Tunneling Hamiltonian description of the atomic-scale 0-pi transition in superconductor/ferromagnetic-insulator junctions

0.19 MB·English

by  Shiro Kawabata

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2 π
Tunneling Hamiltoniandescriptionoftheatomic-scale0- transition

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a insuperconductor/ferromagnetic-insulatorjunctions
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S. Kawabataa,b, Y. Tanakac, A. A. Golubovd, A. S. Vasenkoe, S. Kashiwayaf Y. Asanog
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aNanosystemResearch Institute(NRI), National Instituteof Advanced Industrial Scienceand Technology (AIST), Tsukuba,

] Ibaraki 305-8568, Japan
ll bCREST, Japan Scienceand Technology Corporation (JST), Kawaguchi, Saitama 332-0012, Japan
a cDepartment of Applied Physics, Nagoya University,Nagoya, 464-8603, Japan
h dFaculty of Science and Technology, Universityof Twente, P.O.Box 217, 7500 AE Enschede, The Netherlands
- eInstitut Laue-Langevin, 6rue Jules Horowitz, BP156, 38042, Grenoble, France
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e fNanoelectronics Research Institute(NeRI), AIST, Tsukuba, Ibaraki 305-8568, Japan
m gDepartment of Applied Physics, Hokkaido University,Sapporo, 060-8628, Japan
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Abstract
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WeshowaperturbationtheoryoftheJosephsontransportthroughferromagneticinsulators(FIs).Recentlywehave
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found that the appearance of the atomic scale 0-π transition in such junctions based on numerical calculations. In
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c order to explore the mechanism of this anomalous transition, we haveanalytically calculated theJosephson current
[ using thetunnelingHamiltonian theory and found that thespin dependentπ-phaseshift in theFI barrier gives the
atomic scale 0-π transition.
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v
6 Keywords: Josephson junction, Spointronics, Ferromagnetic insulator,Quantum computer, TunnelingHamiltonian method
0 PACS: 74.50.+r,03.65.Yz, 05.30.-d
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1. Introduction zanov et al. [5] and Kontos et al. [6] Until recently,
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1 however,investigationsontheπ junctionhavebeen
1 In the usual Josephson junctions at equilibrium mainly focusedonthe S/FM/Ssystems.
v: the phase difference of the superconducting order We havepredicteda possibility ofthe π-junction
i parameter on the two superconductor is zero. On formation in Josephson junctions through ferro-
X
the other hand, in the Josephson junctions with magnetic insulators (FIs) by numerically solving
ar ferromagnetic-metalinterlayer(S-MF-S junctions), the Bogoliubov-de Genne equation [7,8] and the
the groundstate may correspondto the phase dif- Nambu Green’s function [9–12]. The formation of
ference[1,2].TheJosephsonπ-junctionformationis the π junction using such an insulating barrier is
relatedwiththedampingoscillatorybehaviorofthe very promising for future qubit application [13–
Cooper pair wave function in a FM [3,4]. In terms 16] because of it’s low decoherence nature [17,18].
of the Josephson relationship I = I sinφ, where More importantly, we have shown that the ground
J C
φisthephasedifferencebetweenthetwosupercon- state of such junction alternates between 0- and π-
ductor layers, a transition from the 0 to π states stateswhenthicknessofFIisincreasingbyasingle
impliesachangeinsignofI frompositivetonega- atomic layer. In this paper in order to understand
C
tive.Experimentallytheexistenceoftheπ-junction the physical mechanism of this atomic scale 0-π
