Dynamic Magnetoelectric Effect in Ferromagnet/Superconductor Tunnel Junctions

Mircea Trif and Yaroslav Tserkovnyak

Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA(Received 21 April 2013; published 20 August 2013)

We study the magnetization dynamics in a ferromagnet/insulator/superconductor tunnel junction and

the associated buildup of the electrical polarization. We show that for an open circuit, the induced voltage

varies strongly and nonmonotonically with the precessional frequency, and can be enhanced significantly

by the superconducting correlations. For frequencies much smaller or much larger than the super-

conducting gap, the voltage drops to zero, while when these two energy scales are comparable, the

voltage is peaked at a value determined by the driving frequency. We comment on the potential utilization

of the effect for the low-temperature spatially resolved spectroscopy of magnetic dynamics.

DOI: 10.1103/PhysRevLett.111.087602 PACS numbers: 76.50.+g, 72.15.Gd, 74.25.F�, 85.30.Mn

The field of spintronics has evolved tremendously overthe last decades, leading to important conceptual and tech-nological advances in spin-based memories, sensing, andlogic [1]. The manipulation and detection of the spindegrees of freedom, such as collective magnetization, liesat the heart of spintronics, with the magnetic field—staticor time dependent—providing a direct way to access it.It turns out, however, that electric rather than magneticcontrol it is often preferred for spintronic manipulations[2], as the former can exert larger torques, act faster, andcan be applied or detected with a finer spatiotemporalresolution [3,4].

There are two main routes for the electrical control ofmagnetization dynamics: one relies on spin-polarized elec-trical currents, which couple to the magnetization via thespin-transfer torque [3], while the other, which is also amore recent development, is based on controlling magneticanisotropies by applying voltage pulses [4]. Focussing onthe former, the most basic implementation for creating spin-polarized currents is to pass unpolarized electrical currentthrough a fixed ferromagnet. An alternative way for gen-erating spin currents relies on the spin Hall effect, whichleverages the spin-orbit interaction in the material and doesnot require any ferromagnetic polarizers. In fact, this hasbeen established as a primary tool both for manipulatingand detecting the magnetization dynamics, transformingspin signals (spin currents) directly into electrical signals(Hall voltages), or vice versa [5]. The spin-Hall-inducedvoltage scales with the lateral dimension of the sample,making itself extremely useful for larger devices [6]. Themain drawback, however, is that it loses its utility when itcomes to detecting local magnetization dynamics.

In this Letter, we study the voltage induced by magne-tization dynamics in a circuit involving a driven metallicferromagnet coupled to an s-wave superconductor via aweak tunnel barrier. It was previously shown that magne-tization dynamics can induce a voltage in tunnel junctionswith normal metals or static reference ferromagnets [7,8],by the process of an adiabatic charge pumping. The

resulting voltage is, however, small, compared to the driv-ing frequency, at typical microwave powers. We show herethat the singularity associated with the quasiparticle den-sity of states can significantly enhance such dynamicallyinduced voltages at driving frequencies corresponding tothe superconducting gap. The underlying pumping behav-ior is thus necessarily nonadiabatic.The Bogoliubov–de Gennes Hamiltonian for our hybrid

ferromagnet/superconductor junction (see Fig. 1) reads [9]

HFISðtÞ ¼�p2

2mþ VðrÞ

��z þ �FðrÞ

2mðtÞ � � þ �SðrÞ

2�x;

(1)

in a certain basis, wheremðtÞ is the magnetization directionin the ferromagnet, �FðrÞ ¼ �F�ð�xÞ is the magnetic

FIG. 1 (color online). A schematic of the system. A metallicferromagnet is driven to precess with frequency ! at angle �with respect to the z axis. The effective splitting in the rotatingframe of reference is ! cos� along the magnetization directionm within the ferromagnet. The superconductor is subjected tothe effective spin splitting ! along the z axis, which shifts thequasiparticle bands in the rotating frame. �S is (twice) thes-wave superconducting gap and � is the chemical potential inequilibrium in the laboratory frame. The magnetic precessionpumps charge current I through the tunnel barrier in the closedcircuit, or a voltage V is measured by a voltmeter in the opencircuit.

