Temperature dependence of the electrical resistivity, thermopower and magnetoresistance of Co-Re superlattices

Journal of Magnetismand MagneticMaterials 137 (1994) 73-88

~ -""

ELSEVIER

Journal of magnetism and magnetic materials

Temperature dependence of the electrical resistivity, thermopower and magnetoresistance of Co-Re superlattices J.B. S o u s a

~'*, R.P.

Pinto

a,

B. A l m e i d a a, M.E. Braga I.G. T r i n d a d e b

a,

p.p. Freitas

b, L . V .

M e l o b,

a Centro de Fisica da Universidade do Porto (INIC) and IFIMUP (IMAT), Pra~a Gomes Teixeira, 4000 Porto, Portugal b INESC, R. Alves Redol 9, 1000Lisboa, Portugal

Received3 February1994;in revisedform22 April 1994

Abstract

Detailed measurements of the electrical resistivity ( p, dp/dT), magnetoresistance ( A p / p ) and thermopower (S) have been performed on magnetron-sputtered (C02oA-ResA)Xl 6 and (C010A-Re16A)x16 superlattices, from 3.7 to 300 K. Resistivity minima were observed at low temperatures due to electron localization/electron-electron interactions, in addition to a magnon resistivity term AT 2 e -a/*r, with A ~ 60 K. The magnetoresistance is negative in the two superlattices, both for H III and H_l.1, but A p / p is one order of magnitude larger in (C02o ,~-Res.~)x 16, where strong antiferromagnetic interlayer coupling exists. In this sample we were able to separate a 'normal' A p / p contribution due to spin-orbit/band effects, displaying the usual cos20 angular dependence. The temperature dependence of the anomalous part of A p / p , due to spin-dependent electron scattering, is also studied. The temperature dependence of the transport saturation field (Hs) has been extracted from such data. The thermopower reveals anomalous small values when compared with pure Co and Co-Cu multilayers. The roles of impurity, magnon and s-d scattering (mediated by phonons) are critically analyzed.

1. Introduction

This paper deals with the temperature dependence of several electrical transport coefficients and of the exchange interlayer coupling in Co-Re supedattices. The two-current model, involving parallel conduction by spin-up and spin-down channels, and spindependent scattering at the interfaces or within the ferromagnetic layers, was successively used to explain the magnetoresistance in antiferromagnetically coupled F e / C r [1], C o / C u [2] and C o / R e [3] superlattices. The observed decrease in magnetoresistance with increasing temperature was first related to spin mixing by electron-magnon scattering [4,5]. This idea was then generalized and used to explain the

* Correspondingauthor.

observed temperature dependence of the magnetothermoelectric power [6]. Finally, but not least relevant is the temperature dependence of the exchange coupling between the magnetic layers (as usually defined [7]). In F e / C r superlattices, the saturation field H s needed to overcome the antiparallel alignment of the Fe layers decreases linearly with increasing temperature [8]. A model involving thermal fluctuations of spins across neighbouring ferromagnetic layers was introduced to account for this linear dependence [8]. Previous measurements of the magnetoresistance and magnetization of Co-Re supedattices [9] show strong antfferromagnetic coupling between the Co layers when the Re spa~r thickness is small, tRe< 9 A. For 9 ,~
0304-8853/94/$07.00 © 1994 ElsevierScienceB.V. All fights reserved SSD! 0304-8853(94)00354-T

J.B. Sousa et al. /Journal of Magnetism and Magnetic Materials 137 (1994) 73-88

74

1 )< 10 - 7 Torr. Co was deposited by rf magnetron sputtering [12,13], at rates ranging from 0.4 to 2 A s -1 and Re was deposited by dc magnetron sputtering at a rate of 0.4 A s -1. These rates were monitored by in situ quartz crystals and the Ar pressure during deposition was kept at 2.5 mTorr. The Co and Re thicknesses were directly calibrated by Rutherford backscattering, profilometry and X-ray diffraction, on superlattices specially grown for that purpose [9-11]. In order to induce [001] crystal ~rowth perpendicular to the plane of the films, a 150 A thick Re buffer film was used. It also reduces spurious effects of the glass substract on the osuperlattice structure. A protective layer of Re, 50 A thick, was deposited on the top of each film. The superlattice samples here investigated contain 16 R e / C o bilayers, having the structure glass/150 A Re/[Co t I .~/Re t 2 .~]×16/50 A Re. Since the

exchange coupling cannot be extracted from such o measurements. For tRe = 16--18 A, a second region with weak antiferromagnetic coupling exists [9,10]. We report a detailed investigation of the temperature dependence of the electrical resistivity (p), its temperature derivative (dp/dT), magnetoresistance (Ap/p) and the thermoelectric power (S, dS/dT), for two superlattices with compositions (C020~Re5.~)×16 and (COlo ~-Re16 ~)×16, falling near the first and second peaks of the oscillatory H s (tRe) curves, respectively [9,11]. 2. Experimental results

2.1. Samplepreparation The films were prepared by magnetron sputtering in a high vacuum system with a base pressure of I

24

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I

I

I

I

19

E ¢.~

14

×

(ResAC°20A)x16

I

Q.. 9"

O O

4-

\

Co "bulk"

0

-

-I

-

~

0 i

0

I

80

I

i

!