in S/FM/S systems has been confirmed by Ryan- transition, we analytically calculate the Josephson
Preprintsubmitted toElsevier 21January 2011
currentbasedonthetunnelingHamiltonianmethod In this paper we focused on the high-T d-wave
C
and show that the spin-dependent π-phase shift of junction witha FI barrier[Fig.1(b)].We note that
the tunneling matrix element of the FI layer is the thequalitativelysameresultcanbeobtainedforthe
originofthis anomaloustransition. caseofconventionals-wavejunctions.
2. Tunneling properties of a ferromagnetic
3. Theory
insulator
In this section, we analytically calculate the
In this section, we briefly describe the electronic
JosephsoncurrentofS/FI/Sjunctionsbasedonthe
structure of a representative of FI materials, i.e.,
tunneling Hamiltonian approach. Let us consider
La BaCuO (LBCO) [19]. The typical DOS of
2 5 a three-dimensional S/FI/S junction as shown in
LBCOforeachspindirectionisshownschematically
Fig. 1(b). The Hamiltonian of the system can be
inFig.1(a).TheexchangesplittingV isestimated
ex describedby
to be 0.34 eV by a first-principle band calculation
using the spin-polarized local density approxima- H =H +H +H +H , (3)
1 2 T Q
tion [20]. Since the exchange splitting is large and
the bands are originally half-filled, the system be- where H and H are Hamiltonians describing the
1 2
comes FI. Based on the band structure [Fig. 1(a)], d-wavesuperconductors:
thespinσ(=↑,↓)dependenttransmissioncoefficient
~2∇2
Tσ forthe FI barriercanbe calculatedas[7] H = dr ψ† (r) − −µ ψ (r)
1 1σ 2m 1σ
ρ t LF Xσ Z (cid:18) (cid:19)
Tσ =αLF (cid:18) gσ (cid:19) , (1) − 12 dr dr′ψ1†σ(r)ψ1†σ′(r′)
σ,σ′Z Z
byusingthetransfermatrixmethod,whereρ = X
↑(↓) ×g (r−r′)ψ (r′)ψ (r), (4)
−(+)1andα isaspin-independentcomplexcon- 1 1σ′ 1σ
LF
stant,tisthehoppingintegralintheFI,gisthegap where µ is the chemicalpotentialand ψ (ψ†) is the
betweentheup-anddown-spinband,andLF isthe fermionfieldoperatorandg(r−r′)istheattractive
layernumber ofthe FI.We immediately find
interaction.ThetunnelingHamiltonianwithaspin-
T↑ =(−1)LF, (2) dependent tunneling matrix element tσ of the FI
barrieris givenby
T
↓
sotherelativephaseofTσbetweenspinupanddown H = dr dr′ t (r,r′)ψ† (r)ψ (r′)
isπ (0)forthe odd(even)numberofL .Nextsec- T σ 1σ 2σ
F
tion,wewillcalculatetheJosephsoncurrentthrough Xσ Z Z h
+ h.c.], (5)
sucha FI analytically.
and
(Q −Q )2
1 2
H = (6)
Q
8C
is the chargingHamiltonian, where C is the capac-
itance of the junction and Q is the operator for
1(2)
the charge on the superconductor 1 (2), which can
be writtenas
Q =e drψ† (r)ψ (r). (7)
1(2) 1(2)σ 1(2)σ
σ Z
X
Fig. 1. (Color online) (a) The density of states for
By using the functional integralmethod [17]and
each spin direction for a ferromagnetic insulator, e.g.,
taking into account the spin dependence of t ex-
LBCO, and (b) schematic picture of c-axis stack high-Tc σ
d-wavesuperconductor/LBCO/high-Tcd-wavesuperconduc- plicitly,thegroundpartitionfunctionforthesystem
tor Josephson junction. canbe writtenasfollows
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Z = Dψ¯ Dψ Dψ¯ Dψ exp −1 ~βdτL(τ) , β(τ)=−~2 t∗↓(k,k′)t↑(k,k′)F1(k,τ)F2† k′,−τ .
1 1 2 2 ~
Z " Z0 # kX,k′ (cid:0) (cid:1)
(8) (15)
where β = 1/k T, ψ(ψ¯) is the Grassmann field The NambuGreenfunctions isgivenby
B
which corresponds to the fermionic field operator
~∆ (k)
[ψ(ψ†)]inEq.(2),andtheLagrangianLisgivenby F (k,ω )= i , (16)
i n (~ωn)2+ξk2 +∆i(k)2
L(τ)= drψ¯iσ(r,τ)∂τψiσ(r,τ)+H(τ). whereξk =~2k2/2m−µisthesingleparticleenergy
Xσ iX=1,2Z relativetotheFermisurfaceand~ωn =(2n+1)π/β
(9) istheMatsubarafrequency(nisaninteger).Inthe
case of the cuprate high-T superconductors with
In orderto removethe ψ4 term in the Hamiltonian c
