Finding order in disorder: Magnetic coupling distributions and competing anisotropies in an amorphous metal alloy

Amorphous metals have unusual magnetic properties that arise due to the disordered atomic arrangement. We show that Co x (Al 70 Zr 30 ) 100 − x (65 < x < 92 at. %) amorphous alloys have a distribution in the local magnetic coupling and ordering temperature, which can be explained by nanoscale composition variations. We use competing anisotropies induced by the substrate and an applied field during growth to probe the Co concentration distribution. Only regions with high enough Co concentration develop a magnetic anisotropy along the magnetic field during growth, whereas regions of low Co concentration have an anisotropy dictated by the substrate. A Gaussian distribution in the Co concentration of width 5.1 at. % is obtained from the variation in anisotropy. The results demonstrate the importance of composition variations for emergent magnetic properties and have far reaching implications for the properties of disordered materials in general


I. INTRODUCTION
Metallic glasses have long been a source of fascination due to their elusive structure and range of unusual properties. 1,2Amorphous materials have a disordered atomic arrangement that makes them difficult to characterize structurally. 3,4Unlike in crystalline materials, which are composed of periodically arranged atoms, the atoms in amorphous materials can be in a range of local environments with different coordination and interatomic distances.This means that linking the local structure of amorphous materials to emergent physical properties can be a highly tedious task.
Significant progress has been made in recent years toward characterizing the local atomic order in amorphous metals.][6][7][8] In addition, direct observations of a local atomic order in amorphous metals have been enabled by nano-beam electron diffraction 9 and atomic electron tomography. 10These measurements have revealed a short-to-medium range order in the form of repeating single or multiple interconnected atomic clusters but without the longrange periodicity of crystalline solids.However, a recent study has shown by atom probe tomography, that local composition variations can exist in a binary metallic glass over a length-scale of several nanometers. 11This far exceeds the typical size of atomic clusters and introduces a new length scale to consider in the study of disorder in materials.
Composition variations on the nanoscale may result in a spatial dependence of local properties that are unique to disordered materials.4][15] Furthermore, competing interactions between these regions of varying composition have the potential to shape the overall response of the material.However, it is challenging to identify such competing interactions and probe the associated composition variations, which explains the limited data on the issue.
The most significant applications of metallic glasses to date are in soft magnets.This is because amorphous metals can have very low coercivity and, therefore, low hysteretic losses. 2 Nonetheless, amorphous metals can have a significant magnetic anisotropy. 15,16Magnetic anisotropy is usually associated with a crystal anisotropy, but in amorphous metals, this cannot be the case.Despite the absence of crystalline order, short-to-medium range structural correlations can result in a substantial magnetostructural anisotropy. 15,17Such correlations can be induced by various means, such as choice of the substrate, interface effects, and growth or annealing in a magnetic field. 15,18,19Large magnetic proximity effects, where a magnetization is induced in a non-magnetic material due to proximity to a magnetic material, [20][21][22] have also been observed in amorphous heterostructures. 23,24In the case of both anisotropy and the proximity effect, local composition variations have been suggested to play a crucial role, [23][24][25] but direct evidence of this has been lacking.
Here, we demonstrate that the magnetic properties of the amorphous alloy CoAlZr are shaped by local variations in its magnetic ordering temperature and anisotropy.We attribute these variations to a distribution in the concentration of the magnetic element Co and use the resulting effective anisotropy to determine the width of the Co atomic concentration distribution.

II. METHODS
The samples were grown using dc magnetron sputtering in a sputtering chamber with a base pressure below 5 × 10 −9 mbar.The sputtering gas was Ar of 99.9999% purity, and the growth pressure was 2.40 × 10 −3 mbar.Si(100) substrates, with the native oxide layer intact, were used with no substrate heating.First, a buffer layer of 2-nm Al 70 Zr 30 was deposited from an Al 70 Zr 30 alloy target.Next, 40 nm of Cox(Al 70 Zr 30 ) 100−x (60 < x < 95 at.%) was grown by co-sputtering from Co (99.9%) and Al 70 Zr 30 (99.9%) alloy targets.Finally, all samples were capped with 5-nm Al 70 Zr 30 .Samples were grown both with and without a uniform external magnetic field of 130 mT, parallel to the film plane.
The magneto-optical Kerr effect (MOKE) and vibrating sample magnetometry (VSM), both in a longitudinal setup, were used for magnetic characterization at room temperature and low temperatures, respectively.Structural characterization was done using a PANalytical X'pert Pro diffractometer, equipped with a Göbel mirror on the incident side and a parallel plate collimator on the diffracted side.The atomic arrangement (amorphous or polycrystalline) was determined using grazing incidence x-ray diffraction (GIXRD) with the incident angle fixed at ω = 1 ○ .Layer thickness and interface roughness were measured using x-ray reflectivity (XRR), and XRR scans were fitted using the X'pert reflectivity software.