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(Stoner or s-d) exchange field, �SðrÞ ¼ �S�ðxÞ is the(real-valued) superconducting pair potential, both writtenin terms of the Heaviside step function �ðxÞ,� ¼ ð�x;�y; �zÞ and � ¼ ð�x; �y; �zÞ are Pauli matrices

operating in spin and particle-hole (Nambu) subspaces,respectively, and VðrÞ is the total effective scalar potentialacting on electrons. We suppose the ferromagnetic andsuperconducting regions are separated by an insulatingbarrier, such that VðrÞ is large near x � 0, where x standsfor the direction normal to the junction placed at x ¼ 0.

For a circular precession, we parametrize the magnetiza-tion direction as mðtÞ¼½sin�cosð!tÞ;sin�sinð!tÞ;cos��with! being the precession frequency and � the precessionangle. The exchange splitting �F is assumed to be muchlarger than both the superconducting gap �S and the pre-cession frequency !: �F��S, ! (setting @¼1 through-out), which is typically the case. For a steady precession, itis convenient to switch to a rotating frame of reference,where the ferromagnet is static. This is achieved by a time-dependent unitary transformation of the Hamiltonian:H0

FIS�UyðtÞHFISðtÞUðtÞ�iUyðtÞ@tUðtÞ¼HFISð0Þ�!�z=2with UðtÞ ¼ expð�i�z!t=2Þ [8,10]. In the ferromagneticbulk, the Hamiltonian in the rotating frame becomesH0

F ¼HFð0Þ � ð!=2Þ�k cos�, where �k � mð0Þ � � is the spin

projection on the magnetization direction mð0Þ. We havedisregarded the component perpendicular tomð0Þ, which iseffectively suppressed for !=�F � 1. On the supercon-ducting side, on the other hand, the spin splitting in therotating frame is simply given by!, with the correspondingHamiltonian H0

SC ¼ HSCð0Þ �!�z=2. The spectrum of

the superconductor in the rotating frame is thus shifted

by the effective magnetic (Larmor) field Eð�ÞS ð�Þ¼

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2þð�S=2Þ2

p �!�=2 corresponding to the quasipar-ticle density of states

Dð�ÞS ðEÞ ¼ D0

jEþ!�=2jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEþ!�=2Þ2 � ð�S=2Þ2p : (2)

Here, D0 is the Fermi-level density of states per spinprojection in the normal state, and � ¼ � for spins up or

down along the z axis. Dð�ÞS ðEÞ diverges at energies E !

ð��S �!�Þ=2. Since the spin-up and spin-down quasi-particle subbands in the superconductor are shifted by !,the gap closes in the rotating frame for !>�S. The spin-dependent spectrumon the ferromagnetic side is shifted too,but by! cos� instead of!. In the tunneling regime, we canassume that the ferromagnetic and superconducting bulksare in their separate equilibria in the laboratory frame. Thisrelies on the fact that the magnetization-dynamics inducedpumping is exponentially weak, such that, in particular, ithas essentially no effect on the self-consistent pairingpotential �S.

We now rewrite Hamiltonian (1) in the tunnelingapproximation, which is given in the second-quantizedform by

HT ¼ Xk;q;�

tk;q;�cyF;k;�cS;q;� þ H:c:; (3)

in the rotating frame. Here, tk;q;� � t is the tunneling

matrix element, which is taken to be constant, for simplic-ity, and cFðSÞ;k;� are the electron annihilation operators in

the ferromagnet (superconductor). k and q label orbitalquantum numbers and � spin projection on the magneticdirection. We do not expect a general spin- and k,q-dependent tunneling to affect the qualitative features ofour final results, apart from modifying parameters associ-ated with the spin-dependent density of states in the ferro-magnet. Hamiltonian (3) can be used to calculate thecharge and spin currents flowing from one metal to theother across the tunnel barrier.We compute the out-of-equilibrium charge current using

Fermi’s golden rule for the transition probabilities involv-ing all electron- and hole-like branches in the supercon-ductor. In the tunneling regime, the superconductor can beeffectively viewed as a simple semiconductor [11,12] (spinsplit in the rotating frame), with a singular quasiparticledensity of states at its band edges, according to Eq. (2). Thetotal current induced by the magnetic driving in the pres-ence of a voltage V reads

