VOLUME 93, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S week ending20 AUGUST 2004

Spin and Charge Pumping by Ferromagnetic-Superconductor Order Parameters

Arne Brataas1 and Yaroslav Tserkovnyak2

1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway2Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

(Received 23 April 2004; published 16 August 2004)

087201-1

We study transport in ferromagnetic-superconductor/normal-metal systems. It is shown that chargeand spin currents are pumped from ferromagnetic superconductors into adjacent normal metals byadiabatic changes in the order parameters induced by external electromagnetic fields. Spin and chargepumping identify the symmetry of the superconducting order parameter, e.g., singlet pairing or tripletpairing with opposite or equal spin pairing. Consequences for ferromagnetic-resonance experiments arediscussed.

DOI: 10.1103/PhysRevLett.93.087201 PACS numbers: 75.70.Cn, 72.25.Mk

Ferromagnetism induces a spin-dependent asymmetryin the densities of itinerant carriers. In contrast, super-conductivity pairs electrons with equal or opposite spinsdepending on the symmetry of its order parameter. Thecoexistence of the two order parameters has been consid-ered to be a rare phenomenon. However, recent experi-mental progress has demonstrated that ferromagnetismand superconductivity coexist in some materials likeRuSr2GdCu2O8 [1], UGe2 [2], ZrZn2 [3], and URhGe[4]. The experiments find triplet pairing in URhGe, andstrong indications of triplet pairing in UGe2 and ZrZn2;they furthermore suggest that the same electrons areresponsible for ferromagnetism as well as superconduc-tivity. Besides, ferromagnetism and superconductivity arepredicted to be simultaneously induced in hybrid ferro-magnet (F)/normal-metal (N)/superconductor (S) sys-tems [5]. These recent experimental discoveries, and thepossibility of tailoring superconductivity and ferromag-netism in nanoscale systems, enable exploring novelphysics involving pairing and spin-related transport pro-cesses. A variety of interesting spin phenomena have al-ready been observed in hybrid F/N and semiconductorsystems: e.g., giant-magnetoresistance —related effects,spin precession, and current-induced magnetization dy-namics [6]. It is therefore natural to expect that interest-ing rich phenomena should also occur in FS/N systems.

This Letter demonstrates how the coexistence of super-conductivity and ferromagnetism is manifested in theadiabatic pumping in hybrid FS/N structures. In particu-lar, we study spin and charge pumping when the magne-tization slowly precesses, which can be achieved inferromagnetic-resonance (FMR) and in current-inducedmagnetization dynamics. FMR experiments have alreadybeen carried out to investigate the magnetism inRuSr2GdCu2O8 [7]. We also consider pumping by slowvariations in the phase of the singlet or triplet orderparameter, or in the direction of the triplet order parame-ter. These can be induced by electric and magnetic fieldsto be discussed below. By pumping we thus mean the spinand charge flows into the adjacent normal conductors inresponse to adiabatic changes in the FS order parameters.

0031-9007=04=93(8)=087201(4)$22.50

Consequently, in the case of pumping by the magnetiza-tion direction, we compute the spin current Is and thecharge current Ic for a given rate of the magnetization-direction change, and for pumping by changing pair-ing, we compute the same quantities as functions of thephase-change or the direction-change rates of the paircorrelations. We employ two approaches giving identicalresults: 1) solving the time-dependent ac problem directlyand 2) using a gauge transformation to obtain a time-independent dc problem. We first explain our model andnotation before proceeding to the derivation and results.Experimental consequences are discussed in the end.

A ferromagnetic superconductor is treated in the mean-field approximation, where ferromagnetism is representedby the average magnetization and superconductivity isdescribed by a pair potential. Our model is phenomeno-logical and we do not discuss the microscopic origin ofthe exchange field or the superconducting pairing. TheBogoliubov–de Gennes (BdG) equation is