160

240

i

320

T(K) Fig. 1. Temperature dependence of the ideal resistivity Pi = P - P0 for 150 ~, Re/[Colo .~/Re16 ~]× 1 6 / 5 0 t~ Re, 150 ,~ Re/[Co20 A / Re 5 ~,]× 16/50 ~k Re, and bulk samples of Co and Re, in the temperature range 4 - 3 0 0 K.

J.B. Sousa et al. /Journal of Magnetism and Magnetic Materials 137 (1994) 73-88

substrate and the coating layer are ke.pt constant, we abbreviate this description to (Co t I A-Re t 2 A)× 16.

75

(dp/dT)ph (see Section 3.1). We see that dp/dT exceeds (dp/dT)ph considerably at intermediate and low temperatures, with a maximum difference at

2.2. Electrical resistivity measurements ( p, d p / dT) Fig. 1 shows the temperature dependence of the ideal resistivity Pi = P -- P0 for (Co10 A - R e J6 ~,)x 16 and (Co20 A-Re5 A)×16 multilayers, in the temperature range 4-300 K, where P0 is the residual resistivity, Po = 67.4 and 40.1 Ixfl-cm respectively. At high temperatures, p is linear with temperature, with the slopes of 0.071 and 0.035 OXI. cm K -1 for (COloA-Rel6 A)x 16 and (Co2oA-Re 5 A)x 16, respectively. We also show the resistivity curves for bulk Re and Co samples [14], having high-temperature slopes of 0.078 and 0.029 p~fl. cm K -1, respectively. Fig. 2 shows the behaviour of the derivative d p/dT, measured directly from 4 to 300 K in the two Co-Re samples, together with the calculated curves for electron-phonon scattering only,

I

0.09

I

T,,,60 K. The low-temperature resistivity data (see expanded scale in Fig. 3 and the negative d p/dT values observed at low temperatures in Fig. 2) show a weak minimum in p, around Tmin = 14 and 8 K for (C°20 A-Re5 A)× 16 and (Co10A-Re16 A)× 16, respectively. This minimum could be due to electron localization and (or) electron-electron interactions (see Section 3.1).

2.3. Thermopower (S, dS /dT) Fig. 4 shows the behaviour of the absolute thermopower (S) from 4 to 320 K, for the two investigated Co-Re films. For comparison, we also show S(T) measured in thick fills of pure Re (420 .~) and pure Co (420 ~, 500 ~,), prepared under the same conditions as the Re-Co multilayer films.

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r

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240

T(K)

Fig. 2. Temperature derivative of the resistivity (dp/dT) as a function of temperature, for the Co-Re multilayers. Also shown are the electron-phonon contribution to (d p / d T ) for both samples, (d p/dT)p, calculated with the Bloch-Griineisen formula and Ooebyc = 450 IC

J.B. Sousa et al. /Journal of Magnetism and Magnetic Materials 137 (1994) 73-88

76

40.3

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u

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~

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0

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, 10

0

, 20

j

67.4 0

40 T(K)

Fig. 3. Low-temperature resistivity of the C o - R e samples, showing a minimum in p around T = 8 K for (Co10-R%6) × 16 and T = 14 K for

(Co20 ~-Re5 ~,)× 1 6 "

21

1

I

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(Re 16~COl0.~)x ! 6 0" ..........................................

.................

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, 0

, 8~0

, 160

,

, 240

,

320 T(K) Fig. 4. Absolute thermopower (S) of the C o - R e multilayers, as a function of temperature. Also shown the S(T) curves for thick films of pure Re (420 ]k) and pure Co (420 A).

J.B. Sousa et aL/Journal of Magnetism and Magnetic Materials 137 (1994) 73-88

(a)

77

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t

(b)

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H(KOe) Fig. 5. Magnetoresistance of Co-Re multilayers as a function of the magnetic field (H), for several temperatures. (a) (Re 5 ~ Co20 ~,)x 16, with H [[ I; (b) (Re s ~ Co2o ~ ) x 16, with H J. I; (c) (Re16 ~ COlO~,)x 16, both for H II I and H .1. I.

78

J.B. Sousa et al. /Journal of Magnetism and Magnetic Materials 137 (1994) 73-88 I

-0.0019

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-0.002

-0.0021

-0.0022 -0.2

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!

