$\theta_0$ thermal Josephson junction

TL;DR

This paper predicts a phase-sensitive thermal Josephson effect in superconductor/ferromagnet/superconductor junctions with non-coplanar magnetic textures, linking heat flow to spin chirality and triplet Cooper pairs.

Contribution

It introduces the concept of a phase shift in thermal Josephson effect caused by magnetic texture and spin chirality, extending understanding of energy transport in magnetic superconducting systems.

Findings

Thermal heat current exhibits phase-sensitive interference proportional to cos(θ - θ_0).

The phase shift θ_0 depends on magnetic spin chirality and is robust against spin relaxation.

Potential applications include coupling heat flow with magnetic precession.

Abstract

We predict the thermal counterpart of the anomalous Josephson effect in superconductor/ferromagnet/superconductor junctions with non-coplanar magnetic texture. The heat current through the junction is shown to have the phase-sensitive interference component proportional to $\cos(\theta - \theta_0)$, where $\theta$ is the Josephson phase difference and $\theta_0$ is the texture-dependent phase shift. In the generic tri-layer magnetic structure with the spin-filtering tunnel barrier $\theta_0$ is determined by the spin chirality of magnetic configuration and can be considered as the direct manifestation of the energy transport with participation of spin-triplet Cooper pairs. In case of the ideal spin filter the phase shift is shown to be robust against spin relaxation caused by the spin-orbital scattering. Possible applications of the coupling between heat flow and magnetic precession are discussed.

Equations

$${\dot{Q}}_{tot}(T_1,T_2,\theta )={\dot{Q}}_{qp}-{\dot{Q}}_{int}\cos \theta ,$$

$${\dot{Q}}_{tot}(T_1,T_2,\theta )={\dot{Q}}_{qp}-{\dot{Q}}_{int}\cos (\theta -{\theta }_0),$$

$$I(\phi )=I_c\sin (\theta -{\phi }_0).$$

$${\check{I}}_{12}=[\check{\Gamma }{\check{g}}_1{\check{\Gamma }}^{†},{\check{g}}_2],$$

$$[i\epsilon {\tau }_3-i(\mathbf{h}\cdot \mathbf{S}){\tau }_3-\check{\Delta }+{\check{\Sigma }}_s,\check{g}]=0$$

$$g^R={\tau }_3\left[g_{03}+g_{33}(\mathbf{\sigma }\mathbf{h})\right]+{\tau }_1\left[g_{01}+g_{31}(\mathbf{\sigma }\mathbf{h})\right]$$

$$R_N\dot{Q}=\frac{1}{16e^2}{\int }_{\infty }^{\infty }d\epsilon \epsilon \mathrm{Tr}({\check{I}}_{12}^K)$$

$$\kappa /{\kappa }_N={\kappa }_{qp}-{\kappa }_c\cos \theta -{\kappa }_s\sin \theta ,$$

$${\kappa }_{qp}={\int }_{-\infty }^{\infty }d\epsilon F\{N_{+}^2+[r({\mathbf{h}}_{1\perp }{\mathbf{h}}_{2\perp })+h_{1\parallel }{h}_{2\parallel }]N_{-}^2\},$$

$${\kappa }_c={\int }_{-\infty }^{\infty }d\epsilon F\times$$

$$\{r(\mathrm{Im}{g}_{01}{)}^2+\left[r{h}_{1\parallel }{h}_{2\parallel }+({\mathbf{h}}_{1\perp }{\mathbf{h}}_{2\perp })\right](\mathrm{Im}{g}_{31}{)}^2\},$$

$${\kappa }_s=\chi P{\int }_{-\infty }^{\infty }d\epsilon F(\mathrm{Im}{g}_{31}{)}^2,$$

$$\frac{R_N{I}_0}{\pi eT}=\sum _{{\omega }_n}\left[r(g_{01}^2+h_{1\parallel }{h}_{2\parallel }{g}_{31}^2)+({\mathbf{h}}_{1\perp }{\mathbf{h}}_{2\perp })g_{31}^2\right]$$

$$\frac{R_N{I}_{an}}{\pi eT}=\chi P\sum _{{\omega }_n}{g}_{31}^2,$$

Full text

[TABLE]

where $\theta$ is the phase difference between superconducting electrodes. Here the first term is the usual quasiparticle heat current while the second one describes the contribution of energy transfer with participation of Cooper pairs. In accordance with Onsager symmetry the heat current is time-reversal invariant since the phase-coherent term in Eq.(1) does not change under the phase inversion $\dot{Q}_{tot}(\theta)=\dot{Q}_{tot}(-\theta)$.