�� �

��� ���

��uv

�� i �h

@@t

�uv

�; (1)

where � � H01� � � �xc is the single-particleHamiltonian, �xc � xcm (with xc > 0) is the ferromag-netic exchange field along the magnetization direction m,and � � �d01� � � d�i�y is the superconducting pairpotential, in terms of the singlet (scalar) part, d0, andthe triplet (vector) part, d � �dx; dy; dz�. uT � �u"; u#� arespin-dependent electron wave functions and vT � �v"; v#�are those of holes; � � ��x; �y; �z� is the vector of Paulimatrices. The single-particle Hamiltonian H0 containsthe kinetic and potential-energy terms. The Fermi energyis taken to be the largest relevant energy scale. The localexchange field can be position dependent close to theinterface, �xc�r�, and the pair potential ��k; r� can alsobe location as well as wave-vector, k, dependent [8]. Forsimplicity, we assume that the exchange field and the pairpotential are uniform inside the superconductor and dropto zero at the FS/N interface: �xc�r� � �xc��z� and

2004 The American Physical Society 087201-1

(t)m(t)∆I s

I c eV=h /2ω

H=γ ω

−e

+h

−e − like q.p.

h+

N

(a)

diso

rder

inte

rfac

e

FS

(b)

− like q.p.k

FIG. 1. (a) A ferromagnetic superconductor coupled to anormal reservoir through a specular FS/N interface in serieswith a normal disordered region. (b) Electrons incident on theFS/N interface from the normal side are reflected as electronsor holes and transmitted as decaying electronlike or holelikequasiparticles.

VOLUME 93, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S week ending20 AUGUST 2004

��k; r� � ��k���z�, where ��z� is the Heaviside stepfunction and z is the coordinate perpendicular to theFS/N interface.

Fermionic statistics dictates ��k� � ��T��k� [9].The singlet (triplet) part of � thus needs to have even(odd) parity: d0�k� � d0��k� and d�k� � �d��k�. Westudy in the following two simple cases of triplet super-conductors: opposite-spin pairing (OSP) along the ex-change field, d�k� �xc � 0 and equal-spin pairing(ESP), d�k� � �xc � 0 [10]. Triplet OSP superconductorsare described by a (complex-valued) vector d�k� �d�k�m along the magnetization direction. We show thatthe transport properties in triplet OSP are similar to thoseof singlet pairing. In ESP, superconductivity occurs inde-pendently for spins along and opposite to the magnetiza-tion direction: by choosing the magnetization along the zaxis, the superconducting pair potential decomposes intotwo terms, corresponding to spins up and down along thez axis, d"�#��k� � �dx�k� � idy�k�. Since superconduct-ing correlations do not mix the spin-up and down sub-systems, two ESP phases can be distinguished: the A�

1phase, where pairing occurs only for spin � (i.e., d� � 0and d�� � 0) and the A2 phase, where pairing occurs forboth spins. A large exchange interaction, xc * jd0j�jdj�,suppresses superconducting singlet (triplet) OSP correla-tions [10]. We therefore only consider xc < jd0j�jdj� forthese systems, so that the quasiparticle excitations have afinite gap. Triplet ESP have a quasiparticle gap in thesuperconducting spin channels independent of the sizeof the exchange interaction [10], and we do not makeassumptions about the ratio of the exchange field to thepair potential in this case.

Let us apply the standard scattering-matrix approach[11–15] to an FS/N system. We assume that theHamiltonian H0 is continuous across the FS/N interfaceand incorporates interfacial disorder and band-structuremismatch into a ‘‘disordered’’ normal region [11].Similarly to Ref. [11], we solve the BdG equation for anelectron (or hole) incident on the specular FS/N interfacefrom the normal-metal side. The total reflection matrix isthen found by concatenating the FS/N reflection with thescattering by the normal disordered region for electronsand holes. The FS layer, in series with the disorderedregion, is viewed as a scatterer for electrons supplied bythe normal reservoir, see Fig. 1. The problem is simplifiedin the clean-superconductor limit, where the FS meanfree path is longer than the superconducting coherencelength �hvF=����, expressed in terms of the Fermi veloc-ity vF and quasiparticle gap �. Focusing on the low-temperature regime, kBT �, we define the M Mspin-dependent electron ! electron and electron ! holereflection matrices r�

ee and r�he, where M is the total

number of quantum channels, or transverse waveguidemodes, at the Fermi level of the normal-metal lead and� is the spin label along the magnetization direction forincident electrons. For singlet or triplet OSP, r�

he describes

087201-2

reflection into holes with spin ��, while for triplet ESP,reflected holes have the same spin � as incident electrons.

