Controllable pi junction with magnetic nanostructures

0.16 MB·English

by  T. Yamashita

Controllable π junction with magnetic nanostructures
T. Yamashita,1 S. Takahashi,1,2 and S. Maekawa1,2
1Institute for Materials Research, Tohoku University, Sendai, Miyagi, 980-8577, Japan
2CREST, Japan Science and Technology Agency (JST), Kawaguchi, Saitama, 332-0012, Japan
(Dated: February 6, 2008)
6
WeproposeanovelJosephsondeviceinwhich0andπstatesarecontrolledbyanelectricalcurrent.
0
In this system, the π state appears in a superconductor/normal metal/superconductor junction
0
due to the non-local spin accumulation in the normal metal which is induced by spin injection
2
from a ferromagnetic electrode. Our proposal offers not only new possibilities for application of

n superconducting spin-electronic devices but also the in-depth understanding of the spin-dependent
a phenomenain magnetic nanostructures.
J

7 PACSnumbers: 74.50.+r,74.45.+c,85.25.Cp

]
n
Nowadays spin-electronics is one of the central topics a function of the control voltage has been demonstrated
o
in condensed matter physics [1, 2, 3]. There has been [22, 23].
c
- considerableinterest in the spin injection, accumulation, In this paper, we propose a new Josephson device in
r
p transport, and detection in ferromagnet/normal metal which the 0 and π states are controlled electrically. In
u (F/N) hybrid structures [4, 5, 6, 7, 8, 9]. Twenty years this device,spinaccumulationis generatedinanonmag-
s ago,JohnsonandSilsbeedemonstratedthespininjection neticmetalbythespin-polarizedbiascurrentflowinginto
.
t anddetectionin a F/N/Fstructure forthe firsttime [4]. the nonmagnetic metal from a ferromagnet. In a metal-
a
m Recently, spin accumulation has been observed at room licJosephsonjunctionconsistingofthespinaccumulated
temperature in all-metallic spin-valve geometry consist- nonmagnetic metal sandwiched by two superconductors,
-
d ing of a F/N/F junction by Jedema et al. [5]. In their the π state appears due to the spin split of the electro-
n system, the spin-polarized bias current is applied at one chemical potential in the nonmagnetic metal. The mag-
o
F/N junction, and the voltage is measured at another nitude of spin accumulation is proportional to the value
c
[ F/N interface for the parallel (P) and antiparallel (AP) ofthespin-polarizedbiascurrent,andthereforethestate
alignments of the F’s magnetizations. They have ob- of the Josephson junction is controlled by the current.
1
served the difference of the non-local voltages between Our proposal leads to an in-depth understanding of the
v
2 the P and AP alignments due to spin accumulation in spin-dependent phenomena in magnetic nanostructures
3 N. Also in a F/I/N/I/F (I indicates an insulator) struc- as well as new possibilities for the application of super-
1 ture,aclearevidenceofspinaccumulationinNhasbeen conducting spin-electronic devices.
1 shown [6]. In hybrid structures consisting of a ferromag- We consider a magnetic nanostructure with two su-
0
net and a superconductor (S), a suppression of the su- perconductors as shown in Fig. 1. The device consists
6
0 perconductivity due to spin accumulation in S has been of a nonmagnetic metal N (the width wN, the thick-
/ studied theoretically and experimentally [10, 11, 12]. ness d ) which is connected to a ferromagnetic metal
t N
a F (the width w , the thickness d ) at x = 0 and sand-
Furthermore, ferromagnetic Josephson (S/F/S) junc- F F
m
wiched by two superconductors S1, S2 located at x=L.
