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Magnon-assisted Andreev transport across ferromagnet–superconductor junctions
Physica E 12 (2002) 938 – 941
www.elsevier.com/locate/physe
Magnon-assisted Andreev transport across ferromagnet–superconductor junctions Edward McCanna; ∗ , Grygoriy Tkachova; b , Vladimir I. Fal’koa a Department
of Physics, Lancaster University, Lancaster LA1 4YB, UK for Radiophysics and Electronics, Kharkiv, Ukraine
b Institute
Abstract We study subgap transport due to Andreev re2ection in a ferromagnet–superconductor junction at low temperature. The mismatch of spin polarized current in the ferromagnet and spinless current in the superconductor results in an additional contact resistance which can be reduced by magnon emission in the ferromagnet. We 4nd that magnon-assisted Andreev re2ection produces a non-linear contribution to the asymmetric part of the conductance, G(V ) − G(−V ), which is related to the density of states of magnons in the ferromagnet. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 73.23.−b; 74.80.Fp; SNS junctions; 73.63.Rt; 72.25.−b Keywords: Andreev re2ection; Spin polarized transport; Magnon-assisted tunnelling
In the past decade there has been intense research into the magnetoresistance e=ect of ferromagnetic tunnelling junctions and multilayers [1]. Ferromagnetic (F) metals have more carriers of one spin polarization (known as majority carriers) present at the Fermi energy EF than of the inverse polarization (minority carriers). An F–F junction with antiparallel spin polarizations has a larger contact resistance than a junction with parallel polarizations due to a mismatch of spin currents at the interface. Spin current mismatch may also a=ect the conductance of a ferromagnet–superconductor junction [2–5]. At low temperatures and small bias voltage, current 2ows through the interface due to Andreev re2ection whereby particles in the ferromagnetic region with excitation energies smaller than the superconducting ∗
Corresponding author. Fax: +44-1524-844037. E-mail address: [email protected] (E. McCann).
gap energy are re2ected from the interface as holes. Since subgap transport in the superconductor (S) is mediated by spinless Cooper pairs the spin current is zero in the superconductor in contrast to the ferromagnet. Spin relaxation processes [1], such as spin–orbit scattering at impurities or magnon emission, can reduce the spin current mismatch. In a given junction, spin–orbit scattering (which is an elastic process) would reduce the value of the additional contact resistance [5] whereas the inelastic process of magnon emission would manifest itself as a modi4cation of the form of the I (V ) characteristics. Previously, magnon assisted Andreev re2ection has been considered for a half-metallic ferromagnet–superconductor junction [6], where a half-metallic ferromagnet has a splitting between the majority and minority conduction bands which is greater than EF measured from the bottom of the majority band so that only majority carriers are present at the Fermi energy. In the
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 4 1 4 - 3
E. McCann et al. / Physica E 12 (2002) 938 – 941
present work we begin by discussing the half-metallic case and then present results for a ferromagnet with an arbitrary degree of polarization. As a particular example, we consider there to be an ‘ideal’ planar tunnelling barrier of area A at the F–S interface so that the momentum parallel to the interface is conserved in the tunnelling process. For simplicity, we discuss the half-metallic case 4rst. Microscopically, the relevant process involves the transfer of a singlet pair of electrons from the superconductor to the ferromagnet where one of them forms an intermediate state at an energy below the bottom of the conduction band for minority electrons. Then, this latter electron can emit a magnon in the ferromagnet which will carry away excess spin allowing the electron to incorporate itself into the majority conduction band. Such a quantum kinetic process generates a non-linear addition to the I (V ) characteristics of the F–S junction, whose di=erential can be related to the magnon density of states (!) on the magnetic side. For a half-metal, where ↑ is the area of the maximal cross section of the Fermi surface of majority (spin ‘up’) electrons in the plane parallel to the interface, we 4nd 2 −1 2 3 4 eV d I 4e A|t| ↑ (2eV ) 1 − = dV 2 h3 S +
0
2eV
− eV d eV −F ;
() F 2
939
Fig. 1. Schematic of the tunnelling process between a superconducting electrode on the left-hand side and a half-metallic ferromagnetic electrode on the right for (a) V ¿ 0 and (b) V ¡ 0. For (a) V ¿ 0 a down spin electron tunnelling into the ferromagnet may emit a magnon and incorporate itself into the majority conduction band. For (b) V ¡ 0 no spin 2ip process is possible at T = 0 because in the initial state (right) there are no thermal magnons for an up spin electron to absorb. See text for details.