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1

1.2

COS2(e) Fig. 6. Ma~etoresistanceundersaturationat292 ~ asa functionofcos20, w h e r e 0 i s t h e a n ~ e b e ~ e e n H ~ d l .

In (Re16 , ~ - C O l 0 / ~ ) × l 6 the thermopower is positive and fairly small (S < 2.5 p,V K- 1) exhibiting a 'normal metal' behaviour at high temperatures: a I

6O0O

w e a k linear temperature d e p e n d e n c e w i t h a positive slope in our sample. A t intermediate temperatures, S has a v e r y shallow h u m p w h i c h fades a w a y for I

I

2~

3~

×

o

5000-

4000-

3000-

o

H par. I

×

H per. I

2000 0

1~

T(K)

Fig. 7. Saturation field (H s) as a function of temperature, for both/-/If I and H _LI.

4~

79

J.B. Sousa et al. /Journal of Magnetism and Magnetic Materials 137 (1994) 73-88

temperatures above ~ 160 K. In the strong antiferromagnetically coupled film (Re 5/~-Co20/~) × 1 6 , S ( T ) is larger, ( I S I ~ 4.5 ~V K-x at room temperature), and negative at all temperatures, exhibiting a negative slope at high temperatures.

this sample this effect dominates the usual spinorbit anisotropic magnetoresistance (AMR): In

AMR = Pll - P ± p(o)

1 + ~P 2 ± ' and Pll( p ±) is the elecwhere p(0) = ~Pll trical resistivity measured with spontaneous magnetisation parallel (perpendicular) to the electrical current. The small AMR effect causes a slight difference in the negative saturation magnetoresistance for parallel and perpendicular fields (Figs. 5a, b), with Pll < P± at all temperatures. The angular dependence of the AMR was investigated under an applied field, producing magnetic saturation for any angle ( 0 ) between the spontaneous magnetization M s and the current I. A simple cos20 dependence [15,16] accounts well for the observed anisotropy, as shown in Fig. 6. From the knee in each (A p / p ) versus H curve at high fields (see Fig.

2.4. Magnetoresistance

The magnetoresistance has been measured in the temperature range 10-300 K, for the two superlattices referred to above, with the applied field in the plane of the film, either parallel [(Ap/p)ll ] or perpendicular [(A p / p ) ± ] to the electrical current. (a) In (Co20 ~,-Re 5 X)×~6 the magnetoresistance is considerably larger and initially parabolic as a function of H, as shown in Fig. 5. We associate this effect with the antiferromagnetic interlayer coupling which leads to a characteristic negative magnetoresistance increasing with the decrease of temperature.

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160

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240

,

,

320 T(K)

Fig. 8. Behaviourof (dp/dT)~g = dp/dT - (d p/dT)ph, as a function of temperature, in the (Re5 ~-Co2o X)x 16 and (Rel6 ~-COlo )i))x 16 multilayers.

80

J.B. Sousa et al. /Journal of Magnetism and Magnetic Materials 137 (1994) 73-88

Re-Co samples, (dp/dT)ph versus T, as shown in Fig. 2. A plot of [(dp/dT) - (dp/dT)ph] versus T (Fig. 8) shows a large excess resistivity (over the electron-phonon resistivity) at intermediate temperatures. The effect is larger in the sample with thinner Co layers. We believe that this enhancement of the p (T) dependence reflects the gradual spin mixing of the two electron conduction channels (T, $) as the temperature rises [18], fading away as spin mixing virtually saturates, above ~ 200 K in our case. (b) The resistivity minimum observed in our films indicates a small enhancement of p at low temperatures. Both electron localization (due to strong disorder) and electron-electron interactions in 3d transition metals can lead to a resistivity enhancement in thin films (2D), with the same logarithmic temperature dependence for both mechanisms [19], giving a combined resistivity (p~):

5) we obtain the saturation field (H~) at each measured temperature. Fig. 7 shows that H~ rises steadily with decreasing temperature (see Section 3.2). (b) In (Co10A-Re16 ~,)x 16 the magnetoresistance is much smaller (Fig. 5c), which corresponds to weak AF coupling, coming from the second peak in the curve of H~ versus Re thickness [9]. The 0-anisotropy of (Ap/p) under saturation is also very small, falling within the limits of experimental resolution.

3. Analysis

of results

3.1. Resistivity data ( p, dp / dT) (a) The dp/dT curves closely approach a constant value for temperatures above ~ 250 K, characteristic of each film. In (Co20 A-Re 5 A)× 16 and (Rel6 .~-COl0 A)x 16 such dp/dT values do not differ significantly from those observed in bulk Co and Re respectively, essentially due to electron-phonon scattering. We thus assume that electron-phonon scattering also dominates pi(T) in our films for measured temperatures T > 250 K. Using the Bloch-Griineisen formula to describe the electron-phonon resistivity, and inserting an average Debye temperature O -- 450 K [17], we calculate the electron-phonon resistivity curves for both

0.49

{a

i

i

i

i

i

i

i

pI(T) = I AI - I B I l n T.

