University of Nebraska - Lincoln
`DigitalCommons@University of Nebraska - Lincoln
`
`Ralph Skomski Publications
`
`Research Papers in Physics and Astronomy
`
`8-1-1997
`
`Magnetism in one dimension: Fe on Cu(111)
`
`J. Shen
`Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany
`
`Ralph Skomski
`University of Nebraska-Lincoln, rskomski2@unl.edu
`
`M. Klaua
`Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany
`
`H. Jenniches
`Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany
`
`S. Sundar Manoharan
`Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany
`
`See next page for additional authors
`
`Follow this and additional works at: http://digitalcommons.unl.edu/physicsskomski
`Part of the Physics Commons
`
`Shen, J.; Skomski, Ralph; Klaua, M.; Jenniches, H.; Manoharan, S. Sundar; and Kirschner, J., "Magnetism in one dimension: Fe on
`Cu(111) " (1997). Ralph Skomski Publications. Paper 21.
`http://digitalcommons.unl.edu/physicsskomski/21
`
`This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska -
`Lincoln. It has been accepted for inclusion in Ralph Skomski Publications by an authorized administrator of DigitalCommons@University of Nebraska
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`Lambeth Magnetic Structures, LLC Exhibit 2007
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`LMBTH-000190
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`Authors
`J. Shen, Ralph Skomski, M. Klaua, H. Jenniches, S. Sundar Manoharan, and J. Kirschner
`
`This article is available at DigitalCommons@University of Nebraska - Lincoln: http://digitalcommons.unl.edu/physicsskomski/21
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`PHYSICAL REVIEW B
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`VOLUME 56, NUMBER 5
`
`1 AUGUST 1997-I
`
`Magnetism in one dimension: Fe on Cu(cid:132)111(cid:133)
`
`J. Shen,* R. Skomski, M. Klaua, H. Jenniches, S. Sundar Manoharan, and J. Kirschner
`Max-Planck-Institut fu¨r Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany
`共Received 21 April 1997兲
`
`The magnetism and the morphology of one-dimensional Fe stripes on a Cu共111兲 vicinal surface with
`perpendicular magnetization are investigated. Scanning tunneling microscopy shows that the Fe stripes have
`nanometer width and are aligned in parallel along the 关011兴 direction. From the magnetization curves it is
`inferred that the stripes exhibit a ferromagnetic behavior well below the nominal thickness of one monolayer.
`In contrast to two-dimensional films, the magnetization of the stripes is not only temperature but also time
`dependent. The dynamics of the stripes have been described by a micromagnetic Ising model with Glauber
`dynamics, yielding an effective anisotropy constant K 1 of 0.65⫾0.15 M J/m3. 关S0163-1829共97兲01230-7兴
`
`The dimensionality of magnetic matter has a strong influ-
`ence on its magnetic properties. Both theory1
`and
`experiment2 indicate that in a two-dimensional magnetic sys-
`tem long-range ferromagnetic order prevails at nonzero tem-
`perature, once anisotropies are present. By contrast, in a one-
`dimensional 共1D兲 magnetic system, theoretical arguments
`show that even a highly anisotropic system does not yield a
`zero-field equilibrium spontaneous magnetization.3 The pro-
`totype model
`is the Ising chain of localized spins with
`nearest-neighbor exchange interaction, which has no long-
`range order at nonzero temperature.4 Experimentally 1D
`magnetism on compound material has been extensively stud-
`ied from the 1970s.5 For 3d metals, a possible experimental
`realization of a 1D system is an array of a large number of
`parallel, nanometer-wide magnetic stripes on nonmagnetic
`substrates. In a pioneering experiment Elmers et al.6 pro-
`duced Fe stripes by step-flow growth on a W共110兲 crystal.
