Scaling of Hall coefficient in CoBi granular thin films

A series of Co-Bi thin films with Co concentrations c=0, 0.05, 0.2, 0.26, 0.3, 0.333, 0.375, 0.545, were grown by magnetron sputtering on Si(100)/SiNX substrates. Resistivity measurements at zero field (!xx) as a function of temperature-T exhibit an exponential variation with T in the region of 240K<T<300K. The Hall coefficient as a function of Co concentration diverges as log|c-0.3| for c<0.333, indicating a scaling of RH nearby a percolation threshold pc=0.3. Only after proper scaling of the anomalous Hall coefficient RS the conventional RS!(!xx) dependence can be satisfied.


Introduction
In magnetic films the Hall resistivity !" is defined by the phenomenological equation: where B is the magnetic induction perpendicular to film surface, R 0 is the ordinary Hall (OH) coefficient, M is the magnetization, and R S is the anomalous Hall (AH) coefficient.Usually, in ferromagnetic metals (FM), R S satisfies a scaling law [1] with the longitudinal film resistivity !xx measured at zero applied magnetic field: where the first term is assigned to skew scattering and the second term to side jump mechanism [1].Generally, three regimes were observed [1] with respect to the dependence of anomalous Hall conductivity (AHC) on the conductivity # xx .Equation-2 is valid within the metallic regime [1] whereas in the insulating regime other scaling exponents were observed [2].Bismuth films may exhibit p-type or n-type conduction properties, depending on film texture [3] and the participation of surface states on the different crystallographic faces [4].In our study Cobalt (FM) and semimetal Bismuth layered thin film structures are used to investigate the effect of FM/semimetal junctions in Hall resistivity measurements.

Experimental Details
A series of thin film Co/Bi structures were grown by magnetron sputtering on Si(100)/SiN X substrates with rectangular shapes of 5x5mm 2 , using a base pressure of Ar-gas about 10  2 are tabulated with increasing order of c-Co.
Resistivity and magneto-transport measurements were performed on the as-made films with a PPMS, using the Van der Pauw method in magnetic fields up to B=9T perpendicular to film-plane.All raw data collected from Hall resistance measurements were corrected according to Ref. [5] to obtain the final loops of !" vs B and Hall coefficient R H vs B shown in Figs. 1, 2. X-ray diffraction (XRD) measurements reveal that all films are polycrystalline.Specifically, the predominant Braggpeaks with (00l) indices (l=3, 6), observed in pure Bi films, decrease towards to zero intensity as c increases, indicating a progressive change in texture of Bi layers with c-Co.

Results
Fig. 1a shows representative !H vs B data and isothermal magnetization M vs B loops observed in Co-rich film at 5K. Fig. 1b shows three selected M vs B loops at 5K that span the c-Co range, indicating that the saturation magnetization M S values are strongly affected by c-Co whereas the saturation field B S (and thus magnetic anisotropy) are not influenced by c-Co at a fixed temperature.Thus, the non linear dependence of the Hall coefficient R H =! " /$ on B, observed in Fig. 2, cannot be attributed to magnetization effects above 2T.In addition, Fig. 2a shows that pure Bi film exhibits non linear R H vs B curves above 2 Tesla for T%100K.As we show in a forthcoming publication, these results cannot be fitted with equations used in the two-band model.The observed R H vs B curves in Fig. 2 can be explained by considering the galvanomagnetic properties of polycrystalline or inhomogeneous metals that depend strongly on the shape and the orientation of the crystallites relative to magnetic field direction [6].The most important effect of crystalline shape and orientation in these films is the observed change of polarity in R H vs B loops, from positive to negative in Fig. 2, that does not depend on a systematic way from c-Co.As shown in Fig. 3, the sign of R H (8T), which is the value of R H at 8T in Fig. 2, and the sign of R S , both change from negative to positive as the percentage of preferable orientation of [00l]-directions in Bi layers change from perpendicular to parallel in film plane.Note that both, the R 0 =d! " /dH and the R H (8T) values have positive sign in pure Bi films (Fig. 3a), indicating that holes are the majority charge carriers [7].In addition, Fig. 3a reveals that R H (8T) becomes negative in between 0.2<c<0.375,indicating a transition from ptype (holes) to n-type (electrons) majority charge carriers in this region.This transition in R H (8T) values with c-Co is plotted together with R 0 vs c-Co in Fig. 4.   A plot of R 0 values instead of R H (8T) in Fig. 3a will exhibit the same trend, but we plot only the R H (8T) values for reasons explained below.Fig. 3b shows that the sign of R S does not follow the systematic sign reversal of R H (8T) or R 0 with c-Co, but depends only on the degree of orientation of [00l]-directions in Bi layers relative to film vertical direction.Since Bi exhibits [4] a strong spinorbit (SO) level splitting along [00l]-directions, then Fig. 3b may imply that the SO Hamiltonian [8] H SO in Bi is negative for example when [00l]-directions are inplane and positive for out-of-plane, thus, reversing from (left to right( direction the effective nonzero transverse electric field created by the SO interaction.In this case a junction with Co may give rise to unequal occupancy of spin-up and spin down states in Co/Bi interfaces.This may contribute a term of interspin band scattering [8] to AH effect (AHE), in addition to intrinsic SO coupling term that contributes to AHE with the scattering of itinerant carriers due either to phonons or to thermal spin disorder.in order to be able to compare the results for all c-Co values.Fig. 6a shows that Eq.2 is not satisfied when the measured value ( ) of the corresponding film (in Fig. 5) is used, and the inset demonstrates that R S values as a function of Co concentration at 5K exhibit a minimum at the same p c =0.3 as the " values observed in Bi(15nm) film (from Fig. 5) then Fig. 6b shows that Eq.2 is satisfied for all the films.
The reason that Eq.2 is satisfied only with