Experimentally the interplay of heat transport and Josephson phase difference has been studied starting from the observations of thermoelectric effects in superconductor/ normal metal/superconductor junctions Ryazanov and Schmidt (1981, 1982); Kartsovnik et al. (1981); Panaitov et al. (1984); Kaplunenko and Ryazanov (1985); Logvenov et al. (1986). Recently the existence of coherent thermal currents (1) has been confirmed in experiments using the Josephson heat interferometry with tunnel contactsGiazotto and Martinez-Perez (2012); José Martínez-Pérez and Giazotto (2014); Martínez-Pérez et al. (2014). Subsequently the number of possible applications has been suggested including heat interferometers José Martínez-Pérez and Giazotto (2014); Giazotto and Martinez-Perez (2012); Fornieri et al. (2016); Guarcello et al. (2016) , diodes Martínez-Pérez et al. (2015), transistors Fornieri et al. (2016); D’Ambrosioa et al. (2015); Giazotto et al. (2014), phase-tunable ferromagnetic Josephson valvesGiazotto and Bergeret (2013); Bergeret and Giazotto (2013) and the probes of topological Andreev bound statesSothmann and Hankiewicz (2016). The direction $\dot{Q}_{int}$ in Eq. (1) can be controlled in experiments providing the realization of $0-\pi$ thermal Josephson junction Fornieri et al. (2017).

In the present paper we report on the possibility to obtain the generalized HCPR of the form

[TABLE]

which can have an arbitrary phase-shift $\theta_{0}$ in contrast to the Eq.(1) studied in all previous works Maki and Griffin (1965); Guttman et al. (1997, 1998); Zhao et al. (2003, 2004). This effect takes place in the systems with broken time-reversal and chiral symmetries such as the S/F/S junctions with non-coplanar magnetic textures or spin-orbital interaction. It can be considered as the thermal counterpart of the anomalous Josephson effect characterized by the generalized current-phase relation (CPR) Braude and Nazarov (2007); Buzdin (2008); Zazunov et al. (2009); Brunetti et al. (2013); Konschelle and Buzdin (2009); Grein et al. (2009); Liu and Chan (2010); Yokoyama et al. (2014); Margaris et al. (2010); Alidoust and Linder (2013); Mironov and Buzdin (2015); Kulagina and Linder (2014); Moor et al. (2015a, b); Mironov et al. (2015); Konschelle and Buzdin (2009); Bergeret and Tokatly (2015); Schrade et al. ; Nesterov et al. (2016); Bobkova et al. (2016); Szombati et al. (2016); Silaev et al. (2017)

[TABLE]

Here $I_{c}$ is the critical current and $\varphi_{0}$ is an arbitrary phase shift which however in the general case is different from that in the generalized HCPR $\theta_{0}\neq\varphi_{0}$.

We demonstrate the phase-shifted HCPR (2) using a generic example of Josephson spin-valveBergeret and Giazotto (2013, 2014); Giazotto and Bergeret (2013); Giazotto et al. (2015) that contains three non-coplanar magnetic vectors, see Fig.(1. It consists of two ferromagnetic layers (F) with exchange fields ${\bm{h}}_{1,2}$ interacting with the superconducting electrodes (S), separated by the spin-filter barrier with the magnetic polarization directed along ${\bm{m}}$. Recently the spin-filter effect in superconductor/ferromagnet structures has been demonstrated by using ferromagnetic insulators (FI) europium chalcogenidesHao et al. (1990); Santos et al. (2008); Miao and Moodera (2009); Li et al. (2013); Wolf et al. (2014); Kolenda et al. (2017) or GdN tunnelling barriersSenapati et al. (2011). The role of outer F1,2 contacts is to induce effective exchange fields in the superconducting electrodes. In case of metallic ferromagnets this can be achieved through the inverse the proximity effect(Tokuyasu et al., 1988; Bergeret et al., 2004, 2005; Moodera et al., 1988). Alternatively F1,2 can be ferromagnetic insulators and induce the effective exchange field in S1,2 as a result of the spin-mixing scattering of conduction electrons Eschrig et al. (2015).

