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Boundary conditions for quasiclassical Green functions at superconductor–ferromagnet interface
Physica B 284}288 (2000) 509}510
Boundary conditions for quasiclassical Green functions at superconductor}ferromagnet interface B.P. Vodop'yanov , L.R. Tagirov * Kazan Physicotechnical Institute, Russian Academy of Sciences, 420029 Kazan, Russia Theoretical Physics Department, Kazan State University, 420008 Kazan, Russia
Abstract The quasiclassical equations of superconductivity for a metal with spin-split conduction band are derived. The boundary conditions for the Green functions at the interface between a ferromagnet and a superconductor are obtained. They are valid for the arbitrary magnitude of exchange splitting of a ferromagnet conduction band. 2000 Elsevier Science B.V. All rights reserved. Keywords: Boundary e!ects; Superconductor}ferromagnet contact
Magnetoelectronics is a new class of electronics which exploits the spin-polarized carriers transport in ferromagnetic metals [1]. The performance of magnetoelectronic devices improves as the polarization of a ferromagnet conduction band increases. Meservey and Tedrow (see, for example, Ref. [2] and references therein) pioneered the superconductor/ferromagnet (S/F) spinpolarized tunneling to probe the electronic spectrum near the Fermi energy. Very recent S/F point contact spectroscopy works [3,4] established the experimental basis for the Andreev spectroscopy of magnetic materials by means of contact with a superconductor. Either tunneling spectroscopy or the point contact spectroscopy deal with transmission of Cooper pairs from a superconductor into the spin-polarized conduction band of a ferromagnet and vice versa. To describe the electronic transport in S/F systems we extend the quasiclassical (QC) theory on superconductor/ferromagnet couples taking explicitly into account the spin dependence of transmission probabilities through the S/F interface and di!erent Fermi momenta of spin-subbands of a ferromagnet.
1. Equations for Green functions Assume that the contact of a superconductor and a ferromagnet is #at. The equations for the QC thermodynamic Green functions of S/F contact are derived by a method close to that one developed by Zaitsev [5]. We suppose that S/F interface does not mix spin channels, as was considered in Ref. [5] for S/N and S/S contacts. The spin-active interface between two superconductors was considered in Ref. [6]. The equations for QC Green function in F and S sides of S/F contact read: vo * *g( v( # (g( v( #v( g( ) v( V V V *x V 2 *oo #KK g( v( !v( g( KK "0, (1) V V * 1 KK "i q ! mvL (vL !q vL q )!DK #eU d(q!q ) X *q 2 V V V V V
#iRK (q,q ),
(2)
where all quantities with hats and q are matrices in X particle}hole space. The spin structures of the Green function and x-component of Fermi velocity v( are the V following:
g g( " ?? !f
* Corresponding author. Tel.: #7-8432-381-573; fax: #78432-381-573. E-mail address: [email protected] (L.R. Tagirov)
f v 0 ?\? , v( " V ? . (3) V g 0 v \?? \?\? V \? Direction x is chosen along the normal to the contact plane, mv "(2m(e #ah)!p , 2h is the spin-splitV ? $ ,
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 1 0 2 - X
510
B.P. Vodop'yanov, L.R. Tagirov / Physica B 284}288 (2000) 509}510
ting energy of conduction band. In F-layer hO0, in S-layer h"0. Symbol `'a means the direction from F to S layer, the equation for the Green function for the reverse direction g( can be written down by changing the sign of the "rst term in Eq. (1). Other de"nitions are standard ones [5].
