- Email: [email protected]
ferromagnet junction
ARTICLE IN PRESS Journal of Physics and Chemistry of Solids 69 (2008) 3257– 3260
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Proximity effects in a superconductor/ferromagnet junction S. Hikino a,, S. Takahashi a,b, M. Mori a, S. Maekawa a,b a b
Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan CREST, Japan Science and Technology Agency (JST), Kawaguchi 332-0012, Japan
a r t i c l e in fo
PACS: 74.45.þc 74.50.þr Keywords: A. Superconductor
abstract A proximity effect in an s-wave superconductor/ferromagnet (SC/F) junction is theoretically studied using the second order perturbation theory for the tunneling Hamiltonian and Green’s function method. We calculate a pair amplitude induced by the proximity effect in a weak ferromagnetic metal (FM) and a half-metal (HM). In the SC/FM junction, it is found that a spin-singlet pair amplitude (Cs ) and spin-triplet pair amplitude (Ct ) are induced in FM and both amplitudes depend on the frequency in the Matsubara representation. Cs is an even function and Ct is an odd function with respect to the Matsubara frequency (on ). In the SC/HM junction, we examine the proximity effects by taking account of magnon excitations in HM. It is found that the triplet-pair correlation is induced in HM. The induced pair amplitude in HM shows a damped oscillation as a function of the position and contains the terms of even and odd functions of on as in the case of the SC/FM junction. We discuss that in our tunneling model the pair amplitude of even function of on only contributes to a Josephson current. & 2008 Elsevier Ltd. All rights reserved.
1. Introduction Rich physics has been found in various experiments of heterostructures with superconductors (SCs). When a normal metal (NM) is in contact with an s-wave SC, the spin-singlet superconducting pair amplitude (Cs ) penetrates into NM and shows an exponential decay as a function of the distance from the interface. This is called the proximity effect, which is important to realize the Josephson effect. In Josephson junctions with a weak ferromagnet (FM), the Josephson critical current oscillates with the thickness (d) of the FM due to the pair amplitude oscillation induced in the FM by the proximity effect [1,2]. The proximity effect in a superconductor/ferromagnet (SC/FM) junction has been studied in terms of the symmetry of the pair amplitude using the quasiclassical method or the Bogoliubov–de Gennes equation [3–5]. It was shown that not only a spin-singlet pair amplitude but also a spin-triplet pair amplitude are induced in the F, although the SC is an s-wave one. Moreover, the spin-triplet pair amplitude is an odd function with a frequency, which is different from the spin-singlet pair amplitude. We have also obtained the similar results, which will be discussed in the following. In previous works, we studied the Josephson effect in SC/FM/SC and SC/HM/ SC junctions using the tunneling Hamiltonian method, where HM is a half-metal ferromagnet [6–9]. It is important to study the nature of the penetrating superconducting pair amplitude in FM and HM. Corresponding author. Tel.: +81 22 215 2009.
E-mail address: [email protected] (S. Hikino). 0022-3697/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2008.06.063
In this paper, we study the proximity effect in the SC/FM and SC/HM junctions using tunneling Hamiltonian method. This method does not need boundary conditions used in the quasiclassical method at the interface.
