Semiclassical theory of spin transport in magnetic multilayers

Spin Dynamics and Relaxation II: Multilayers

and Thin Films Robert McMichael, Chariman

Semiclassical theory of spin transport in magnetic multilayers

R. Urban,^a)B. Heinrich, and G. Woltersdorf

Simon Fraser University, 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada 共Presented on 15 November 2002兲

A semiclassical model of the spin momentum transfer in ferromagnetic film 共FM兲/normal metal 共NM兲 structures is presented. It is based on the Landau–Lifshitz equation of motion and the exchange interaction in FM, and on the spin diffusion equation in NM. The internal magnetic field is treated by employing Maxwell’s equations. A precessing magnetization in FM creates a spin current which is described by spin pumping proposed by Tserkovnyak et al. The back flow of spins from NM into FM is assumed to be proportional to the spin accumulation in NM as proposed by Silsbee et al. These theoretical calculations are tested against the experimental results obtained by different groups. A good agreement was found for Py/Cu samples, but spin pumping is significantly enhanced in Py/Pt systems. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1555374兴

In our recent ferromagnetic resonance共FMR兲studies^{1– 4} it was shown that the transfer of the spin momentum across ferromagnetic共FM兲/normal metal共NM兲interfaces can result in nonlocal interface Gilbert damping G⬘. The generation of spin momentum in magnetic ultrathin films was theoretically described by Tserkovnyak et al.⁵ and the effect was called

‘‘spin pumping.’’ The presence of a second magnetic layer creates a spin sink.^3,4,6,7The combination of spin pump and spin sink in the ballistic limit leads to an additional interface Gilbert damping. In this article we extend the spin pump and spin sink mechanisms to the nonballistic electron transport which includes a full treatment of the Landau–Lifshitz共LL兲 equation of motion in FM and diffusion equation in NM and Maxwell’s equations accounting for a finite penetration of the rf fields.

The coordinate system was chosen in such a way that the sample normal is parallel to the z axis. The external dc field, H, lies in the sample plane and is parallel to the y axis, and the internal electromagnetic rf fields are h⫽(h,0,0), e

⫽(0,e,0). The LL equations of motion in FM and NM layers can be written as

1

␥

⳵^MF

⳵^{t ⫽⫺共MF⫻HeffF 兲⫹}

G₀

␥^2Ms

2

冉

^{MF⫻⳵M⳵tF}

冊

^{, 共1兲}

1

␥

⳵^MN

⳵^{t ⫽⫺共MN⫻HeffN兲⫹}

D

␥^ⵜ2␦^MN⫺␦^MN

␥␶s f

, 共2兲

where ␥ is the absolute value of the electron gyromagnetic ratio, M_s is the saturation magnetization of FM, G₀ is the intrinsic Gilbert damping, D is the diffusion constant in NM (D⫽vF

2␶el/6,vF is the Fermi velocity and␶elis the electron momentum relaxation time兲, ␶s f is the spin–flip relaxation

time, and ␦^MN⫽M^N⫺␹Ph is the excess magnetization in NM, where ␹P is the Pauli susceptibility. The effective field H_eff^F is derived from the total Gibbs free energy which con- tains the external fields, magnetocrystaline anisotropies, and exchange interaction.⁸ The effective field H_eff^N in NM con- tains only the external dc, internal field H, and the demag- netizing field perpendicular to the sample plane. Equations 共1兲and共2兲were solved in a small angle approximation using M^F⫽共m_x^F, M_s,m_z^F兲, 共3兲

␦^MN⫽共m_x^N⫺␹Ph,␹PH,m_z^N兲, 共4兲 where H is the external applied magnetic field. The time and spatial variations of the rf components were assumed to be

⬃exp(i␻^t⫺kz), where k is the propagation wave vector and

␻is the rf angular frequency. Maxwell’s equations in Gauss- ian units neglecting the displacement current for this geom- etry are

⫺4␲^k0

2im_x⫹共k²⫺ik₀²兲h⫽0, 共5兲 e

h⫽k c

4␲␴^{, bz⫽0, 共6兲}

where␴ is the appropriate conductivity, c is the velocity of light in free space, and k₀²⫽(4␲ⁱ␴␻^)/c2. The skin depth␦

⫽c/

冑

⁽²␲␴␻^).

