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Anisotropic super-paramagnetism in cobalt implanted rutile-TiO2 single crystals

We study the magnetic properties of single crystals of rutile TiO2 implanted with cobalt for various fluences. The temperature variation of zero field cooled(ZFC) and field cooled (FC) magnetization shows a much higher blocking temperature (TB) along [1-10]. Similarly the scaling of magnetization isotherms above TB is seen only when the field is parallel to [1-10] direction. With field along this direction, the magnetization shows near saturation at a much smaller field compared to that of[001] direction. The Co nanoclusters possess an"easy"and"hard axis"of magnetization coupled by the magneto crystalline anisotropy of secondary phases of cobalt with TiO2. In addition, at T=2 K we observe a crossover in the magnetization vs field isotherms between the two field directions in the samples which has been attributed to the anisotropic paramagnetism arising from cobalt present in 2+ ionic state with S = 3/2.


I. INTRODUCTION
The study of magnetism in nano-particles has gained enormous interest in last two decades from technological as well as fundamental perspectives 1 . In nanoscale systems, magnetic nature can be drastically different, compared to a bulk, since surface effects play crucial role in determining this behavior such that properties as diverse as ferromagnetism, anti-ferromagnetism, super-paramagnetism (SPM) or spin-glass(SG) like behavior are observed [1][2][3][4] . Among these, SPM is a property that crucially depends on the size of the nano-particle and shows magnetic moment that is proportional to the particle-volume 3 . SPM nano-sized cluster can exhibit giant magnetic moment (sometimes as high as few thousand µ B ) which is randomly oriented in the absence of external field. The particles are non-interacting, except for a weak dipole interaction. Thus, an ideal super-paramagnet should exhibit paramagnetic behavior, i.e follow Curie -Weiss law, but with a large effective moment and a non-hysteretic M − H curve up to 0 K. However, effects of magneto-crystallinity as well as surface and shape anisotropy alter this behavior such that a real super-paramagnet shows deviations from the Curie-Weiss law at the non-zero temperatures 5 . The super-paramagnetic clusters possess a uniaxial anisotropic direction which is random in direction for each SPM particle. Thus every individual nano-particle has its corresponding easy axes of magnetization.
As the system is cooled through the SPM state, there comes a characteristic temperature called the blocking temperature (T B ). Above T B , the magnetic moment of the individual SPM particle is oriented randomly like a normal paramagnet, which can rotate freely under the influence of external field. Below T B , the individual SPM particle has its magnetic moment blocked along its respective easy anisotropy axis. This temperature is prominently seen as a bifurcation of zero field cooled (ZFC) and field cooled (FC) magnetization. Associated with this temperature is an energy barrier U , which is the energy required for the individual magnetic moment to flip its direction along the two easy axes directions. The time for flip is giving by the characteristic equation 3 , Here τ 0 is the limiting relaxation time, k B is the Boltzmann constant, T is the temperature and U is the potential barrier. The blocking temperature is affected by individual particle volume and their distribution 6 (also see table 1). At the highest fluence 1×10 17 ions/cm 2 (sample C), the difference in T B of 80 K between H and H ⊥ , indicates that this system is very anisotropic.
For H field, magnetization in samples B and C appears nearly constant, with temperature, above T B . The decrease in magnetization is much slower than the expected Curie-Weiss like decrease in SPM systems. However for H ⊥ field, a decrease in magnetization with temperature is observed simalr to that expected in SPM systems. Fig. 2 Thus the magnetic moments of the clusters in this cobalt implanted system are rotatable, due to external field and temperature, only when the field is along H ⊥ direction. In a typical SPM, well below T B , the ZFC susceptibility increases with increasing temperature, suggesting that more nano-particles get unblocked and contribute to the susceptibility. Most of the particles get unblocked near T B and the system becomes an SPM for T > T B . Thus irrespective of field direction all SPM particles get unblocked above T B and display a Curie-Weiss like behavior with increasing temperature. Such a behavior is observed, above T B , in our system only for H ⊥ . For field along H , the magnetization above T B remains nearly constant with increasing temperature. This suggests that along this direction, only the smaller nano-particles get unblocked above T B , whereas the larger particles remain blocked.
Combined anisotropy due to the Co clusters and Ti 1−x Co x O 2 will be responsible for this observation, as will be discussed below.
C. Magnetization vs. Field  fig. 3(a)) for H and H ⊥ . At 300 K, the magnetization rises linearly as expected for a paramagnetic TiO 2 , without attaining any saturation. A small (< 20 Oe) coercivity has also been observed. Moreover, the magnetization is higher along H (than H ⊥ ) indicating this to be the easy axis of magnetization in the pristine sample. Similar behavior is also observed at 2 K. The magnetization curve for Sample A at 100 K along H ⊥ is shown in fig. 3(a) and shows a saturation-like behavior near 1 T. This curve also displays a very small coercivity of 30 Oe. At this fluence the cobalt concentration in TiO 2 lattice is small and consequently the Co induced magnetic moment is comparable to the pristine. Assuming that the magnetization from the host lattice and from implanted ions are independent in any implanted system, so the the contribution to magnetization of the former has been subtracted from that of all the implanted samples. After subtracton of the pristine magnetization, the magnetization plots show a saturation like behaviour for all temperatures, which is the true behaviour of the SPM system.

