P. Khatua IIT Kanpur. D. Temple MCNC, Electronic Technologies. A. K. Majumdar, S. N. Bose National Centre for Basic Sciences, Kolkata

Transcription

1 The scaling law and its universality in the anomalous Hall effect of giant magnetoresistive Fe/Cr multilayers A. K. Majumdar S. N. Bose National Centre for Basic Sciences, Kolkata & Department of Physics Indian Institute of Technology, Kanpur

2 Collaborators P. Khatua IIT Kanpur D. Temple MCNC, Electronic Technologies and C. Pace Division, North Carolina P. Khatua et al., Phys. Rev. B 68, (2003), Phys. Rev. B 73,, (2006), Phys. Rev. B 74,, (2006).

3 Introduction Motivation Experimental Outline Giant magnetoresistance (GMR) Hall Effect Results and Discussion Conclusions Sample preparation and characterization Experimental techniques Interpretation of the Hall data

4 Magnetoresistance Introduction I. Ordinary or normal magnetoresistance (OMR) Due to the Lorentz force acting on the electron trajectories in a magnetic field. MR ~ B 2 at low fields. MR is significant only at low temperatures for pure materials at high fields & is positive. II. Anisotropic magnetoresistance (AMR) in a ferromagnet In low field LMR is positive and TMR is negative. Negative MR after saturation is due to quenching of spin-waves by magnetic fields. FAR is an inherent property of FM materials originating from spin-orbit interaction of conduction electrons with localized moments. FAR(ferromagnetic anisotropy of resistivity) = ( ρ //s - ρ s )/ ρ 0 Fig. 1

5 Introduction Giant magnetoresistance (GMR) Fe-Cr is a lattice matched pair : Exchange coupling of ferromagnetic Fe layers through Cr spacers gives rise to a negative giant magnetoresistance (GMR) with the application of a magnetic field. RKKY interaction ~ cos(2k f r)/(k f r) 3. Established by experiments on light scattering by spin waves. [ P. Grünberg et al., Phys. Rev. Lett. 57, 2442(1986).] At low fields the interlayer antiferromgnetic coupling causes the spins in adjacent layers to be antiparallel and the resistance is high At high fields the spins align with the field (saturating at H sat ) and the resistance is reduced. Magnetoresistance is negative!

6 Introduction Magnetoresistance is defined by MR = ρ ( H, T ) ρ (0, T ρ (0, T ) ) 100 %. (1) Fe-Cr

7 Introduction Calculation of GMR (Spin-dependent scattering) Bulk scattering only, assuming MFP > Cr spacer thickness. Spin-dependent scattering in ferromagnetic metals 2 current conduction model of Fert & Campbell. [ A. Fert and I. A. Campbell, J. Phys. F : Metal Phys., 6, 849(1976).] In Fe ( weak ferromagnet ) minority band has a much higher conductivity as seen from σ = neµ and its band structure. [ P. B. Visscher and Hui Zhang, Phys. Rev. B 48, 6672(1993).]

8 Introduction Interface scattering Local spin density approximation to density functional theory, treating the disordered atomic planes by KKR-CPA, was used by Butler et al. to calculate the electronic structure, magnetic moments, scattering rate and electrical conductivity in many GMR systems. [W. H. Butler et al, J. Magn. Magn. Mater. 151, (1995)] Fig. 2: Calculated majority and minority valence electrons per atom for a trilayer system of 10 atomic layers of Cr embedded in Fe. Note the matching at the minority channel. Electrons/atom Fe Cr Fe Layer Number

9 Introduction So minority spin electrons at the Fermi level would see hardly any difference between the atomic potentials of Fe and Cr & experience weak reflections from interfaces and weak impurity scattering at interdiffused Fe-Cr zone. Calculations also show that the density of states at the matched spin channel is low. majority Fe Cr Fe minority Layer Number Fig. 3: DOS in Fe-Cr trilayer shows large values for the majority carriers which show no band matching.

10 Introduction Impurity scattering relaxation time τ si of electrons with spin s goes as 1 ~ N ( ) 2 s E i F Vs s τ where N s (E F ) is the DOS at E F and V s is the difference in potential between the host and the impurity. So in the case of Fe-Cr, the resistivity is much larger for the majority band due to its larger potential mismatch and higher DOS. Electron-phonon scattering, dominant at higher temperatures, also behave similarly. Using the above picture one can show that GMR = ρ / ρ 1 / 1 ρ ρ + It is the imbalance between ρ and ρ which produces GMR. Alloying of Fe and Cr at the interface is also spin-dependent and contribute to GMR, so does the spin-flip scattering at high temperatures. 2 [ Magnetic Multilayers and Giant Magnetoresistance, Ed. by Uwe Hartmann,Springer Series in Surface Sciences, Vol. 37, Berlin (1999).],

