Theory on the giant magnetoresistance in magnetic layered structures at finite temperatures

MATERIALS SCIENCE & ENGINEERING ELSEVIER

Materials Science and Engineering B31 (1995) 31-39

B

Theory on the giant magnetoresistance in magnetic layered structures at finite temperatures Hideo Hasegawa Department of Physics, Tokyo Gakugei University, Koganei, Tokyo 184, Japan

Abstract

The temperature dependence of the giant magnetoresistance (GMR) in M I/N/M 2 multilayers, consisting of magnetic M i (i = 1, 2) and non-magnetic N layers, is discussed using the finite temperature band theory in which the effect of spin fluctuations is taken into account by the static functional integral method. When the temperature is increased, the magnetoresistance (MR) ratio (AR/R) decreases in the normal MR, but increases in the inverse MR where the MR ratio is negative, although its absolute magnitude decreases. In a multilayer, where normal and inverse MRs coexist, AR/R may show a maximum vs. temperature. In the compensated MR, the temperature coefficient of AR/R becomes small because of the cancellation between the two contributions over a fairly wide temperature range; this would be beneficial for practical applications.

Keywords: Multilayers; Giant magnetoresistance; Spin fluctuations; Finite-temperature theory

1. Introduction

The giant magnetoresistance (GMR) [1,2] in layered structures has been studied extensively in recent years. One of the important aspects of the G M R is its temperature dependence. A careful study of the temperature dependence of G M R is not only important in understanding its mechanism, but also beneficial for potential applications. Most of the magnetic multilayers are fabricated with transition metals, such as Fe, Ni and Co. Before we study the temperature dependence of the G M R or the resistivity of transition metal multilayers, it is instructive to discuss the resistivity of bulk transition metals at finite temperatures. It has been reported that, when the temperature is increased from T = 0 K, the resistivity of Fe or Ni gradually increases up to the Curie temperature, where it exhibits a cusp [3]. This characteristic temperature dependence of the resistivity has been classically interpreted as being due to spin disorder scattering using the s - d model [4]. Recently, a m o d e r n theory on itinerant electron magnetism has accounted for the temperature dependence in terms of spin fluctuations [5]. It is well known that d electrons in transition metals show both localized and itinerant character: the 0921-5107/95/$09.50 © 1995 Elsevier Science S.A. Al! rights reserved SSDI 0 9 2 1 - 5 1 0 7 ( 9 4 ) 0 8 0 0 6 - 2

Curie-Weiss susceptibility and large specific heat peak near the Curie temperature are easily explained by the localized spin model, whereas the non-integral ground state moment and the large linear-specific heat coefficient favour the band model. It has been realized that the effect of spin fluctuations plays an essential role in reconciling the duality of d electrons [6]. The finite temperature band theory, which has been independently proposed by Hasegawa [7] and H u b b a r d [8], includes the effect of spin fluctuations using the static functional integral method combined with the coherent potential approximation (CPA). This approach has proved useful in understanding various finite temperature properties of transition metals, alloys and multilayers. By employing the finite temperature band theory [7], we have previously discussed [9-11] the temperature dependence of the magnetoresistance (MR) ratio (AR/R) of M / N multilayers consisting of magnetic M and non-magnetic N layers. Our model calculations [10] have shown the following features: (a) the M R ratio is significantly more temperature dependent than the (average) layer moment; (b) the temperature dependence of the MR ratio is greater in a multilayer with a larger ground state M R ratio; (c) the M R ratio is quasi-linear near the Curie temperature. These

H. Hasegawa / Materials Science and Engineering B31 (1995) 31-39

32

features are commonly observed in many transition metal multilayers [12-17]. It has been shown [17] that our theory explains well the temperature dependence of the MR ratio of NiFe/Cu, NiCo/Cu and C o F e / C u multilayers. Our theory has recently been generalized [11] to account for the temperature and layer thickness dependences of AR/R of F e / C r multilayers [16]. The purpose of the present paper is to apply our theory [10] to M ~ / N / M 2 multilayers where M i ( i = 1, 2) and N denote magnetic and non-magnetic layers respectively. In Section 2, we present the formulation for applying finite temperature band theory to MR. Based on this formalism, a semi-phenomenological approach is employed for numerical calculations, the results of which are reported in Section 3. Supplementary discussions are given in Section 4.