(1) where F(x) = x=(1 − x2 )2 , the factor |t|4 stands for the probability of the simultaneous tunnelling of two particles involved in the process of Andreev re2ection, S is the spin of the localized moments, h is the Planck constant and e is the absolute value of the elementary charge. The factor of two in (2eV ) arises because an elementary Andreev re2ection process involves a net transfer of charge 2e across the biased junction. We consider there to be an anisotropy energy !0 responsible for a gap in the magnon density of states so that (!) is non-zero for !0 ¡ ! ¡ !m and zero otherwise, where !m is the magnon bandwidth. A detailed derivation of the I (V ) characteristics (1) will be presented in a long paper [7]. The non-linear contribution to the current (1) is asymmetric with respect to bias voltage V : it is zero for
negative bias and 4nite for V ¿ !0 =(2e). The asymmetry can be explained using the sketch in Fig. 1 which illustrates the tunnelling process between an S electrode on the left-hand side and an F electrode on the right for (a) V ¿ 0 and (b) V ¡ 0. We have adopted the convention that positive (negative) voltage results in a Fermi energy EF in the ferromagnet that is lower (higher) by energy |eV | than the Fermi energy ES in the superconductor. For V ¿ 0, Fig. 1(a), Andreev re2ection results in the injection of both a majority (spin ‘up’) and a minority (spin ‘down’) electron into the ferromagnet. At zero temperature T = 0 the core spins are all aligned in the majority, up direction and, if EF , then only spin up conduction electrons are present at EF in the ferromagnet. A dynamic process which allows a spin down electron
940
E. McCann et al. / Physica E 12 (2002) 938 – 941 1.0 0.8 0.6
0.6
2
0.4 0.2 0.0 2.0
2
d I/dV (arb.)
2
2
0.8
d I/dV (arb.)
1.0
1.0
0.4
0.2
0.0 −1.0
0.0 1.0 eV/ωm
2.0
T=0
−0.5
0.0 eV/∆
0.5
1.0
Fig. 2. Typical form of d 2 I=dV 2 for a half-metallic F–S junction with various choices of material parameters. From top to bottom, =W = 5:0 (dot-dashed), 1:0 (dashed), and 0:2 (solid). As an example, we choose !0 = 0:2 and !m = 4:0. For comparison, inset shows d 2 I=dV 2 for an F–F junction with antiparallel spin polarizations.
to enter the ferromagnet is shown schematically in Fig. 1(a). The spin down electron tunnels from the superconductor into a virtual, intermediate spin down state above EF , then it emits a magnon and incorporates itself into an empty state in the majority conduction band. In Fig. 1(a) the magnon is depicted as a 2ip in the spin of one of the localized magnetic moments. On the other hand, for V ¡ 0, Fig. 1(b), Andreev re2ection would result in the injection of a spin up and a spin down electron from F into S, but initially there are no spin down states near EF in the ferromagnet. Since a spin up electron cannot emit a magnon (left side of Fig. 1(b)), due to conservation of total spin in the exchange interaction, the only possibility would be that a spin up electron in the ferromagnet would absorb a magnon to form an intermediate spin down state before tunnelling into the superconductor. However there are no thermally excited magnons at T = 0 in the initial state of the ferromagnet (right side of Fig. 1(b)) so it is impossible for magnon-assisted Andreev re2ection to contribute to current formation in negatively biased junctions. A graphic representation of the result of Eq. (1) at zero temperature T = 0 is given in Fig. 2, in comparison with a similar analysis for an F–F junction in an antiparallel con4guration (inset). It illustrates how the
relation between the conduction electron bandwidth W and the exchange splitting energy may a=ect the result, since for W the contribution of large energy magnons to the lifting of spin current mismatch is not so eJcient as for W . Note that at T ¿ !0 , the absorption of thermal magnons would allow for a similar contribution to the current at V ¡ 0. Now let us consider a ferromagnet of arbitrary polarization where the area of the maximal cross section of the Fermi surface of minority (spin ‘down’) electrons in the plane parallel to the interface ↓ is non-zero, so that, in general ↑ ¿ ↓ ¿ 0. In this case an additional inelastic contribution to the current is present at T =0 for negative bias V ¡−!0 =(2e) which is due to the magnon-assisted transport of minority carriers. As the magnon-assisted Andreev re2ection processes may co-exist with conventional elastic Andreev re2ection, the asymmetric part of the conductance, G(V ) − G(−V ), could be used to separate the e=ect of the magnon-assisted process: d [G(V ) − G(−V )] dV
−1 2 eV 4e3 A|t|4 [ − ]
(2eV ) 1 − = ↑ ↓ h3 S +
2eV
0
− eV d eV −F ;
() F 2 (2)
which is written for V ¿ 0. Our result suggests a technique to image the low frequency part of the magnon density of states in magnetic multilayers and superlattices, especially in those revealing giant magnetoresistance properties [1]. The authors thank I.L. Aleiner, B.L. Altshuler, N.R. Cooper, D.M. Edwards, C.J. Lambert, and Yu.V. Nazarov for discussions. This work was supported by EPSRC, NATO and the Royal Society.
