# Novel Josephson Effect with Triplet Superconductors

A fundamental aspect of the superconducting state is the orbital symmetry of the order parameter which is often regarded as a key indicator of the underlying physical mechanism for Cooper-pairing. Of particular interest is the case of an odd-parity (*p*-wave) orbital state, as it implies that the Cooper-pair is in a triplet state characterized by a ** d**-vector. In a junction where two triplet superconductors are separated by a barrier of sufficiently small width, surface states overlap forming so-called Andreev bound states. If the tunneling barrier is chosen to be magnetic one expects on general grounds both a Josephson charge and spin current. We have studied this new type of Josephson effect in various types of superconducting junctions. Since the prototype, i.e. a Triplet-Superconductor--Ferromagnet--Triplet-Superconductor (TFT) has not been fabricated yet, all our calculations are predictions.

**Figure 1:** Schematic picture of a 1dim TFT Josephson junction along the *z*-direction. Δ_{αβ}(*z*,*z*') = [Δ(*z*,*z*')(** d** ×

**σ**)

*i*σ

_{2}]

_{αβ}is the superconducting gap with Δ and

*= (cosθ,sinθ,0) describing its orbital and spin structure. The tunneling barrier is described by*

**d****M**= M

_{0}(cosα,sinα,0)δ(

*z*)δ(

*z'*) and by a potential scattering U

_{p}.

** Charge current:** We have calculated a rich dependence of the conventional Josephson charge current I

_{J}on the relative orientation between the ferromagnetic moment

**M**and the

*d*_{L,R}vectors on the left and right triplet superconductor as described by the angles α and θ in Fig.1. One important example is a novel 0-π transition as a function of the orientation of the ferromagnetic moment. Furthermore, special orientations can be used to create Josephson current switches in which small changes of α or θ can tune the junction between two current states in which I

_{J}is either 'on' (I

_{J}≠ 0) or 'off' (I

_{J}≈ 0) or differs in its direction. Such two-level systems are of great interest in quantum information technology. We also predict that I

_{J}will exhibit an unconventional sign (i.e. direction) change with temperature for certain orientations of

**M**and

*d*_{L,R}.

** Spin current: **We have identified three distinct mechanisms: (a) spin filtering, (b) misalignment of the

*d*-vectors, and (c) spin-flipping by a non-conserving moment.

*case (a):*The most direct way to produce a spin current is by the spin filtering effect. This requires both roughness (characterized by its strength U

_{p}) and a magnetic moment

**M**┴

*d*_{L,R}. In that situation, one finds a spin-dependent effective scattering potential U

_{p}± σ|

**M**|. This implies that the transparency of the tunneling barrier is different in the two spin sectors, thus preferentially allowing the transmission of one spin species of Cooper-pairs over the other.

*case (b):*In analogy to superfluid

^{3}He, the gradient of the order parameter in spin space produces a spin current. This situation can easily be achieved if the

**-vectors are not parallel any longer yielding to a non-zero value of**

*d*

*d*_{L}x

*d*_{R}. When this

**-vector misalignment is combined with the spin filter effect, the charge currents do not cancel at a phase difference 0 or π and the junction is then is a so-called fractional state. In this state the free energy lies neither at 0 nor π; flux quanta which are non-integer multiples of Φ_0 = hc / 2e may consequently be found at the barrier.**

*d**case (c):*The third fundamental mechanism for the generation of a spin current requires that one has a non-conserving moment at the tunneling barrier, i.e.

**M**×

*d*_{L,R}≠ 0. Then, the spin of the tunneling Cooper-pairs is flipped by the transverse moment. This also yields to a spin-dependent reflection process is which reflected quasiparticles acquire a phase shift. All three mechanisms opens the route for the inclusion of triplet Josephson junctions in spintronics applications.

## Local experts:

- Dirk Manske
- Gaetano Annunziata
- Wei Chen

Our group enjoys collaboration with

- Philip Brydon (now U Maryland),
- Mario Cuoco (U Salerno), and
- Manfred Sigrist (ETH Zürich).