{5,, = ( 1 / 4 ) 2 Uj(v~ + ~2)

(3)

J

exp(-/3~b l) = Tr exp(-/3Herf)

(4)

Her f = ~2 ~2 [(ej - (i/2)Ujvj)nj- (1/2)sUj[jm/] + HI} I

(5) Here nj=njT +nj+, rnj=njt -nJ; and H '0 denotes the second (hopping) term in Eq. (1). We can evaluate the partition function by calculating the partition function of the effective one-electron system given by Her f including the random charge (vj) and exchange (~'j) fields with the gaussian weight, exp(-/3~b0). We take account of the charge field by the saddle point approximation and the exchange field by the alloy analogy approximation with the CPA. When the decoupling approximation is employed, the modified CPA equation is given by [7]

2. Formulation

(T~,~) = 0 2.1. Spin fluctuation theory on MR

with

We adopt an Nf layer thin film with a simple cubic (001) interface. The layer parallel to the interface is assigned by the index n ( = 1 - Nf). The film has a sandwich structure with two kinds of magnetic layers, M~ and M2, which are separated by a non-magnetic N layer. The magnetic MI and M 2 layers predominantly consist of magnetic A~ and A 2 a t o m s respectively, but they are allowed to include a non-magnetic B atom, particularly near their interfaces. The magnetic A i (i = 1, 2) atom and non-magnetic B atom are assumed to distribute randomly on the layer n with the concentrations of x~, and y, respectively (x~,, + y, = 1). The film is described by the single-band Hubbard model as t s

s

jl

j

g2 -s(U2/2)(;~2) -~,,~ + [(U2/2)2((~2) z) (X,,,- g2)2]I% [ 1 - ( g 2 )2,,,)F,.] 2 (U2/2)2((~2) 2)

(7) A

A

A

where g• = e,, + ( U , / 2 ) ( N , ) and the angular bracket stands for the configuration average. The coherent potential for an s-spin electron on the layer n (£,,,) is determined by Eqs. (6) and (7), and is a function of A A A 2 a ~,,, (~,,), A ( ( ; , , ) ) and ( N , ) ( = - i ( v,)). The selfconsistent equations for ( [ 2 ) , ((~.,2)2) and (U2) are given by

= fd , ¢,C,*,(d,) ((~2) 2) =

where cj, is an annihilation operator of an electron t with s p i n s ( = 1' , ,~) on the lattice site j, nj,=c~cj~ and tit is the hopping integral. The atomic potential ej and the on-site interaction Uj are assumed to be given by { A and U ~ when the lattice site j is occupied by a A ( = A I , A2, B) atom. In order to study the finite temperature properties of the magnetic film, we apply the functional integral method within the static approximation to the model hamiltonian given by Eq. (1). The partition function is given by [7,8] Z = f [ I d~) [ I d~'j exp[-/3(~b0 + 4h)] with

T~=

(8)

¢

j

J

(6)

J

(2)

f

d~'j ~;C,(~j) "~ A

(N2) = fdE f ( E ) ~ ( - 1 / T r ) Im F,%(e)

(9) (10)

where f(E) is the Fermi distribution function. We should note that F~s(e ), the local Green function of an s-spin electron at a h atom on layer n, and C~(~'j), the distribution of the potential of (U~/2)~'j when a h atom occupies the layer n, are functions of the coherent potentials N,s, which depend on {~'~), ( ( ,~.~ , ) 2) and (N~). Thus we have to simultaneously solve these quantities; the details are reported elsewhere [7]. Once these are determined, the average of the magnetic moment and the root-mean-square (RMS) value of a h atom on the n layer are given by (M~) = ( ~ )

(11)

33

H. Hasegawa / Materials Science and Engineering B31 (1995) 31-39 ( ( M , ,A) 2 )~ /2

=[(((~)2)-(2T/U~)]1/2

(12)

The conductivity for currents parallel to the film layer is given by [9] 0" = U ~ 7 ~ 0",,

T

(13)

n

with 0". = (e/h)Zrr

(

de -

M1 Vsa"lsr"ls (A.~ + A;.,)

(14)

N,J

I I i

N2

M1

(a)

N1

M2

N2

(b)

Fig. 1. A model film consisting of magnetic (M 1 and M2) and non-magnetic (N~ and N2) layers, with thicknesses M and N respectively. The moments on the magnetic layers align in the (a) antiferromagnetic (AF) and (b) ferromagnetic (F) configurations.