Equations共1兲,共2兲,共5兲, and共6兲provide the secular equa- tion for k. In both cases, FM and NM, the secular equation results in a cubic equation in k² which leads to six k wave numbers with corresponding six waves of propagation. The rf magnetization and electromagnetic field components are given by a linear superposition of six waves. The coefficients are evaluated by matching the boundary conditions at the film interfaces.

a兲Electronic mail: rurban@sfu.ca

JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 10 15 MAY 2003

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We assume no direct exchange interaction between the FM and NM layers. The coupling between FM and NM is caused by spin currents across the FM/NM interface. We consider three contributions to the net spin flow:

I_FM→NM⫽ បg^↑↓

4␲^Ms

2

冉

^{MF⫻⳵M⳵tF}

冊

^{, 共7兲}

I_NM→FM⫽v_Ft_NM

4␥ ␦^MN, 共8兲

I_diff⫽Dk

␥ ␦^MN. 共9兲

I_FM→NM is described by the spin pumping model proposed by Tserkovnyak et al.^5,9Parameter g^↑↓represents is the num- ber of conducting channels per unit area⁹ which is directly related the interface mixing conductance G^↑↓ by

g^↑↓⫽h

e²G^↑↓, 共10兲

where e is the electron charge, and h is Planck’s constant.

G^↑↓ were evaluated for different interfaces by first principle band calculations by Xia et al.¹⁰

I_NM→FM was proposed by Silsbee et al.^11,12 from a simple kinematic argument. t_NM is the transmission coeffi- cient for conduction electrons from NM into FM. Tserk- ovnyak et al. used for I_NM→FM a similar term (I_s^backin their notation兲. The transmission coefficient t_NM can be deter- mined by direct comparison of I_NM→FMand I_s^back共Ref. 9兲and is found to be

t_NM⫽␲^g↑↓

k_F² , 共11兲

where k_F is the Fermi wave vector. Note, that the coefficient in Eq.共7兲and t_NMare proportional to the number of conduct- ing channels, which reduces the number of free fitting pa- rameters. Since g^↑↓⬇k_F²/4␲^,13the transmission coefficient is

⬇0.25.

I_diffis present only in NM. It represents the flux of non- equilibrium spins away from the FM/NM interface into the NM bulk. ␦M relaxes back to equilibrium with the rate of 1/␶s f.

At each interface there are two electromagnetic bound- ary conditions 共continuity of h and e兲. In addition, the fol- lowing four boundary conditions satisfy the magnetic and spin flow requirements at the FM/NM and NM/FM inter- faces.

FM:

冉

^{⫺2AMsk⫺MKss}

冊

^{Mz⫹IFM共x兲→NM⫽INM共x兲→FM,}

共12兲 2A

M_sk M_x⫹I_FM^共z兲_→NM⫽I^共_NM^z兲_→FM,

where A is the strength of the bulk exchange coupling and K_s is the interface perpendicular uniaxial anisotropy (E_s⫽

⫺K_scos²(␪⁾关erg/cm²兴), see Ref. 8. The term in the round

bracket arises from the interface torques generated by the exchange coupling and the interface perpendicular uniaxial anisotropy.

NM:

I_FM→NM⫽I_NM→FM⫹I_diff. 共13兲 The calculations were carried out for symmetric driving.

This means that the rf components of h at both outer surfaces are equal.

It is interesting to explore the following aspects of the above theory:

共A兲 The strength of g^↑↓:g^↑↓can be found in Ref. 10 and ranges between 1 and 2⫻10¹⁵cm^⫺2. In the limit of t_NM

→0 there is no backflow of the spin momentum from NM into FM. This corresponds to a ‘‘perfect spin sink’’ and gives the maximum effect regardless of d_NM共thickness of NM兲, D, and␶s f.

共B兲 FMR linewidth,⌬H vs d_FM: Figure 1共a兲shows the total FMR linewidth as a function of the FM layer thickness d_FM. The dotted line does not include spin pumping (g^↑↓

⫽0). In this case, there are two regions: 共i兲 For d_FM

⬍300 Å⌬H is dominated by the intrinsic damping G₀ of a single layer; 共ii兲 for d_FM⬎500 Å the additional broadening arises from eddy currents. The solid line includes spin pump- ing (g^↑↓⫽1⫻10¹⁵cm^⫺2). Amazingly, the additional broad- ening always scales like 1/d_FM. For d_FM⬎500 Å the addi- tional interface damping is negligible (⌬H with and without g^↑↓ are within 1 Oe兲.