`These stripes showed a strong in-plane anisotropy along the
`stripe axis and a two-dimensional Ising-like behavior, despite
`their quasi-1D appearance. In our experiment Fe stripes of
`1–2 atom height by 5–15 atom width are produced by step-
`edge decoration of a stepped Cu crystal 共see Fig. 1兲. Because
`of their perpendicular magnetization axis and their high as-
`pect ratio between cross section and length, the Fe stripes
`come close to an array of 共finite兲 Ising chains. Yet, these
`quasi-1D stripes show all the usual signs of ferromagnetism:
`hysteresis loops and a critical temperature, above which the
`magnetization disappears. We will show below that this ap-
`parent conflict is resolved by a statement put forward by
`Jacobs and Bean7 in 1963: ‘‘ . . . the one-dimensional Ising
`chain 共or any anisotropic chain兲 is not ferromagnetic in equi-
`librium but will, in fact, show all the usual characteristics of
`ferromagnetic matter owing to the difficulty of reaching equi-
`librium.’’ It is only today, several decades later, that the
`experimental means and tools have become available to test
`the validity of their predictions on 3d pure metals.
`The experiments were performed in a multichamber sys-
`tem including a molecular beam epitaxy 共MBE兲 preparation
`chamber, a scanning tunneling microscopy 共STM兲 chamber,
`an analysis chamber equipped with cylindrical mirror ana-
`lyzer 共CMA兲 based Auger electron spectroscopy 共AES兲 and
`low-energy electron diffraction 共LEED兲, and a magneto-
`optical Kerr effect 共MOKE兲 chamber. The base pressure of
`the individual chambers is better than 5⫻10⫺11 mbar. The
`characteristic features of the Fe growth on Cu共111兲 are its fcc
`
`structure inherited from the copper8–11 and its tendency to-
`wards step decoration.12 Using this effect we have prepared
`fcc Fe stripes on a vicinal Cu共111兲 substrate, which has a
`miscut of 1.2° and an average terrace width of about 10 nm.
`Prior to film preparation the copper substrate was cleaned by
`Ar⫹ sputtering and annealing cycles to 700 K. The crystal-
`lographic quality and the cleanliness of the substrate were
`monitored by LEED and AES, respectively. The Fe stripes
`were prepared in the analysis chamber from an iron wire 共
`5N in purity兲 heated by e-beam bombardment. At a typical
`Fe evaporation rate of 0.2 ML/min the pressure increased
`from 5⫻10⫺11 mbar to 1⫻10⫺10 mbar. To suppress the in-
`terdiffusion the copper substrate was kept at 0 °C during
`deposition.13 Afterwards the sample was further cooled down
`to 170 K in order to avoid a temperature rise above 0 °C
`during transferring. MOKE measurements were carried out
`in the sample temperature range between 90 and 270 K, and
`STM data were recorded at room temperature after the
`MOKE measurements.
`Figure 1 is a STM topography image showing the typical
`morphology of 1D Fe stripes on Cu共111兲 at nominal cover-
`ages of 0.3 ML 关Fig. 1共a兲兴 and 0.8 ML 关Fig. 1共b兲兴. The Fe
`stripes are located on the upper terrace of the Cu共111兲 steps
`and aligned along the 具011典 direction. The marked line pro-
`file in Fig. 1共a兲 indicates that at 0.3 ML the Fe stripes are
`divided into segments which are mainly one monolayer in
`height and about 10–20 nm in average length 共the dashed
`line stands for the height level of the substrate兲. The presence
`of a small fraction of bilayer high segments 共marked as 2兲 is
`due to both the fact that the stripes at the edges of the wider
`terraces tend to collect more Fe adatoms and the fact that
`some copper diffuses onto the top of the Fe stripes during the
`STM experiments. The diffusion of the copper leaves some
`one-monolayer-deep holes in the regions between the Fe
`stripes. This finding is consistent with a previous STM study
`of the Fe/Cu共111兲 system.9 However, it is worthwhile men-
`tioning here that our magnetic measurements were per-
`formed on samples without any significant interdiffusion. At
`higher thickness of 0.8 ML in Fig. 1共b兲, the Fe stripes be-
`come much more continuous as indicated by the marked line
`profile. The edges of the stripes are rough due to the ten-
`dency of the Fe edge atoms aligning along all three 具011典
`directions. The edge roughness together with some defects
`may cause some structural discontinuity along the stripes as
`shown in the line profile.