tions have been studied actively in recent years [13, 14,
-
d 15, 16, 17, 18]. In the S/F/S junctions, the pair poten-
n tial oscillates spatially due to the exchange interaction I
o in F [13, 14]. When the pair potentials in two S’s take I
c different sign, the direction of the Josephson current is (cid:1) (cid:2)(cid:3) (cid:4)
:
v reversed compared to that in ordinary Josephson junc- (cid:9)
i
X tions. This state is called the π state in contrast with
r the 0 state in ordinary Josephson junctions because the (cid:5)(cid:7) (cid:8) (cid:0)
a current-phase relation of the π state is shifted by “π” I
compared to that of the 0 state. The observations of (cid:10) (cid:2)(cid:11)
the π state have been reported in various systems ex-
perimentally [15, 16, 17, 18]. The applications of the π (cid:5)(cid:6)
statetothequantumcomputingalsohavebeenproposed
[19, 20, 21]. Another system to realize the π state is a FIG. 1: Structure of a controllable π junction with mag-
S/N/S junction with a voltage-control channel [22, 23]. netic nanostructures. The bias current I flows from a
In the system, the non-equilibrium electron distribution ferromagnet (F) to the left side of a normal metal (N).
inNinducedbythebiasvoltageplaysanimportantrole, The Josephson current IJ flows in a superconductor/normal
metal/superconductor (S1/N/S2) junction located at x=L.
and the sign reversalof the Josephsoncriticalcurrent as
2
In this device, the electrode F plays a role as a spin- (cid:17)(cid:16)(cid:22)(cid:26) (cid:27)(cid:28)(cid:29) (cid:30)(cid:31)
injector to the electrode N, and the S1/N/S2 junction
!"(cid:29) (cid:30)(cid:31)#
is a metallic Josephson junction. The spin-diffusion (cid:17)(cid:16)(cid:22)(cid:25)
length λ in N is much longer than the length λ in F
N F
[4, 5, 6, 7, 8], and we consider the structure with dimen-
(cid:17)(cid:16)(cid:22)(cid:24)
sions of λ (w ,d ) λ which is a realistic
F N(F) N(F) N
≪ ≪
geometry [5, 6].
(cid:17)(cid:16)(cid:22)(cid:23)
In the electrodes N and F, the electrical current with
spin σ is expressed as (cid:16) (cid:17) (cid:18) (cid:19) (cid:20) (cid:21)
(cid:14)(cid:13)(cid:15)
j = (σ /e) µ , (1) (cid:12)
σ − σ ∇ σ
where σ and µ are the electrical conductivity and FIG. 2: Spatial variation of the split of the electrochemical
σ σ
the electrochemical potential (ECP) for spin σ, respec- potentialinN.Thesolidlineisforthetunnel-limitcase(R≫
tively. Here ECP is defined as µ = ǫ +eφ, where ǫ ℜN,ℜF),thedashed,dotted,anddot-dashedlinesareforthe
is the chemical potential of electσrons σwith spin σ andσ metallic-limitcases(R=0)withr=ℜF/ℜN =0.01,0.1,and
0.2, respectively.
φ is the electric potential. From the continuity equa-
tion for charge, (j +j ) = 0, and that for spin,
↑ ↓
∇ ·
∇·(j↑−j↓) = e∂(n↑−n↓)/∂t (nσ is the carrier den- and δµN = (eλNIspin/2σNAN)e−|x|/λN. In the electrode
sity for spin σ), we obtain [8, 9] F, the spin split of ECP, δµσ, decays in the z-direction
F
because the thickness of F and the dimension of the in-
2(σ µ +σ µ )=0, (2)
∇ ↑ ↑ ↓ ↓ terface are much larger than the spin-diffusion length in
∇2(µ↑−µ↓)=(µ↑−µ↓)/λ2, (3) F (dF,wN,wF ≫λF) [8]. In a similar way to the case of
N, ECP in F is obtained from the continuity conditions
where λ = Dτ is the spin diffusion length with the
sf for charge and spin currents. ECP in F is expressed as
diffusion copnstant D = (N↑+N↓)/(N↑D↓−1 +N↓D↑−1) µσF(z) = µF +σδµσF, where µF = (eI/σFAJ)z+eV and
(Nσ and Dσ are the density of states and the diffusion δµσF = (eλF(pFI − Ispin)/2σFσAJ)e−z/λF with the con-
constantforspinσ,respectively)andthescatteringtime tact area A = w w , the voltage drop at the interface
J N F
of an electronτsf =2/(τ↑−↓1+τ↓−↑1) (τσσ¯ is the scattering V =(µF−µN)/e, and the polarization of the current in
time of an electron from spin σ to σ¯). In order to derive F, p = (σ↑ σ↓)/σ . The influence of the electrodes
Eqs. (2) and (3), we take the relaxation-time approxi- S1 aFnd S2 Fon−ECFP inF N may be neglected. When the
mation for the carrier density, ∂nσ/∂t= δnσ/τσσ¯, and superconducting gap in S1 and S2 is much larger than
−
use the relationsσσ =e2NσDσ andδnσ =Nσδǫσ, where the spin split δµN at x = L, almost no quasiparticle is
δnσ and δǫσ are the carrier density deviation from equi- excited above the gap at low temperature. Therefore,
librium and the shift in the chemical potential from its the spin current does not flow into S1 and S2, and the
equilibrium value for spin σ, respectively. In addition, behavior of ECP in N is not modified by the connection
the detailed balance equation N↑τ↑−↓1 = N↓τ↓−↑1 is also to the electrodes S1 and S2.