2. Model and formulation We examine the proximity effect in hybrid structures of SC/FM and SC/HM junctions. The SC and FM are connected by tunneling Hamiltonian. The total Hamiltonian in the SC/FM junction is written as HSF ¼ HSC þ HFM þ HFM t ,
(1)
and the total Hamiltonian in the SC/HM junction is written as HSH ¼ HSC þ HHM þ HHM t , HSC ¼
(2)
Z
y ^ dr cSC;s ðrÞxc SC;s ðrÞ Z y y dr cSC;" ðrÞcSC;# ðrÞ þ h:c:; þD r2SC
(3)
r2SC
HFM ¼
Z
HHM ¼
dr cFM;s ðrÞ½x^ shex þ UðrÞcFM;s ðrÞ, y
r2FM
Z r2HM
y dr cHM;" ðrÞ½x^ þ UðrÞcHM;" ðrÞ,
(4)
(5)
ARTICLE IN PRESS 3258
S. Hikino et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3257–3260
HFM ¼ t ¼ HHM t
Z
y
r2SC;r0 2FM
dr dr0 t 0 ðr; r0 ÞcSC;s ðrÞcFM;s ðr0 Þ þ h:c:;
Z
y
r2SC;r0 2HM
dr dr0 t 0 ðr; r0 ÞcSC;" ðrÞcHM;" ðr0 Þ
Z
y
þ r2SC;r0 2HM
x^ ¼
(6)
dr dr0 t 1 ðr; r0 ÞSþ ðr0 ÞcSC;" ðrÞcHM;" ðr0 Þ þ h:c:;
(7)
where CFM and CFM are the spin-singlet and spin-triplet pair s t amplitudes, respectively. The second order perturbation in the tunneling Hamiltonian yields the pair amplitudes, ! D r r FM 2 Cs ðr; on Þ / t0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos exp þ , (10) xFM xFM o2n þ D2
1 2 5 m, 2m
(8)
where ci;s ðrÞ is the field operator of an electron with spin s at position r for i ði ¼ SC; FM; HMÞ, HBCS is the BCS mean field Hamiltonian in SC, and D is a superconducting gap. HFM and HHM are the Hamiltonians of FM and HM, respectively, which include non-magnetic impurities. hex is the magnetic exchange energy in FM, and UðrÞ is a non-magnetic impurity potential. HFM t is the tunneling Hamiltonian in the SC/FM junction, in which t 0 is a tunnel matrix element without spin flip. HHM is the tunneling t Hamiltonian in the SC/HM junction, in which t 1 is a tunnel matrix element with spin flip accompanied with a magnon. Only the spin-flip tunneling due to the magnon excitation is allowed in the SC/HM junction. Sþ ¼ Sx þ iSy is a spin operator of a local moment in HM. Sx and Sy are the x and y components of the local spin operator, respectively. m is the mass of an electron and m is the chemical potential. Here we make use of the fact that the tunneling matrix element differs from 0 only for rr0 in each boundary of the SC/FM and SC/HM junction, i.e., t 0ð1Þ ðr; r0 Þ ¼ t 0ð1Þ dðr r0 Þdðr RÞ, where R is the position of the interface between SC and FM or SC and HM. The essence of the proximity effect is a non-vanishing pair amplitude penetrated in a normal metal which is in contact with a SC. In a tunneling Hamiltonian method, the proximity effect is described by the proximity diagram as shown in Fig. 1 [10], where the up and down arrows represent up and down spin electrons, respectively. The proximity diagram means that the Cooper pair in SC tunnels to FM, where the two solid lines in FM represent the correlated propagation of up- and down-spin electrons which are scattered by impurities to form a diffusive motion of Cooperon in FM. First, we calculate the pair amplitude induced in FM in the SC/FM junction using the proximity diagram of Fig. 1. The pair amplitude in FM is defined by
!
