EXCHANGE THROUGH A TUNNELING BARRIER

(1)

HAL Id: jpa-00228986

https://hal.archives-ouvertes.fr/jpa-00228986

Submitted on 1 Jan 1988

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

EXCHANGE THROUGH A TUNNELING BARRIER

J. Slonczewski

To cite this version:

(2)

JOURNAL DE PHYSIQUE

Colloque C8, Supplement au no 12, Tome 49, decembre 1988

EXCHANGE THROUGH A TUNNELING BARRIER J. C. Slonczewski

IBM Research Division, Thomas J. Watson Research Center, P.O. B o x 218, Yorktown Heights, N Y 10598, U.S.A.

Abstract. - A tunneling junction consisting of two ferromagnets separated by an insulating barrier is treated in the Stoner limit. One thus derives expressions for Julliere's magnetic valve effect and Heisenberg-type interfacial exchange coupling. In addition, a voltage-dependent irreversible form of exchange coupling is predicted.

This paper summarizes a theory [I] of three re- Region 3 (ferromagnet B, d

5

<) has only the trans- lated phenomena involving two ferromagnets sepa- mitted wave

rated by a non-magnetic tunneling barrier (film or vat- _$'

- C, eike(<-d)

uum): 1) a magnetic valve effect, according to which ~3

-

,

g=T,

L .

(4) the tunneling conductance G = Go (1

+

E cos 0) varies

~h~ axis of spin quantization for the above waves is with angle between the vectors

[21;

2, a _{(direction h}₌_{h A ) in regions 1}_{and 2, and}_z'_(direction Heisenberg-type interfacial exchange coupling energy _h₌_{hB) in region}_3. _Matching_$, _{and d$,}

_/

_{dJ at}

-3 cos 6

PI;

3) a newly predicted dissipative interfacial

<

₌₀_{and d, including}_{a spinor rotation through angle} exchange interaction proportional to external voltage ₀_at

₆

_{= d,}_{determines the coefficients}_R,

_,

_A,,

_B,

_,

_C,

_.

v.

1. T h e o r y

Consider two ferromagnetic conductors separated by a plane diamagnetic tunneling barrier. The longitudi- nal part of the effective one-electron Hamiltonian may be written

In this section, our units are such that the electron mass and Planck constant are each unity. Equation (1) includes a potential U (<) and internal exchange - h - a where -h (<) is the molecular field and a ( = 2 s) is the conventional Pauli spin operator. Although transverse momentum lcll is omitted from the above notations, the effects of summation over kll will be accounted for in the results.

We assume a rectangular potential U = Uo for 0

<

d and U = 0 otherwise, as indicated in figure 1. By assumption, h = 0 inside the barrier. But h = h~ or hg is constant, with l h ~ l = l h ~ l = ho, within each semi-infinite ferromagnet. However, the directions of hA and hg differ by angle 8.

Consider a spin-up incident plane wave having unit incident particle %ux in region 1 (ferromagnet A, J

<

0

in Fig. 1). Including the reflections, the eigenfunction of ?iE (eigenvalue EE) here has spin-up (t) and spin- down

(1)

components

Fig. 1. - Schematic potential for two metallic ferromagnets separated by a barrier.

One may substitute the wave function into the fol- lowing expressions for particle (T,) and spin T = (Tz, T,,

T,)

transmissivity

In region 2 (barrier, 0

5 <

5

d), the evanescent wave Summations of -Tp and T / 2 over occupied states

is give the total charge (I) and spin (IS = -SA = SB)

$,2 = A, e-"(

+

B, en<, a

=t,

.

( 3 ) currents per u n i t area flowing from A to B.

(3)

JOURNAL DE PHYSIQUE 2. Predictions

Evaluation of I t o first order in Vpredicts a magnetic valve with conductance of the form

where the effective polarization

Pfi

of the magnet- barrier interface is 1 for the one-band model (see Fig. 2) and

Pfi

= (n2

-

Wl)

(kt

-

kl)

(n2

+

ktkl) (kt

+

k ~ ) (9) otherwise. Spin polarization is also measured in tunneling between a ferromagnet and a superconductor [41.

k 4 0 k,

Fig. 2. - Fermi energy for one-band (EF1) and two-band

(EF2) models of a ferromagnet.

2.2 HEISENBERG COUPLING

When V = 0, only the component Is, = - S A ~ of spin current does not vanish. In the two-band case, we find the reversible exchange coupling

J =

Sky

/

sin B = (UO

-

EF) b i / 87r2d2, (10)

In the presence of external voltage V = VA

-

VB

#

0, x and z components of I s no longer vanish. The con- sequent i~reversible exchange contribution to the component SA,, called transverse, is the principal effect. In coordinate-free form it is written (see Fig. 3)

s A ~ . = (VB

-

VA) ~ 2 g A X ($A X 3B)

,

₍₁₂₎

where designates unit vector, and

where q is the electron charge.

Fig. 3. - Spin-vector dynamics due to the transverse ex- change relaxation induced by a voltage

3. Discussion

According to Stearns [5], the electron states in Fe, Co and Ni which are most important for tunneling closely resemble free electrons. Accordingly, our model provides interesting estimates of valve and exchange effects in such metals.

Notable is the sensitivity of our results, for given nd, to the height of the tunneling barrier (K rc2) above EF. The valve (9) and both exchange expressions (11) and (13) have the zero K~ = ktkb Thus varying K, can effect sign changes in all three effects under discussion. In addition, the changes of sign with band splitting (ho or kl), also made evident by the same zeros, is striking.

The transverse exchange relaxation newly predicted by equations (12) and (13) resembles Landau-Lifshitz damping, except it may have either sign depending on the sign of V. To illustrate, assume that tunnel- exchange coupling with J

>

0 dominates the dynamics of a very thin film A. Then equation (10) describes precession of S A (and h A ) about an assumedly static

SB

at small-amplitude frequency wo = sA, /SAB. ~t

follows that the valtage induces a (*) contribution to linewidth Awv = W O S A ~ /S*,. The foregoing relations

predict

observe voltage-pumped oscillation, one may ad- just d so that wo attains a value in the desired range for FMR or Brillouin scattering. When Awv [= 4woV (Volts) for an Fe-C-Fe junction] predicted by equation (14) is sufficiently negative t o overcome the posi- tive linewidth contribution due to conventional losses, spin precession will be excited.

[I] Slonczewski, J. C., unpublished.

[2] Julliere, M., Phys. Lett. 54A (1975) 225; Maekawa, S. and Gavert, U., IEEE Trans. Magn. MAG-18 (1982) 707.

[3] Pomerantz, M., Slonczewski, J. C. and Spiller, E.,

J. Appl. Phys. 6 1 (1987) 3747.

[4] Tedrow, P. M. and Meservey, R., Phys. Rev. B 7

(1973) 318.

[5] Stearns, M. B., J. Magn. Magn. Mater., 5 (1977)