11 Introduction Hall Effect In ferromagnetic metals and alloys ρ H = E y / j x = R 0 B z + µ 0 R S M, (2) where R 0 = ordinary Hall constant (OHC), R S = extra-ordinary or spontaneous Hall constant (EHC), B = magnetic induction and M = magnetization. µ 0 R s M s ρ H µ 0 M s Fig. 4 Slope = R 0 B R 0 ( Ordinary Hall constant ) :1/ne for a single band. R S ( Extra-ordinary or spontaneous Hall constant ) : a) Classical Smit asymmetric scattering(as). b) Non-classical transport (side-jump).

12 Introduction R S is caused by AS of electrons by impurities in the presence of spin- orbit interaction in a ferromagnet. a) Classical Smit asymmetric scattering (AS) Boltzmann Eq. is correct to the lowest order in h τe F («1) (τ = relaxation time), realized for pure metals and dilute alloys at low temperatures. Using classical Boltzmann transport equation in the relaxation time r dp r = ene + dt r r m r j B [ s]j e approximation and the boundary condition: j y = 0 one can show that R S = aρ.

13 Introduction b) Non-classical transport (side-jump mechanism) h τe is not small F Concentrated and disordered alloys, high temperatures. Boltzmann transport is no more valid. Karplus-Luttinger-Kohn quantum mechanical transport theory yields ρ H = R 0 B z + µ 0 R S M S, with R S = b ρ 2. [R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 (1954).] A simple intuitive theory was proposed by Berger. The next diagram represents both classical (skew) and non-classical scattering mechanisms.

14 Introduction t» 0 t» 0 δ y t = 0 t = 0 Y Y t «0 t «0 a) b) Skew scattering Side jump scattering [L. Berger and G. Bergmann, in the Hall effect and its applications, edited by C. L. Chien and C. R. Westgate (Plenum, New York, 1980), p.55 and references therein.] S

15 Theory of Hall effect in inhomogeneous ferromagnets Hall effect in GMR multilayers All the theories discussed earlier are valid for homogeneous ferromagnets. Scaling law is valid only in the local limit; the mean free path λ «d, the layer thickness. It is invalid in the long mean free path limit; λ» d. Zhang had shown the failure of scaling law in composite magneticnonmagnetic systems: The standard boundary condition used for calculating ρ H, j y (z) = 0, is not valid here for all z, but j y (z), integrated over z, is zero. The two-point local Hall conductivity is given by σ yx (z, z') (m λ SO M Z σ CIP (z, z'))/τ(z), (3) where σ CIP (z, z') is the CIP two-point Ohmic conductivity. To get σ yx, integrate σ yx (z, z') over z and z' and sum over the spin variables. [S. Zhang, Phys. Rev. B 51, 3632(1995).]

16 Theory of Hall effect in inhomogeneous ferromagnets For homogeneous magnetic materials,σ CIP (z, z'), is proportional to τ(z) and ρ yx is simply proportional to the square of the ordinary resistivity ρ 2. For inhomogeneous magnetic systems: σ yx is found in terms of average relaxation times (τ) and thickness (t) of the components σ s yx = λ SO M Z A t m s (t m + t nm τ ms / τ nm ) -1. (4) Thus Hall conductivity depends on the ratio of relaxation times & ρ xy ~ ρ 2 is no more valid. Experiments give n between ~ 1 to ~ 4. Finally, Zhang had shown that n could be < 2, > 2, or = 2 if MFP of only the magnetic layer or the non-magnetic layer or both but in a fixed ratio, is temperature dependent.

17 Motivation To investigate Hall effect in inhomogeneous magnetic systems like Fe/Cr multilayers for better understanding of GMR Q1.Why is Hall effect important in GMR systems? An interplay between magnetic state and electrical transport GMR! Both magnetic behavior & electrical transport are involved. Hall effect! Q2. Does any scaling relation exist for GMR systems? Is it the same or is it different from that of homogeneous ferromagnets?

18 Sample structure Cr ( 50-t )Å 30 bi-layers of Fe/Cr Cr 50 Å Cr(t Å) Fe(20 Å) bi-layer Si Substrate Sample details Si/Cr(50Å)/[Fe(20Å)/Cr(tÅ)] 30/Cr((50-t)Å) Varying Cr thickness t = 6, 8, 10, 12, and 14 Å Fe/Cr multilayers prepared by ion-beam sputtering technique. Ar and Xe ions were used. Beam current 20 ma /30 ma and energy 900eV/1100eV.