Conclusion
In summary, magneto-transport measurements in polycrystalline Co/Bi layered thin film structures reveal that f H R or R 0 values exhibit (Fig. 4) a critical scaling close to a percolation threshold p c =0.3, with a critical exponent g=0.3.It was observed that between 0.2<c<0.375:(i) both, the OH coefficient R 0 and f H R values become negative, evidencing a transition from ptype to n-type majority charge carriers in this region, and (ii) a similar behaviour appears in the plot (Fig. 6a

Fig. 1 .
Fig.1.(a) Indicative Hall resistivity !H (&) and magnetization-M (o) loops that explain the estimation of R 0 from the high-field slope and R S from the point of intersection with !H -axis.(b) Selected isothermal magnetization loops, showing that saturation of magnetization M S occurs at B S '2T for T=5K.

Fig. 3 .
Fig. 3. Semi-log plot of XRD intensities from Bi (00l) Bragg peaks (l=3 or l=6) as a function of: (a) R H (8T) values (dash line) and R S values (solid line) at 5K, and (b) R S values in the range of 5K%T%200K.The (003) and (006) indices are plotted with open and dark symbols, respectively.Their intensities were normalized to observed (012) Bragg-peak intensity of Bi.

Fig. 4 .
Fig. 4. Scaling of (a) f H R and (b) R 0 values with c-Co.Solid lines are fitting curves to equation shown with arrow.Insets show the temperature dependence of Bi H R and 0 Bi R values.
Fig.4a shows normalized Hall coefficient f H R values (obtained from Fig.2 at B=8T for every film) to corresponding Bi film values Bi H R (see the inset) as a function of c-Co.The notation is that superscripts f and Bi on R H , R S and !xx parameters indicate that each one is measured on a film of given c-Co) 0 and on pure Bi film

Fig. 5 .
Fig. 5. Temperature dependence of the longitudinal film resistivity !xx observed at zero external field in films with different Co concentrations.These !xx (T) curves are normalized to the corresponding film !xx value observed at T=10K.Lines are guides to the eye.

"
(c-Co=0), respectively.The f H R values were found to diverge as log|c-0.3| 0.3 for c<0.333, evidencing a critical behaviour[9] nearby a percolation threshold p c =0.3, with critical exponent g=0.3.The physical origin of p c =0.3 might be related to energy balance between the work functions[10] of Co and Bi layers and their Fermi level matching at Co/Bi interfaces, that may create[11] a depletion layer (defining a critical length scale) of holelike or electron-like carriers in Bi as a function of c-Co.In addition, the same trend of f H R vs c-Co appears in all ( ) f H R B values determined for B!5T.However, a plot of the ordinary Hall coefficient R 0 with c-Co in Fig.4b (determined by the high-field slope in Fig.2, with B>3T) reveals a slightly shifted p c value at a value of c-Co' 0.333.A comparison of c-Co values in Figs.3, 4 and 5(see the inset), where a minimum of I(00l), f H R or R 0 , and R S values occurs, indicates that the p c value is more likely to be nearby c-Co=0.3.In addition, the same function, log|c-0.3| 0.3 , fits (solid line) both data sets in Figs.4a and 4b for c-Co*0.3 in the region of 5K*T*250K.Fig.5 shows the temperature dependence of !xx observed in all films at zero applied field.Remarkably, all these films exhibit an exponential variation of !xx (T) at 240K<T<300K whereas in some films (c=0.05,0.2, 0.375) appears a !xx 'T n behavior for T<240K.The exponential dependence in !xx (T) curves arises from the dominant contribution of Bi layers to film resistivity, curves of pure Bi films can be simulated with a single exponential function of T in the region of 5K%T%300K.The dominant contribution of ( ) Bi xx T " in the magnetotransport properties of our films can be clearly demonstrated if we try to satisfy Eq.2 for every c-Co by plotting the R S (T)/! xx (T) ratio against the corresponding values of !xx (T) from Fig.5.Fig.6 includes two such plots where all parameters are normalized to their values at 5K 12002-p.3

R
in Fig.4a.However, when we keep the actual R S (T) values for each c-Co and replace the ( ) clear.Preliminary results from analysis of the experimental data shown in Figs.5 and 6 provide evidence that the conventional scaling of R S in Eq.2 is maintained by substituting ( ) Bi xx T" because of a proper scaling[12] that involves a critical length scale in Bi side of Bi/Co junctions.

Fig. 6 .
Fig. 6.(a) Normalized AHE coefficients R S (T) to R S (5K) of the films are divided by the normalized film resistance at zero field !xx (T)/! xx (5K), and plotted as a function of !xx (T)/! xx (5K) of the film.Lines are guides to the eye.The inset shows R S values at 5K as a function of c-Co.(b) The same R S (T)/R S (5K) values divided by the normalized resistance of Bi(15nm) film at zero field, as a function of the temperature dependence of the normalized resistance of Bi(15nm) film.Lines are linear fits satisfying Eq.2.Different symbols correspond to c-Co in different films and are the same in both plots.
, inset) of R S values as a function of c-Co at 5K.However, the sign of R S depends on the degree of orientation of [00l] crystallographic directions in Bi layers relative to film vertical direction, and is not related with Co concentration.It was shown that the obtained R S values satisfy Eq.2 only if ( ) Bi(15nm) film.The last two results provide evidence that the dominant contributions to AHE are related with intrinsic and extrinsic mechanisms in Bi side of Bi/Co interfaces.