PHYSICAL REVIEW B VOLUME 58, NUMBER 5 1 AUGUST 1998-I Magnetic behavior and resistivity of the domain-wall junction GdFe 1000 Å.../TbFe/GdFe 500 Å... S. Mangin Laboratoire de Magne´tisme Louis Ne´el, CNRS, BP166, 38042 Grenoble cedex 9, France and Laboratoire de Physique des Mate´riaux, Universite´ Henri Poincare´-Nancy I, BP 239, 54506 Vandoeuvre les Nancy cedex, France G. Marchal and C. Bellouard Laboratoire de Physique des Mate´riaux, Universite´ Henri Poincare´-Nancy I, BP 239, 54506 Vandoeuvre les Nancy cedex, France W. Wernsdorfer and B. Barbara Laboratoire de Magne´tisme Louis Ne´el, CNRS, BP166, 38042 Grenoble cedex 9, France Received 6 August 1997; revised manuscript received 14 January 1998 A GdFe/TbFe/GdFe trilayer constitutes a magnetic nanostructure: the domain wall junction. With this device, we studied the propagation of 180° domain walls from one GdFe layer to the other, through a single planar defect the thin TbFe layer that acts as an artificial energy barrier. Before crossing the energy barrier, by thermal activation, due to the applied magnetic field, the domain walls are compressed against the TbFe layer. Nucleation, compression, and propagation phenomena of 180° domain walls are presented. The behavior of domain walls is followed from the electrical resistivity of the sample. A parallel between the domain-wall decompression and the exchange biasing problem in ferromagnetic/antiferromagnetic bilayers is proposed. S0163-1829 98 01629-4 I. INTRODUCTION magnetic material and the associated potential-energy bar- rier. The propagation of magnetic domain walls in ferromag- In fact, the motion of the domain wall is achieved by the netic or ferrimagnetic materials is known to be of consider- application of an external magnetic field that produces a able importance for magnetic applications. Indeed, depend- magnetic pressure on the domain wall.1 The displacement ing on the ability of its domain walls to move, a material is induced by the magnetic pressure is such that the magnetic classified among the hard or among the soft magnetic domains orientated parallel to the field grow. As shown in materials.1 Moreover, the propagation of the domain walls is the potential-energy diagram of Fig. 1 b , the magnetic field of great interest for theoreticians studying the dynamics of adds a linear Zeeman contribution 2MSH x to the zero- assemblies of spins: at high temperature, where the thermal field potential.4 activation plays an important role, or at low temperature, The second effect of the magnetic field is to reduce the where quantum effects could arise.2­5 Usually, domain walls height of the potential barrier due to the hard magnetic ma- interact with magnetic or nonmagnetic defects precipitates, grain boundaries, etc. . These defects act as potential wells or as potential barriers that the domain walls have to cross to travel inside the material and to reverse the magnetization. However, in most of the usual macroscopic samples, the de- fects are very complex and present a large distribution of shape and of size, which makes the analysis very compli- cated. In a recent paper, Gunther and Barbara proposed a device called a domain-wall junction DWJ ,4 which can be consid- ered as a model system suitable for the study of the interac- tion between a 180° magnetic domain wall and a well- defined potential-energy barrier. This device consists of a ferromagnetic or ferrimagnetic trilayer system, with an in- plane uniaxial anisotropy, in which a layer of hard magnetic material separates two layers of soft magnetic material. As the potential energy per surface unit of the domain wall is FIG. 1. Potential barrier created by a layer of hard magnetic simply the usual domain-wall energy 4 AK, where K is material deposited between two layers of soft magnetic mate- the uniaxial anisotropy constant and A is the exchange stiff- rial. a under zero magnetic field, b with an applied magnetic ness, the hard magnetic material, which presents the larger field. The black ball represents the domain wall, m* is its mass, anisotropy constant, acts as a potential barrier Fig. 1 a . and v its velocity. E0 is the height of the potential barrier in zero Therefore, to move from one layer of soft magnetic material field. When the field is applied, a 2MSH x component is added to the second one, the domain wall has to cross the hard and the height of the barrier is reduced. 0163-1829/98/58 5 /2748 10 /$15.00 PRB 58 2748 © 1998 The American Physical Society PRB 58 MAGNETIC BEHAVIOR AND RESISTIVITY OF THE . . . 2749 terial Fig. 1 b . The aim of the study of the DWJ is the determination of the field and temperature conditions under which the potential barrier is crossed and by which mecha- nism: thermal activation or quantum tunneling. In a preliminary paper,6 we proposed a DWJ system made of a very thin layer of amorphous Tb45Fe55 alloy deposited between two layers of Gd62Fe38 ferrimagnetic amorphous alloy.7 The amorphicity of the layers generates a structural continuity of the sample and prevents structural defects in the form of grain boundaries, which could generate parasitic potentials. The structural continuity is all the better that Tb and Gd atoms are very similar same size, same chemical behavior and form with isomorphic amorphous alloys iron.8 The large anisotropy of terbium compared to that of gado- linium the second-order anisotropy constant K of terbium is three orders of magnitude larger than that of gadolinium9 makes the TbFe layer a potential barrier for a domain wall FIG. 2. General view of the sample. The length L of the sample that has to move from the first GdFe layer to the second one. is 2 cm and its width is 0.5 cm. The easy magnetization axis is The compositions of the alloys were chosen according to along 0z. The slab ``Fig. 5'' will be enlarged for the description of several criteria. First of all, the alloys had to be amorphous. the magnetic configurations in Fig. 5. The electrical current flowing Then, they had to exhibit a large magnetization density for a along 