New Journal of Physics The open–access journal for physics
The chiral superconductor–ferromagnet–chiral superconductor Josephson junction P M R Brydon1,2,4 , C Iniotakis3 and Dirk Manske1 1 Max-Planck-Institut für Festkörperforschung, Heisenbergstreet 1, 70569 Stuttgart, Germany 2 Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany 3 Institut für Theoretische Physik, ETH Zürich, CH-8093 Zürich, Switzerland E-mail:
[email protected] New Journal of Physics 11 (2009) 055055 (15pp)
Received 28 November 2008 Published 29 May 2009 Online at http://www.njp.org/ doi:10.1088/1367-2630/11/5/055055
Abstract. We study a Josephson junction between two chiral p-wave superconductors separated by a magnetically active tunneling barrier. We find that the interaction of the barrier magnetic moment with the spin of the tunneling triplet Cooper pairs is responsible for a number of unconventional Josephson effects. For example, we show that the critical current depends on the orientation of the moment, and that a finite spin current is possible in the ground state of the junction. In the case where the two superconductors have opposite chirality, we have also calculated the chiral Andreev bound state spectrum and the associated interface currents at the barrier, which have similar dependence on the magnetic moment to the Josephson currents.
4
Author to whom any correspondence should be addressed.
New Journal of Physics 11 (2009) 055055 1367-2630/09/055055+15$30.00
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2 Contents
1. Introduction 2. The quasiclassical Green’s function 3. The equal chirality junction 3.1. Spin-filtering barrier . . . . . . 3.2. Spin-flipping barrier . . . . . . . 4. The opposite chirality junction 4.1. Spin-filtering barrier . . . . . . 4.2. Spin-flipping barrier . . . . . . . 5. Conclusions Acknowledgments References
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2 3 6 7 7 11 11 12 14 14 15
1. Introduction
The discovery of superconductors with unconventional (i.e. not s-wave) pairing symmetries is one of the great highlights of the past quarter century of condensed matter physics. All the cuprates fall into this category [1, 2], as well as many organic [3] and heavy fermion systems [4, 5] and perhaps also the newly found FeAs-based superconductors [6]. In the vast majority of these materials, the parity of the pairing symmetry is even in the orbital momentum (e.g. d-wave, extended s-wave) and even in the frequency, implying a spin-singlet Cooper pair. Much rarer are the systems with an odd-parity orbital (e.g. p-wave, f-wave) and even-parity frequency pairing states, which requires a spin-triplet Cooper pair. There are currently no known examples of odd-frequency pairing states, for which the association between orbital parity and spin state is reversed, although odd-frequency correlations have been predicted at superconductor interfaces due to a proximity-like effect [7]. In this paper, we will focus exclusively on the even-frequency odd-orbital spin-triplet pairing state, which is likely realized in Sr2 RuO4 [8], UGe2 [9], UPt3 [5] and URhGe [10]. Of these, the most well-characterized is Sr2 RuO4 [11]: the absence of the Knight shift implies that only Cooper pairs with ±1 spin projections in the a–b plane are present [12], while muon spin relaxation experiments indicate a time-reversalsymmetry breaking state [13]. By symmetry requirements, this fixes the superconducting state of Sr2 RuO4 to be chiral p-wave, 1± (k) ∝ (k x ± ik y ) [14]. The ‘chiral’ nature q of this state is reflected in the k-dependent phase of the condensate. Writing k x ± ik y = k x2 + k 2y exp[±i arctan(k y /k x )], we see that the phase winds about the positive z-axis in the anti-clockwise direction for the + state, and in the clockwise direction for the − state. One of the most remarkable consequences of this intrinsic chirality is the existence of surface currents at impenetrable barriers [15]: due to the k-dependent phase, specularly reflected Cooper pairs at non-normal incident trajectories acquire a phase difference 1φ 6= 0, π, driving a surface current that flows parallel to the interface. The Andreev surface states [16] that form at such an interface with normal vector along the x-axis therefore have a dispersion ∝ k y , where the sign is determined by the chirality of the superconductor. The two chiral pairing states are degenerate, however, and so they may coexist as distinct domains within a bulk Sr2 RuO4 sample. The surface bound states that form at the edge of each domain hybridize, New Journal of Physics 11 (2009) 055055 (http://www.njp.org/)