used. We use the notations σ = 2σ↑ = 2σ↓ in N and Inordertoobtaintherelationbetweenthebiascurrent
N N N
σFA=t tσhF↑e+inσteF↓rf(aσcF↑e6=beσtwF↓)eeinnNF hanerdeaFft,etrh.e interfacial cur- wIeansudbtshteitsuhtiefttohfeEoCbtPa,iδnµedN,µaσtathnedrµigσhftosridteheinexNp(rxes>sio0n)s,
N F
rent Iσ flows due to the difference of ECPs in N and of I and Ispin, and eliminate V. As a result, we obtain
F: Iσ = (Gσ/e)(µσF|z=0+ − µσN|z=0−), where Gσ is the the relation between I and Ispin, and finally we get the
spin-dependent interfacial conductance. We define the relation between I and δµN as follows:
interfacial charge and spin currents as I = I + I
↑ ↓
δµ (x)=
and I = I I , respectively. The spin-flip ef- N
spin ↑ ↓
−
fect at the interface is neglected for simplicity. In the PJ R pF F
+ ℜ
electrode N with the thickness and the contact dimen- 1 P2 (cid:18) (cid:19) 1 p2 (cid:18) (cid:19)
e I − J ℜN − F ℜN e−x/λN, (4)
sions being much smaller than the spin-diffusion length ℜN 2 R 2
F
(tdioNn,w[8N].,wTFhe≪chλaNrg)e, aµnσNd vsapriniescuornrleyntindetnhseitiexs dinireNc-, 1+ 1−PJ2 (cid:18)ℜN(cid:19)+ 1−p2F (cid:18)ℜℜN(cid:19)
j = j + j and j = j j , are derived from where = λ /(σ A ) and = λ /(σ A ) indicate
↑ ↓ spin ↑ ↓ N N N N F F F J
− ℜ ℜ
Eqs. (1) (3), and satisfy the continuity conditions at the non-equilibrium resistances of N and F, respectively,
the interf−ace: j = I/A and j = I /A , where R = G−1 = (G +G )−1 is the interfacial resistance,
N spin spin N ↑ ↓
A = w d is the cross-sectional area of N. From these and P = (G G )/G is the polarization of the in-
N N N J ↑ ↓
−
conditions, we obtain ECP in N, µσ(x) = µ +σδµ , terfacial current. When the F/N interface is the tunnel
N N N
where µ = (eI/σ A )x for x < 0, µ = 0 for x > 0, junction(R , ),Eq. (4)reducestoasimpleform
N N N N ≫ℜN ℜF
3
junction as follows [13, 14, 15, 16, 17, 18]: In the
1
S/F/S systems, Cooper pairs are formed by the An-
./ - dreev reflection of spin-σ electrons with the wave
%& ,
$ number kF (√2m/~)√E +σE and holes with
. 0 kF (√e2,σm/≈~)√E σE Fat theeexnergy E 0. In
h,σ ≈ F − ex ≈
the case that the exchange interaction is much weaker
than the Fermi energy (E E ), the stable state
ex F
≪
(0 or π) in the system depends on the dimensionless
’ () (+
* parameter α = (E /E )(k d ), where d is the
F ex F F F F
thickness of F and k is the Fermi wave number [17].