(1)

From a fit to our low temperature data (T < 7 K) we obtain:

(C°20 A-Re5 A)× 16:

/

p l ( T ) = 40.11 - 0.0178 × In T

(~xl)-cm),

(C°10 A-Re16 ~) × 16: p I ( T ) = 67.44 - 1.0567 × In T

( ~ f l . cm).

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=3_ [-.-

+ + ++

0.09. •. ~ . . , ~ ' ¢ r ~

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( R e I 6 A C ° 1 0 A ) x 16

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=5_

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(b)

20A)x 16

;o

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llO

210

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310

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40

T(K) T(K) Fig. 9. (a) Resistivity P2 = Pph(T) + Pmag(T) as a function of temperature, for the (Re5 ,~-C020 A)x 16 and (Rc16A-C010A)× 16 multilayers. (b) Corresponding temperature derivatives (dpe/dT).

J.B. Sousa et al. /Journal of Magnetism and Magnetic Materials 137 (1994) 73-88

Localization effects are expected in fairly disordered regions, as could be the case in samples with very thin layers. In our ftlms each layer corresponds to a selected number of A (not atomic diameters), thus it cannot contain a complete set of atomic planes. This could explain the increase in the minimum depth as we go from ( C o 1 0 A - R e 1 6 A ) x 1 6 to ( C o 2 0 / ~ Re5A)×16(thinner Re layers; see Fig. 3). A significant amount of structural defects is also consistent with the high P0 values observed in these samples. The Kondo effect could in principle lead to a logarithmic temperature dependence of the electrical resistivity at low temperature. However, the standard effect requires the presence of dilute magnetic impurities in a nonmagnetic matrix, producing spin-dependent electron scattering by such essentially isolated impurities in a paramagnetic state. It seems unlikely that this effect dominates in our samples, since the resistivity minimum occurs both for antiferro- and ferromagnetic interlayer coupling, in the later case paramagnetic-like behaviour of the magnetic impurities (Co in Re) is not expected, due to the strong internal field. L

0.4

I

I

At this stage we have not definitively identified the dominant (temperature-dependent) physical mechanism. Further work is needed, particularly a careful study of the effect of an applied magnetic field on the resistance minimum, under both isofield and isothermal conditions. At higher temperatures the phonon and magnon resistivities become important, giving a specific contribution, P2 = Pph + Pmag, to the total resistivity: /9 ----P l q-/92

=lAl-ieilnT+pph(T)+Pmag(T ).

0.3-

I~

(2)

Fig. 9(a) shows the behaviour of P2(T) (phonon + magnon scattering), separated from p(T) through the use of P2 = P - PI, with pl(T) given by Eq. (1). In an analogous manner we obtain the temperature derivative curves for the two samples, d p2/dT versus T, as displayed in Fig. 9(b). Both types of curves (/32, dp2/dT) rapidly attain very small values for temperatures below ~ 20 K. The similarities between the P2 and dp2/dT temperature dependences suggest a dominant exponential factor in both cases. We performed a least-squares fit of our data to the I

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I

/

9.2

Study of the quality of fit with regard to the choice ofn value~: n=2 gives the best fit (minimum of AO)

9

81

8.8

~ g.6

J g.4

0.2-

Fit

0 8"21.4

2.2

1.$

2.,1

2.6

1,[, /

.; /,~

¥.." 0.1-

0

5

10

15

20

35

40

T (K) Fig. 10. The magnetic contribution to the resistivity (P2 term; see text) is well described by p 2 = A T " e - A / t r (dotted line), at low temperatures. Inset: best fit obtained with n = 2.

J.B. Sousa et al. /Journal of Magnetism and Magnetic Materials137 (1994) 73-88

82

standard expression for electron-magnon scattering, but with a damping factor (A): p2(T) = A T " e - a / k r .

(3)

To avoid significant (spurious) effects of electronphonon scattering, the fit has been made only for temperatures below 35 K (see Fig. 2 for the small magnitude of phonon effects). As shown in Fig. 10 for (Re s i-Co20 A)x 16, expression (3) completely describes our data from the lowest measured temperature (3.7 K) up to ~ 38 K, with n = 2 and A / k = 60.9 K. The inset of Fig. 10 shows a study of the improvement of the fit (smaller variance ~ ) as n approaches 2, from either above or below. The decrease in P2 at low temperatures could be caused, in principle, by (i) damping of spin waves due to an energy gap (A) [20], or (ii) a cut-off in the electron-magnon scattering processes [18,21]. Mechanism (i) is very unlikely, due to the relatively low magnetic anisotropy of Co. Mechanism (ii) could be enhanced by the presence of transition metal impurities like Re [21]. Due to the incomplete d-bands, spin $ and $ electrons are scattered differently, generating two distinct magnetic resistivities p 1"(0), p $(0) at T = 0 K. As T rises, spin-flip collisions gradually increase (they require a finite exchange energy, A) producing an additional resistivity p 1" $. For appreciable impurity scattering and low temperatures, we have p $ $ << p 1" + p $, p $ = p 1' (0), p J, vl p $ (0), leading to an approximate electron-magnon resistivity [18,21]: p ( T ) - p(O) = A T 2 e - a / k r ,