We find that for both dc and adiabatic ac transport, spinand charge currents are uniquely determined by two real-valued conductances and one complex-valued conduc-tance, representing transport of spins aligned to the mag-netization, antialigned to the magnetization, andtransverse to the magnetization, respectively:

g"" � Tr�

r"he�r"he�

y

�; g## � Tr

�r#he�r

#he�

y

�; (2)

g"# � Tr�1� r"ee�r

#ee�y � �r"he�

yr#he

�: (3)

It is convenient to define the total conductance g � g"" �g## and the polarization p � �g"" � g##�=g. We will belowinterpret how these conductances determine charge andspin flow, and discuss their values in various limits.

Let us first fix the magnetization direction m and con-sider pumping by an adiabatically varying phase "�t� ofthe superconducting pair potential, @t" � ! � xc;��= �h. As is well known, this time-dependent prob-lem can be transformed into a dc problem by a gaugetransformation U�t� � ei"�t�=2: varying the phase resultsin nonequilibrium transport corresponding to a dc re-sponse at a voltage bias �h!=�2e�. The pumped currentis thus nothing more than a response to a voltage V ��h!=�2e� applied between the superconducting condensateand the normal-metal, and we compute

Ic �e2�

g! and Is � ��h4�

pg!m: (4)

g is thus the usual Andreev-reflection conductance, inunits of e2=�� �h�. The spin current is determined by theconductance polarization p. Equation (4) can also bederived in the ac pumping framework of Ref. [15].

087201-2

VOLUME 93, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S week ending20 AUGUST 2004

Pure spin flow is generated by the variations in themagnetization direction m, induced by, e.g., a resonant rfmagnetic field. In Ref. [14] we computed the correspond-ing pumped spin flow into the normal-metal,

I s ��h

4�

�Reg"#m

dmdt

� Img"# dmdt

�; (5)

for a general F/N contact with no superconducting corre-lations. Here g"# � Tr�1� r"ee�r

#ee�y� is the F/N mixing

conductance for transverse spins expressed in terms ofthe normal-side reflection matrices [13]. The pumped spinflow induces an enhanced Gilbert damping when thenormal-metal is a good spin sink, so that g"# is experi-mentally measurable [14]. We have generalized Eq. (5) tospin pumping by a superconducting ferromagnet: thepumped spin flow remains of the form (5) with a rede-fined mixing conductance g"# (3). There is no accompany-ing charge pumping. We derive Eqs. (3) and (5) by twodifferent methods: first, we extend the pumping approachof Ref. [14], where the scattering matrix is time-dependent in spin space due to a slow variation in themagnetization direction m, to include electron ! holereflection. Secondly, and much simpler, the calculationis reformulated as a dc problem in the spin frame ofreference which is moving together with m�t�. The latteris achieved by (instantaneously) applying a spin-rotationoperator around the vector m @tm and correspondinglyadding a new term in the Hamiltonian: �0 � �� �h=2��m @tm� � �. This term corresponds to an equilibriumtransverse spin accumulation which, in turn, induces thespin current (5). The mixing conductance (3) is obtainedafter extending the F/N dc theory of Ref. [13] to accountfor electron ! hole reflection. By unitarity of the scatter-ing matrix [11], the real part of g"# is bounded from aboveby twice the number of channels: Reg"# � 2M.

For triplet OSP superconductors, the derivation leadingto Eq. (5) assumes that the triplet pair-potential directionmoves together with m, while its wave-vector dependenceis locked to the atomic lattice: d�k; t� � d�k�m�t�. Fortriplet ESP, the situation is more complex since the vectorpair potential d is perpendicular to m, resulting in anadditional dynamic degree of freedom. In particular, themagnetization motion does not uniquely determine thetrajectory of d. In deriving Eq. (5), we assumed that drotates together with m around m’s instantaneous rota-tion axis. We can get more complex trajectories by com-bining (instantaneous) rotations of m and d with‘‘twisting’’ of d around m and the overall phase variationof the pair potential. The induced currents will then begiven by adding the corresponding contributions topumping. Calculating the exact trajectories for realisticsystems, which might be governed by spin-orbit interac-tions in the lattice field or other microscopic details, liesbeyond the scope of this paper. A simple example is theclockwise rotation of d around m with frequency !which induces additional currents

087201-3

Ic � �e2�

pg! and Is ��h

4�g!m: (6)

Note that while the preceding equations are general,Eq. (6) applies to the case of triplet ESP only. For a FS/N system disconnected from an Ohmic circuitry, the low-frequency charge flow vanishes, so that the overall phasevariation of d and its twisting around m must result incanceling charge currents, Ic’s, in Eqs. (4) and (6), but afinite net spin current for p < 1. Equation (6) can bederived similarly to Eq. (5), either as a time-dependentpumping problem or a dc problem in the gauge-transformed frame of reference rotating with d.