3 resulting in dispersive bound states at domain walls which carry a current flowing parallel to the interface [15, 17]. A major concern in the study of unconventional superconductors is establishing the symmetry of the pairing state. This can be most directly probed by tunneling experiments, and considerable effort has been directed at obtaining [1, 2] and interpreting [16, 18] the tunneling spectra of these systems. Alternatively, the signatures of unconventional pairing states can also be detected in the characteristics of Josephson junctions constructed from such superconductors [16]. Much work has recently been performed to identify the distinguishing features of devices involving triplet superconductors: normal metal–triplet superconductor junctions [19]–[21], singlet–triplet Josephson junctions [22]–[24] and triplet–triplet Josephson junctions [17, 22], [25]–[30] have all been studied. A particularly promising approach for the unambiguous identification of a triplet pairing state involves the use of magnetically active elements in the tunnel junction [24, 27, 29, 30]. This allows us to directly access the new degree of freedom provided by the spin of the triplet Cooper pair. It also suggests the possibility of unconventional spin transport properties of such devices, e.g. a spontaneous spin current [26, 28, 30, 31]. Moreover, the study of the interplay between triplet superconductivity and magnetism at their mutual interfaces may be expected to lead to a deeper understanding of their relationship. In this work, we use a quasiclassical method to investigate the transport properties of a Josephson junction between two chiral superconductors with a ferromagnetic tunneling barrier. For both a junction between superconductors of equal and opposite chirality, we demonstrate that the Josephson currents are controlled by the magnitude and orientation of the barrier magnetic moment. In particular, we are able to classify the junction on the basis of whether or not the magnetic moment has a component parallel to the vector order parameters of the triplet superconductors. We provide a physical understanding of the differences between these two cases in terms of the interaction of a tunneling Cooper pair with the barrier moment. We argue that the unconventional dependence of the Josephson charge current on the moment orientation may be considered as a test of the triplet superconducting state. When the junction is constructed from superconductors of opposite chirality, we demonstrate that chiral Andreev bound states form at the tunneling barrier, and we calculate the resulting interface charge and spin currents. 2. The quasiclassical Green’s function
At the core of our method is the construction of the 4 × 4 matrix quasiclassical Green’s function in Nambu-spin space, G(r, k, ωn ) . This may be written in terms of the Riccati parameterization [32, 33] as −1 1 − γ γ˜ 0 1 + γ γ˜ 2γ G(r, k, ωn ) = , (1) 0 1 − γ˜ γ −2γ˜ −(1 + γ˜ γ ) where the so-called coherence factors γ = γ (r, k, ωn ) and γ˜ = γ˜ (r, k, ωn ) are 2 × 2 spin matrices. These are determined by integrating the Riccati equations ˜ k (r)γ − 1k (r), ih¯ vF · ∇γ = −2ωn γ + γ 1 (2) ˜ k (r), ih¯ vF · ∇ γ˜ = 2ωn γ˜ + γ˜ 1k (r)γ˜ − 1
(3)
where vF is the Fermi velocity along the chosen trajectory k, and in a triplet system 1k (r) = ˜ k (r) = −iσˆ y (d∗k (r) · σ ) are the spatially dependent gaps. The d-vector is i(dk (r) · σ )σˆ y and 1 New Journal of Physics 11 (2009) 055055 (http://www.njp.org/)
4 Left superconductor
∼l
γo
γ
l i
Right superconductor
(γ ol )
(γ or )
∼
γ ro
k’
k
k
k’
γ∼ l
γ ri
γ∼ r
( i)
( i) Barrier
z Figure 1. The incoming (blue) and outgoing (red) trajectories along which the