References [1] G.A. Prinz, Science 282 (1998) 1660. [2] M.J.M. de Jong, C.W.J. Beenakker, Phys. Rev. Lett. 74 (1995) 1657.
E. McCann et al. / Physica E 12 (2002) 938 – 941 [3] V.I. Fal’ko, A.F. Volkov, C.J. Lambert, Phys. Rev. B 60 (1999) 15394; E. McCann, V.I. Fal’ko, A.F. Volkov, C.J. Lambert, Phys. Rev. B 62 (2000) 6015. [4] F.J. Jedema, B.J. van Wees, B.H. Hoving, A.T. Filip, T.M. Klapwijk, Phys. Rev. B 60 (1999) 16549.
941
[5] V.I. Fal’ko, C.J. Lambert, A.F. Volkov, JETP Lett. 69 (1999) 532. [6] E. McCann, V.I. Fal’ko, Europhys. Lett. 56 (2001) 583. [7] G. Tkachov, E. McCann, V.I. Fal’ko, in preparation.
www.elsevier.com/locate/physe
Magnon-assisted Andreev transport across ferromagnet–superconductor junctions Edward McCanna; ∗ , Grygoriy Tkachova; b , Vladimir I. Fal’koa a Department
of Physics, Lancaster University, Lancaster LA1 4YB, UK for Radiophysics and Electronics, Kharkiv, Ukraine
b Institute
Abstract We study subgap transport due to Andreev re2ection in a ferromagnet–superconductor junction at low temperature. The mismatch of spin polarized current in the ferromagnet and spinless current in the superconductor results in an additional contact resistance which can be reduced by magnon emission in the ferromagnet. We 4nd that magnon-assisted Andreev re2ection produces a non-linear contribution to the asymmetric part of the conductance, G(V ) − G(−V ), which is related to the density of states of magnons in the ferromagnet. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 73.23.−b; 74.80.Fp; SNS junctions; 73.63.Rt; 72.25.−b Keywords: Andreev re2ection; Spin polarized transport; Magnon-assisted tunnelling
In the past decade there has been intense research into the magnetoresistance e=ect of ferromagnetic tunnelling junctions and multilayers [1]. Ferromagnetic (F) metals have more carriers of one spin polarization (known as majority carriers) present at the Fermi energy EF than of the inverse polarization (minority carriers). An F–F junction with antiparallel spin polarizations has a larger contact resistance than a junction with parallel polarizations due to a mismatch of spin currents at the interface. Spin current mismatch may also a=ect the conductance of a ferromagnet–superconductor junction [2–5]. At low temperatures and small bias voltage, current 2ows through the interface due to Andreev re2ection whereby particles in the ferromagnetic region with excitation energies smaller than the superconducting ∗
Corresponding author. Fax: +44-1524-844037. E-mail address: [email protected] (E. McCann).