as observed in P2 at lower temperatures. At higher temperatures, spin-flip collisions mix the two spin channels ($, $) and the anomalous resistivity temperature dependence fades away. 3.2. Magnetoresistance 3.2.1. Magnetic resistivity contributions

(i) In bulk ferromagnets we have an isotropic term arising from spin-disorder scattering [22], usually negligible at room temperature and an anisotropic magnetoresistivity (AMR) arising from spin-orbit and electron band effects on the electrical resistivity [15,23]:

=8(T) cos20(hr),

(5)

where B(T)o~ M2(T), and 0 is the angle between the electrical current I and the direction of the spontaneous magnetization (or H, at magnetic saturation). The normal magnetoresistance arising from the cyclotronic motion of the conduction electrons around the internal field direction [24] can become important at low temperatures if impurity scattering is not dominant (which is not the present case). (ii) In multilayer films we have a periodic magnetic potential seen by the conduction electrons as they cross the layer interfaces. An anomalous magnetoresistivity arises when the conduction electron scattering is spin dependent [25,26]. This occurs in magnetic transition metals where the internal exchange field splits the d-band into spin up and down subbands. Also, nonmagnetic d-impurities (like Re in Co) can produce virtual bound states in one of the d-electron spin subbands. Electron-magnon interactions can also lead to spin-dependent electron collisions. It can be shown [25] that the transmission coefficient of an electron across nearest magnetic layers with spontaneous magnetizations M 1 and M 2 (angles 01 and 02 with I ) is proportional t o c o s 2 [ ( 0 1 02)/2]. If this angular dependence remains in the electrical resistivity [16,27,28], the magnetoresistance takes the form:

Apm(T, H ) ~ - 9 1 F ( T ) [ c o s 2 ( 0 1 - 02 ]

cos(0 ]

2

]n

9t contains an oscillatory term due to the interlayer exchange coupling, depending on the barrier thickness and the Fermi surface topology (and g F) [2931]. It also depends on the final density of states available for scattering, in the case of s-d transitions. F ( T ) contains the effect of temperature on the spontaneous magnetization and on the conduction electron scattering, namely through spin-ffip processes ( p $ $ ). For ferromagnetic interlayer coupling (01 = 02), no magnetoresistance arises from (6). However, for initial antiferromagnetic coupling, (01 - 0 2 ) 0 = 7r, A Pm gets larger and negative as H increases, giving Apm = - - 9 I F ( T ) at saturation. This can lead to the giant magnetoresistance (GMR) found in some antiferromagnetically coupled multilayer films [31,32].

83

J.B. Sousa et al./Journal of Magnetismand MagneticMaterials137 0994) 73-88

Adding expressions (5) and (6), we obtain the total saturation magnetoresistance: Ap~ t = 2CM2( T) [cos281( H ) - cos281(0)] +~F(T)/I_

2[/81-82]

1~

cos I t - - T - ) o i l '

(7)

where we have assumed equal width (and composition) magnetic layers and use B =- 2CM 2. 3.2.2. Analysis of ((?.02oz-Res X) x 16 data Here the magnetic layers are antiferromagnetically coupled in zero field, so (01 - 82) 0 = xr. (i) Using expression (7) for the cases H III and H ± I, we obtain the saturation values of the 'normal' and 'anomalous' magnetoresistance:

Ap~t(normal ) --- CM2(T) =l[APll(Hs)-Ap.(as) m Psat ( a n o m a l o u s )

],

(8)

- fit F (T)

=

a,:,,,(Hs)+ a,o i (as)], (9)

I

1.2

where H s is the saturation magnetic field. Due to the very small difference P l l - P .L , Eq. (8) does not give sufficiently reliable information on Ms(T). However, the anomalous magnetoresistance (Eq. (9)) enables us to extract the F(T) dependence, as illustrated in Fig. 11. (ii) Eq. (7) justifies the cos20 dependence observed in A psat for different directions (0) of the saturation magnetic field, as shown in Fig. 6. (iii) The observed initial parabolic shape of the magnetoresistance as a function of H is associated with the strong antiferromagnetic coupling between the C~. layers in this sample, which causes an (initial) linear field dependence of the technical magnetization, M ~ ( M J H s) H [12]. On the other hand, when the spontaneous magnetization in nearest magnetic layers makes an angle 81 -- 8 2 [ M 1 = M s U l , M 2 = M s u 2 , u l ' u 2 --cos(01 - 0 2 ) ] , the resulting average magnetization ( M ) can be written M = (Ms u 1 + MsU2)/2 , giving M 2 = [M 2 + M 2 coS(81 -- 82)]/2. We then have cos[(01 - 82)/2] = M/Ms, so Eq. (6) immediately leads to Ap at H E.