A. Sample structure
A series of thin films of Cox(Al 70 Zr 30 ) 100−x with x in the range 60-100 at.% was studied.The films were bounded by thin layers of amorphous AlZr to ensure identical top and bottom interfaces and prevent oxidation of the magnetic film.The sample structure is shown schematically in the inset of Fig. 1(a).XRR measurements were carried out to examine the layering of the samples, as shown in Fig. 1(a).Kiessig fringes are observed up to at least 2θ = 8 ○ confirming the well-defined layer thickness characteristic of amorphous films.Fitting the XRR data allows us to determine the thickness of each layer as well as the layer density and root-mean-square interface roughness, which is of the order of 0.5 nm. Figure 1(b) shows GIXRD measurements for several samples with the Co content ranging from 72 to 100 at.%.For the Co content 92% and above, there are small peaks present corresponding to hcp Co crystallites.The peaks are superimposed onto the broad peak centered at 2θ = 45 ○ .For x ≤ 90, no sharp peaks are observed, and the GIXRD only shows the single broad peak typically found for amorphous materials. 23,26

B. Macroscopic magnetic properties
Figure 2 shows two hysteresis loops for a sample with 90% Co content measured along two orthogonal in-plane directions, which are characteristics for the sample series.During growth, we apply a uniform magnetic field parallel to the film plane that has been shown to induce a uniaxial anisotropy in thin magnetic films, with the easy axis direction parallel to the applied field. 15,19,24,27,28The hysteresis loop parallel to the growth field is square, with a sharp single switch and a very low coercive field of Hc ≈ 0.2 mT.In the perpendicular inplane direction, the loop has zero remanence but saturates at a low field.The inset shows a polar plot of the remanent magnetization Mrem as a function of the azimuthal angle φ.The uniaxial anisotropy is clear, with the easy directions where Mrem/Msat = 1 separated by 180 ○ and a hard axis perpendicular to the easy axis.This can be described by a periodic function, as shown in Fig. 2. Here, α is the offset of the easy axis with respect to the direction of the growth field (φ = 0) and Msat is the saturation magnetization.The close fit demonstrates that the film exhibits a small, but well-defined, uniaxial anisotropy.This applies to the entire composition range studied.Figure 3 shows the saturation magnetization Msat at 20 K (right, green circles) and Curie temperature Tc (left, blue squares) as a function of the Co content, measured by VSM.In the amorphous composition range below 92 at.% Co, the magnetization decreases linearly with the decrease in the Co content.This is consistent with the magnetization of other amorphous Co alloys. 29An extrapolation of the linear dependence to 100 at.% Co reveals a magnetization of 12.3 × 10 5 A/m, which is equivalent to an effective moment of 1.47 μ B per Co atom.This is slightly below the magnetization of hcp Co.For dilution of Co with a fixed moment, one might expect the magnetization to go to zero at 0% Co.Here, however, the magnetization is reduced to zero at 66% Co.This is because the magnetic interaction in transition metals is through the itinerant electrons.Replacing a Co atom with non-magnetic Al or Zr not only removes the magnetic moment of a single atom but also decreases the magnetic moment of neighboring atoms.This results in a net decrease of 4.3 μ B per Co atom, which is 2.7 times larger than the magnetic moment per atom in hcp Co.Similarly, we observe a linear decrease in the Curie temperature Tc with the decrease in the Co content.The decrease in magnetization as a function of temperature is consistent with that of a typical ferromagnet (see, for example, Ref. 24).We define Tc as the temperature where the remanent magnetization becomes zero, and this is determined from full hysteresis loop measurements as a function of temperature.The Curie temperature can be viewed as a measure of the magnetic coupling strength J, since J is proportional to Tc according to the Weiss model of a ferromagnet. 11,30We, therefore, also assign a unit of temperature to J for convenience.With the decrease in the Co content, the effective magnetic coupling between Co atoms is, therefore, reduced.A linear extrapolation of the Curie temperature to 100% Co gives Tc = (1200 ± 100) ○ C.This can be compared to Tc of hcp Co, which is 1130 ○ C. Thus, one can assume a weak or negligible impact of disorder on the effective Co-Co interactions, in stark contrast to what is observed for Fe. 11oth the magnetization and Tc cross zero at (66.5 ± 0.5) at.%, below which the alloy is not ferromagnetic at any temperature.This demonstrates the high degree of magnetic tunability of the CoAlZr alloy and also how sensitive its magnetic properties are to small changes in composition.