## Key publications:

- P.M.R. Brydon, B. Kastening, D. Morr, and D. Manske
**Interplay of ferromagnetism and triplet superconductivity in a Josephson junction**Phys. Rev. B**. 77,**104504 (2008), 19 pages - P.M.R. Brydon and D. Manske
**0-π transition in magnetic triplet superconductor Josephson junction**Phys. Rev. Lett.**103**, 147001 (2009), 4 pages - P.M.R. Brydon, C. Iniotakis, D. Manske, and M. Sigrist
**Functional superconductor interfaces from broken time-reversal symmetry**Phys. Rev. Lett.**104**, 197001 (2010), 4 pages - P. Gentile, M. Cuoco, A. Romano, C. Noce, D. Manske, and P. M. R. Brydon
**Spin-orbital coupling in a triplet superconductor – ferromagnet junction**Phys. Rev. Lett.**111**, 097003 (2013), arXiv 1208.5871, 5 pages - D. Terrade, P. Gentile, M. Cuoco, and D. Manske
**Proximity effect in spin-triplet superconductor – ferromagnet heterostructure with spin-active interface**Phys. Rev. B**88**, 054516 (2013), arXiv 1210.5160, 14 pages

# Density-Matrix Theory for Ultrafast Dynamics of Superconductors

As the nature of the interactions between quasiparticles in the cuprates and pnictides is still under debate, it is interesting to directly observe the characteristic dynamics of condensate depletion and Cooper-pair recombination. This can be done by time-resolved spectroscopy. in which the sample is excited with an intense femtosecond laser (pump) pulse, and after a delay time Δt, spectra are measured by a second, less intense probe pulse. We have used density-matrix theory to investigate this ultrafast dynamics of superconductors and study the role of phonons at a full quantum kinetic level.

For the case of incoherent phonons we have calculated the time evolution of the optical conductivity as well as pump-probe spectra for a *d*-wave order parameter. We have compared our results with an effective T* model and find good agreement in the low-energy limit, but the inclusion of nonequilibrium effects yields deviation at higher energies. We also have derived the widely used rate equations from our microscopic formalism. Finally, we have quantitatively compared our theory with data obtained from a novel time-resolved pump-probe Raman experiment done on Bi_{2}Sr_{2}CaCu_{2}O_{8+}_{d} in which a two-component dynamics of the superconducting order parameter was obtained. We were able to reproduce one part of the spectrum by calculating the condensate dynamics due to incoherent phonons acting during the Cooper-pair recombination process.

In order to investigate coherent phonons one needs to generalize our formalism. In particular, we find that for ultrafast excitations pulses the superconductor exhibits a non-adiabatic regime (short pulse duration), in which the superconducting order parameter oscillates Δ(t→∞) = Δ_{∞}. In this nonadiabatic regime traditional approaches for computing nonequilibrium dynamics, such as the time-dependent Ginzburg-Landau theory or the Boltzmann kinetic equation, are not applicable, since the full dynamics of both the normal and the anomalous quasiparticle densities, as well as that of the coherent-phonon amplitudes needs to be accounted for. Therefore we resort to the density-matrix formalism to numerically compute the coherent response of the model system after excitation by a short pump pulse In this regime we find that the generation of coherent phonons is resonantly enhanced when the frequency of these oscillations is tuned to the phonon energy, a condition that can be achieved in experiment by varying the fluence, see Fig. 2. Recently, such order parameter oscillations seems to be observed for the first time.

**Figure 2:** Calculated lattice displacement *U(0,t)* as a function of time *t* for resonant (dotted red) and off-resonant conditions (solid black). The gray envelope curve shows the *t* dependence predicted by us.

## Local experts:

- Dirk Manske
- Andreas Schnyder

Our group enjoys collaboration with

- A. Avella (U Salerno)
- Ilya Eremin (U Bochum),
- Götz Uhrig (TU Dortmund) and
- Michael Rübhausen (U Hamburg).

