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Cu multilayers
Journal of Magnetism and Magnetic Materials 389 (2015) 169–175
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Spin-dependent scattering of conduction electrons in Co/Cu multilayers I.D. Lobov n, M.M. Kirillova, A.A. Makhnev, M.A. Milyaev, L.N. Romashev, V.V. Ustinov M.N.Miheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences, 620137 Ekaterinburg, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 21 November 2014 Received in revised form 13 April 2015 Accepted 16 April 2015 Available online 18 April 2015
The magnetotransport properties of giant magnetoresistive Co/Cu multilayers with Cu layer thickness tCu varied from 8 to 27 Å are studied using the magnetorefractive effect. Interfacial relaxation times τi↑ (↓) and scattering probabilities Pi↑ (↓) , and interface scattering asymmetry γCo/Cu of conduction electron were obtained from the magnetoreflection in the intraband absorption region. The greatest changes of τi and Pi occur in a spin-up current channel with increasing tCu . Due to a size quantization of transverse component of a quasi-momentum of free electrons in Cu layers, the magnitude of τi↑ oscillates reaching the maximum values at tCu corresponding to the first two peaks of antiferromagnetic interlayer exchange coupling, while the τi↓ experiences only slight changes keeping low values at all thicknesses of Cu layer. It was found that condition τi↑ E τCu is critical to minimize the scattering of spin-up conduction electrons at Co/Cu interface. In this case, in the region of the first peak of antiferromagnetic interlayer exchange the spin asymmetry coefficient γ ≈ 0.86 , and the probability of electron scattering Pi↑ is approximately equal to 0.02, which ensures the maximum values of giant magnetoresistance and magnetorefractive effect. & 2015 Elsevier B.V. All rights reserved.
Keywords: Co/Cu multilayers Magnetotransport properties Electron scattering Magnetoreflection
1. Introduction Among the magnetic metallic multilayers, Co/Cu system is of particular interest because of the giant magnetoresistance (GMR) at room temperature [1–7]. There is an evidence for oscillation of an indirect exchange interaction and magnetoresistance (MR) in Co/Cu(tx) [1,3,6]. Theoretical analysis within the RKKY model showed that oscillation period of the exchange coupling corresponds with the size and topology of the Fermi surface of the spacer [8–10]. Oscillations of an electronic density of states at the Fermi level were also observed in Co/Cu(tx) systems [11–13]. It is considered that the microscopic nature of GMR in magnetic metallic superlattices is due to spin-dependent scattering of conduction electrons in the bulk and at the interfaces of ferromagnetic (F) layers with the dominant contribution of the latter. The effect of the spin-polarized band structure on the magnetoresistance and conductivity in Co/Cu was studied in theory [14,15]. In particular, the realistic description of GMR in CIP (current in plane) and CPP (current perpendicular to plane) geometry was achieved taking into account spin polarization of energy bands of fcc Co, (s, p-d) hybridization of electronic states, and contribution of d electrons to conduction at the Fermi level. The estimates of scattering asymmetry of conduction electrons at Co/Cu interface were n
Corresponding author. E-mail address: [email protected] (I.D. Lobov).
http://dx.doi.org/10.1016/j.jmmm.2015.04.063 0304-8853/& 2015 Elsevier B.V. All rights reserved.
obtained from the first principle calculations of the band structure as well as from CPP-MR measurements [16,17]. Recently, we employed the optical method of studying electron scattering [18] using the magnetorefractive effect (MRE) [19]. MRE characterizes the effect of an applied magnetic field on a complex refractive index n˜ (n˜ 2 = ε , ε is a dielectric constant) being a high-frequency analog of GMR in an intraband absorption region. Both GMR and MRE are due to the asymmetry of conduction electrons scattering for spin-up (↑) and spin-down (↓) directions relative to spontaneous magnetization in bulk and at interfaces of F layers.Correlation of GMR and MRE is confirmed experimentally [18–21]. The contactless method of control of magnetoresistive properties using MRE is proposed [22,23]. However, the key problem of MRE exploring is getting an information of spin-dependent transport. The main objective of this study was to determine the fundamental characteristics of spin-dependent scattering of conduction electrons at Co/Cu interface. The paper is organized as follows: the details concerning experimental methods are presented in Section 2. Section 3 provides the optical and electronic characteristics of our samples (Section 3.1), magnetorefractive effect and the corresponding magnetoresistive effect (Section 3.2), modeling of MRE (Section 3.3), and parameters of conduction electrons responsible for magnetotransport properties (Section 3.4). Brief summary is done in Section 4.
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2. Experimental methods Samples were prepared by magnetron sputtering at the Ulvas MPS 4000 C6 sputtering system on glass substrates with Fe (50 Å) buffer layer and Cr (20 Å) cover layer. We studied the following structures: [Co(15 Å)/Cu(tCu , Å)]10, nominal tCu = 8.25–27 A˚ (series A); [Co(15 Å)/Cu(9.75 Å)]10 (sample B); [Co(15 Å)/Cu(9.0 Å)]30 (sample C). The structural properties of the samples were studied elsewhere [24]. The MRE infrared reflection spectra were measured in transverse geometry in p-polarized light using the Frontier IR Fourier spectrometer with VeeMax II Specular Reflectance accessory and grid polarizer. The spectral region was between 1.2 and 28 μm . The angle of incidence of light was set at 70° with respect to the surface normal. Magnetorefractive effect was defined as MRE(H) ¼[R(0) R(H)]/R(H), where R(0) and R(H) are the reflectivities of a sample in nonmagnetized state (H¼ 0) and in an applied magnetic field H (H ≠ 0), respectively. The magnetic field was varied from 0 to 9 kOe. The total number of scans was several thousands for each value of R(0) and R(H). The spectra were analyzed in the wavelength range of 2.5–22.5 μm . The effective refractive neff and absorption k eff indices were obtained by the ellipsometric method [25] in the wavelength range of 0.3–13 μm in two structures: sample with tCu = 23.75 A˚ (series A) and sample C. In the studied region of spectrum, the relation δ ≤ L (δ is the penetration depth of light, L is the total thickness of layered medium) is valid for these structures. The magnetoresistance was measured in a magnetic field of H ≤ 32 kOe by a standard four-contact method in CIP geometry. The magnetoresistance is defined as r ¼ [ρ(H) ρ(0)]/ρ(H), where ρ(H) and ρ(0) are the resistances in a magnetic field H and in nonmagnetized state, respectively. All measurements were carried out at room temperature.
3. Discussion 3.1. Optical and electronic characteristics We measured optical properties of two samples with Cu layer thicknesses close to the first two maxima of antiferromagnetic (AF) interlayer exchange coupling. Fig. 1 shows the plots of effective optical conductivity σ eff (ω) = neff k eff ω/4π (ω is a cyclic fre-
quency of light). Real ε1eff (λ) and imaginary ε2eff (λ) parts of
Fig. 2. Dielectric functions ε1 and ε2 of Co/Cu multilayers: 1,3 [Co(15 Å)/Cu(23.75 Å)]10; 2,4 – [Co(15 Å)/Cu(9.0 Å)]30. Lines are guide for eye.