the dx2−y2 symmetry [21], the order parameter is
H(τ), we will use the Hubbard-Stratonovichtrans-
givenby
formationwhichintroducesacomplexpairpotential
field ∆ (k)=∆ cos2θ. (17)
i 0
∆(r,r′;τ)=|∆(r,r′;τ)|exp[iφ(r,r′;τ)]. (10) Below we will calculate the Josephson critical cur-
rent I analytically and discuss the possibility of
C
Theresultingactionisonlyquadraticinthe Grass- the atomicscale0-π transition.
mannfield, sothat the functional integraloverthis
number can readily be performed explicitly. The
4. Josephsoncritical current
functionalintegraloverthe modulusofthepairpo-
tential field is taken by the saddle-point method.
The Josephson critical current I for a c-axis d-
Then the partition function is reduced to a single C
wave junction through the ferromagnetic insulator
functional integral over the phase difference φ =
[Fig.1(b)]canbe expressedas
φ −φ . To second order in the tunneling matrix
1 2
element,onefinds 2e2
I = t∗(k,k′)t (k,k′)F(k,ω )F(k′,ω )
C ~β ↓ ↑ n n
Z = Dφ(τ)exp −Seff[φ] , (11) kX,k′Xωn
~
(18)
Z (cid:20) (cid:21)
wherethe effectiveactionisgivenby
by assuming ∆ = ∆ = ∆ cos2θ and thus F =
1 2 0 1
F = F. We also assume that the tunneling matrix
Seff[φ]=Z0~βdτ"C2 (cid:18)2~e∂φ∂(ττ)(cid:19)2− 2~eICcosφ(τ)#. ehle2ermenetnttutnσn(kel,ikng′)in∝wThσicihs tghiveemnoinmteenrtmums okfkthpearcaol--
(12) leltothelayerisconserved[22,23]andhasthesame
L dependenceonthetransmissioncoefficientT as
F σ
In the calculation we have ignored the irrelevant Eq.(1)[7],i.e.,
quasiparticletunneling termforsimplicity.Here
2e ~β t∗↓(k,k′)t↑(k,k′)=|t0|2(−1)LFδkkk′k, (19)
IC =− ~ dτβ(τ) (13) we obtain an analytical expression of IC for T = 0
Z0 K as
is the Josephson critical current, and then the
∆ G
Josephsoncurrentis givenby IC =(−1)LF 0 N, (20)
2πe
I (φ)=I sinφ, (14) where
J C
4πe2
wherethenegative(positive)IC correspondstothe GN = ~ |t0|2N02, (21)
π (0) junction. The Josephson kernel β(τ) is given
intermsoftheoff-diagonalNambuGreen’sfunction isthenormalconductancewithN beingthedensity
0
fortwosuperconductorF (i=1,2), i.e., of states at E . The sign of I becomes negative
i F C
3
forthe oddnumberofLF andpositiveforthe even [14] S. Kawabata, S. Kashiwaya, Y. Asano, Y. Tanaka, A.
number of L as was numerically found in [7–12]. A.Golubov, Phys. Rev. B 74(2006) 180502(R).
F
Therefore, the spin dependent π-phase shift of the [15] S.Kawabata,A.A.Golubov,PhysicaE40(2007)386.
[16] S. Kawabata, Y. Asano, Y. Tanaka, S. Kashiwaya, A.
tunneling matrix element t in the FI barrier gives
σ A.Golubov, Physica C468(2008) 701.
riseto the atomicscale0-π transition. [17] G.Sch¨on, A.D.Zaikin,Phys.Reports 198(1990) 237.
[18] T. Kato, A. A. Golubov, Y. Nakamura, Phys. Rev. B
76(2007) 172502.
5. Summary
[19] F. Mizuno, H. Masuda, I. Hirabayashi, S. Tanaka, M.
Hasegawa, U.Mizutani, Nature 345(1990) 788.
To summarize, we have studied the Josephson [20] V.Eyert,K.H.H¨oc,P.S.Riseborough,Europhys.Lett.
31(1995) 385.
effect in S/FI/S junction by use of the tunneling
[21] S.KashiwayaandY.Tanaka,Rep.Prog.Phys.63(2000)
Hamiltonian method. We have analytically calcu-
1641.
lated the Josephson current and showed the possi- [22] S.Kawabata,S.Kashiwaya,Y.Asano,Y.Tanaka,Phys.
bility ofthe formationofthe atomicscale0-π tran- Rev. B 70(2004) 132505.
sition in such systems. This observation is consis- [23] T.Yokoyama,S.Kawabata, T.Kato,Y.Tanaka, Phys.
Rev. B 76(2007) 134501.
tentwithpreviousnumericallyresults.Wehopthat
suchFIbasedπ-junctionsbecomeanelementinthe
architectureofquantuminformationdevices.
Acknowledgements
This workwas supported by CREST-JST,and a
Grant-in-Aid for Scientific Research from the Min-
istry of Education, Science, Sports and Culture of
Japan(GrantNo.22710096).
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