I ¼ 2�ejtj2X��0

ZdEDð�Þ

F Dð�0ÞS ðEÞjh�j�0ij2

½fð�ÞF ðE� eVÞ � fð�0Þ

S ðEÞ�; (4)

where V is the electrochemical potential (applied and/orinduced) of the ferromagnet relative to the superconduct-

ing condensate, e < 0 is the electron’s charge, Dð�ÞF is

the spin-� Fermi-level density of states in the ferromag-

net, and fð�ÞF ðEÞ ¼ fexp½ðEþ!� cosð�Þ=2Þ=kBT� þ 1g�1,

fð�ÞS ðEÞ ¼ fexp½ðEþ!�=2Þ=kBT� þ 1g�1 are, respec-

tively, the Fermi-Dirac distributions in the ferromagnetand superconductor at temperature T. For the spin matrixelements, � labels spin in the ferromagnet along mð0Þand �0 in the superconductor along z, so that we havejh�j�0ij2 ¼ cos2ð�=2Þ���0 þ sin2ð�=2Þ� ���0 , where �� ���. We are now equipped to calculate the resultant currentin the presence of the magnetization dynamics, at anarbitrary temperature. In contrast to band semiconductors,the gap �S ! �SðTÞ itself depends on T, closing at Tc �1:76�Sð0Þ within the s-wave BCS model [12], while theFermi level is pinned, in the laboratory frame, midgap bythe superconducting condensate.We start by computing the charge current as a function

of ! and V at T ¼ 0. This regime allows for an analyticalevaluation of the current, as well as capturing qualitativefeatures that extrapolate to finite T. Writing I ¼ P

��0I��0 ,we arrive at

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I��0 ¼ Ið�Þ0 ð1þ��0 cos�Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½!ð���0 cos�Þ=2�V�2 � 1

q

� Xs¼�1

�½!ð���0 cos�Þ=2�Vþ s� � 1

�; (5)

where Ið�Þ0 ¼ �e�Sð0Þjtj2Dð�ÞF D0. Here, we are measuring

all the energies in units of the zero-temperature gap�Sð0Þ=2. In the following, we focus on the open-circuitcase (see Fig. 1), so that the current flowing through theheterostructure is zero (I ¼ 0), while the voltage V inducedby the magnetization dynamics is measured by a voltmeter.We are specifically interested in regimes where this voltagecan be significantly enhanced by the superconductor. In thetunneling regime, the Andreev processes are strongly sup-pressed, so that transport is governed by the excited qua-siparticles [13]. This means that for !< 1 (in units of�S=2) and T ¼ 0, the quasiparticles are gapped out andV � 0, as long as the microwave power is low enough sothat multimagnon processes do not contribute [15].

In Fig. 2, we plot the open-circuit voltage V at T ¼ 0, asa function of frequency ! and precession angle �. We findthree different regimes in the dependence of V on !,corresponding to the activation of new spin channels inthe tunneling current (5) with increasing !. The voltagedepends on! nonmonotonically, reaching a maximum Vc1

which, as shown later, can be as large as 1. More specifi-cally, we find that below a minimum frequency !0 ¼2=ð1þ cos�Þ the voltage is always zero (assuming onlypositive frequencies, i.e., !> 0), while for !>!0 thereis a finite voltage drop in the system. The total current in

this case is given initially by Iþ� þ I�þ for 0< �< �=2and Iþþ þ I�� for �=2< �< �. At certain higher fre-quencies (whose specific values are discussed below), theterms Iþþ and I�� get activated for 0< �<�=2 and Iþ�and I�þ for �=2< �<�. Successive activation of thesedifferent tunneling channels defines voltage landscapesas shown in Fig. 2. Note that the symmetry ofour system dictates that Vð�!Þ ¼ �Vð!Þ as well asVð�� �Þ ¼ �Vð�Þ, allowing us to henceforth restrictour discussion to � 2 ð0; �=2� and !> 0. While it ispossible to extract analytical expressions for the voltage(and the activation frequencies) as a function of frequencyat T ¼ 0, these are too long and unilluminating. Instead,we will derive approximate expressions for the inducedvoltage in different frequency ranges.At frequencies! above!0 but still sufficiently low such

that j!cos2ð�=2Þ � Vj> 1 and j!sin2ð�=2Þ � Vj< 1[where V � Vð!; �; PÞ needs to be solved self-consistentlyfor, in the open circuit], the voltage V increases monotoni-cally with !, as shown in Fig. 2, given approximatively[as an expansion in (1� P)] by the following expression:

V ��!cos2

�

2� 1

��1� 2!