Above Blocking temperature
The magnetization plots for samples B and C, at 300 K, are shown in fig. 3 fig. 3 show that the SPM -like behavior is observed in samples B and C only, when the field is applied along H ⊥ .
In a superparamagnet, there exists a distribution of magnetic moments due to the variations in the particle size of the nanoclusters. Hence, the net magnetization is given as a weighed sum of the Langevin function 32 , where L(x) = coth(x) − 1/x is the Langevin function, f (µ) is the distribution of magnetic moments, given by a log-normal distribution 5 , Here µ 0 is the median of distribution and σ is the width of this distribution. The mean magnetic moment µ M = where the role of anisotopy can be neglected.
In many SPM systems, equation (2) has been shown to yield good results for magnetization well above T B . The fittings for the magnetization curves, using this eqn. (2), have also been shown in fig. 3(b,c) for samples B and C when the fields is along H ⊥ . Fittings of the magnetization curves have been utilized to obtain average magnetic moment (µ M ), particle size (D) and standard deviation (σ) and are listed in table 1. With increasing fluence there is a systematic increase in the average magnetic moment as well as the particle diameter. In addition, the deviation σ also considerably increases from sample A to C. This is also observed in the ZFC-FC plots, wherein sample A displays a sharper transition while sample C shows a broader transition. The magnetization curves for the implanted samples below T B , at 2 K, are presented in fig. 4. Inset shows the magnetization curves of sample A for both H ⊥ and H . Small coercivity is observed along both these fields.
Magnetization curves for the pristine at 2 K are similar to those of sample A. Also as demonstrated by sample A ( fig. 4 inset), the magnetization in pristine does not saturate even at high fields (2 T) but rather continues to increase, indicating a paramagnet-like behavior.  4). Moreover, the slope (dM/dH) here at 2 K is higher than that observed above T B at 300 K ( fig. 3(b,c)). A crossover in M between the two field directions (indicated by the arrows in fig. 4) is also observed at 2 K for both the samples. This suggests an additional contribution to the magnetization at low temperatures and high fields which will be discussed below in the section on anisotropic paramagnetism. The hysteresis loop appears like a "parallelogram" for H ⊥ , indicative of the easy axis of magnetization, but narrow "ribbon-like" for H , similar to the hard axis of a ferromagnet.

IV. DISCUSSION
A. Anisotropy

Super-paramagnetic region
We briefly discuss the effect of anisotropy on the magnetic behaviour of Co nano clusters in the super-paramagnetic regime. We demonstrate that the Co nano-particles are not randomly oriented, like usual SPM clusters, rather have a fixed easy axis direction even above the blocking temperature. When T >> T B , effects of uniaxial anisotropy can be neglected and magnetization can be completely described by eqn. crystallographic directions is exactly equivalent to applying magnetic fields along the "easy" and "hard" axis of the Co clusters.
This confirms that the magnetic anisotropy directions of the clusters, in the present system, are not entirely random as expected in a super-paramagnetic system but are effectively along the <110> crystallographic direction.
This also suggests that the anisotropy of the Co clusters is coupled to the magneto-crystalline anisotropy of the TiO 2 /Ti 1−x Co x O 2 lattice which would be subsequently discussed.

Anisotropy Below the Blocking Temperature
The M − H plots of samples B and C for 2 K show a large anisotropic character similar to that of the bulk metallic Co. The uniaxial anisotropy of the system is given by K µ = M S H K /2, where H K is the anisotropy field 33 , which is also the saturation field for the hard axis. In both the samples, the saturation along the hard axis (H ) is The net magnetic moment ∼0.7 µ B obtained here is lower than 1 µ B expected for the S=1/2 system, suggesting an itinerant character. Among the three spin configurations, the system has the lowest energy when spin is along H ⊥ , while energy is highest along H direction. This is in agreement with the experimental results observed here. The difference in energy, in these two directions, is approximately -0.5 meV/Co i.e. nearly ∼5 K. These results indicate that the magnetocrystalline anisotropy of Co in Ti 1−x Co x O 2 determines the easy and hard axis and leads to highly anisotropic super-paramagnetism in TiO 2 .