D r r 2 CFM exp þ , t ðr; on Þ / i sgnðon Þt 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin x 2 x 2 FM FM on þ D
(11)
with
x FM
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u DFM u ¼ tqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 on þ h2ex on
(12)
where DFM is the diffusion constant in FM, and on ¼ ð2n þ 1ÞpT is the Matsubara frequency. In the casepofffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hex bo ffi n , the decay length of þ Þ, so CFM and CFM are short ðxFM DFM =h ex s t ffi that both pair pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi amplitudes rapidly decrease as expðr hex =DFM Þ with increasing hex . In a completely spin-polarized ferromagnetic metal (HM), the proximity effect does not occurs because the spin-singlet Cooper pair cannot tunnel into HM. In the SC/HM junction, a new type of proximity effect arises from the magnon assisted tunneling at the interface. The pair amplitude in HM is induced by the spin-flip scattering at the interface and the magnon excitation in HM. We introduce the pair amplitudes induced in HM, 0 1 CHM ðr; t2 t1 Þ ¼ 2hT t cHM" ðr; tÞcHM" ðr; t Þ
½Sþ ðr; t1 Þ ðSþ ðr; t2 Þi,
(13)
which express the expectation value of the composite operator [11] at distance r from the interface and correspond to the proximity diagram shown in Fig. 2, where the wavy line represents the magnon propagation and the two solid lines represent the correlated propagation of up-spin electrons which are scattered by impurities to form a diffusive motion of a Cooperon in HM. A spin-triplet Cooper pair in HM couples with a magnon to construct a spin-singlet composite state. The magnon assisted pair amplitude is induced in HM and this is a new type proximity effect. The second order perturbation yields the pair
0 0 1 CFM sðtÞ ðr; t t Þ ¼ 2½hT t cFM;" ðr; tÞcFM;# ðr; t Þi
hT t cFM;# ðr; tÞcFM;" ðr; t0 Þi,
(9)
t0 t0
t1 t0
SC
FM
Fig. 1. Proximity diagram in the second order perturbation. t 0 is the matrix element of non-spin-flip tunneling. Up and down arrows represent up- and downspin electrons, respectively.
SC
HM
Fig. 2. Proximity diagram in the second order perturbation. t 0 is the matrix element of non-spin-flip tunneling and t 1 is that of spin flip tunneling with magnon excitation. The wavy lines represent the magnon propagator. Up and down arrows represent up- and down-spin electrons, respectively.
ARTICLE IN PRESS S. Hikino et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3257–3260
amplitudes,
DZ ‘¼1
cos
r
x HM
exp
0.6
!
r
xþ HM
ðf þ þ f Þ,
FM
Ψs (14)
CHM ðr; on Þ / t 0 t 1
1 DT X
DZ ‘¼1
sin
r
x HM
exp
r
xþ HM
! ðf þ f Þ,
(15)
with 2
f
3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1 1 j2on n‘ j 6 7 ¼ 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 exp r , DHM 2 2 2 2 on þ D ðon n‘ Þ þ D
Ψ in FM [arb. unit]
CHM þ ðr; on Þ / t 0 t 1
1 DT X
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u 2xM ; ¼u tqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2‘ þ 20 0
xM ¼
pffiffiffiffiffiffiffiffiffiffiffi Z=D0 ,
0.2 FM
Ψt -0.2
(16)
x HM
3259
-5
(17)
where DHM , Z, 0 , and n‘ ¼ 2‘pT are the diffusion constant in HM, the stiffness of spin wave, the gap of spin wave excitation, and the Matsubara frequency of boson, respectively.
0
5
ωn/Δ0
Fig. 3. The pair amplitude C in FM as a function of the Matsubara frequency, on . D0 is a superconducting gap at T ¼ 0.