0y will be used to measure the resistance of the sample in the better sensitivity in magnetic measurements. It was also pref- transverse geometry see Sec. IV . erable to avoid alloy compositions presenting a compensa- tion temperature, which could have added complications. Fi- GdFe layers are e1 1000 Å, e2 500 Å, and the thickness nally, from the data reported by Hansen et al.,10 we chose the of the TbFe layer ranges from e 3 Å to e 15 Å. compositions Gd The alloys were obtained by high vacuum coevaporation 62Fe38 and Tb45Fe55, which exhibit close Curie temperatures and thus exchange coupling constants of of the elements from separate crucibles. The deposition rates the same order of magnitude. Nevertheless, the composition were monitored by quartz oscillating systems, previously of the alloys is probably not critical and a change of the calibrated by optical methods.11 The compositions of the al- compositions of the alloys should not significantly alter the loys were subsequently checked by x-ray analysis and were results. found within 1% of the nominal values. As the quartz oscil- In this paper, we confirm our first data6 and show the lators are sensitive to the mass, the thicknesses of the layers efficiency of the GdFe/TbFe/GdFe domain-wall junction. We were deduced from the density of the amorphous alloys, in present in detail the mechanisms of nucleation, compression, the same way as in Ref. 12. The substrates were glass plates propagation, double compression, and annihilation of the do- on which 100 Å silicon buffer layers were deposited at main walls, as deduced from the magnetization measure- 150 °C just before the cooling down of the substrates, which ments and confirmed by resistivity measurements. The very were kept at 77 K during the coevaporation process. The demonstrative resistivity data displayed in this paper are ob- pressure in the chamber was maintained at 10 8 Torr during tained from a single-domain wall. the deposition of the alloys. The amorphicity of the alloys We also present a diagram indicating at which tempera- was checked by electron microscopy and observation of the ture and under which magnetic field the energy barriers gen- typical diffuse rings consistent with the interference func- erated by very thin TbFe layers from 3 to 9 Å are crossed tions of rare-earth transition metals and amorphous alloys.13 by the domain walls. These very small thicknesses of the The magnetization measurements were performed with a hard magnetic layers should classify the GdFe/TbFe/GdFe conventional superconducting quantum interference device DWJ in the regime II-b of the classification proposed by magnetometer. Each of the curves shown in Figs. 3, 4, and 6 Gunther and Barbara.4 In that regime, the thickness of the were obtained after cooling down the sample from 100 K to hard magnetic layer as well as s, the width of the transition the measurement temperature under a 1000 Oe magnetic region between the two materials, are smaller than the width field applied along the easy axis of magnetization. With this of the domain wall. procedure, the magnetization was kept saturated along the Finally, we focus on the compression of the domain walls cooling field, even after canceling the field. The magnetiza- against the TbFe layers that precedes the crossing of the tion curves presented below were obtained while increasing barriers. This stage is probably typical of systems in which the field in the direction antiparallel to the cooling field. the domain-wall energy of the hard magnetic layer is very Electrical resistivity measurements were performed with a high. A parallel between the domain-wall compression and four-point method in a magnet-equipped cryostat in the same the exchange biasing problem is proposed. way as measurements reported in Ref. 14. A general view of a sample is pictured in Fig. 2. The length L of the sample is 2 cm and its width is 0.5 cm. 0x is II. EXPERIMENTAL PROCEDURE perpendicular to the plane of the sample and, as shown be- low, the easy magnetization axis is along 0z. In the follow- The data presented in this study have been collected from ing, the slab ``Fig. 5'' will be enlarged for the description of GdFe/TbFe/GdFe samples, in which the thicknesses of the the magnetic configurations. The electrical current flowing 2750 S. MANGIN et al. PRB 58 FIG. 3. Hysteresis loops of a single 1000 Å GdFe layer mea- sured at 10 K. The field is applied along the easy axis (M/M FIG. 4. Hysteresis loops collected from a GdFe 1000 Å / S) and perpendicularly to the easy axis (M/M TbFe 3 Å /GdFe 500 Å sample at different temperatures. The first S) . HK is the saturation field of the Stoner and Wohlfarth model Refs. 15 and 16 . parts negative fields of the loops have been obtained after cooling the sample under a 1000 Oe field. The second parts have been collected after the application of a 1000 Oe field at the measure- along 0y will be used to measure the resistivity of the sample ment temperature. H in the transverse geometry see Sec. IV . n1 is the nucleation field for the GdFe 1000 Å layer. Hp is the propagation field passage of the domain wall through the potential barrier . III. MAGNETIC BEHAVIOR change constants J A. In-plane anisotropy axis GdGd , JGdFe , and JFeFe by a relation given for the ferrimagnetic amorphous alloys by Hasegawa19 and First of all, the presence of an uniaxial anisotropy axis in Mimura et al.20 Using the exchange constants published by the plane of the sample is necessary for the creation of 180° Hansen et al.,10 and the interatomic distance given by domain walls. The occurrence of such an axis is clearly dem- Cargill,21 the exchange stiffness constant A of the Gd62Fe38 onstrated in Fig. 3 that represents two hysteresis loops col- amorphous alloy can be estimated to 15 10 8 erg/cm. In lected from a single Gd62Fe38 1000 Å layer with, in both order to evaluate K, we used the Nimura relation with nGd cases, the magnetic field applied in the plane of the sample. 