## Key publications:

- J. Unterhinninghofen, D. Manske, and A. Knorr
**Theory of ultrafast nonequlibrium dynamics in d-wave superconductors**

Phys. Rev. B**77**, 180509(R) (2008), 4 pages - A.P. Schnyder, D. Manske, and A. Avella
**Resonant generation of coherent phonons in a superconductor by ultrafast optical pump pulses**

Phys. Rev**. B**84, 214513 (2011), 8 pages - P. P. Saichu, I. Mahns, A. Goos, S. Binder, P. May, S. G. Singer, B. Schulz, A. Rusydi, J. Unterhinninghofen, D. Manske, P. Guptasarma, M.S. Williamsen, and M. Rübhausen
**Two component dynamics of the superconducting order parameter revealed by time-resolved Raman scattering**

Phys. Rev. Lett.**102**, 177004 (2009), 4 pages - A. Akbari, A. P. Schnyder, D. Manske, and I. Eremin
**Theory for nonequilibrium dynamics of multiband superconductors**

Europhysics Letters**101**, 17002 (2013), 6 pages

# Unconventional Properties of Non-Centrosymmetric Superconductors

Superconductivity in materials without inversion symmetry reveals intriguing properties, since a strong antisymmetric spin-orbit coupling (ASOC) splits the Fermi surface and invalidates the classification of the superconducting order parameter with respect to spin singlet/even parity and spin triplet/odd parity. This, in turn, allows unconventional Cooper-pairing in a so-called mixed parity form which is a linear combination of both spin singlet and spin triplet on the splitted bands. Since the **g**-vector of the ASOC and the ** d**-vector of the triplet state turn out to be parallel, the resulting order parameter reads:

Δ_{k}_{±} = *Ψ* ± *d*|**g _{k}**| =

*Ψ*(1±

*p*|

**g**|) = Δ

_{k}_{±}

where the parameter *p* = *d* / *Ψ* represents the unknown triplet-singlet ratio. In order to analyze the so-called mixed parity form we have formulated a kinetic theory for low temperatures and in the clean limit. The transport equations are solved in three dimensions for any kind of ASOC in an extended momentum and frequency range. Our result is a particle-hole symmetric, gauge-invariant and charge conserving description that allows to calculate various response functions

As an application of our kinetic theory, the polarization-dependent electronic Raman response in non-centrosymmetric superconductors (NCS) is studied for two important cases of the ASOC in CePt_{3}Si and Li_{2}Pd_{x}Pt_{3-x}B. New analytical expressions for the Raman vertices are derived and the low-energy power laws as well as characteristic pair-breaking peaks are calculated. The total Raman response is a sum of both bands:

Χ_{γγ}^{total} = Χ_{γγ}(Δ_{-}) + Χ_{γγ}(Δ_{+}).

In general, we predict a two-peak structure due to both singlet and triplet contributions of the superconducting order parameter. Our predictions will help to detect the unknown relative magnitude of the singlet and triplet gap in these materials, see Fig. 3.

**Figure 3:** Theoretical prediction of the Raman spectra Χ_{γγ}(Δ_{-}) [blue] and Χ_{γγ}(Δ_{+}) [red] for A_{1g} (solid lines) and for B_{1,2} (dashed lines) polarizations for the point group C_{4v}. The polar diagrams in the insets demonstrate the four qualitative different cases in which the triplet component increases.

NCS are also interesting in view of the involvement of magnetic fluctuations for the microscopic Cooper-pairing mechanism. In this context, an efficient numerical method is introduced in order to calculate the dynamical spin and charge response of CePt_{3}Si using an itinerant description for the electrons. From the spin- and charge susceptibility, a superconducting pairing interaction is constructed for the weak-coupling gap equation. A sign analysis of the decoupled gap equation supports the experimental evidence of a strong triplet contribution to the order parameter. For CePt_{3}Si it can be shown that an increasing ASOC strengthens this triplet contribution. Finally, we also study interface properties and the Josephson effect involving NCS and other superconductors.

## Local experts:

- Dirk Manske
- Gaetano Annunziata

Our group enjoys collaboration with

- D. Einzel (WMI Garching)
- J. Linder (U Trondheim) and
- Manfred Sigrist (ETH Zürich).