%£~ = 6,, + (1 - 6~,) -[(A, s _ At.,)2 + ( A + A/s)2] (15) which is valid within the Born approximation. In Eqs. (13)-(15), A., -- Re £,,(a), A = Jim £~,(E)[, £~, is the coherent potential of an s-spin electron on layer n and a.t ~ and v~ are specified by the electronic structure of the film (see Eqs. (19) and (20) in Ref. [9]). The advantage of our formalism is that we can self-consistently calculate the one-electron properties (such as the density of states and local magnetic moment) and the two-electron properties (such as the conductivity) because the CPA preserves the Ward identity [18]. It should be noted that the so-called spin-flop process is implicitly included through the spin fluctuation term responsible for a decrease in the layer magnetization, as will be shown shortly. We have employed our formalism in microscopic calculations of the ground state MR of multilayers with the simple cubic (001) structure [9] and of the temperature-dependent local magnetic moments in F e - C r multilayers [19]. Alternatively, our theory has been employed in the semi-phenomenological way for the study of the MR ratio at finite temperatures [10]. We will adopt the semi-phenomenological analysis on M R in the following sections.

After some manipulation, the conductivity in the antiferromagnetic configuration is given by 0-

=2CMM

(Air

+A25

) + (AI$

+1~2 T

(19) where 0"i and 0"~ stand for the configuration-independent terms given by , ( 1 1 1 0"1=4CMN\AI~. + A o + AI~ + A o + A 2 t + A o 1

)

-{- A2 ~ -J- A 0

2CNN + --

(20)

A0

, dM{ 1 1 1 1 ) 2d N 0 " 2 = 2 - \ ~ - ~ - ~ +&~-~{ + A2~-+~-~2~ + A ~ -

CMM=Nfl(e/t~)

2"lrv Z

Z

anm'rnm

a M = N f ' (e/h)2"n'~ Z

Z

a.m%.,

(23)

n @M 2

(17)

• AF = A o - iA o for n ~ N 1, N 2 - tA,,,

(18)

A F , _ . zA..~ F = Als - iAl,,, for n ~ M 1

• AF --tA,, s = A l s - i A c ~ for n E M 1

AriAsF -- lA,,s . AF = A 2 AnAF

(22)

nEM 1 m ~ M 2

and CNN, CMN and d N are given by similar expressions, the spin and configuration dependence in a ..... and %m, being neglected. In Eqs. (19)-(23), subscripts MM, NN and MN denote the contributions from the interlayer scattering between magnetic layers, between non-magnetic layers and between magnetic and nonmagnetic layers respectively. On the contrary, the single subscript M (N) expresses the contribution from the intralayer scattering within magnetic (non-magnetic) layers. We employed the T = 0 limit of Eqs. (13)-(15) because the relevant temperature is much less than the Fermi energy. In the ferromagnetic (F) configuration (Fig. l(b)), on the other hand, the real and imaginary parts of the coherent potentials are given by [10]

We assume a system consisting of magnetic (M 1, M2) and non-magnetic (N1, N2) layers (see Fig. 1), whose thicknesses are M and N respectively. Within the M; layer, the concentration of the constituent atoms is assumed to be uniform: x;, = x; and Yn = Y~. When moments on the magnetic layers are in the antiferromagnetic (AF) configuration as shown in Fig. l(a), the real and imaginary parts of the coherent potentials of an s-spin electron are assumed to be given by [10] AnAsv

(21)

with

n ~ M 1 mCM 1

2.2. A semi-phenomenological study on M R

+0-1 +0"2

s - iA2-.~ f o r

(16)

(24)

34

H. Hasegawa / Materials Science and Engineering B31 (199.5) 31-39

A F

.... - - I•A , ,F, ,

=

A2,

-

iAz~ for

n

E

M 2

A,Fs _ tA,,., • F = A o - iA o for n C N 1, N 2

(25)

2~i, = Ai,• + Ais~ + AP

(26)

with

The conductivity for the ferromagnetic configuration is given by [

1 1 )] 2CMM •(Al'r JT-a2"f) + (AI,, + A25

0-F =

,

q- O" 1 -~ 0- 2

By using Eqs. (19) and (27), we obtain the MR ratio, & R / R , given by [10] (R,W R e ) _ ( o . V _ 0-AF) Rv AF 0-

