig. 21(a) . By using a FM electrode, a spin-polarized current is injected in a NM strip (N). It propagates to both sides away from the injection point and decays within the spin-diffusion length λsd. A laterally induced voltage VSH, which results from the conversion of the injected spin current into charge imbalance owing to the ISHE, is then measured using a Hall-cross structure. The magnitude of VSHis proportional to the anomalous Hall operator σSHσ×Es, where σSHdenotes the spin Hall conductivity, σis the direction of the spin polarization injected from the FM electrode, and Esis an effective spin-dependent “electric ”field, which follows from the spin-dependent electrochemical potential μsalong the NM strip, i.e., EsðrÞ¼−∇μsðrÞ. In the device of Fig. 21(b) , the injector FM electrode (F1) is made of CoFe, while the NM strip material is Al of thickness tAl. The entire structure is fabricated without breaking vacuum using electron-beam evaporation and shadow evaporation techniques. An Al 2O3tunnel barrier is used for spin-current injection. The purpose of the barrier is twofold. First, it enhances the polarization of the injected electrons and, second, it assures a uniform current injection. The latter isessential because it suppresses the flow of charge current toward the Hall cross, preserving the nonlocal character of the measurements and eliminating the previously mentioned spurious effects. The FM electrode is magnetized in-plane at zero magnetic field due to shape anisotropy and thus an out-of-plane magnetic field B ⊥is used to generate a perpendicularly polarized spin current at the Hall cross [Fig. 21(b) ]. Spin imbalance in the Al film occurs with a defined spin direction given by the magnetization orientation of the F1 electrode. Consequently, VSHis expected to vary when B⊥is applied and the magnetization Mof the electrode is tilted out of the substrate plane. Defining θas the angle between Mand the electrode axis, it follows from the cross product in theanomalous Hall operator that V SHis proportional to sin θ, correlating with the component of Mnormal to the substrate [Fig. 21(b) ]. FIG. 20 (color online). Nonlocal spin detection and spin accu- mulation. (a) Schematic illustrations of the device layout. Aninjected current Ion the source (F1) generates spin accumulation in the NM (N) which is quantified by the detector (F2) voltageV NL. (b) Schematic representation of the spin splitting in the electrochemical potential induced by spin injection. The splittingdecays over characteristic lengths λ sdover the N side. (c) Detector behavior for an idealized Stoner FM with a full spin subband forthe parallel magnetization orientation (top) and for the antipar-allel magnetization orientation (bottom). FIG. 21 (color online). (a) Spin-current-induced Hall effect or inverse spin Hall effect. Schematic representation of an actualdevice where the pure spin current is generated by spin injectionthrough a FM (F) with out-of-plane magnetization. (a) Devicefabricated with CoFe electrodes (light gray) and an Al channel(dark gray). Adapted from Valenzuela and Tinkham, 2006 .1238 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015The device layout in Fig. 21(b) is more sophisticated than the schematics in Fig. 21(a) , where only F1 is required. The second FM electrode (F2) together with the injection FM electrode (F1) and the NM strip form a spin injection anddetection device [Fig. 22(a) ] for calibration purposes. Calibration procedures are necessary to demonstrate consis- tency with standard nonlocal methods. Explicitly, this device can be utilized to measure the spin accumulation in the NMand then determine its associated spin-diffusion length λ sd, the amplitude of the spin polarization of the injected electrons P, and the magnetization orientation of the FM electrodes θin the presence of an external magnetic field (perpendicular tothe substrate). For this purpose, batches of samples are commonly used where the distance between the two FMs L Fis modified and the spin-precession signal acquired [Fig. 22(b) ]. The distance of F1 relative to the Hall cross LSHis also modified in order to test the consistency of the spin-diffusion results. Subsequent measurements [Fig. 22(d) ] in the configuration of Fig. 22(c) , performed in Al of different tAl, and thus different λsd, yielded σSH∼20−40Ωcm−1and αSH∼ð1–3Þ×10−4, which compares well with theoretical estimates based on extrinsic mechanisms ( Shchelushkin and Brataas, 2005 ). Olejník et al. (2012) used the same geometry to detect the ISHE in n-GaAs using epitaxial ultrathin Fe/GaAs injection contacts with strong in-plane magnetic anisotropy. Hybridsemiconductor –metal FM structures suffered for a long time from the resistance mismatch problem ( Schmidt et al. , 2000 ). Since the spin transport relies on different conductivities for spin-up and spin-down electrons and is governed by the leastconductive part of the device, the effects are weak in devices inwhich the nonmagnetic semiconductor with equal spin-up and spin-down conductivities dominates the resistance of thedevice ( Rasbha, 2000 ). The introduction of a highly resistive tunnel barrier between the FM metal electrode and the semiconductor channel solved the problem ( Rasbha, 2000 ; Lou et al. , 2007 ). The device of Olejník et al. (2012) , shown in Fig. 23(a) , comprised the n-GaAs channel, a Hall cross, and two ferromagnetic (Fe) electrodes [as described in Fig. 21(a) ] with Fe Schottky injection contact. The Fe =n-GaAs hetero- structure was grown epitaxially in a single molecular-beam epitaxy chamber without breaking ultrahigh vacuum. Theheterostructure contained 250 nm of low Si-doped GaAs(5×10 16cm−3), 15 nm of GaAs with graded doping, and 15 nm of highly Si-doped GaAs ( 5×1018cm−3). The purpose of the doping profile was to create a narrow tunnelSchottky barrier between GaAs and Fe favorable for spininjection or detection. It was then possible to simultaneously detect the spin current in n-GaAs generated by nonlocal injection from a Fe contact by using the ISHE and the spinaccumulation by using the additional Fe contact [Figs. 23(b) and 23(c) ]. The spins were manipulated by spin precession with an external magnetic field combined with drift using an external bias ( Huang, Monsma, and Appelbaum, 2007 ). In this case, the magnetic field was applied in-plane ( xdirection) to precess the spin accumulation into the out-of-plane direction, so that it could be detected by the ISHE. The signal first increases at low fields but then is suppressed due to spindephasing [Fig. 23(c) ]. I+I- V+V-F1F2 3 2 1 0-202 0.4 0.2 0.0 -0.2 -0.4 -202V/I(m) B(T)LFM=2 mV/I(m)B(T) I+I- V+V-F1F2 -101 -4 -2 0 2 4-0.10.00.1 B(T)RSH(m) sinRSH(a) (c) (b) (d) FIG. 22 (color online). Observation of the ISHE (right) in a metal device with an electrical spin injection from a FM, compared withthe spin detection by the nonlocal spin valve effect (left). Thelight gray FM electrodes in the micrographs are made of a CoFealloy. The dark gray Hall cross is made of Al. (a) and (c) representthe measurement configurations; (b) and (d) show typical spinprecession and ISHE signals, respectively. From Valenzuela and Tinkham, 2006. FIG. 23 (color online). (a) Schematic of the device used to detect the ISHE in n-GaAs. Current is injected on the Fe electrode on the right, the voltage generated by ISHE and by spin accumu-lation are detected simultaneously with the Hall cross and the Feelectrode on the left, respectively. The spin transport can befurther modified by a drift current applied between the outermostAu electrodes. (b),(c) The experimental symmetrized nonlocalspin injection or detection signal and the antisymmetrized ISHEsignal in the in-plane hard-axis field for constant spin-injectionbias current ( 300 μA) and for three different drift currents. From Olejník et al. , 2012 .Jairo Sinova et al. : Spin Hall effects 1239 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015The devices described required the application of a magnetic field for observing the ISHE. Seki et al. (2008) used a FM (FePt) with an out-of-plane anisotropy, which enabled them to inves- tigate the ISHE in Au without magnetic fields. The device was fabricated with the geometry in Fig. 21using Ohmic contacts. The measurements presented a rather large background voltage, which is likely due to the flow of charge current at the position of the Hall cross ( Mihajlovic et al. , 2009 ). The use of Ohmic contacts, as opposed to tunnel barriers, results in inhomo- geneous in-plane current injection. Because the width of the Au wire and the distance of the Hall cross were comparable, somecurrent reached the Hall cross contributing to the background; therefore the experimental artifacts discussed previously cannot be ruled out completely. By considering that the voltage wasindependent of the magnetization of the injector electrodes, Seki et al. deduced α SH¼0.113for Au at 295 K, which was weakly dependent on temperature. This large αSHwas first attributed to resonant scattering in the orbital-dependent Kondo effect of Fe impurities in the Au host metal ( Guo, 2009 ). In the follow-up work, Sugai, Mitani, and Takanashi (2010) found that αSH∼ 0.07was approximately independent of the Fe concentration. Seki et al. (2010) further observed a reduction of αSHfrom 0.1 to 0.03 when the Au thickness was increased from 10 to 20 nm.Additionally, Guet al. (2010) obtained similar results in Pt- doped Au by codeposition of Pt and Au with magnetron sputtering (1.4% Pt). These results in combination with ab initio and quantum Monte Carlo calculations for the skew scattering due to a Pt impurity led to the proposal of a much larger α SHin the surface of Au than in the bulk ( Guet al. ,2 0 1 0 ). 3. Nonlocal detection of spin Hall effects with vertical spin current The approach described in the previous section enables proper quantification of the spin Hall angle because of thedirect measurements of the spin-diffusion length and the fact that no additional interfaces are required. However, it is suitable for materials that have spin-diffusion lengths beyond tens ofnanometers. For smaller spin-diffusion lengths, Kimura et al. (2007) used a similar approach with the device structure shown in Fig. 24(a) . The structure comprises a Hall cross where the material of the transverse arm is the large spin-orbit couplingNM with short λ sd, which acts as a spin-current absorber that induces VSHvia the ISHE. The longitudinal arm (N1), on the other hand, is made of a NM with long spin-diffusion length andfulfils the purpose of transporting spin information between theFM electrode (F) and N2. The way the measurements are performed is sketched in Fig.24(b) (left). A charge current is injected from F into N1 that induces a spin current toward N2 polarized in-plane in the direction parallel to the N1 arm. When the distance between Fand the cross is smaller than the spin-diffusion length in N1, the spin current is preferably absorbed into the transverse arm N2 because of the strong spin relaxation in N2. The injectedvertical spin current into N2 vanishes in a short distance from the N1-N2 interface because of the short spin-diffusion length of N2 and generates a transverse voltage via the ISHE. Owing to the broken inversion symmetry at the N1-N2 interface, the crystal field can induce an interface contribution tothe spin-orbit coupling ( Linder and Yokoyama, 2011 ) and therefore an additional source of SHE as well as a contribution of the spin to charge conversion from SGE ( Rojas-Sánchez et al. , 2013b ;Zhang et al. , 2015 ). In the following discussion, the SGE is not invoked to explain the results when using the device in Fig. 24. This is rooted in the fact that the relevance of SGE-ISGE in metals was brought to the attention of thecommunity in the recent SOT studies, only after most of the SHE experiments discussed here were carried out. Future experiments might require a more careful analysis to determinethe relative weight of the ISHE and SGE in the measurements. The device shown in Fig. 24can be also used to measure the SHE (or the ISGE). The bias configuration is modified as shown in Fig. 24(b) (right). Here N2 acts as a spin-current source, which induces a spin accumulation in N1 that is detected withthe FM electrode F, as originally proposed by Zhang (2000) . Kimura et al. (2007) used permalloy (Py) as the FM source, and Cu and Pt as N1 and N2, respectively (see Fig. 25). The FIG. 24 (color online). (a) Schematic illustration of a nonlocal device to measure the direct and inverse spin Hall effect inmaterials (N2) with short spin relaxation length λ N2 sd. (b) Sche- matic illustration of the charge accumulation process in N2 (left)due to the ISHE when a spin current is injected from F as in (a).Schematic illustration of the charge to spin-current conversiondue to the SHE when a current is applied to N2. This processgenerates spin accumulation that is detected by measuring thevoltage at which F floats. See Kimura et al. (2007) . Adapted from Valenzuela and Kimura, 2012 . FIG. 25 (color online). (a) Scanning electron microscope (SEM) image of the fabricated spin Hall device to measure the SHE in Pttogether with a schematic illustration of the fabricated device.(b) Signal due to the ISHE at 77 K. The black and grey curvesshow measurements for the two opposite sweeps of the magneticfield. Spin-accumulation signal generated by SHE at 77 K. Insets:measurement setup. NiFe, Cu, and Pt are in different colors. FromKimura et al. , 2007 .1240 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015materials were deposited by electron-beam evaporation. The devices were fabricated with transparent interfaces between Py and Cu and between Pt and Cu. Ar ion beam etching was done prior to depositing Cu in order to clean the surfaces of Pyand Pt, a method that has been repeated in the other studiesdescribed below. The long spin-diffusion length of Cu (about 500 nm) assured that the spin current reached Pt, which was 4 nm thick. The measurements were interpreted with a one-dimensional model by assuming that the induced spin current at the Cu-Pt interface was completely absorbed by the Pt and was uniform along the vertical direction. The spin relaxationlength for Py was assumed (not measured) to be 3 nm. Kimura et al. (2007) then obtained that σ SH∼2.4×102Ωcm−1 andαSH¼3.7×10−3. Over the last few years some of the initial simplifications that are mentioned have been removed, leading to more reliable quantitative interpretations of the experimental results. Vila, Kimura, and Otani (2007) noted that the absorption efficiency of the spin current may depend onthe device geometry and temperature. They modified the design of Fig. 25to a conventional nonlocal spin injection or detection structure where a Pt electrode was inserted betweenthe FM Py electrodes (see Fig. 26). This change enabled them to determine explicitly the magnitude of the absorbed spin current. By comparing with reference devices without the Pt insertion, they observed that the ratio between the spin signalwith and without Pt varied from 0.35 at 5 K to 0.2 at roomtemperature, irrespective of the Pt thickness. They then performed systematic spin-absorption studies as a function of the Pt thickness, obtaining that λ sdfor Pt was 10 and 14 nm at room temperature and at 5 K, respectively. The Pt thickness dependence of the ISHE signal resulted in somewhat lower λsdfor Pt of 7 and 8 nm at room temperature and at 5 K, respectively. The obtained value of σSH∼ 3.5×102Ωcm−1was larger than that in the Kimura et al.(2007) experiment; this is because the assumption of the complete spin-current absorption into the Pt wire led to underestimating the spin Hall conductivity. Additionally, Vila, Kimura, and Otani (2007) found that the spin Hall conductivity was nearly constant as a function of temperature, indicating that the spin Hall resistivity likely evolves in a quadratic form with the Pt resistivity in the analyzed temperature range, which was initially associated to a side-jump origin of the SHE. However, this resistivitydependence can also be associated with the intrinsic mecha- nism ( Tanaka et al. , 2008 ;Kontani et al. , 2009 ). Niimi et al. (2011) further included a correction factor 0< x<1that accounted for the fact that the transverse charge current induced by the ISHE is partially shunted by the wire N1 above the N1-N2 interface or, conversely, that the charge current that induces the spin current via the SHE does not only flow through N2 but also leaks into N1 [see also Liuet al. (2011) ]. In order to determine xexperimentally, they mea- sured the voltage drop of two identical N2 nanowires with and without shunting N1 bridges. Within a one-dimensional circuit model, the current flowing into the N2 wire I 0was assumed to divide into two components at the N1-N2 inter-face:xI 0for the N2 wire and ð1−xÞI0for the N1 bridge. With this,xwas estimated to be 0.36/C60.08for Cu (N1), when using a number of transition metals and alloys as N2 ( Morota et al. , 2011 ;Niimi et al. , 2011 ), therefore appearing to be rather insensitive to the resistivity of N2. Because of thiscorrection, former reports underestimated σ SHby a factor x−1∼2.8. Such large correction is to be expected given that the N1 wire (usually Cu or Ag) is highly conductive[conductivity ∼ð3−5Þ×10 7Ωm−1) and thick ( ∼100nm), when comparing with N2 ( ∼105−107Ωm−1and∼10nm). In addition, Morota et al. (2011) andNiimi et al. (2011) pointed out that the spin currents injected in N2 should dilute when its thickness tN2is larger than the spin-diffusion length in N2 leading to smaller spin Hall signals. To correct for this effect, they obtained an aggregate spin current in N2 by integrating over tN2, which was then divided by tN2; they also forced the spin current to be zero at the bottom surface of N2. Niimi et al. (2011) reported αSH¼0.021/C60.006for the skew scattering off Ir in a Cu matrix, which is consistent with experimental work relying on spin-polarized currents gener- ated by dilute Mn impurities, for which αSH¼0.026(Fert, Friederich, and Hamzic, 1981 ;Fert and Levy, 2011 ). The spin Hall angle was extracted with CuIr wires that were preparedwith different Ir concentrations (0%, 1%, 3%, 6%, 9%, and 12%) using magnetron sputtering. They measured ρ Hof CuIr as a function of the resistivity induced by the Ir impurities,defined as ρ imp¼ρCuIr−ρCu, finding a simple linear depend- ence up to Ir concentration of 12%. This was presented as a proof that the dominant mechanism of the extrinsic SHE induced by the Ir impurities is the skew scattering, with αSH¼ρH=ρimp. Morota et al. (2011) investigated the ISHE and SHE in 4d and5dtransition metals, Nb, Ta, Mo, Pd, and Pt. Nb, Ta, and Mo wires were deposited by magnetron sputtering while Pd and Pt wires were grown by electron-beam evaporation. In particular, for Pt, they obtained a spin Hall angle σSH¼ 0.021/C60.005that was roughly 6× larger than that in Kimura et al. (2007) . Such a difference can be explained with the FIG. 26 (color online). (a) SEM image of the typical device for SHE measurements and an illustration of the device. (b) Directand inverse SHE (SHE and ISHE) recorded at T¼10K using a device with a Pt thickness of 20 nm, altogether with the AMRfrom the Py wire measured on the same condition. SHEmeasurement corresponds to V BC=IAE, and ISHE to VEA=IBC; withVthe voltage, Ithe applied current; A,B,C, and Eare the contact leads as denoted in the SEM image. From Vila, Kimura, and Otani, 2007 .Jairo Sinova et al. : Spin Hall effects 1241 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015above corrections. They also found that the sign of the spin Hall conductivity changes systematically depending on thenumber of delectrons, a tendency that is in good agreement with theoretical calculations based on the intrinsic SHE(Kontani et al. , 2009 ). More recently, Niimi et al. (2012) studied the ISHE and SHE by introducing a small amount of Bi impurities in Cu. The alloysof Cu 1−xBixwere deposited by magnetron sputtering from Bi-sintered Cu targets with different Bi concentrations (0%,0.3%, 0.5%, 1%, 3%, and 6%). The spin Hall resistivity wasderived by 1D and 3D calculations as a function of the resistivityinduced by the Bi impurities. As for the case with Ir impurities,the experimental results follow the linear variation of the spinHall resistivity, characteristic of skew scattering by diluteimpurities but only at the lowest concentrations ( <1%). At larger concentrations, inhomogeneous distribution on Bi resultsin the departure from the dilute impurity regime. From the slopeρ H=ρimpin the linear regime, αSHwas estimated with the standard 1D analysis above, and with more accurate 3Dcalculations, resulting in α SHð1DÞ¼−ð0.12/C60.04Þand αSHð3DÞ¼−ð0.24/C60.09Þat 10 K. The 3D calculations yield a larger αSHbecause spin accumulation is observed to spread at the side edges of theCuBi/Cu junction, which is not taken into account in the 1Dmodel. For the calculations with the 1D model, the spincurrent is considered to flow vertically into the CuBi wire,therefore, they cannot take into account the spin escape bylateral spreading. In general, the correction is observed tobecome important when the spin-diffusion length in N2 islonger than t N2. For the cases of CuIr or Pt, it produces a small additional error because the spin-diffusion length in N2 isusually shorter than t N2. For Pt, αSHwas estimated to increase from 0.021 (1D model) to 0.024 (3D model). Nonlocal methods have been used to estimate spin Hall angles in a number of other materials, including IrO 2 (Fujiwara et al. , 2013 ) and Bi ( Fan and Eom, 2008 ). It was also applied to determine the sign of the spin-injectionpolarization of FMs by using materials with a well-establishedspin Hall angle, which is not possible with standard nonlocalspin injection and detection methods using the same FMmaterial for the two electrodes. This procedure was demon-strated for the Heusler alloy Co 2FeSi ( Okiet al. , 2012 ). The ISHE in nonlocal geometries was also used as a probe of spinfluctuations in weak FM NiPd alloys ( Wei et al. , 2012 ). An anomaly near the Curie temperature was explained by thefluctuation contributions to skew scattering via spin-orbitinteractions; the total magnetic moment involved in theexperiment was extremely small (less than 10 −14emu), high- lighting the very high sensitivity of the technique. 