HM
3. Results and discussion
Ψ in HM [arb. unit]
In the SC/FM junction, it is found from Eqs. (10) and (11) that the pair amplitudes depend on the position r in FM and frequency on. The spin-singlet pair amplitude CFM shows the oscillation of s cosine function as a function of r and is an even function of on (this pair-amplitude is called an even frequency pairing function). On the other hand, although SC is an s-wave superconductor, the spin-triplet pair amplitude CFM has a finite value in FM due to hex , t and shows the oscillation of sine function as a function of r and is an odd function of on due to sgnðon Þ differently from CFM (this s amplitude is called an odd frequency pairing function). If we take hex ¼ 0, CFM is identically zero as seen in Eq. (11). This situation t corresponds to the SC/NM junction case, where NM represents a normal metal. Fig. 3 plots the on -dependence of CFM and CFM for s t ~ FM ¼ k2 DFM =D0 ¼ 10 000; T=T SC ¼ 0:01, the parameters, kF r ¼ 5; D F hex =D0 ¼ 100, where kF is the Fermi wave length, and T SC is the superconducting transition temperature. In the SC/HM junction, it is found from Eqs. (14) and 15 that the pair amplitudes depend on r in HM HM and on , and that CHM þ and C are an odd and even functions with respect to on , respectively. Fig. 4 shows the on -dependence of HM CHM þ and C , where we choose the following parameters, kF r ¼ 2, ~ xM ¼ 10; DHM ¼ k2F DHM =D0 ¼ 10 000; T=T SC ¼ 0:01 in numerical calculation. It is seen that the amplitude of CHM is much larger HM than that of CHM compared with CHM þ . The small values of Cþ come from the term ðf þ f Þ in Eq. (16). Here, we discuss the relation between the Josephson effect and the proximity effect in the tunneling Hamiltonian method. As mentioned in the Introduction, the Josephson effect occurs due to the pair amplitude penetrated from one SC to the other SC through a normal conducting metal sandwiched by SCs by the proximity effect. Therefore, the nature of the pair amplitude is reflected in that of the Josephson effect. In a previous work, we studied the Josephson effect in SC/FM/SC and SC/HM/SC junctions in the tunneling Hamiltonian method [6–9]. Using these results, we discuss the relation between the Josephson effect and the proximity effect. From Refs. [6–8], the Josephson critical current in the SC/FM/SC and SC/HM/SC junctions is given by IFM / c þ þ expðd=xFM Þ cosðd=xFM Þ and IHM / expðd=xHM Þ cosðd=xHM Þ, rec spectively. Comparing these Josephson critical currents with the pair amplitudes in the SC/FM and SC/HM junctions, it is found that
Ψ+
HM
Ψ− 0
× 100 -5
0
ωn/Δ0
5
Fig. 4. The pair amplitude C in HM as a function of the Matsubara frequency, on . D0 is a superconducting gap at T ¼ 0.
the even frequency pairs CFM and CHM contribute to the s Josephson effect in the SC/FM/SC and SC/HM/SC junctions, respectively, because of the same behavior in their spatial dependence. Therefore, the odd frequency component of the pair amplitude does not contribute to the Josephson effect within present tunneling models. In summary, we have studied the proximity effect in the SC/FM and SC/HM junctions using the tunneling Hamiltonian method. In the SC/FM junction, it is found that the pair amplitude CFM and s CFM are induced in FM and both amplitudes depend on the t position r in FM and the Matsubara frequency on. We have obtained that CFM is the even function and CFM is the odd s t function of on . In the SC/HM junction, the spin-flip tunneling with the magnon excitation at the interface is crucial for including the HM proximity effect. The induced pair amplitude CHM þ and C show the damped oscillation as a function of r. It is found that CHM þ and CHM are the even and odd functions of on , respectively. Moreover, we have investigated the effect of even and odd frequency components on the Josephson effect in the SC/FM/SC and SC/HM/SC junctions. As a result, the even frequency component
ARTICLE IN PRESS 3260
S. Hikino et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3257–3260
only contributes to the Josephson effect in the both cases within present models.
Acknowledgment This work is supported by a Grant in Aid for Scientific Research from MEXT, the Next Generation Supercomputer Project of MEXT. References [1] A.A. Golubov, M.Yu. Kupriyanov, E. Ilfichev, Rev. Mod. Phys. 76 (2004) 411. [2] A.I. Buzdin, Rev. Mod. Phys. 77 (2005) 935.
