Competitive 0 and π states in S/F/S trilayers: Multimode approach
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Citations
Supergap and subgap enhanced currents in asymmetric S 1 FS 2 Josephson junctions
Hybrid helical state and superconducting diode effect in superconductor/ferromagnet/topological insulator heterostructures
Anomalous current-voltage characteristics of SFIFS Josephson junctions with weak ferromagnetic interlayers.
Angular Dependence of the Superconducting Transition Temperature in Ferromagnet-Superconductor-Ferromagnet Trilayers
Proximity effect in [Nb(1.5 nm)/Fe(x)]10/Nb(50 nm) superconductor/ferromagnet heterostructures.
References
Magnetization Control and Transfer of Spin-Polarized Cooper Pairs into a Half-Metal Manganite
Superconducting transition in Nb/Gd/Nb trilayers.
Spin-controlled coexistence of 0 and π states in S F S F S Josephson junctions
Current-phase relations in SIsFS junctions in the vicinity of 0-π transition
Quasiclassical theory of magnetoelectric effects in superconducting heterostructures in the presence of spin-orbit coupling
Related Papers (5)
Frequently Asked Questions (15)
Q2. What are the future works in this paper?
One of the interesting problems would be to extend the MMA to the nonequilibrium case by using the KeldyshUsadel Green ’ s function approach [ 84 ]. It is also interesting to study more complex phases in S/F multilayers in the MMA, extending the results of Ref. [ 63 ] obtained in the SMA.
Q3. What is the critical temperature in S/F/S trilayers?
In S/F/S trilayers, only the state with highest Tc is realized at certain d f , i.e., when increasing d f the dashed red line appears above the solid black line, the 0-π transition occurs, and the structure switches to the π phase state.
Q4. What is the effect of the inverse proximity effect on the critical temperature of the S/F?
With decrease of the S-layer thickness ds in S/F/S trilayers, the critical temperature is suppressed due to the inverse proximity effect, which becomes more profound in the case104502-6of small ds.
Q5. What is the critical temperature of a S/F interface?
For fully transparent S/F interfaces, γb = 0, the critical temperature appears at d f ∼ ξh (we note that in their case ξh = 0.54ξ f , since h = 6.8πTcs), reaches a maximum at a particular d f , and with further increase in d f eventually drops to zero.
Q6. What are the possible extensions of this work?
Other possible extensions will include spin-orbit coupling effects in equilibrium [72] and nonequilibrium cases [90] and considering Tc in S/F/S junctions in the presence of an equilibrium supercurrent [91].
Q7. What is the critical temperature of a ferromagnetic layer?
In this case, the critical temperature Tc vanishes when the π phase state becomes energetically unfavorable in a certain interval of d f , and at d f ξ f the Tc eventually tends to zero.
Q8. what is the usadel equation for the ferromagnetic layer?
2s πTcs d2Fs dx2− |ωn|Fs + = 0. (1) In the F layer (−d f /2 < x < d f /2), the Usadel equation can be written asξ 2f πTcs d2Ff dx2− (|ωn| + ih sgn ωn)Ff = 0.
Q9. What is the Usadel Eq. for F+s?
(13) According to the Usadel Eqs. (1)–(3), there is a symmetry relation F (−ωn) = F ∗(ωn), which implies that F+ is a real while F− is a purely imaginary function.
Q10. What is the resistance of the S/F boundary?
At the borders of the S layers with a vacuum, the authors naturally havedFs(±ds ± d f /2) dx= 0. (6) The solution of the Usadel equation in the F layer depends on the phase state of the structure.
Q11. what is the boundary condition for the s/f/s structure?
The self-consistency Eq. (14) and boundary conditions Eqs. (15), together with the Usadel equation for F+s ,ξ 2s πTcs d2F+s dx2− ωnF+s + 2 = 0, (17) will be used for finding the critical temperature of the S/F/S structure both in 0 and π phase states.
Q12. How can the authors explain the reentrant behavior of the phase state?
This situation corresponds to an S/F/S structure enclosed in a ring, where the π phase shift can be fixed by applying the magnetic flux quantum for any d f .
Q13. What is the importance of using the MMA in a wide range of parameters?
Thus the authors confirm the importance of using the MMA in a wide range of parameters in the case of S/F/S trilayers, where 0-π phase transitions are possible.
Q14. what is the boundary condition for f+s?
Using the boundary condition Eq. (10) the authors arrive at the effective boundary conditions for F+s at the boundaries of the right S layer,ξs dF+s (d f /2)dx = W 0,π (ωn)F+s (d f /2), (15a)dF+s (ds + d f /2) dx= 0, (15b) where the authors used the notationsW 0,π (ωn) = γ As( γb + Re B0,πf ) + γ As∣∣γb + B0,πf ∣∣2 + γ (γb + Re B0,πf ) , As = ksξs tanh(ksds), ks = 1ξs√ ωnπTcs .
Q15. How do the authors calculate the critical temperature in S/F/S trilayers?
To provide complete behavior of the critical temperature in S/F/S trilayers, the authors calculate Tc(d f ) dependencies in both 0 and π phase states by using the MMA and show them on the same plot, see Fig.