R

-- (O~1 ~-C1!2)(/31 "~ /32) X

(28t

with X = [1 + K o + K I ( N / M ) Ko = go (a, +/31 +

//

1

0~2 -I- /32) ~ 7

1 "1-

-

1 -{-

-

(%+/3,)(a2+[32) ( 1 (O¢l q_ /31_[_ OC2q_ /32) \ ~

l

+

A,,s

(39)

)

1 ~2+1

1

(,,1 +/3,)(~,2 +/32)

(311

K2 = 2g2( a, +/31 + a2 +/32)

(32)

ai=Ai;/Ao,

(33)

/3i=Ai~/A0

In Eqs. (29)-(32), go, g~ and g2 are defined by

where ~ and ~0 are the spin-independent H a r t r e e Fock potentials of atoms Ai and B respectively and p, is the density of states at the Fermi level of an s-spin electron. The first term (A[~) in Eq. (37) arises from the scattering due to random Hartree-Fock potentials s for an s-spin electron, the second term (As.~) from the effect of spin fluctuations and the third term (A~) from the electron-phonon scattering. On the other hand, the real and imaginary parts of the coherent potential in the non-magnetic (N~ or Nx) layer are assumed to be given by A0 = 0

(40)

A0 = &,~ + &P

(41)

Eq. (40) defines the origin of the energy scale, and the first and second terms in Eq. (41) denote the contributions from random potentials and phonons respectively. For simplicity of our model calculation, we neglect the phonon contributions given by dx~[ and &P in Eqs. (37) and (41), and assume that PT =P+ = P in Eqs. (38) and (39). Using Eqs. (33) and (37)-(41), we obtain a~ and /3~ given by

/3i = A i [ ( B i - rni) 2 + Y ; ' ( t z 2 - rn~)l

(43)

Yi

(44)

m i = (m i)/mio

which come from the following relations (35)

AS will be shown shortly, ~+ and /3~ ( i = 1,2) are temperature dependent, whereas go, g~ and g2 are temperature independent because they are determined by the geometry of the film under consideration. In order to discuss the temperature dependence of the MR, we obtain the coherent potentials, which can be evaluated by solving Eqs. (6) and (7). The real and imaginary parts of the coherent potential in the magnetic (M l or M2) layer are given within the Born approximation by [10] A,~; = xi[ ~ - s ( U J Z ) ( M~) ] + Yigo

l

(42)

tx i = ~

CNN ~ d N ~e N 2, CMM a: d M oc M 2

+

([a,~ - rn2)]

ai = A i[(Bi + m i ) 2

with

dM 2CM~ (-~-) 4CMM -- go, CMM -- gx ,

CMN ~ M N ,

v ,rp,.x,(U+/2) 2 [{(M,)-) - (M,) 2]

1\

(30) K,=g~

(38)

(29)

+ K 2 ( N / M ) 2] l

(O{ 1 q- /31)(~£2 -[- ~ 2 )

a,~ = rrp,x y , [ ~ - e~, - s ( U , / 2 ) ( M , ) ] 2

, (27)

AR

(37)

(36)

/Mio

(45)

A , = rrpx,y,(U~M,o/2) 2/zX,,

(46)

B i = ( 2 / U , M , , , ) ( g o - ~)

(47)

where { M i ) and X / ( ( M i ) 2) are the average and RMS values of the layer moment on the Mi layer at the temperature T, and M~0 is its ground state value (A 1 and A 2 in Eq. (46) should not be confused with A 1 and A 2 expressing the atomic species of the magnetic metals). 2.3. M R d u e to interface scattering

We consider the case in which the interface scattering is predominant. Setting X = 1 in Eq. (28), we obtain

35

H. Hasegawa / Materials Science and Engineering B31 (1995) 31-39

AR

h ( a , - 1)(a 2 - 1)

R

(1 + h ) ( a I + h a 2 )

10

(48)

~.

f

'/

'

uI , , v

/

//

I

,

with

/

I

/

a~ = A~ T/Ai ~. h = A 2 ~,/m 1 ~,

/

(49) /

/

W h e n a~ > 1 and a 2 > 1 (or a~ < 1 and a 2 < 1) in Eq. (48), we obtain the n o r m a l , positive M R ratio. In particular, w h e n a] = a 2 = a and h = 1, Eq. (48) becomes AR

(a

1) 2

--

4a

yi(Bi + mi) 2 + (~

: ( o ) h=l.O

0.1

1 I I

0.1

/

I I

I

I

h

]~2 y2(B2 - m2) 2 + (/x~ - m~) h =--~l = d y l ( B , - m , ) e + (tz~ - m ~ )

(52)

.