The boundary conditions at S/F interface are written down for the following combinations of GF: (4)
(5)
In the above equations h "h !h , h (h ) are phases of PB P B P B re#ection (transmission) amplitudes [5], which are spinchannel dependent in our case. They have the same spin structure as x-component of the Fermi velocity. The boundary conditions read I1 #q I1 q "I$#q I$q , ? X ? X ? X ? X I>f #f I>#I\f #f I\"2( f !f ), ? ? ? ? I>f #f I>#I\f #f I\"2( f !f ). ? ? ? ? In the above equations f "A? (I\)#A? (q I\) A X A #A? (I>)#A? (q I>), A X A
f "I\A? I\#I\A? q I\q A A A X A X #I>A? I>#I>A? q I>q , A A A X A X #I\A? I>#I\A? q I>q , A A A X A X
1 1 1#(D?D\?#(R?R\? A? " $ , 2 (R? (R?#(R\? 1 1!(D?D\?#(R?R\? A? " (R?$ , 2 (R?#(R\?
(v( 1 I1 " V (e\ FB g( e FB $e\ FPB g( e FPB )(v( 1 , A ? 1 1 V 2 (v( $ I$ " V (g( $e\ FP g( e FP )(v( $ . A ? $ $ V 2
#A? I>I\#A? q I>q I\, A A X A X A
f "I>A? I\#I>A? q I\q A A A X A X
2. Boundary conditions at S/F interface
I!"I1$I$, I!"I1 $I$, A A A ? ? ? where
f "*A? I\I>#A? q I\q I> A A X A X A
R? and D? are the re#ection and transmission coe$cients for electron with spin projection a. If h"0 on the F side of a contact our boundary conditions (6) and (7) match Zaitsev's ones. The above relations solve the problem of "nding the quasiclassic GF in S/F contact.
References
(6)
(7)
[1] G.A. Prinz, Science 282 (1998) 1660. [2] R. Meservey, P.M. Tedrow, Phys. Rep. 238 (1994) 173. [3] R.J. Soulen Jr., J.M. Byers, M.S. Osofsky et al., Science 282 (1998) 85. [4] S.K. Upadhyay, A. Palanisami, R.N. Louie, R.A. Buhrman, Phys. Rev. Lett. 81 (1998) 3247. [5] A.V. Zaitsev, Sov. Phys. -JETP 59 (1984) 863. [6] A. Millis, D. Rainer, J.A. Sauls, Phys. Rev. B 38 (1988) 4504.
Boundary conditions for quasiclassical Green functions at superconductor}ferromagnet interface B.P. Vodop'yanov , L.R. Tagirov * Kazan Physicotechnical Institute, Russian Academy of Sciences, 420029 Kazan, Russia Theoretical Physics Department, Kazan State University, 420008 Kazan, Russia
Abstract The quasiclassical equations of superconductivity for a metal with spin-split conduction band are derived. The boundary conditions for the Green functions at the interface between a ferromagnet and a superconductor are obtained. They are valid for the arbitrary magnitude of exchange splitting of a ferromagnet conduction band. 2000 Elsevier Science B.V. All rights reserved. Keywords: Boundary e!ects; Superconductor}ferromagnet contact
Magnetoelectronics is a new class of electronics which exploits the spin-polarized carriers transport in ferromagnetic metals [1]. The performance of magnetoelectronic devices improves as the polarization of a ferromagnet conduction band increases. Meservey and Tedrow (see, for example, Ref. [2] and references therein) pioneered the superconductor/ferromagnet (S/F) spinpolarized tunneling to probe the electronic spectrum near the Fermi energy. Very recent S/F point contact spectroscopy works [3,4] established the experimental basis for the Andreev spectroscopy of magnetic materials by means of contact with a superconductor. Either tunneling spectroscopy or the point contact spectroscopy deal with transmission of Cooper pairs from a superconductor into the spin-polarized conduction band of a ferromagnet and vice versa. To describe the electronic transport in S/F systems we extend the quasiclassical (QC) theory on superconductor/ferromagnet couples taking explicitly into account the spin dependence of transmission probabilities through the S/F interface and di!erent Fermi momenta of spin-subbands of a ferromagnet.