Equations (4)–(6) are general expressions for pumpedcharge and spin flows by varying FS order parameters.These currents are all governed by two real-valued andone complex conductance parameters: g"", g##, and g"#,which can be evaluated in microscopic models. A finitevalue of the electron ! hole reflection coefficient, r�

he,requires that an electron incident from the normal-metal reservoir gets transmitted through the disorderedregion, converted into holes at the interface, and trans-mitted back into the normal lead as a hole, see Fig. 1. Forsmall (characteristic) transmission eigenvalues T of thenormal disordered region, g"" and g## therefore scale as T2

rather than as T, as in the Landauer-Buttiker formula foran N/N contact. Indeed, the Andreev conductance for asinglet nonmagnetic superconductor/normal-metal con-tact was shown to be gS=N �

Pm2T2

m=�2� Tm�2, where

m labels the transmission eigenvalues for scattering in thenormal-metal [11]. In the limit of no disorder (and noband-structure mismatch), Tm � 1 and gS=N � 2M, i.e.,Andreev reflection causes a doubling of the conductance,as compared to the N/N interface. We generalize gS=N tothe case of a magnetic singlet superconductor with ans-wave symmetry of the pair-potential, d0�k� � jd0je

i":

g �Xm

2T2m

�2� Tm�2 � 4�1� Tm�� xc=jd0j�

2 (7)

and p � 0. The mixing conductance (3) of s-wave mag-netic superconductors can also be expressed in terms ofthe scattering matrix of the disordered region, by gen-eralizing the formalism of Ref. [11]. For that we needto concatenate the transfer matrix of the normal disor-dered region with electron $ hole conversion at the inter-face: r"eh�he� � ei�'�"� and r#eh�he� � e�i�'�"�, where ' �

arccos� xc=jd0j�. The situation is even more complicatedfor the k-dependent pair potential (which is always thecase for the odd-parity triplet pairing) and we do notpursue it here. The exception is the case with no disor-dered region in our model, Fig. 1. The conductance pa-rameters for the singlet and triplet OSP in such ballisticsystems are then given by

g �Ak2F2�

; p � 0; and (8)

087201-3

P H Y S I C A L R E V I E W L E T T E R S week ending20 AUGUST 2004

g"# � AZ d2k?

�2��2

�1� exp

��2i arccos

xc

��k�

�; (9)

where the � sign corresponds to singlet (triplet) pairingand � � jd0j or jdj, respectively. k is the incident-electron wave vector at the Fermi level, jkj � kF, andk? is its transversal projection in the lead with crosssection A. r"he � ei�'�"� and r#he � �e�i�'�"�, where ' �arccos� xc=jdj� for the triplet OSP with d � jdjei", for agiven k.

It is instructive to discuss the mixing conductance g"#

(3) values in some special cases of a clean interface withmatched band structures since it can be measured experi-mentally in FMR. For an F/N interface, r�

ee � r�he � 0,

while g"# � M is large determined by the number oftransverse wave-guide channels M, assuming that the Flayer is thicker than the ferromagnetic coherence length�hvF=�� xc�. For a perfect electron ! hole reflection offthe singlet superconductor, we find r"he � �r#he, jr�

hej � 1,in the limit when xc �, resulting in a vanishing mix-ing conductance, as follows from Eq. (9). This is easy tounderstand since r"he � �r#he means that the transversespin- " electrons get reflected as the spin- # holes whichexactly cancel the incident transverse spin current. In theanalogous limit of the triplet OSP, r"he � r#he, jr�

hej � 1,doubling the F/N mixing conductance. We thus find thatg"# � M, 0, 2M for F/N, singlet FS/N, and triplet OSP FS/N interfaces. Since g"# is a direct measure of the ferro-magnetic Gilbert-damping enhancement [14], the onsetand the nature of the superconducting pairing has non-trivial consequences in FMR experiments [7]. It is worth-while noting that g"# ! 2M is a rather special limit forthe mixing conductance in multilayer magnetoelectroniccircuit theories [16,17]. For example, in the case of asymmetric FS/N/FS structure, it should result in a dy-namic locking of the two magnetizations which can bemeasured experimentally [17].

