Magnetic Moment Softening and Domain Wall Resistance in Ni Nanowires J. D. Burton1,3,R. F. Sabirianov2,3, S. S. Jaswal1,3, and E. Y. Tsymbal1,3* 1Department of Physics and Astronomy, University of Nebraska, Lincoln, Nebraska 68588-0111, USA 2Department of Physics, University of Nebraska, Omaha, Nebraska 68182-0266, USA 3Center for Materials Research and Analysis, University of Nebraska, Lincoln, Nebraska 68588-0111, USA O. N. Mryasov Seagate Research, Pittsburgh, Pennsylvania 15222, USA Magnetic moments in atomic scale domain walls formed in nanoconstrictions and nanowires are softened which affects dramatically the domain wall resistance. We perform ab initio calculations of the electronic structure and conductanceof atomic-size Ni nanowires with domain walls only a few atomic lattice constants wide. We show that the hybridizationbetweennoncollinearspin states leads to a reduction of the magnetic moments in the domain wall. This magnetic moment softening strongly enhances the domain wall resistance due to scattering produced by the local perturbation of the electronic potential. DOI: PACS number(s): The study of domain walls (DWs) in bulk magnitude of the magnetic moment across the DW. It is ferromagnetic materials began in the early 20th century with well established, however, that the magnitude of the contributions from Bloch1, Landau and Lifshitz2, and Néel.3 magnetic moments in itinerant magnets can depend strongly In bulk3d metal ferromagnets DWs are wide(~100 nm) on on the orientation of the neighboring moments.17 This the scale of the lattice spacing due to the strong exchange effect is relatively weak in well localized ferromagnets like interaction which tends to align neighboring regions of Fe, but can reduce and even destroy the atomic magnetic magnetization, compared to anisotropic effects which prefer moment of the itinerant ferromagnets, like Ni. the magnetization to lie along specific directions. With the The origin of this phenomenon is the presence of recent interest in magnetic nanostructuresit has been shown hybridization between noncollinear spin states. In the that DWs can be quite thin due to the enhanced anisotropic uniformly magnetized material with no spin-orbit coupling effect of constrained geometry in nanowires and the minority and majority spin bands are independent. nanoconstrictions.4 For example, Prokop et al.5 have However, in a noncollinear state, such as a DW, this is no recently observed DWs of only a few lattice constants wide longer the case and the two spin bands are hybridized. This insingle monolayer Fenanowires grown on Mo. spin mixing leads to charge transfer and level broadening, The DW width controls the DW resistance. The origin which results in the reduction of the overall exchange of the DW resistance is known to be the mixing of up- and splitting between majority and minority states on each atom, down-spin electrons due to the mistracking of the electron’s and hence the atomic moments are reduced. In bulk DWs of spin in passing through the DW.6 The narrower DW width any ferromagnetic material, this effect is small due to the results in a larger angle between the magnetization slow variation of the direction of the magnetization. For the directions of successive atomic layers thereby lowering the atomic scale DWs formed in magnetic nanostructures, electron transmission. In bulk ferromagnets DWs do not however, there is a large degree of canting between affect appreciably the resistancebecause the DW wall width neighboring magnetic moments. This results in a significant is much larger than the Fermi wave length, and hence hybridization between spin stateswhichleads to a reduction, electrons can follow adiabatically the slowly varying or softening, of the magnetic moments within the magnetization direction within the DW. In magnetic constrained DW. The magnetic moment softening affects nanoconstrictions, where an atomic scale DW can beformed the DW resistance due to the local perturbation in the between two electrodes magnetized antiparallel to one electronic potential. another, the DW resistance may be appreciable. The We note that the spatial variation of the magnetization ballistic transport across such a DW leads to interesting in DWs has been suggested before, but only in the context magnetoresistive phenomena.7,8 of finite temperature magnetic disorder of the (fixed The theoretical description of the DW resistance in the magnitude) atomic magnetic moments in a