1. Equations for Green functions Assume that the contact of a superconductor and a ferromagnet is #at. The equations for the QC thermodynamic Green functions of S/F contact are derived by a method close to that one developed by Zaitsev [5]. We suppose that S/F interface does not mix spin channels, as was considered in Ref. [5] for S/N and S/S contacts. The spin-active interface between two superconductors was considered in Ref. [6]. The equations for QC Green function in F and S sides of S/F contact read: vo * *g( v( # (g( v( #v( g( ) v( V V V *x V 2 *oo #KK g( v( !v( g( KK "0, (1) V V * 1 KK "i q ! mvL (vL !q vL q )!DK #eU d(q!q ) X *q 2 V V V V V
#iRK (q,q ),
(2)
where all quantities with hats and q are matrices in X particle}hole space. The spin structures of the Green function and x-component of Fermi velocity v( are the V following:
g g( " ?? !f
* Corresponding author. Tel.: #7-8432-381-573; fax: #78432-381-573. E-mail address: [email protected] (L.R. Tagirov)
f v 0 ?\? , v( " V ? . (3) V g 0 v \?? \?\? V \? Direction x is chosen along the normal to the contact plane, mv "(2m(e #ah)!p , 2h is the spin-splitV ? $ ,
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 1 0 2 - X
510
B.P. Vodop'yanov, L.R. Tagirov / Physica B 284}288 (2000) 509}510
ting energy of conduction band. In F-layer hO0, in S-layer h"0. Symbol `'a means the direction from F to S layer, the equation for the Green function for the reverse direction g( can be written down by changing the sign of the "rst term in Eq. (1). Other de"nitions are standard ones [5].
The boundary conditions at S/F interface are written down for the following combinations of GF: (4)
(5)
In the above equations h "h !h , h (h ) are phases of PB P B P B re#ection (transmission) amplitudes [5], which are spinchannel dependent in our case. They have the same spin structure as x-component of the Fermi velocity. The boundary conditions read I1 #q I1 q "I$#q I$q , ? X ? X ? X ? X I>f #f I>#I\f #f I\"2( f !f ), ? ? ? ? I>f #f I>#I\f #f I\"2( f !f ). ? ? ? ? In the above equations f "A? (I\)#A? (q I\) A X A #A? (I>)#A? (q I>), A X A
f "I\A? I\#I\A? q I\q A A A X A X #I>A? I>#I>A? q I>q , A A A X A X #I\A? I>#I\A? q I>q , A A A X A X
1 1 1#(D?D\?#(R?R\? A? " $ , 2 (R? (R?#(R\? 1 1!(D?D\?#(R?R\? A? " (R?$ , 2 (R?#(R\?
(v( 1 I1 " V (e\ FB g( e FB $e\ FPB g( e FPB )(v( 1 , A ? 1 1 V 2 (v( $ I$ " V (g( $e\ FP g( e FP )(v( $ . A ? $ $ V 2
#A? I>I\#A? q I>q I\, A A X A X A
f "I>A? I\#I>A? q I\q A A A X A X
2. Boundary conditions at S/F interface
I!"I1$I$, I!"I1 $I$, A A A ? ? ? where
f "*A? I\I>#A? q I\q I> A A X A X A
R? and D? are the re#ection and transmission coe$cients for electron with spin projection a. If h"0 on the F side of a contact our boundary conditions (6) and (7) match Zaitsev's ones. The above relations solve the problem of "nding the quasiclassic GF in S/F contact.
References
(6)
(7)
[1] G.A. Prinz, Science 282 (1998) 1660. [2] R. Meservey, P.M. Tedrow, Phys. Rep. 238 (1994) 173. [3] R.J. Soulen Jr., J.M. Byers, M.S. Osofsky et al., Science 282 (1998) 85. [4] S.K. Upadhyay, A. Palanisami, R.N. Louie, R.A. Buhrman, Phys. Rev. Lett. 81 (1998) 3247. [5] A.V. Zaitsev, Sov. Phys. -JETP 59 (1984) 863. [6] A. Millis, D. Rainer, J.A. Sauls, Phys. Rev. B 38 (1988) 4504.