�1� P

1þ P

�2cos2

�

2

�: (6)

Here, we defined the polarization P ¼ ðD#F �D"

FÞ=ðD#

F þD"FÞ and assumed 0< P ’ 1 (large polarization).

We see that in this limit the voltage increases roughly linearlywith!, consistent with the exact result shown in Fig. 2, untilthe frequency reaches a critical value !c1 ¼ 2½ð1þ PÞ2 þð1� PÞ2 cos��=½4Pþ ð1� PÞ2sin2�� corresponding to thecondition j!sin2ð�=2Þ þ Vj ¼ 1. At this frequency, the termIþþ starts contributing to the total current I, and the voltagestarts decreasing monotonically with increasing !, asdepicted in Fig. 2. The critical voltage in the circuit at! ¼ !c1 reaches a value Vc1 given by

Vc1 ¼ 4P cos�

4Pþ ð1� PÞ2sin2� ; (7)

which approaches unity as � ! 0. This is the maximumvoltage achievable in the circuit, which, in physical units, isbounded by the superconducting gap �S=2. Note that Vgenerally increases with decreasing angle �, as shown inFig. 2, being in stark contrast to the typical F=I=F magnetictunnel junction with one ferromagnet being free and onepinned, where V / sin2�, i.e., vanishing with microwavepower, as the precession angle � ! 0. Moreover, in thepresent setup, the voltage does not depend on the orientationof the precession axis, as opposed again to the F=I=F mag-netic tunnel junction, where the induced voltage is sensitiveto the relative orientation of the axis of precession withrespect to the pinned reference ferromagnet [7,8].For !>!c1, we find that V decreases with increasing

!, and it is given approximatively by the followingexpression:

FIG. 2 (color online). Dependence of the induced voltage V onfrequency ! and cos� at zero temperature. The black (back)dashed curve shows the activation frequency !0 ¼2=ð1þ cos�Þ, while the red (middle) dashed and blue (front)dashed curves correspond to the voltages Vc1 [Eq. (7)] and Vc2

[Eq. (9)] that signal the activation of new transport channels inthe current I. In these plots, we set P ¼ 2=3 and expressed V and! in units of �S=2.

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V �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

�2P

1þ P!sin2

�

2

�2

s�!sin2

�

2; (8)

where we neglected corrections of the order of1=!cos2ð�=2Þ � 1. Moreover, !sin2ð�=2Þ 1 since, inthis regime, j!sin2ð�=2ÞþVj>1 and j!sin2ð�=2Þ � Vj<1, while 0< V < 1 for all frequencies !.

As the frequency increases further beyond !c1, thevoltage continues to decrease according to Eq. (8) untilthe term I�� in the total current I is activated, whichhappens when j!sin2ð�=2Þ � Vj ¼ 1. This determinesthe second transition frequency !c2 � ð1þ PÞ2=ð1þ 2PÞsin2ð�=2Þ, at which the voltage is given by

Vc2 � P2=ð2Pþ 1Þ; (9)

neglecting corrections of order 1=!cos2ð�=2Þ. Accordingto Eq. (9), the maximum voltage at this transition point isVc2 � 1=3, corresponding to P ¼ 1.

Finally, for !>!c2, all terms in Eq. (5) contributeto the charge current I, and the voltage tends to zero as! ! 1. Specifically, for ! � !c2, we find the followingapproximation for the voltage:

V � Pcsc2ð�=2Þ=!: (10)

The vanishingly small voltage at large! reproduces the nilresult of ferromagnet/normal-metal junctions [8], sincethe superconducting correlations become unimportant atfrequencies ! � �S.