3 and nFe 2. From these A and K values, the domain-wall The field was applied either along the easy axis (M loop or thickness in the GdFe samples is evaluated to about 650 Å. perpendicularly to the easy axis (M loop . Such a behavior This value has to be considered as an estimate at 10%. It is quite consistent with the classical Stoner and Wohlfarth depends on several parameters that are very difficult to ob- model of uniform rotation:15 the M hysteresis loop is rect- tain with accuracy. They are deduced from a mean-field ap- angular, whereas the M hysteresis loop is linear from HK proximation and require the evaluation of the Curie tempera- to HK , where HK is the anisotropy field. HK is at the ture of a lot of amorphous alloys. More significant will be intercept of the linear part and of the saturation magnetiza- the values determined below from the compression of the tion for a review, see, for example, Ref. 16 . As a matter of domain wall. fact, the evaporation of Gd and Fe from two separate cru- cibles positioned symmetrically with respect to the substrate leads spontaneously to the occurrence of an easy magnetiza- B. Nucleation and propagation tion axis in the plane of the sample. The anisotropy axis is Magnetization data collected at different temperatures perpendicular to the vertical plane containing the sources.17 from a GdFe 1000 Å /TbFe 3 Å /GdFe 500 Å sample are The existence of such an anisotropy axis has been observed shown in Fig. 4. When the temperature is lower than 7 K, with different preparation procedures.18 each curve exhibits two magnetization steps. The first step Following Stoner and Wolfarth, the uniaxial anisotropy occurs at a field Hn1 18 Oe, almost independent of the tem- constant K can be deduced from the anisotropy field HK by perature, where the magnetization drops from the saturation the relation K MSHK/2,15,16 where MS is the saturation magnetization MS to a value slightly larger than zero. The magnetization. From very low field 0.25 Oe ac susceptibil- second step occurs at a field Hp(T), where the magnetization ity measurements, the Curie temperature of the Gd62Fe38 falls to MS . In contrast to Hn1 , Hp(T) is strongly tem- 1000 Å sample has been found to be 325 K, which is very perature dependent and increases when the temperature de- close to the value given by the reference paper of Hansen creases. Between the two steps, the magnetization exhibits a et al.10 for this composition. From these authors, the satura- slow decrease on a kind of plateau. tion magnetization MS is close to 1400 emu/cm3, which has As it will be justified in the next sections from the ampli- been checked with an accuracy of 5%. With HK close to 70 tudes of the magnetization drops and from the behavior of Oe, the uniaxial anisotropy constant can be estimated to K the electrical resistivity, the interpretation of the two steps 4.9 104 erg/cm3. In fact, the 180° domain wall is due to and of the intermediate decrease of magnetization is sketched the competition between the exchange and the uniaxial an- in Fig. 5 part of Fig. 2 . In these figures, each arrow is isotropy energies. As a result of this competition, the thick- representative of the magnetization of a plane parallel to the ness of the domain wall is given by A/K, where the surface (0y,0z) of the sample. In each plane referred by its exchange stiffness constant A is related to the usual ex- position x, all the magnetic moments are supposed to be PRB 58 MAGNETIC BEHAVIOR AND RESISTIVITY OF THE . . . 2751 parallel to each other and oriented along the arrow represen- tative of the magnetization of the plane. They form an angle (x) with the direction 0y. 0x is the axis along which the domain wall propagates. However, it is likely that the mag- netic moments do not rotate in the same way along the 0x direction in the whole sample. There are certainly parts of the sample in which the magnetic moments turn clockwise along 0x, whereas in other parts they turn anticlockwise, which makes that the angle is in fact (x) in some parts of the sample and (x) in other parts. At the present time, we have no information on these parts which turn out to constitute ``in-plane domains'' but we believe that they are large enough to neglect the defects that should occur at their boundaries. In the following, we will forget the ``in-plane domains'' and the only domain wall we will consider is that which is parallel to the plane of the sample and that propa- gates along 0x. We will describe the data as if an unique domain wall is present in the whole sample although it is likely that ``identical'' domain walls move simultaneously and in the same way except for the chirality inside each ``in-plane domain.'' Figure 5 a represents the saturated magnetization along the direction of the cooling field. This saturated configuration is maintained until the field Hn1 is applied in the direction opposite to the cooling field. The drop of magnetization at Hn1 corresponds to the nucleation of a magnetic domain in the thicker GdFe 1000 Å layer. The reversal of the magne- tization starts from the outer surface but it is blocked by the TbFe layer Fig. 5 b . The slow continuous decrease of the magnetization between Hn1 and Hp is attributed to the ``compression'' of the domain wall against the TbFe layer Fig. 5 c . It is the result of the magnetic pressure on the domain wall, which is blocked on its other side by the mag- netization of the TbFe layer, stuck itself by its very strong anisotropy. The second drop of the magnetization, which oc- curs at Hp(T), corresponds to the crossing of the potential barrier by the domain wall. The domain wall first reverses the magnetization of the TbFe layer and then that of the FIG. 5. Modelized magnetization configurations in the trilay- GdFe 500 Å layer Fig. 5 d . Hp is called the ``propagation er: a For an applied magnetic field H, 0 H Hn1 ; the magne- field.'' Beyond Hp , the magnetization is completely re- tization of the sample is saturated. b At H Hn1 a domain wall is versed Fig. 5 f . We notice that, when the temperature is created in the thicker GdFe layer. It