4. Direct detection of the spin Hall induced spin accumulation As discussed in Sec. II.C,Zhang (2000) proposed to detect the spin accumulation induced by the SHE via a FM probedirectly attached in the side of a thin conductor. Themagnetization of the FM points to the direction perpendicularto the plane of the film. The method is based on measuring thevoltage at which the FM floats depending on the direction ofits magnetization, which gives direct information of the spinaccumulation at the edge of the conductor (see Sec. IV.C.1 ).The implementation of the method took several years because of the local currents that circulate nearby the FM, which result in spurious signals that are avoided by the nonlocal methods, as described above. Garlid et al. (2010) implemented devices based on epitaxial Fe=In xGa1−xAs heterostructures (Fig. 27). The active layers consisted of a 2.5-μm-thick Si-doped [ ð3−5Þ×1016cm−3] channel, a highly doped Schottky tunnel barrier(5×10 18cm−3), and a 5-nm-thick Fe layer. Heterostructures with In concentrations 0, 0.03, 0.05, and 0.06 were processedusing lithographic and etching techniques into devices with30-μm-wide channels oriented along the [110] direction, which is the xdirection in Fig. 27. It is technically difficult to fabricate a thin film with a FM attached at its edge with the magnetization orientationproposed by Zhang (2000) . To circumvent this obstacle, Garlid et al. (2010) patterned pairs of Fe electrodes so that the centers of the contacts in each pair are 2, 6, or 10μm from the edges of the channel. However, since the contacts aremagnetized along x, and the spin polarization generated by the SHE is oriented along z, a magnetic field along ywas applied to precess the spin accumulation into the xdirection so that it could be detected. The spin accumulation is identified through FIG. 27 (color online). (a) Micrograph of a spin Hall device with Fe contacts located 10μm from the edges of the GaAs channel. The contact pairs aband cdare used to measure the spin accumulation. (b) Nonlocal spin valve (colored lines) and Hanleeffect (black dots) data obtained on a GaAs device at T¼60K for injection current 8.2×10 2A=cm2. (c) Measured voltage Vab−Vcdfor a GaAs device with Fe contacts 2μm from the edges at T¼30K for a channel current 5.7×103A=cm2.A n offset voltage of 13.2 mV has been subtracted from the data. In(b) and (c), data are shown for both parallel and antiparallel statesof injector and detector. (d) Spin Hall signal for both positive (fullcircle) and negative (open circle) currents, after removing back-ground and extracting antisymmetric signal. The spin Hall signalin the antiparallel state is shown as the solid red line. From Garlid et al. , 2010.1242 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015the observation of the Hanle effect in the voltage measured between the pairs of FM contacts. The voltage first increasesat low fields but then is suppressed due to spin dephasing inlarge fields. The local character of the measurement causes a large background signal due to (i) imperfect cancellation of thebackground HE voltage induced by the applied magnetic field,(ii) local HEs due to fringe fields generated by the FMcontacts, and (iii) because a small fraction of the channelcurrent is shunted through the Fe contacts. The HE voltageswere eliminated by using the expected symmetries of thesignal, while the shunting effect was reduced by subtractingthe voltages for the two current directions. The results showed that the magnitude of the spin Hall conductivity was in agreement with models of the extrinsicSHE due to ionized impurity scattering. The bias and temper-ature dependences of the SHE indicated that both skew andside-jump scattering contribute to the total spin Hall conduc-tivity. By analyzing the dependence of the SHE on channelconductivity, which was modified with the In content, Garlid et al. (2010) determined the relative magnitudes of the skew and side-jump contributions to the total spin Hall conductivity. Ehlert et al. (2012 ,2014) reported measurements of the SHE using a similar structure based on n-GaAs layers with relatively low carrier concentration ( 5×10 16cm−3) and corresponding low conductivity. The FM voltage probes wereimplemented with (Ga,Mn)As/GaAs Esaki diode structures.The heterostructures were grown by molecular-beam epitaxyand consisted of a 1-μm-thick n-type transport channel, a 15- nm-thick n→n þGaAs transition layer ( 5×1018cm−3), a 2.2-nm Al 0.36Ga0.64As diffusion barrier, and a 15-nm-thick layer of Ga 0.95Mn 0.05As. The highly doped (Ga,Mn)As/GaAs p-njunction forms an Esaki diode. This structure was covered on the top by 2 nm of Fe and 4 nm of Au. The purpose of Fewas to make the contacts harder magnetically, which helped tokeep the magnetization aligned along their long axes duringHanle measurements. The values of spin Hall conductivitiesthat were extracted are consistent with those calculated byEngel, Halperin, and Rashba (2005) but smaller than those observed by Garlid et al. (2010) .Ehlert et al. (2012 ,2014) observed that the combined results of these two experimentsshow that both the skew and side-jump contributions to thespin Hall conductivity cannot be treated as fully independentof the conductivity of the channel. 5. Spin Hall injection and detection without ferromagnets Spin injection by the SHE combined with spin detection by the ISHE in one device ( Hirsch, 1999 ) was successfully implemented by using a device geometry proposed byHankiewicz et al. (2004) . The original Hirsch (1999) proposal required a transverse strip connecting the edges of a slab onwhich spin accumulation was generated due to the SHE. Aspin current would circulate in the transverse strip whichwould then generate a measurable voltage transverse to it [seealso Sec. IIand Fig. 2(c)]. The fabrication of such structure is challenging, albeit not impossible. Hankiewicz et al. (2004) considered the same concept but on a planar structure shapedas an H, which is much simpler to fabricate. The device andmeasurement principle is shown in Fig. 28[see also Sec. IIand Fig. 2(b)]. An electric current is applied in one of the legs of the H-shaped structure and generates a transverse spin current owing to the SHE. The spin current propagates toward the other leg through the connecting part and produces ameasurable voltage via the ISHE. This nonlocal voltage in the second leg dominates local contributions if the separation between the legs is large enough. The structure was imple-mented experimentally by Mihajlovic et al. (2009) in Au; although no spin Hall signal was observed, the experiment set an upper bound for the spin Hall angle in this material. Thefirst results associated with spin Hall signals were reported a year later by Brüne et al. (2010) . Brüne et al. (2010) used devices based on high-mobility HgTe/(Hg,Cd)Te quantum wells with a top gate electrode. The H structures consisted of legs 1μm long and 200 nm wide, with the connecting part being 200 nm wide and 200 nm long. The estimated mean free path in the system was ≥2.5μm, i.e., the samples were well within the quasiballistic regime.Sweeping the gate voltage in the sample allowed one to vary the strength of the Rashba spin-orbit coupling by a variation of both the electrical field across the quantum well and the Fermilevel in the quantum well. In the sample it was possible to V+ V- I-I+(a) (b) FIG. 28 (color online). (a) Scanning electron micrograph of an H-shape device and probe configuration for spin injection viaSHE and spin detection via ISHE. (b) The inset indicates themeasurement configuration for current injection (arrows) andvoltage probes. The black curve in the main panel shows thenonlocal ISHE resistance signal. The solid curve indicates theresidual voltage owing to current spreading. From Brüne et al. , 2010 .Jairo Sinova et al. : Spin Hall effects 1243 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015electrically tune the carrier density from strongly n-type, through insulating, down to a p-type regime. This resulted in a strong modulation of the ISHE voltage, as shown in Fig. 28.I n thepregime, where the spin-orbit coupling is strong, the signal is at least 1 order of magnitude larger than in the weaklyspin-orbit-coupled nregime. Detailed numerical calculations confirmed that the observed spin Hall signals had the ballisticintrinsic origin ( Brüne et al. , 2010 ). An H-bar structure was also used in graphene devices (Abanin et al. , 2011 ). Here a large Hall response was observed near the graphene neutrality point in the presence of anexternal magnetic field. The results were ascribed to spincurrents that resulted from the imbalance of the Hall resistivity for the spin-up and spin-down carriers induced by the Zeeman interaction; a process that does not involve a spin-orbitinteraction, i.e., is not of the SHE origin, and that is largestin the cleanest graphene samples ( Abanin et al. , 2011 ). More recently, the controlled addition of small amounts of cova-lently bonded hydrogen atoms has been reported to induce an enhancement of the spin-orbit interaction by 3 orders of magnitude in graphene ( Castro Neto and Guinea, 2009 ; Balakrishnan et al. , 2013 ). Such large enhancement was estimated from nonlocal signals of up to 100Ω, which are observed at zero external magnetic fields and at room temper-ature. From the magnetic field and the length dependence ofthe nonlocal signal, a spin-orbit strength of 2.5 meV was extracted for samples with 0.05% hydrogenation. Similar results were observed by Balakrishnan et al. (2014) in graphene grown using chemical vapor deposition (CVD).Estimations of a spin-orbit coupling as high as 20 meV andspin Hall angle ∼0.2were reported. They argued that the observations are due to the presence of copper contaminationin CVD graphene, which act as local spin-obit scattering centers in the resonant limit. This hypothesis is tested independently by introducing metallic adatoms, such ascopper, silver, and gold on exfoliated graphene samples.Even though the nonlocal signal is absent in exfoliatedgraphene, it is clearly observed once any of the previousadatoms is introduced, resulting in calculated spin Hall angles ∼0.2in all cases, rivaling the largest values observed in metals. 6. Spin Hall magnetoresistance In bilayer FM-NM systems, a new type of magnetoresist- ance has been recently discovered which is directly associated with the SHE ( Huang et al. , 2012 ;Weiler et al. , 2012 ). The observed magnetoresistance is given by ρ¼ρ0þρ1½ˆm·ðˆj×ˆzÞ/C1382; ð4:1Þ where ρ0is the normal resistance, ρ1is the anisotropic resistance amplitude, and ˆj,ˆm, and ˆzare the directional vectors of the current, the magnetization, and the normal to the interface. This means that the magnetoresistance depends on the in-plane component of the magnetization perpendicular tothe current. In contrast, the conventional noncrystalline AMR(McGuire and Potter, 1975 ) has the form of ρ¼ρ 0þρ1ðˆj·ˆmÞ2; ð4:2Þwith ˆj·ˆm¼cosðθj−mÞ, where θj−mis the angle between the current and the magnetization. This phenomenon has been termed the spin Hall magneto- reristance (SHMR) ( Weiler et al. , 2012 ;Chen, Matsukura, and Ohno, 2013 ;Hahn et al. , 2013 ;Nakayama et al. , 2013 ; Vlietstra et al. , 2013 ;Isasa et al. , 2014 ). Its origin is illustrated in Fig. 29. When a current flows parallel to the FM-NM interface, a SHE spin current is generated in the NM directedto the interface. If the magnetization is parallel to thepolarization of the spin current generated by the SHE, it getsreflected at the interface and a spin current back flows, asshown in Fig. 29(a) . This backflow spin current then gets transformed into a charge current via the ISHE in the directionof the longitudinal current. If the magnetization is insteadperpendicular to the polarization of the spin current generatedby the SHE, it can enter the FM and dephase, as shown inFig.29(b) . In this case there is no spin-current backflow and no contribution via the ISHE to the longitudinal current inthe NM. The typical experimental results are illustrated in Fig. 30, where the bilayer system was YIG/Pt. The magnetoresistancetraces are measured as a function of the magnetization angle inthex-yplane parallel to the interface, and in the z-yandz-x planes that are perpendicular to the interface. The measuredangular dependencies are consistent with the SHMR phe-nomenology described by Eq. (4.1) and are inconsistent with the AMR expression (4.2). The theory of the effect was derived by Chen, Matsukura, and Ohno (2013) based on the scattering formalism and the spin-charge drift-diffusionequations. D. Spin Hall effect coupled to magnetization dynamics When the SHE is studied by coupling to magnetization dynamics three different FMR-based techniques can be found:(i) ferromagnetic resonance-spin pumping (FMR-SP),(ii) modulation of damping (MOD) experiments, and (iii) spinHall effect-spin-transfer torque (SHE-STT). The general underlying principle for the three methods is similar. In a bilayer NM-FM structure, the FM is used to inject or absorb adynamic spin current into or from the NM. Note that thesestudies have been also extended to replacing the SHE-ISHEgenerating NM with another FM ( Freimuth, Blügel, and M \|\| σ Je Jsback´ M ⊥σ Je JsbackJsabs ´Pt = NYIG = FJs JsM MJsabs(a) (b) FIG. 29 (color online). Illustration of the SHE magnetoresist- ance. (a) When the magnetization aligns with the polarization ofthe SHE spin current, its backflow reflection generates an ISHEcurrent that contributes to the longitudinal current. (b) When themagnetization is perpendicular to polarization of the SHE spincurrent, the spin current is absorbed and no ISHE current affectsthe longitudinal current. From Nakayama et al. , 2013 .1244 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015Mokrousov, 2010 ;Miao et al. , 2013 ;Azevedo et al. , 2014 ; Mendes et al. , 2014 ) or antiferromagnet ( Zhang et al. , 2014 ). In FMR-SP, a spin current is injected from the FM into the NM. The injected spin current is a pure ac spin current which is not accompanied by a charge current but which neverthelesscan be detected electrically since it is converted into a chargecurrent by means of the ISHE in the NM ( Saitoh et al. , 2006 ). The efficiency of the conversion process can be quantified by the spin Hall angle. Since in the process of spin injectionangular momentum is lost in the FM, the FMR-SP leads to abroadening of the FMR line ( Mizukami, Ando, and Miyazaki, 2001 ;Urban, Woltersdorf, and Heinrich, 2001 ;Heinrich et al. , 2003 ), whereas the backflow of spin current into the ferro- magnet generates a dc voltage that can also be used to detectSP (Wang et al. , 2006 ) as was experimentally demonstrated by Costache, Watts, Sladko, van der Wal (2006) andMiao et al. (2013) . Note that an additional contribution that might have to be taken into account in special cases arises fromthe SGE as has recently been demonstrated for the Ag/Bi interface ( Rojas-Sánchez, Vila et al. , 2013 ;Zhang et al. , 2015 ) and in FM-topological insulator surface state system ( Fan et al. , 2014 ;Mellnik et al. , 2014 ). In MOD experiments, the direct SHE induced in the NM by a dc electrical current is used to modify the damping in the FMwhich is concomitantly driven into FMR by the application of an rf magnetic field. In this approach, the dc spin current generated by the SHE and injected across the NM-FMinterface leads to a damping or anti-damping-like torqueacting on the precessing magnetization of the FM. Modulation of the damping is observed as a function of the applied dc charge current and a detailed line-width analysisallows extraction of the spin Hall angle ( Saitoh and Ando, 2012 ). Note that the pure dc spin current is generated in the bulk of the NM and that in order to quantitatively determine the spin Hall angle it is important to know the transmissibilityof the NM-FM interface for the pure spin current.In the SHE-STT, a spin current is used to transfer spin angular momentum and thus to exert a torque on the magneticmoments. In these experiments an ac current sent along theNM-FM interface can create an rf excitation of the magneti- zation of the FM via the SHE-STT. In conventional STT junctions, an electrical current is sent perpendicular to a stackwith two FM electrodes to transfer angular momentum fromone FM to the other FM ( Ralph and Stiles, 2008 ). SHE-STT experiments, on the other hand, exploit the use of a perpendicular pure spin current generated by an in-planeelectrical current in the attached NM via the SHE. In both the MOD experiments and the SHE-STT, the torques in the FM that are generated by the SHE in theNM would be in addition to the ISGE-related SOTs present atthe inversion asymmetric FM-NM interface ( Garello et al. , 2013 ;Freimuth, Blügel, and Mokrousov, 2014 ;Kurebayashi et al. , 2014 ). Hence, in these experiments the spin Hall angle is in reality a parametrization of the total torques generated bythe currents and therefore it should be considered instead asthe effective spin Hall angle for the specific bilayer system. As mentioned above, similarly spin pumping and detection of the spin Hall angle via the ISHE may be affected by the SGE arising from the specific bilayer interface. In the rest of the section we expand on the details and recent results of each of these FMR-based techniques. FMR-SP is the more widely used technique to measure the effective spin Hallangle thus we detail this technique more extensively. 