C. Competing anisotropies
The simple linear dependence of magnetization and Curie temperature on composition hides a more complex picture of the amorphous film structure, revealed by a closer look at the magnetic anisotropy.As the Co content of the films is reduced below 85 at.%, we observe an anomalous rotation of the easy axis direction away from the direction of the imprinting (growth) field.Figure 4 shows a polar plot of the remanent magnetization of the sample with 75 at.% Co (blue dots).The anisotropy is still clearly uniaxial, but the easy axis is rotated by an angle of α = 17 ○ with respect to the growth field.Furthermore, we find that the angle of the easy axis (with respect to the growth field) increases smoothly as the Co content is reduced.The angle α is shown as a function of composition in Fig. 5(a).For a high Co content, α = 0 (easy axis parallel with the growth field), whereas for the lowest Co content, α = 45 ○ .
To determine the cause of this rotation of the easy axis, a series of reference samples was grown without a growth field.The films grown without field are also uniaxial but their easy axis is fixed at α = 45 ○ , irrespective of the composition.This is shown by the red   1).The direction of the growth field is shown by the blue arrow below the graph.The easy axis angle α depends on whether or not a growth field is applied.
squares in Fig. 4, also for 75 at.% Co.The uniaxial anisotropy can be eliminated by increasing the thickness of the amorphous AlZr buffer layer to 8 nm or by using a thermally oxidized Si substrate, as shown in the supplementary material.We, therefore, conclude that we have a hierarchy of effects governing the anisotropy in the amorphous films: (i) A magnetic field during growth induces an anisotropy parallel to the field.(ii) In the absence of a growth field, the uniaxial anisotropy is dictated by the Si(100) substrate and the easy axis is parallel to the Si[110] in-plane direction.Without a growth field or substrate effects, the films are isotropic.We stress that all the films are x-ray amorphous, regardless of the thickness of the AlZr buffer layer or the presence of a growth field.
A material with two competing uniaxial anisotropy axes, A and B, will not, in general, exhibit two easy axes but rather a single easy axis C, where the easy axis will point in a direction between the two competing axes. 31The easy axis angles between α = 0 ○ and α = 45 ○ observed here are, therefore, a result of the competition between the anisotropy induced by the growth field and the anisotropy induced by the substrate.The size of the two anisotropies is equal as evidenced by the equal saturation fields along the hard axis for samples grown with and without field (see the supplementary material).The difference lies in the direction of the easy axis.
Magnetic field imprinted anisotropy relies on a strong magnetic response of the growing film, i.e., the effective temperature during growth must be lower than the Curie temperature of the film.This means that field imprinted anisotropy only develops in films with sufficiently high Curie temperature.If the Tc of an alloy is below the effective growth temperature, the anisotropy will be determined by the substrate.For a Tc above the growth temperature, the anisotropy will be determined by the growth field.If the magnetic coupling between Co atoms in the CoAlZr alloy was homogeneous such that there was a single uniform Curie temperature, we would expect a sharp transition in the direction of the anisotropy.The smooth change in α as a function of composition in Fig. 5(a), therefore, shows that Tc is not as well-defined as shown in Fig. 3.