C)~/

/

~/

/ J/

J

,

, , Jt

1.0

10

0 1

I0 : / ~

p

I/U

/

,,~.

/

/

s

/ -, !/

/

1.0

I I

I

./

/

. i /

/

AR/R o.o

.I

cl

I

I I

/

~ /

I

I

ii

/ /

I

I

t J / I

/

/ -

Ill;u

/

(b) h-:O.5 /

/

\ (51)

---:.

/ / t /

t t

-- m ~ )

ai = fi~ = y i ( B i - m i ) 2 + ( 1 ~ - rn~)

1

/

/

//

which is the result o b t a i n e d previously [10,20,21]. O n the o t h e r hand, w h e n a I > 1 and a 2 < 1 and vice versa, we obtain the negative A R / R , which has b e e n recently p o i n t e d out in Ref. [22] and is referred to as the inverse M R hereafter. Fig. 2 shows the c o n t o u r m a p s of A R / R as functions of a 1 and a 2 for h = 1.0 and 0.5. T h e M R ratio is positive in the first and third q u a d r a n t s , whereas it is negative in the s e c o n d and fourth q u a d r a n t s , as m e n t i o n e d a b o v e . It is s y m m e t r i c for h -- 1.0 but asymmetric for h =0.5. T h e t e m p e r a t u r e d e p e n d e n c e of A R / R arises f r o m the t e m p e r a t u r e d e p e n d e n c e s of a l, a 2 and h, which are given by OLi

A R/R 0.0

(50)

m

R

1.0

I

I

f

/

/

//i-

/

0.1

ii

0.1

I

I

I

/~.%// / / I

I/

I

I

I

I

I I

1.0

0

Ol with d = /\ X / 2{| |

(53)

e2M2°)2

\x, / \ UIMIo/ A t T = 0 K, w h e r e m~ = p~i = 1, Eqs. (51) and (52) become a . , = a , ( T = 0) = [(B, + 1)/(B, - 1)] 2

h o = h(T=

O) = d ( y 2 / y ) ) [ ( B

2 - a)/(B~ - 1)12

Fig. 2. The contour map of the MR ratio, A R / R ( = (R AF- RF)/ RF), as a function of a~ and a z for (a) h - 1.0 and (b) h =0.5. The solid (broken) contours show the positive (negative) A R / R with a step of 0.2 (0.1).

d : ho(Y2/Yl)[(B1

- 1 ) / ( B 2 - 1)]2

(54)

Substituting Eqs. (56) and (57) into Eqs. (51) and (52), we can express a/ and h in terms of ai0, h 0 and Yi

(55)

as

{[(~

f r o m which the coefficients B~ and d are given by

+ 1)/(X/h~,0 - 1)] + rn,} 2 + y,-,,"~/~,2 - m~)

a, = {[(V~,J + 1)/(X/h~m _ 1)] - rn,} z + y ; ' ( / ~ B i = ( V ~ 0 + 1)/(V~-~,.0 - 1)

(57)

(56)

- mr) (58)

H. Hasegawa / Materials Science and Engineering B3I (1995) 3 1 - 3 9

36

.....

)<

{[(X/~2o + 1) / (X/-d~2o- 1)1 - m2} 2 + Y;~(P~22 - m~) {[(~o.~ + 1)/(~o~o - 1)] - m, }2 + y ~ 1(/z21 _ m21)

,

.••

rr

~o= ~

(59) ot3. Calculated results

tY

m , ( T ) = V1 - ( T / T c ) 2, /.ti(T) = 1

(60)

Alternatively, we can discuss AR/R, treating h o, aio and y, (i = 1,2) as input parameters, by using Eqs. (48) and (58)-(60) I . We present some numerical calculations of the normal and inverse M R separately in the following two subsections.