1. Ferromagnetic resonance spin pumping As described in the theory section (Sec. III.D ), Tserkovnyak, Brataas, and Bauer (2002a) andTserkovnyak, Brataas, and Halperin (2005) showed that the precessing magnetization in a FM generates a spin current strictly at the FM-NM interface, as sketched in Fig. 31. The spin current generated at the interface propagates into the NM andconsequently decays on a length scale connected to the effective spin-diffusion length λ sdof the NM. As mentioned in the theory section, we note that the term effective is used α: yyz x, J eαH β: z Hβ y x, J e γ: zHγ y x, J e α, β, γ-90° 0° 90° 180° 270°-101 -101exp. SMR calc. AMR calc. (j)(i)MRxx (γ) MRxx (β)-11(g)MRxx (α) [sample 2] α, β, γ (a) (c)(b) FIG. 30 (color online). Magnetoresistance curves as a function of the angles (a) α, (b) γ, and (c) β, illustrated in the right panel. The key contrast to conventional AMR is the trace in (b), where nodependence is observed, while conventional AMR would give thesinusoidal form illustrated in the dashed line. From Nakayama et al. (2013) . FIG. 31 (color online). A spin current is generated by SP at the FM-NM interface (gray arrows). The time-dependent spin polari-zation of this current (indicated as a dark gray arrow) rotatesalmost entirely in the y-zplane. The small time-averaged dc component (small upward arrow) appears along the xaxis. Both components lead to charge currents in NM and can be convertedinto ac and dc voltages by placing probes along the xand the y direction, respectively. From Wei et al. , 2014 .Jairo Sinova et al. : Spin Hall effects 1245 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015here, since the determination of the spin-diffusion length for a NM interfaced with a FM may also be connected to spinmemory loss and proximity polarization at the interface. In the case of Pt and Pd in contact with a FM metal, proximity effects are well known from x-ray magnetic circular dichroismexperiments. The direction of the injected pure spin current points from the FM to the NM and its polarization is time dependent. Its projection onto the static magnetization direction of the FMleads to a small dc component of the injected spin current intothe NM. Performing time averaging one obtains a net dc spin current given by Eq. (3.26) from Sec. III.D , j s;dc¼ℏω 4π~Arsin2Θ; where ωis the driving rf and Θis the cone angle of precession. Here ~Aris the effective SP conductance. If the thickness of the NM is smaller than the spin-diffusion length, the build-up of spin accumulation will yield a backflow spin current which will reduce the total spin current into the NM. The SPconductance ~A ris proportional to the real part of the mixing conductance, discussed in Sec. III.D , and is reduced by this backflow. The reduction depends on the ratio τtr=τsf, the reduction being strongest as this ratio increases. Hence, theeffective spin-mixing conductance may become small eventhough a pure spin current is efficiently transferred across the FM-NM interface. Recently, spin-flip scattering near the FM-NM interface has been divided up into a spin memoryloss occurring directly at the interface (interface scattering)and the decay of the spin polarization as described above (Rojas-Sánchez et al. , 2014 ). The ISHE is used to electrically detect pure spin currents generated by the SP ( Saitoh et al. , 2006 ), as shown Fig. 32.I n spin-orbit-coupled NMs like Pt or Pd, the ISHE converts the pure spin current into a detectable charge current given byEq.(3.34) , Sec. III.D , j c¼αSH2e ℏjs×σðtÞ: Here the vector of the spin-current density jspoints perpendicular to the NM-FM interface into the NM. Note that the vector of the spin-current polarization σðtÞis a time varying quantity, which we do not average here, since it hasnow been demonstrated that the ac component is alsomeasurable ( Wei et al. , 2014 ;Weiler et al. , 2014 ). In Fig.32, only the dc component of the spin-current polarization is depicted. To measure the effect of the injected spin current via ISHE, i.e., to measure the generated charge current, contact electro- des have to be attached to the sample. If the coordinate system of Fig. 31is considered, placing electrodes along the y direction allows detecting the small dc component of theSP-induced ISHE. In contrast, if the contact electrodes are attached along the xdirection, the much larger ac component in the GHz frequency range can be detected when highfrequency lines are used. In case of dc detection the time-averaged dc component of the injected spin current pointing along the xdirection(arrow in Fig. 31) leads to a charge current which is converted to a potential drop across the resistance of the NM and can bemeasured as a voltage signal. When performing FMR-SP experiments not only voltages due to ISHE are generated, but also due to the AMR or the AHE. Thus, great care has to betaken to disentangle these contributions. In the geometry sketched in Fig. 31the propagation direction of the spin current is along zand its polarization is along the xdirection. Equation (3.35) is then used to convert between this spin current and the measured voltage. In the original experiments by Saitoh et al. (2006) , the bilayer is placed in a FMR cavity in which the magnetic-field component of the microwave mode with frequency 9.45 GHzis maximized while the electric-field component is minimized.The voltage probes are placed on the sides of the millimeter-sized sample (see Fig. 32). A similar setup was used by Azevedo et al. (2011) . Here the sample is rotatable in the cavity and the cavity (i.e. the direction of the rf excitationfield) is kept fixed with respect to the dc external magneticfield. This experimental geometry has advantages and dis- advantages. The main advantage is that it is possible to find an in-plane angle between excitation rf field and angular positionof the voltage probes where the AMR contribution to thesignal vanishes exactly while ISHE is detectable. Second, inthe in-plane excitation geometry typically used, the sensitivity is large due to the large in-plane susceptibility at FMR. A major disadvantage is that it is not easy to perform frequencydependent measurements and that due to the use of a cavity theexact amplitude of the excitation field, and thus the cone angleof precession which enters Eq. (3.26) in Sec. III.D , is usually not well known. Finally, since typically large, millimeter-sized samples are used in the experiments, spurious rf electric fields (a) (b) (c) FIG. 32 (color online). Observation of the ISHE in a metal device with spin injection from a FM by FMR-SP. (a) Schematic illustration of the NiFe/Pt sample system used in the study and of the SP effect and the ISHE. (b) Magnetic-field dependenceof the FMR signal for the NiFe/Pt bilayer film and a bare NiFefilm.Idenotes the microwave absorption intensity. (c) Magnetic- field dependence of dVðHÞ=dH for the NiFe/Pt sample. V denotes the electric-potential difference between the electrodeson the Pt layer. From Saitoh et al. , 2006 .1246 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015may lead to additional contributions due to the AHE. It is therefore not straightforward to obtain an exact quantitative value of the spin Hall angle from cavity FMR-type measurements. In the experiments shown in Fig. 32, the measured FMR spectrum of the NiFe/Pt sample is compared to a reference NiFe sample [see Fig. 32(b) ]. The FMR linewidth of the NiFe/ Pt sample is larger than that of the reference NiFe film which demonstrates the presence of the SP effect in the NiFe/Pt. Theinduced voltage signal measured simultaneously across the sample along an axis parallel to the NiFe/Pt interface is shown in Fig. 32(c) .Saitoh et al. (2006) andAndo et al. (2008) demonstrated that the signal is present only when the spin- polarization vector of the injected spin current has a compo- nent perpendicular to the measured electric field across the sample, consistent with the ISHE. Ando et al. (2009) reported electrical detection of a spin wave resonance in nanostructured NiFe/Pt samples. Electrical tuning of the spin signal in a semiconductor has been recently demonstrated also by Ando, Takahashi, Ieda, Kurebayashi et al. (2011) . In the experiment, spins were injected from NiFe into GaAs through a Schottky contact using the FMR-SP.Tuning of the SP efficiency was achieved by applying a bias voltage across the NiFe/GaAs Schottky barrier and interpreted as a consequence of a suppressed or enhanced spin coupling across the interface. The FM-semiconductor SP experiments inAndo, Takahashi, Ieda, Kurebayashi et al. (2011) were performed also on samples with an Ohmic contact between NiFe and GaAs. The measurements indicate that the resistance mismatch problem in Ohmic metal-semiconductor spin-injection devices can be circumvented by using the FMR-SP technique. Similar experiments have recently been performed also for spin injection into Si ( Ando et al. , 2010 ), Ge (Jain et al. , 2012 ) and organic semiconductors ( Watanabe et al. , 2014 ). The latter one shows a surprisingly long spin coherence in the hopping regime and opens new possibilities in organic spintronics. A second possibility to quantify the spin Hall angle has been pioneered by Mosendz, Pearson et al. (2010) . They used a microstructured coplanar wave guide (CPW) with integratedbilayer structure on top of the center wave guide. This geometry allows excitation of FMR in the FM layer over a wide frequency range while the driving rf field is in the plane of the bilayer at 90° to the long axis of the several hundred micrometer long device (see Fig. 34). The use of a wave guide structure allows precise knowledge of the amplitude of the rf fields and thus the cone angle of the precessing magnetization. Voltage pickup at the ends of the wire are used, perpendicularto the direction of the rf driving field. Mosendz, Pearson et al. (2010) applied the external magnetic bias field at an angle of 45° to the long axis of the wave guide. In this experimental geometry both ISHE and AMR signals are detected at the voltage probes as can be seen directly in the recorded voltagetraces (see Fig. 33). AMR leads to a parasitic dc voltage signal at FMR due to the mixing of the time-dependent resistivity (AMR and precessing magnetization) with a capacitively or inductively coupled microwave current IðtÞ in the bilayer. The AMR of the bilayer can be taken into account by considering the orientation of the magnetization with respect to the currentdirection: RA¼R∥−R⊥. The general formula describing the parasitic voltage pick-up due to the AMR is given byhVðtÞ i¼h IðtÞR AαipðtÞsinð2φHÞi,(Costache, Sladko, Watts, van der Wal et al. , 2006 ;Mecking, Gui, and Hu, 2007 ; Obstbaum et al. , 2014 ) and it follows that this time-averaged dc voltage is to first order proportional to the in-plane dynamiccone angle of the magnetization α ipðtÞ. The cone angle of precession can easily be calculated from the simultaneouslymeasured susceptibility at FMR in the exactly known geom-etry of the CPW structure. The angle φ His defined in Fig. 34. Note that according to Baiet al. (2013) , spurious effects due to the AMR can be excluded by carefully analyzing the highfrequency characteristics of the CPWs used in the experimentswith in-plane excitation, leading to a quantitative determi-nation of the spin Hall angles. Another possibility is to place the bilayer in the gap of the CPW (see Fig. 34). Now the in-plane dynamic cone angle relevant for the AMR is given by α ipðtÞ¼χy0y0hxðtÞsinðφHÞþ χy0zhzðtÞ. The formula contains both in-plane and out-of-plane magnetic fields, together with the corresponding tensorelements of the susceptibility ( χ ij). Since the out-of-plane FIG. 33 (color online). (a),(b) Derivative of FMR spectra for Py/ Pt (open circles) and Py (black triangles). The solid lines are fitsto a Lorentzian FMR absorption function. (c) Voltage along thesamples vs field dc magnetic field (Py/Pt: open circles; Py: blacktriangles). Dotted and dashed lines show the decomposition of thespectrum into a symmetric (ISHE) and antisymmetric (AMR)contribution. The solid line shows the combined fit for the Py/Ptsample. From Mosendz et al. , 2010a .Jairo Sinova et al. : Spin Hall effects 1247 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015field produced by the CPW is about 3 orders of magnitude larger than its in-plane component, one is tempted to simply neglect the terms arising form the in-plane field. This approach is justified as long as only a single layer is studied.However, as soon as a FM-NM bilayer with a highlyconductive NM is used, the inductively or capacitively coupled microwave current largely flows in the NM and therefore generates an in-plane Oersted field of the samefrequency and phase and with an amplitude comparable to the rf field generated by the CPW. Hence the rf current distribu- tion in the bilayer has a significant effect on the magnetizationdynamics in the FM layer and can even be the dominating source of dc voltage generation by the AMR ( Obstbaum et al. , 2014 ). Using standard electromagnetic wave simulation codes, the rf magnetic-field contribution can be calculatedrather accurately. When performing angular-dependent measurements, the symmetric and antisymmetric contributions due to the ISHEand the AMR can be traced [see Figs. 34(a) and3(b)]. While for in-plane excitation the signal shows the same angulardependence, for the out-of-plane excitation case the antisym- metric contribution can be suppressed completely at an angle ofφ H¼0[see Fig. 34(d) ]. The voltage contribution at this angle is thought to arise from ISHE exclusively and allowsquantitative determination of the spin Hall angle. Note that inthese measurements both symmetric and antisymmetric con- tributions can be observed in a bare FM layer when the angle is set to φ H¼45° [see Fig. 34(c) ]. 2. Spin Hall effect modulation of magnetization damping A MOD experiment that is the inverse of the FMR-SP was proposed by Ando et al. (2008) . In the MOD described in Fig.35, a FM-NM bilayer (in this case Py/Pt) is placed in a microwave cavity (frequency 9.4 GHz) and subjected to an rf driving field. By adjusting the external field, the bilayer can bebrought into FMR. A typical FMR trace dIðHÞ=dH is shown in Fig. 35(b) . The direction of the external magnetic field encloses an angle θwith the direction of current flow. Since the mm-sized sample consists of 10 nm NiFe and 10 nm Pt,the effect of SP which contributes to the relaxation of theprecessing magnetization can be observed as a linewidth broadening when comparing to the data obtained for a plain NiFe film. Figure 35(c) illustrates the effect of a dc current sent through the bilayer sample due to the combined action ofthe SHE and STT. Because of the SHE a spin current is generated in the Pt layer and enters the NiFe film. Its flow direction is perpendicular to the interface and its polarizationdirection σdepends on the direction of the current flow. The FIG. 34 (color online). Symmetric (dots) and antisymmetric (open squares) voltage signals amplitudes at FMR (at 12 GHz)for a Py/Pt bilayer as a function of angle φ H. In (a) the magnetic excitation field is in-plane placing a Py/NM bilayer on top of thesignal line of a CPW. Both symmetric and antisymmetricamplitudes obey a sin ðφ HÞsinð2φHÞbehavior. (b) The magnetic excitation field generated by the CPW is out-of-plane with respectto the Py/Pt layers. The amplitudes of the antisymmetric partfollow a ½asinðφ HÞþb/C138sinð2φHÞbehavior. The symmetric part obeys ½csinðφHÞþd/C138sinð2φHÞþecosðφHÞ, which reflects the fact that the symmetric part is due to AMR and ISHE. (c) Voltageat FMR for φ H¼45°, and (d) φH¼0° for a single Py layer and a Py/Pt bilayer. From Obstbaum et al. , 2014 . FIG. 35 (color online). (a) A schematic illustration of the MOD experiment to determine the spin Hall angle. His the external magnetic field, and Jcrepresents the applied electric current density. (b) Magnetic-field dependence of the FMR signal for aNiFe/Pt bilayer film and a pure NiFe film. Note the linewidthbroadening for NiFe/Pt due to SP. (c) Schematic illustration of thespin Hall and the spin torque effects. M,J s, and σdenote the magnetization, the flow direction of the spin-current density, andthe spin-polarization vector of the spin current, respectively. FromAndo et al. , 2008 .1248 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015spin current exerts a torque on the precessing magnetization which either adds to the damping torque or opposes it. Theeffect is maximized when the external magnetic field pointsperpendicular to the direction of current flow. For the situationsketched here, the spin-current density can be written as j s¼αSHðℏ=2eÞjjcjσ. The effect of the injected spin current on the precessing magnetization can be modeled in terms of anadditional STT contribution to the Landau-Lifshitz Gilbertequation ( Ando et al. , 2008 ;Liuet al. , 2011 ) that has to be added on top of the SP contribution, τ STT¼−μ0γαSHηℏ 2ejc μ0M2sdPyM×ðM×σÞ.ð4:3Þ Here dPyis the thickness of the Py layer. For simplicity, the factor κ¼αSHηℏ 2ejc μ0M2sdPyð4:4Þ is introduced. Note that this factor is dimensionless and κ<0 forjc>0due to the negative electron charge. The parameter ηdefines the so-called injection efficiency and contains the effects of spin-current losses near the interface. There is noconsensus on the exact ingredients for this parameter, so itcould be useful to use η×α SHas an effective quantity parametrizing the STT efficiency. Figure 36shows the MOD experimental findings. When a current flows through the FM-NM bilayer, the STT generatedby the spin current traversing the NM-FM interface due to theSHE alters the FMR linewidth when the current flow direction and the external magnetic-field direction enclose an angle of 90° while no effect is observed for collinear orientation,consistent with the theoretical expectation. Similar experiments have been performed by Demidov, Urazhdin, Edwards, Demokritov (2011) and Demidov, Urazhdin, Edwards, Stiles et al. (2011) using Brillouin light scattering methods. The key finding in these experiments isthe control of the FMR linewidth of the FM film by employingthe SHE which generates a pure spin current in the adjacentNM. Ultimately, in suitable nanostructured materials, theapplication of a large enough charge current density should lead to the generation of coherent auto-oscillations in the FM nano object due to a dc charge current ( Demidov et al. , 2012 ; Liu, Pai, Ralph, Buhrman, 2012b ;Liu, Lim, and Urazhdin, 2013 ;Duan et al. , 2014 ;Hamadeh et al. , 2014 ). These magnetization induced nano-oscillators are of particular inter-est in the research of tunable microwave sources. 