D. Composition distribution
To understand how the competing anisotropy axes arise, we must look to the disordered structure of the amorphous films.Gemma et al. 11 have recently revealed the inhomogeneous composition of amorphous alloys on the nanometer scale by means of atom probe tomography on sputtered FeZr amorphous films.They showed that the elements are not evenly distributed resulting in regions with higher or lower concentration of the magnetic element, extending over a length scale of several nanometers.This, in turn, means that there is a variation in the effective magnetic coupling strength J on this length scale.3][34] At the global Tc, there may still exist disconnected regions that are magnetically ordered, but they do not exhibit long range ordering in the absence of an external field.These are essentially superparamagnetic regions that are easily polarizable and can mediate exchange coupling and proximity effects over several tens of nanometers. 24,27As shown in Fig. 3, there is a strong change in effective coupling strength with composition in CoAlZr, meaning that there can potentially be a large variation in the local Curie temperature within each sample.
As mentioned previously, a material must have a strong magnetic response during growth to be susceptible to magnetic field imprinting of anisotropy.Due to the local variations in J (Tc), only the regions with high enough J (Tc) will be susceptible to field imprinting.Regions with a lower Co content (lower J, Tc) will have their anisotropy dictated by the substrate.For films with sufficiently high mean Co concentration, the growth field induced anisotropy will dominate, whereas for low Co concentration films, the substrate induced anisotropy is dominant.In the intermediate composition region, the two mechanisms will coexist and the width of this composition region is determined by the width of the Co concentration distribution and the slope of the J(x) dependence.
To test this hypothesis, we can analyze the angular dependence of the anisotropy.The anisotropy constant describing the uniaxial in-plane anisotropy can be written as where Hsat is the saturation field along the in-plane hard axis and Msat is the saturation magnetization.We denote the anisotropy constant due to the substrate by K A and the anisotropy constant due to the growth field as K B and the angle between the two is 45 ○ .The total anisotropy energy for the two competing anisotropies K A and K B can be written as where M B is the part of the magnetization with anisotropy parallel to the growth field and M A is the magnetization with anisotropy at 45 ○ .Hsat is the saturation field measured along the respective hard axis, which, in this case, has the same magnitude for both K A and K B .
The direction of the easy axis, α, can then be found by minimizing the total energy with respect to α, giving The direction of the easy axis is, therefore, governed by the ratio of the partial magnetizations with the two anisotropy axes.
To determine the partial magnetizations, we assume that the distribution of Co concentration can be described by a Gaussian function, as shown by the solid black line in Fig. 5(b), centered at the mean Co concentration x and with a standard deviation σ.A similar approach has been taken to describe a distribution in blocking temperature due to disorder in a ferromagnetic-antiferromagnetic exchange coupled bilayer. 35The magnetization distribution is then found by scaling the Gaussian Co concentration distribution with the magnetization as a function of composition M(x), which is given by a linear fit of the magnetization in Fig. 3 (see the supplementary material).Figure 5(b) shows the magnetization distribution corresponding to a concentration distribution with a mean composition of x = 73 at.%.The area under the curve equals the total magnetization of the sample (at 20 K).Each composition is also associated with a Curie temperature (through Fig. 3), which we can compare to the effective growth temperature T G .The effective growth temperature determines the regions within the sample, which are affected by the applied magnetic field during growth.Regions that have (J, Tc) < T G will couple with the substrate, whereas regions with (J, Tc) > T G will align with the growth field.The partial magnetizations, M A and M B , in Eq. ( 4), are, therefore, the areas under the curve below (orange) and above (yellow) T G , respectively.
The solid line in Fig. 5(a) is calculated from Eq. ( 4) based on partial magnetizations deduced from the Gaussian concentration distribution in Fig. 5(b).How steeply α changes depends on σ of the Gaussian distribution, and thus, the width can be determined as σ = (5.1 ± 0.8) at.%.The horizontal position of the curve is determined by the effective growth temperature that is found to be T G = (450 ± 90) K.This can be considered as the effective temperature of the condensing vapor as the atomic arrangement is formed.The model is in good agreement with the data, as shown by the close fit in Fig. 5(a).