I

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I

I

I

i 1~

|

i 2~

|

i 3~

i

F(T) F(0) 1

0.8

06

0.4

0.2

0

i

0

400

T(K) Fi8.

11.

~/I ap~,,(II)

F(T)/F(O) +

Ap,,(.L) I.

as

function of temperature, experimentally obtained from We relation

F( T ) = const.

84

J.B. Sousaet al./Journal of Magnetismand MagneticMaterials137 (1994) 73-88

(iv) The systematic increase in the saturation field ( H s) with decreasing temperature (Fig. 7; ~ 20% decrease in H s from room temperature to ~ 10 K) can be understood if we recall that the exchange energy between spins in nearest Co layers (J12), is much less than the intralayer (ferromagnetic) exchange coupling J. The equivalent temperature for J12 is typically less than 1 K (whereas J ~ 103 K), so we have kT >> J12 in our temperature range. This leads to important spin fluctuations between consecutive magnetic layers, as was shown by Cullen and Hathaway [8]. For superlattices with thin magnetic layers, i.e. with a number of atomic planes (N) in each magnetic layer less than J/J12, these authors obtained: 4J12 Hs(T) = -
(10)

where (S 1 • S2 )T represents a suitable surface spin (layer 1)~surface spin (layer 2) spin correlation function (see Ref. [8] for details). Due to the weakness of the interlayer exchange coupling J12, this correlation function falls significantly as T increases, outweighing the opposite effect from 1/M(T). We therefore observe a decrease in H~. The interlayer spin fluctuations also lead to a rapid decrease in the magnetization as T increases [8]. (v) We have seen that ( A p / p ) ± <(Ap/p)ll (Figs. 5a, b), implying a negative AMR for our (Co20 ~-Res.~)×l 6 film. Such negative values have been found previously in COl_xlrx [34] and Co l_xRu x [35] films (0.02 < x < 0.15), as well as in several diluted bulk alloys where the impurity develops a virtual bound state close to the Fermi level of the majority spin band [36]. In our case, the negative AMR suggests Co-Re interface mixing (up to 10 and 20% Re incorporation at interface Co layers), since the AMR in pure Co films, due to phonon and defect scattering [37] is positive.

3.2.3. Analysis of (Colo x-Re16 X)x16 data This sample exhibits weak magnetic coupling between the Co layers, corresponding to the second peak in the H~ versus tRe oscillatory curve [9], indicative of AF interlayer coupling. (i) The magnetoresistance is about 30 times smaller than in the (C0203,-Res~,)Xl 6 multilayer. The sharp decrease in the magnetoresistance of the

Co-Re superlattice as tRe increases was first explained by modelling both the magnetoresistance and the resistivity ( p ) dependences on tRe and tco thicknesses, by solving the Boltzmann equation for the finite multilayer structure, assuming interface scattering only. We obtained electron mean free paths h(Re) = 26 .~ and h(Co) = 76 .~, in agreement with the resistivity ratios for pure Re and Co thin films at room temperature, p = (45 _+ 5) and (16 _+ 1) Ix~" cm, respectively [38]. We also notice that the spinorbit contribution is larger in (Co203-Re5~,)×16, representing about 15% of the total magnetoresistance (see Figs. 5a, b). These features are due to the thicker Co layers in this sample. (ii) The magnetoresistance is negative both for H II I and H l I . A possible explanation is the persistence of weak antiferromagnetic interlayer coupling, as evidenced by the observed reduction in the remanence of samples with Re thicknesses of 16-18 ,~ [12]. In addition to an intrinsic effect (oscillatory behaviour of the interlayer magnetic coupling), the small negative magnetoresistance could also be partially related to the fact that our layers are not formed by sets of perfect atomic planes and, in particular, the interlayer separation is not strictly constant across the facing planes [39].