Note that at each transition point from one regime toanother, the slope of V with respect to ! is discontinuous,as opposed to the case of the normal magnetic junctions,where it is constant for typical microwave frequencies! � �F [8]. The resulting voltages are limited by thedriving frequency, achieving values ! when ! 1 (or!�S=2, in physical units) which are well within theexperimental reach for typical superconducting gaps onthe order of a few Kelvin, corresponding to a fraction ofa milli-electron-volt. This is significantly larger than the1 �eV voltages induced by magnetic dynamics innormal metals [7].

At finite temperatures, we expect rounding off of thesharp features present at the transitions between differentaforementioned regions as well as a reduction of the in-duced voltage, due to the diminished superconductivity. Inorder to calculate explicitly the pumped current at T � 0using Eq. (4), we need to account not only for the thermallybroadened Fermi distribution but also for the T dependenceof the gap �SðTÞ, which closes at Tc. We write �SðTÞ ¼�Sð0ÞFðTÞ, where FðTÞ is a dimensionless function, whichcan be found numerically from the self-consistent gapequation in the BCS theory [12]. In Fig. 3, we plot thecorresponding dependence of V on! (left) and cos� (right)at different temperatures, with the black curves showingthe T ¼ 0 result. We see that the signal becomes visiblyreduced as the temperature increases from zero and,

moreover, it singularly changes the cos� ! 1 behaviorsuch that V ! 0 for any T � 0 instead of a finite V ! 1at precisely T ¼ 0.In the left panel of Fig. 4, we plot the induced voltage as

a function of T for different frequencies ! (left) and cos�(right). As expected, all sharp features are smoothed out byfinite T, with the signal eventually vanishing as T ! Tc. Atsubcritical temperatures, the voltage shows a nonmono-tonic behavior as a function of T, for a wide range offrequencies and angles, surprisingly reaching values inexcess of the T ¼ 0 result. This is attributed to thermalactivation of otherwise closed pumping channels. Had thegap been the same at all temperatures, the voltage wouldhave exhibited an even more dramatic increase as thetemperature is raised from T ¼ 0, which is suppressed bythe thermal reduction of the gap.

1.0 0.5 0.0 0.5 1.01.0

0.5

0.0

0.5

1.0

0 5 10 15

0.2

0.4

0.6

0.8

1.0

FIG. 3 (color online). Dependence of the induced voltage V on! (left panel), and cos� (right panel) for different temperatures.The increasingly lighter gray curves correspond to T=Tc ¼ 0:1,0.3, 0.5, 0.7, and 0.9 with cos� ¼ 0:8 (left) and ! ¼ 3 (right).Here c1 and c2 label the critical points at which new chargetransport channels are activated [see Eqs. (7) and (9)] at T ¼ 0.In these plots, we set P ¼ 2=3, and expressed V and! in units of�Sð0Þ=2.

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

FIG. 4. Dependence of the induced voltage V on temperaturefor different values of ! (left panel) and angle � (right panel).Left: angle dependence of the induced voltage for ! ¼ 1:1, 2, 3,5, and 10, with lighter gray corresponding to increasing fre-quency, with cos� ¼ 0:9. Right: The increasingly lighter graycurves correspond to cos� ¼ 0:9, 0.7, 0.5, 0.3, and 0.2, with! ¼ 3. In these plots, we set P ¼ 2=3 and expressed V and ! inunits of �Sð0Þ=2.

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In conclusion, we analyzed the dynamics of a ferromag-net tunnel coupled to a conventional superconductor. Wefind a large (compared to the normal junction) nonadia-batic electrical polarization induced in an open circuit,when ! �S. We speculate that it could be useful, inpractice, as a local probe for magnetic dynamics. Thisvoltage can be easily understood to stem from the factthat the superconductor effectively behaves as a staticreference ferromagnet in the rotating frame, and we predictthe maximum induced voltage of order of �S for smallprecession angles and temperatures T < Tc.

This work was supported in part by FAME (an SRCSTARnet center sponsored by MARCO and DARPA), theNSF under Grant No. DMR-0840965, and GrantNo. 228481 from the Simons Foundation.

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[15] Although absent from our model with circular precession,we expect the multimagnon processes to become relevantfor general elliptical precession at sufficiently high micro-wave powers.

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