`
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`FIG. 2. 共a兲 Magnetic hysteresis of a 0.8-ML Fe film measured at
`different temperatures. 共b兲 Saturation and remanent magnetization
`plotted as a function of temperature. The solid lines are only for
`guiding the eye.
`
`The magnetization of the Fe stripes is not only tempera-
`ture but also time dependent. Figure 3 shows the time depen-
`dence of the magnetization of the 0.8-ML stripes. The zero
`level refers to the demagnetized state of the Fe stripes. After
`switching on the field (⬃0.6 T) the magnetization of the
`stripes quickly saturates. When the field is removed, at low
`temperature of about 100 K the magnetization drops rapidly
`to about 70% of the M s and then it decays slowly. The slow
`decay of the magnetization at 100 K guarantees the ferro-
`magnetic behavior of the system in any practical measuring
`time. At higher temperature of about 160 K the magnetiza-
`tion relaxation process becomes significantly faster. It is in-
`
`FIG. 3. Time dependence of the magnetization of a 0.8-ML film.
`After switching on the field 共about 0.6 T兲,
`the magnetization
`quickly saturates. About 30 sec later the external field has been
`removed leading to the magnetization decaying slowly at low tem-
`perature 共100 K兲 but very quickly at elevated temperature 共160 K兲.
`
`FIG. 1. STM topography image of 0.3-ML 共a兲 and 0.8-ML 共b兲
`Fe stripes on Cu共111兲 vicinal surface. The marked line profiles are
`shown in the upper-right corner with the dashed lines indicating the
`height level of the substrate.
`
`The MOKE measurements show that these Fe stripes ex-
`hibit hysteresis with a coercivity depending on the tempera-
`ture. Figure 2共a兲 displays magnetic hysteresis loops for a
`0.8-ML film measured in the polar geometry. No magnetic
`signal was observed in the longitudinal geometry, irrespec-
`tive of the external field being parallel or perpendicular to
`the stripes, indicating that the easy magnetization axis is
`along the surface normal. Figure 2共a兲 qualitatively shows
`that with increasing temperature the saturation magnetization
`(M s) is rather stable as compared to the quick decay of the
`remanent magnetization (M r). This is quantitatively demon-
`strated in Fig. 2共b兲, which displays the temperature depen-
`dence of M s and M r of the Fe stripes. Note in this case it is
`M s rather than M r that roughly follows the two-dimensional
`Ising model 共the solid line in M s curve兲. Such phenomena
`have already been observed and discussed in a previous pa-
`per on a system of a monolayer of Co on Cu共111兲.14 Upon a
`further increase of the temperature above 250 K the satura-
`tion drops down quickly, indicating that the system becomes
`nonmagnetic.
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`FIG. 4. Theoretical hysteresis curve 共solid line兲 from Eq. 共4兲
`fitted to experimental data 共full circles兲 from the stripes of 0.3 ML
`at 50 K, 0.6 ML at 163 K, 0.8 ML at 182 and 200 K, respectively.
`
`teresis indicates that the films are in a frozen or blocked
`superparamagnetic state. The blocking temperature T b can be
`estimated from the vanishing of the coercivity, which is
`about 200 K for the 0.8-ML film. Above T b the system be-
`haves like a typical superparamagnet exhibiting no hysteresis
`but having essentially the same M s in an external field. At
`even higher temperatures, above about 270 K, the individual
`spin blocks become magnetically disordered as indicated by
`the vanishing of M s 关Fig. 2共b兲兴.
`To calculate the equilibrium magnetization of the Fe
`stripes, we generalize the one-dimensional Ising model to
`complex block spins. We start from the one-dimensional
`Ising model and obtain21
`
`.
`
`共1兲
`
`sinh共h/k BT 兲
`冑sinh2共h/k BT 兲⫹exp共⫺4J/k BT 兲
`Here h⫽␮0M sHV 0 , where H is the external field, V 0 is the
`volume of the block-spin segments, and J is the exchange
`coupling between the segments. If there was no hysteresis,
`Eq. 共1兲 could be used to fit the experimental magnetization
`curves. This fails, however, since Fig. 2 shows that the hys-
`teresis loops are not restricted to very low temperatures.