coherence functions must be evaluated. The γ functions are integrated in the direction of the arrow, whereas the γ˜ functions are integrated in the opposite direction. The functions to be integrated along each trajectory are indicated, with those to be integrated from the barrier in brackets. defined as dk = 21 (1↑↑ (k) − 1↓↓ (k))ˆx − 2i (1↑↑ (k) + 1↓↓ (k))ˆy + 1↑↓ (k)ˆz, where 1σ σ 0 (k) is the triplet pairing amplitude for electrons in the states |k, σ i and | − k, σ 0 i. The spin of the Cooper pair formed from these states must lie in the plane perpendicular to dk . The d vector points along the same axis for all k in Sr2 RuO4 , a so-called equal-spin-pairing state where the spins of all Cooper pairs lie in the same quantization plane. To evaluate the Green’s function for a given k, we must determine the coherence factors along the four paths: two incoming towards the barrier and two outgoing away from it. This is shown in figure 1 for the case studied here, where the Fermi surfaces on either side of the barrier are the same. The integration of γ (γ˜ ) is stable along the direction (opposite direction) indicated by the arrows. The initial conditions for the functions to be integrated from the bulk are: q .h i 2 + |d∞ |2 γil (−∞, k, ωn ) = −i(d∞ · σ ) σ ˆ ω + ω (4a) y n L,k n L,k γ˜ol (−∞, k0 , ωn )
=
γir (∞, k0 , ωn )
−i(d∞ R,k0
γ˜or (∞, k, ωn )
= =
∗ −iσˆ y ((d∞ L,k0 )
· σ )σˆ y
∗ −iσˆ y ((d∞ R,k )
q i .h ∞ 2 2 · σ ) ωn + ωn + |dL,k0 |
.h
·σ)
ωn +
.h
q
ωn +
ωn2
q
2 + |d∞ R,k0 |
ωn2
i
2 + |d∞ R,k |
(4b) (4c)
i
,
(4d)
where k0 = (k x , k y , −k z ) is the wave vector of a specularly reflected quasiparticle with incident wave vector k = (k x , k y , k z ) and d∞ ν,k is the d vector in the bulk superconductor on the ν-hand side. The initial conditions for the coherence factors to be integrated from the barrier, γol , γ˜il , γor and γ˜ir , are given in terms of the coherence factors integrated from the bulk and the scattering
New Journal of Physics 11 (2009) 055055 (http://www.njp.org/)
5 matrix describing the barrier. The exact expressions are rather complicated [34, 35], and so we therefore do not state them in full here, but rather only give the illustrative example of γol : † † † ∗ r −1 † ∗ r −1 † −1 ∗ r −1 l ∗ γol = [(S12 − γil S12 γ˜o ) S11 − (S11 − γir S11 γ˜o ) S12 ] (S12 − γil S12 γ˜o ) γi S11 † † † ∗ r −1 † ∗ r −1 † −1 ∗ r −1 r ∗ − [(S12 − γil S12 γ˜o ) S11 − (S11 − γir S11 γ˜o ) S12 ] (S11 − γir S11 γ˜o ) γi S12 .
(5)
Note that all the coherence factors in (5) are assumed to be evaluated at the barrier. We assume the tunneling barrier to be atomically thin, which we model as a δ-function. We allow both magnetic scattering off a moment M = (M⊥ cos(η), M⊥ sin(η), Mz ) and charge scattering off a potential U P at the barrier (see figure 2). The scattering matrices are then written i(k z /kF )geiη (k z /kF )2 + i(k z /kF )(Z + g 0 ) −1 + g 2 + g 0 2 − (−ik z /kF + Z )2 g 2 + g 0 2 − (ik z /kF + Z )2 S11 = (6a) , −iη 2 0 i(k z /kF )ge (k z /kF ) + i(k z /kF )(Z − g ) −1 + g 2 + g 0 2 − (ik z /kF + Z )2 g 2 + g 0 2 − (ik z /kF + Z )2 S12 = 1ˆ + S11 ,
(6b)
where Z = U P m/h¯ 2 kF , g = M⊥ m/h¯ 2 kF and g 0 = Mz m/h¯ 2 kF are dimensionless constants parameterizing the scattering off the charge potential, the moment perpendicular to the z-axis and the moment parallel to the z-axis, respectively. m is the effective mass and kF is the Fermi momentum. Having obtained G(r, k, ωn ), we may calculate several physical quantities of interest. The local angular-resolved density of states is written as N (0) Im Tr G(r, k, + i0+ ) , N (r, k, ) = − (7) 4π where N (0) is the density of states at the Fermi energy. For an interface lying in the plane z = 0, the Andreev bound state energies for the trajectory k are given by the poles of N (z = 0± , k, ). The charge current is found from 1X hEvF Tr {G(r, k, ωn )}i F S (8) IE(r) = −πieN (0) β ω n
while the µ-component of the spin current is * ( " #)+ X ˆ h 1 σˆ 0 ¯ IES,µ (r) = iπ N (0) vEF Tr G(r, k, ωn ) ˆµ 0 σˆ µ∗ 2 β ω n
.
(9)
FS
Since it is not in general possible to perform the Matsubara frequency summation analytically, we numerically evaluate the sum by restricting it to ω < 4kB Tc . In general, the currents have a Josephson component normal to the interface (charge and spin currents I J and I JS,z , respectively), and a surface current component parallel to the interface (charge and spin currents IscC and IscS,z , respectively). With respect to the µ-component of spin, the charge current carried by the spin-σ sector is ( IE − 2eσ IES,µ h¯ )/2. In what follows, the charge and spin currents are evaluated at the barrier. Note that although the charge current is of course the same at z = 0+ and 0− , the spin currents on either side of the junction can differ since the Cooper pair’s spin is not necessarily conserved during tunneling. New Journal of Physics 11 (2009) 055055 (http://www.njp.org/)
6
(a)
(b)
Triplet Triplet Ferromagnetic superconductor superconductor barrier
x
dL y (p +ip ∆ z y) z0
z0
Figure 2. Schematic diagrams of the Josephson junctions studied in this paper.
