Document text
o
o
3
>—>
ON
Spin-relaxation and magnetoresistance in FM/SC/FM tunnel junctions
S. Takahashi a T. Yamashita a , H. Imamura b , and S. Maekawa a
a Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
b Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan
The effect of spin relaxation on tunnel magnetoresistance (TMR) in a ferromagnet/superconductor/ferromagnet
(FM/SC/FM) double tunnel junction is theoretically studied. The spin accumulation in SC is determined
by balancing of the spin-injection rate and the spin-relaxation rate. In the superconducting state, the spin-
relaxation time t s becomes longer with decreasing temperature, resulting in a rapid increase of TMR. The TMR
of FM/SC/FM junctions provides a useful probe to extract information about spin-relaxation in superconductors.
cd
T3
G
O
o
>
(N
O
o
c3
o
o
X
Spin-polarized tunneling plays an important
role in the spin-dependent transport of magnetic
nanostructures GJ. The spin-polarized electrons
injected from ferromagnets (FM) into nonmag-
netic metals (NM) such as a normal metal, semi-
conductor, and superconductor creates a nonequi-
librium spin polarization in NM &-§. The effi-
cient spin injection and weak spin-relaxation dur-
ing transport are required for practical applica-
tions. A number of experiments for observing the
spin relaxation time r s in SCs has been reported
by using a spin-injection device || and by the
conduction electron spin resonance PjlQ].
A double tunnel junction FM/SC/FM contain-
ing superconductor (SC) sandwiched between two
FMs is a unique system to investigate nonequi-
librium phenomena caused by spin injection, es-
pecially the magnetoresistive effects by compe-
tition between superconductivity and spin accu-
mulation |]ll]— [l^] . The pronounced magnetoresis-
tance effects is brought about by a long spin relax-
ation time t s in SC, which corresponds to a long
spin-diffusion length. In this article, we take into
account the coherence effect of superconductivity
on the spin-relaxation due to spin-orbit scattering
by impurities |lq | , and demonstrate that the tun-
nel magnetoresistance (TMR) of the FM/SC/FM
junction exhibits a large enhancement due to the
increase of t s in the superconducting state.
We consider a FM/SC/FM double tunnel junc-
tion. The left and right electrodes are made of a
ferromagnet, and the central one is a supercon-
ductor with thickness d smaller than the spin-
diffusion length As . The magnetizations of FMs
are aligned either parallel or antiparallcl. Using
the Fermi's golden rule, we calculate the spin-
dependent tunnel currents across the junctions
jll| . |Rql . In the following we consider the case that
the bias voltage V is much smaller than the su-
perconducting gap parameter A. In this case, the
shift of chemical potential Sfi for up-spin (— 6fi for
down-spin) electrons due to spin accumulation is
much smaller than A, so that the tunnel current
If across the «th junction (z = 1, 2) becomes
ll(V) = Glx(T)[V/2-6n/e], (1)
li(V) = G[x(T)[V/2-
4(V)=Glx(T)[V/2-
lhV) = Gix(T)[V/2
<We] , (2)
Sii/e] , (3)
- We] ■ (4)
Here, Gfx(T) (i — 1, 2) is the tunnel conductance
for electrons with spin <r in the superconducting
state, Gf is that in the normal state, and
E k ( df
X(T) = 2 f
Ja
<M
dE v
dE^
(5)
where fo(Ek) is the Fermi distribution function
and Ek — y^ + A 2 the dispersion of quasipar-
ticles, £k being one-electron energy relative to the
chemical potential.
The spin density S accumulated in SC is de-
termined by balancing the spin injection rate
{dS/dt) h - with the spin relaxation rate S/t s :
{I 1] -I n )-{I 2] -hi) = 2eS/T s
(6)
where t s is the spin relaxation time and
S = \ E [/T (^k) - h (#k)] « N(0) X {T)Sn, (7)
k
where / <T (£ , k ) ~ /oC%) - (df /dE k )cj5n is the
distribution function of quasiparticles with spin
a and iV(0) is the normal-state density of states
in SC.
It follows from Eqs. |l])-(0) that the tunnel cur-
rents for the parallel (P) and antiparallel (AP)
alignments are given by
Iap
X (T)V/R T ,
i - p 2 + r s
i + r s
X (T)V/Ri
(8)
(9)
where R T = 1/G T (G T = Gj + GJ) is the tunnel
resistance and T s is the relaxation parameter
T s = e 2 N(0)R T Ad/r s , (10)
with A being the junction area. Therefore, we
have the TMR ratio at low bias (V< A)
I P - J AP P 2
Iai
l-P 2
11
where P = {G\ - G\)/{G]+G\) is the tunnel-
ing spin polarization. For a weak spin relaxation
(r s « 1), TMR = P 2 /(1-P 2 ), while for a strong
spin-relaxation (T s > 1), TMR = P 2 /T s < 1.
In SC, the spin relaxation is caused by the spin-
orbit scattering from impurities or grain bound-
aries. The spin-orbit interaction TL so via impurity
potential V; mp (r) is given by
H a0 = -i(h/2mc) 2 a • [VMmp(r) X V] , (12)
where a is the Pauli spin matrix. The scattering
matrix elements over quasiparticle states |kcr) has
the form:
(kV|W so |ker) = i\ so Vk<u ovv • (k X k'
(13)
where A so is the spin-orbit coupling parameter,
Vk'k = (wk'^k - Wk'^k) V imp , |w k | 2 = 1 - M 2 =
\ (1 + £k/Ek), and k = k/|k|. Using the golden
rule formula, we obtain the spin-relaxation rate
due to the spin- flip scattering by H. so :
f) =Y ni £ l(k ' i|Ko|kT>|2 ^ k
/ sf k / k
Ek>)
0.5 1
Temperature (T/T c )
Figure 1 . Temperature dependence of the spin-
relaxation time Ty. Inset shows x(T) and 2/o(A)
versus T, which are used to calculate r s .