3. Spin Hall effect: Spin-transfer torque Finally, a third FMR technique has been employed that allows accessing the spin Hall angle experimentally. Liuet al. (2011) applied a microwave frequency charge current in the plane of a NiFe/Pt sample and observed the FMR in NiFe.Because of the action of the SHE a transverse spin current isgenerated in the NM, in this case Pt, which is injected intothe FM layer. Consequently, an oscillatory STT acts on the magnetic moments in the FM, inducing precession of the magnetization (see Fig. 37). The oscillatory magnetization inthe FM leads to an oscillatory AMR which in turn leads to an oscillatory resistance. This high frequency resistance mixeswith the rf current and leads to a detectable dc voltage acrossthe device which can be picked up using a bias tee [Fig. 37(c) ]. FIG. 36 (color online). FMR spectra for the NiFe/Pt bilayer measured at various electric current density values Jcwhen the magnetic-field direction is (a) 90° and (b) 0°. The inset showsmagnified views around the peaks of the spectra, wherethe solid and dashed curves are the FMR spectra measuredwith electric current densities J cand−Jc, respectively. From Ando et al. , 2008 . FIG. 37 (color online). (a) Schematic of Pt/Py bilayer thin film illustrating the STT induced by the SHE rising from the rf currentthrough NM as well as the damping torque and the torque due to theOersted field when the magnetization of FM is aligned in anexternal magnetic field. (b) The dimensions of the sample and theOersted field due to a current flowing through FM. (c) The electricalmeasurement scheme. From Liuet al. , 2011 .Jairo Sinova et al. : Spin Hall effects 1249 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015In these experiments the external magnetic field is typically fixed at an angle of 45° and swept in the plane of the films to achieve the FMR condition. In the setup, different torques act on the magnetization of the FM which is aligned along themagnetic-field direction as depicted in Fig. 37(a) . The torques include all the STTs due to the SHE in the NM, the torque induced by the Oersted field due to the rf current through the device, and the torque already modified by SP. We alsoemphasize that the torques generated by the SHE in the NM would be in addition to the ISGE-related SOTs present in the FM near the interface ( Garello et al. , 2013 ;Freimuth, Blügel, and Mokrousov, 2014 ;Kurebayashi et al. , 2014 ). Landau-Lifshitz-Gilbert equations including all relevant torques can be used to model the dc voltage response ofthe bilayer device and the result shows that the mixing voltagecontains the contributions of symmetric and antisymmetric Lorentzian lines ( Liuet al. , 2011 ). According to Liuet al. (2011) , the detailed analysis of the resonance properties of this voltage enables a quantitative measure of the spin currentabsorbed by the FM and of the spin Hall angle. Liuet al. (2011) showed that the ratio of the symmetric to antisym- metric components of the resonance curve, when scaledproperly by material parameters like the saturation magneti-zation, thickness and width of the FM, and the external magnetic field, is linked to the ratio of spin and charge currents and thus to the spin Hall angle. They emphasized thatthe measurement method is (in a reasonable thickness regimeof the FM and the NM) self-calibrating since the strength of the torque from the spin current is measured relative to the torque from the rf magnetic field, which can be calculatedfrom the geometry of the sample. The same method has beenapplied to various combinations of FMs and NMs ( Liuet al. , 2011 ;Liu, Pai, Li et al. , 2012 ;Paiet al. , 2012 ). Also in these types of experiments the tunability of the effective damping parameter has been demonstrated by Liu et al. (2011) andKasai et al. (2014) . An example is illustrated in Fig. 38for the case of Py/Pt where the effective damping parameter is shown to be tunable as a function of the current direction and amplitude ( Liuet al. , 2011 ).4. Spin Hall effect induced switching of the magnetization For sufficiently large current densities pushed through the NM and large spin Hall angles, it is possible to even reverse themagnetization in a FM nanoelement placed on top of the NMcurrent carrying line, as has been demonstrated by Miron, Garello et al. (2011) andLiu, Pai, Li et al. (2012) (see Figs. 39 and40). In these experiments it is important to use a NM-FM combination where, when placing the NM in contact with theFM, the induced damping due to SP remains negligible. This is the case for CoFeB/Ta (Fig. 40). On the one hand, β-Ta shows a giant spin Hall angle ( Liu, Pai, Li et al., 2012 ), on the other hand, enhancement of damping due to SP is not observed in theCoFeB layer. Furthermore, due to the large resistivity of theCoFeB layer a large portion of the applied current is pushedthrough the Ta layer where it produces the pure spin current dueto the SHE. Another important feature is that the bilayer iscapped with MgO to induce a large perpendicular anisotropy inCoFeB. The thin layer of MgO (1.6 nm) is used as a tunnelbarrier between the thin CoFeB free layer (1.6 nm) and thethicker CoFeB reference layer (3.8 nm) so that the tunnelingmagnetoresistance (TMR) effect can be used to determine the relative orientation of their magnetization. In earlier experiments using ultrathin FM layers, Miron et al. (2010) andMiron, Garello et al. (2011) demonstrated similar results. Their devices are based on the Pt =Co=AlO xsystem with ultrathin Co layers of a thickness of only 0.6 nm sandwichedbetween Pt (3 nm) and AlO x(1.6 nm). The use of ultrathin Co in contact with Pt leads to a strong perpendicular anisotropy. Whena current is driven through the Pt layer, switching of the Comagnetization can be observed by monitoring the AHE of the device (see Fig. 39). In the original interpretation the driving force for the observed switching was thought to arise mostlyfrom the Rashba symmetry ISGE due to the broken inversionsymmetry along the growth direction of the layer stack. FIG. 38 (color online). Effective damping as a function of current density through the Pt layer in a Py ð4nmÞ=Ptð6nmÞbilayer. From Liuet al. , 2011 . FIG. 39 (color online). Top left: device schematic and current- induced switching. Hall-cross geometry. Black and white arrowsindicate the up and down equilibrium magnetization states of theCo layer, respectively. Bottom left: scanning electron micrographof the sample and electric circuitry used in the measurements.Shown are the terminals for the Hall voltage measurements aswell as the current line where a pulsed current is applied for theswitching experiments. Middle: the state of the perpendicularmagnetization is measured via the anomalous Hall resistance as afunction of applied field B. After injection of positive (squares) and negative (circles) current pulses of amplitude I p¼52.58mA the Hall resistance is measured. The data are reported during asingle sweep of B. Right: the measurement schematics and pulse sequence. From Miron, Garello et al. , 2011 .1250 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015However, detailed analysis in later three-dimensional vector measurement ( Garello et al. , 2013 ) point toward significant contributions from the SHE. The results of these experiments may be viewed as a paradigm change in the mechanism for switching magnetic nanoelements in spintronic devices since here switching is driven by a purely in-plane electrical current and not via acurrent perpendicular to the layer stack. Similar results havebeen obtained for W/CoFeB layers ( Paiet al. , 2012 ). One should note that, while the exact value for the spin Hall angleextracted from these experiments is still under debate, the factthat switching can be achieved for these devices underpins not only the technological relevance, but also that a sizeable SHE (possibly in combination with other ISGE-related SOTs) mustbe generated in these structures. Similarly, the interpretation of the results of current-driven domain-wall motion experiments in the same type of layer stacks has to be revisited ( Miron et al. , 2011 ). Current and even field-induced domain-wall motion experiments in layerstacks where ISGE, SHE, and proximity polarization of theNM can contribute are complicated for interpretation, anddisentangling the relative strength of these contributions is not straightforward. Experimentally, however, it has been observed that the inclusion of relativistic torques, of eitherthe SHE or ISGE origin, leads to a large increase of domain-wall velocities for optimally tuned materials which is poten-tially of technological interest ( Emori et al. , 2013 ;Ryu et al. , 2013 ).Spin-orbit coupling together with broken inversion sym- metry introduces yet another important aspect into the physics of these systems. To fully understand the underlying mech- anisms in these experiments one needs to take into accountalso the fact that these domain walls are chiral due to the Dzyaloshinski-Moriya interaction present at the FM-NM interface. This opens a new field connecting spintronics with the skyrmion physics. We conclude by discussing in more detail that in the NM- FM bilayer systems the relativistic torques inducing magneti- zation dynamics are, in general, not only due to the SHE but the ISGE-induced SOTs may also contribute ( Chernyshov et al. , 2009 ;Manchon and Zhang, 2009 ;Miron et al. , 2010 ; Fang et al. , 2011 ). The ISGE originate from spin-orbit coupling which, combined with broken inversion symmetry in the crystal, can produce spin polarization when electrical current is driven through a NM. In combination with FMs, theISGE and the SHE can drive magnetization dynamics in devices with similar geometries. Disentangling these contri- butions in NM-FM bilayer systems and engineering them for maximal effect is at present a highly active field in spintronics. However, the discrimination of the SHE and ISGE-based microscopic mechanisms between the field-like and the anti- damping-like torque components is difficult to achieve for several conceptual reasons. The original theoretical proposals (Aronov and Lyanda-Geller, 1989 ;Edelstein, 1990 ; Malâshukov and Chao, 2002 ) and experimental observations (Ganichev et al. , 2004 ;Kato et al. , 2004b ;Silov et al. , 2004 ; Wunderlich et al. , 2004 ,2005 ) of the ISGE were made in NMs with no FM component in the structure. The correspondingnonequilibrium spin density, generated in the ISGE by inversion-asymmetry terms in the relativistic Hamiltonian, has naturally no dependence on magnetization. Hence, in the context of magnetic semiconductors ( Bernevig and Vafek, 2005 ;Chernyshov et al. , 2009 ;Endo, Matsukura, and Ohno, 2010 ;Fang et al. , 2011 ) or FM-NM structures ( Manchon et al. , 2008 ;Miron et al. , 2010 ;Piet al. , 2010 ;Miron, Garello et al. , 2011 ;Suzuki et al. , 2011 ), the ISGE may be expected to yield only the field-like component of the torque ∼M×ζ, where the vector ζis independent of the magnetization vector M. However, when carriers experience both the spin-orbit coupling and magnetic exchange coupling, the inversion asymmetry can generate a nonequilibrium spin-density com- ponent of extrinsic, scattering-related ( Pesin and MacDonald, 2012 ;Wang and Manchon, 2012 ) or intrinsic, Berry-curvature (Garate and MacDonald, 2009 ;Freimuth, Blügel, and Mokrousov, 2014 ;Kurebayashi et al. , 2014 ) origin which is magnetization dependent and yields an anti-damping-liketorque ∼M×ðM×ζÞ. Experiments in (Ga,Mn)As confirmed the presence of the ISGE-based mechanism ( Chernyshov et al. , 2009 ;Endo, Matsukura, and Ohno, 2010 ;Fang et al. , 2011 ) and demonstrated that the field-like and the Berry- curvature anti-damping-like SOT components can have com-parable magnitudes ( Kurebayashi et al. , 2014 ). The STT is dominated by the anti-damping-like component (Ralph and Stiles, 2008 ) in weakly spin-orbit-coupled FMs with τ ex≪τs, where τexis the precession time of the carrier spins in the exchange field of the FM and τsis the spin lifetime in the FM. This, in principle, applies also to the case when the spin current is injected to the FM from a NM via the SHE. (a) (b) (c)(d) FIG. 40 (color online). SHE-induced switching for an in-plane magnetized nanomagnet at room temperature. (a) Schematic of the three-terminal SHE devices and the circuit for measurements. (b) TMR minor loop of the magnetic tunnel junction as a functionof the external applied field B extapplied in-plane along the long axis of the sample. Inset: TMR major loop of the device. (c) TMRof the device as a function of applied dc current I dc. An in-plane external field of −3.5mT is applied to set the device at the center of the minor loop. (d) Switching currents as a function of theramp rate for sweeping current. Squares indicate switching fromantiparallel (AP) to parallel (P) magnetizations; triangles indicateswitching from P to AP. Solid lines represent linear fits ofswitching current versus log (ramp rate). Error bars are smallerthan the symbol size. From Liu, Pai, Li et al. , 2012 .Jairo Sinova et al. : Spin Hall effects 1251 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015TABLE III. Experimental spin Hall angles and related parameters. SP ¼spin pumping, NL ¼nonlocal, STT þSHE ¼spin transfer torque combined with spin Hall effect, LSA ¼local spin accumulation, MR ¼magnetic resonance, KRM ¼Kerr rotation microscopy. Values marked with /C3 are not measured but taken from the literature. T(K) λsd(nm) σNM(106S=m) αSH(%) Comment Reference Al 4.2 455/C615 10.5 0.032/C60.006 NL (12-nm-thick films) Valenzuela and Tinkham (2006 ,2007) 4.2 705/C630 17 0.016/C60.004 NL (25-nm-thick films) Valenzuela and Tinkham (2006 ,2007) Au 295 86/C610 37 11.3 NL (10-nm-thick films) Seki et al. (2008 ,2010) 295 83 37 3 NL (20-nm-thick films) Seki et al. (2010) 4.5 65/C348.3 <2.3 NL (SHE-ISHE) Mihajlovic et al. (2009) 295 36/C325.7 <2.7 NL (SHE-ISHE) Mihajlovic et al. (2009) 295 35/C64 28 7.0/C60.1 NL Sugai, Mitani, and Takanashi (2010) 295 27/C63 14 7.0/C60.3 NL (0.95 at. % Fe) Sugai, Mitani, and Takanashi (2010) 295 25/C63 14.5 12/C64 NL (1.4 at. % Pt, 10-nm-thick films) Guet al. (2010) 295 50/C68 16.7 0.8/C60.2 NL (1.4 at. % Pt, 20-nm-thick films) Guet al. (2010) <10 40 /C616 25 1.4/C60.4 NL Niimi et al. (2014) 295 35/C63/C325.2 0.35/C60.03 SP Mosendz, Pearson et al. (2010) 295 35 20 0.25/C60.1 SP Vlaminck et al. (2013) 295 35/C63/C35.25 1.6/C60.1 SP Hung et al. (2013) 295 35/C63/C37 0.335/C60.006 SP Hung et al. (2013) 295 35/C31.1/C60.3 SP Obstbaum et al. (2014) 295 60 20.4 8.4/C60.7 SP Wang, Pauyac, and Manchon (2014) AuW 295 1.9 1.75 >10 NL and SP (7 at. % W concentration in Au host, 10 K) Laczkowski et al. (2014) Ag 295 700 15 0.7/C60.1 SP Wang, Pauyac, and Manchon (2014) Bi 3 0.3/C60.1 - >0.3 Local, signal decreases with ρN Fan and Eom (2008) 295 - 2.4/C60.3ðIÞ −ð7.1/C60.8ÞðIÞ SP as a function of Bi thickness Hou et al. (2012) 50/C612ðVÞ 1.9/C60.2ðVÞ Volume ( V) and interfacial ( I) parameter Cu 295 500 16 0.32/C60.03 SP Wang, Pauyac, and Manchon (2014) CuIr 10 5 –30 2.1/C60.6 NL (Ir concentrations from 0% to 12%) Niimi et al. (2011) CuMn xTy 0.7(Ta); 2.6(Ir) T ¼Lu, Ta, Ir, Au, Sb [ y∼ð1–20Þ×10−4] Fert, Friederich, and Hamzic (1981) 1.35(Au); 1.15(Sb) Mn [ x∼ð1–2Þ×10−4] creates Is −1.2ðLuÞ Note a factor of 2 in the definition of αskew SH Ref. [15] in Fert and Levy (2011) CuBi 10 ∼100;∼30 −11 NL (Bi ¼0.3%; 0.5%), αskew SH¼−ð24/C69Þon Bi Niimi et al. (2012) ∼10;∼7 Similar in AgBi ( Niimi et al. , 2014 ) n-GaAs 4.2 2200 0.0056 0.15 NL, n≈1017cm−3Olejník et al. (2012) 4.2 8500 0.00137 0.08 LSA, n≈1016cm−3Ehlert et al. (2012) 30 0.0036 0.08 LSA, n≈3–5×1016cm−3Garlid et al. (2010) 2 ≈−0.001 MR, αSHT-dependent, sign change at ≈10K Chazalviel (1975) 295 0.027 0.00044; 0.001 SP, not annealed and annealed values; n-type Rojas-Sánchez et al. (2013a) n-InGaAs 30 ∼3000 ∼0.002 ≈0.02 KRM, x¼0.07,n≈3×1016cm−3Kato et al. (2004a) (Si doped) 30 0.003 –0.005 ≈0.1;≈0.25;≈0.38 LSA, x¼0.03;0.05;0.06,n≈ð3–5Þ×1016cm−3Garlid et al. (2010) InSb 1.3 −0.026/C60.005 MR, n≈1014cm−3,μ≈2.2×104cm2=Vs Chazalviel and Solomon (1972) 1.3 0.003 MR n≈1014cm−3,μ≈4×104cm2=Vs Chazalviel and Solomon (1972) IrO 2 300 3.8(P) 0.5(P); 0.18(A) 4(P); 6.5(A) NL, polycrystalline (P), amorphous (A) Fujiwara et al. (2013) (Table continued)1252 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015T(K) λsd(nm) σNM(106S=m) αSH(%) Comment Reference Mo 10 10 3.03 −0.20 NL Morota et al. (2009) 10 10 0.667 −0.075 NL Morota et al. (2009) 10 8.6/C61.3 2.8 −ð0.8/C60.18Þ NL Morota et al. (2011) 295 35/C63/C34.66 −ð0.05/C60.01Þ SP Mosendz, Pearson et al. (2010) Nb 10 5.9/C60.3 1.1 −ð0.87/C60.20Þ NL Morota et al. (2011) Pd 10 13/C62 2.2 1.2/C60.4 NL Morota et al. (2011) 295 9/C31.97 1.0 SP Ando et al. (2010) 295 15/C64/C34.0 0.64/C60.10 SP Mosendz, Pearson et al. (2010) 295 5.5/C60.5 5 1.2/C60.3 SP Vlaminck et al. (2013) 295 2.0/C60.1 3.7 0.8/C60.20 STT þSHE Kondou et al. (2012) Pt 295 6.41 0.37 NL Kimura et al. (2007) 5 8 8.0 0.44 NL ( λN¼14nm from spin absorption) Vila, Kimura, and Otani (2007) 295 7 5.56 0.9 NL ( λN¼10nm from spin absorption) Vila, Kimura, and Otani (2007) 10 11/C62 8.1 2.1/C60.5 NL Morota et al. (2011) 10 ∼10 8.1 2.4 NL [3D corrected ( Morota et al. , 2011 )] Niimi et al. (2012) 295 7/C36.4 8.0 SP Ando et al. (2008) 295 10/C62/C32.4 1.3/C60.2 SP Mosendz, Pearson et al. (2010) 295 10/C32 4.0 SP Ando, Takahashi, Ieda, Kajiwara (2011) 295 3.7/C60.2 2.42 8/C61 SP Azevedo et al. (2011) 295 8.3/C60.94 .3/C60.21 .2/C60.2 SP Feng et al. (2012) 295 7.7/C60.71 .3/C60.11 .3/C60.1 SP Nakayama et al. (2012) 295 1.5−10/C32.45/C60.13þ4 −1.5 SP, spin Hall magnetoresistance Hahn et al. (2013) 295 4 4 2.7/C60.5 SP Vlaminck et al. (2013) 295 8/C61/C31.02 2.012/C60.003 SP Hung et al. (2013) 295 1.3/C32.4 2.1/C61.5 SP Baiet al. (2013) 295 1.2 8.6/C60.5 SP Zhang et al. (2013) 295 1.4/C312/C64 SP Obstbaum et al. (2014) 295 3.4/C60.4 6.0 5.6/C60.1 SP Rojas-Sánchez et al. (2014) 295 7.3 2.1 10/C61 SP Wang, Pauyac, and Manchon (2014) 295 1.2/C60.1 3.6 2.2/C60.4 STT þSHE Kondou et al. (2012) 295 3ð<6Þ 5.0 7.6þ8.4 −2.0 STT þSHE Liuet al. (2011) 295 2.1/C60.2 3.6 2.2/C60.8 STT þSHE Ganguly et al. (2014) 295 2.1/C60.2 3.6 8.5/C60.9 STT þSHE, modulation of damping Ganguly et al. (2014) 295 2.4/C31.2 ∼4 Spin Hall magnetoresistance Nakayama et al. (2013) 295 1.5/C60.5 0.5–3 11/C68 Spin Hall magnetoresistance (variable Pt thickness) Althammer et al. (2013) p-Si 295 ≈0.01 SP,τs∼10psn≈2×1019cm−3Ando and Saitoh (2012) Ta 10 2.7/C60.4 0.3 −ð0.37/C60.11Þ NL Morota et al. (2011) 295 1.9 0.34 −7.1/C60.6 SP Wang, Pauyac, and Manchon (2014) 295 1.8/C60.7 0.08–0.75 −ð2þ0.8 −1.5Þ SP, spin Hall magnetoresistance (variable Ta thickness) Hahn et al. (2013) 295 0.53 −ð12/C64Þ STT þSHE ( β-Ta) Liuet al. (2012a) 295 1.5/C60.5 0.5 −ð3/C61Þ SP (β-Ta) Gómez et al. (2014) W 295 2.1 0.55 −14/C61 SP Wang, Pauyac, and Manchon (2014) 295 0.38/C60.06 −ð33/C66Þ STT þSHE ( β-W, lower in α-WαSH) Paiet al. (2012)TABLE III. ( Continued )Jairo Sinova et al. : Spin Hall effects 1253 Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015However, at finite τs, the STT also acquires a field-like component ( Ralph and Stiles, 2008 ). Experiments in W/Hf/ CoFeB structures confirmed the presence of the SHE-basedmechanism in the observed torques and showed that the SHE- STT can have both anti-damping-like and field-like compo- nents of comparable magnitudes ( Paiet al. , 2014 ). In the commonly studied polycrystalline transition-metal- FM-NM samples, the dependence of the torques on the angle ofthe driving in-plane current also does not provide the directmeans to disentangle the two microscopic origins. The lowest order inversion-asymmetry spin-orbit terms in the Hamiltonian have the Rashba form for which the vector ζis in the plane parallel to the interface and perpendicular to the current,independent of the current direction. The same applies to thespin polarization of the SHE spin current propagating from theNM to the FM. The Mandζfunctional form of the field-like and anti-damping-like SHE-STTs is the same as