(a) The equal chirality junction, where both left and right superconductors are in the pz + i p y pairing state, and (b) the opposite chirality junction, where the left superconductor is in the pz + i p y pairing state while the right superconductor is in the pz − i p y pairing state. The phase of the chiral superconductor is k dependent, and the circular arrows indicate the direction about the x-axis of increasing phase. We simplify our analysis by neglecting the spatial variation of the superconducting gaps, i.e. we assume dν,k (r) = d∞ ν,k = dν,k . This approximation may be justified on the basis that the effects we are most interested in are related to the spin pairing state of the superconductors. We nevertheless note that there can be a signficiant reconstruction of the pairing state of a chiral superconductor at interfaces due to inducement of a subdominant pairing component with opposite chirality [15, 21]. Since we find that the Josephson behaviour in junctions between superconductors with equal and opposite chirality are qualitatively similar, it seems likely that including this effect will only lead to quantitative revisions of our predictions. 3. The equal chirality junction
We first examine a Josephson junction between two chiral superconductors with the same chirality, i.e. both left and right superconductors are in the pz + i p y -symmetry pairing state, dL,k = dR,k e−iφ = 1(T )(k z + ik y )ˆx. The gap magnitude 1(T ) is assumed to have Bardeen–Cooper–Schrieffer (BCS) temperature dependence and a T = 0 magnitude 10 . Although it is usual to choose d to lie along the z-axis [15, 17, 22, 25], it is more convenient to assume that the superconductors are in an equal-spin-pairing state with respect to the z-axis when discussing the spin current, and so we require d ⊥ zˆ . The results for the usual choice of coordinates may be recovered by appropriate rotation. As the suggested pairing state for Sr2 RuO4 has the d vector pointing along the axis of the cylindrical Fermi surface, we restrict the momentum integrals to a circle |k| = kF in the y–z plane. The junction is schematically illustrated in figure 2(a). The Andreev bound states in the equal chirality junction do not show a chiral dispersion, as they result from the hybridization of surface bound states with opposite chirality. There is therefore no component of the current parallel to the interface of the junction, as in a Josephson junction between non-chiral superconductors. Although the (Josephson) charge and New Journal of Physics 11 (2009) 055055 (http://www.njp.org/)
7 Total, g′=0.5 Spin−↑, g′=0.5 Spin−↓, g′=0.5 Total, g′=0
0.3
IJ (2πe∆0/h)
(b)
0.4
0.2
0.12 0.08
I JS,z (∆0)
(a)
0.1 0.0 −0.1
Total, g′=0.5 Spin−↑, g′=0.5 Spin−↓, g′=0.5
0.04 0.00 −0.04
−0.2 −0.08
−0.3 −0.4 0
0.5
1
φ/π
1.5
2
−0.12 0
0.5
1
φ/π
1.5
2
Figure 3. (a) Charge current versus phase and (b) spin current versus phase
relationships for the spin filter with Z = 1, g = 0 and T = 0.3Tc . We show the contributions to both currents from each spin sector. In (a) we also show the charge current for a non-magnetic barrier (g = 0); the spin current for a nonmagnetic barrier is vanishing. spin currents normal to the barrier may be written in terms of the Andreev bound states, it is not necessary to consider them in detail to obtain a physical understanding of the junction’s behaviour. 3.1. Spin-filtering barrier The behaviour of the junction may be classified by whether or not the magnetic moment M has a component parallel to the two d vectors. The simpler case is when M ⊥ dL,R , where the moment defines a special basis within the quantization plane of the two superconductors. It is convenient to work within this basis, as the corresponding spin quantum numbers are conserved during scattering at the barrier. Without loss of generality, we take M to lie along the z-axis; the results for other orientations require only a rotation in spin space. If the barrier also possesses a charge scattering term