Anisotropic paramagnetism
In addition to the anisotropic effects of super-paramagnetic Co clusters, an additional anisotropy from paramagnetic Co ions has also been observed here. In the magnetization isotherms of samples B and C above T B (fig. 3), magnetization is observed to be nearly constant for fields higher than 1 T, especially for field along H ⊥ . However in the blocked region, at 2 K, an increase in dM/dH for samples B and C is observed (see fig. 4). This is in contrast to the usual SPM systems where M remains almost constant with increasing H. A possible reason for this could be the presence of uncompensated paramagnetic Co spins. In addition, a cross over in magnetization between the two field directions (shown by arrow in fig. 4) is also observed. The cobalt that occupies Ti sites in TiO 2 lattice should be in a 4 + state corresponding to the S=1/2 system, as also observed in the first principles calculations carried out here.
In S=1/2 system, the single ion anisotropy does not affect the magnetic behavior. The unusual cross-over behavior can be explained qualitatively by considering the presence of Co 2+ ions, i.e. S=3/2 system. The crossover observed here then arises due to presence of the single ion anisotropy of this 3/2 spin state. Origin of Co 2+ is via formation of CoTiO 3 nanoclusters whose presence in sample C has been observed by XRD ( fig. 1). In addition, Co 2+ also exists as XPS studies have shown that along with a Ti 4+ state, a small percentage of Ti 3+ also develops for samples B and C and increases with the fluence 28 . Still, the main source of Co 2+ spins here is CoTiO 3 and though it has been observed only for sample C, it is likely that smaller amounts will be present in sample B also.
Similar paramagnetic anisotropy has been observed in the magnetization of Co:ZnO thin films 37 . Ney et al. 37,38 have discussed the magnetization in terms of an effective spin Hamiltonian with an anisotropy along the crystallographic c axis (referred as z axis) of Zn 1−x Co x O films. Here, effective spin Hamiltonian has been applied to understand the anisotropic paramagnetic behavior observed in fig. 4. Remarkably, no crossover was observed when anisotropy was alongẑ axis (i.e. along H : <001>), rather it was observed whenx direction (along H ⊥ : <110>) of the spin was considered to be the axis of single ion anisotropy (geometry of the present system is shown in fig. 5). Hence, a modified S = 3/2 spin Hamiltonian was used here and is discussed below: Here the magnetic state is characterized by two g factors, g (2.238) and g ⊥ (2.276), and the zero field splitting constant Q 37 . In the above equation, Q corresponds to single ion anisotropy of S=3/2 system, due to Co 2+ , along thê  Similar to the reduced remenance, the temperature dependent coercivity H C (T ) for a system of randomly oriented and non interacting particles displays a behavior given by 5 , with H CO = K ub /M sb , where K ub and M sb are the anistropy and saturation magnetization of bulk cobalt. T 0 B is the Blocking temperature at zero field, T 0 B = Kµ<V > kB ln(τm/τ0) , where K µ is the uniaxial anisotopy constant, < V > is the average particle volume, τ 0 and τ m are the characteristic limiting relaxation time and the measurement time. K B is the Boltzman constant. Fig. 7 shows H c (T ) as a function of reduced temperature k B T /K u <V> for samples B and C along the H ⊥ . Here < V > corresponds to the average volume and has been evaluated using the size(D) of the clusters as mentioned in table 1. In order to directly observe the linear relation, inset of fig. 7 also displays a plot of H C against T 1/2 for samples B and C. For sample B, a linear relation has been observed up to T ∼ 50 K. However above this temperature, deviations occur as T approaches T B (=65 K). In sample C also, H C shows deviations beyond ∼ 50 K, a temperature which is much lower than its T B (=150 K). Also, the extrapolation of H C (T ) to 0 yields values of T B ∼ 64 K and 100 K for samples B and C, respectively, which are lower than the observed blocking temperatures.
The deviation from linearity suggests a strong presence of inter-particle interactions between the Co clusters The most prominent interaction among the clusters is the inter-particle dipole interaction. In addition, there also exists presence of inter-particle exchange interactions. Using Monte-Carlo (MC) simulations, Kechrakos and Trohidou etal. 40,41 have investigated the role of inter-particle dipole and exchange interactions in determining the coercivity and remanence of the magnetization, in the blocked region, for a system of SPM nano-particles with random uniaxial 11 anisotropic directions. The variations in m R and H C have been studied as a function of increasing volume fraction of the magnetic nano-particles. Since the size and number of the Co nano-clusters formed inside TiO 2 , increase with fluence, we expect an increase in the volume fraction of nano-particles. Thus with fluence, the inter-particle dipolar strength increases, which has a demagnetizing effect on the magnetization, causing a decrease in m R and H C . This suggests that in addition to the dipole interaction, there exists a strong inter-particle exchange interaction between the clusters which can even cause complete ferromagnetic alignment between the clusters. The itinerant nature of the Co occupying Ti sites, can produce this carrier mediated exchange interaction between the clusters. Presence of competing dipolar-interactions has also been observed via ZFC-FC curves. In addition at T=2 K, surprisingly a crossover in the magnetization for two field directions in sample B and C is observed. The origin of this crossover is the anisotropic paramagnetism arising from the 2+ ionic state of Cobalt in a S = 3/2 system. Role of dipole-interactions and inter-cluster exchange interactions have also been discussed.