## Key publications:

- L. Klam, D. Einzel, and D. Manske
**Electronic Raman scattering in noncentrosymmetric superconductors**

Phys. Rev. Lett.**102**, 027004 (2009), 4 pages - L. Klam, D. Manske, and D. Einzel
**Kinetic Theory for Response and Transport in Non-Centrosymmetric Superconductors**(32 pages)

review article in "Non-centrosymmetric superconductors", Lecture Notes in Physics**847**(2012); Springer Verlag, Berlin, Heidelberg; Eds.: E. Bauer and M. Sigrist; - arXiv 1003.0407
- G. Annunziata, D. Manske, and J. Linder
**Proximity effect with noncentrosymmetric superconductors**

Phys. Rev. B**86**, 174514 (2012), 11 pages

# Theory for Raman Scattering on high-Tc Superconductors

Polarized Raman scattering is an important tool to study elementary excitations in cuprate superconductors. We have formulated a theory that is able to describe both the electronic and phononic Raman response in orthorhombic YBa_{2}Cu_{3}O_{7-}_{d}_{.} Using general arguments we find new so-called screening terms in the *B _{1g}* scattering channel which are required by gauge invariance. As a result, we obtain a small but measurable softening of the pair-breaking peak, whose position has been attributed for a long-time to twice the superconducting gap maximum. Furthermore, we predicted characteristic superconductivity-induced changes in the phonon lineshape due to a small

*s*-wave admixture to the

*d*

_{x2-y2}-wave pair function. We employed our theory to detwinned, slightly overdoped YBa

_{2}Cu

_{3}O

_{6.95}and moderately overdoped Y

_{0.85}Ca

_{0.15}Ba

_{2}Cu

_{3}O

_{7-}

_{δ}single crystals. The best agreement with the Raman data was obtained for calculations based on admixtures of 15% ± 5% and 10% contribution

*s*-wave, respectively. This agrees with values by other experiments and confirms the previously reported trend of an increase in the

*s*-wave contribution in the underdoped regime of cuprates. In general, our microscopic theory allows to disentangle the electronic Raman signal from the phononic part and to identify corresponding interference terms (Fano effect).

We have also calculated the high-energy Raman response in which we find two-magnon-like excitations in the paramagnetic phase. In experiment a pronounced amplitude enhancement of a high-energy peak related to two-magnon excitations in insulating cuprates upon cooling below the critical temperature T_{c} is observed, see Fig 4. This effect is accompanied by the appearance of the superconducting gap and a pairing peak above the gap in the Raman spectrum, and it can be understood as a feedback effect on the high-energy magnetic fluctuations due to the Cooper pairing interaction. This implies a direct involvement of the high-energy magnetic fluctuations in the pairing mechanism. We have compared our theory quantitatively with data taken from the model single-layer cuprate superconductor HgBa_{2}CuO_{4+}_{δ}. Our analysis allows to extract the antiferromagnetic superexchange coupling *J* for various doping concentrations.

**Figure 4**: (a) Measured low-temperature raw Raman spectra for three doping concentrations (overdoped, T_{c}=90K; underdoped, T_{c}=94K and T_{c}=77K, respectively) and our model calculations (symbols) up to 4*J*. (b) Raman difference spectra (10K-300K) after subtracting phonon peaks. Inset: Enlarged view of the two-magnon peak.

## Local experts:

- Dirk Manske
- Andreas Schnyder

We enjoy collaboration with the group of Berhard Keimer (MPI-FKF).

## Key publications:

- A.P. Schnyder, C. Mudry, and D. Manske
**Screening in (d+s)-wave superconductors: Application to Raman scattering**

Phys. Rev. B**75**, 174525 (2007) - M. Bakr, A. Schnyder, L. Klam, D. Manske, C.T. Lin, B. Keimer, M. Cardona and C. Ulrich
**Anisotropic electronic Raman scattering and phonon anomalies in detwinned YBa**_{2}Cu_{3}O_{6.95}crystals:*s*-wave admixture to the*d*_{x2-y2}-wave order parameter

Phys. Rev. B**80**, 064505 (2009), 11 pages - Yuan Li, M. Le Tacon, M. Bakr, D. Terrade, D. Manske, R. Hackl, L. Ji, M. K. Chan, N. Barisic, X. Zhao, M. Greven, B. Keimer
**Feedback Effect on High-Energy Magnetic Fluctuations in the Model High-Temperature Superconductor HgBa**_{2}CuO_{4+δ}Observed by Electronic Raman Scattering

Phys. Rev. Lett.**108**, 227003 (2012), 6 pages