Z 6= 0, we then find that the effective barrier potential for a spin-σ quasiparticle is Z − σ g 0 . For Z , g 0 > 0, the transparency of the junction in the spin-↑ sector is greater than that in the spin-↓ sector. This therefore favours the transmission of spin-↑ Cooper pairs over spin-↓ Cooper pairs, i.e. the barrier acts to ‘filter out’ tunneling spin-↓ Cooper pairs [30]. Since the currents flowing through each spin sector are no longer equal, we obtain a net Josephson spin current through the barrier. We show the charge and spin currents through such a spin filter in figures 3(a) and (b), respectively. We note that the critical current of the junction, I J c = maxφ {I J (φ)}, can be greater for a magnetic barrier than for a non-magnetic barrier (dotted line in figure 3(a)). This occurs only for sufficiently small g 0 , where the increased transparency in the spin-↑ sector overcompensates for the decreased transparency in the spin-↓ sector, thus leading to an over-all enhancement of the critical current. 3.2. Spin-flipping barrier When the magnetic moment has a component outside of the quantization plane of the two superconductors, i.e. M · dL,R 6= 0, the spin of the tunneling Cooper pair is no longer a conserved quantity. In such a junction, we therefore have spin-flipping processes occurring at the barrier. New Journal of Physics 11 (2009) 055055 (http://www.njp.org/)
8
(a) 0.30
(b) 0.30
0.25
η/π
η/π
0.20
0.15 0.10
0′
0.05 0.00 0
0.2
0.15 0.10
π
π′ 0.4
0.6
0.8
0.05
0′
π′
0.00 0
1
2 3
0.2
(d) 0.30
0
0.25
0.25
0′
0.20
η/π
η/π
0.10
π′
0.00 0
0.2
0.4
0.6
0.15 0.10
π
0.05
0.6
0.8
1
0.8
1
c
0.30
0.15
0.4
T/T
c
0.20
π
4
T/T (c)
0
1
0.25
0
0.20
0 0′
π′
π
0.05 0.8
1
T/T
c
0.00 0
0.2
0.4
0.6
T/T
c
Figure 4. Phase diagram of the equal chirality junction as a function of η and
the reduced temperature T /Tc . (a) g = 1, (b) g = 1.5, (c) g = 2 and (d) g = 3. In all figures we take g 0 = Z = 0. Definitions of the phases are given in the text. Current versus phase relationships at the points labelled 1, 2, 3 and 4 in panel (b) are shown in figure 5. As we shall see, these processes are directly responsible for several unconventional Josephson effects. We begin by constructing the ‘phase diagram’ of the junction. The state of the junction can be classified in terms of the location of the free energy minimum. The free energy F(φ) of the junction for a phase difference φ is defined as Z φ h¯ F(φ) = I J (χ ) dχ . (10) 2e 0 In figure 4, we show the phase diagrams for the junction in η–T space for four different values of g and fixed g 0 = Z = 0. We identify four distinct phases: a 0 state, where a free energy minimum is found only at φ = 0; a 00 state, where free energy minima are found at both φ = 0 and π , with the former giving the global minimum; a π 0 state where free energy minima are found at both φ = 0 and π , with the latter the global minimum; and a π state, where a free energy minimum is found only at φ = π . Note that the 0 state extends to η = 0.5π , and that the phase diagram for 0.5π 6 η 6 π is obtained from the 0 6 η 6 0.5π phase diagram by reflection about the line η = 0.5π . The phase diagram is also periodic with period π . In figure 5, we plot representative current versus phase relations from each of the four phases at g = 1.5, g 0 = Z = 0 and T = 0.3Tc . We observe that the 00 and π 0 phases are characterized by a contribution to the current ∝ |1|4 sin(2φ), which is comparable to the usually dominant ∝ |1|2 sin(φ) term. This higher-order harmonic in the current versus phase relationship arises from the coherent New Journal of Physics 11 (2009) 055055 (http://www.njp.org/)
9 0.06
η=0.25π, (1) η=0.19π, (2) η=0.14π, (3) η=0.07π, (4)
0.02
J
0
I (2πe∆ /h)
0.04
0.00 −0.02 −0.04 −0.06 0.0
0.5
1.0
1.5
φ /π
2.0
Figure 5. Current versus phase relationships at the four points indicated in
figure 4(b). These are characteristic of the four distinct phases of the junction. In all figures we take g = 1.5, g 0 = Z = 0 and T = 0.3Tc .