[3] F.S. Bergeret, A.F. Volkov, K.B. Efetov, Rev. Mod. Phys. 77 (2005) 1321. [4] M. Eschrig, T. Lo¨fwander, Th. Champel, J.C. Cuevas, G. Scho¨n, J. Low. Temp. Phys. 147 (2007) 457. [5] Y. Asano, Y. Tanaka, A.A. Golubov, Phys. Rev. Lett. 98 (2007) 107002. [6] S. Hikino, M. Mori, S. Takahashi, S. Maekawa, J. Magn. Magn. Mater. 310 (2007) 2456. [7] M. Mori, S. Hikino, S. Takahashi, S. Maekawa, J. Phys. Soc. Jpn. 76 (2007) 054705. [8] S. Hikino, S. Takahashi, M. Mori, J. Martinek, S. Maekawa, Physica C 463–465 (2007) 198–201. [9] S. Takahashi, S. Hikino, M. Mori, J. Martinek, S. Maekawa, Phys. Rev. Lett. 99 (2007) 057003. [10] T. Dittrich, P. Ha¨nggi, G.-L. Ingold, B. Kramer, G. Scho¨n, W. Zwerger, Quantum Transport and Dissipation, Wiley-VCH Verlag GmbH, Weinheim. [11] E. Abrahams, A. Balatsky, D.J. Scalapino, J.R. Schrieffer, Phys. Rev. B 54 (1995) 1271.
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Proximity effects in a superconductor/ferromagnet junction S. Hikino a,, S. Takahashi a,b, M. Mori a, S. Maekawa a,b a b
Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan CREST, Japan Science and Technology Agency (JST), Kawaguchi 332-0012, Japan
a r t i c l e in fo
PACS: 74.45.þc 74.50.þr Keywords: A. Superconductor
abstract A proximity effect in an s-wave superconductor/ferromagnet (SC/F) junction is theoretically studied using the second order perturbation theory for the tunneling Hamiltonian and Green’s function method. We calculate a pair amplitude induced by the proximity effect in a weak ferromagnetic metal (FM) and a half-metal (HM). In the SC/FM junction, it is found that a spin-singlet pair amplitude (Cs ) and spin-triplet pair amplitude (Ct ) are induced in FM and both amplitudes depend on the frequency in the Matsubara representation. Cs is an even function and Ct is an odd function with respect to the Matsubara frequency (on ). In the SC/HM junction, we examine the proximity effects by taking account of magnon excitations in HM. It is found that the triplet-pair correlation is induced in HM. The induced pair amplitude in HM shows a damped oscillation as a function of the position and contains the terms of even and odd functions of on as in the case of the SC/FM junction. We discuss that in our tunneling model the pair amplitude of even function of on only contributes to a Josephson current. & 2008 Elsevier Ltd. All rights reserved.
1. Introduction Rich physics has been found in various experiments of heterostructures with superconductors (SCs). When a normal metal (NM) is in contact with an s-wave SC, the spin-singlet superconducting pair amplitude (Cs ) penetrates into NM and shows an exponential decay as a function of the distance from the interface. This is called the proximity effect, which is important to realize the Josephson effect. In Josephson junctions with a weak ferromagnet (FM), the Josephson critical current oscillates with the thickness (d) of the FM due to the pair amplitude oscillation induced in the FM by the proximity effect [1,2]. The proximity effect in a superconductor/ferromagnet (SC/FM) junction has been studied in terms of the symmetry of the pair amplitude using the quasiclassical method or the Bogoliubov–de Gennes equation [3–5]. It was shown that not only a spin-singlet pair amplitude but also a spin-triplet pair amplitude are induced in the F, although the SC is an s-wave one. Moreover, the spin-triplet pair amplitude is an odd function with a frequency, which is different from the spin-singlet pair amplitude. We have also obtained the similar results, which will be discussed in the following. In previous works, we studied the Josephson effect in SC/FM/SC and SC/HM/ SC junctions using the tunneling Hamiltonian method, where HM is a half-metal ferromagnet [6–9]. It is important to study the nature of the penetrating superconducting pair amplitude in FM and HM. Corresponding author. Tel.: +81 22 215 2009.
E-mail address: [email protected] (S. Hikino). 0022-3697/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2008.06.063
In this paper, we study the proximity effect in the SC/FM and SC/HM junctions using tunneling Hamiltonian method. This method does not need boundary conditions used in the quasiclassical method at the interface.