共C兲 G vs d_NM, influence of ␭s f: In Fig. 1共b兲 the solid lines represent calculated total Gilbert damping G assuming a perfect mirror at the back side of NM关d(␦^M)/dz⫽0兴. For d_NMⰆ␭s f⫽v_F

冑

␶el␶s f/6 the rf magnetization accumulates in NM and the spin current I_NM→FM compensates the spin pumping current I_FM→NMresulting in zero interface damping (G⫽G₀兲. When d_NM becomes comparable to ␭s f the spin current I_NM→FM is not sufficient to compensate I_FM→NMre- sulting in increased Gilbert damping. This increase eventu- ally saturates and its final value depends on the ratio of

␶el/␶s f. The dashed line in Fig. 1共b兲represents a perfect spin sink at the back side of NM (␦^M⫽0). Note, that in this case

FIG. 1. 共a兲Total FMR linewidth⌬H as a function of dFMat f⫽10 GHz.

Calculations were carried out for FM using permalloy 共Py, Ni80Fe20, 4␲^Ms⫽10.7 kOe, G₀⫽0.7⫻10⁸s^⫺1, ␳⫽15 ␮⍀cm兲 and NM using Cu 共␳⫽1 ␮⍀cm, ␶el⫽2.5⫻10^⫺14s, D⫽95 cm²/s). The dashed line corre- sponds to a single Py layer and therefore⌬H is caused by the bulk proper- ties. The solid line shows the linewidth which includes spin pumping (g^↑↓

⫽1⫻10¹⁵cm^⫺2), assuming a perfect sink at the FM/NM interface (t_NM

⫽0).共b兲 ⌬H as a function of the NM thickness for two different values of

␶s f. The solid lines correspond to a perfect mirror at the back side of NM 关d(␦^{M )/dz}⫽0兴. The dashed lines correspond to a perfect spin sink at the back side of NM (␦M⫽0).

8281 J. Appl. Phys., Vol. 93, No. 10, Parts 2 & 3, 15 May 2003 Urban, Heinrich, and Woltersdorf

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for d_NMⰆ␭s f one obtains a perfect spin sink共equivalent to t_NM⫽0) and for a large Cu thickness there is no difference between the Cu/ perfect-sink and Cu/ perfect-mirror.

共D兲 Influence of the skin depth ␦: It is interesting to discuss the limit when the skin depth␦becomes comparable or even less then ␭s f. Figure 2 simulates the effect of de- creasing temperature. The solid line corresponds to Py/Cu at room temperature共RT兲, and the dashed line corresponds to a cryogenic temperature共CT兲with the resistivity ratio equal to 10. In this calculation the ratio between ␶el and ␶s f was assumed to be temperature independent. The spin diffusion lengths for RT and CT are 0.1 and 1␮m, respectively. The corresponding skin depths are 0.5 and 0.2␮m, respectively.

For RT the ratio R⫽␦^/␭s f⬇5 while for CT R⬇1/6. Note, that in both cases the additional linewidth increases when d_NM becomes comparable to ␭s f, and saturates for d_NM Ⰶ␭s f.

In the remainder of this article some recent experimental results will be discussed. Mizukami et al.¹⁴ and Invarsson et al.¹⁵ investigated the FMR linewidth in Py films which were surrounded by NM layers. In both cases they observed an interface damping. Their studies were carried out at dif- ferent microwave frequencies. The strength of the interface damping in the same type of samples共Pt/Py/Pt兲scaled with the microwave frequency. It is, therefore, appropriate to in- terpret their results using the spin pumping theory as outlined above. The strength of the interface damping in the Py films surrounded by Pt and Pd is surprisingly high. Even for the case when these layers act as perfect sinks (t_NM⫽0) one needs to use g^↑↓⫽2.5 and 1.4⫻10¹⁵cm^⫺2 for Pt and Pd, respectively. The number of transversal channels for light electrons共m*/m_el⬇2兲of the sixth band¹⁶is the same for Pt and Pd and leads to g^↑↓⫽0.7⫻10¹⁵ cm^⫺2. This number can be perhaps enhanced by a factor of 2 by including the heavy holes. Therefore, one can expect g^↑↓ to be in a range be- tween 0.7 to 1.4⫻10¹⁵ cm^⫺2. This is clearly at variance with the experimental value for Pt (2.5⫻10¹⁵cm^⫺2). A pos- sible explanation is being offered by the Stoner enhancement factor which enhances the strength of spin pumping, see Si- manek and Heinrich.¹⁷

Recently we studied the increase of the Gilbert damping in GaAs/16Fe/10Pd/20Au共001兲, where integers represent the number of atomic layers. This sample was prepared by mo- lecular beam epitaxy 共MBE兲 where atomic intermixing be-

tween Fe and Pd is kept at its minimum. The additional Gilbert damping at f⫽24 GHz was found to be 0.3