"-<\

\

o

~

T h e t e m p e r a t u r e dependence of the M R ratio can be calculated for given band parameters, such as ~o, ~i and U~, with the use of Eqs. (48) and (51)-(53), if we adopt simple analytical expressions for m~(T) and /~(T) given by [10]

~,,',/y=0.002

~

\"~..

0.1

,

0

T/Tc

1

l

3

\

\

o 0

TIT C

1

Fig. 3. The temperature dependence of the MR ratio. A R / R , for various a, ( a~,~=a2~,) with Y ( = Y t y 2 ) = 0 . I. The inset shows A R / R for various y with a 0 = 5, the broken curve denoting the result with no spin fluctuation contributions.

3.1. Normal M R We first present the calculated result for multilayers in which magnetic A I and A 2 atoms are identical. Fig. 3 shows the calculated M R ratio for various values of a0 ( = al0 = a20 ) with a fixed value of y( =Yl = Y 2 ) = 0.1. The t e m p e r a t u r e dependence of A R / R is more significant than that of the magnetic m o m e n t , rag(T). We note that a film with a larger ground state M R ratio, a0, has a more significantly t e m p e r a t u r e - d e p e n dent M R ratio, and that the M R ratio is quasi-linear near the Curie temperature. The inset shows the calculated M R ratio for various y values with fixed a 0 = 5. A film with a smaller y has a more significantly t e m p e r a t u r e - d e p e n d e n t M R ratio. In the limit of y--~ 0, only the spin fluctuation term contributes to A (see Eqs. (51) and (52)). The t e m p e r a t u r e dependence of the M R ratio in the opposite limit, where A includes only the r a n d o m potential term, is shown by the broken curve in the inset. The t e m p e r a t u r e dependence of M R is studied in m o r e detail for a typical case of a 0 = 5 and y = 0.1 in Fig. 4. At T = 0 K, the resistivity in the A F configuration (R AF) is larger than that in the F configuration (Rr). When the t e m p e r a t u r e is raised, both R AF and R F increase up to Tc, above which they become identical. This arises from the difference between the The values of a o ( - a,~ = a20) adopted for an analysis of the temperature-dependent MR ratio of NiFe/Cu, N i C o / C u and CoFe/ Cu multilayers are in good agreement with those estimated from the band parameters (e and U) for bulk Fe, Co, Ni and Cu (Ref. [17]).

t e m p e r a t u r e dependence of A~ ( = A IT = A 2 T ) and A ,, ( = A I ~ = A25 ). The decomposition of A = A~ + As in Fig. 4(b) shows that, when the t e m p e r a t u r e is r raised, A~ increases while A t decreases because the magnetization m i decreases. On the contrary, the spin fluctuation term A~ monotonicallly increases. The increase in R AF and R v is mainly due to the contribution from the spin fluctuation term, In order to study the case in which the two magnetic atoms A~ and A 2 a r e different, we show in Fig. 5 the calculated result for various a20 values with fixed values of al~ 3, h 0 = 1 and Y l = Y2 = 0.1. W h e n a20 is increased, the M R ratio increases and its t e m p e r a t u r e dependence becomes more significant. We show, in the inset, the M R ratio with changing Y2 and fixed values of a~o, a2o, h o and y~.

3.2. Inverse M R We calculated the M R ratio for various a2o values with a u ~ = 3 , h o = l and y l = y 2 - - O . 1 (Fig. 6). At T = 0 K, A R / R is negative as mentioned before. When the t e m p e r a t u r e is raised, A R / R increases, although its absolute value decreases. The inset shows the calculated A R / R values with changing Y2 but fixed values of a~0 - 3, a~0 --- 0.2 and Yl = 0.1. The t e m p e r a ture dependence of the M R ratio with smaller Y2 is m o r e significant, as shown in the inset of Fig. 5 for the normal MR.

H. Hasegawa / Materials Science and Engineering B31 (1995) 31-39

37

i

(a)

(b)

Ar

!

<3

oE

RF. /

_---

I

0

I

0

As

I

I

I

1

TIT C

I

0

I

I

I

I

TI TC

Fig. 4. (a) The temperature dependence of the resistivity in A F (RAv) and F (R v) states and of the imaginary parts of s-spin coherent potentials for a 0 = 5 and y = 0.1 (R and A normalized by their values at Tc). (b) The decomposition of A into A = A[ + A~.

!