–
dielectric constant are shown in Fig. 2. Label “eff” will be omitted at further discussion of optical characteristics. The contribution of interband absorption to optical conductivity becomes appreciable at photon energies of =ω > 0.3 eV and is manifested in the formation of the band “A” and broad “plateau”. According to the firstprinciple calculations [26], the interband transitions of (d,p–p,d)type in bulk fcc Co begin at IR frequencies in spin-down bands (↓). However, the photon energy of =ω = 0.3 eV (Fig. 1) is not a lowenergy threshold of interband absorption in Co. Performed by LMTO method, calculations of optical conductivity of fcc Co predict the formation of an additional band of interband absorption [27] in the spectral region of 0.15–0.32 eV (8.2–3.8 μm ) due to hybridization of the bands with the opposite spin direction taking into account spin–orbit interaction of conduction electrons. Drude contribution to the optical conductivity disguises the low-energy interband absorption (Fig. 1). The optical conductivity s(ω) appreciably increases at t Cu = 23.75 A˚ (curve 2), thus the interband absorption edge in Cu becomes distinct at photon energy of 2.1 eV. The dominating contribution of intraband absorption to the dielectric functions ε1 and ε2 with increasing wavelength is demonstrated in Fig. 2. From the analysis of frequency dependence of dielectric constant it is possible to estimate the effective plasma opt of conduction frequency ωp and effective relaxation time τeff electrons which are given in Table 1. Here, Neff is the effective concentration of conduction electrons defined as (ωp )2 = 4πNeff e2/m0 (e and m0 denote the charge and mass of a free electron, respectively). Notice the increase of ωp and Neff with increasing tCu . The values of ωp are used further in MRE simulations. 3.2. Spectra of magnetorefractive effect The MRE spectra in p-polarized light for series A are shown in Fig. 3. The maximum magnitude of the effect and its spectral position change not monotonically with increasing tCu . The positive sign of effect at the wavelengths less than 8 μm is due to the contribution of interband absorption to the real part of optical Table 1 Parameters of conduction electrons from optical ellipsometry: plasma frequency opt . =ωp , effective concentration Neff , and effective relaxation time τ eff
Fig. 1. Spectra of optical conductivity of Co/Cu multilayers: 1 – [Co(15 Å)/Cu(9.0 Å)]30, 2 – [Co(15 Å)/Cu(23.75 Å)]10. Low-energy band is marked as “A”. Lines are guide for eye.
tCu (Å)
=ωp (eV)
Neff (1028 m 3)
opt (10 15 s) τ eff
9.0 23.75
3.02 3.37
0.67 0.83
9.0 9.3
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εsal = εst −
ω p2 ω2
2 ⎞ m2 βsal iωτsal ⎛⎜ ⎟. 1+ ⎜ ⎟ 2 2 2 1 + iωτsal ⎝ (1 + iωτsal ) − βsal m ⎠
171
(1)
There εst is the frequency independent contribution to εsal , ωp is the plasma frequency of conduction electrons, and m is the normalized magnetization of layers (values of m we substituted by normalized magnetoresistance, m2 ¼(r /rs ), rs is the saturation value of magnetoresistance). The average relaxation time of conduction electrons τsal in a zero field and the parameter of average spin asymmetry βsal are obtained in the self-averaging limit (SAL) [19,29] of probabilities of electron scattering over a period of a multilayer structure T = tCo + tCu + 2ti , where ti is the thickness of the interface layer. Copper and cobalt have quite similar lattice parameters. Therefore, contributions to the interfacial layer from Cu and Co layers are assumed to be the same and the interface thickness ti is assumed to be equal to two monolayers (MLs). τsal and βsal were defined as in Fig. 3. Experimental magnetorefractive spectra in p-polarized IR light for Co/Cu multilayers. Cu layer nominal thicknesses are marked in angstroms by numbers on the curves. Vertical segment specifies measurement error.
conductivity of fcc Co. The effect of interband excitation of electrons on the MRE spectra is discussed in theory [28]. The correlation between the GMR and the MRE in Co/Cu(tx) structures is shown in Fig. 4 in saturation magnetic fields Hs. The maximum values of magnetorefractive effect within the λ of 8–22 μm were used in constructing the MRE(tCu ) graph in Fig. 4. The similarity of GMR and MRE dependences on the thickness of copper layers is clearly seen. The maximum values of GMR and MRE occur in the regions close to the first two maxima of AF exchange coupling, tCu = 9–10 A˚ and 22–24 Å. Their minimum values fall on the region of the interlayer exchange coupling of the F-type at tCu = 16–18 A˚ . The oscillation period of two functions is approximately equal to 12.7 Å. The largest magnetoresistive ( 50%) and magnetorefractive ( 7.25%) values for sample B are also shown in Fig. 4. 3.3. MRE spectra simulation The following considerations were taken into account in simulating the MRE. The cobalt layers have a fcc structure due to a small layer thickness and direct contact with a fcc copper. Very low mutual solubility of the components of the cobalt–copper system produces well delimited layers. Modeling consisted in calculating the intensity of reflected light using the Fresnel formulas. We used the expression [19] for the modified complex Drude dielectric function εsal :
Fig. 4. Magnetoresistive and magnetorefractive effects as functions of the Cu layer thickness. Squares denote the values of MR (filled) and MRE (unfilled) in the sample B. Lines are guide for eye.
Ref. [20]: and (τsal )−1 = 2ci /τi + cCo/τCo + (1 − 2ci − cCo )/τCu βsal = τsal × (γ (2ci )/τi + βcCo/τCo ). Here, cCo and ci are the volume fractions of Co and interface layers; τi, τCo , and τCu are the relaxation times of conduction electrons at interfaces in Co and Cu layers, respectively; γ and β are the coefficients of spin asymmetry at an interface and in Co layer. The following parameters were chosen: τCo = 1.2 × 10−14 s, τCu = 2.0 × 10−14 s (τCu was evaluated from the ellipsometric data [30]), ωp = 4.6 × 1015 s−1 for tCu = 8–18 A˚ and
ωp ¼5.13 1015 s 1 for tCu = 20–27 A˚ , β ¼0.67 corresponding to the density-of-states ratio nF↑/nF↓ = 0.2 in fcc Co at the Fermi level [31]. Thus, the numerical simulation of the MRE spectra is reduced to finding γ and τi. It was shown that spectral position of the MRE maximum is mainly controlled by relaxation time τi and the magnitude of MRE is controlled by γ and τi [18], therefore the magnitude and spectral position of the MRE maximum are the critical parameters in this procedure. We matched the model MRE curves to the measured spectra manually using the coordinatewise descent method. The criteria of the quality of τi and γ fit to obtain a spectral position and an amplitude of calculated MRE maximum in exp ± 0.5) μm and the range of values bounded by segments of (λmax exp (MREmax ± 0.1)% . In Fig. 5 the uncertainties for our fitting criteria of γ and τi are shown using contour plots. Experimental and calculated as described above, the MRE spectra of sample B are given in Fig. 6. It can be seen that contribution to the effect from free carriers tends to zero at the short wavelengths, and gradually rises with increasing λ. The difference between the experimental (1) and the calculated (2) MRE characterizes the interband contribution (3) to
Fig. 5. Contours of deviations of spectral positions and amplitudes of calculated ˚, MRE maxima as functions of τi and γ for four samples: 1 − tCu = 16 A ˚ , 3 − tCu = 22.75 A ˚ , 4 – sample B. Contour line indicates the combi2 − tCu = 27 A calc nation of τi and γ corresponding to deviation of position and amplitude of MREmax exp exp not more than (λmax ± 0.5) μm and (MREmax ± 0.1)% .