`Thus, to achieve a comprehensive description of the stripes it
`is necessary to include nonequilibrium effects. The starting
`point of the consideration of nonequilibrium superparamag-
`netism is the master equation22
`
`⫽
`
`M M
`
`s
`
`teresting to note that the magnetization of the stripes with
`shorter segments 共i.e., 0.3 ML兲 decays to zero rapidly at even
`much lower temperature of 50 K.
`We explain the magnetic behavior in terms of a model
`involving interacting Ising block spins. In particular, the
`measured magnetization hysteresis loops have been fitted by
`the suggested model, which yields fitting parameters such as
`the anisotropy constant K 1 and the volume of the block spin
`units. Although an Ising chain 共or any anisotropic chain兲 is
`nonmagnetic in equilibrium, the validity of this result is
`restricted to a single stripe and the absence of magnetostatic
`interactions. In a real system which consists of a large num-
`ber of parallel stripes, there is a coupling between neighbor-
`ing stripes caused by the magnetostatic and RKKY interac-
`tions. Compared to the in-plane magnetized Fe/W共110兲
`stripes,6 the magnetostatic interaction could be much more
`in the perpendicular magnetized Fe/Cu共111兲
`important
`stripes. The minimization of the total magnetostatic energy
`would lead to an antiferromagnetic interaction between
`the neighboring stripes. The strength of the magnetostatic
`interaction is given by H 0⫽(1/4␲)M st兰(1/r 3)dA, where
`M s is the spontaneous magnetization and t is the thickness.15
`A short calculation yields a simplified form, H 0⫽␲wtM s/
`8␭ 2, where w is the stripe width and ␭ is the distance
`between the neighboring stripes. Taking the typical values
`t⫽0.2 nm, ␭⫽10 nm, and literature data
`w⫽5
`nm,
`␮0M s⫽0.75 T,16 we estimate that the magnetostatic interac-
`tion is of the order of 2 mT. This value is much smaller than
`the coercivity of the measured hysteresis loop, about 100 mT
`at 100 K, and is invisible in the measured hysteresis loops.
`A more difficult problem is the RKKY-type interaction
`mediated by the substrate. Taking a typical coupling strength
`of the order of 0.4 mJ/m2 at a copper interlayer distance of
`1.3 nm 共Ref. 17兲 and rescaling this value according to the
`1/r 2 distance dependence for coupled layers yields interac-
`tion fields comparable to those caused by the magnetostatic
`interaction. The total coupling strength is much lower, since
`the RKKY interaction involves functions with oscillatory pe-
`riods not exceeding a few tenths of a nanometer, whereas the
`scale of the morphological inhomogeneities clearly exceeds a
`few nanometers. In other words, due to the nonideal, patchy
`shape of the stripes and the slightly varying distance between
`the stripes, the positive and negative contributions are largely
`canceled out. Therefore, the RKKY interaction in the quasi-
`one-dimensional Fe/Cu共111兲 films is unlikely to play an im-
`portant role.
`Another possible coupling mechanism between the stripes
`is magnetic quantum tunneling. But the temperature where
`the magnetic tunneling is effective has been observed below
`10 K, and in the tunneling regime the magnetization relax-
`ation should be temperature independent.18–20 Since the de-
`caying rate of the magnetization of the Fe stripes is strongly
`temperature dependent, the magnetization relaxation appears
`to be a thermally activated process rather than the magnetic
`quantum tunneling.