F
FIG. 3: Schematic diagram of energy vs. momentum in the At α =0 the system is in the 0 state, and the first 0-π
F
AndreevreflectionwhenthereisspinaccumulationinN.The transition occurs at α = π/2, and then the system is
F
filled and open circles represent an electron and a hole, re- in the π state at α = π [17]. Because the value of E
F ex
spectively. In N, the solid and dashed lines denote electron
is fixed in the S/F/S system, the 0 and π states change
andholebands,respectively,theshadedareaindicatesanoc-
periodically with the period 2π(E /E ) as a function
cupation by electrons. In the Andreev reflection, a spin-up F ex
of d . As a result, the d dependence of the Josephson
electron (a) injected into S’s captures another electron with F F
spin down (b),and aspin-uphole(b’) is reflected backtoN. critical current shows a cusp structure and the critical
current becomes minimum at the 0-π transition [16, 17].
In analogy with the case of the S/F/S junction dis-
cussed above, when there is spin accumulation in N
δµ (x)=(e IP /2)e−x/λN. On the other hand, when
N N J
ℜ as shown in Fig. 3, the 0 or π state is realized in
the F/N junction is of metallic contact (R = 0), Eq.
the S1/N/S2 junction depending on the parameter α =
(4) becomes δµN(x) = eℜNIpFre−x/λN/(2r+(1−p2F)), (δµ /E )(k w ). In this case, the width w is fixed,
where r = / is a mismatch factor of the resis- N F F N N
F N
ℜ ℜ and the 0 and π states are controlled through the value
tances in F and N. Figure 2 shows the spacial variation
ofδµ whichisproportionaltothebiascurrentI (seeEq.
of δµ (x) both for the tunnel- and metallic-limit cases N
N
(4)). The N part of the system is in the non-equilibrium
with P = 0.4 and p = 0.6 [1, 24]. As shown in this
J F
statebythespincurrentincontrastwithFintheequilib-
figure, in the case of the metallic contact, δµ becomes
N
rium state of the S/F/S junction. However,one can dis-
larger with decreasing the resistance mismatch [8].
cuss the critical current in the non-equilibrium S1/N/S2
Next we consider how spin accumulation affects the
junctioninthe samewayasthe equilibriumS/F/Sjunc-
JosephsoncurrentI flowing throughthe S1/N/S2junc-
J tion because the critical current is dominated by the en-
tionlocatedatx=L(Fig. 1). InthemetallicJosephson
ergy of the quasiparticles in N, not by the flow of the
junction, the Andreev bound state plays a key role for
current [22, 23].
the Josephson effect [17, 25]. The Andreev bound state
From the point of view of more detailed description,
is formed by a multiple Andreev reflectionof anelectron
the free energy in the system is obtained by the summa-
with the wave number k = (√2m/~)√E +E and a
e F tionofthe energyofthe Andreevbound states[19]. The
hole with k = (√2m/~)√E E, respectively, where
h F − bound state energyis calculatedfrom the Bogoliubov-de
E is the energy of the electron and hole measured from
Gennes equation [26], and the free energy is minimum
the Fermi energy E . As shown in Fig. 3, when there is
F for the phase difference 0 (π) for the 0 (π) state. In
the spin split δµ in N, a spin-up (-down) electron with
N the S1/N/S2 junction with no spin accumulation in N
the energy E δµ ( δµ ) is injected into S’s from
≈ N − N (δµN = 0), the bound states with the energy E > 0
N at low temperatures. The injected electron captures
contribute to the free energy. On the other hand, when
another electron with the energy E δµ (δµ ) from
≈− N N spinaccumulationexistsinN,thespin-up(-down)bound
the opposite spin band in order to form a Cooper pair
stateswiththeenergyE >δµ ( δµ )contributetothe
N N
in S’s. Therefore, a spin-up (-down) hole with energy −
free energy because ECP is shifted by δµ ( δµ ) in N.