DW.18,19,20 The ballistic transport regime has attracted much attention. The effect of spatial variation of the magnetization on DW approaches which were used are based on either free- resistance was addressed previously, but only within the electron models9,10,11,12 in which the DW is represented by diffusive transport regime and a free electron model.21 an appropriate potential profile or first-principles In this Letter we illustrate the importance of magnetic calculations13,14,15,16 in which the DW is typically described moment softening by performing ab initio calculations of by a spin-spiral structure. All these models assume that the the electronic structure and ballistic conductance of atomic DW is rigid, i.e. they neglect any spatial variation of the scale DWs in Ni nanowires. We show that the magnetic 1 region were similar to those in the remaining semi-infinite section of the leads in the uniformly magnetized state. A DW was modeled by a finite spin spiral with the relative angle between neighboring atomic layers of magnetic moments being 180°/(N + 1), where N is the number of atomic layers in the DW. In all the calculations the a orientation of the magnetic moments was held fixed. First we consider an N = 1 DW in amonatomic Ni wire (Fig 1a). Fig. 1 displays the results of our calculations for 30 the N = 1 DW. The electronic potentials of the antiparallel majority magnetized lead sites (blue) were frozen, while on the 20 centralfivesites (red) the electronic structure was calculated self-consistently. We find a 16% reduction of the magnetic 10 ) moment on the central site and a 7% reduction on the two 1 -V neighboring sites. The reduction in the magnetic moment e 00 S ( on the central site is due to the hybridization with the O 10 different spin states of the non-collinear neighbors which D results in the reduction of the exchange splitting, as L 20 described earlier. This fact is evident from Fig. 1b which b minority shows the local DOS for the uniformly magnetized 30 monatomic Ni wire and for the central atom of an N = 1 DW. The reduction in the moment of the nearest neighbors -2 -1 0 1 of the central atom is only about half of that of the central E - E (eV) F atom because they are noncollinear with only one of the two Fig. 1 (Color online) (a) A monatomic wire with an N = 1 DW. The neighbors. Also apparent in Fig. 1b is an overallbroadening arrows show the orientation and relative magnitude of calculated magnetic of the DOS on the central site compared to the uniformly moments. (b) DOS per atom of the uniformly magnetized monatomic wire (dashed curve) and the LDOS at the center of an N = 1 DW (solid curve). magnetized state produced by the spin mixing. Fig. 2 shows the magnetic moment profile for several DW widths in the monatomic wire. The reduction in the moments within DWs only a few lattice constants wide can moment is strongly dependent on the width of the DW. For be significantly reduced compared to the magnetic moments N = 0 and N = 1 DWs the effect of softening is the largest in a uniformly magnetized wire due to the presence of owing to the fact that the degree of noncollinearity is the substantial hybridization between spin states. We find that largest in these two cases. As the width of the DW the magnetic moment softening strongly enhances the DW increases the softening decreases because of the reduction of resistance due to additional scattering resulting from the the angle between the nearest neighbor magnetic moments. local perturbation in the electronic potential. Fig. 3a shows the 5×4 wire which has five atoms in one Density functional calculations of the spin-dependent layer and four in the next layer resulting in three electronic structure of atomic scale DWs in Ni nanowires were performed using the tight-binding linear muffin-tin- orbitalmethod22 in the atomic sphere approximation and the ) 1.0 N = 5 µB local spin density approximation for the exchange- correlation energy. We used the real space recursion nt ( 0.8 e method23 with a Beer-Pettifor terminator24 to calculate the m 1.0 N = 3 o local density of states (DOS). Ultrathin domain walls were M 0.8 examined for two different free-standing Ni wires based on c the bulk fcc structure: (110) monatomic chains and 5×4 eti 1.0 N = 1 n wires (described below). The wires were surrounded by a g a 0.8 few layers of empty spheres to accurately describe the M charge density. Allthestructures used thelattice constanta 1.0 N = 0 =3.52 Åofbulk fcc Ni. 