of the correspond-ing ISGE-SOT components. In the observed lowest order torqueterms in Pt/Co and Ta/CoFeB structures ( Garello et al. ,2 0 1 3 ) the ISGE-based and the SHE-based mechanism remained,therefore, indistinguishable. The simultaneous observation ofhigher order torque terms in these samples pointed to SOTs dueto structural inversion-asymmetry terms beyond the basicRashba model. From the Ta thickness dependence measure- ments in the Ta/CoFeB structure it was concluded that in these samples both the ISGE-based and the SHE-based mechanismscontributed to both the field-like and the anti-damping-liketorques ( Kim et al. , 2013 ). In another experiment, the effective spin-orbit field was found to be diminished with increasing theferromagnetic layer thickness and to persist even with theinsertion of a copper spacer ( Fanet al. , 2013 ), suggesting that the spin torque does not rely in the studied structure on theheavy-NM-FM interface. To separate the two model microscopic origins, experi- ments were performed in epitaxial Fe/(Ga,Mn)As bilayers(Skinner et al. , 2015 ). The structures allowed one to simulta- neously observe ISGE-based and SHE-based torques ofcomparable amplitudes. Designed magnetization-angle andcurrent-angle symmetries of the Fe/(Ga,Mn)As single-crystalstructure with Dresselhaus-like inversion asymmetry allowedone to split the two microscopic origins between the ISGEdominated field-like torque and the SHE dominated anti-damping-like torque. E. Spin Hall angles In this section, we present in Table IIIexperimental measurements of the SHE in different materials. The list, insuch an active and evolving field, is by no means exhaustive. As discussed in this experimental section, as more things are learned about the techniques and systematic errors are better understood and corrected, the measurements begin to con- verge for several materials, particularly for the transitionmetals. In Table IIIwe show the material, the temperature the measurement was taken at, the spin-diffusion length either measured or used in the analysis, the conductivity, the spinHall angle, the reference of the work, and the type of technique as well as relevant comments.V. FUTURE DIRECTIONS AND REMAINING CHALLENGES We conclude this review with our personal view of possible interesting directions and remaining challenges. As such, reflects our own preference and intuition. We apologize for any omissions of the many interesting possibilities that others may consider. We only know for certain that such future outlook is bound to always fail in a field that continues to bring surprises. Transition metals have traditionally played a dominant role in spintronics both in basic research and, in particular, in applications. It is therefore not surprising that the SHE field has gained new momentum when bringing nonmagnetic transition metals in the game. And they have played their role particularly well. When brought out of equilibrium by an applied electric field, the SHE (or ISGE) in some nonmagnetictransition metals can generate sufficient flux of spin angular momentum to reorient magnetization in an adjacent transition metal FM. Entirely new concepts for writing information in magnetic tunnel junctions or domain-wall based spintronic devices have emerged from this discovered large strength of the SHE in this common, and technologically relevant, family of materials. Ta, W, Ir, and Pt are examples among the nonmagnetic transition metals with large SHE. The strength of the effect is derived from the large spin-orbit coupling in these heavy elements. Apart from the new opportunities for applications, this brings also new challenges for the basic research of the SHE in transition metals. We have mentioned in the review the pitfalls in attempting to microscopically describe the SHE in structures comprising heavy transition metals from theories of spin transport in weakly spin-orbit-coupled systems. Theproper description and microscopic understanding of the SHE structures in the strong spin-orbit coupling regime is among the key remaining challenges in the SHE field. The flurry of recent SHE studies in transition metals may give an impression that the field is forgetting its semicon- ductor roots. Robust FMs are typically dense-moment systems and their switching requires comparably large electricalcurrent densities generating the SHE spin current. Highly conductive transition metals are clearly favorable from this perspective when compared to semiconductors. Moreover, the reported spin Hall angles in semiconductors do not reach the record values in transition metals. We nevertheless foresee semiconductors playing a vital role in future SHE research, in particular, when consideringspintronics concepts without FMs. In the transition metal context, the SHE is used as an efficient spin-current generator or detector but these studies rarely consider spin manipulation in the nonmagnetic SHE system. Especially in the strongly spin-orbit-coupled heavy metals, the spin-diffusion length is of the order of nanometers, too short for implementing any spin manipulation tools along the nonmagnetic transport channel. For semiconductors, on the other hand, we have mentioned in the review several examples of electricalmanipulation of the output SHE signal. A gate electrode can be used to control coherent spin precession along the channel, additional drift current was shown to modify the spin-current profile along the channel, and nonlinear1254 Jairo Sinova et al. : Spin Hall effects Rev. Mod. Phys., Vol. 87, No. 4, October –December 2015intervalley transport can strongly enhance the spin Hall angles bringing the values close to their heavy metal counterparts. The physics is, however, no different in principle between metals and semiconductors. Large SHE requires large spin-orbit coupling which, on the other hand, tends to suppress spin coherence or diffusion length. Semiconductors with their simpler electronic structure and model spin-orbit fields offer unique ways how to get around this problem. As demon- strated, a proper tuning of the Rashba and Dresselhaus spin-orbit fields can significantly enhance spin coherence in the presence of strong spin-orbit coupling. Experiments outside the SHE field have recently made major progress in control-ling these two canonical spin-orbit fields in common semi- conductor structures and we envisage new developments in semiconductor SHE devices utilizing the coherent spin- manipulation techniques. Combining optical selection rules with SHE makes semiconductors also favorable materials for exploring new concepts in optospintronics. These may include optical spin-torque structures, electrical polarimeters, spin- photovoltaic cells, switches, invertors, and interconnects. The optospintronic subfield of the SHE research is still atits infancy and we expect growing activity in this direction in the future. The fascinating feature of the SHE is that it can generate a large spin current, and a resulting large spin accumulation by bringing weakly out of equilibrium a nonmagnetic system. Itis, therefore, natural that nonmagnetic materials have been traditionally at the center of the SHE research. However, limiting ourselves to paramagnetic or diamagnetic materials,whether metallic or semiconducting, is not necessary when considering the spin Hall phenomena. Recently, several transition-metal FMs and antiferromagnets were demon- strated to act as efficient ISHE spin-current detectors, which opens a new broad area of future materials research inthe SHE. It also brings us back to the opening paragraphs of this section, where we mentioned SHE-induced spin torques in NM-FM heterostructures. Since in strongly spin-orbit-coupled systems these torques are limited to a few atomic layersaround the NM-FM interface, and considering the likely material intermixing at the interface, it is not meaningful to speak strictly about a nonmagnetic layer SHE in these structures. The difference then becomes blurred whether including magnetism via intermixing or proximity polariza-tion at heterointerfaces, or directly considering the SHE in bulk FM or antiferromagnetic materials. Within this notion, an important challenge arises not only for the normal-metal – magnet interfaces but also for monolayer magnets to identify the microscopic origin of the observed spin torques. It remains an open question whether the current-induced torques in the magnet are better linked to a SHE-induced spin-current origin or to one of the variants of the ISGE-induced nonequilibriumspin polarization. Resolving these contributions is an impor- tant academic exercise with potentially large implications for the utility of these spin-orbit-coupling phenomena in spin- tronic information technologies. We conclude by emphasizing that the field of SHE does not live in a vacuum. Its interconnections to other emerging fields, e.g., graphene and other 2D systems, topological insulators,and spin caloritronics, make its growth and possibilities very difficult to predict. Many things that we have discussed hereand that have emerged from its link to these fields were not known or expected a few years ago. It is a rapidly evolving field that produces discoveries at a neck-breaking speed andwe all look forward to its exciting future. LIST OF SYMBOLS AND ABBREVIATIONS 2DEG Two-dimensional electron gas 2DHG Two-dimensional hole gas ac Alternating current AHE Anomalous Hall effect AMR Anistropic magnetoresistance CPW Coplanar wave guide dc Direct current FM Ferromagnet FMR Ferromagnetic resonance HE Hall effect ISGE Inverse spin galvanic effect MOD Modulation of damping MRAM Magnetic random access memory NM Nonmagnetic material QHE Quantum Hall effect QSHE Quantum spin Hall effect rf Radio frequency SGE Spin galvanic effectSHE Spin Hall effect SOT Spin-orbit torque SP Spin pumping STT Spin-transfer torque TMR Tunneling magnetoresistance ACKNOWLEDGMENTS We thank all colleagues who have given us the permission to show their results in this review. We also thank all ourcolleagues within the spintronics community that engaged usin many fruitful interactions and spirited discussions. J. S. acknowledges partial support by the Alexander von Humboldt Foundation, the European Research Council (ERC) SynergyGrant No. 610115, and the German Research Foundation(DFG) through Program No. SPP 1538. S. O. V. acknowl-edges partial support from the European Research Councilunder Grant Agreement No. 308023 SPINBOUND and fromthe Spanish Ministry of Economy and Competitiveness (MINECO) under Contract No. MAT2013-46785-P and Severo Ochoa Program No. SEV-2013-0295. J. W. acknowl-edges partial support from the European Metrology ResearchProgramme Joint Research Project (EMRP JRP) IND08MetMags and the ERC Synergy Grant No. 610115. C. B.acknowledges partial support from the DFG throughPrograms No. SFB 689 and No. SPP 1538 and from theEuropean Research Council (ERC) through starting Grant No. 280048 ECOMAGICS. T. 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PhysRevB.103.184403.pdf	PHYSICAL REVIEW B 103, 184403 (2021) Spin-wave wells revisited: From wavelength conversion and Möbius modes to magnon valleytronics K. G. Fripp and V. V. Kruglyak* University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom (Received 11 January 2021; accepted 20 April 2021; published 3 May 2021) We have used micromagnetic simulations to model backward-volume dipole-exchange spin waves in graded proﬁles of the bias magnetic ﬁeld. We demonstrate spin-wave wavelength conversion upon the wave’s reﬂectionfrom turning points due to the two characteristic minima (“ Upoints”) occurring in their dispersion at ﬁnite (nonzero) wave vectors. As a result, backward-volume dipole-exchange spin waves conﬁned in “spin-wavewells” either form Möbius modes, making multiple real-space turns for each reciprocal-space round trip, or splitinto pairs of degenerate modes in the valleys near the two Upoints. The latter modes may therefore be assigned a pseudospin. We show that the pseudospin can be switched by scattering the spin wave from decreases of the biasmagnetic ﬁeld, while it is immune to scattering from ﬁeld increases. Pseudospin creation and read-out can beaccomplished using chiral spin-wave transducers, as described in Au et al. [Appl. Phys. Lett. 100, 182404 (2012)] and Au et al. [Appl. Phys. Lett. 100, 172408 (2012)] , respectively. Taken together, the possibility of pseudospin creation, manipulation, and read-out suggests a path to development of a spin-wave version of valleytronics(“magnon valleytronics”), in which the pseudospin (rather than amplitude or phase) of spin waves would be usedto encode data. Our results are not limited to graded bias magnetic ﬁeld but can be generalized to other magnonicmedia with spatially varying characteristics, produced using the toolbox of graded index magnonics. DOI: 10.1103/PhysRevB.103.184403 I. INTRODUCTION AND BACKGROUND The phenomena of wave propagation, scattering, and quan- tization play the pivotal role in both classical [ 1] and quantum [2] physics. In the one-dimensional (1D) case, the quanti- zation is observed when a wave is conﬁned between twobarriers, causing the wave to be backscattered. When thebarriers are smoothly graded rather than sharp (i.e., haveproperties slowly rather than abruptly varying in space), thewave does not “feel” the magnetic gradients until it reachesthe turning points, which causes its backscattering. Beyondthe turning points, the wave’s propagation is forbidden (forthe given frequency): The wave number turns imaginary, andso, the wave decays exponentially in space. At the turningpoint itself, the wave number is (usually) equal to zero, whichcorresponds to the “ /Gamma1point” in the reciprocal space. Conse- quently, the wave numbers of the incident and backscatteredwaves have opposite signs. This typical scenario is realized,e.g., for light, sound, and quantum-mechanical particles. Very different behavior is expected for waves whose dis- persion relation has frequency minima at ﬁnite (nonzero)wave vectors. Such a dispersion relation is found, e.g., forrotons in liquid helium [ 3–6], for electrons in graphene near the Fermi level [ 7–10], and for dipole-exchange spin waves—wavelike excitations of the magnetization in mag-netically ordered materials [ 11–13]—in the backward-volume dipole-exchange spin-wave (BVDESW) geometry [ 12,14]. The propagation, backscattering, and conﬁnement of this lat-ter type of spin wave is studied theoretically in this paper. *V.V.Kruglyak@exeter.ac.ukTheir dispersion, plotted in Fig. 1, can have two “valleys” around frequency minima at ﬁnite wave numbers (whichwe call “ Upoints”). Importantly, such spin waves can be not only modeled numerically (as we do here) but also de-tected and imaged experimentally [ 15,16]. This promises a direct experimental veriﬁcation of the effects predictedin this paper—backscattering-induced wavelength conversionand Möbius-like conﬁnement—and gives credibility to theproposed development of a spin-wave version of valleytronics(“magnon valleytronics”). As an illustration, let us consider a BVDESW wave packet with central frequency f. The wave packet propagates in nonuniform bias magnetic ﬁeld H(x) that varies smoothly with coordinate x. In the commonly used Wentzel-Kramers- Brillouin (WKB) approximation [ 2], applied to spin waves, e.g., in Refs. [ 17–26], it is assumed that the wave’s dispersion relation, which may be written for BVDESWs as [ 14] F(k,H) =μ 0γ/radicalBigg/parenleftbigg H+2A μ0Mk2/parenrightbigg/bracketleftbigg H+2A μ0Mk2+M1−exp(−kd) kd/bracketrightbigg , (1) may still be introduced and is satisﬁed for each value of x. Then, the packet’s central wave number k(x) follows the ﬁeld variation adiabatically, while the value of fis conserved [ 27]: f=F[k(x,f),H(x)]=const. (2) The discrete frequencies fnof modes conﬁned in regions of reduced magnetic ﬁeld can be calculated using the Bohr- 2469-9950/2021/103(18)/184403(9) 184403-1 ©2021 American Physical SocietyK. G. FRIPP AND V. V. KRUGLYAK PHYSICAL REVIEW B 103, 184403 (2021) Sommerfeld quantization formula [ 2], /contintegraldisplay k(x,fn)dx =2πn, (3) where n=1,2,3,···, and k(x,fn) is found from Eq. ( 2). The integration is performed along the wave packet’s closedtrajectory, yielding the area enclosed by the trajectory in thephase space. Usually, Eq. ( 3) is equivalent to /integraldisplay x1 x0k(x)dx =πn, (4) where x0and x1are the coordinates of the two turning points between which the wave is conﬁned. In Refs. [ 19,20,22– 24,26], Eq. ( 4) was also applied to the analysis of the quan- tization of BVDESWs. The spin waves were conﬁned in thedemagnetized regions (“spin-wave wells”) near the edges ofpatterned magnetic elements (e.g., stripes) orthogonal to thedirection of the applied magnetic ﬁeld. However, the useof Eq. ( 4) instead of Eq. ( 3) could only be justiﬁed if the wavelength of the spin waves was preserved upon backscat-tering from the turning points, which was not veriﬁed in Refs.[19,20–24,26] and is therefore done here. The paper is organized as follows. Section IIis devoted to the methods used in our numerical micromagnetic simulationsand data analyses. In Sec. III, we describe our results, focusing on the BVDESW conﬁnement and wavelength conversionin different ﬁeld proﬁles, and on the concepts of BVDESWpseudospin and magnon valleytronics. Section IVcontains a qualitative discussion of the implications of our ﬁndings forthe use of the Bohr-Sommerfeld quantization formulas forBVDESWs. In Sec. V, we summarize the implications of our ﬁndings to various other research directions in magnonics. II. METHODS We simulate the scattering and conﬁnement of BVDESW wave packets in 1D proﬁles of the bias magnetic ﬁeld us-ing the MUMAX 3 software [ 28]. The magnonic medium has permalloylike properties: the saturation magnetization of 800kA/m, the exchange stiffness of 13 pJ/m, and zero magne-tocrystalline anisotropy. To make the wave effects clearer,we assume the Gilbert damping parameter to have a verylow value of 0.0001, inherent to high-quality ferrites ratherthan permalloy. The samples have dimensions of l×10 nm × 80 nm ( x×y×z), where lis typically in the range of sev- eral tens of micrometers. One-dimensional periodic boundaryconditions are used in the ydirection, to remove any in-plane shape anisotropy. The simulation volume is discretized intocuboidal cells each dimension of which is 5 nm or smaller.The bias magnetic ﬁeld is applied in the xdirection, with the equilibrium magnetization conﬁguration established prior todynamical simulations. All dynamic magnetic ﬁelds in our simulations here are uniform in the yand zdirections. To calculate the spin-wave dispersion (Fig. 1), a uniform ﬁlm in a uniform bias mag- netic ﬁeld is excited by a dynamic magnetic ﬁeld the proﬁleof which is given by the sinc function both in time (cutofffrequency 25 GHz) and along the xdirection (cutoff wave number π/20 rad/nm) [ 29]. To excite BVDESW wave packets, FIG. 1. The lines show the dispersion relation of BVDESWs plotted using Eq. ( 1) for the indicated values of the bias magnetic ﬁeld. The curve for 2 mT exempliﬁes the dispersion relation of “conventional waves,” which has the frequency minimum at the /Gamma1point. The background shows the dispersion for the ﬁeld of 140.5 mT obtained by Fourier-transforming results of the micromag- netic simulations (brighter color corresponds to greater spin-wave amplitude). we use dynamic magnetic ﬁelds with running-wave proﬁles [30,31]: h=h0exp/bracketleftbigg −(x−x0)2 2σ2x/bracketrightbigg exp/bracketleftbigg −(t−t0)2 2σ2 t/bracketrightbigg ×cos( ±k0x−2πft). (5) Here, h0/lessorequalslant0.1 mT is the maximum transient ﬁeld ampli- tude, fis the central frequency of the wave packet, and k0is the corresponding wave number, obtained from the prelimi-narily computed dispersion relation, such as the one shown inFig.1. The wave packets are excited centered at position x 0 at time t0relative to the beginning of the simulation, to suit the goal of each particular simulation. The size and durationof the spatial and temporal envelopes of the ﬁeld are deﬁnedby parameters σ xandσt, respectively. Their values are chosen so as to excite a small band of frequencies and wave numbersaround the desired central values while also ensuring that thepacket ﬁts and has room for movement within the well formedby the ﬁeld proﬁle. The spin-wave well sizes of about 10–20μm considered here represent a compromise of this sort. Very similar results are obtained for larger-scale ﬁeld proﬁles.However, it becomes impossible to form wave packets withwell-deﬁned trajectories in wells with sizes of about 2 μm or smaller, which will be addressed elsewhere. To obtain thespin-wave dispersion relations and wave-packet trajectories,we employ standard Fourier transform techniques, described,e.g., in Refs. [ 29–31]. 