⫻10⁸s^⫺1. This value is small compared to the increase in G (1.7⫻10⁸s^⫺1) that was measured by Mizukami et al.¹⁴ for the same FM thickness. In interpretation of our data we have to invoke a finite spin diffusion length. The required value is ␭sf⫽70 Å. However, that needs␶sf⫽␶elwhich we find unrealistic. The mean free path exceeds significantly the Pd thinkness; therefore, we are in the ballistic limit where our theory does not apply. In the ballistic limit it is more reasonable to interpret the measured data by determining the fraction of I_FM→NM which was absorbed in Pd. In Mizuka- mi’s experiment everything is absorbed, in our measure- ments only 20% is lost in Pd.

In separate experiments Mizukami et al.¹⁸ studied the Gilbert damping as a function of d_Cu共from 10 nm to over 1

␮^m兲 in glass/Cu(5 nm兲/Py/Cu(d_Cu) and glass/

Cu(5 nm兲/Py(3 nm兲/Cu(d_Cu兲/Pt samples. Their results are similar to those shown in Fig. 1共b兲 for ␶s f⫽200␶el (␭s f

⫽0.2␮m). Notice, that Cu on its own is a poor spin sink even for d_CuⰇ␭s f. In glass/Cu/Py/Cu(d_Cu兲/Pt structures one was able to explore the role of the Pt layer when separated from Py by a variable thickness of Cu. The experimental results were possible to explain by assuming that the Cu/Pt interface acted as a perfect spin sink and therefore the in- crease in the Gilbert damping can be explained by the maxi- mum strength of spin pumping in Cu.

The authors thank Y. Tserkovnyak, E. Simanek, and J. F.

Cochran for valuable discussions. Financial support from the Natural Sciences and Engineering Research Council of Canada 共NSERC兲 and Canadian Institute for Advanced Re- search 共CIAR兲is gratefully acknowledged. G.W. thanks the German Academic Exchange Service 共DAAD兲 for a gener- ous scholarship.

1R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 共2001兲.

2B. Heinrich, R. Urban, and G. Woltersdorf, J. Appl. Phys. 91, 7523共2002兲.

3B. Heinrich, R. Urban, and G. Woltersdorf, IEEE Trans. Magn. 38, 2496 共2002兲.

4B. Beinrich, G. Woltersdorf, R. Urban, and E. Simanek, J. Appl. Phys.共to be published兲.

5Y. Tserkovnyak, A. Brataas, and G. Bauer, Phys. Rev. Lett. 88, 117601 共2002兲.

6M. Stiles and A. Zangwill Phys. Rev. B 66, 014407共2002兲.

7B. Heinrich, G. Woltersdorf, R. Urban and E. Simanek, J. Magn. Magn.

Mater.共in press兲.

8B. Heinrich and J. F. Cochran, Adv. Phys. 42, 523共1993兲

9Y. Tserkovnyak, A. Brataas, and G. Bauer, e-print cond-mat/0208091.

10K. Xia, J. Kelly, G. B. A. Brataas, and I. Turek, Phys. Rev. B 65, R220401 共2002兲.

11R. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19, 4382共1979兲.

12P. Sparks and R. Silsbee, Phys. Rev. B 35, 5198共1987兲.

13A. Brataas, Y. Tserkovnyak, G. Bauer, and B. Halperin e-print cond-mat/0205028.

14S. Mizukami, Y. Ando, and T. Miyazaki, Jpn. J. Appl. Phys., Part 1 40, 580 共2001兲.

15S. Ingvarsson, L. Ritchie, X. Liu, G. Xiao, J. Slonczewski, P. Trouilloud, and R. Koch, e-print cond-mat/02008207.

16C. Lehmann, S. Sinning, P. Zahn, H. Wonn, and I. Mertig, Fermi surface database共1996-1998兲, URL http://www.physik.tu-dresden.de/⬃fermisur/.

17E. Simanek and B. Heinrich, e-print cond-mat/0207471.

18S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. Magn. Mater. 239, 42 共2002兲.

FIG. 2. ⌬H for Py共20 Å兲covered by Cu(dNM) for RT共solid line兲and a cryogenic temperature 共dashed line兲with the resistivity ratio equal to 10.

Calculations were carried out at f⫽10 GHz. For RT␶el⫽2.5⫻10^⫺14s and

⑀⫽␶s f/␶el⫽100.⑀was assumed to be temperature independent.

8282 J. Appl. Phys., Vol. 93, No. 10, Parts 2 & 3, 15 May 2003 Urban, Heinrich, and Woltersdorf

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