0.8 In the calculations we constructed a central region -8 -6 -4 -2 0 2 4 6 8 containing the DW, surrounded by two uniformly Atomic Position magnetized leads aligned antiparallel relative to each other. Fig. 2 Magnetic moments in the monatomic Ni wire as a function of The self-consistent calculations were carried out for the distance from the DW center for several DW widths. The N = 1 plot central region and a large enough portion of the surrounding corresponds to Fig. 1a. leads so that the outermost atoms of the self-consistent 2 determine the contribution of magnetic moment softening to the DW resistance. Fig. 4a shows the spin-resolved conductance, G , of a FM uniformly magnetized 5×4 Ni wire as a function of energy E. The conductance is quantized in steps of e2/h, corresponding to the number of bands crossing the energy a E. The bands are seen in Fig. 4b as being bound by the sharp peaks corresponding to the band edges. The conductance in the zero-bias limit is given by the value at 0.8 the Fermi energy, EF, located at the center of the narrow minority band. For the 5×4 Ni wire it is 14 e2/h. Fig. 4c ) B 0.4 N = 5 displays the conductance through an N = 1 DW, G . For a m( DW nt “rigid” DW (no softening of the magnetic moments) the me 0.8 Hamiltonian is constructed from the self-consistent o potentials of the uniformly magnetized wire, while for the M 0.4 N = 3 c “soft” DW we use the potentials that yield reduced magnetic eti 0.8 momentsin the DW region, as seenin Fig. 3. We find that n g the magnetic moment softening in the DW leads to a a M 0.4 b N = 1 ree2/dhu ctoti o3n.2 i1n eth2/eh .co Tndhuisc tiamnpclei east tthhee sFigernmifii ceannetrgenyh farnocme m5.e3n8t 0.0 of the DW magnetoresistance, (GFM – GDW)/GDW, from -8 -6 -4 -2 0 2 4 6 8 160% to about 340%. Atomic Layer Theorigin of this phenomenon can be understoodusing Fig. 3 (Color Online) (a) The 5×4 nanowire showing the three a simple tight-binding model of a monatomic chain with an nonequivalent sites and an N = 3 DW. Each arrow represents the abrupt N = 0 DW. We model the majority states by a wide magnitude and orientation of the average magnetic moment of the plane band, characterized by the (large) nearest neighbor hopping with 5 atoms. (b) The self consistent magnetic moments for several DW parameter t , to emulate the wide band near the Fermi widths. The color in the plot corresponds to the site of the same color in maj (a). energy (see Fig. 4b). The minority states are modeled by a narrow band with (small) hopping t , offset from the min nonequivalent sites, each indicated by a different color. Although the average magnetic moment of the uniformly magnetized 5×4 wire, µ = 0.7µ , is lower than that of the 12 B a E majority monatomic wire, µ = 1.1µB, we find that in the presence of h) 8 F ultra-thin DWs the spatial variation of the magnetization / 4 2 displays qualitatively similar behavior. Fig. 3b shows the (e 0 M magnetic moments found for three DW widths in the 5×4 4 F G wire. The largest reduction in magnetic moment is nearly 8 minority 90% for the central site of the N = 1 DW, and the effect is still substantial for the N = 5 DW where the magnetic -1V) 20 b majority moment of the central layer is softened by about 10%. The 10 e effect is significantly larger in the 5×4 wire than that in the S ( 00 monatomic chain due to the enhanced hybridization O reflectingan increase in the number of neighboring atoms. D 60 minority The magnetic moment softening affects dramatically 120 the conductance across DWs. We calculate the conductance of the 5×4 Ni wire with an N = 1 DW using the standard h) 6 c / tight-binding technique described in detail in Ref. 16. It 2e 4 ( should be noted, however, that in Ref. 16 the Hamiltonian W 2 D for the DW region was built by simply rotating (in spin G 0 space) the self-consistent potentials obtained for the 0.0 0.1 0.2 uniformly magnetized wire. Here we derive the E- E (eV) Hamiltonian from the self-consistent potentials found in the F presence of the N = 1 DW, i.e. taking into account magnetic Fig. 4 (a) Spin-resolved conductance,GFM, and (b) density of states near the Fermi energy, E , as a function of energy for the uniformly magnetized moment softening. For comparison we also calculate the F 5×4 Ni wire. (c) Conductance of the N = 1 DW, G , as a function of conductance in the same way as Ref. 16 which allows us to DW energy for the rigid DW (dashed curve) and for the soft DW (solid curve). 