184403-2SPIN-WA VE WELLS REVISITED: FROM WA VELENGTH … PHYSICAL REVIEW B 103, 184403 (2021) FIG. 2. The wave-packet trajectories (top row) and correspond- ing magnetic ﬁeld proﬁles (bottom row) are shown in the real (left column) and reciprocal (right column) space for the centralfrequency of 8.4 GHz. The brighter color corresponds to greater spin-wave amplitude. The dashed lines and points at their crossings show the correspondence between features from the different panels. See the Supplemental Material [ 33] for an animated version of this ﬁgure. III. RESULTS A. Backward-volume dipole-exchange spin-wave (BVDESW) dispersion Figure 1compares the BVDESW dispersion relation com- puted from the results of our micromagnetic simulations withthe analytical curve plotted using Eq. ( 1), for a uniform bias ﬁeld of 0.1405 T. The numerical and analytical results agreewell at small wave numbers. The discrepancy developing forshorter wavelengths may be attributed to the approximatecharacter of either Eq. ( 1)[14], or micromagnetic simulations [29], or both. For 0.1405 T, the two characteristic frequency minima (the Upoints) correspond to the wave numbers of ± 35 rad/ μm and the frequency of 8.4 GHz. As the bias ﬁeld is reduced, the Upoints shift toward the /Gamma1point and eventually disappear. B. Symmetric spin-wave well: Low-frequency conﬁnement The propagation and conﬁnement of BVDESW wave packets differs qualitatively depending on whether the wavepacket’s central frequency is below or above that of the fer-romagnetic resonance (FMR) at the lowest value of the biasmagnetic ﬁeld. Here, we consider the case of “low-frequencyconﬁnement,” i.e., when the central frequency is below theFMR value but is above the U-point frequency at the ﬁeld minimum. The top left panel in Fig. 2shows the real-space trajectory P0-P4 of the envelope of a Gaussian magnonic wavepacket (“wave-packet trajectory”) at the central frequencyof 8.4 GHz conﬁned in a symmetric spin-wave well. Thiswell is created by the parabolic magnetic-ﬁeld proﬁle witha minimum of 0.1 T, shown in the bottom left panel. Thecorresponding reciprocal-space trajectory R0-R4 (top right)reveals that the turning points in real (P1, P3) and recipro-cal (R0, R2, R4) space do not coincide: The wave packetcontinues moving forward in reciprocal (real) space when it is turned back in real (reciprocal) space. This is due to thedifference between the group and phase velocities of spinwaves, apparent from Fig. 1. The character of the trajectories can be explained in greater detail by considering the variationof the packet’s central wave number with the ﬁeld (bottomright). The packet begins its journey from points P0 and R0in the real and reciprocal space, respectively, at the bias ﬁeldof 0.1 T. At this ﬁeld, the frequency of 8.4 GHz correspondsto the wave numbers of ±61.5 rad/ μm (shorter wavelength) and ±4.5 rad/ μm (longer wavelength). We launch the wave packet with the central wave number of −4.5r a d/μm, where the negative sign ensures that the group velocity is positive[32], so the packet propagates from left to right. The ﬁrst real-space turning point (P1) corresponds to the magnetic ﬁeld of about 0.1405 T, at which the Upoint is reached in the reciprocal space at −35 rad/μm (R1). Notably, although the wave packet reverses its direction of travel inthe real space, it continues to propagate in the same directionthrough the Upoint in the reciprocal space. Hence, when the packet returns to the start of the journey in the real space at P2,its central wave number is converted from the original valueof−4.5r a d/μm (R0) to a new value of −61.5r a d/μm( R 2 ) . As the packet propagates further, gets backscattered from theother real-space turning point at P3, and returns to the startagain at P4, its central wave number is converted back to theoriginal value of −4.5r a d/μm (R4). Importantly, the wave number never switches its sign, so that the packet remainsin the same valley, near the same Upoint in the dispersion relation. This is because the frequency of 8.4 GHz is below theFMR and above the U-point frequencies at the ﬁeld of 0.1 T corresponding to the bottom of the spin-wave well. As a result,conﬁned modes belonging to the same valley have “traveling”proﬁles, as discussed, e.g., in relation to dipole-exchange spinwaves in obliquely magnetized ﬁlms [ 34,35]. C. Symmetric spin-wave well: High-frequency conﬁnement and Möbius modes Figure 3shows wave-packet trajectories for the case when its central frequency of 10.5 GHz exceeds the FMR value atthe bottom of the well. Then, the wave packet is not restrictedto any valley. To ensure conﬁnement within a real-space re-gion similar to that in Fig. 2, the parabolic spin-wave well has steeper walls. Also, for the sake of illustration, we launch awave packet of short, exchange dominated spin waves: i.e., awave packet with a wave number of 82 rad/ μm( R 0 )i se x - cited at point P0. The shape of the reciprocal-space trajectoryR0-R8 is qualitatively similar to that in Fig. 2. However, the real-space trajectory P0–P8 changes drastically. Upon the ﬁrstbackscattering from the “high-ﬁeld” turning point at P1, thewave packet propagates through the Upoint (R1); its wave- length continues to increase as the magnetic ﬁeld decreases;and once the /Gamma1point is reached (R2), the wave packet is backscattered at P2 [ 36–41]. The latter “low-ﬁeld” turning point occurs even before the wave packet reaches the bottomof the spin-wave well in the real space. In the reciprocal space,unlike transitions through the Upoints at high-ﬁeld values, this transition through the /Gamma1point leads to switching of the valley pseudospin. When the central frequency exceeds the 184403-3K. G. FRIPP AND V. V. KRUGLYAK PHYSICAL REVIEW B 103, 184403 (2021) FIG. 3. The wave-packet trajectories (top row) and correspond- ing magnetic ﬁeld proﬁles (bottom row) are shown in the real (left column) and reciprocal (right column) space for the centralfrequency of 10.5 GHz. The brighter color corresponds to greater spin-wave amplitude. The dashed lines and points at their crossings show the correspondence between features from the different panels.See the Supplemental Material [ 33] for an animated version of this ﬁgure. FMR value by a smaller amount, a fraction of the wave packet tunnel through the bottom of the well [ 36–41] (not shown). Upon another backscattering from the same high-ﬁeld turningpoint at P3, the wave packet propagates through the other U point (R3), and its wavelength continues to decrease (ratherthan to increase) as the magnetic ﬁeld decreases. Therefore, in contrast to its long-wavelength counterpart (P2, R2), theshort-wavelength packet propagates through the bottom ofthe well without any scattering (P4). As the ﬁeld begins toincrease, the wavelength also begins to increase (R4), and thewhole scenario repeats itself in the left half of the spin-wavewell (P4–P8, R4–R8). Overall, the wave packet changes itsdirection of travel in the real space six times (P1, P2, P3, P5,P6, and P7) during just a single round trip in the reciprocalspace. In Ref. [ 42], nonlinear spin waves needing to make two real-space round trips to meet the initial phase condition weretagged “Möbius solitons.” The behavior of linear conﬁnedmodes exempliﬁed in Fig. 3is more complex than that in Ref. [ 42]. However, this behavior justiﬁes such spin waves to be called “Möbius modes.” D. Möbius modes in an asymmetric spin-wave well Figure 4presents BVDESW wave-packet trajectories in the real (P0–P9, top left) and reciprocal (R0–R9, top right) spaceobtained from simulations with an antisymmetric, slopy mag-netic ﬁeld proﬁle (bottom left). This proﬁle is qualitativelysimilar to the one from Refs. [ 19–24]: the wave packet is conﬁned by the ﬁeld increase and the physical edge of themagnetic sample on the right- and left-hand sides, respec-tively. The region of nearly constant ﬁeld around x=0i s introduced to simplify the excitation of regular Gaussian wavepackets. The wave packet has a central frequency of 8.3 GHzand central wave number of −3r a d/μm at the bias ﬁeld of 0.1 T (P0, R0). In the real space, the wave packet propagates FIG. 4. The wave-packet trajectories (top row) and corresponding magnetic ﬁeld proﬁles (bottom row) are shown in the real (left column) and reciprocal (right column) space for the frequency of 8.3 GHz. The brighter color corresponds to greater spin-wave amplitude. The dashed lines and points at their crossings show the correspondence between features from the different panels. See the Supplemental Material [ 33]f o r an animated version of this ﬁgure. 184403-4SPIN-WA VE WELLS REVISITED: FROM WA VELENGTH … PHYSICAL REVIEW B 103, 184403 (2021) to the right, where it encounters the parabolic ﬁeld increase and is eventually backscattered (P1). In the reciprocal space,the packet continues its journey in the same direction throughpoint R1 and then R2, so that it returns to its real-space startposition (P2) with a much-shortened wavelength. Then, itbounces back from the physical edge of the sample (pointP3/P4), changing the sign but retaining the absolute valueof the central wave number, which corresponds to a jumpfrom R3 to R4 in the reciprocal space. This jump is followedby a one-directional evolution of the wave number (R4–R9).During this, the wave packet experiences two real-space re-ﬂections: from the high-ﬁeld (P6) and low-ﬁeld (P8) turningpoints, corresponding to a Upoint (R6) and the /Gamma1point (R8), respectively. Overall, the wave packet changes its real-spacedirection of travel four times (P1, P3/P4, P6, and P8) during asingle round trip in the reciprocal space. E. Magnon valleytronics The results shown in Fig. 2suggest that BVDESWs with frequencies below the FMR and above the U-point fre- quencies could be used to develop a spin-wave version of“valleytronics”—a paradigm exploiting the wave’s afﬁliationwith a particular valley as an additional degree of freedom,i.e., as a “pseudospin” [ 7]. Inspecting Fig. 1, we see that the different BVDESW pseudospins can be easily distinguished.Indeed, at each frequency, copropagating modes (i.e., modeswith group velocities of equal sign) of opposite pseudospin(i.e., from the different valleys) have different wavelengths,while counterpropagating modes of the same wavelength haveopposite pseudospins. Also, the pseudospin can be associatedwith the sign of the phase velocity (wave number), whichremains the same (irrespective of the group velocity) for eachvalley mode. Finally, the traveling proﬁles inherent to theconﬁned modes with a particular pseudospin could be inter-preted as spin current [ 34,35]. The wave number locked, chiral resonant microwave to spin-wave transducers from Ref. [ 43] and spin-wave valves (or phase shifters) from Ref. [ 44] could be used to create and to read out, respectively, magnonic statesdepending on their pseudospin [ 43–45]. To substantiate the claim of valleytronics, one must also demonstrate how the pseudospin can be “switched.” Figure5shows results of simulations in which a BVDESW wave packet is incident on a region of reduced bias magneticﬁeld. Panel (a) shows the ﬁeld proﬁle, while the trajecto-ries from rows (b) and (c) correspond to wave packets ofopposite pseudospin. The long wavelength/negative phase ve-locity wave packet [Fig. 5(b)] partly tunnels through and partly is reﬂected from the ﬁeld decrease, as reported earlier[36–41]. The tunneled part of the wave packet retains its pseudospin. In contrast, the packet’s reﬂected portion goesthrough the /Gamma1point and therefore switches its pseudospin. As an aside, we note that the same pseudospin switching eventoccurs also at points P2/R2 and P6/R6 in Fig. 3and at point P8/R8 in Fig. 4. The short wavelength/positive phase velocity wave packet [Fig. 5(c)] passes through the ﬁeld decrease ballistically, experiencing negligible reﬂection, if any, andretains its pseudospin. Similar simulations for scattering fromregions of increased bias magnetic ﬁeld (not shown) haverevealed that the pseudospin is immune to both backscatter- FIG. 5. The trajectories of a BVDESW wave packet passing through the bias magnetic ﬁeld in (a) are shown in the real (left column) and reciprocal (right column) space for the frequency of 8.4GHz. The brighter color corresponds to greater spin-wave amplitude. The initial values of the central wave number are −6r a d /μma n d 62 rad/ μm in rows (b), (c), respectively. The arrows show corre- spondence between the real-space and reciprocal-space wave-packet envelopes. See the Supplemental Material [ 33] for an animated ver- sion of this ﬁgure. ing from and transmission/tunneling through regions of local enhancement of the bias magnetic ﬁeld. Together, the possibility to create, to switch and to read out the pseudospin constitutes the backbone of the concept ofmagnon valleytronics. However, any more extensive investi-gation and exploitation of the concept goes beyond the scopeof this paper. IV . DISCUSSION The wavelength conversion occurring as a result of backscattering from magnetic-ﬁeld increases mandates theuse of the full version of the Bohr-Sommerfeld formula,Eq. ( 3), when employing the WKB approximation to de- scribe conﬁnement of BVDESWs. Hence, the analyses ofmode conﬁnement in spin-wave wells from Refs. [ 19–26], in which Eq. ( 4) was used, need to be revisited, even if qualita- tively. To illustrate the arising peculiarities, Fig. 6shows the phase-space trajectories corresponding to the three cases ofspin-wave conﬁnement presented in Figs. 2–4, which we dis- cuss in the following. In this discussion, we deliberately avoidusing the just introduced notion of the BVDESW pseudospin:it is too new and could therefore create more confusion ratherthan clarify things. However, we note that the pseudospin isswitched every time a phase-space trajectory in Fig. 6crosses line k=0. We begin from a symmetric spin-wave well, which is relevant, e.g., to the FMR force microscopy (FMRFM) mea-surements reported by Chia et al. in Ref. [ 26]. At each frequency f nconstrained by the Upoint and FMR values 184403-5K. G. FRIPP AND V. V. KRUGLYAK PHYSICAL REVIEW B 103, 184403 (2021) FIG. 6. The wave-packet trajectories for the ﬁeld proﬁle from Fig. 2(curves 1 and 2) and its antisymmetric counterpart (curve 3) are shown in the phase space for the frequencies of 8.4 GHz (curves 1a, 1b, and 3) and 10.5 GHz (curve 2). The packets movecounterclockwise starting from the points indicated by circles. The green dashed arrow shows the reciprocal-space jump of the packet due to its reﬂection from the edge at −10.24 μm. The shaded areas show the difference between the accumulated phase calculated using Eqs. ( 3)a n d( 4). (at the ﬁeld corresponding to the bottom of the well) and satisfying Eq. ( 3), there are two degenerate normal modes: one in each dispersion valley. The modes have oval-shapedphase-space trajectories, labeled “1a” and “1b” in Fig. 6. Each trajectory contains two reﬂections from ﬁeld increasesin the real space. In the reciprocal space, the reﬂections cor-respond to the same Upoint albeit approached from opposite sides. Hence, the wavelength is converted twice and reachesthe same value after one round trip in the well. Then, thequantization condition given by Eq. ( 3) corresponds to the constructive interference of the mode with itself, which occursif the phase accumulated after the round trip is equal to 2 πn. As the frequency reaches the FMR value from below, the twoovals touch at the /Gamma1point; at frequencies above the FMR value, only one mode, with an hourglasslike trajectory, labeled“2” in Fig. 6, remains. The trajectories at any frequency have a symmetry axis at x=0. At frequencies above the FMR value, line k=0 is also a symmetry axis. The integration in the full Bohr-Sommerfeld formula [Eq. ( 3)] yields the accumulated phase as the area enclosed by a phase-space trajectory (Fig. 6). At the same time, the shortcut given by Eq. ( 4), a variant of which was used by Chia et al. [26], would additionally include the shaded areas. Hence, the use of Eq. ( 4) leads to an overestimation of the accumulated phase and therefore to an underestimation ofthe mode frequencies. This is likely to be one of the factorscontributing to the failure of the WKB approximation to ex-plain the FMRFM results in Ref. [ 26], with the conﬁnement in the orthogonal in-plane direction and the intermodal couplingbeing others [ 26,46]. For the antisymmetric well [ 19–24], the phase-space tra- jectory, labeled “3” in Fig. 6, has the only symmetry axis at k=0. During the ﬁrst real-space round trip across the well, the wave is reﬂected once from the high-ﬁeld turning point,which corresponds to the negative Upoint in the recipro- cal space, and once from the sample edge, which induces areciprocal-space jump from −kto+k. Therefore, the wave- length conversion occurs only once as a result of this roundtrip. Hence, the two copropagating waves at x=0h a v ed i f - ferent wavelengths and cannot amplify or cancel each other,regardless of their relative phase. The interference is reenabledafter the second real-space round trip, which adds anotherwavelength converting reﬂection from the high-ﬁeld turningpoint (now corresponding to the other, positive Upoint) and a wavelength preserving reﬂection from the ﬁeld decrease(corresponding to the /Gamma1point in the reciprocal space). As for the symmetric well, the accumulated phase yielded by thefull Bohr-Sommerfeld formula [Eq. ( 3)] is equal to the area enclosed by the trajectory but is overestimated by Eq. ( 4), used in Refs. [ 19–24], by the amount equal to the shaded area. This could explain the need for additional phase jumps introduced but not explained in Ref. [ 23]. It is also clear that non-WKB-based analytical models put forward, e.g., inRefs. [ 24,46–49] to describe spin-wave conﬁnement need to be revised to account for the behavior revealed here. The Möbius-like and valley-conﬁned wave-packet trajec- tories shown in Fig. 6cannot be resolved explicitly for spin-wave wells with sizes below that of the BVDESW wavepacket, although they should continue to have effect on themode quantization and spatial character. We speculate thatone piece of evidence for formation of Möbius and valley-conﬁned modes could be found in the character of theiramplitude and phase proﬁles, respectively. Indeed, the am-plitude proﬁle of a standard standing mode contains nodesbetween antinodes, with nodes separating regions with phasediffering by π. The traveling proﬁles inherent to valley- conﬁned modes