gap energy are re2ected from the interface as holes. Since subgap transport in the superconductor (S) is mediated by spinless Cooper pairs the spin current is zero in the superconductor in contrast to the ferromagnet. Spin relaxation processes [1], such as spin–orbit scattering at impurities or magnon emission, can reduce the spin current mismatch. In a given junction, spin–orbit scattering (which is an elastic process) would reduce the value of the additional contact resistance [5] whereas the inelastic process of magnon emission would manifest itself as a modi4cation of the form of the I (V ) characteristics. Previously, magnon assisted Andreev re2ection has been considered for a half-metallic ferromagnet–superconductor junction [6], where a half-metallic ferromagnet has a splitting between the majority and minority conduction bands which is greater than EF measured from the bottom of the majority band so that only majority carriers are present at the Fermi energy. In the
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 4 1 4 - 3
E. McCann et al. / Physica E 12 (2002) 938 – 941
present work we begin by discussing the half-metallic case and then present results for a ferromagnet with an arbitrary degree of polarization. As a particular example, we consider there to be an ‘ideal’ planar tunnelling barrier of area A at the F–S interface so that the momentum parallel to the interface is conserved in the tunnelling process. For simplicity, we discuss the half-metallic case 4rst. Microscopically, the relevant process involves the transfer of a singlet pair of electrons from the superconductor to the ferromagnet where one of them forms an intermediate state at an energy below the bottom of the conduction band for minority electrons. Then, this latter electron can emit a magnon in the ferromagnet which will carry away excess spin allowing the electron to incorporate itself into the majority conduction band. Such a quantum kinetic process generates a non-linear addition to the I (V ) characteristics of the F–S junction, whose di=erential can be related to the magnon density of states (!) on the magnetic side. For a half-metal, where ↑ is the area of the maximal cross section of the Fermi surface of majority (spin ‘up’) electrons in the plane parallel to the interface, we 4nd 2 −1 2 3 4 eV d I 4e A|t| ↑ (2eV ) 1 − = dV 2 h3 S +
0
2eV
− eV d eV −F ;
() F 2
939
Fig. 1. Schematic of the tunnelling process between a superconducting electrode on the left-hand side and a half-metallic ferromagnetic electrode on the right for (a) V ¿ 0 and (b) V ¡ 0. For (a) V ¿ 0 a down spin electron tunnelling into the ferromagnet may emit a magnon and incorporate itself into the majority conduction band. For (b) V ¡ 0 no spin 2ip process is possible at T = 0 because in the initial state (right) there are no thermal magnons for an up spin electron to absorb. See text for details.
(1) where F(x) = x=(1 − x2 )2 , the factor |t|4 stands for the probability of the simultaneous tunnelling of two particles involved in the process of Andreev re2ection, S is the spin of the localized moments, h is the Planck constant and e is the absolute value of the elementary charge. The factor of two in (2eV ) arises because an elementary Andreev re2ection process involves a net transfer of charge 2e across the biased junction. We consider there to be an anisotropy energy !0 responsible for a gap in the magnon density of states so that (!) is non-zero for !0 ¡ ! ¡ !m and zero otherwise, where !m is the magnon bandwidth. A detailed derivation of the I (V ) characteristics (1) will be presented in a long paper [7]. The non-linear contribution to the current (1) is asymmetric with respect to bias voltage V : it is zero for
negative bias and 4nite for V ¿ !0 =(2e). The asymmetry can be explained using the sketch in Fig. 1 which illustrates the tunnelling process between an S electrode on the left-hand side and an F electrode on the right for (a) V ¿ 0 and (b) V ¡ 0. We have adopted the convention that positive (negative) voltage results in a Fermi energy EF in the ferromagnet that is lower (higher) by energy |eV | than the Fermi energy ES in the superconductor. For V ¿ 0, Fig. 1(a), Andreev re2ection results in the injection of both a majority (spin ‘up’) and a minority (spin ‘down’) electron into the ferromagnet. At zero temperature T = 0 the core spins are all aligned in the majority, up direction and, if EF , then only spin up conduction electrons are present at EF in the ferromagnet. A dynamic process which allows a spin down electron
940
E. McCann et al. / Physica E 12 (2002) 938 – 941 1.0 0.8 0.6
0.6
2
0.4 0.2 0.0 2.0
2
d I/dV (arb.)
2
2
0.8
d I/dV (arb.)