3 8 a S 1 F.Bloch, Z. Phys. 74, 295 (1932). O 4 2 L. Landau and E. Lifshitz, Phys. Z. Sowjet Union 8,153 (1935). D 3 L. Néel, J. Phys. Radium 17, 250 (1956). 4 P. Bruno, Phys. Rev. Lett.83,2425 (1999). 0 5 J. Prokop, A. Kukunin, and H. J. Elmers, Phys. Rev. Lett. 95, 2G (e/h) 12 b 2G (e/h) 12 c 67 1G(C18..9 7S7G24.0. )2Y.C (aa2nb0gr0e, 5rCa) .a Zndh aLn.g ,M J.. FRaeldiceopve,n nPihnygs,. aSntdat.B S. oDl.o (ubd)in6,1A, p5p3l9. 0 0 Phys. Lett. 84, 2865 (2004). -1 0 1 0.0 0.2 0.4 8 H. Chopra, M. Sullivan, J. Armstrong, and S. Hua, Nature E - E d Materials4,832 (2005). F 9 H. Imamura, N. Kobayashi, S. Takahashi, and S. Maekawa, Fig.5 Results ofa simple tight-binding model. (a) DOS ofthe uniformly Phys. Rev. Lett. 84, 1003 (2000). magnetized state versus energy for majority- (wide band) and minority- 10L. R. Tagirov, B. P. Vodopyanov, and K. B. Efetov, Phys. Rev. (narrow band) spin electrons. (b) Conductance versus energy for a rigid B65, 214419 (2002). DW (d = 0, dashed line) and for a soft DW (d = 0.2, solid line). (c) 11V. K. Dugaev, J. Berakdar, and J. Barnas, Phys. Rev. B 68, Conductance versus d for E=E . F 104434 (2003). 12J. D. Burton, A. Kashyap, M. Ye. Zhuravlev, R. Skomski, E. Y. center of the majority band by energy D . To model Tsymbal, S. S. Jaswal, O. N. Mryasov, and R. W. Chantrell, magnetic moment softening we assume that the onsite Appl. Phys. Lett. 85, 251 (2004). energies of the two interface sites are shifted relative to the 13J. B. A. N. van Hoof, K. M. Schep, A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rev. B 59, 138 (1999). uniformly magnetized leads. Majority states are shifted up 14J. Kudrnovsky, V. Drchal, C. Blaas, P. Weinberger, I. Turek, by energy d and minority states are shifted down by the and P. Bruno, Phys. Rev. B 62, 15084 (2000). sameamount d . This corresponds effectivelyto a reduction 15B. Yu. Yavorsky, I. Mertig, A. Ya. Perlov, A. N. Yaresko, and of the local exchange splitting by 2d on the two interface V. N. Antonov, Phys. Rev. B 66, 174422 (2002). sites, and hence softening of the magnetic moment. The 16R. F Sabirianov, A. K. Solanki, J. D. Burton, S. S. Jaswal, and parametersfor our model are chosenas follows:t = 1, t E. Y. Tsymbal, Phys. Rev. B 72, 054443 (2005). maj min = 0.05, D = 1. The results of the model are shown in Fig. 17J. Hubbard, Phys. Rev. B 23, 5974 (1981); S. A. Turzhevskii, A. I. Likhtenshtein, and M. I. Katsnelson, Sov. Phys. Solid State 5. As is seen from Fig. 5b,c, the model predicts a drastic 32, 1138 (1990); O. N. Mryasov, V. A. Gubanov, and A. I. reduction in the conductanceas a result of magnetic moment Liechtenstein, Phys. Rev. B 45 12330 (1992). softening. The origin of this reduction is the local 18V. A. Zhirnov, Zh. Eksp. Teor. Fiz. 35, 1175 (1959) [Sov. Phys. perturbation in the electronic potential which leads to JETP8, 822 (1959)]. stronger scattering of transport electrons by the DW. When 19L. N. Bulaevskii and V. L. Ginzburg, Sov. Phys. JETP 18, 530 the onsite energies of the interface atoms are shifted with (1964). respect to the narrow minority band, which determines the 20N. Kazantseva, R. Weiser, and U. Nowak, Phys. Rev. Lett. 94, energy window for conductance, these sites act as additional 037206 (2005). 21R. P. van Gorkom, A. Brataas, and G. E. W. Bauer, Phys. Rev. scatterersthat hinder conductance. Lett.83, 4401 (1999). In conclusion, we have shown that in atomic scale 22 O. K. Andersen and O. Jepsen, Phys Rev. Lett. 53, 2571 (1984). domain walls of Ni nanowires the magnetic moments are 23R. Haydock, in Solid State Physics (Academic Press, New York. softened due to the noncollinearity with the neighboring 1980), Vol. 35. pp. 215-294. magnetic moments. This effect is stronger in wires of larger 24N. Beer and D. G. Pettifor, in The Electronic Structure of cross-section due to the enhanced hybridization but falls off Complex Systems, NATO ASI Series, Physics, Ser. B, ed. P. quickly with increasing DW width due to a decreasing Phariseau and W. M. Temmerman (Plenum, New York, 1984), degree of noncollinearity. The magnetic moment softening p.769. significantly enhances the DW resistance as a result of scattering produced by the local perturbation of the electronic potential. This work is supported by Seagate Research, the NSF (Grant Nos. DMR-0203359 and MRSEC: DMR-0213808), and the Nebraska Research Initiative. The calculations were performed using the Research Computing Facility of the University of Nebraska-Lincoln. *Electronic address: [email protected] 4