8A 2 o 7V(0)
9r;
x [fi(Ev) - / T (£ k )]
/ [/ T (^k)-/i(^k)]^k, (14)
J A
where 1/Th
imp J A
(2ir/h) ni V 2 N(0) is the scat-
'imp -- v~"/ "v'"i ' lmp '
tering rate by impurities and 7ij is the impurity
concentration.
From Eqs. ([7j) and (|14|), we determine the re-
laxation time t s from (dS/dt) si = —S/t s , and
obtain
r s r
/:
A ^E't-A-
[f^E)-f l (E)]dE
J™lME)-fl(E)]dE
(15)
where t s { = 9T imp /8A 2 is the spin-flip scattering
time in the normal state. Note that the expres-
sion of Eq. ( |15| ) is valid for vpr s f 3> £o — frvp/ir/S.0
Q. For 5/x < A, Eq. @ reduces to
[ X (T)/2/ (A)]t s
sf.
(16)
which is the same as the result of Yafet |15| , but
differs from the result of Zhao and Hershfield [Hj .
Equation ([15|) is a generalization of Yafet to the
case of arbitrary value of Sfi.
The temperature dependence of the spin-
relaxation parameter T s is scaled to the normal-
ized spin-relaxation time t s /t s { by the relation
T s = (r sf /r s )rf, where T^ = e 2 N{0)R T Ad/T sf
is the relaxation parameter in the normal state.
Figure 1 shows the temperature dependence of
H
H
1 1 1 1 1
— ^ZL_ ' '
■
" \\ 8 10
N. - E
i \\ s
I \V Hi
normal state \
:
■ \ V\co
(T=T C ) \e
:
0.1 1 10 100 ■
r N
1 s
L \20\x
-
:\. 5^C^
c>..
^-^2^^^
~~ ^^5^ ~
: r^
^^^^^
P = 0.5 :
■
0.5
T/X-
Figure 2. Tunnel magnetoresistance as a func-
tion of temperature for different value of the re-
laxation parameter. Inset shows the TMR versus
T^ in the normal state.
r s /r s f. Above T = T c , the spin relaxation time
r s coincides with the spin- flip scattering time r S £ .
As temperature T is lowered below T c , t s be-
comes longer with decreasing T and behaves as
t s ~ (nA/2k B T) 1 / 2 T si at low T.
Figure 2 shows the temperature dependence of
the normalized TMR for different values of T™.
The inset shows the TMR versus T^ in the nor-
mal state. In the case of T^ > 1, which corre-
sponds to the case that the spin-relaxation rate
is larger than the spin-injection rate in the nor-
mal state, the TMR above T c is suppressed com-
pared with the optimal value 33% for T™ = and
P = 0.5 as shown in the inset of Fig. 2. How-
ever, in the superconducting state below T c , the
TMR increases rapidly with decreasing T due to
the increase of r s , and recovers the optimal TMR
in the limit of T — ► 0. If one uses the values
of R T A = 100 ft/im 2 , r sf = 10- 10 sec, d = 10
nm, and N(Q) = 10 22 /(eVcm 3 ), then one obtains
r^ 1 = 10. Notice that in the case of strong spin-
relaxation (r^ 3> 1), the TMR becomes propor-
tional to t s , so that the temperature dependence
of TMR/TMR(T C ) coincides with that of t s /t s[
as shown by the dashed curve in Fig. 2. The result
indicates that the TMR of FM/SC/FM junctions
provides a method to extract important informa-
tion about spin-relaxation in superconductors.
The authors are grateful to A. Fert and
M. Johnson for fruitful discussions. A part
of this work was done during stay (S.T.) in
CNRS/Thomson-CSF, France. This work is sup-
ported by a Grant-in- Aid for Scientific Research
from Ministry of Education. S.M. acknowledges
support of the Humboldt Foundation.
REFERENCES
1. Spin dependent transport in magnetic nanostruc-
tures, edited by S. Maekawa and T. Shinjo (Gor-
don and Breach Sci. Pub., London) (in press).
2. M. Johnson and R. H. Silsbee, Phys. Rev. Lett.
55 (1985) 1790.
3. T. Varet and A. Fert, Phys. Rev. B 48 (1993)
7099.
4. M. Jemeda et al, Nature 410 (2001) 345.
5. M. Johnson, Appl. Phys. Lett. 65 (1994) 1460.
6. V. A. Vas'ko et al. Phys. Rev. Lett. 78 (1997)
1134.
7. Z. W. Dong et al. Appl. Phys. Lett. 71 (1997)
1718.
8. T. Daibo et al. (unpublished).
9. D.C. Vier and S. Schultz, Phys. Lett. 98 (1983)
283.
10. N. M. Nemes et al., Phys. Rev. B 61 (2000) 7118.
11. S. Takahashi, H. Imamura, and S. Maekawa,
Phys. Rev. Lett. 82 (1999) 3911.
12. S. Takahashi et al, J. Appl. Phys. 87 (2000) 5227;
ibid. 89 (2001) 7505.
13. S. Takahashi, H. Imamura, and S. Maekawa,
Physica C 341-348 (2000) 1515.
14. K. Maki, Phys. Rev. B 8 (1973) 191; L. R.
Tagirov et al., J. Phys. F 17 (1987) 695.
15. Y. Yafet, Phys. Lett. A 98 (1983) 287.
16. H. L. Zhao and S. Hershfield, Phys. Rev. B 52
(1995) 3632. In their result, x(T) is missing in
Eq. @.