is stopped by the TbFe layer. larger 7 K , the magnetization switches from MS to MS c For Hn1 H Hp the domain wall is compressed against the at a field referred as Hc1 . At this temperature, the TbFe layer TbFe layer. d At H Hp the TbFe layer magnetization is reversed seems to be inefficient, which is consistent with the fact that, and the domain wall propagates into the thinner GdFe layer. e At as it will be shown below, the crossing of the barrier is a H Hn2 a second domain wall is nucleated in the thinner GdFe thermal activated process. layer. f Beyond Hp or Ha the magnetization is completely re- versed. C. Double compression and annihilation However, below 20 K, the ``compression'' stage does not With the same procedure, hysteresis loops have been col- end at Hp(T) by a full drop of the magnetization to MS . lected from the GdFe 1000 Å /TbFe 9.5 Å /GdFe 500 Å There is instead a limited drop of the magnetization at a field trilayer Fig. 6 . In the temperature range from 30 to 50 K, Hn2 close to 60 Oe. This limited drop is followed by another the same magnetization behavior as above is observed, with slow decrease of the magnetization, which finally ends by a two steps at Hn1 18 Oe and at Hp(T). Above T 50 K, more rapid decrease of the magnetization to MS at a field there is an unique step, where the magnetization switches Ha(T). from MS to MS as in the GdFe 1000 Å /TbFe 3 We attribute the step occurring at Hn2 to the nucleation of Å /GdFe 500 Å sample above T 7 K. The comparison be- a second domain wall in the thinner GdFe 500 Å layer Fig. tween Figs. 4 and 6 shows that the propagation field is trans- 5 e and to the propagation of this new domain wall up to lated towards the highest temperatures when the thickness of the TbFe layer. As it will be explained below, the nucleation the TbFe layer increases. of magnetic domains and the corresponding nucleation fields 2752 S. MANGIN et al. PRB 58 IV. ELECTRICAL BEHAVIOR A. The anisotropy of magnetoresistance It is now well known that the magnetic state of a material can significantly contribute to its electrical resistivity via several different effects. In return, the electrical resistivity can be used to determine or confirm the magnetic structure of a sample.22 In this section, we show how the variation of the electrical resistance of the sample with the field allowed us to ascertain that the magnetic configurations sketched in Fig. 5 represent the actual magnetization processes, giving rise to the behav- ior reported in Figs. 4 and 6. For this purpose, we took ad- FIG. 6. Hysteresis loops from the GdFe 1000 Å /TbFe 9.5 vantage of the significant anisotropy of magnetoresistance Å /GdFe 500 Å sample at different temperatures. The measure- AMR of iron, which has been interpreted by Smit23 and ments were performed after cooling down the sample under a 1000 Fert.24 As shown by these authors, the resistivity of the tran- Oe magnetic field. Only the first part of the cycles has been pre- sition metals and of their alloys depends significantly on the sented. Hn2 is the nucleation field for the GdFe 500 Å layer. Ha is angle between the direction of the magnetization and the the annihilation field. direction of the electrical current used to measure the resis- tivity. The resistivity is enhanced and equal to are related to the thickness of the GdFe layers. The thicker when the electrical current is parallel to the magnetic moments and, at the layer, the higher the nucleation field. After the second the opposite, it is reduced and equal to nucleation, the two domain walls are simultaneously com- when it is perpen- dicular. Finally, when in an homogeneously magnetized tran- pressed against the TbFe layer, which results in a slow de- sition metal sample, the angle between the magnetic mo- crease of the magnetization with the increase of the field ments and the electrical current is , the resistivity can be ``double-compression stage'' . expressed as ( ) Here it is important to emphasize that without the nucle- cos2 where .22 In pure iron, the order of magnitude of ( ation delay in the thinner GdFe layer (H ) / is about n2 Hn1), we never 0.5%.23 could have observed the propagation of the domain wall In the following, we will refer to the geometry in which through the TbFe layer. The asymmetry of the trilayer sys- the electrical current flows along the direction of the applied tem, with two different thicknesses of the GdFe layers, is magnetic field parallel or antiparallel as parallel and to the necessary for the device to operate. geometry in which the electrical current and the magnetic Finally, the last drop of the magnetization is related to the field always applied along the easy magnetic axis 0z in the reversal of the TbFe layer magnetization, which occurs with plane of the sample are perpendicular as transverse the the annihilation of the domain walls at a field Ha(T), the transverse geometry is that represented in Fig. 2 . This so-called annihilation field. Ha is not as well defined as the technique permits for example the determination of the co- other characteristic fields and the transition spreads out on ercitive field of thin films, with the occurrence of peaked several Kelvin. It is about 80 Oe at 20 K and 180 Oe at 10 K. maxima in the parallel geometry or of peaked minima in These fields are relatively high and we can point out the the transverse geometry .25,26 The occurrence of these peaks unexpected stability of the configuration of Fig. 5 e where expresses that, at the coercitive field, a maximum number of the TbFe layer is squeezed between two domain walls. In- spins are oriented rather perpendicularly to the magnetic deed, under the pressure of two domain walls on each side of field. This technique was recently used to show the aligned it, the magnetization of the TbFe layer should switch very and twisted phases in GdFe multilayers.14 It was possible to easily. This is no longer true if the domain walls tend to detect the magnetic field at which the iron moments leave the rotate the magnetization of