Finally, in triplet ESP with no disordered region,

g �Ak2F4�

; p � �; and g"# � g (10)

in the A�1 phase, and

g �Ak2F2�

; p � 0; and (11)

g"# � AZ d2k?

�2��2f1� exp�i�""�k� � "#�k���g (12)

in the A2 phase. r�he � ie�i"�

, for each superconduct-ing spin channel with d� � jd�jei"�

� 0. Note that inderiving Eq. (5), we made a convention for the instanta-neous coordinate system of x / m @tm, y / �@tm, andz � m. (It is necessary to specify the coordinate-systemconvention in the case of the A2 phase, because d"�#� �

�dx � idy and, since d transforms as a vector, the relativephase of d"�#� entering Eq. (12) depends on the choice of

VOLUME 93, NUMBER 8

087201-4

the x and y axes.) It then follows that the second term inthe curly brackets of Eq. (12) is modulated for a small-angle precession of m, unless d twists around m with itsprecession frequency (i.e., following the instantaneousrotation axis). In the former case, the theory thereforepredicts an anisotropic Gilbert-damping parameter. Forthe triplet ESP A2 phase, the symmetry of the super-conducting order parameter may thus be seen in an addi-tional anisotropic FMR line width, as one changes thetemperature across the F-to-FS transition.

In summary, we have studied the interplay betweenferromagnetism and superconductivity in adiabaticpumping by varying order parameters in FS materialsin contact with normal metals. We demonstrate that thesymmetry of the superconducting pair potential is re-flected in the conductance parameters which govern thepumped spin and charge flows. An experimental quantityof a particular interest, which encodes information aboutboth ferromagnetic and superconducting correlations, isthe mixing conductance g"# which governs the Gilbert-damping of the magnetization dynamics [14]. Conse-quently, signatures of the FS order parameter can be mea-sured in thin film FS/N or FS/N/FS FMR experiments.

We are grateful to G. E.W. Bauer, B. I. Halperin, andA. Sudbø for stimulating discussions. This work wassupported in part by the Norwegian Research Council,Nanomat Grants No. 158518/143 and No. 158547/431, andthe Harvard Society of Fellows.

[1] J. Tallon et al., IEEE Trans. Appl. Supercond. 9, 1696(1999).

[2] S. S. Saxena et al., Nature (London) 406, 587 (2000).[3] C. Pfleiderer et al., Nature (London) 412, 58 (2001).[4] D. Aoki et al., Nature (London) 413, 613 (2001).[5] D. Huertas-Hernando, Yu. V. Nazarov, and W. Belzig,

Phys. Rev. Lett. 88, 047003 (2002).[6] Applications of Magnetic Nanostructures, edited by S.

Maekawa and T. Shinjo (Taylor and Francis, New York,2002); Semiconductor Spintronics and QuantumComputing, edited by D. D. Awschalom, D. Loss, andN. Samarth (Springer-Verlag, Berlin, 2002).

[7] A. Fainstein et al., Phys. Rev. B 60, R12597 (1999).[8] C. Bruder, Phys. Rev. B 41, 4017 (1990).[9] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991).

[10] B. J. Powell, J. F. Annett, and B. L. Gyorffy, J. Phys. A36, 9289 (2003).

[11] C.W. J. Beenakker, in Mesoscopic Quantum Physics,edited by E. Akkermans et al. (Elsevier Science B.V.,New York, 1995), p. 279.

[12] P.W. Brouwer, Phys. Rev. B 58, R10135 (1998).[13] A. Brataas, Yu. V. Nazarov, and G. E.W. Bauer, Phys. Rev.

Lett. 84, 2481 (2000).[14] Y. Tserkovnyak, A. Brataas, and G. E.W. Bauer, Phys.

Rev. Lett. 88, 117601 (2002).[15] M. Blaauboer, Phys. Rev. B 65, 235318 (2002).[16] G. E.W. Bauer et al., Phys. Rev. B 67, 094421 (2003).[17] B. Heinrich et al., Phys. Rev. Lett. 90, 187601 (2003).

087201-4