3.2.4. Magnitude of GMR in the two-current model A solution of transport in multilayer systems, using the Boltzmann equation, with both spin-dependent bulk and interface scattering, and the two current model [18] has been previously reported [1-3]. In this section only bulk scattering is assumed, with the current density and the electrical conductivity given by the sum of the corresponding contributions from conduction electrons with spin 1' and spin $ (with respect to a global quantization axis for the multilayer, e.g. imposed by an external magnetic field): j=j('~)+j($), or= o-(]') + ( $ ) . (11) In addition, the symbols + ( - ) are used to designate electrons with spin parallel (antiparallel) to the local magnetization where they move. Thus, for ferromagnetic interlayer coupling, we have p+ = O+ and p ~ = p_; therefore [36]: PT P~ P+Pr . . . . , (12) PF PT +P,t p++p_ = P + l + r

J.B. Sousa et al./Journal of Magnetismand MagneticMaterials137 (1994) 73--88

where r = p_/p+. In an antiferromagnetically coupled multilayered thin film with the electron mean free path much longer than an individual layer, an electron with a given spin spends, on average, equal periods of time in ( + ) and ( - ) local environments, so p t = P ~ = ( P+ + P - ) / 2 , leading to a total resistivity [27]: PT P~ PAF = - = 1( p + + p_) = p+(1 + r ) / 4 . P~ + P ~

(13) From (12) and (13) we obtain a magnetoresistivity: ApPF--PAF P PAF

(l--r) ~

2, (14)

which is always negative. It is different from zero when r #: 1, i.e. p r ~ p ~, thus for spin-asymmetric electron scattering. According to (14), I A p / p l is always smaller than 1, but for r either very large ( p_ >> p+) or very small ( p_ << p+) the magnetoresistance gets close to 100%. Spin-dependent electron scattering may occur both inside the layers or at the interfaces [40]. Experimentally, interracial scattering seems to dominate in several important cases, but the clear identification of the microscopic mechanisms underlying the spin-scattering asymmetry is still an unsettled question. Some authors claim that it could result from a spin-dependent electron scattering potential, e.g. through electron-magnon interactions [4,5], but large spin-scattering asymmetry can also occur (even for a spin-independent electron scattering potential), when the final density of available states for the scattered electrons is spin-dependent, as occurs in transition metals with the s-d electron transitions mediated by phonons (see Section 3.3.1). 3.2.5. Valve effect in GMR Recently Hasegawa developed a general theory for the magnetoresistance in multilayers [41], using the so-called functional-integral method in the static approximation, which treats the interface and bulk scattering on the same footing. The temperature variation of the GMR is also derived in this model, from T = 0 up to Tc. Besides the two-current model term given by Eq. (14), another contribution is found describing the so called valve effect. Under simplifying assumptions, the total expression for A p / p can

85

be written in a transparent form displaying the main factors underlying the valve effect: Ap

PF - PAF

PAl: ~ l+r]

tAF

where the second term on the r.h.s, of this equation represents the valve effect magnetoresistance. Here t F (tAF) plays the role of a valve in the interlayer transmission process for ferro (antiferro) interlayer coupling. One sees that even for symmetric bulk scattering (r = 1; no MR in Eq. (14) for the two-current model) we can still have a sizeable magnetoresistance, if there is an asymmetry in the interlayer transmission process (tF * tAF), for which interracial scattering can play a central role. 3.3. Thermopower

The small values of the positive thermopower in

(COlo/~-Re16/~)×16 and its linear high-temperature dependence resemble the usual behaviour of the diffusion thermopower in normal metals [42]. This is due to the dominance of the nonmagnetic metal in this sample (Re = 16/~ layer thickness). In contrast, S(T) in (Co20 A-Re5 A)x 16 is larger and negative over the whole temperature range, reflecting the dominance of Co in this sample. In fact S(T) qualitatively resembles the curves obtained in magnetron-sputtered films of Co (420 and 500 ,~ thick), showing a gradual increase in IS[ with film thickness (Fig. 4). 3.3.1. Effect of s - d transitions In bulk cobalt (also in Ni and Fe) the large S values are due to s ~ d electron transitions mainly caused by phonon scattering [43]. The non-flip electron scattering rates ~.~1( 1' ) and ~.~1( $ ) are proportional to the density of available states in the 3d subbands at the corresponding energies, Nd(l') and Nd($), leading to a thermopower [44]: S~ffi +

~r2 kT

1

3 lel Na(EF)

X[ Nd'~) Nd(t-------2)+Nd(t-------)aNd(~)] (15) Nd(t) OE Nd(~) OE gffigF"

J.B. Sousa et aL /Journal of Magnetism and Magnetic Materials 137 (1994) 73-88

86

I S~ I reaches a maximum when the top of the majority subband (3d 1') is close to the Fermi level, since then (0Nd ~/OE)eF is very large. This occurs in Co at around 400-450°C [43]. Below this temperature range the 3d$ subband is virtually full, and S is governed by s-d transitions into the minority subband (3d $ ), decreasing in magnitude as T decreases. The characteristic negative sign of S is associated with dNd( ~)/dE < O.

3.3.2. Two-current model for the thermopower in multilayers This model (Section 3.2.4) can be adapted to describe the thermopower in multilayers [28,45]. We recall that, if we apply an electric field under isothermal conditions, S can be expressed in terms of the heat (jQ) and the charge (j) current densities flowing in the medium,

S =j°/rj.