N N
E δµ ( δµ ) is reflected back to N (Andreev reflec- −
≈ N − N The 0-π transition occurs due to the shift of the energy
tion [25]). In other words, the spin-up (-down) electron
region of the Andreev bound states which contribute to
with k (√2m/~) E +( )δµ and the spin-up (-
e ≈ F − N the free energy.
down) hole with kh p(√2m/~) EF (+)δµN mainly As an example, we consider the case that the F/N
≈ −
contribute to the formation of apCooper pair. Note that interface consists of a tunnel junction. The material pa-
the values of the wave numbers ke and kh differ due to rameters PJ = 0.4, ρN = σN−1 = 2µΩcm, λN = 1µm,
the spin split δµN in contrast with the case of no spin wN = 800nm, and dN = 10nm, which lead to N =
split (δµN =0) in which ke kh. 2.5Ω, are taken. The distance between F andℜS’s is
≈
The split δµ corresponds to the exchange en- taken to be L = 500nm. When no bias current is ap-
N
ergy E of a ferromagnet in a superconduc- plied between F and N (I = 0), the S1/N/S2 junction
ex
tor/ferromagnet/superconductor (S/F/S) Josephson is in the ordinary 0 state because there is no spin split
4
of ECP (δµ = 0). With increasing the bias current, supported by NAREGI Nanoscience Project, Ministry
N
themagnitudeoftheJosephsoncriticalcurrentdecreases of Education, Culture, Sports, Science and Technology
because the parameterα increases due to the increaseof (MEXT) of Japan, and by a Grant-in-Aid from MEXT
the spin split. When the bias current reaches the value and NEDO of Japan.
I =I 3mAwhichinducesthespinsplitδµ 1meV
0 N
≈ ≈
at x = 500nm, the parameter α π/2 and the first
≈
transition to the π state from the 0 state occurs (the
values of E = 5eV and k = 1˚A−1 are taken [27]).
F F
[1] Spin Dependent Transport in Magnetic Nanostructures,
As a result, the magnitude of the Josephsoncritical cur-
editedbyS.MaekawaandT.Shinjo(TaylorandFrancis,
rent takes its minimum at I =I , and increases with in-
0 London and New York,2002).
creasing the bias current I >I . When the bias current
0 [2] Concepts inSpinElectronics,editedbyS.Maekawa(Ox-
attains I = 2I0, the magnitude of the Josephson criti- ford Univ.Press, 2006).
cal current becomes maximum because of α π, and [3] I. Zˇuti´c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.
decreases with increasing the bias current I >≈2I . For 76, 323 (2004).
0
I =3I correspondingtoα 3π/2,thesecondtransition [4] M. Johnson and R.H.Silsbee, Phys. Rev.Lett. 55, 1790
0 ≈ (1985); M. Johnson, ibid. 70, 2142 (1993).
to the 0 state from the π state occurs.
[5] F.J.Jedema,A.T.Filip,andB.J.vanWees,Nature(Lon-
Here we discuss the effect of spin accumulation on the
don) 410, 345 (2001).
superconducting gap [10]. The spin split δµN at x = L [6] F.J. Jedema et al.,Nature (London) 416, 713 (2002).
in N causes the split of ECP of S’s by δµN near the S/N [7] Y.Otaniet al.,J.Magn.Magn. Mater.239,135(2002);
interfaces. The spin split in S’s decreases exponentially T. Kimura, J. Hamrle, and Y. Otani, Phys. Rev. B 72,
with the spin-diffusion length λ from the interface. In 014461 (2005).
S
thesuperconductors,thesuperconductinggapisnotsup- [8] S. Takahashi and S. Maekawa, Phys. Rev. B 67, 052409
(2003).
pressedbyspinaccumulationuntilδµ exceedsthecriti-
N
[9] P.C.vanSon,H.vanKempen,andP.Wyder,Phys.Rev.
calvalueofthespinsplitδµ [10]. Atlowtemperatures
Nc Lett. 58, 2271 (1987).
much lower than the superconducting critical tempera-
[10] S. Takahashi, H. Imamura, and S. Maekawa, Phys. Rev.
ture (T Tc), the critical value of the spin split is ob- Lett. 82, 3911 (1999).