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Fig. 6. Magnetorefractive spectra of sample B: 1 – experimental spectrum; 2 – model MRE spectrum; 3 – interband contribution to MRE. Parameters of simulation: =ωp = 3.02 eV ; m¼ 0.778; τ sal = 6.38 × 10−15 s ; βsal = 0.745.
Fig. 8. Coefficient of scattering asymmetry γ vs tCu . Square denotes the value of γ in the sample B. The γ values obtained from the CPP-MR are indicated by segments: 1 – nanowires [16]; 2 – superlattices [17]. Dashed line is guide for eye. Error bars specify the uncertainties in the determination of γ.
(tCu = 16–18 A˚ ), the value of γ is sharply reduced to 0.1. In the sample B the interface scattering asymmetry γCo/Cu reaches the value of 0.86. For γ, the oscillation period Λ ≈ 12.7 A˚ which is si-
Fig. 7. Dependence of effective relaxation time τ sal on Cu layer thickness. Line is guide for eye.
magnetorefractive response in the free electron model [19]. The other samples show a similar correspondence between the model and experimental curves. The probability of an electron scattering P at passing one period T of multilayer is determined by the effective relaxation rate of electrons (τsal )−1 = 〈νF 〉P /T (〈νF 〉 is the average electron velocity on the Fermi surface). The values of the spinindependent relaxation time τsal obtained by modeling MRE spectra are shown in Fig. 7. τsal has two maxima in the region from 8 to 27 Å of tCu . The behavior of τsal indicates the complex nature of conduction electrons scattering in Co/Cu(tx) multilayers due to the electronic structure of Co and Cu, and testifies to the unambiguous relationship with quantum oscillations in the electron density of states at the Fermi level in copper layer with increasing tCu . It should be noted that the difference between the values of τsal in Fig. 7 and τ opt obtained from the ellipsometric measurements (Table 1) is not more than 22%. 3.4. Interfacial parameters of conduction electron scattering, relationship with the electronic structure Fig. 8 exhibits the variation of interface scattering asymmetry γ with Cu layer thickness. The shape of γ vs tCu curve has two peaks with oscillation amplitude of 0.6–0.7 formed at 9–10 and 22–24 Å of Cu layer thickness corresponding to the first two peaks of AFexchange coupling. In the region of F-type of exchange coupling
milar to that of MRE and MR (Fig. 4) and to oscillation period of an AF-exchange coupling in Co/Cu multilayers [1,3,6], where Λ ¼10– 13 Å. In the ferromagnetic fcc Co the energy bands E(k ) 3d↑ and E (k ) 3d↓ are shifted relative to each other on an energy E E1.5 eV. As a result, the Fermi level passes above the top of 3d↑ bands and intersects the 3d↓ bands in their middle part. The difference in populations of electronic states with spin-up and spin-down directions leads to a difference in the Fermi surface topologies and in the values of conductivities σ ↑ (↓) . The s, p-electrons make the main ↑ conductivity [14,15] (of about 80%), whereas dcontribution to σCo ↓ electrons hybridized with s, p-electrons play a major role in σCo conductivity. The spin-up electrons can relatively freely pass through the interface experiencing only a partial reflection due to the close identity of the Fermi surface topology of (s–p)-type in copper with fcc Co (↑) one in cobalt. On the contrary, due to the large difference in symmetry of the electronic states E(k )(↓) at the Fermi level in Co with s–p states in Cu, the spin-down electrons experience the significant scattering at interfaces. Taking into account only the electronic structure of contacting metals, the first principle calculations of spin asymmetry coefficient γ in the case of specular reflection give the results [32] quite close to that obtained from the MRE spectra: γCo/Cu(111) = 0.64 , and γCo/Cu(100) = 0.76. The values of γCo/Cu is useful to compare with that obtained from the magnetoresistive measurements in CPP geometry for multilayer films and electrodeposited wires with nanometer and micrometer thicknesses of Co and Cu layers (tCu > 60 A˚ ). In such structures an interlayer exchange coupling is usually greatly weakened. The data analysis [16,17] of CPP-MR in the framework of theory [33] yielded the values of γCo/Cu(100) = 0.76 (4.2 K) and 0.85 70.1 (77 K) which are quite close to that obtained in the present study (Fig. 8). Calculations of an interface resistance in layered films [34] indicate that imperfection of interfaces pulls together the values of conductivity in spin-up and spin-down current channels and, consequently, reduces the coefficient of the spin asymmetry γ. The increased value of γCo/Cu = 0.86 testifies to a more perfect microstructure of interfaces in the sample B compared with the series A (Fig. 8). Fig. 9 shows the graphs of spin-independent τi(tCu ) and spin-dependent τi↑ (↓) (tCu ) interfacial relaxation times, here
τi↑ (↓) = τi /(1 ∓ γm) and m¼ 1. The
τi values vary within the limits of
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Fig. 9. Relaxation times τi and τ i↑ (↓) of conduction electrons at Co/Cu interface depending on Cu layer thickness. Triangles denote the values of relaxation times in sample B: τ i↑ – filled triangle, τi – half-filled triangle, and τ i↓ – unfilled triangle. Dashed line represents relaxation time of conduction electrons in bulk Cu. Solid lines are guide for eye.