0.8

i

~a20= 10 8

T/TC 0

0.5 - \~'/Y2-0001 ~. \./~ /o.1

0.4

0.0

~

1T T

1

1

o.o "~ - 0 . 2 n.-

-~. 0.1 0.0

'

~4

I/ /--J~o.1 I [ ~ 0[01

'

0

1

TITC

- 0.4

Fig. 5. The temperature dependence of the MR ratio, AR/R, for various a2,, with a m = 3 , h o = l and y] - y 2 = 0 . 1 . The inset shows A R / R for various Y2 with al0 = 3 , a2o = 5, h 0 = 1 and YL =0.1.

Fig. 6. The temperature dependence of the MR ratio, AR/R, for various a2o with alo = 3 , h o = 1 and y~ =Y2 = 0 . 1 . The inset shows AR/R for various Y2 with a]0 = 3, a20 h, = 1 and y~ = 0.1.

3.3. Coexistence o f normal and inverse M R

and a 2 > 1 in region I, a 1 < 1 and a 2 < 1 in region II and a I > 1 and a 2 < 1 and vice versa in region III. These three regions are a s s u m e d to coexist with the probabilities of (1 _ p ) 2 , p2 and 2 p ( 1 - p ) respectively, where the p a r a m e t e r p stands for the d e g r e e of coexistence of the region with the inverse M R (p = 0 or 1 c o r r e s p o n d s to the n o r m a l M R only). W h e n the three regions are a s s u m e d to yield additive contributions to the total conductivity, we obtain A R / R for various p values with a]0 = 3, a20 = 0.3, h 0 = 1, y~ =

W e have shown in the previous two subsections that, w h e n the t e m p e r a t u r e is raised, A R / R increases in the inverse M R but decreases in the n o r m a l M R . W e m a y expect that the t e m p e r a t u r e d e p e n d e n c e of A R / R will show a variety of b e h a v i o u r if a given multilayer includes b o t h the n o r m a l and inverse M R . In o r d e r to investigate this possibility, we assume that t h e r e are three regions in a given multilayer: a I > 1

=0.2,

38

H Hasegawa / Materials Science and Engineering B31 (1995) 31-39

0A

i

i

|

i

.p=O.O 0.3 " .._.',x/,,' /0.05

>.

~_. 0.2 ,,~

~ 0 ->~>z',,.\\

T/Tc

1

y2=o.oo

- o.o o 2 o.o

0.007 ~/

00 0

1 TIT C

Fig. 7. The temperature dependence of AR/R of the multilayer in which the normal and inverse MR coexist, for various y, values with a,. 3, a_,. 0.3, h . = l , y ~ - 0 . 1 a n d p = 0 . 1 (p denotes the degree of the presence of the inverse MR region). The inset shows the MR ratio with changing p and au,=3, a..=0.2, h.= 1, y,=0.1 and y, = 0.001 (see text).

0.1 and Y2 = 0.001, which is shown in the inset of Fig. 7. W h e n p = 0, A R / R behaves as the n o r m a l M R . If the p r e s e n c e of the region of the inverse M R is allowed to s o m e extent, A R / R increases at low temp e r a t u r e s and then decreases at higher temperatures. T h e m a x i m u m in A R / R is shown to arise from that in AR ( = R AF - R F) [23]. Fig. 7 shows A R / R for various Y2 values with a m = 3, a20 = 0.3, h 0 = 1, yj = 0.1 and p=0.1. We note that, when y 2 = 0 . 0 0 7 , A R / R is almost constant below T / T c ~ 0 . 3 , where a decrease in the n o r m a l M R as the t e m p e r a t u r e is raised is nearly c o m p e n s a t e d by an increase in the inverse M R . This c o m p e n s a t e d M R with a small t e m p e r a t u r e coefficient would be beneficial for practical application. A m a x i m u m in A R / R was recently o b s e r v e d in C u N i / C o [24] and N i C o / C u multilayers [251, although it is not clear at the m o m e n t w h e t h e r the observed p h e n o m e n o n is due to the m e c h a n i s m discussed above.