0.24 0.20
Jc
0
I (2πe∆ /h)
(b)
0.28 g=1 g=1.5 g=2 g=3
IJc (2πe∆0/h)
(a)
0.16 0.12 0.08
0.4
0.3
Z=0 Z=0.5 Z=1 Z=1.5 Z=2
0.2
0.1
0.04 0.00 0.0
0.1
0.2
η/π
0.3
0.4
0.5
0.0 0.0
0.1
0.2
η/π
0.3
0.4
0.5
Figure 6. Critical current I J c as a function of the angle η for (a) various values
of g at g 0 = Z = 0 and T = 0.3Tc ; (b) various values of Z at g = 1.5, g 0 = 0 and T = 0.3Tc . tunneling of two Cooper pairs across the barrier. Due to the closing of the gap, such multiple tunneling processes are strongly suppressed as T → Tc , and so the boundaries of the phases converge to a single point in this limit. In figure 6(a), we plot the critical current of the junction as a function of the angle η at fixed T = 0.3Tc and g 0 = Z = 0. The 0–π transition is visible as the sharp minimum. Note that I J c does not go to zero at this point, as the sin(2φ) contribution to the current does not change sign through the 0–π transition (see figure 5). The 0–π transition is very sensitive to potential scattering at the barrier, and so it is likely only to be realized if the tunneling region is a strong half-metal. I J c nevertheless still displays a strong η-dependence even when Z 6= 0, as shown in figure 6(b); this corresponds to a barrier constructed from a magnetic insulator. The origin of the η-dependence of the current can be understood as the competition between spin-preserving and spin-flipping tunneling processes, i.e. the diagonal and offdiagonal components of the scattering matrix S12 , respectively. In the former, the Cooper pair tunnels through the barrier without a spin-flip, and the effective phase difference across the New Journal of Physics 11 (2009) 055055 (http://www.njp.org/)
10
(a)
0.12
(b)
LHS
S,z
(∆0)
I
η=0, 0.5π η=0.1π η=0.25π η=0.4π
−0.04 −0.08 0
η=0, 0.5π η=0.1π η=0.25π η=0.4π
0.04
0.5
1
φ/π
1.5
IJ
0.00
0
(∆ )
0.04
S,z J
0.08
0.08
0.00 −0.04 −0.08
2
−0.12 0
RHS 0.5
1
φ /π
1.5
2
Figure 7. z-component of the spin current on (a) the left-hand side and
(b) the right-hand side of a spin-flipping barrier. We take g = 1.5, g 0 = Z = 0 and T = 0.3Tc . junction for this process is simply φ. As can be seen from the off-diagonal terms of (6b), a tunneling spin-σ quasiparticle undergoing a spin-flip acquires a phase shift of sign(k z )π/2 + σ η; a spin-2σ Cooper pair undergoing a spin-flip therefore acquires the phase shift 2σ η. Furthermore, for our choice of d-vector orientation, the gaps in the spin-↑ and spin-↓ sectors of each superconductor differ by a π phase shift. Since a spin-flipping Cooper pair tunnels between left and right condensates of opposite spin, the effective phase difference for this process is φ + 2σ η + π . At η = 0, the spin-flipping Cooper pairs therefore experience an extra π phase shift as compared to the spin-preserving Cooper pairs; as spin-flip processes dominate at large g and g 0 = Z = 0 (i.e. the magnitude of the off-diagonal terms of (6b) are much larger than the diagonal terms), the junction is in a π state. For finite η, the spin-dependent phase shifts of the spin-flipping Cooper pairs interfere, leading to a reduction of the current due to spinflip processes. As η is increased, this interference eventually reverses the sign of the spin-flip current, driving the junction into the 0 state. The 0–π transition disappears above some critical value of Z , as a finite Z suppresses spin-flip tunneling in favour of spin-preserving tunneling. The dependence of the charge current on η represents a major point of difference with singlet–ferromagnet–singlet junctions, where the critical current only depends on the width of the tunneling barrier or the magnitude of the magnetization [36, 37]. Because the barrier moment can couple to the spin of the tunneling triplet Cooper pair, in contrast, here the orientation of the magnetic moment is important. This unconventional dependence of the Josephson charge current is a signature property of triplet superconductor junctions, and its experimental observation in a magnetic junction would be clear evidence of a bulk equal-spin-pairing triplet state. Since the phase shift experienced by a spin-flipped Cooper pair depends on its initial spin, the currents carried in each spin sector of the superconductors are not in general equal. As before for the spin filter, this produces a Josephson spin current. Due to the spin-flip processes, however, here the current in the spin-↑ (↓) sector of the left superconductor is equal to the current in the spin-↓ (↑) sector of the right superconductor. The spin current therefore has opposite sign on either side of the barrier, as shown in figure 7. Note that the spin current vanishes at η = nπ/2 for n ∈ N: at these values of η, the effective phase differences in each spin channel for spin-flip processes are identical, and hence so too is the current in each spin sector. It is interesting to note that a spin current flows at φ = 0, π , where the charge current New Journal of Physics 11 (2009) 055055 (http://www.njp.org/)
11 is vanishing, i.e. when 0 < η < π/2 a finite spin current is intrinsic to the two possible ground states of the junction. Furthermore, since the sign of the spin current is opposite at these two phase differences, the 0–π transition in the charge sector also corresponds to a sign reversal of the ground state spin current. Orbital symmetry effects clearly play a non-trivial role, as the presence of a finite spin current at φ = 0 is in contrast with the results for a pz -wave junction, where the spin current at zero phase difference is vanishing [30]. 4. The opposite chirality junction