(16)

The two-current model for the heat current density is:

jO =jO( $ ) +jQ(,L ).

(17)

At constant temperature the ($, $) heat current densities are thermodynamically related to the corresponding electric current densities and Peltier coefficient ~" (it = TS) [42,43] by the simple relations: jQ(,)

=

= rs(,)j(,),

jo(,l,) = rr(,l,)j(,[,) = TS( $ )j(,],).

(18)

(19)

Inserting in (16) we get: S=

S( , ) j( ? ) + S( ~ ) j( +) J

(20)

In an antiferromagnetically coupled sample like (Coz0/~-Res~)xl 6, we have j($)=j(~,)=j/2, S( 1") = S( $ ); therefore:

S~

= ½(S,

+S+) =S t =s~

.

(21)

For thin enough layers, an electron spends, on average, equal periods of time in ( + ) and ( - ) magnetization-oriented regions. Piraux et al. [45] calculated S+ and S_, assuming that electron-magnon scattering is dominant:

,rr2k2 T S+ = -T-~ R ( T ) ,

where R(T) is a coefficient of the order of unity that varies slowly with temperature (except at very low temperatures where R vanishes exponentially). Since the ( + ) and ( - ) characters are symmetrically mixed in each current, Piraux et al. argue that a balance occurs between positive and negative contributions to the thermopower (S+, S_), leading to SAF = 0. However, from expression (14) giving PAF in terms of p+ and p_, and Mott's expression [43], one can explicitly write SAF in terms of S+ and S_ [28]:

++rS_ ~r2 k2T ( Oln PAF) SEF= a----E-l+r

SAF= 3 lel

Using S + = - S for electron-magnon scattering (Eq. (22)), we get 1-r SAF ~-- l + r

S+,

which shows that SAF due to electron-magnon is not necessarily zero in general. Our recent results on the magneto-thermopower AS (as defined in Ref. [46]) of a [C022 x-Re5 X]× 16 multilayer [47] indicate a very small electron-magnon contribution to the thermopower in this case, since the total change in S when H changes from 0 up to H s is below 0.2 IxV K -1 at any temperature between 300 and 4.2 K. We have not measured AS(H) in the [Co20~-Res~,]Xl 6 sample, but the similar behaviour of the magnetoresistance observed in both samples and the linear relation found between Ap/p and AS (at different fields) [46] justifies a similar behaviour of AS(H) in both samples. Therefore, (non-spin-flip) s-d transitions are likely to dominate the temperature dependence S(T) in these multilayers. In this case, using Mott's formula for the S+ and S_ thermopower [42,43],

S+

~r2 k2T[~ln P± 1 3 lel O ~ E,'

and noting that p+ ct Nd(+) and p_ ot Nd(--), we obtain immediately [28,48]:

7r2 k2T [ SAF(s-d scattering)

dNd( + )

3 lel [ Nd(+ ) 1

(22)

1

+ NO(----~)

dE

dNd( - ) ] d----E-] eF"

J.B. Sousa et al. /Journal of Magnetism and Magnetic Materials 137 (1994) 73-88

Due to the subband splitting below the Curie point, non-negligible s-d contributions can arise from the density of states energy derivatives, as has recently been emphasized for multilayers [28]. We believe that this mechanism is partly responsible for the zero-field temperature dependence of S in our film of (Co20),-Re 5 h)× 16, leading to the qualitative similarities with S(T) in Co films. It also explains the 'high temperature' increase of Is! in the range of our measurements (T << 400°C), since in cobalt [ONd(-)/OE]eF increases with increasing temperature. A final remark is necessary on the small magnitude of the thermoelectric power in Co/Re multilayers, about one order of magnitude smaller than in C o / C u multilayers and Co ( ~ - 3 0 IxV K -1 at room temperature). This is likely to be due to impurity scattering caused by Co-Re mixing. Our previous work on the structural characterization of Co/Re multilayers [13,49], and the evidence shown here for the case of the AMR, indicate that interface Co-Re mixing occurs, at most, over about 5 ,~ in each 20 ,~ Co layer. Nevertheless, as shown by Cadeville and Roussel [50], 1% Re impurity in Co is sufficiently to reduce the room-temperature thermopower from about - 3 0 to - 2 p,V K-1. Such a dramatic reduction is caused by the formation of a virtual bound state associated with Re in Co, giving a density of states (N) with a maximum near E F, so d N / d E ~ 0 leads to a small thermopower.

Acknowledgements One of us (B. Almeida) gratefully acknowledges a PhD research grant (BD 2217/92-IC) from Junta Nacional de Investiga~.,~o Cientffica e Tecnol6gica, Portugal. This work has been partially supported by Stride Project no. STRDA/ C / C E N / 522/ 92 (Porto Group).

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