19 Small Angle X-ray Reflectivity Intensity(a.u.) fit data Electron Density ( 10-5 ) air 30 bilayers of Fe-Cr Cr cap layer (37.6 A 0 ) Cr buffer layer (48 A 0 ) z (Å) Si Substrate [Fe(20A )/Cr(10A )] q z (Å -1 ) Fig. 5: Grazing angle x-ray reflectivity.

20 Experimental Techniques Hall effect and Transverse/Longitudinal Magnetoresistance measured by 5-probe and 4-probe dc methods, respectively using a home-made made variable temperature cryostat and a 7 T superconducting magnet Accuracy of V H 1 in 3000,,,, V R 1 in Magnetization measured with a Quantum Design SQUID Magnetometer.

21 Experimental Set-up Helium Level Detector Magnet Power Supply Keithley DMM Prema DMM Keithley Voltage Source Prema DMM Helium container Hall Effect Fig. 6: Hall & Magnetoresistance Set-up B Sample Decade Box Keithley Current Source Lakeshore Temperature Controller I + V + V + H I - V - V H -

22 Hall Effect Hall Resistivity (10-9 Ωm) Hall resistivity at 4.2K (Fe/Cr) 10 F10 F12 F14 F8 F6 Fe (5nm) Field(tesla) Hall Resistivity (10-9 Ωm) Hall resistivity at 4.2K (Fe/Cr) 30 F10 F12 F14 F8 F Field (tesla) Fig. 7: Hall resistivity (ρ H ) vs. applied magnetic field (µ 0 H applied ). In Hall geometry B = µ 0 [H applied + (1-N) M], N=demagnetization factor.

23 Magneto resistance MR Ratio (Fe/Cr) 10 F6 F8 F10 F12 F14 MR Ratio (Fe/Cr) 30 F6 F8 F10 F12 F at 4.2K Field (tesla) at 4.2K Field (tesla) Fig. 8: MR ratio vs. applied magnetic field (µ 0 H applied ).

24 Hall Effect Hall Resistivity (10-9 Ωm) F8 (30L) Peak position T = 300K T = 4.2K Field (tesla) Fig. 9: Hall resistivity (ρ H ) vs. applied magnetic field (µ 0 H applied ). Q3. Why do the humps appear in the EHE just before the saturation field in these GMR systems?

25 Magnetoresistance 0.00 Magnetor resistance Ratio F8 (30L) T = 300K T = 4.2K Field (tesla) Fig. 10: MR ratio vs. applied magnetic field (µ 0 H applied ).

26 Hall Effect Analysis of Hall effect in multilayer systems (Finding R s ) First, R s is determined from the initial (small magnetic fields where magnetoresistance is negligibly small) slope χ R s of ρ H vs. B curves (after subtracting R 0 B term of Eq.(3)) and correlated with the zerofield Ohmic resistivity (ρ). Second, R s is calculated at any given intermediate magnetic field below the saturation field and is correlated with the Ohmic resistivity(ρ), measured at the same field. Third, R s is obtained by fitting ofρ H vs. B data to Eq. (2) beyond saturation field (where the magnetoresistance is almost constant) and then correlated with the Ohmic resistivity measured at the same magnetic field. All three methods need transverse M(B) at each temperature.

27 Hall Effect Explanation of anomalous humps in EHE lies in its correlation with GMR Fig. 11: Plots of R s M, M and ρ against applied magnetic field (µ 0 H applied ). All show saturation around 3 tesla.

28 Hall Effect F8(30L) R S (10-9 Ωm/T) T = 5K T = 5K F10(30L) F12(30L) 1.5 T = 5K Field (tesla) Fig. 12: R s vs. applied magnetic field (µ 0 H applied ). Q5.Why does R s decrease with magnetic field at a given temperature in these GMR systems? ρ H = R 0 B z + µ 0 R S M, R S is a constant for homogeneous FM s. Instead here it decreases with field. R s and ρ go hand in hand: both affected by the modified current density in Fe layers as the magnetic field gradually brings them to a FM alignment from the AF one.

29 Hall Effect ln R s R s ~ ρ n n = 3.07 ± 0.04 F10(10L) F10(30L) n = 3.45 ± ln ρ (at B = 0) Fig. 13: Plot of ln R s vs. ln ρ (ρ being the resistivity at B = 0). Q6. Why has the scaling law failed?

30 Hall Effect (Fe/Cr) N F8 F10 F12 F14 Scaling exponent n for N = 10 Scaling exponent n for N = 30 Table I N is the number of Fe/Cr bi-layers. The exponent is larger for samples having higher GMR!!!