the TbFe layer in opposite direc- field direction and exhibit a spin-flop-like transition. tions clockwise and anticlockwise and as a consequence The resistance data presented in this section have been cancel each other. A first explanation to this high stability would be that the domain walls located on each side of the collected from GdFe 1000 Å /TbFe 15 Å /GdFe 500 Å TbFe layer are systematically of opposite chirality in all the samples. Golden electrodes have been deposited on the glass ``in-plane domains'' . It could be due to the effect of the substrate prior the elaboration of the sample. The orientation dipolar fields of the first layer, in which domains are already of the substrate during the deposition of the sample had been present, on the second one, in which the nucleation is taking chosen for the easy magnetization axis to be parallel parallel place. A second possibility would be that the shape and the geometry or perpendicular transverse geometry to the cur- distribution of the ``in-plane domains'' in the two GdFe lay- rent flow during the electrical measurement. As above, the ers are uncorrelated. There are some places where the chirali- magnetic field is always applied in the direction of the easy ties of the domain walls located on each part of the TbFe axis direction 0z of Figs. 2 and 5 , whereas the current layer would be opposite and some places where they would flows along the direction 0y. The data presented in Figs. 7 a be identical. The parts where the chiralities would be oppo- and 7 b show the relative variation of the resistance in the site would maintain strongly the magnetization of the TbFe transverse geometry see Fig. 2 . The data have been normal- along the direction of the cooling field. ized to the resistance value measured in the saturated state. PRB 58 MAGNETIC BEHAVIOR AND RESISTIVITY OF THE . . . 2753 resistance of the sample with the resistivity . From C to D, the electrical resistance decreases slowly because of the compression of the domain wall against the TbFe layer. The reason of the decrease of the resistance is that the compres- sion leaves less and less spins in the direction of the electri- cal current. At the propagation field Hp(T), which in the GdFe 1000 Å /TbFe 15 Å /GdFe 500 Å is close to 50 Oe at T 60 K, the domain wall crosses the barrier, and after its traveling through the GdFe 500 Å layer, all the magnetic moments become aligned along the direction of the magnetic field 0z Fig. 5 f . They are again perpendicular to the electrical current point E, Fig. 4 c . As in zero field, the resistivity is uniformly and the resistance recovers its minimal value. The cycling of the field leads to a symmetri- cal behavior about the vertical axis of Fig. 7 a . From resistance measurements performed on a second sample in the parallel geometry we could not use the same sample because the electrodes had to be placed differently , we found that the electrical variations were opposite and symmetrical about the horizontal axis of Fig. 7 a . There is a drop of the resistance at Hn1 , a slow increase between Hn1 and Hp , and a sharp increase at the field Hp beyond which the initial resistance is recovered. In this geometry, all the magnetic moments are parallel to the electric current when there is no domain wall the resistance is the highest and a part of them are perpendicular to the electric current, when a domain wall is present. In that sample, the compression of the domain wall manifests by a slow increase of the resis- tance. FIG. 7. Evolution at 60 K a and at 30 K b of the resistance of a GdFe 1000 Å /TbFe 15 Å /GdFe 500 Å sample with the mag- netic field applied along the easy axis, perpendicularly to the elec- C. Double nucleation and annihilation trical current see Fig. 2 . The first parts negative field have been A second set of resistance curves obtained in the trans- obtained after cooling down the sample under a 1000 Oe field. The second part quite symmetrical about the vertical axis has been verse configuration shows, at 30 K Fig. 7 b , both the obtained after the application of a 1000 Oe field at the measurement double nucleation and the annihilation processes. On that temperature. The notation H curve, the points A, B, C, and D represent similar situations n1 , H p , Hn2 , and Ha is the same as in Figs. 4 and 6. as in curve 7 a . From A to B, the magnetization is uniform along the cooling field and is perpendicular to the electrical Because the TbFe thickness is larger in this sample than current. The step B-C corresponds to the nucleation of the in samples presented above Figs. 4 and 6 , the different domain wall in the thicker GdFe layer. From C to D, the magnetization processes are shifted to higher temperatures. resistance decreases slowly because the domain wall, nucle- ated at Hn1 , compresses against the TbFe layer Fig. 5 c . B. Nucleation and propagation However, beyond D, the evolution of the resistance turns out to be different. First of all, there is a second step D-F, which The field dependence of the electrical resistance measured corresponds to the nucleation of the second domain wall in at 60 K in the transverse geometry after the cooling under a the GdFe 500 Å layer Fig. 5 d . With two domain walls, 1000 Oe field is shown in Fig. 7 a . At point A, the electrical more magnetic moments exhibit a component along the elec- resistance is minimum: all the magnetic moments are par- tric current and as a consequence, the electrical resistance is allel to the magnetic field and thus, because of the transverse still higher. From F to G, the resistivity decreases slowly in geometry, the magnetization is perpendicular to the electrical the same way as between C and D, which is typical of the current. The resistivity is in the whole sample. From A to compression of domain walls: here, two domain walls are B, the resistance is kept unchanged because the saturated compressed simultaneously. Finally, beyond G, the resistiv- state is maintained. Then, when the magnetic field reaches ity decreases rapidly with the