We now turn our attention to a Josephson junction where the two chiral superconductors have opposite chirality i.e. dL,k = 1(T )(k z + ik y )ˆx (pz + ip y symmetry) and dR,k = eiφ 1(T )(k z − ik y )ˆx (pz –ip y symmetry). Such a junction is shown in figure 2(b). The behaviour of the Josephson charge and spin currents in the opposite chirality junction are qualitatively identical to those in the equal chirality junction. In particular, we can still classify a magnetically active barrier as either spin filtering or spin flipping, and in the latter case we also find a 0–π transition as a function of the moment orientation. We will therefore not discuss the Josephson current in the opposite chirality junction. What distinguishes the opposite chirality junction from the equal chirality junction is the presence of chiral Andreev bound states at the interface, which result from the hybridization of surface bound states with the same chirality. The chiral nature of the Andreev bound states implies that there will be a spontaneous current parallel to the barrier interface (i.e. along the y-axis) in each superconductor. These surface currents are suppressed within the bulk by screening, which requires a fully self-consistent treatment of the junction [15, 17]. Our results are therefore only valid very close to the interface. 4.1. Spin-filtering barrier Two distinct spin-degenerate chiral Andreev bound states form at a non-magnetic interface due to the hybridization of the surface bound states of each superconductor. When Z 6= 0, the effect of introducing a magnetic moment M ⊥ dL,R is to lift the spin degeneracy of these states. For the case when the magnetic moment lies along the z-axis, we have an analytic expression for the bound state energies p p E ±,σ (φ, θ) = |1| sin (θ ) 1 − Dσ cos2 (φ/2) ± cos (θ) Dσ cos (φ/2) , (11) where Dσ = cos2 (θ )/[cos2 (θ ) + (Z − σ g 0 )2 ] is the angular-dependent transparency in the spin σ sector and −π/2 < θ = arctan(k y /k z ) < π/2 is the angle between the z-axis and the incident classical trajectory. Note that the spin quantum number σ enters into this expression only through the effective height of the tunneling barrier. We plot the chiral bound states at φ = 0 and π/2 in figuers 8(a) and (b), respectively: when g 0 , Z 6= 0, we see that the spin-↑ states repel, whereas the spin-↓ states move towards one another. Because of the spin splitting of the Andreev bound states, in addition to the usual charge surface current IscC there is also a surface spin current IscS,z . These are shown as a function of the phase difference in figures 8(c) and (d). Note that the value of the charge current at φ = π does not change as the moment is turned on. As can be seen from (11), at this phase difference the Andreev bound states are degenerate and do not depend on the barrier height: it therefore follows that the charge current at φ = π is the same for any value of Z or g 0 , and that the spin current must vanish here. We also note that the current through the spin-↓ sector has much weaker φ dependence than that through the spin-↑ New Journal of Physics 11 (2009) 055055 (http://www.njp.org/)
12
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Figure 8. Comparison of chiral Andreev bound states for a non-magnetic barrier
and a spin filter with g 0 = 0.5 at (a) φ = 0 and (b) φ = π/2. The current versus phase relationships at T = 0.3Tc for the surface charge and spin currents are shown in (c) and (d), respectively. We take Z = 1 and g = 0 in all plots. sector when g 0 6= 0. This is a consequence of the greater effective barrier height in the spin-↓ sector. As the barrier height increases, we approach the limit of an impenetrable barrier: in this limit, there is no hybridization between the surface states of each superconductor, and the chiral Andreev bound states are therefore degenerate and independent of the phase difference φ. 4.2. Spin-flipping barrier As with the Josephson current in the equal chirality junction, we find unconventional behaviour of the surface currents in the opposite chirality junction when M · dL,R 6= 0. We start by considering the chiral Andreev bound state spectrum of such a junction. Although explicit analytic expressions for the Andreev bound states are possible, these are unfortunately much too complicated to reproduce here. In figure 9, we instead plot the Andreev bound states as a function of the angle θ for various values of η and φ. As with the spin-filter case, the coupling to the magnetic barrier lifts the spin degeneracy of the Andreev bound states. This results from the direct hybridization of the Andreev states in each spin sector by the x-component of M, as evidenced by the increased splitting of the bound states as M is rotated out of the y–z plane. Because of this hybridization, we are unable to assign a definite spin quantum number to each bound state. Interestingly, we find that the states have a very different dependence on φ compared to the spin-filtering case: the bound states are independent of η at φ = 0 (not shown), while at φ = π the four-fold degeneracy of the η = 0.5π state is reduced to two-fold when 0 6 η < 0.5π . New Journal of Physics 11 (2009) 055055 (http://www.njp.org/)
13
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Figure 9. Chiral Andreev bound states for a spin-flipping moment with g = 1.5
and (a) η = 0.4π , φ = 0.5π, (b) η = 0.2π, φ = 0.5π , (c) η = 0, φ = 0.5π, (d) η = 0.4π, φ = π , (e) η = 0.2π, φ = π and (f) η = 0, φ = π. The solid blue lines give the actual states; the red dashed lines show the bound states for the same value of φ but η = 0.5π. In all figures we take Z = g 0 = 0.