31 Problems of Hall effect analysis in composite systems like Fe/Cr multilayers Homogeneous ferromagnets like Bulk Fe Hall Effect Inhomogeneous ferromagnets like Fe/Cr (GMR) R s and Ohmic resistivity (ρ) have hardly any field dependence In ln R s vs. ln ρ plot, it does not matter whether one considers the resistivity(ρ) in zero field or saturation field (the scaling law remains unaltered). Only Fe causes AHE, not Cr Both R s and Ohmic resistivity (ρ) are field dependent The antiferromagnetic coupling plays a crucial role in modifying the extraordinary Hall effect & the GMR and make both of them field dependent. GMR systems and homogeneous bulk systems should not be treated on equal footings!!!

32 Hall Effect ln R s R s (calculated from the intercept of the straight line above saturation) R s (calculated from the initial slope) F6(30L) R s ~ ρ n n = 1.84 ± ln ρ Fig. 14: Plot of ln R s vs. ln ρ (B=0).

33 Hall Effect F10 (10L) 0.5T 1T 2T 3T ln R s F10(30L) 0.5T 1T 0.3 2T 3T ln ρ Fig. 15: Plot of ln R s (B) vs. ln ρ (B).

34 Hall Effect Table II Scaling exponent n for B = 0.5T B = 1T B = 2T B = 3T F8 F10 F12 F14 N = N = N = N = N = N = N = N =

35 Hall Effect Master plot for (Fe/Cr) 10 Master plot for (Fe/Cr) R s ~ ρ n n = 1.95 ± R s ~ ρ n n = 2.05 ± 0.03 ln R s F8 F10 F12 F14 ln R s ln ρ ln ρ Fig. 16: Plot of ln R s (B) vs. ln ρ (B). Scaling exponent n = 1.95 for 10 bi-layer & 2.05 for 30 bi-layer series. Each plot has 4 samples x 4 fields x 12 temperatures = 192 points. Consitent with Zhang s theory for inhomogeneous ferromagnets where n could be ~ 2 if the MFP s of both the layers are temperature dependent, but in a fixed ratio F8 F10 F12 F

36 Hall Effect Fig. 17: R S / ρ vs. ρ for all 11 samples. Fits to to the most general case R S = a ρ + b ρ 2 gives b = m/ω 2 C and a = T 1. Very small intercept (a) indicates insignificant contribution from classical asymmetric scattering. b = m/ω 2 C in dilute Fe-Cr alloys Some mixing of Fe & Cr at the interfaces possible.

37 Hall Effect A few remarks The anomalous Hall effect occurs only in individual ferromagnetic Fe layers (not in Cr) ; its ultimate manifestation depends on the effective resistivity of all the Fe layers as a whole but not the entire sample. ρ (T) of Fe & Cr are similar. The values of R s and ρ, measured at the same given field only, follow the scaling relation as the two processes occur together & governed by the same physics. So R s, measured at saturation field, should NOT be correlated with ρ at zero fields, as done commonly. The validity of the scaling law here suggests that it is the same quantum mechanical scattering mechanism which is responsible for the Hall effect in GMR systems as well. However, now it is restricted only to the individual ferromagnetic layers.

38 Hall effect A realization of the charge confinement in a Quantum Well Spin-polarized quantum well states are formed in the nonmagnetic Cr spacer layer due to multiple reflections of electrons from the interfaces of adjacent magnetic Fe layers. They mediate the exchange coupling between the FM layers. (R S (0)-R S (H S ))/R S (0) = 62 % & 20.4 % at 4.2 & 300 K as against 31.8 % & 10.3 % of (ρ(0)- ρ(h S ))/ρ(0) R s ~ ρ 2. Fig. 18 Majority electrons get gradually confined to the Cr spacer layer as H increases. This reduces the current in the Fe layers which, in turn, causes the decrease of both R s & ρ..

39 Conclusions A clear understanding of the magnetic state as well as the physical properties is necessary for proper interpretation of the ferromagnetic Hall effect leading to a better understanding of the GMR effect. The occurrence of humps in the Hall resistivity of GMR systems is quite inevitable. It manifests itself only for samples having higher GMR. The EHE is strongly correlated with the GMR effect over the entire field & temperature regions and shows the universal scaling R s ~ ρ 2. This confirms that the side-jump scattering is still the predominant scattering mechanism. Quantum-well picture: As H increases the majority electrons get gradually confined to the Cr spacer layer where spin-polarized quantum well states are formed. They mediate the exchange coupling between the FM layers. This confinement reduces the current in the Fe layers and consequently both R s & ρ. The decrease of GMR with temperature is due to the attenuation of the quantum-well.

40 Thanks