2. Model and formulation We examine the proximity effect in hybrid structures of SC/FM and SC/HM junctions. The SC and FM are connected by tunneling Hamiltonian. The total Hamiltonian in the SC/FM junction is written as HSF ¼ HSC þ HFM þ HFM t ,
(1)
and the total Hamiltonian in the SC/HM junction is written as HSH ¼ HSC þ HHM þ HHM t , HSC ¼
(2)
Z
y ^ dr cSC;s ðrÞxc SC;s ðrÞ Z y y dr cSC;" ðrÞcSC;# ðrÞ þ h:c:; þD r2SC
(3)
r2SC
HFM ¼
Z
HHM ¼
dr cFM;s ðrÞ½x^ shex þ UðrÞcFM;s ðrÞ, y
r2FM
Z r2HM
y dr cHM;" ðrÞ½x^ þ UðrÞcHM;" ðrÞ,
(4)
(5)
ARTICLE IN PRESS 3258
S. Hikino et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3257–3260
HFM ¼ t ¼ HHM t
Z
y
r2SC;r0 2FM
dr dr0 t 0 ðr; r0 ÞcSC;s ðrÞcFM;s ðr0 Þ þ h:c:;
Z
y
r2SC;r0 2HM
dr dr0 t 0 ðr; r0 ÞcSC;" ðrÞcHM;" ðr0 Þ
Z
y
þ r2SC;r0 2HM
x^ ¼
(6)
dr dr0 t 1 ðr; r0 ÞSþ ðr0 ÞcSC;" ðrÞcHM;" ðr0 Þ þ h:c:;
(7)
where CFM and CFM are the spin-singlet and spin-triplet pair s t amplitudes, respectively. The second order perturbation in the tunneling Hamiltonian yields the pair amplitudes, ! D r r FM 2 Cs ðr; on Þ / t0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos exp þ , (10) xFM xFM o2n þ D2
1 2 5 m, 2m
(8)
where ci;s ðrÞ is the field operator of an electron with spin s at position r for i ði ¼ SC; FM; HMÞ, HBCS is the BCS mean field Hamiltonian in SC, and D is a superconducting gap. HFM and HHM are the Hamiltonians of FM and HM, respectively, which include non-magnetic impurities. hex is the magnetic exchange energy in FM, and UðrÞ is a non-magnetic impurity potential. HFM t is the tunneling Hamiltonian in the SC/FM junction, in which t 0 is a tunnel matrix element without spin flip. HHM is the tunneling t Hamiltonian in the SC/HM junction, in which t 1 is a tunnel matrix element with spin flip accompanied with a magnon. Only the spin-flip tunneling due to the magnon excitation is allowed in the SC/HM junction. Sþ ¼ Sx þ iSy is a spin operator of a local moment in HM. Sx and Sy are the x and y components of the local spin operator, respectively. m is the mass of an electron and m is the chemical potential. Here we make use of the fact that the tunneling matrix element differs from 0 only for rr0 in each boundary of the SC/FM and SC/HM junction, i.e., t 0ð1Þ ðr; r0 Þ ¼ t 0ð1Þ dðr r0 Þdðr RÞ, where R is the position of the interface between SC and FM or SC and HM. The essence of the proximity effect is a non-vanishing pair amplitude penetrated in a normal metal which is in contact with a SC. In a tunneling Hamiltonian method, the proximity effect is described by the proximity diagram as shown in Fig. 1 [10], where the up and down arrows represent up and down spin electrons, respectively. The proximity diagram means that the Cooper pair in SC tunnels to FM, where the two solid lines in FM represent the correlated propagation of up- and down-spin electrons which are scattered by impurities to form a diffusive motion of Cooperon in FM. First, we calculate the pair amplitude induced in FM in the SC/FM junction using the proximity diagram of Fig. 1. The pair amplitude in FM is defined by
!