(1.8–3.4) 10 15 s at changing tCu from 8 to 27 Å. Spin-down (↓) electrons persist the lowest values of relaxation time for all tCu . On the contrary, the τi↑ parameter experiences the substantial changes with increasing tCu . Depending on an ordering of magnetic moments of adjacent Co layers, the values of τi↑ vary with increasing tCu from 11 10 15 s (antiparallel orientation) to 2 10 15 s (parallel orientation), and increase again to 5.3 10 15 s in the second peak of AF-exchange coupling (Fig. 9). In sample B the relaxation time of spin-up (↑) electrons reaches a maximum value of 19.6 10 15 s (Fig. 9). The value of relaxation time of conduction electrons in the bulk Cu layer τCu = 20 × 10−15 s was obtained analyzing the ellipsometric data of [30] in the spectral region of 1–10 μm . Thus, the interfacial parameter τi↑ in the sample B almost reached its limit, equaling in magnitude to the relaxation time of conduction electrons in bulk Cu layer. This is due to the close similarity of the Fermi surface topology of nonmagnetic Cu and that of magnetic fcc Co (↑) which corresponds to the free-electron (s–p) character of conductivity. The maximum difference in the interfacial scattering of conduction electrons with different spin directions in the sample B is characterized by the ratio τi↑/τi↓ ≈ 13. The probabilities of electron scattering P at passing one period T of ↑ (↓) multilayer were obtained using the relation P ↑ (↓) = PCo + PCu+ ↑ (↓) , PCu , and Pi↑ (↓) are the scattering probabilities of 2Pi↑ (↓) , where PCo conduction electron in the layers of Co and Cu, and at the Co/Cu interface, respectively. The values of Pi↑ (↓) for Co/Cu and Cu/Co interfaces were assumed to be the same. Probabilities were calcu↑ (↓) lated using the formula Pa↑ (↓) = ta /(τa↑ (↓) 〈νG( a) 〉), where index (a) denotes Co, Cu, and interface. Here are the quantities not mentioned until now: tCo = 11.35 A˚ (net of interface), tCu = tx − 3.65 A˚
(where tx are the nominal Cu layer thicknesses), ti ¼3.65 Å ↑ (↓) were estimated (2 MLs). The relaxation times in cobalt layer τCo ↑ (↓) by the relation τCo = τCo/(1 ∓ βCo ). The average electron velocities on the Fermi surfaces were assumed to be the following: ↑ (↓) 〈νF(Co) 〉 = 8 × 107 (5 × 107) cm/s ; 〈νF(Cu) 〉 = 1 × 108 cm/s. And we did not take into account a possible spin polarization of conduction electrons in copper, βCu = 0. Fig. 10 exhibits the probability of electron scattering at passing one period of multilayer as a function of the Cu layer thickness. The scattering probability of spin-up electron is less than unity for all tCu . Another case occurs for spin-down electron which scattering probability when passing one bilayer always reaches its physical limit (P max = 1). This means a scattering of spin-down electron within the thickness of one bilayer at all tCu . We are also interested in the interfacial scattering
Fig. 10. (a) Scattering probability P ↑ of conduction electron when passing one period of Co/Cu multilayer. (b) Difference between the scattering probabilities at interface ΔPi↓ (↑) = Pi↓ − Pi↑ . Squares denote the values of parameters for sample B. Lines are guide for eye.
Fig. 11. Mean free path l↑ (↓) of conduction electrons depending on Cu layer thickness. Squares denote l↑ (↓) for sample B: l↑ – filled square, l↓ – unfilled square. Dashed line denotes the bilayer period of Co/Cu multilayer. Solid lines are guide for eye.
probabilities of conduction electrons in the regions of the first two maxima of AF interlayer exchange coupling at 9–10 Å and 22–23 Å of Cu layer thickness. According to our estimates the interface scattering probability of spin-up electrons is only 0.02–0.07 in these regions,while for the spin-down electrons it remains high (Pi↓ = 0.35–0.60). Fig. 10(b) demonstrates the ΔPi↓ (↑) function which is equal to the difference between the scattering probabilities of conduction electrons with opposite spins at an interface. With increasing tCu , the two maxima of ΔPi↓ (↑) (tCu ) are observed which coincide with the maxima of MRE(tCu ) and MR(tCu ). The amplitudes of maxima are close to 0.5. In Fig. 11 we estimate the mean-free ↑ (↓) , where path (mfp) of conduction electrons: l↑ (↓) = 〈νF↑ (↓) 〉τsal ↑ (↓) 〈νF↑ (↓) 〉 = (1/T )(〈νF(Co) 〉tCo + 〈νF(Cu) 〉tCu ) [there, tCu and tCo are nom-
inal]. The l↓ value is approximately equal to multilayer period T while the l↑ oscillates reaching the maximum values at tCu corresponding to the positions of maxima of AF-exchange coupling and becoming the minimum in the F-exchange region. In sample B we
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Thus, the conductivity of spin-up current channel is further limited by the bulk properties of structure, not by interface. The scattering probability of conduction electrons in the spin-down channel reaches its maximum (P ↓ = 1) already over one period of multilayer. This means that the spin-down electrons are restricted in their moving in the direction perpendicular to the sample plane while the spin-up electrons move freely through all the layers implementing a predominantly “single-spin-channel” transport at the parallel ordering of magnetic moments of adjacent magnetic layers. The spin asymmetry coefficient of the interface scattering was determined and its oscillation with Cu layer thickness was found. The amplitude values of spin asymmetry coefficient γCo/Cu in the range of 8–27 Å thick of Cu layer coincide with the values obtained by CPP-MR measurements at hundred nanometers of layer thicknesses, which indicate the dominant role of the electronic structure of contacting metals in the interface scattering.
Acknowledgments
Fig. 12. (a) Spin-dependent scattering probability of conduction electrons at one period of Fe/Cr superlattice; (b) mean free path of conduction electrons versus tCr . Dashed line denotes the period of Fe/Cr superlattice. Solid lines are guide for eye.
˚ . This means obtained the following values: l↑ ≈ 220 A˚ and l↓ ≈ 26 A that the spin-down electrons are restricted in their movement while the spin-up electrons can travel long distances from the substrate up to the top layer of a multilayer structure. Comparison of scattering probabilities and mfp of electrons in Co/Cu multilayers with that in Fe/Cr superlattices [35] (Fig. 12) shows that the difference between spin-up and spin-down electronic characteristics in Co/Cu system is more pronounced. In particular, the probability of scattering when passing through the bilayer period in Fe/Cr system is always less than unity, and the ΔPi↑ (↓) value in the region of the first AF-peak of exchange coupling is equal to 0.14 in Fe/Cr as compared with ΔPi↑ (↓) = 0.46 in Co/Cu. Similarly, the mfp ratio for conduction electrons in the region of the first maximum of AF exchange constitutes l↑/l↓ = 8.6 in Co/Cu, and l↓/l↑ = 1.6 in Fe/Cr.
4. Conclusions Optical ellipsometry of Co(15 Å)/Cu(tx) multilayers with Cu layer thicknesses corresponding to the first two peaks of AF interlayer exchange coupling showed that the Drude-type optical absorption becomes dominating in the wavelength range of λ ≥ 8 μm , and in simulating it is necessary to take into account the growth of plasma frequency with increasing Cu thickness. We got a good correlation between the GMR and MRE effects depending on Cu layer thickness, which gave us a confidence in correctness of our results and allowed to advance onward in obtaining the parameters of electron scattering. From the analysis of the MRE spectra it follows that the major changes of the interface scattering of conduction electrons occur with increasing thickness of Cu layer in the spin-up current channel. In the sample with the maximal magnetoresistance the value of interfacial relaxation time τi↑ is virtually equal to relaxation time of electrons in bulk copper, which is a consequence of the similarity of the Fermi surface topology of (s–p)-type in copper with that of fcc Co (↑) in cobalt.
We thank V. Proglyado and N. Bannikova for preparation and characterization of the samples, and we thank N. Bebenin for fruitful discussions. This work was performed as part of the state assignment “Spin” No. 01201463330 (Project 12-P-2-1051) with the support of the Russian Ministry of Education and Science (Contract nos. 14.Z50.31.0025 and 14.120.14.1540-SS), and the Russian Foundation for Basic Research (Project 13-02-00749).