should lead to a continuous rather than abruptvariation of phase. The very different wavelengths inherent toMöbius modes during different passages of the same regionsshould lead to a nonmonotonic amplitude variation (withoutnodes) within regions of the same or slowly varying phase.However, these effects may be obscured by the Schlömannemission [ 17] of spin waves from magnonic index nonunifor- mities [ 50] and from sample boundaries [ 51]. Hence, we leave the more detailed analysis of submicrometer spin-wave wellsto future studies. V . CONCLUSIONS AND OUTLOOK We conclude by outlining other research directions that are likely to be affected by our main ﬁndings (the BVDESWwavelength conversion and Möbius mode formation) and bygiving some further remarks on our observations. (i) Our considerations should apply to other magnonic systems, e.g., van der Waals heterostructures discussed in thecontext of magnon valleytronics [ 52–54]; vice versa, some of the effects from Refs. [ 52–54] could be transferred to BVDESWs. (ii) The spin-wave wavelength conversion (albeit probably not the Möbius mode formation) should occur in magnonicsystems featuring a single valley at a ﬁnite wave number,with the shift induced, e.g., by the Dzyaloshinskii-Moriyainteraction (DMI) interaction [ 55] or electric ﬁeld [ 56]. (iii) The magnonic valley modes with different pseu- dospins form two different Bose-Einstein condensates [ 57], which can be spatially separated by a transient graded mag-netic ﬁeld [ 58]. Our results suggest that a local transient ﬁeld 184403-6SPIN-WA VE WELLS REVISITED: FROM WA VELENGTH … PHYSICAL REVIEW B 103, 184403 (2021) decrease could induce mixing of the condensates of opposite pseudospin. (iv) The wave amplitude increase (seen as brightening of the real-space trajectories in Figs. 2–4) near the high-ﬁeld turning points is due to the wave shoaling effect (i.e., energyconcentration due to reduction of the group velocity), as dis-cussed earlier for other types of spin waves [ 59,60]. (v) The real-space relative shift between the trajectories of the incident and tunneled wave packets is a manifestationof the Hartman effect [ 61], demonstrated here for dipole- exchange spin waves. Its discussion for purely exchange spinwaves can be found in Ref. [ 62]. (vi) A promising avenue for further research lies in combining spatial magnetic gradients, e.g., such as those con-sidered here, with magnetic transients of the kinds consideredin Refs. [ 63–66]. (vii) Our simulations are performed for graded 1D pro- ﬁles of the bias magnetic ﬁeld. However, Maxwell equations impose certain limitations on the feasibility of the proﬁle’sexact implementation [ 67]. Moreover, the assumed ﬁeld vari-ation of 0.1 T over 10 μm length scales is also quantitatively challenging to implement, although the strong demagnetizing ﬁelds in patterned magnetic structures [ 19,20] and the stray ﬁeld from the magnetic tip in FMRFM [ 26] suggest realistic ways forward. 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PhysRevApplied.11.044093.pdf	PHYSICAL REVIEW APPLIED 11,044093 (2019) Chaos and Relaxation Oscillations in Spin-Torque Windmill Spiking Oscillators R. Matsumoto,1,S. Lequeux,2,3H. Imamura,1and J. Grollier2 1National Institute of Advanced Industrial Science and Technology (AIST), Spintronics Research Center, Tsukuba, Ibaraki 305-8568, Japan 2Unité Mixte de Physique, CNRS, Thales, Univ. Paris-Sud, Université Paris-Saclay, Palaiseau 91767, France 3Univ. Grenoble Alpes, CEA, CNRS, Grenoble INP, INAC-SPINTEC, 38000 Grenoble, France (Received 3 October 2018; revised manuscript received 27 February 2019; published 30 April 2019) Spintronic neurons that emit sharp voltage spikes are required for the realization of hardware neural networks enabling fast data processing with low power consumption. In many neuroscience and com-puter science models, neurons are abstracted as nonlinear oscillators. Magnetic nano-oscillators called spin-torque nano-oscillators are interesting candidates for imitating neurons at the nanoscale. These oscillators, however, emit sinusoidal waveforms without spiking while biological neurons are relaxationoscillators that emit sharp voltage spikes. Here, we propose a simple way to imitate neuron spiking in high-magnetoresistance nanoscale spin valves, where both magnetic layers are free and thin enough to be switched by spin torque. Our numerical-simulation results show that the windmill motion induced by spintorque in the proposed spintronic relaxation oscillator gives rise to spikes whose shape and frequency, set by the charging and discharging times, can be tuned through the amplitude of injected dc current. We also ﬁnd that these devices can exhibit chaotic oscillations. Chaoticlike neuron dynamics has beenobserved in the brain, and it is desirable in some neuromorphic computing applications, whereas it should be avoided in others. We demonstrate that the degree of chaos can be tuned in a wide range by engineering the magnetic stack and anisotropies and by changing the dc current. The proposed spintronic relaxationoscillator is a promising building block for hardware neuromorphic chips, leveraging nonlinear dynamics for computing. DOI: 10.1103/PhysRevApplied.11.044093 I. INTRODUCTION Neuromorphic chips need several millions of neu- rons to run state-of-the-art neural networks [ 1]. Keeping these chips small therefore requires developing nanoscale artiﬁcial neurons. In many neuroscience and computer science models, neurons are abstracted as nonlinear oscil- lators [ 2–5]. Memristive oscillators (also called neuristors) [6], Josephson junctions [ 7], nanoelectromechanical sys- tems [ 8], and magnetic nano-oscillators called spin-torque nano-oscillators [ 9–11] are interesting candidates for imi- tating neurons at the nanoscale. In particular, it has been shown experimentally that spin-torque nano-oscillators can implement hardware neural networks and perform cognitive tasks with high accuracy due to their large signal- to-noise ratio, their high nonlinearity, and enhanced ability to synchronize [ 12,13]. However, the microwave voltage signals delivered by these spin valves driven by spin torque are typically sinusoidal. In contrast, biological neurons are relaxation rie-matsumoto@aist.go.jposcillators, based on two time scales: a long charging period followed by a short discharge period [ 14,15]. Their output consists of sharp voltage spikes of ﬁxed amplitude with a frequency that depends on the amplitude of the inputs. Therefore, it is interesting to exploit the multifunc- tionality and tunability of spin-torque to imitate the sharp neuron spikes. Here, we propose a simple way to imitate neuron spik- ing in high-magnetoresistance nanoscale spin valves where both magnetic layers are free and thin enough to be switched through spin torque [ 16–18]. We study these devices through macrospin and micromagnetic simula- tions [ 19]. We show that the windmill motion induced by spin torque [ 20] in these structures gives rise to spikes whose shape and frequency, set by the charging and discharging times, can be tuned through the ampli- tude of injected dc current as well as the materials and thicknesses of the ferromagnetic layers. We observe that these devices with many coupled degrees of freedom can exhibit chaotic oscillations. In our paper, “chaos” means the dynamics whose behavior after a long period of time is unpredictable because of the sensitivity to ini- tial conditions, although the behavior is determined by 2331-7019/19/11(4)/044093(10) 044093-1 © 2019 American Physical SocietyMATSUMOTO, LEQUEUX, IMAMURA, and GROLLIER PHYS. REV. APPLIED 11,044093 (2019) scientiﬁc laws or equations [in our case, Eqs. (1)and(2)] [21]. The time evolution of the magnetization direction is deterministic, but it is diﬃcult to predict the direc- tion after a long period of time. Only after the numerical simulations with the equations does the magnetization direction become predictable. On the other hand, the noisy dynamics is not deterministic because of its stochastic behavior. Chaoticlike neuron dynamics has been observed in the brain and dynamics at the edge of chaos has been con- sidered to optimize information processing [ 15,22,23]. In this paper, the term “chaoticlike” dynamics indicates the dynamics at the edge of chaos or at a critical point, i.e., dynamics between ordered dynamics and chaotic dynamics [ 23]. It is predicted by the “criticality hypoth- esis” that neural networks will self-organize to operate near a critical point that is close to the chaotic regime [15,22,23]. This phenomenon is called self-organized crit- icality. Self-organized criticality is often pointed out in network chaos and the degree of chaos in a neural net- work should depend on the degree of chaos in a single neuron. We point out that the dipolar coupling between magnetic layers is the main source of chaos in spin-torque wind- mill relaxation oscillators. We demonstrate that the degree of chaos can be tuned in a wide range by engineering the magnetic stack and anisotropies. The proposed spiking windmill spin-torque relaxation oscillator with control- lable chaos is a promising building block for hardware neuromorphic chips, leveraging nonlinear dynamics for computing. Disordered dynamics is desirable in some neuromorphic computing applications [ 24–31]. For example, in recur- rent neural networks, the information processing capacity is maximized at the edge of chaos [ 24,25]. The disordered spiking oscillators in the scheme of the array enhanced stochastic resonance (AESR) can be useful for faint-signal detection and phase-locked loops [ 27–30]. In this scheme, noises composed of multiple frequencies enhance the abil- ity for the signal detection and transmission [ 29,30]. Its sensitivity is higher in the network where each oscilla- tor is equipped with an individual noise source than in the network where the oscillators are exposed to a com- mon noise source [ 28]. The disordered spiking dynamics we propose can be used as an internal noise source. Also, the sensitivity is maximized by tuning the amount of noise and/or the number of the oscillators [ 27–30]. Even in the case of a single neuron, disordered spiking dynamics can be beneﬁcial to the faithful transmission of high-frequency inputs, although it requires analysis using many cycles of the inputs [ 31]. On the other hand, chaotic and chaoti- clike dynamics should be avoided in other neuromorphic computing applications [ 32]. The above-mentioned facts supports the importance of being able to tune the degree of chaos in a system.II. WINDMILL RELAXATION OSCILLATIONS: PRINCIPLE The structure of the proposed windmill relaxation oscil- lator, illustrated in Fig. 1(a), is a spin valve, consisting of a nonmagnetic spacer layer sandwiched between two ferro- magnetic layers. The spacer layer can be either a metallic layer in giant magnetoresistance devices [ 33,34] or a thin insulating tunnel barrier layer in magnetic tunnel junctions [35–39]. The two magnetizations, m1and m2,h a v ep r e f - erential directions due to magnetic anisotropy. However, contrary to typical spin-valve stacks, both layers are free to switch: neither of them is pinned. In the absence of spin torque, the magnetization directions are either parallel (P) or antiparallel (AP). They can point in-plane (IP) [ 40] or out-of-plane (OOP) [ 41], depending on the dominant source of anisotropy. When a dc current is injected in the spin valve, perpendicularly to the layer planes, the torques on the two magnetizations tend to induce rotations in the same direction, as illustrated in Fig. 1(b). The direction of rotation is set by the sign of the applied dc current. III. MODEL It has been predicted, as well as experimentally observed, that this torque conﬁguration can gener- ate a windmill-like motion of the two magnetizations [16–18,20]. The equations of motion of the magnetizations are given by the Landau-Lifschitz-Gilbert-Slonczewski (LLGS) equation [ 9,10,42]: ∂ˆm1 ∂t=−γˆm1× Heﬀ1+αˆm1×∂ˆm1 ∂t −γτst1ˆm1×(ˆm1×ˆ m2),( 1 ) (a) z xye–J > 0 m2P favoredAP favored m1(b) m2m1 FIG. 1. (a) Schematic of the spin-torque windmill spiking oscil- lator: a spin valve with two free layers. The double-headed arrow in cyan (magenta) represents the magnetization unit vector, m1 (m2), in the equilibrium states. The axis zis parallel to the OOP direction. When the current density ( J) is positive, electrons ( e−) ﬂow from m1to m2. (b) Schematic of the windmill spin-torque conﬁguration of the two magnetizations m1and m2. In the case ofJ>0, m1switches away from m2,a n d m2follows m1as indicated by the arrows. In other words, at J>0, m1favors an antiparallel (AP) conﬁguration and m2favors a parallel (P) conﬁguration, as indicated in (a). 044093-2CHAOS AND RELAXATION OSCILLATIONS. . . PHYS. REV. APPLIED 11,044093 (2019) ∂ˆm2 ∂t=−γˆm2× Heﬀ2+αˆm2×∂ˆm2 ∂t +γτst2ˆm2×(ˆm2×ˆ m1).( 2 ) Here, tandγare the time and the electron gyromagnetic ratio. The second term on the right-hand side of Eqs. (1) and(2)is the damping-torque term, where αis the Gilbert damping constant. In this article, α=0.01 is assumed. Hereafter, iin the subscript represents the quantities of the layer having milayer where i=1o r2 . τst1andτst2 represent the coeﬃcient of the Slonczewski torque: τsti=/planckover2pi1 21 μ0Ms1 diJ \|e\|P.( 3 ) Here, /planckover2pi1is the Dirac constant, μ0is the vacuum permeabil- ity,Msis the saturation magnetization, diis the thickness of layer i,eis the electron charge, Jis the current density, and Pis the spin polarization. In the rest of the paper, we take P=0.6. Heﬀiis the eﬀective ﬁeld, expressed as Heﬀ= Hanis+ Hdip.( 4 ) Hereafter, the layer index, i, is abbreviated. Hanisrepre- sents the anisotropy ﬁeld, expressed as Hanis=2K μ0Msmz.( 5 ) Here, Krepresents the anisotropy constant. In the spin valve shown in Fig. 1(a),K=115 kJ /m3is assumed in m1,a n d K=70 kJ/m3is assumed in m2.Hdiprepresents the dipolar ﬁeld, expressed as Hdip=− Ms/parenleftbig Nxmx,Nymy,Nzmz/parenrightbig .( 6 ) Here, Nx,Ny,a n d Nzare the demagnetization coeﬃcients [43]. IV . RESULTS To highlight the principle of windmill relaxation oscil- lator, we ﬁrst neglect the dipolar-ﬁeld interactions between the two magnetic layers in Fig. 1(a) and consider that they behave as macrospins with uniform magnetizations. Figure 2(a) shows macrospin simulations of magnetic switching in these conditions, for out-of-plane magnetized layers that diﬀer only through their anisotropy constants: Km1=115 kJ /m3and Km2=70 kJ/m3(the other mag- netic parameters are indicated as SV OOP1 in Table I). The windmill motion induces sustained switching of the mag- netizations one after the other at J≥Jth=1.0 MA /cm2, where Jthis the threshold current density for sustained windmill switching. The repeated magnetic switches give rise to changes in the device resistance ( R) through mag- netoresistance (MR) eﬀects: R=(RAP+RP)/2−[(RAP− Resistance ( Ω)(b) 100(a) mz 150200 10 ns Time (ns)m1m2 FIG. 2. (a) Times traces of magnetization switching (macrospin simulations for a spin valve with OOP magnetized layers without dipolar-ﬁeld interaction between the two ferromagnetic layers,SV OOP1) (b) Corresponding resistance time trace. In (a) and (b), Jis positive and the normalized current density is J/Jth=1.5. RP)/2] cos ˆθ12, where RP(RAP) is the resistance in the parallel (antiparallel) conﬁguration and ˆθ12is the angle between m1and m2. Since the injected current is dc, the resulting voltage variations across the spin valve, i.e., V= R×Idc, are proportional to the resistance variations. In this paper, we always consider a MR ratio (RAP−RP)/RP of 100% and RP=100/Omega1, assuming that spin valves are magnetic tunnel junctions. The resistance variations corresponding to the magnetic switches in Fig. 2(a) are plotted in Fig. 2(b). A spiking behavior similar to neu- ron responses is observed. The low-resistance states just after spikes correspond to the refractory periods. At the low-resistance states where R∼RP=100/Omega1,ˆθ12∼0◦.ˆθ12 gives the spin-torque strength, which is proportional to \|sinˆθ12\|. The spin-torque strength is zero at ˆθ12=0◦and 180◦. Thus, at ˆθ12∼0◦, magnetizations hardly move. The time scales of these relaxation oscillations are set by the switching times of the two layers. Here, the long charging period corresponds to the switching of m1and the short discharge period to the switching of m2.A t\|J\|<\|Jth\|, on the other hand, no spike appears in the resistance time trace. The asymmetry of the switching times comes from the diﬀerent anisotropy constants of the layers used in the sim- ulations ( Km1=115 kJ /m3and Km2=70 kJ/m3). Indeed, the magnetization switching time TSWunder spin torque is 044093-3MATSUMOTO, LEQUEUX, IMAMURA, and GROLLIER PHYS. REV. APPLIED 11,044093 (2019) TABLE I. Structures under study (SV OOP1,S V OOP2,S V IP1,S V IP2,a n dS V IP3); simulation method; and threshold current density ( Jth). mA1,m 1 ,m 2a n dm A2represent the magnetic layers which have magnetization mA1,m1,m2and mA2, respectively. OOP1, OOP2, IP1, IP2 are the labels of magnetic layers whose parameters are summarized in Table II. In all structures, the spacer layer between m1 and m2 has the thickness of 1 nm and its JRKKY=0. In (d) SV IP2, the spacer layer between m1 and m A1(m2 and m A2) has the thickness of 0.7 nm and its JRKKY=−0.1 mJ /m2. Structure (a) SV OOP1 (b) SV OOP2 (c) SV IP1 (d) SV IP2 (e) SV IP3 mA1 ··· ··· ··· IP1 ··· m1 OOP1 OOP1 IP1 IP1 IP1 m2 OOP2 OOP2 IP2 IP2 IP2m A2 ··· ··· ··· IP2 ··· Simulation method MacrospinaMicromagneticsbMacrospinaMicromagneticsbMicromagneticsb Jth(>0)(MA/cm2)1 . 0 1 . 1 ···c21 ···c Jth(<0)(MA/cm2) −1.0 −1.1 −6.0 −23 −34 aMacrospin-model simulations were conducted without dipolar coupling. bMicromagnetic simulations were conducted with dipolar coupling. cIn (c) SV IP1and (e) SV IP3, positive current induces continuous spin-torque oscillations of m1and m2, which does not result in a spiking time trace of resistance. proportional to 1 /(J−J(0) th)[44], where J(0) this the indi- vidual threshold current density for switching [ 17]. Layers with higher anisotropy Kare more diﬃcult to switch and have a larger threshold current density J(0) th. In our case, we ﬁnd through simulations that J(0) th1and J(0) th2are respectively e q u a lt o0 . 9 5M A /cm2a n d0 . 4 9M A /cm2, where m2(m1) is ﬁxed at the equilibrium state during the evaluation of J(0) th1(J(0) th2). The switching times during the windmill motion for the two magnetic layers as a function of current density are plotted in Fig. 3(a)(solid curves), together with the cor- responding ﬁts in TSWi=ci/(J−J/prime thi)(dotted curve and dotted-dashed curve). Here, ciand J/prime thiare ﬁtting param- eters. The agreement between the analytical prediction of Ref. [ 44] and our simulations is excellent. The ﬁt- ting yields J/prime th1=0.954±0.002 MA /cm2,c1=4.088± 0.215 ns MA /cm2,J/prime th2=0.471±0.015 MA /cm2,a n d c2=1.305±0.031 ns MA /cm2. The threshold currents extracted from the switching times J/prime th1(J/prime th2) agree wellwith the previously determined threshold currents J(0) th1 (J(0) th2). These results show that the response of the wind- mill relaxation oscillator can be tuned by dc current. Tracesat diﬀerent dc current densities are shown in Fig. 3(b), and the evolution of the frequency as a function of cur-rent is plotted in Fig. 3(c). As determined experimentally and numerically in previous studies [ 16–18], the frequency increases with an increase of \|J\|. Note that the shape of spikes can also be tuned by controlling the switching timeratio through materials engineering of the two layers ( M s, P, etc.). Figure 3(b) shows reversed spikes at negative cur- rent density. This feature can allow coupling relaxation oscillators in both an excitatory and inhibitory way using adder circuits for the summation of input signals. Another way of excitatory and inhibitory coupling is to use synap- tic connections. By adjusting the resistance of a memristive device that corresponds to synaptic weights, the coupling of spiking oscillators can be tuned [ 45,46]. (c) Current density (MA/cm2)Resistance ( Ω)(b) 100 Ω –1.75 MA/cm2–1.25 MA/cm2+1.25 MA/cm2+1.75 MA/cm2 5 ns Current density (MA/cm2)(a) TSW1 (ns), TSW2 × 10 (ns)m1 m2 FittingFitting Time (ns) Frequency (GHz) FIG. 3. (a) Average switching times ( TSW1and TSW2)f o r m1(solid triangles on solid curve) and m2(solid circles on solid curve) as a function of current density, with ﬁts by ci/(J−J/prime thi)(dotted curve and dotted-dashed curve). (b) Resistance time traces at J= +1.75 MA /cm2(thick solid curve), +1.25 MA /cm2(thin solid curve), −1.25 MA /cm2(thin dotted curve), −1.75 MA /cm2(thick dotted curve). (c) Frequency as a function of current density for negative and positive current densities. 