D r r 2 CFM exp þ , t ðr; on Þ / i sgnðon Þt 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin x 2 x 2 FM FM on þ D
(11)
with
x FM
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u DFM u ¼ tqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2 on þ h2ex on
(12)
where DFM is the diffusion constant in FM, and on ¼ ð2n þ 1ÞpT is the Matsubara frequency. In the casepofffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hex bo ffi n , the decay length of þ Þ, so CFM and CFM are short ðxFM DFM =h ex s t ffi that both pair pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi amplitudes rapidly decrease as expðr hex =DFM Þ with increasing hex . In a completely spin-polarized ferromagnetic metal (HM), the proximity effect does not occurs because the spin-singlet Cooper pair cannot tunnel into HM. In the SC/HM junction, a new type of proximity effect arises from the magnon assisted tunneling at the interface. The pair amplitude in HM is induced by the spin-flip scattering at the interface and the magnon excitation in HM. We introduce the pair amplitudes induced in HM, 0 1 CHM ðr; t2 t1 Þ ¼ 2hT t cHM" ðr; tÞcHM" ðr; t Þ
½Sþ ðr; t1 Þ ðSþ ðr; t2 Þi,
(13)
which express the expectation value of the composite operator [11] at distance r from the interface and correspond to the proximity diagram shown in Fig. 2, where the wavy line represents the magnon propagation and the two solid lines represent the correlated propagation of up-spin electrons which are scattered by impurities to form a diffusive motion of a Cooperon in HM. A spin-triplet Cooper pair in HM couples with a magnon to construct a spin-singlet composite state. The magnon assisted pair amplitude is induced in HM and this is a new type proximity effect. The second order perturbation yields the pair
0 0 1 CFM sðtÞ ðr; t t Þ ¼ 2½hT t cFM;" ðr; tÞcFM;# ðr; t Þi
hT t cFM;# ðr; tÞcFM;" ðr; t0 Þi,
(9)
t0 t0
t1 t0
SC
FM
Fig. 1. Proximity diagram in the second order perturbation. t 0 is the matrix element of non-spin-flip tunneling. Up and down arrows represent up- and downspin electrons, respectively.
SC
HM
Fig. 2. Proximity diagram in the second order perturbation. t 0 is the matrix element of non-spin-flip tunneling and t 1 is that of spin flip tunneling with magnon excitation. The wavy lines represent the magnon propagator. Up and down arrows represent up- and down-spin electrons, respectively.
ARTICLE IN PRESS S. Hikino et al. / Journal of Physics and Chemistry of Solids 69 (2008) 3257–3260
amplitudes,
DZ ‘¼1
cos
r
x HM
exp
0.6
!
r
xþ HM
ðf þ þ f Þ,
FM
Ψs (14)
CHM ðr; on Þ / t 0 t 1
1 DT X
DZ ‘¼1
sin
r
x HM
exp
r
xþ HM
! ðf þ f Þ,
(15)
with 2
f
3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1 1 j2on n‘ j 6 7 ¼ 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 exp r , DHM 2 2 2 2 on þ D ðon n‘ Þ þ D
Ψ in FM [arb. unit]
CHM þ ðr; on Þ / t 0 t 1
1 DT X
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u 2xM ; ¼u tqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2‘ þ 20 0
xM ¼
pffiffiffiffiffiffiffiffiffiffiffi Z=D0 ,
0.2 FM
Ψt -0.2
(16)
x HM
3259
-5
(17)
where DHM , Z, 0 , and n‘ ¼ 2‘pT are the diffusion constant in HM, the stiffness of spin wave, the gap of spin wave excitation, and the Matsubara frequency of boson, respectively.
0
5
ωn/Δ0
Fig. 3. The pair amplitude C in FM as a function of the Matsubara frequency, on . D0 is a superconducting gap at T ¼ 0.