(b) (∆0)
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η=0 η=0.15π η=0.25π η=0.4π η=0.5π
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η =0, 0.5π η =0.15π η =0.25π η =0.4π
−0.02 2
−0.03 0
0.5
1
φ /π
1.5
2
Figure 10. (a) Interface charge current versus phase and (b) interface spin current
versus phase relationships for the opposite chirality junction with a spin-flipping barrier. In both panels we take g = 1.5, g 0 = Z = 0 and T = 0.3Tc . Note that the interface spin current is evaluated on the left-hand side of the barrier (i.e. z = 0− ); the interface spin current on the right-hand side has the opposite sign. We plot the interface charge and spin currents associated with these chiral bound states in figure 10. As we expect from the discussion of the Andreev bound states, the charge current shows no η-dependence at φ = 0. The charge current at φ = π , however, changes from the global maximum value at η = 0.5π to the global minimum value at η = 0. It is interesting to compare this with the 0–π transition in the Josephson current found in section 3.2: in both cases the sign of the φ-dependent term reverses as η is rotated out of the y–z plane. The origin of New Journal of Physics 11 (2009) 055055 (http://www.njp.org/)
14 the effect is in fact the same: as the φ-dependent term in the interface current clearly originates from the tunneling of Cooper pairs across the barrier, we may again naturally separate these into contributions from spin-flip and spin-preserving tunneling processes. As before, the Cooper pairs undergoing a spin flip acquire extra phase shifts relative to those that preserve their spin, which leads to a sign reversal of the current when spin-flip tunneling dominates. Here, however, spin-flip reflection also plays an important role: the η-independence of the interface current at φ = 0 implies that the φ-independent contribution to the current from reflection must also be dependent on the moment orientation. As can be seen from (6a) and (6b), the matrix elements, and so also the additional phase shift, for spin-flip reflection and transmission are identical, leading to the exact cancellation of the spin-flip tunneling and reflection contributions at φ = 0. This interpretation of the charge current naturally also explains the spin current results, which have a similar η-dependence to the Josephson spin current in the equal chirality junction. As before, the spin currents flow in opposite directions on either side of the barrier, suggesting that they may in fact cancel at the interface. This nevertheless requires a self-consistent treatment of the junction, which we have not attempted here. 5. Conclusions
In this paper, we have employed a quasiclassical technique to study a Josephson junction involving chiral triplet superconductors and a ferromagnetic tunneling barrier. This was motivated by the realization of such a superconducting state in Sr2 RuO4 , and the likely unconventional Josephson effects resulting from the coupling of the barrier moment to the spin of the tunneling Cooper pairs. We have demonstrated that the behaviour of the junction may be classified as spin filtering or spin flipping on the basis of the orientation of the magnetic moment with respect to the d vectors of the two superconductors. Physical explanations for the transport properties of these two cases were given. We have also argued that the highly unconventional dependence of the critical current on the orientation of the magnetic moment may be used to identify spin-triplet superconductors. The chiral Andreev bound state spectrum at a barrier between superconductors with opposite chirality was calculated, and it was shown that our classification scheme also holds for the resulting interface currents. Our results have been obtained under two assumptions: the approximation of the complicated Fermi surface of Sr2 RuO4 as a single cylinder, and the treatment of the gap as spatially constant. It seems unlikely that adopting a more realistic band structure would qualitatively alter our results, while significantly enhancing the complexity of our calculations. This is certainly true for the cuprates [2, 16], where tunneling experiments can be adequately understood in terms of a model dx 2 – y 2 -wave order parameter on a cylindrical Fermi surface. More interesting is the possibility of exotic proximity effects associated with the magnetic scattering. This requires a more thorough study of our model junctions, which we must leave as an open problem deserving further work. Our study nevertheless opens the way for the understanding of realistic triplet superconductor junctions, and illustrates the importance of the Cooper pair spin as a novel degree of freedom in Josephson physics. Acknowledgments
The authors thank Y Asano, J Linder, D K Morr, M Sigrist and Y Tokura for useful discussions, and J Sirker for his critical reading of the manuscript. PMRB is grateful for the hospitality of New Journal of Physics 11 (2009) 055055 (http://www.njp.org/)
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