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Spin-dependent scattering of conduction electrons in Co/Cu multilayers I.D. Lobov n, M.M. Kirillova, A.A. Makhnev, M.A. Milyaev, L.N. Romashev, V.V. Ustinov M.N.Miheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences, 620137 Ekaterinburg, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 21 November 2014 Received in revised form 13 April 2015 Accepted 16 April 2015 Available online 18 April 2015
The magnetotransport properties of giant magnetoresistive Co/Cu multilayers with Cu layer thickness tCu varied from 8 to 27 Å are studied using the magnetorefractive effect. Interfacial relaxation times τi↑ (↓) and scattering probabilities Pi↑ (↓) , and interface scattering asymmetry γCo/Cu of conduction electron were obtained from the magnetoreflection in the intraband absorption region. The greatest changes of τi and Pi occur in a spin-up current channel with increasing tCu . Due to a size quantization of transverse component of a quasi-momentum of free electrons in Cu layers, the magnitude of τi↑ oscillates reaching the maximum values at tCu corresponding to the first two peaks of antiferromagnetic interlayer exchange coupling, while the τi↓ experiences only slight changes keeping low values at all thicknesses of Cu layer. It was found that condition τi↑ E τCu is critical to minimize the scattering of spin-up conduction electrons at Co/Cu interface. In this case, in the region of the first peak of antiferromagnetic interlayer exchange the spin asymmetry coefficient γ ≈ 0.86 , and the probability of electron scattering Pi↑ is approximately equal to 0.02, which ensures the maximum values of giant magnetoresistance and magnetorefractive effect. & 2015 Elsevier B.V. All rights reserved.
Keywords: Co/Cu multilayers Magnetotransport properties Electron scattering Magnetoreflection
1. Introduction Among the magnetic metallic multilayers, Co/Cu system is of particular interest because of the giant magnetoresistance (GMR) at room temperature [1–7]. There is an evidence for oscillation of an indirect exchange interaction and magnetoresistance (MR) in Co/Cu(tx) [1,3,6]. Theoretical analysis within the RKKY model showed that oscillation period of the exchange coupling corresponds with the size and topology of the Fermi surface of the spacer [8–10]. Oscillations of an electronic density of states at the Fermi level were also observed in Co/Cu(tx) systems [11–13]. It is considered that the microscopic nature of GMR in magnetic metallic superlattices is due to spin-dependent scattering of conduction electrons in the bulk and at the interfaces of ferromagnetic (F) layers with the dominant contribution of the latter. The effect of the spin-polarized band structure on the magnetoresistance and conductivity in Co/Cu was studied in theory [14,15]. In particular, the realistic description of GMR in CIP (current in plane) and CPP (current perpendicular to plane) geometry was achieved taking into account spin polarization of energy bands of fcc Co, (s, p-d) hybridization of electronic states, and contribution of d electrons to conduction at the Fermi level. The estimates of scattering asymmetry of conduction electrons at Co/Cu interface were n
Corresponding author. E-mail address: [email protected] (I.D. Lobov).
http://dx.doi.org/10.1016/j.jmmm.2015.04.063 0304-8853/& 2015 Elsevier B.V. All rights reserved.
obtained from the first principle calculations of the band structure as well as from CPP-MR measurements [16,17]. Recently, we employed the optical method of studying electron scattering [18] using the magnetorefractive effect (MRE) [19]. MRE characterizes the effect of an applied magnetic field on a complex refractive index n˜ (n˜ 2 = ε , ε is a dielectric constant) being a high-frequency analog of GMR in an intraband absorption region. Both GMR and MRE are due to the asymmetry of conduction electrons scattering for spin-up (↑) and spin-down (↓) directions relative to spontaneous magnetization in bulk and at interfaces of F layers.Correlation of GMR and MRE is confirmed experimentally [18–21]. The contactless method of control of magnetoresistive properties using MRE is proposed [22,23]. However, the key problem of MRE exploring is getting an information of spin-dependent transport. The main objective of this study was to determine the fundamental characteristics of spin-dependent scattering of conduction electrons at Co/Cu interface. The paper is organized as follows: the details concerning experimental methods are presented in Section 2. Section 3 provides the optical and electronic characteristics of our samples (Section 3.1), magnetorefractive effect and the corresponding magnetoresistive effect (Section 3.2), modeling of MRE (Section 3.3), and parameters of conduction electrons responsible for magnetotransport properties (Section 3.4). Brief summary is done in Section 4.
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2. Experimental methods Samples were prepared by magnetron sputtering at the Ulvas MPS 4000 C6 sputtering system on glass substrates with Fe (50 Å) buffer layer and Cr (20 Å) cover layer. We studied the following structures: [Co(15 Å)/Cu(tCu , Å)]10, nominal tCu = 8.25–27 A˚ (series A); [Co(15 Å)/Cu(9.75 Å)]10 (sample B); [Co(15 Å)/Cu(9.0 Å)]30 (sample C). The structural properties of the samples were studied elsewhere [24]. The MRE infrared reflection spectra were measured in transverse geometry in p-polarized light using the Frontier IR Fourier spectrometer with VeeMax II Specular Reflectance accessory and grid polarizer. The spectral region was between 1.2 and 28 μm . The angle of incidence of light was set at 70° with respect to the surface normal. Magnetorefractive effect was defined as MRE(H) ¼[R(0) R(H)]/R(H), where R(0) and R(H) are the reflectivities of a sample in nonmagnetized state (H¼ 0) and in an applied magnetic field H (H ≠ 0), respectively. The magnetic field was varied from 0 to 9 kOe. The total number of scans was several thousands for each value of R(0) and R(H). The spectra were analyzed in the wavelength range of 2.5–22.5 μm . The effective refractive neff and absorption k eff indices were obtained by the ellipsometric method [25] in the wavelength range of 0.3–13 μm in two structures: sample with tCu = 23.75 A˚ (series A) and sample C. In the studied region of spectrum, the relation δ ≤ L (δ is the penetration depth of light, L is the total thickness of layered medium) is valid for these structures. The magnetoresistance was measured in a magnetic field of H ≤ 32 kOe by a standard four-contact method in CIP geometry. The magnetoresistance is defined as r ¼ [ρ(H) ρ(0)]/ρ(H), where ρ(H) and ρ(0) are the resistances in a magnetic field H and in nonmagnetized state, respectively. All measurements were carried out at room temperature.
3. Discussion 3.1. Optical and electronic characteristics We measured optical properties of two samples with Cu layer thicknesses close to the first two maxima of antiferromagnetic (AF) interlayer exchange coupling. Fig. 1 shows the plots of effective optical conductivity σ eff (ω) = neff k eff ω/4π (ω is a cyclic fre-
quency of light). Real ε1eff (λ) and imaginary ε2eff (λ) parts of
Fig. 2. Dielectric functions ε1 and ε2 of Co/Cu multilayers: 1,3 [Co(15 Å)/Cu(23.75 Å)]10; 2,4 – [Co(15 Å)/Cu(9.0 Å)]30. Lines are guide for eye.