reverse of the magnetization of Hn1 , the resistance increases sharply and reaches its maxi- the TbFe layer and the annihilation of the domain walls. mum value at C. At this point, the magnetic moments lo- Again, the cycling of the field gives symetrical curves about cated in the domain wall Fig. 5 c are no longer perpen- the vertical axis of the figure. dicular to the electrical current and they exhibit a parallel As at 60 K, the transverse resistivity measurements per- component. The magnetic moments of a layer confined be- formed on a second sample where the electrodes had been tween x and x dx form a (x) angle with the current flow placed appropriately exhibit opposite curves. The steps of that is along 0y. Each layer contributes to the total electrical increase of resistivity are replaced by steps of decrease, 2754 S. MANGIN et al. PRB 58 which once more supports the interpretation of the data by the mechanism of magnetoresistance anisotropy. D. Resistivity of a domain wall Beyond the qualitative aspects developed above to under- stand the behavior of domain walls, it would be of interest to perform a quantitative analysis of the resistivity of one do- main wall. It certainly requires more measurements from dif- ferent samples in parallel and transverse geometry, to see for example if other electron scattering mechanisms are involved.26­28 However, as the AMR is undoubtedly the dominant effect, a first simple analysis of the variation of the resistance with the domain-wall compression can be pre- FIG. 8. Variation of the propagation field Hp as a function of the sented. temperature for a selection of GdFe 1000 Å /TbFe/GdFe 500 Å As it is well known, in a thin-film sample whose length samples. The values indicated on each curve correspond to the direction of the current flow is L, whose width and thick- thickness of the TbFe layers. ness are w and e, respectively, and in which the resistivity is a function of x 0x is the direction perpendicular to the plane thickness , the resistance of the rest of GdFe layer whose of the sample , the resistance is given by thickness is e1 and the GdFe layer. The resistivity of the last two parts is . 1 w e dx This gives R L . 0 x 1 w e 1 1 e2 , If the spatial variation of the resistivity is due to the ori- R L RDW entation of the spins AMR , which are supposed to be par- or allel to each other in the layer comprised between x and x dx, but which rotate along the 0x direction, the resistivity 1 1 is (x) 1 1 cos2 (x). (x) is the angle between the R R e direction of the spins and the electrical current. The resis- 0 1 e2 tance of the sample can be expressed by with , R ( ) is an increasing function of . The nucleation of a domain wall produces a positive step of re- 1 w 1 e dx sistance, its disappearance a negative step and the domain- R L . 0 wall compression, a slow decrease of the resistance. 1 cos2 x V. DISCUSSION Let us now define a ``single domain-wall sample.'' It is a hypothetical sample in which a domain wall spreads from a In this section, we first present and discuss the thermal surface of the sample (x 0) to the other one (x e ). In dependence of the propagation field Hp(T) on the thickness such a sample, the spins located on the first surface, at x of the TbFe layer. In the second part, we focus on the nucle- 0, are parallel to the field direction. Those in contact with ation fields Hn1 and Hn2 and propose an estimate of these the second surface at x are oriented antiparallel to the quantities. For this purpose, we use a very simple model field. Between x 0 and x , we can assume that, as in a derived from those which have been proposed to evaluate the first approximate in domain walls,29 the rotation varies lin- exchange bias field in ferromagnet-antiferromagnet bilayers. early with x. Therefore, in the longitudinal geometry where We show that, in our system, as in exchange biased layered the electrical current is parallel to the field, (x) x/ . At structures, a shift of the hysteresis loop appears. In the third the opposite, in the transverse geometry where the electrical part, we present an analysis of the compression and double- current is perpendicular to the field: (x) /2 x/ . compression stages of the magnetization curves and evaluate As a result, because in both geometries the distribution of the thickness of the compressed domain wall as a function of the spins spreads with an equal weight in all the directions of the applied magnetic field. the space, the resistance of a ``single domain-wall sample'' of thickness is given by the same expression: A. Thermal dependence of the propagation field The propagation fields Hp(T) deduced from a set of dif- L RDW ferent GdFe 1000 Å /TbFe/GdFe 500 Å samples whose L w 1 w . TbFe thicknesses were between 3 and 9.5 Å are plotted in Fig. 8. As it can be seen, the propagation fields are restricted If the sample is a GdFe/TbFe/GdFe trilayer where e e1 to the range 20­ 60 Oe. The reason is now obvious: the and e2 and in which a domain wall of thickness has devel- propagation fields can only be determined beyond the first oped see Fig. 5 c , the total resistance R ( ) in the trans- nucleation field Hn1 and below Hn2 , the field at which the verse geometry can be considered as the result of three re- propagation is bypassed by the nucleation of a domain wall sistances in parallel: the resistance of the domain wall of in the thinner GdFe layer. PRB 58 MAGNETIC BEHAVIOR AND RESISTIVITY OF THE . . . 