044093-4CHAOS AND RELAXATION OSCILLATIONS. . . PHYS. REV. APPLIED 11,044093 (2019) 10 ns(b) Resistance ( Ω) 5 ns(a) Resistance ( Ω) Time (ns) Time (ns) FIG. 4. Resistance time traces for (a) in-plane and (b) out-of- plane magnetized layers at negative Jwith J/Jth=1.5. V . OCCURRENCE OF CHAOS Figure 4compares resistance versus time traces sim- ulated through macrospin equations of motion for in- plane [Fig. 4(a)] and out-of-plane magnetized spin valves [Fig. 4(b)] [the structure of the in-plane magnetized spin valve, SV IP1, and its parameters are shown in Fig. 6(c) and Tables Iand II]. As can be seen, the trace in the out-of-plane case is highly regular, whereas ﬂuctuations aﬀect the periodicity of switching in the in-plane case, even if temperature-induced ﬂuctuations are not included in the simulations. The reduced periodicity, that is, the presence of chaos in SV IP1, will be more clearly seen in the quantitative analyses shown in Table IIIof Sec. VI. This chaotic switching of in-plane spin valves [ 47] under windmill motion can be interpreted in the following way. For windmill motion, the switching of one layer tog-gles the switching of the other. Indeed, magnetization m 1 wants to achieve the AP conﬁguration, whereas m2wants to maintain a P conﬁguration (and inversely for a reversed sign of the current density); therefore, the P and AP con- ﬁgurations become consecutively unstable. However, the switching trajectories are very diﬀerent for in-plane and out-of-plane magnetized samples. As shown in Fig. 5(a), for in-plane magnetized samples, the strong anisotropy distorts the trajectories in a clamshell shape. During the dynamics, the spin torque competes with the damping torque, which makes magnetizations relax to the direction of Heﬀwhile dissipating magnetic energy. In in-plane magnetized samples, the damping torque makes the magnetization relax to (mx,my,mz)=(±1, 0, 0) when spin torque is zero. Larger spin torque makes the clamshell trajectory larger, but the energy dissipation by the damping TABLE II. Parameters of magnetic layers: OOP1, OOP2, IP1, and IP2. Sis the area of the base. Magnetic layer OOP1 OOP2 IP1 IP2 S(nm2)1 6 ×16×π 16×16×π 30×10×π 30×10×π d( n m ) 111 0 . 5 Ms(kA/m) 200 200 1300 1300 K(kA/m3) 115 70 0 0(b) (a) FIG. 5. Sketch of magnetization orbits of m1for (a) in-plane and (b) out-of-plane magnetized layers at negative Jth. torque becomes larger at a larger trajectory [ 42]. When the trajectory reaches the bifurcation points at (mx,my,mz)=(0, ±1, 0), the magnetization can switch. Let us consider the situation where one of the magne- tizations, m2, is close to equilibrium and the other one, m1, is switching toward m2. The switching of m1from one hemisphere to the other is strongly determined by the exact magnetization dynamics in the narrow window highlighted in Fig. 5(a). In this dotted window, the angle between magnetizations ˆθ12that gives the torque strength is also strongly varying. The spin-torque strength is pro- portional to \|sinˆθ12\|. It is zero at ˆθ12=0◦and 180◦and maximized at ˆθ12=90◦. The closer to 90◦ˆθ12is, the eas- ier becomes the switching of m1from the dotted window. Therefore, small variations in the direction of m2strongly inﬂuence the switching of m1. This high coupling between degrees of freedom induces a high sensitivity of magne- tization reversal to initial conditions and can favor the appearance of chaos. The situation is diﬀerent for out-of-plane magnetized samples. Here, the competition between spin torque and damping torque is most severe at (mx,my,mz)=(0, 0,±1) [48]. Once m1deviates from (0, 0, ±1), precessions of m1 remain mostly circular during the whole switching of m1 [Fig. 5(b)]. Therefore, precessions are much less sensitive to ﬂuctuations of m2. Until now, we have not included the dipolar-ﬁeld inter- action between the magnetic layers in the simulations. The dipolar-ﬁeld interaction is expected to enhance strongly the chaoticity of the system because it increases the coupled degrees of freedom. Indeed, if the dipolar-ﬁeld interaction 044093-5MATSUMOTO, LEQUEUX, IMAMURA, and GROLLIER PHYS. REV. APPLIED 11,044093 (2019) TABLE III. Evaluated degree of chaos: quality factor ( Qfactor), [ DET ,L,Lmax,ENTR ] of Recurrence Quantiﬁcation Analysis (RQA), and Lyapunov exponent of SV OOP1,S V OOP2,S V IP1,S V IP2,a n dS V IP3atJ/Jth=1.5. Method (a) SV OOP1 (b) SV OOP2 (c) SV IP1 (d) SV IP2 (e) SV IP3 Qfactor >104440 11 2.5 3.7 DET(%) 0.88 0.66 0.10 0.066 0.11 L 5.4 4.5 2.4 2.2 2.3 Lmax 3100 650 40 8 9 ENTR 1.9 1.7 0.81 0.44 0.69 Lyapunov exponent (Gbit/s) 0.14 0.89 4.5 5.0 6.9 exists, the switching of m2will strongly depend on the direction of m1(and reciprocally), yielding an increased sensitivity of the repeated magnetization switching events on initial conditions. VI. TUNING CHAOS BY STRUCTURE The strength of the dipolar-ﬁeld interaction between layers (dipolar coupling) can be controlled by tuning the anisotropy and by tuning the stack. In this section, we compare the windmill dynamics in the diﬀerent structures sketched in Fig. 6. Figure 6(a) shows an out-of-plane bilayer macrospin spin valve simulated without dipolar-ﬁeld interaction (SV OOP1). The dipolar-ﬁeld interaction is included in the micromagnetic simulations of the out-of-plane bilayer (SV OOP2)o fF i g . 6(b) (micromagnetic simulations are described in the Appendix). However, the dipolar-ﬁeld interaction in the out-of-plane conﬁguration is expected to be small because of the small Ms. Figure 6(c) shows an in-plane bilayer macrospin spin valve without the dipolar- ﬁeld interaction (SV IP1). Figure 6(d) shows a structure with a more complicated stack, where the free layers are each composed of two antiferromagnetically coupled lay- ers (SV IP2). The dipolar ﬁeld between the two free layers is expected to be strongly minimized in this conﬁguration thanks to ﬂux closure. Finally, Fig. 6(e) shows an in- plane bilayer spin valve (SV IP3), including the dipolar-ﬁeld interaction, which is expected to be strong in this con- ﬁguration. Because of the dipolar-ﬁeld interaction, SV IP3 favors AP magnetization conﬁguration. As a result, theswitching from the AP to the P conﬁguration is often interrupted and the resistance oscillates in a higher range around 150–200 /Omega1. Typical time traces are shown below each structure. As can be seen, the degree of chaos seems to increase when the anisotropy changes from out-of-plane [Figs. 6(a) and 6(b)] to in-plane [Fig. 6(c)]. It also increases in the in-plane conﬁguration when the strength of dipolar-interaction between layers increases [Figs. 6(c)–6(e)]. In order to evaluate more thoroughly the degree of chaos in structures shown in Fig. 6, we have used three methods: quality factor ( Qfactor), recurrence quantiﬁcation analy- sis (RQA) [ 49–52], and Lyapunov exponent [ 53]. Low Qfactor and low DET ,L,L maxand ENTR in RQA indicate a high degree of chaos, and a high Lyapunov exponent indicates a high degree of chaos. Here, DET is Determin- ism being the percentage of recurrence points which form diagonal lines, Lis averaged diagonal line length, Lmax is the length of the longest diagonal line, and ENTR is Entropy being the Shannon entropy of the probability dis- tribution of the diagonal line lengths. The evaluated values atJ/Jth=1.5 are summarized in Table III. First, we have extracted a quality factor ( Qfactor) for the interspike time interval from each time trace. In Figs. 6(a),6(b),a n d 6(d) [Figs. 6(c) and6(e)], each time interval where R≤150/Omega1(R≥150/Omega1) is deﬁned as the interspike time interval. The Qfactor is evaluated as TI/σI during 10 sets of switching of m1and m2, where TI(σI)i s the average value (standard deviation) of interspike time interval. As we see the enhanced chaotic magnetization dynamics in the in-plane magnetized spin valve in Sec. V, theQfactor decreases when the anisotropy changes from out-of-plane [(a) SV OOP1 and (b) SV OOP2] to in-plane [(c) SVIP1,( d )S V IP2,a n d( e )S V IP3]. In (a) SV OOP1, the Qfactor exceeds our analyzable upper limit of 104because σI=0 in our simulation with a time step of 1 ps. The Qfactor also decreases by the introduction of dipolar-ﬁeld interac- tion between layers [(a) SV OOP1 versus (b) SV OOP2 and (c) SVIP1versus (d) SV IP2and (e) SV IP3]. However, the ﬂux- closure structure in Fig. 6(d) hardly improves the Qfactor. Nevertheless, it recovers the full amplitude of resistance oscillation compared to Fig. 6(e)and reduces the threshold current Jth. Then we have conducted recurrence quantiﬁcation anal- ysis of R(t)for each structure. Recurrence plots for each structure are shown at the bottom of Fig. 6. A recur- rence plot [ 49,50] is a square matrix, in which the matrix elements correspond to those times at which a similar resis- tance state recurs, i.e., a plot of Rι,κ=/Theta1(/epsilon1ι−/bardblR(tι)− R(tκ)/bardbl). Here, tιand tκare time during about 10 periods of resistance oscillation shown in the middle panels of Fig. 6. /epsilon1ιis a threshold distance and /epsilon1ι=0.05/Omega1is chosen in Fig.6./Theta1is the Heaviside function, and the elements where Rι,κ=1 are dots in the recurrence plots. In other words, the elements where R(tι)∼R(tκ)appear as dots in the plots. Trivial dots at the matrix diagonal elements at tι= tκare removed. A perfectly periodic oscillator has dots 044093-6CHAOS AND RELAXATION OSCILLATIONS. . . PHYS. REV. APPLIED 11,044093 (2019) (e) 12 10 8 6 4 2 012 10 8 6 4 2 0Time (ns) 2 nsResistance ( Ω)J < 0 m2m1Dipolar coupling 15 10 5 015 10 5 0Time (ns) 5 nsResistance ( Ω)m2m1(d) J > 0Dipolar coupling 20 15 10 5 020 15 10 50Time (ns) 5 nsResistance ( Ω)m2m1(c) J < 0(b) 40 30 20 10 040 30 20 10 0Time (ns) 10 nsResistance ( Ω)m2m1Dipolar coupling J > 0 50 40 30 20 10 050 40 30 20 10 0Time (ns) Time (ns) Time (ns) Time (ns) Time (ns) Time (ns) Time (ns) Time (ns) Time (ns) Time (ns) Time (ns) 10 nsResistance ( Ω)(a) z xye–e– e– e–e–J > 0 m2m1 FIG. 6. Sketch of the diﬀerent structures: (a) macrospin OOP spin valve without dipolar coupling (SV OOP1), (b) micromagnetic OOP spin valve with dipolar coupling (SV OOP2), (c) macrospin IP spin valve without dipolar coupling (SV IP1), (d) micromagnetic IP spin valve with antiferromagnetically coupled layers (m A1and m A2) and with dipolar coupling (SV IP2), (e) micromagnetic IP spin valve with dipolar coupling (SV IP3). The x(z) axis is parallel to the major axis of the ellipse (out-of-plane direction). Typical traces of the resistance versus time and the corresponding recurrence plots are shown below each case for J/Jth=1.5. mainly along the diagonal. In Figs. 6(d) and6(e), the plots show patterns with reduced regularity reﬂecting their high degree of chaos compared to the cases of Figs. 6(a)–6(c). Results of recurrence quantiﬁcation analysis [ 50–52], i.e., DET ,L,Lmax,a n d ENTR , are summarized in the middle of Table III.DET ,L,Lmax,a n d ENTR are quantities character- ized by the diagonal lines in a recurrence plot. The lengths of diagonal lines are directly related to the ratio of pre- dictability inherent to the system. Suppose that the states at times tιand tκare neighboring. If the system exhibits predictable behavior, similar situations will lead to a sim- ilar future; i.e., the probability for R(tι+δt)∼R(tκ+δt) is high. For perfectly periodic systems, this result leads to inﬁnitely long diagonal lines. In contrast, if the system is chaotic, the probability for R(tι+δt)∼R(tκ+δt)is small and we only ﬁnd single points or short lines. In accordance with the evaluated Qfactors, DET ,L,Lmax,a n d ENTR decrease when the anisotropy changes from out-of-plane to in-plane. They also decrease by the introduction of dipolar interaction between layers. Then we have determined the Lyapunov exponent from each time trace [ 53]. The Lyapunov exponent is a quantity that characterizes the rate of separation of inﬁnitesimally close trajectories in dynamic systems.Lyapunov exponents are evaluated with about 100 periods of resistance oscillation for each structure. As we expected, the Lyapunov exponent, characterizing the degree of chaos, increases when the anisotropy changes from out-of- plane to in-plane. It also increases in the in-plane conﬁg- uration when the strength of dipolar interaction between layers increases. These results show that the degree of chaos can be tuned in a wide range by engineering the mag- netic stack and anisotropies, which is suitable for various neuromorphic computing applications. VII. TUNING CHAOS BY CURRENT We also checked the tunability of chaos by current. The evaluated current density dependence of quality fac- tors ( Qfactors) and Lyapunov exponents is shown in Fig. 7.J/\|Jth\|represents the current density normalized by the threshold current density for each polarity of cur- rent. The Lyapunov exponents at J/\|Jth\|=± 1 are not shown because the too long interspike time interval againstpulse width makes evaluation of the Lyapunov exponent itself impossible and long simulations of 100 periods are not possible with our computational capacity. Both trends in Figs. 7(a) and7(b) show that the degree of chaos is 044093-7MATSUMOTO, LEQUEUX, IMAMURA, and GROLLIER PHYS. REV. APPLIED 11,044093 (2019) J/\|Jth\|(a) J/\|Jth\|(b) SVIP2SVIP1SVOOP2SVOOP1 SVIP3 FIG. 7. (a) J/\|Jth\|dependence of the Qfactor of the interspike time interval and (b) Lyapunov exponent. Open circles on dotted lines are for SV OOP1. Solid circles on solid lines are for SV OOP2. Open squares on dotted lines are for SV IP1. Solid triangles on solid lines are for SV IP2. Solid squares on solid lines are for SVIP3. increased by increasing the magnitude of J/\|Jth\|. One might notice the deviation from the trend in SV OOP1,S V IP2, and SV IP3, whose degrees of chaos at small \|J/Jth\|are larger than those at medium J/\|Jth\|.A ts m a l l \|J/Jth\|, that is, around ±1, just after the switching of m1or m2,a n almost complete P or AP conﬁguration is made. Around ˆθ12∼0◦and 180◦, the spin-torque strength, which is pro- portional to \|sinˆθ12\|, is approximately 0, but it is very sen- sitive to ˆθ12. The small diﬀerence of ˆθ12just after switching of m1(or m2), results in a large diﬀerence in switching time of m2(or m1). Therefore, the deviation from the trend occurs at small \|J/Jth\|. A cause of the increased degree of chaos at large \|J/Jth\|can be the increased instability of m2during the interspike time interval. Fluctuations of m2 strongly vary the angle between magnetizations ˆθ12, which gives the torque strength. Therefore, the switching of m1is complex through the dynamics of m2. The trend in Fig. 7 means that the degree of chaos can be tuned in a wide range by the dc current. The tunability of chaos by the current is quite beneﬁcial because it enables the control of chaos in real time in a ready-made circuit. VIII. TOW ARD SPINTRONIC NEURONS In this paper, we propose spintronic relaxation oscilla- tors that show spiking dynamics whose degree of chaos can be tuned by structures and current. In order for the spintronic relaxation oscillators to be used as hardware neurons in applications, they should be equipped with functions such as temporal coding and excitability [ 54]a n d an ability to synchronize with other spintronic relaxation oscillators [ 45,46]. They should also ensure high fan-out and fan-in. In the case of spintronic relaxation oscilla-tors, they can couple with each other through couplings other than electronic connections, which require wires. For example, spintronic relaxation oscillations can be cou- pled through dipolar couplings, through spin waves, andthrough microwaves [ 55,56]. Because these couplings do not require electrical wires, spintronic relaxation oscilla- tors can be advantageous for ensuring high fan-out and fan-in. IX. CONCLUSION We propose a simple way to imitate neuron spiking in high-magnetoresistance nanoscale spin valves where both magnetic layers are free and can be switched by spin torque. Our numerical-simulation results show that the windmill motion induced by spin torque in the proposed spintronic relaxation oscillators gives rise to spikes whose shape and frequency can be tuned through the amplitude of injected dc current. We also ﬁnd that these devices can exhibit chaotic oscillations. By evaluating the quality fac- tors of interspike time intervals and Lyapunov exponents, as well as conducting recurrence quantiﬁcation analysis for the time evolutions of resistance, we demonstrate that the degree of chaos can be tuned in a wide range by engineering the magnetic stack and anisotropies and bychanging the dc current. The degree of chaos increases when the anisotropy of the free layer changes from out-of- plane to in-plane. It also increases when the dipolar-ﬁeld interaction between the free layers increases. The pro- posed spintronic relaxation oscillator is a promising build- ing block for hardware neuromorphic chips leveraging complex nonlinear dynamics for computing. ACKNOWLEDGMENTS This work was partly supported by JSPS KAKENHI Grant No. JP16K17509 and JP19K05259 and the Euro- pean Research Council (ERC) under Grant No. bioSPIN- spired 682955. APPENDIX: MODEL IN MICROMAGNETIC SIMULATIONS In micromagnetic simulations, ˆm i=(mxi,myi,mzi)of Eqs. (1)and(2)means the unit magnetization vector of a unit cell at the position ri,ˆmi(ri). The simulations are conducted with the simulation code SPINPM [19]. In the micromagnetic simulations of this paper, each magnetic layer is divided into unit cells with the area of 4 ×4n m . In the third term on the right side of Eqs. (1)and(2), i.e., the Slonczewski-torque term, the xand ycomponents of r1 and r2are the same. In Eqs. (1)and(2),Heﬀis the eﬀective ﬁeld, expressed as Heﬀ= Hexch+ Hanis+ Hdip+ HRKKY . (A1) Hexchrepresents the exchange ﬁeld, expressed as Hexch=2A μ0Ms∇2ˆm. (A2) 044093-8CHAOS AND RELAXATION OSCILLATIONS. . . PHYS. REV. APPLIED 11,044093 (2019) Ais the exchange constant. In the micromagnetic simula- tions, it is assumed to be A=2×10−11J/m in this paper. Note that ˆmin Eqs. (A2) and(A3) represents ˆmiorˆmAi. Hdiprepresents the dipolar ﬁeld. Hdipon the position ris expressed as Hdip=−Ms 4π/integraldisplay Vol/bracketleftbiggˆm(r/prime) \|r− r/prime\|3 −3[ˆm(r/prime)·(r− r/prime)] \|r− r/prime\|5(r− r/prime)/bracketrightbigg dr/prime. (A3) Here, the integral is performed over the volume (Vol) including all magnetic layers. HRKKY represents the RKKY coupling ﬁeld, expressed as HRKKY=JRKKY μ0Msd∇/parenleftbig ˆmi·ˆmAi/parenrightbig . (A4) Here, JRKKY is the exchange coupling constant. 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