To calculate the currents across spin-filtering barriers are we use generalized Kuprianov-Lukichev boundary conditions Kuprianov and Lukichev (1988), that include spin-polarized tunnelling at the SF interfaces Bergeret et al. (2012); Eschrig et al. (2015). The matrix tunnelling current from S1 to S2 is given by

[TABLE]

where $\check{g}_{k}$ for $k=1,2$ are the matrix Green’s functions (GF) in the superconducting electrodes Sk. The spin-polarized tunnelling matrix has the form $\check{\Gamma}=t_{+}\hat{\sigma}_{0}\hat{\tau}_{0}+t_{-}(\bm{m\hat{\sigma}})\hat{\tau}_{3}$, where ${\bm{m}}$ is the direction of barrier magnetization, $t_{\pm}=\sqrt{(1\pm\sqrt{1-P^{2}})/2}$ and $P$ being the spin-filter efficiency of the barrier that ranges from [math] (no polarization) to $1$ (100% filtering efficiency). The matrix GF is given by \check{g}=\left(\begin{array}[]{cc}g^{R}&g^{K}\\ 0&g^{A}\\ \end{array}\right), where $g^{K}$ is the Keldysh component and $g^{R(A)}$ is the retarded (advanced) GF determined by the equation Bergeret et al. (2005)

[TABLE]

Here $\varepsilon$ is the energy, $\check{\Delta}=\Delta\tau_{1}e^{i\tau_{3}\varphi}$ is the order parameter with the amplitude $\Delta$ and phase $\varphi$, ${\bm{h}}$ is the exchange field, ${\bm{S}}=(\sigma_{1},\sigma_{2},\sigma_{3})$, $\sigma_{1,2,3}$ and $\tau_{1,2,3}$ are the Pauli matrices in spin and Nambu spaces respectively. We include the spin-orbital (SO) scattering process which lead to the spin relaxation described by Abrikosov and Gorḱov (1961); Bergeret et al. (2005) $\check{\Sigma}_{s}=({\bm{S}}\cdot\check{g}{\bm{S}})/8\tau_{so}$, where $\tau_{so}$ is the SO scattering time. Due to the normalization condition $\check{g}^{2}=1$ the Keldysh component can be written as $g^{K}=(g^{R}-g^{A})f_{L}$, where $f_{L}=f_{L}(\varepsilon)$ is the distribution function. We assume that it has an equilibrium form $f^{(1,2)}_{L}=\tanh(\varepsilon/2T_{1,2})$ characterized by the different temperatures $T_{1,2}$ in the electrodes S1,2.

Proximity of the outer ferromagnetic layers shown in Fig.(1) induces Zeeman splitting of electronic states which acts as an effective exchange field in the superconducting electrodes. We assume that superconducting layers are thin enough to neglect the spatial variations of the spectral GFs (retarded and advances) so that up to leading order they retain their bulk values in the presence of a homogeneous exchange field

[TABLE]

and $g^{A}=-\tau_{3}g^{R{\dagger}}\tau_{3}$. The terms diagonal in Nambu space ($\tau_{3}$) correspond to the normal correlations which determine the total density of states (DOS) and the DOS difference between the spin-up and spin-down subbands is given by the components $N_{+}={\rm Re}g_{03}$ and $N_{-}={\rm Re}g_{33}$ respectively. The off-diagonal components ($\tau_{1}$) describe spin-singlet $g_{01}$ and spin-triplet $g_{31}$ superconducting correlations which appears as the result of the exchange splitting Bergeret et al. (2001).