–
dielectric constant are shown in Fig. 2. Label “eff” will be omitted at further discussion of optical characteristics. The contribution of interband absorption to optical conductivity becomes appreciable at photon energies of =ω > 0.3 eV and is manifested in the formation of the band “A” and broad “plateau”. According to the firstprinciple calculations [26], the interband transitions of (d,p–p,d)type in bulk fcc Co begin at IR frequencies in spin-down bands (↓). However, the photon energy of =ω = 0.3 eV (Fig. 1) is not a lowenergy threshold of interband absorption in Co. Performed by LMTO method, calculations of optical conductivity of fcc Co predict the formation of an additional band of interband absorption [27] in the spectral region of 0.15–0.32 eV (8.2–3.8 μm ) due to hybridization of the bands with the opposite spin direction taking into account spin–orbit interaction of conduction electrons. Drude contribution to the optical conductivity disguises the low-energy interband absorption (Fig. 1). The optical conductivity s(ω) appreciably increases at t Cu = 23.75 A˚ (curve 2), thus the interband absorption edge in Cu becomes distinct at photon energy of 2.1 eV. The dominating contribution of intraband absorption to the dielectric functions ε1 and ε2 with increasing wavelength is demonstrated in Fig. 2. From the analysis of frequency dependence of dielectric constant it is possible to estimate the effective plasma opt of conduction frequency ωp and effective relaxation time τeff electrons which are given in Table 1. Here, Neff is the effective concentration of conduction electrons defined as (ωp )2 = 4πNeff e2/m0 (e and m0 denote the charge and mass of a free electron, respectively). Notice the increase of ωp and Neff with increasing tCu . The values of ωp are used further in MRE simulations. 3.2. Spectra of magnetorefractive effect The MRE spectra in p-polarized light for series A are shown in Fig. 3. The maximum magnitude of the effect and its spectral position change not monotonically with increasing tCu . The positive sign of effect at the wavelengths less than 8 μm is due to the contribution of interband absorption to the real part of optical Table 1 Parameters of conduction electrons from optical ellipsometry: plasma frequency opt . =ωp , effective concentration Neff , and effective relaxation time τ eff
Fig. 1. Spectra of optical conductivity of Co/Cu multilayers: 1 – [Co(15 Å)/Cu(9.0 Å)]30, 2 – [Co(15 Å)/Cu(23.75 Å)]10. Low-energy band is marked as “A”. Lines are guide for eye.
tCu (Å)
=ωp (eV)
Neff (1028 m 3)
opt (10 15 s) τ eff
9.0 23.75
3.02 3.37
0.67 0.83
9.0 9.3
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εsal = εst −
ω p2 ω2
2 ⎞ m2 βsal iωτsal ⎛⎜ ⎟. 1+ ⎜ ⎟ 2 2 2 1 + iωτsal ⎝ (1 + iωτsal ) − βsal m ⎠
171
(1)
There εst is the frequency independent contribution to εsal , ωp is the plasma frequency of conduction electrons, and m is the normalized magnetization of layers (values of m we substituted by normalized magnetoresistance, m2 ¼(r /rs ), rs is the saturation value of magnetoresistance). The average relaxation time of conduction electrons τsal in a zero field and the parameter of average spin asymmetry βsal are obtained in the self-averaging limit (SAL) [19,29] of probabilities of electron scattering over a period of a multilayer structure T = tCo + tCu + 2ti , where ti is the thickness of the interface layer. Copper and cobalt have quite similar lattice parameters. Therefore, contributions to the interfacial layer from Cu and Co layers are assumed to be the same and the interface thickness ti is assumed to be equal to two monolayers (MLs). τsal and βsal were defined as in Fig. 3. Experimental magnetorefractive spectra in p-polarized IR light for Co/Cu multilayers. Cu layer nominal thicknesses are marked in angstroms by numbers on the curves. Vertical segment specifies measurement error.
conductivity of fcc Co. The effect of interband excitation of electrons on the MRE spectra is discussed in theory [28]. The correlation between the GMR and the MRE in Co/Cu(tx) structures is shown in Fig. 4 in saturation magnetic fields Hs. The maximum values of magnetorefractive effect within the λ of 8–22 μm were used in constructing the MRE(tCu ) graph in Fig. 4. The similarity of GMR and MRE dependences on the thickness of copper layers is clearly seen. The maximum values of GMR and MRE occur in the regions close to the first two maxima of AF exchange coupling, tCu = 9–10 A˚ and 22–24 Å. Their minimum values fall on the region of the interlayer exchange coupling of the F-type at tCu = 16–18 A˚ . The oscillation period of two functions is approximately equal to 12.7 Å. The largest magnetoresistive ( 50%) and magnetorefractive ( 7.25%) values for sample B are also shown in Fig. 4. 3.3. MRE spectra simulation The following considerations were taken into account in simulating the MRE. The cobalt layers have a fcc structure due to a small layer thickness and direct contact with a fcc copper. Very low mutual solubility of the components of the cobalt–copper system produces well delimited layers. Modeling consisted in calculating the intensity of reflected light using the Fresnel formulas. We used the expression [19] for the modified complex Drude dielectric function εsal :
Fig. 4. Magnetoresistive and magnetorefractive effects as functions of the Cu layer thickness. Squares denote the values of MR (filled) and MRE (unfilled) in the sample B. Lines are guide for eye.
Ref. [20]: and (τsal )−1 = 2ci /τi + cCo/τCo + (1 − 2ci − cCo )/τCu βsal = τsal × (γ (2ci )/τi + βcCo/τCo ). Here, cCo and ci are the volume fractions of Co and interface layers; τi, τCo , and τCu are the relaxation times of conduction electrons at interfaces in Co and Cu layers, respectively; γ and β are the coefficients of spin asymmetry at an interface and in Co layer. The following parameters were chosen: τCo = 1.2 × 10−14 s, τCu = 2.0 × 10−14 s (τCu was evaluated from the ellipsometric data [30]), ωp = 4.6 × 1015 s−1 for tCu = 8–18 A˚ and
ωp ¼5.13 1015 s 1 for tCu = 20–27 A˚ , β ¼0.67 corresponding to the density-of-states ratio nF↑/nF↓ = 0.2 in fcc Co at the Fermi level [31]. Thus, the numerical simulation of the MRE spectra is reduced to finding γ and τi. It was shown that spectral position of the MRE maximum is mainly controlled by relaxation time τi and the magnitude of MRE is controlled by γ and τi [18], therefore the magnitude and spectral position of the MRE maximum are the critical parameters in this procedure. We matched the model MRE curves to the measured spectra manually using the coordinatewise descent method. The criteria of the quality of τi and γ fit to obtain a spectral position and an amplitude of calculated MRE maximum in exp ± 0.5) μm and the range of values bounded by segments of (λmax exp (MREmax ± 0.1)% . In Fig. 5 the uncertainties for our fitting criteria of γ and τi are shown using contour plots. Experimental and calculated as described above, the MRE spectra of sample B are given in Fig. 6. It can be seen that contribution to the effect from free carriers tends to zero at the short wavelengths, and gradually rises with increasing λ. The difference between the experimental (1) and the calculated (2) MRE characterizes the interband contribution (3) to
Fig. 5. Contours of deviations of spectral positions and amplitudes of calculated ˚, MRE maxima as functions of τi and γ for four samples: 1 − tCu = 16 A ˚ , 3 − tCu = 22.75 A ˚ , 4 – sample B. Contour line indicates the combi2 − tCu = 27 A calc nation of τi and γ corresponding to deviation of position and amplitude of MREmax exp exp not more than (λmax ± 0.5) μm and (MREmax ± 0.1)% .