HM
3. Results and discussion
Ψ in HM [arb. unit]
In the SC/FM junction, it is found from Eqs. (10) and (11) that the pair amplitudes depend on the position r in FM and frequency on. The spin-singlet pair amplitude CFM shows the oscillation of s cosine function as a function of r and is an even function of on (this pair-amplitude is called an even frequency pairing function). On the other hand, although SC is an s-wave superconductor, the spin-triplet pair amplitude CFM has a finite value in FM due to hex , t and shows the oscillation of sine function as a function of r and is an odd function of on due to sgnðon Þ differently from CFM (this s amplitude is called an odd frequency pairing function). If we take hex ¼ 0, CFM is identically zero as seen in Eq. (11). This situation t corresponds to the SC/NM junction case, where NM represents a normal metal. Fig. 3 plots the on -dependence of CFM and CFM for s t ~ FM ¼ k2 DFM =D0 ¼ 10 000; T=T SC ¼ 0:01, the parameters, kF r ¼ 5; D F hex =D0 ¼ 100, where kF is the Fermi wave length, and T SC is the superconducting transition temperature. In the SC/HM junction, it is found from Eqs. (14) and 15 that the pair amplitudes depend on r in HM HM and on , and that CHM þ and C are an odd and even functions with respect to on , respectively. Fig. 4 shows the on -dependence of HM CHM þ and C , where we choose the following parameters, kF r ¼ 2, ~ xM ¼ 10; DHM ¼ k2F DHM =D0 ¼ 10 000; T=T SC ¼ 0:01 in numerical calculation. It is seen that the amplitude of CHM is much larger HM than that of CHM compared with CHM þ . The small values of Cþ come from the term ðf þ f Þ in Eq. (16). Here, we discuss the relation between the Josephson effect and the proximity effect in the tunneling Hamiltonian method. As mentioned in the Introduction, the Josephson effect occurs due to the pair amplitude penetrated from one SC to the other SC through a normal conducting metal sandwiched by SCs by the proximity effect. Therefore, the nature of the pair amplitude is reflected in that of the Josephson effect. In a previous work, we studied the Josephson effect in SC/FM/SC and SC/HM/SC junctions in the tunneling Hamiltonian method [6–9]. Using these results, we discuss the relation between the Josephson effect and the proximity effect. From Refs. [6–8], the Josephson critical current in the SC/FM/SC and SC/HM/SC junctions is given by IFM / c þ þ expðd=xFM Þ cosðd=xFM Þ and IHM / expðd=xHM Þ cosðd=xHM Þ, rec spectively. Comparing these Josephson critical currents with the pair amplitudes in the SC/FM and SC/HM junctions, it is found that
Ψ+
HM
Ψ− 0
× 100 -5
0
ωn/Δ0
5
Fig. 4. The pair amplitude C in HM as a function of the Matsubara frequency, on . D0 is a superconducting gap at T ¼ 0.
the even frequency pairs CFM and CHM contribute to the s Josephson effect in the SC/FM/SC and SC/HM/SC junctions, respectively, because of the same behavior in their spatial dependence. Therefore, the odd frequency component of the pair amplitude does not contribute to the Josephson effect within present tunneling models. In summary, we have studied the proximity effect in the SC/FM and SC/HM junctions using the tunneling Hamiltonian method. In the SC/FM junction, it is found that the pair amplitude CFM and s CFM are induced in FM and both amplitudes depend on the t position r in FM and the Matsubara frequency on. We have obtained that CFM is the even function and CFM is the odd s t function of on . In the SC/HM junction, the spin-flip tunneling with the magnon excitation at the interface is crucial for including the HM proximity effect. The induced pair amplitude CHM þ and C show the damped oscillation as a function of r. It is found that CHM þ and CHM are the even and odd functions of on , respectively. Moreover, we have investigated the effect of even and odd frequency components on the Josephson effect in the SC/FM/SC and SC/HM/SC junctions. As a result, the even frequency component
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only contributes to the Josephson effect in the both cases within present models.
Acknowledgment This work is supported by a Grant in Aid for Scientific Research from MEXT, the Next Generation Supercomputer Project of MEXT. References [1] A.A. Golubov, M.Yu. Kupriyanov, E. Ilfichev, Rev. Mod. Phys. 76 (2004) 411. [2] A.I. Buzdin, Rev. Mod. Phys. 77 (2005) 935.
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