`If the interaction between the stripes is negligible, the
`equilibrium of the system is superparamagnetic. There are
`interacting and therefore correlated block spins having par-
`allel spin alignments inside each block, but due to thermal
`activation there is no long-range ferromagnetic order. The
`fact that the magnetization curves of the Fe films show hys-
`
`兲
`
`dP共s Zi
`dt
`
`⫽W 共⫺s Zi
`
`!s Zi
`
`兲P共⫺s Zi
`
`兲⫺W 共s Zi
`
`!⫺s Zi
`
`兲P共s Zi
`
`兲,
`共2兲
`⫽⫾1 indicate whether the magnetization of the
`where s Zi
`ith block-spin segment points in the ⫹ez or ⫺ez directions,
`!⫺s Zi
`respectively. P(s Zi
`) denote probabilities
`) and W(s Zi
`and transition rates, respectively. The Ising model has no
`inherent dynamics, so that we choose22,23
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`W共s Zi
`
`!⫺s Zi
`
`兲⫽
`
`冊
`
`⌫ 0
`2
`
`exp冉 ⫺
`⫻冉 1⫺s Zi
`
`tanh
`
`冊 ,
`
`共3兲
`
`V 0K 1
`k BT
`h⫹Js Zi⫹1
`⫹Js Zi⫺1
`k BT
`where the attempt frequency ⌫ 0 is of order 109 – 1012 s⫺1
`共see, e.g., Ref. 24兲. After some calculation we obtain for
`small J and narrow hysteresis loops
`
`also compatible to the actual geometry of the segments. Tak-
`ing the average width of 4 nm and height of 0.2 nm, for the
`0.3-ML stripes (V 0⫽25 nm3) we estimate the length of the
`block-spin unit to about 30 nm. This agrees, as shown by the
`line profile in Fig. 1共a兲, well with the average length of the
`segments in the stripes.
`Therefore, the fitting of Fig. 4 shows that the model gives
`a fair description of the magnetism of the stripes. The fact
`that J has a nonzero value indicates the Fe stripes are differ-
`ent from a system consisting of isolated clusters or islands,
`whose J should have zero value. The restriction of the model
`to small J is reasonable for not too large coverages. Physi-
`cally, the smallness of this value reflects the fact that the
`width of the stripes is not constant but has pronounced
`minima at which the switching of the magnetization is ener-
`getically favorable. There is, in fact, no major difficulty in
`generalizing Eq. 共4兲 to large J, but then there may be other
`magnetization processes such as domain-wall motion along
`the stripes which are not included in the transition rates Eq.
`共3兲.
`In conclusion we have prepared and investigated 1D fcc
`Fe stripes on Cu共111兲. The stripes are characterized by a
`perpendicular anisotropy of order of 0.6 MJ/m3 and exhibit a
`pronounced temperature and time dependence of the magne-
`tization. The magnetic behavior of the stripes is reproduced
`by a micromagnetic generalization of the Ising model, which
`considers coupled block-spin segments. The nonequilibrium
`behavior of the stripes, that is the magnetic hysteresis, is
`explained by Glauber block-spin transitions.
`
`The authors are grateful to Professor U. Gradmann for
`helpful discussions. We also acknowledge F. Pabisch, G.
`Kroder, and J. Barthel for technical support.
`
`冊册
`冉 1⫺tanh2 h
`冉 1⫺tanh2 h
`冉 1⫺4 tanh 2
`
`k BT
`
`冊
`冊册,
`
`k BT
`
`共4兲
`
`r ⌫
`
`h
`k BT
`
`⫺
`
`⫻tanh
`
`⫻冋 1⫺
`
`2J
`k BT
`
`⫽冋 1⫹
`
`M M
`
`s
`
`4Jr
`J
`⌫k BT
`k BT
`where ⌫⫽⌫ 0exp(⫺V0K1 /kBT) and r⫽(1/M s)(⳵H/ ⳵t).
`In
`agreement with Fig. 3,
`it has been observed that
`larger
`sweeping rates r lead to wider loops. Using Eq. 共4兲 we have
`fitted four typical hysteresis loops of 0.3 ML at 50 K, 0.6 ML
`at 163 K, 0.8 ML at 182 K and 200 K, respectively. For all
`four curves the sweeping rate r is about 0.04 T/s. The fitted
`curves as well as the fitting parameters 共K 1 , J, and V 0兲 are
`shown in Fig. 4. The anisotropy constant K 1 is in the range
`between 0.55 and 0.80 MJ/m3. These values are an order of
`than the bulk anisotropy of ␣-Fe
`magnitude
`larger
`(0.05 MJ/m3), indicating that the surface anisotropy is the
`dominant contribution to K 1 . The fitted V 0 increases with
`increasing thickness, which is consistent with the STM im-
`ages 共see Fig. 1兲 showing that the segments in the stripes
`become longer with increasing thickness. Moreover, V 0 is
`
`*Corresponding author.