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Fig. 6. Magnetorefractive spectra of sample B: 1 – experimental spectrum; 2 – model MRE spectrum; 3 – interband contribution to MRE. Parameters of simulation: =ωp = 3.02 eV ; m¼ 0.778; τ sal = 6.38 × 10−15 s ; βsal = 0.745.
Fig. 8. Coefficient of scattering asymmetry γ vs tCu . Square denotes the value of γ in the sample B. The γ values obtained from the CPP-MR are indicated by segments: 1 – nanowires [16]; 2 – superlattices [17]. Dashed line is guide for eye. Error bars specify the uncertainties in the determination of γ.
(tCu = 16–18 A˚ ), the value of γ is sharply reduced to 0.1. In the sample B the interface scattering asymmetry γCo/Cu reaches the value of 0.86. For γ, the oscillation period Λ ≈ 12.7 A˚ which is si-
Fig. 7. Dependence of effective relaxation time τ sal on Cu layer thickness. Line is guide for eye.
magnetorefractive response in the free electron model [19]. The other samples show a similar correspondence between the model and experimental curves. The probability of an electron scattering P at passing one period T of multilayer is determined by the effective relaxation rate of electrons (τsal )−1 = 〈νF 〉P /T (〈νF 〉 is the average electron velocity on the Fermi surface). The values of the spinindependent relaxation time τsal obtained by modeling MRE spectra are shown in Fig. 7. τsal has two maxima in the region from 8 to 27 Å of tCu . The behavior of τsal indicates the complex nature of conduction electrons scattering in Co/Cu(tx) multilayers due to the electronic structure of Co and Cu, and testifies to the unambiguous relationship with quantum oscillations in the electron density of states at the Fermi level in copper layer with increasing tCu . It should be noted that the difference between the values of τsal in Fig. 7 and τ opt obtained from the ellipsometric measurements (Table 1) is not more than 22%. 3.4. Interfacial parameters of conduction electron scattering, relationship with the electronic structure Fig. 8 exhibits the variation of interface scattering asymmetry γ with Cu layer thickness. The shape of γ vs tCu curve has two peaks with oscillation amplitude of 0.6–0.7 formed at 9–10 and 22–24 Å of Cu layer thickness corresponding to the first two peaks of AFexchange coupling. In the region of F-type of exchange coupling
milar to that of MRE and MR (Fig. 4) and to oscillation period of an AF-exchange coupling in Co/Cu multilayers [1,3,6], where Λ ¼10– 13 Å. In the ferromagnetic fcc Co the energy bands E(k ) 3d↑ and E (k ) 3d↓ are shifted relative to each other on an energy E E1.5 eV. As a result, the Fermi level passes above the top of 3d↑ bands and intersects the 3d↓ bands in their middle part. The difference in populations of electronic states with spin-up and spin-down directions leads to a difference in the Fermi surface topologies and in the values of conductivities σ ↑ (↓) . The s, p-electrons make the main ↑ conductivity [14,15] (of about 80%), whereas dcontribution to σCo ↓ electrons hybridized with s, p-electrons play a major role in σCo conductivity. The spin-up electrons can relatively freely pass through the interface experiencing only a partial reflection due to the close identity of the Fermi surface topology of (s–p)-type in copper with fcc Co (↑) one in cobalt. On the contrary, due to the large difference in symmetry of the electronic states E(k )(↓) at the Fermi level in Co with s–p states in Cu, the spin-down electrons experience the significant scattering at interfaces. Taking into account only the electronic structure of contacting metals, the first principle calculations of spin asymmetry coefficient γ in the case of specular reflection give the results [32] quite close to that obtained from the MRE spectra: γCo/Cu(111) = 0.64 , and γCo/Cu(100) = 0.76. The values of γCo/Cu is useful to compare with that obtained from the magnetoresistive measurements in CPP geometry for multilayer films and electrodeposited wires with nanometer and micrometer thicknesses of Co and Cu layers (tCu > 60 A˚ ). In such structures an interlayer exchange coupling is usually greatly weakened. The data analysis [16,17] of CPP-MR in the framework of theory [33] yielded the values of γCo/Cu(100) = 0.76 (4.2 K) and 0.85 70.1 (77 K) which are quite close to that obtained in the present study (Fig. 8). Calculations of an interface resistance in layered films [34] indicate that imperfection of interfaces pulls together the values of conductivity in spin-up and spin-down current channels and, consequently, reduces the coefficient of the spin asymmetry γ. The increased value of γCo/Cu = 0.86 testifies to a more perfect microstructure of interfaces in the sample B compared with the series A (Fig. 8). Fig. 9 shows the graphs of spin-independent τi(tCu ) and spin-dependent τi↑ (↓) (tCu ) interfacial relaxation times, here
τi↑ (↓) = τi /(1 ∓ γm) and m¼ 1. The
τi values vary within the limits of
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Fig. 9. Relaxation times τi and τ i↑ (↓) of conduction electrons at Co/Cu interface depending on Cu layer thickness. Triangles denote the values of relaxation times in sample B: τ i↑ – filled triangle, τi – half-filled triangle, and τ i↓ – unfilled triangle. Dashed line represents relaxation time of conduction electrons in bulk Cu. Solid lines are guide for eye.