4. Discussion In o r d e r to m a k e a multilayer showing the inverse M R , we must adopt p r o p e r elements which satisfy the conditions: a~ > 1 and a 2 < 1 (or vice versa); this implies (~'1 ? -- G ) ) 2 p ~, > (El ; -- ~ ) ) 2 p $ a n d (g2,g,)2p7 < (g2; - g,)2p~ . O n e of the candidates would

be a c o m b i n a t i o n of M~ = Ni, M 2 = Fe and N = Pt (or Pd), for which IgN~; -- ~,l ~ ] g F ~ -- g0[ ~ 0 , because the n u m b e r s of d electrons with spin s per a t o m (Nd,) a r e N a ,L(Ni) ~ Na(Pt ) = N d t (Fe) ~- 4.5. G e o r g e et al. [22] have a d o p t e d a sophisticated F e / C r / F e multilayer as M t with M e = Fe and N = Cu. T h e y have claimed that the global spin a s y m m e t r y a~ m a y be greater than unity because a huge av,~c r at inner F e - C r interfaces o v e r c o m e s that at the o u t e r F e - C u interface (avecu = a, < 1) of M 1 . It is not casy to d e t e r m i n e the simple transition metal elements which satisfy the condition m e n t i o n e d above. Despite such a difficulty, it is beneficial for practical applications to fabricate a multilayer with a c o m p e n s a t e d M R whose A R / R value has a small t e m p e r a t u r e coefficient over a fairly wide t e m p e r a t u r e range. We have assumed in our m o d e l calculation that the t e m p e r a t u r e d e p e n d e n c e of the magnetic m o m e n t s follows m~(T) = [1 - (T/Tc)2] ' ' ' (Eq. (60)). This is not so different from the Brillouin function for spin 1/2, which simulates well the overall t e m p e r a t u r e d e p e n dence of magnetization in f e r r o m a g n e t s such as bulk Fe, Ni and Co. It is, h o w e v e r , well k n o w n that magnetic m o m e n t s on the free surface decrease m o r e rapidly than in the bulk, particularly at low t e m p e r a tures. This is also e x p e c t e d for m o m e n t s in the interfaces of layered structures, as the recent b a n d calculation has shown for F e / C r multilayers [19]. In o r d e r to examine how the t e m p e r a t u r e d e p e n d e n c e of interface m o m e n t s reflects that of A R / R , we repeat the calculation of A R / R , a d o p t i n g the functional f o r m for m~(T) given by rn,(T) = [1 - (T/T,,)2] k

(61)

where k is a p a r a m e t e r (Eq. (61) with k == 0.5 corresponds to Eq. (60)). W h e n the k value is increased, the t e m p e r a t u r e d e p e n d e n c e of m i ( T ) b e c o m e s significant, as shown in the inset of Fig. 8. T h e M R ratios calculated with alo - a2o = 3 and Yl = Y2 = 0.1 are shown in Fig. 8. We note that the t e m p e r a t u r e d e p e n dence of A R / R is rather sensitive to m s ( T ) because it d e p e n d s on [/xi(T) 2 - m~(T)2], although all the calculated results show qualitatively similar behaviour. It m a y be possible that, in very thin films, the t e m p e r a ture d e p e n d e n c e of A R / R provides information on the interfacial local m o m e n t s of the film. We have discussed the cases in which interface scattering is assumed to be p r e d o m i n a n t . In actual multilayers, h o w e v e r , both interface and bulk scattering are considered to play i m p o r t a n t roles. We can discuss both types of scattering on the same footing if our s e m i - p h e n o m e n o l o g i c a l analysis is e x t e n d e d to include p a r a m e t e r s relevant to interface as well as bulk scattering [10,11], although the n u m b e r of adjustable p a r a m e t e r s is increased considerably. For a b e t t e r

H. Hasegawa

0.4

i

i

rr

'\\\ \~0

~_. 0.2

Materials Science and Engineering B3I (1995) 31-39

-_\

0

k" : o . 5

1.0~ 1.5 /

,

-'4 T/Tc 1

,

,

\

3, \

"\\\

0

\

1

TIT C Fig. 8. The temperature dependence of AR/R, calculated using m ( T ) = [1 - (T/Tc)2] ~ with k = 0.5 (full curve), 1.0 (broken curve) and 1.5 ( - - - ) for a~0 = a2o = 3 and y~ = 0.1; the inset shows m~(T) for relevant parameters.

understanding of the observed data, it is desirable to perform the microscopic calculation employing the formalism presented in Section 2.1.

Acknowledgment This work was partly supported by a Grant-in-Aid for Scientific Research on Priority Areas from the Japanese Ministry of Education, Science and Culture.

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39

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