2755 FIG. 9. Evolution of H FIG. 10. Decompression of the domain wall in a GdFe 1000 n1 , the nucleation field in the GdFe(e1), as a function of e Å /TbFe 4.5 Å /GdFe 500 Å sample. The minor cycle has been 1 and of Hn2 , the nucleation field in the GdFe(e obtained after reversing the field at point A in the compression 2), as a function of e2 . stage. At a given temperature T, the field to be applied for the crossing of the barrier increases with the thickness of the At this stage, we propose a parallel between the problem TbFe layer. On the other hand, under a given magnetic field, of nucleation in the GdFe layer with formation of domain a larger temperature is required for the crossing of a thicker wall against the TbFe layer and the exchange biasing prob- TbFe layer. From these observations, we can conclude that lem, which is currently a subject of great interest.30­32 The the height of the barrier increases with the TbFe thickness exchange biasing occurs in ferromagnetic-antiferromagnetic and that the crossing of the barrier is a thermally activated bilayers where the reversal of the magnetization of the fer- phenomenon. romagnetic layer is accompanied by the occurrence of an interface energy. The exchange biasing field that is actually B. Nucleation of the domain walls and exchange close to our nucleation field is the result of the balance biasing problem between the Zeeman energy and the interface energy. It is usually given by H From a lot of samples prepared with different thicknesses EB /2M Se, where e is the thickness of the ferromagnetic layer and is the interface energy. e1 , e2 , and e, we have observed that the nucleation field This relation assumes that the reversal of the magnetic mo- Hn1 depends actually only on the thickness e1 of the layer in ments of the ferromagnetic layer is complete and that the which the nucleation occurs. Likewise, the nucleation field interface energy is rather localized in the antiferromagnetic Hn2 depends mainly on e2 . It is important to note that Hn1 layer, for example with the formation of a domain wall in- and Hn2 are independent of the TbFe layer thickness. Hn1 side that layer.33 In our system, where the soft antiferromag- and Hn2 increase when e1 and e2 , respectively, decrease netic layer is replaced by a very hard ferromagnetic layer, the and, as a matter of fact, Hn1(e1) and Hn2(e2) follow about interface energy is simply the domain wall energy that the same curve Fig. 9 . Such a behavior can be easily un- is localized inside the soft magnetic material. The difference derstood if we consider that, in both cases, a domain wall is between the expressions of H created against the TbFe layer whose magnetization is at first EB and Hn1 stands in the sub- stitution of e by (e /2). The substraction of /2 is due to kept unchanged Fig. 5 b . the space occupied by the domain wall inside the GdFe 1000 For the nucleation process, two energies have to be con- Å layer instead of inside the antiferromagnetic layer in the sidered: i the Zeeman energy and ii the domain-wall ferromagnetic/antiferromagnetic bilayers . energy. Consider Fig. 5 the trilayer system submitted to the Here, likewise in the exchange biasing problem, the do- field H antiparallel to the cooling field, before the nucleation main wall acts as an elastic spring compressed by the mag- Fig. 5 a and after the nucleation of the domain wall Fig. netic pressure. When the magnetic field is lowered, the mag- 5 c . The occurrence of the domain wall lowers the Zeeman netic pressure is reduced and the energy stored in the domain energy because of the complete reversal of the magnetization wall is large enough to reverse the magnetization of the layer in the thickness (e1 ) and because of the semireversal of in the direction of the cooling field, even before the field the magnetic moments of the domain wall. The variation of returns to zero. As a consequence, in both cases, the hyster- Zeeman energy per surface unit is 2MSH (e1 /2). On esis loop is shifted. Figure 10 shows the decompression ef- the other hand, the occurrence of the domain wall increases fect which occurs in a GdFe 1000 Å /TbFe 4.5 Å /GdFe 500 the energy by 4 AK. As a consequence, a critical field Å sample, when the field is reversed from point A, located larger than /2MS(e1 ) is required for the nucleation of a before the propagation field. It appears as a minor cycle be- domain. With e1 1000 Å, 700 Å, M 1400 emu and cause only the magnetization of the GdFe 1000 Å is con- 0.35 erg/cm3, we find Hn1 28 Oe. The estimated value cerned. During this cycle, the magnetization of the GdFe 500 is of the right order of magnitude, even if this relation rep- Å layer is unchanged. resents a first approximate. It has to be improved to take into account the fact that the shape of the domain wall is modified by the field and that and themselves depend on the mag- C. Thickness and compression of the domain walls netic field. However it clearly shows that the critical field We focus now on the compression of the domain walls Hni (i 1,2) increases when ei decreases. against the TbFe layer. The thickness of the domain walls 2756 S. MANGIN et al. PRB 58 at 10 K, between 70 and 180 Oe in Fig. 6. If we assume that the two domain walls are identical on each side of the TbFe layer, their thicknesses are related to the magnetization by (H) M(H) (e1 e2) (e1 e2) /2MS. The values de- duced from experimental data, by using this relation, are reported in Fig. 11. They are quite in agreement with the values deduced from a single compression. Finally, the experimental width (H) has been compared with the wall thicknesses deduced from the simulation of a linear chain of spins continuous line in Fig. 11 , in which each spin is submitted to an external field H, to an uniaxial anisotropy K and interacts with its neighbors via an ex- change constant A. In this simulation, the last spin of the FIG. 11. Evolution of the thickness of the domain wall as a chain was fixed antiparallel to the external field and the other function of the applied magnetic field. Symbols are used to plot data ones were free. An agreement between experimental determined from the experimental magnetization measure- data and the simulation leads to an A value of about ments: Circles: GdFe 1000 Å /TbFe 9.5 Å /GdFe 500 Å at 5 K 17 10 8 erg/cm, which is quite consistent with the value for 20 Oe H 60 Oe in the compression domain. Triangles: deduced from the exchange constants given by Hansen GdFe 1000 Å /TbFe 9.5 Å /GdFe 200 Å at 5 K for 20 Oe et al.10 H 140 Oe in the compression domain. Squares: GdFe 1000 Å /TbFe 9.5 Å /GdFe 500 Å at 5 K for 70 Oe