The tunnelling heat current $\dot{Q}$ across the Josephson junction (JJ) is given by the general expression

[TABLE]

where $R_{N}$ is the normal-state resistance of the tunnelling barrier and $e$ is the electron charge. At first we calculate the heat current using (7). Assuming that the temperature difference is small we expand the distribution functions in S1,2 electrodes $f_{L}^{(1,2)}=f_{0}+(T_{1,2}-T)\frac{\partial f_{0}}{\partial T}$ where $T=(T_{1}+T_{2})/2$ and introduce the total heat conductance as $\kappa=\dot{Q}/(T_{1}-T_{2})$. In accordance with the Eq.(7) it can be written as the superposition of three terms

[TABLE]

where $\kappa_{N}=\pi^{2}T/(3e^{2}R_{N})$ is the normal state thermal conductance of the junction. We will refer to the different contributions in (8) as the quasiparticle $\kappa_{qp}$, the usual $\kappa_{c}$ and phase-shifting $\kappa_{s}$ interference terms. Assuming S1,2 superconductors to be identical we get expressions for components in Eq.(8):

[TABLE]

where $\chi={\bm{m}}\cdot(\bm{h}_{1}\times\bm{h}_{2})$ is spin chirality, $r=\sqrt{1-P^{2}}$, $h_{\parallel}=({\bm{m}}{\bm{h}})$ and ${\bm{h}}_{\perp}={\bm{h}}-h_{\parallel}{\bm{m}}$ are the exchange field components parallel and perpendicular to the FI barrier polarization, $F(\varepsilon)=(6\varepsilon^{2}/\pi^{2}T^{2})\partial f_{0}/\partial\varepsilon$.

The quasiparticle and the usual phase-sensitive contributions (9,10) have been analysed for the coplanar magnetic configuration Bergeret and Giazotto (2013). The term $\kappa_{s}$ (11) is non-zero only in the non-coplanar case $\chi\neq 0$ and it produces the phase shift of HCPR in Eq.(2) given by $\theta_{0}=\arctan(\kappa_{s}/\kappa_{c})$. Comparing different parts of the conductance (9,10,11) one can see that $\kappa_{s}$ is qualitatively different from the others since it stems exclusively from the triplet part of the condensate associated with the GF component $g_{31}$. In the general case of a non-ideal spin-filter $P\neq 1$ the usual interference part $\kappa_{c}$ has contributions from both spin-singlet and spin-triplet Cooper pairs. Thus one can conclude that the non-trivial phase shift $\theta_{0}\neq 0$ of the HCPR is a direct experimentally measurable evidence of the transport of spin-triplet Cooper pairs across the tunnel junction.

Physically the phase-shifting term $\kappa_{s}$ appears as a result of the additional phase picked up by the spin-triplet Cooper pairs when tunnelling between two superconductors with non-collinear exchange fields through the spin-polarising barrier. To understand this phenomenon on a qualitative level let us consider the magnetic configuration ${\bm{h}_{1}}=h_{1}{\bm{z}}$, ${\bm{h}_{2}}=h_{2}{\bm{x}}$ and ${\bm{m}}={\bm{y}}$. The spin-triplet condensates in S1 and S2 are described by the wave functions $\Psi_{t1}\sim|\uparrow,\downarrow\rangle_{z}+|\downarrow,\uparrow\rangle_{z}$ and $\Psi_{t2}\sim e^{i\varphi}(|\uparrow,\downarrow\rangle_{x}+|\downarrow,\uparrow\rangle_{x})$ respectively, where the spin quantization axes are set by the directions of exchange fields ${\bm{h}_{1,2}}$. Assuming the spin filter to be ideal $P=1$ we find for the tunnelling amplitude $\langle\Psi_{2t}|\hat{P}_{y}|\Psi_{1t}\rangle\sim ie^{i\varphi}$, where $\hat{P}_{y}$ is the projection operator acting on each of the single-electron states $\hat{P}_{y}|\uparrow\rangle_{z}=\frac{1}{2}|\uparrow\rangle_{y}$ and $\hat{P}_{y}|\downarrow\rangle_{z}=\frac{-i}{2}|\uparrow\rangle_{y}$. Thus one can see that the spin-filtering provides an additional $\pi/2$ phase in the tunnelling amplitude of spin-triplet Cooper pairs, which is the origin of the anomalous Josephson effect Silaev et al. (2017) and the phase-shifted HCPR studied here.