≪
tained as δµNc . ∆0 by solving the gap equation [10], [11] V.A. Vas’koet al.,Phys.Rev. Lett.78, 1134 (1997).
where ∆ is the superconducting gap for δµ = 0 at [12] Z.W. Donget al., Appl.Phys.Lett. 71, 1718 (1997).
0 N
T = 0. In the case discussed in the above paragraph, [13] A.I.Buzdin,L.N.Bulaevskii, and S.V.Panyukov,JETP
Lett. 35, 178 (1982).
δµ 1meV at the first 0-π transition (α π/2). For
N
≈ ≈ [14] E.A. Demler, G.B. Arnold, M.R. Beasley, Phys. Rev. B
example, ∆ 1.5meV for niobium [28], and therefore
0 ≈ 55, 15174 (1997).
the superconducting gap is almost not affected by spin [15] V.V. Ryazanov et al.,Phys. Rev.Lett. 86, 2427 (2001).
accumulation at the first 0-π transition. When super- [16] T. Kontos et al., Phys. Rev. Lett. 86, 304 (2001); ibid.
conductors with the higher value of Tc, e.g., MgB2 (Tc 89, 137007 (2002).
39 K) [29] or High-T materials (T is several 10 K’s) [17] H. Sellier et al.,Phys. Rev.B 68, 054531 (2003).
c c
[≈28], are used as the electrodes S1 and S2, the supercon- [18] A. Bauer et al., Phys.Rev.Lett. 92, 217001 (2004).
[19] T.Yamashitaet al.,Phys.Rev.Lett.95,097001 (2005).
ductivity is robust even at the second (δµ 3meV,
N ≈ [20] T. Yamashita, S. Takahashi, and S. Maekawa,
α 3π/2) and higher 0-π transitions.
≈ cond-mat/0507199.
In summary, we have proposed the novel Josephson [21] L.B. Ioffe et al., Nature (London) 398, 679 (1999); G.
device in which the 0 and π states are controlled elec- Blatter, V.B. Geshkenbein, and L.B. Ioffe, Phys. Rev.B
trically. The spin split of the electrochemical potential 63, 174511 (2001).
is induced in the electrode N by the spin-polarized bias [22] J.J.A.Baselmansetal.,Nature(London)397,43(1999).
current flowing from F to N. The π state appears in the [23] J.J.A. Baselmans et al., Phys. Rev. Lett. 89, 207002
(2002);J.J.A.Baselmans,B.J.vanWees,andT.M.Klap-
S1/N/S2junctionduetothenon-localspinaccumulation
wijk, Phys. Rev.B 65, 224513 (2002).
inN.Becausethemagnitudeofspinaccumulationispro-
[24] J.BassandW.P.PrattJr.,J.Magn.Magn.Mater.200,
portional to the value of the spin-polarized bias current, 274 (2000); J. Bass and W.P. Pratt Jr., Physica B 321,
the 0 and π states of the Josephson junction are con- 1 (2002).
trolled by the current. Our proposal provides not only [25] A.F. Andreev,Sov. Phys.JETP 19, 1228 (1964).
new possibilities for the application of superconducting [26] P.G. de Gennes, Superconductivity of Metals and Alloys
spin-electronic devices but also the deeper understand- (W. A. Benjamin, NewYork, 1966), chap. 5.
[27] N. Ashcroft and N. Mermin, Solid State Physics (Saun-
ing of the spin-dependent phenomena in the magnetic
ders College Publishing, NewYork,1976).
nanostructures.
[28] C.Kittel,IntroductiontoSolidStatePhysics(JohnWiley
We are grateful to M. Mori and G. Montambaux for
& Sons, Inc.,New York,1996).
fruitful discussion. T.Y. was supported by JSPS Re- [29] J. Nagamatsu et al.,Nature(London) 410, 63 (2001).
search Fellowships for Young Scientists. This work was