`1R. P. Erickson and D. L. Mills, Phys. Rev. B 43, 11 527 共1991兲.
`2U. Gradmann, J. Magn. Magn. Mater. 54-57, 733 共1986兲.
`3G. F. Newell and E. W. Montroll, Rev. Mod. Phys. 25, 159
`共1953兲; 25, 353 共1953兲.
`4E. Ising, Z. Phys. 31, 253 共1925兲.
`5L. J. de Jongh and A. R. Miedema, Adv. Phys. 23, 1 共1974兲; M.
`Steiner, J. Villain, and C. G. Windsor, ibid. 25, 87 共1975兲.
`6H. J. Elmers, J. Hauschild, H. Ho¨che, U. Gradmann, H. Bethge,
`D. Heuer, and U. Ko¨hler, Phys. Rev. Lett. 73, 898 共1994兲.
`7I. S. Jacobs and C. P. Bean, in Magnetism, edited by George T.
`Rado and Harry Suhl 共Academic Press, New York, 1963兲, Vol.
`III, p. 300.
`8U. Gradmann and P. Tillmanns, Phys. Status Solidi A 44, 539
`共1977兲.
`9Y. Darici, J. Marcano, H. Min, and P. A. Montano, Surf. Sci. 195,
`566 共1988兲.
`10D. Tian, F. Jona, and P. M. Marcus, Phys. Rev. B 45, 11 216
`共1992兲.
`11M. T. Kief and W. F. Egelhof, Jr., J. Vac. Sci. Technol. A 11,
`1661 共1993兲.
`12A. Brodde, K. Dreps, J. Binder, Ch. Lunau, and H. Neddermeyer,
`Phys. Rev. B 47, 6609 共1993兲.
`13M. Klaua, H. Ho¨che, H. Jenniches, J. Barthel, and J. Kirschner,
`Surf. Sci. 共to be published兲.
`
`14J. Kohlhepp, H. J. Elmers, S. Cordes, and U. Gradmann, Phys.
`Rev. B 45, 12 287 共1992兲.
`15There is a logarithmically diverging self-energy contribution in
`this expression, which describes the interaction inside the stripes
`but does not contribute to the interaction between stripes.
`16U. Gradmann, W. Ku¨mmerle, and P. Tilmans, Thin Solid Films
`34, 249 共1976兲.
`17M. T. Johnson, S. T. Purcell, N. W. E. McGree, R. Coehoorn,
`J.aan de Stegge, and W. Hoving, Phys. Rev. Lett. 68, 2688
`共1992兲.
`18Eugene M. Chudnovsky, J. Appl. Phys. 73, 6697 共1993兲.
`19J. Tejada, X. X. Zhang, and LI. Balcells, J. Appl. Phys. 73, 6709
`共1993兲.
`20B. Barbara, L. C. Sampaio, J. E. Wegrowe, B. A. Ratnam, A.
`Marchand, C. Paulsen, M. A. Novak, J. L. Tholence, M. Uehara,
`and D. Fruchart, J. Appl. Phys. 73, 6703 共1993兲.
`21J. M. Yeomans, Statistical Mechanics of Phase Transitions 共Uni-
`versity Press, Oxford, 1992兲.
`22K.-H. Fischer and A. J. Hertz, Spin Glasses 共Cambridge Univer-
`sity Press, Cambridge, England, 1991兲.
`23Except the anisotropy prefactor, which is necessary to describe
`the perpendicular anisotropy of the film, Eq. 共3兲 gives the well-
`known Glauber transition rates.
`24P. Gaunt, J. Appl. Phys. 59, 4129 共1986兲.
`
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