H 200 Oe in the double compression-domain. The continuous line corresponds to a VI. CONCLUSION simulation of a linear chain of spins with HK 70 Oe and A 17 10 8 erg/cm. With the GdFe 1000 Å /TbFe/GdFe 500 Å trilayer sys- and their compression are shown by the amplitudes of the tem, this paper reports on the elaboration of the domain-wall magnetization steps that occur at H junction as suggested in a theoretical paper by Gunther and n1 and Hn2 and by the slow decrease of the magnetization in the ``compression'' Barbara.4 We first studied the conditions in which a domain stage between H wall could be nucleated in one of the GdFe layers and pro- n1 and H p(T) or in the double-compression stage between H posed an asymmetrical device with two different GdFe layer n2 and Ha . The first important point to con- sider for the evaluation of the thickness of the domain wall is thicknesses. This disymmetry leads to two different nucle- the amplitude M of the magnetization step at the nucle- ation fields Hn1 and Hn2 in the two GdFe 1000 Å and ation field H GdFe 500 Å layers, and leaves the 20­60 Oe magnetic-field n1 . Indeed, the reversal of the magnetization of the whole GdFe(e1) layer at Hn1 without formation of do- range available to study the propagation of the domain wall main wall , would produce a reduced magnetization step nucleated in the GdFe 1000 Å . On the other hand, we M/2MS e1 /e1 e2 . After the step, the magnetization showed that, when the propagation field Hp is ``virtually'' should be stabilized up to Hn2) on a plateau at a reduced larger than 60 Oe, two domain walls were nucleated on each magnetization M/MS (e2 e1 )/(e1 e2) . With e1 side of the TbFe layer and the propagation was bypassed. 1000 Å and e2 500 Å, the magnetization plateau should The occurrence of the two domain walls has been estab- be at M/MS 0.33. But, as seen in Figs. 4 and 6, the lished on one hand by the amplitude of the magnetization magnetization step which occurs at Hn1 is significantly steps that occur at the nucleations, and on the other hand by smaller. The missing part of amplitude of the step is due to the anisotropy of magnetoresistance that is related to the the occurrence of the domain wall and to the fact that, in amount of magnetic moments along the directions parallel average, the variation of the magnetization of the spins in- and perpendicular to the electrical current. From the magne- volved in the domain wall is MS instead of 2MS . The am- tization data, we could determine the variation of the width plitude of the step becomes M/2MS (e1 )/(2/e1 e2) of the domain walls as a function of the field in the compres- and, between Hn1 and Hp , the width of the domain wall is sion as well as in the double-compression stage. then related to the magnetization M(H) by the relation Then, we established that the height of the energy barrier (H) M(H) (e1 e2) (e1 e2) /MS. Therefore, the fact due to the presence of the TbFe layer in GdFe/TbFe/GdFe that in each sample where e1 1000 Å and e2 500 Å, M is increases with the TbFe thickness and that the crossing of the equal to 0 at H 35 Oe, which means that for this field 35 barrier was thermally activated. The height and the shape of Oe is equal to 500 Å. The thickness of the domain wall is the potential barrier as a function of the TbFe thickness and largely independent of the TbFe thickness and of the tem- of the applied magnetic field is still to determine quantita- perature. tively. This is currently performed by relaxation measure- Thus, from the field dependence of M(H) in the ``com- ments of the magnetization. On the other hand, it has been pression'' stage and by using the above relation, we have shown that at very low temperatures, a crossing of the barrier deduced the field dependence of (H). In Fig. 11, we by the tunnel effect was possible.34 present the results obtained from the GdFe 1000 An important problem is the way the domain wall propa- Å /TbFe 9.5 Å /GdFe 500 Å sample. gates through the TbFe layer. In fact, we suggest that, re- Let us now come to the simultaneous compression of the garding Gunther and Barbara's analysis,4 our system consti- two domain walls represented in Fig. 5 e , which occurs be- tutes an unconventional domain-wall junction because of the tween Hn2 and Ha . Such a double compression explains the sperimagnetic structure of the TbFe amorphous alloy and of slow decrease of the magnetization, as it occurs, for example, the large random anisotropy of terbium in this amor- PRB 58 MAGNETIC BEHAVIOR AND RESISTIVITY OF THE . . . 2757 phous alloy.15 We believe that at Hp , under the pressure of An important feature of this system is the occurrence of a the domain wall and because of the low temperature, the kind of ``exchange biasing field'' referred to here as nucle- magnetization of the TbFe layer ``switches'' between two ation field. It seems clear that a parallel between the ex- states. This is related to the shape of the hysteresis loops of change biasing and the domain-wall junction problems has to the bulk TbFe amorphous alloy. In such systems with a be made. strong random anisotropy, the loops are pretty square and are Finally, our system can be fruitfully considered for the interpreted by the collective spin flop of spins along the local study of the magnetoresistivity of the domain walls that is easy axis. In a recent simulation, Saslow and Koon35 showed currently a field of intense interest. In our study, we used the that half of the spin flips occured at the vicinity of the coer- anisotropy of magnetoresistance to get information on the citive field. This means that the GdFe/TbFe/GdFe system magnetic configuration of the sample. We are currently could be analyzed in a different way and that an extended studying the resistivity in detail to quantify the anisotropy of classification of the domain-wall junctions has to be consid- resistivity and determine if other fundamental magnetoresis- ered. tivity mechanisms are involved and could be shown. 1 D. 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