On a quantitative level let us analyse the particular configuration ${\bm{h}}_{1,2}\perp{\bm{m}}$ when the expressions (10,11) yield the interference contributions which are proportional to each other $\kappa_{int}=\int_{-\infty}^{\infty}d\varepsilon F(\varepsilon)({\rm Im}g_{31})^{2}$ so that $\kappa_{c}=-\cos\theta_{h}\kappa_{int}$ and $\kappa_{s}=\sin\theta_{h}\kappa_{int}$, where $\theta_{h}$ is the angle between ${\bm{h}_{2\perp}}$ and ${\bm{h}_{1\perp}}$ shown schematically in Fig.(1). Hence for the ideal spin-filter $P=1$ the phase shift of HCPR is determined by the geometry of magnetic configuration $\theta_{0}=\theta_{h}$ although the overall amplitude of $\kappa_{int}$ is strongly suppressed by the spin relaxation. To demonstrate this we find the order parameter and spectral functions (6) self-consistently taking onto account the presence of exchange field and SO scattering rate which can vary in wide limits corresponding toPoli et al. (2008) $(\Delta_{0}\tau_{so})^{-1}\approx 0.2$ in Al and to Wakamura et al. (2014) $(\Delta_{0}\tau_{so})^{-1}\approx 500$ in Nb, where $\Delta_{0}$ is the bulk superconducting gap at $h=0$, $\tau_{so}=\infty$ and $T\to 0$.

Let us consider the calculated dependencies of quasiparticle $\kappa_{qp}(T)$ and interference $\kappa_{int}(T)$ parts shown in Figs. (2)a and (2)c respectively for the fixed exchange field $h=0.5\Delta_{0}$. First there is a non-monotonic dependence $\kappa_{qp}=\kappa_{qp}(T)$ which increases above the normal state value as the temperature goes down below $T_{c}$. This behaviour is explained by the DOS enhancement near the gap edge. At lower temperatures $T\ll T_{c}$ the quasipartiles are frozen out which results in the exponential drop of the heat conductance. Of interest is the evolution of the peak amplitude in the $\kappa_{qp}(T)$ dependence with increasing spin relaxation rate. Initially with increasing $\tau_{so}^{-1}$ from zero to small values the peak of $\kappa_{qp}(T)$ is suppressed while at the larger values of $\tau_{so}^{-1}$ it is restored. This tendency reveals the the evolution of DOS $N_{+}(\varepsilon)$ and the anomalous function ${\rm Im}g_{01}(\varepsilon)$ with increasing $\tau_{so}^{-1}$ shown in Fig.(3). For $\tau_{so}^{-1}\ll T_{c0}$ the singularities of spectral functions are smeared. However at larger values of $\tau_{so}^{-1}>T_{c0}$ both $N_{+}$ and ${\rm Im}g_{01}$ again develop the peaks although without the spin-splitting features. At the same time the spin-triplet components $g_{33}$ and $g_{13}$ are strongly suppressed by the SO scattering.

The temperature dependence of interference thermal conductance $\kappa_{int}(T)$ is also non-monotonic due to the similar mechanism as discussed above. As shown in Fig.(2)c the maximum of $\kappa_{int}(T)$ is strongly suppressed by the SO scattering which tends to remove the spin-dependent components $g_{31}$ of the GFs. However for the fully-polarizing ideal spin filter $P=1$ according to the equitation (10) the suppression of $\kappa_{int}$ does not affect the phase shift of HCPR which is fixed by the angle between exchange fields ${\bm{h}_{1,2}}$ as discussed above.

The situation is different for $P<1$ when $\kappa_{c}$ and $\kappa_{s}$ are no longer proportional to each other. The qualitative difference between the usual $\kappa_{c}$ and phase-shifting $\kappa_{s}$ contributions in this case is determined by the transport of spin-singlet Cooper pairs which is only possible if both the spin projections can pass the spin filter. This contribution is described by the first term in the r.h.s of Eq.(10) which yields a non-zero contribution since $r\neq 0$. In this case the phase shift of HCPR is suppressed by the SO scattering which leads to the decrease of spin-triplet correlations so that $\kappa_{s}\to 0$. At the same time the contribution of spin-singlet Cooper pairs survives keeping $\kappa_{c}\neq 0$ in the limit $\tau_{so}\to 0$ as shown in the Fig.(2)b.