IV. CONCLUSIONS
We have shown that unusual magnetic properties arise in amorphous thin films due to their disordered atomic arrangement.Local composition variations result in competing interactions that influence the development of the short-to-medium range structural order during growth.By taking advantage of competing uniaxial anisotropies induced by the substrate and growth field, we are able to map out the distribution in the local magnetic coupling.Only high Co concentration regions with local ordering temperature above the effective growth temperature are affected by the magnetic field during growth, whereas regions with lower ordering temperature are affected by the substrate.We show that these observed magnetic properties of the amorphous alloy can be attributed to a Gaussian distribution in the Co concentration.
The standard deviation of the distribution is a measure of how much the composition of the material varies on the length scale of the magnetic interactions.A small spatial variation in Tc(J) would be associated with a narrow peak (small standard deviation) centered around the mean composition.On the other hand, a large spatial difference in Tc(J) would be a result of a wide composition distribution (large standard deviation).From our model, we find that the composition distribution has a standard deviation of 5.1 at.%, which is similar to that measured in the FeZr system. 11However, it should be noted that any assessment of the composition distribution is always affected by the size of the probe.Indeed, for large enough probe sizes, the films in question are highly homogeneous as shown by macroscopic material characterization techniques.
The magnetic properties of CoAlZr are just one example of how the nanoscale composition variations inherent in amorphous alloys can affect their macroscopic properties.Such composition variations can explain the variability in the properties reported for seemingly identical amorphous alloys.The specific growth conditions, including the growth technique, temperature, and pressure, will influence the width of the distribution and, in turn, define the competing interactions that shape the emergent properties of the material.This can result in unexpected variability but may also be used to our advantage to tune the material performance.Therefore, it is crucial to be aware of this defining characteristic of disordered materials.

SUPPLEMENTARY MATERIAL
See the supplementary material for additional details of the substrate induced anisotropy, thickness dependence of α, size of the anisotropy induced by the substrate and growth field, and the composition distribution model.

FIG. 1 .
FIG. 1. Sample design and structure.(a) X-ray reflectivity measurements representative of the sample series (blue dots), including a fit (red line), demonstrating the well-defined layering.The inset shows a schematic of the sample structure.(b) Grazing incidence x-ray diffraction of samples with composition ranging from 72 to 95 at.% Co.The data have been shifted for clarity.Samples with more than 8% AlZr do not show any crystal peaks and are x-ray amorphous.

FIG. 2 .
FIG. 2. Uniaxial magnetic anisotropy.Hysteresis loops of a film with 90% Co, measured parallel (φ = 0 ○ ) and perpendicular (φ = 90 ○ ) to the applied growth field.The inset shows the polar plot of the normalized remanent magnetization (Mrem/M sat ) as a function of azimuthal angle φ, showing the well-defined uniaxial anisotropy.

FIG. 3 .
FIG. 3. Temperature dependence of magnetic properties.(Left) The Curie temperature Tc, as a function of Co atomic percentage x.Tc can be viewed as a measure of the magnetic coupling strength J. (Right) The saturation magnetization M sat (at 20 mT) measured at 20 K as a function of x.Amorphous compositions are denoted by circles and polycrystalline compositions by diamonds.Both Tc and M sat show a linear dependence on composition.

FIG. 4 .
FIG. 4.Competing anisotropies.Polar plot of the normalized remanent magnetization of the sample with 75 at.% Co as a function of the azimuthal angle φ for a sample growth with (blue circles) and without (red squares) an external growth field B, including a fit using Eq.(1).The direction of the growth field is shown by the blue arrow below the graph.The easy axis angle α depends on whether or not a growth field is applied.

FIG. 5 .
FIG. 5.The magnetization and Co concentration distributions.(a)The easy axis angle α as a function of Co content including a fit using Eq.(4).For a high Co content, the easy axis is aligned with the growth field, whereas for a low Co content, it is aligned with the Si substrate [110] direction.(b) The Co concentration (black line) and magnetization distributions (orange-yellow shaded region) for the sample with x = 73%.The area under the curve corresponds to the total magnetization of the sample.At temperatures above T G (yellow area), the magnetization interacts with the growth field.The inset is an illustration of the regions with competing anisotropy axes.We find that the effective growth temperature is T G = (450 ± 90) K and σ = (5.1 ± 0.8) at.%.