1.0
1.0
0.4
0.2
0.0 −1.0
0.0 1.0 eV/ωm
2.0
T=0
−0.5
0.0 eV/∆
0.5
1.0
Fig. 2. Typical form of d 2 I=dV 2 for a half-metallic F–S junction with various choices of material parameters. From top to bottom, =W = 5:0 (dot-dashed), 1:0 (dashed), and 0:2 (solid). As an example, we choose !0 = 0:2 and !m = 4:0. For comparison, inset shows d 2 I=dV 2 for an F–F junction with antiparallel spin polarizations.
to enter the ferromagnet is shown schematically in Fig. 1(a). The spin down electron tunnels from the superconductor into a virtual, intermediate spin down state above EF , then it emits a magnon and incorporates itself into an empty state in the majority conduction band. In Fig. 1(a) the magnon is depicted as a 2ip in the spin of one of the localized magnetic moments. On the other hand, for V ¡ 0, Fig. 1(b), Andreev re2ection would result in the injection of a spin up and a spin down electron from F into S, but initially there are no spin down states near EF in the ferromagnet. Since a spin up electron cannot emit a magnon (left side of Fig. 1(b)), due to conservation of total spin in the exchange interaction, the only possibility would be that a spin up electron in the ferromagnet would absorb a magnon to form an intermediate spin down state before tunnelling into the superconductor. However there are no thermally excited magnons at T = 0 in the initial state of the ferromagnet (right side of Fig. 1(b)) so it is impossible for magnon-assisted Andreev re2ection to contribute to current formation in negatively biased junctions. A graphic representation of the result of Eq. (1) at zero temperature T = 0 is given in Fig. 2, in comparison with a similar analysis for an F–F junction in an antiparallel con4guration (inset). It illustrates how the
relation between the conduction electron bandwidth W and the exchange splitting energy may a=ect the result, since for W the contribution of large energy magnons to the lifting of spin current mismatch is not so eJcient as for W . Note that at T ¿ !0 , the absorption of thermal magnons would allow for a similar contribution to the current at V ¡ 0. Now let us consider a ferromagnet of arbitrary polarization where the area of the maximal cross section of the Fermi surface of minority (spin ‘down’) electrons in the plane parallel to the interface ↓ is non-zero, so that, in general ↑ ¿ ↓ ¿ 0. In this case an additional inelastic contribution to the current is present at T =0 for negative bias V ¡−!0 =(2e) which is due to the magnon-assisted transport of minority carriers. As the magnon-assisted Andreev re2ection processes may co-exist with conventional elastic Andreev re2ection, the asymmetric part of the conductance, G(V ) − G(−V ), could be used to separate the e=ect of the magnon-assisted process: d [G(V ) − G(−V )] dV
−1 2 eV 4e3 A|t|4 [ − ]
(2eV ) 1 − = ↑ ↓ h3 S +
2eV
0
− eV d eV −F ;
() F 2 (2)
which is written for V ¿ 0. Our result suggests a technique to image the low frequency part of the magnon density of states in magnetic multilayers and superlattices, especially in those revealing giant magnetoresistance properties [1]. The authors thank I.L. Aleiner, B.L. Altshuler, N.R. Cooper, D.M. Edwards, C.J. Lambert, and Yu.V. Nazarov for discussions. This work was supported by EPSRC, NATO and the Royal Society.
References [1] G.A. Prinz, Science 282 (1998) 1660. [2] M.J.M. de Jong, C.W.J. Beenakker, Phys. Rev. Lett. 74 (1995) 1657.
E. McCann et al. / Physica E 12 (2002) 938 – 941 [3] V.I. Fal’ko, A.F. Volkov, C.J. Lambert, Phys. Rev. B 60 (1999) 15394; E. McCann, V.I. Fal’ko, A.F. Volkov, C.J. Lambert, Phys. Rev. B 62 (2000) 6015. [4] F.J. Jedema, B.J. van Wees, B.H. Hoving, A.T. Filip, T.M. Klapwijk, Phys. Rev. B 60 (1999) 16549.
941
[5] V.I. Fal’ko, C.J. Lambert, A.F. Volkov, JETP Lett. 69 (1999) 532. [6] E. McCann, V.I. Fal’ko, Europhys. Lett. 56 (2001) 583. [7] G. Tkachov, E. McCann, V.I. Fal’ko, in preparation.