(1.8–3.4) 10 15 s at changing tCu from 8 to 27 Å. Spin-down (↓) electrons persist the lowest values of relaxation time for all tCu . On the contrary, the τi↑ parameter experiences the substantial changes with increasing tCu . Depending on an ordering of magnetic moments of adjacent Co layers, the values of τi↑ vary with increasing tCu from 11 10 15 s (antiparallel orientation) to 2 10 15 s (parallel orientation), and increase again to 5.3 10 15 s in the second peak of AF-exchange coupling (Fig. 9). In sample B the relaxation time of spin-up (↑) electrons reaches a maximum value of 19.6 10 15 s (Fig. 9). The value of relaxation time of conduction electrons in the bulk Cu layer τCu = 20 × 10−15 s was obtained analyzing the ellipsometric data of [30] in the spectral region of 1–10 μm . Thus, the interfacial parameter τi↑ in the sample B almost reached its limit, equaling in magnitude to the relaxation time of conduction electrons in bulk Cu layer. This is due to the close similarity of the Fermi surface topology of nonmagnetic Cu and that of magnetic fcc Co (↑) which corresponds to the free-electron (s–p) character of conductivity. The maximum difference in the interfacial scattering of conduction electrons with different spin directions in the sample B is characterized by the ratio τi↑/τi↓ ≈ 13. The probabilities of electron scattering P at passing one period T of ↑ (↓) multilayer were obtained using the relation P ↑ (↓) = PCo + PCu+ ↑ (↓) , PCu , and Pi↑ (↓) are the scattering probabilities of 2Pi↑ (↓) , where PCo conduction electron in the layers of Co and Cu, and at the Co/Cu interface, respectively. The values of Pi↑ (↓) for Co/Cu and Cu/Co interfaces were assumed to be the same. Probabilities were calcu↑ (↓) lated using the formula Pa↑ (↓) = ta /(τa↑ (↓) 〈νG( a) 〉), where index (a) denotes Co, Cu, and interface. Here are the quantities not mentioned until now: tCo = 11.35 A˚ (net of interface), tCu = tx − 3.65 A˚
(where tx are the nominal Cu layer thicknesses), ti ¼3.65 Å ↑ (↓) were estimated (2 MLs). The relaxation times in cobalt layer τCo ↑ (↓) by the relation τCo = τCo/(1 ∓ βCo ). The average electron velocities on the Fermi surfaces were assumed to be the following: ↑ (↓) 〈νF(Co) 〉 = 8 × 107 (5 × 107) cm/s ; 〈νF(Cu) 〉 = 1 × 108 cm/s. And we did not take into account a possible spin polarization of conduction electrons in copper, βCu = 0. Fig. 10 exhibits the probability of electron scattering at passing one period of multilayer as a function of the Cu layer thickness. The scattering probability of spin-up electron is less than unity for all tCu . Another case occurs for spin-down electron which scattering probability when passing one bilayer always reaches its physical limit (P max = 1). This means a scattering of spin-down electron within the thickness of one bilayer at all tCu . We are also interested in the interfacial scattering
Fig. 10. (a) Scattering probability P ↑ of conduction electron when passing one period of Co/Cu multilayer. (b) Difference between the scattering probabilities at interface ΔPi↓ (↑) = Pi↓ − Pi↑ . Squares denote the values of parameters for sample B. Lines are guide for eye.
Fig. 11. Mean free path l↑ (↓) of conduction electrons depending on Cu layer thickness. Squares denote l↑ (↓) for sample B: l↑ – filled square, l↓ – unfilled square. Dashed line denotes the bilayer period of Co/Cu multilayer. Solid lines are guide for eye.
probabilities of conduction electrons in the regions of the first two maxima of AF interlayer exchange coupling at 9–10 Å and 22–23 Å of Cu layer thickness. According to our estimates the interface scattering probability of spin-up electrons is only 0.02–0.07 in these regions,while for the spin-down electrons it remains high (Pi↓ = 0.35–0.60). Fig. 10(b) demonstrates the ΔPi↓ (↑) function which is equal to the difference between the scattering probabilities of conduction electrons with opposite spins at an interface. With increasing tCu , the two maxima of ΔPi↓ (↑) (tCu ) are observed which coincide with the maxima of MRE(tCu ) and MR(tCu ). The amplitudes of maxima are close to 0.5. In Fig. 11 we estimate the mean-free ↑ (↓) , where path (mfp) of conduction electrons: l↑ (↓) = 〈νF↑ (↓) 〉τsal ↑ (↓) 〈νF↑ (↓) 〉 = (1/T )(〈νF(Co) 〉tCo + 〈νF(Cu) 〉tCu ) [there, tCu and tCo are nom-
inal]. The l↓ value is approximately equal to multilayer period T while the l↑ oscillates reaching the maximum values at tCu corresponding to the positions of maxima of AF-exchange coupling and becoming the minimum in the F-exchange region. In sample B we
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Thus, the conductivity of spin-up current channel is further limited by the bulk properties of structure, not by interface. The scattering probability of conduction electrons in the spin-down channel reaches its maximum (P ↓ = 1) already over one period of multilayer. This means that the spin-down electrons are restricted in their moving in the direction perpendicular to the sample plane while the spin-up electrons move freely through all the layers implementing a predominantly “single-spin-channel” transport at the parallel ordering of magnetic moments of adjacent magnetic layers. The spin asymmetry coefficient of the interface scattering was determined and its oscillation with Cu layer thickness was found. The amplitude values of spin asymmetry coefficient γCo/Cu in the range of 8–27 Å thick of Cu layer coincide with the values obtained by CPP-MR measurements at hundred nanometers of layer thicknesses, which indicate the dominant role of the electronic structure of contacting metals in the interface scattering.
Acknowledgments
Fig. 12. (a) Spin-dependent scattering probability of conduction electrons at one period of Fe/Cr superlattice; (b) mean free path of conduction electrons versus tCr . Dashed line denotes the period of Fe/Cr superlattice. Solid lines are guide for eye.
˚ . This means obtained the following values: l↑ ≈ 220 A˚ and l↓ ≈ 26 A that the spin-down electrons are restricted in their movement while the spin-up electrons can travel long distances from the substrate up to the top layer of a multilayer structure. Comparison of scattering probabilities and mfp of electrons in Co/Cu multilayers with that in Fe/Cr superlattices [35] (Fig. 12) shows that the difference between spin-up and spin-down electronic characteristics in Co/Cu system is more pronounced. In particular, the probability of scattering when passing through the bilayer period in Fe/Cr system is always less than unity, and the ΔPi↑ (↓) value in the region of the first AF-peak of exchange coupling is equal to 0.14 in Fe/Cr as compared with ΔPi↑ (↓) = 0.46 in Co/Cu. Similarly, the mfp ratio for conduction electrons in the region of the first maximum of AF exchange constitutes l↑/l↓ = 8.6 in Co/Cu, and l↓/l↑ = 1.6 in Fe/Cr.
4. Conclusions Optical ellipsometry of Co(15 Å)/Cu(tx) multilayers with Cu layer thicknesses corresponding to the first two peaks of AF interlayer exchange coupling showed that the Drude-type optical absorption becomes dominating in the wavelength range of λ ≥ 8 μm , and in simulating it is necessary to take into account the growth of plasma frequency with increasing Cu thickness. We got a good correlation between the GMR and MRE effects depending on Cu layer thickness, which gave us a confidence in correctness of our results and allowed to advance onward in obtaining the parameters of electron scattering. From the analysis of the MRE spectra it follows that the major changes of the interface scattering of conduction electrons occur with increasing thickness of Cu layer in the spin-up current channel. In the sample with the maximal magnetoresistance the value of interfacial relaxation time τi↑ is virtually equal to relaxation time of electrons in bulk copper, which is a consequence of the similarity of the Fermi surface topology of (s–p)-type in copper with that of fcc Co (↑) in cobalt.
We thank V. Proglyado and N. Bannikova for preparation and characterization of the samples, and we thank N. Bebenin for fruitful discussions. This work was performed as part of the state assignment “Spin” No. 01201463330 (Project 12-P-2-1051) with the support of the Russian Ministry of Education and Science (Contract nos. 14.Z50.31.0025 and 14.120.14.1540-SS), and the Russian Foundation for Basic Research (Project 13-02-00749).
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