It is instructive to compare the phase-shifted HCPR (2) and CPR (3) calculated for the spin-valve shown in Fig.(1) using the general matrix current (4). The usual $I_{0}=I_{c}\cos\varphi_{0}$ and anomalous $I_{an}=I_{c}\sin\varphi_{0}$ Josephson currents through tunnel barrier are given by

[TABLE]

where $\chi={\bm{m}}({\bm{h}_{1}\times\bm{h}_{2}})$ is the spin chirality, $g_{01}$ and $g_{31}$ are the spin-singlet and spin-triplet components of the Matsubara GF written in the form (6) analytically continued to the imaginary frequencies $\varepsilon\to i\omega_{n}$ with $\omega_{n}=(2n+1)\pi T$.

Expressions for the Josephson current (12,13) are dual to that of the interference heat conductance (10,11). Similar to the phase-shifting term $\kappa_{s}$ the anomalous current $I_{an}$ is mediated by spin-triplet component $g_{31}$. Therefore $\varphi_{0}$ Josephson effect is the directly observable signature of the spin-triplet superconducting current across the junction. For the ideal spin filter $P=1$ Eqs.(12,13) yield a temperature-independent phase shift of CPR $\varphi_{0}=\theta_{0}=\theta_{h}$, although in the general case $\theta_{0}$ and $\varphi_{0}$ can be quite different, as shown in Fig.(3)a,b.

The predicted effect of phase-shifted HCPR can be experimentally observed using the Josephson heat interferometer Giazotto and Martinez-Perez (2012); Martínez-Pérez et al. (2014) consisting of the temperature-biased superconducting quantum interference device (SQUID ) with the usual JJ in the one part and $\theta_{0}$-JJ in the other part. In this case following the derivation in Ref.(Martínez-Pérez et al., 2014) one can show that the interference pattern of the heat current across the SQUID $\dot{Q}_{int}$ contains a spontaneous shift as a function of the external magnetic flux $\Phi$ so that $\dot{Q}_{int}=\dot{Q}_{int}(\Phi-\Phi_{e})$. For the ideal spin filter when $\varphi_{0}=\theta_{0}=\theta_{h}$ we get $\Phi_{e}=\theta_{h}\Phi_{0}/2\pi$

The $\theta_{0}$-shifted HCPR (3) provides an interesting possibility to couple the heat transport with magnetization dynamics. Oscillations of moments ${\bm{h}_{1,2}}$ and ${\bm{m}}$ driven by the Larmour precession around the effective fieldKonschelle and Buzdin (2009) in the generic thermomagnetic circuit Fig.(1) produce the time-dependent spin chirality $\chi=\chi(t)$ and hence generate the non-stationary phase shifts $\varphi_{0}=\varphi_{0}(t)$ and $\theta_{0}=\theta_{0}(t)$. Thus according to Eqs.(2,3) one can generate alternating heat and charge currents at the Larmour frequency which can be controlled by an external magnetic an the anisotropy field. The other possible application is based on the effective conversion of spin currents inside the ferromagnet or ferromagnetic insulator into the electronic heat and charge currents across the attached Josephson junctions. Based on the discussed effect it is in principle possible to implement the superconducting JJ detector of magnetic precession associated with magnons in FI layer Kajiwara et al. (2010); Uchida et al. (2010); Bauer et al. (2012) or the skyrmion motion inside ferromagnets which can be used for the racetrack magnetic memory applicationsFert et al. (2013).

To summarize, we have found the thermal counterpart of the anomalous Josephson effect. Under the conditions of broken time-reversal and chiral symmetries the interference heat current acquires an arbitrary phase shift $\theta_{0}$ which substantially generalizes the previously found forms of HCPR. For the generic example of the non-coplanar Josephson spin valve (Fig.1) $\theta_{0}$ is determined by the non-zero spin chirality $\chi={\bm{m}}\cdot(\bm{h}_{1}\times\bm{h}_{2})\neq 0$. The phase shift is demonstrated to be the direct and experimentally measurable evidence of the heat transport with participation of spin-triplet Cooper pairs. In view of possible applications the proposed effect allows to change the heat conductance of the system in a continuous way by rotating magnetic vectors. For this purpose it is preferable to use magnetic elements with different coercivity fields Li et al. (2013) or anisotropies.

We thank T. T Heikkilä , F.S. Bergeret, A. Mel’nikov, I. Bobkova and A. Bobkov for stimulating discussions. The work was supported by the Academy of Finland.