Source: https://www.groundai.com/project/theory-of-nonequilibrium-dynamics-of-multiband-superconductors/
Timestamp: 2019-04-21 22:34:18+00:00

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We study the nonequilibrium dynamics of multiband BCS superconductors subjected to ultrashort pump pulses. Using density-matrix theory, the time evolution of the Bogoliubov quasiparticle densities and the superconducting order parameters are computed as a function of pump pulse frequency, duration, and intensity. Focusing on two-band superconductors, we consider two different model systems. The first one, relevant for iron-based superconductors, describes two-band superconductors with a repulsive interband interaction V12 which is much larger than the intraband pairing terms. The second model, relevant for MgB2, deals with the opposite limit where the intraband interactions are dominant and the interband pair scattering V12 is weak but attractive. For ultrashort pump pulses, both of these models exhibit a nonadiabatic behavior which is characterized by oscillations of the superconducting order parameters. We find that for nonvanishing V12, the superconducting gap on each band exhibits two oscillatory frequencies which are determined by the long-time asymptotic values of the gaps. The relative strength of these two frequency components depends sensitively on the magnitude of the interband interaction V12.
The discovery of iron-based superconductors  and MgB2  , where superconductivity is characterized by more than one order parameter, has lead to a renewed interest in multiband superconductors. The existence of multigap superconductivity was predicted more than fifty years ago in the pioneering works by Moskalenko  and Suhl et al.  , where it was argued that a band-dependent interaction potential can give rise to different order parameters on different Fermi surface sheets. The presence of multiple superconducting order parameters has many interesting physical consequences, most of which are related to the relative amplitudes and phase differences between the gaps on different Fermi surfaces.
In recent years, numerous studies of the nonequilibrium dynamics of multiband superconductors have been performed using femtosecond time-resolved spectroscopy [5, 6, 7, 8, 9, 10] . The relaxation kinetics measured in these experiments gives important information on the electronic band structure, on the electron-phonon coupling strengths, as well as on the symmetry of the superconducting order parameters. For example, time-resolved measurements on the iron pnictide superconductor Ba0.6K0.4Fe2As2  have revealed two distinct quasiparticle relaxation processes: a fast one, whose decay rate increases linearly with excitation density, and a slow one, which is independent of excitation density. This behavior has been attributed to the multigap structure of the superconductor. Furthermore, from a careful analysis of the temperature dependence of the decay rates, the authors of Ref.  have been able to deduce the pairing symmetries of the superconducting state.
In this paper, we shall not be concerned with the relaxation dynamics, but rather with the nonequilibrium evolution of multiband superconductors at times shorter than the relaxation time, i.e., with the nonadiabatic behavior of these system. It has been shown that the nonequilibrium response at these short time scales exhibits interesting properties, such as order parameter oscillations [11, 12, 13, 14] . In a pioneering work, B. Mansart et al.  have been able to detect these coherent oscillations of the order parameter in a high-Tc cuprate superconductor via transient broad-band reflectivity measurements. The recently discovered multiband superconductors MgB2 and iron pnictide superconductors, which all have a relatively large superconducting gap, offer an excellent opportunity to analyze this nonadiabatic dynamics also in multiband systems, where the interplay of the intraband pairing interactions and interband pair scattering terms play an important role. One goal of the present work, is to provide guidance for experimental studies of these interesting nonadiabatic response properties in multigap superconductors.
where aklσ is the electron (hole) annihilation operator with band index l=1,2, spin σ, and momentum k. The bare dispersions of the two bands, εk1 and εk2, are assumed to be quadratic with εkl=ℏ2k2/(2ml)−EFl, where ml and EFl represent the effective electron (hole) mass and Fermi energy of band l, respectively. The intraband pairing interactions are denoted by V11kk′ and V22kk′, while the interband pair scattering is represented by V12kk′=(V21k′k)∗. Assuming that the splitting of the bands is much larger than the superconducting energy scales, we neglect all interband pairing terms, i.e., we set ⟨a−kl↓akl′↑⟩=0 for all l≠l′. Furthermore, we restrict ourselves to momentum-independent pairing interactions with V11kk′=V11, V22kk′=V22, and V12kk′=V12=|V12|eiθ. The intraband pair couplings are assumed to be attractive, i.e., V11,V22>0. The phase θ of V12 is taken to be either 0 or π, depending on whether the interband pair scattering is attractive or repulsive.
From this condition, the transition temperature Tc can be obtained.
Figure 1: (Color online) Calculated time evolution of |Δσ(t)| and |Δπ(t)| for the case of MgB2, for V12=0.2V11 and three different pump pulse widths τ=0.4 ps (solid green), 1.0 ps (dotted red), and 5.0 ps (dashed blue). The central energy of the pump pulse is ℏω0=8.0 meV, i.e., it is slightly larger than 2Δπ(ti), but smaller than 2Δσ(ti). We take A20τ≈5.3×10−27J2s3C2m2.
Hence, the temporal evolution of the gaps is fully determined by the time-dependence of the Bogoliubov quasiparticle densities ⟨α†klαk′l⟩, ⟨β†klβk′l⟩, ⟨αklβk′l⟩, and ⟨β†klα†k′l⟩. As shown in the Appendix, it is straightforward to derive a closed set of equations of motion for these quantities, see Eq. (0.2).
We numerically solve the equations of motion for the Bogoliubov quasiparticle densities (see Appendix) to determine the time evolution of the order parameter amplitudes |Δl(t)|. As is evident from Eq. (0.2), off-diagonal elements in the quasiparticle density matrices, such as, e.g., ⟨α†klαk+nq0l⟩, are of order |A0|n. Thus, for sufficiently small |A0|, the off-diagonal entries decrease rapidly as n increases. Therefore we set all off-diagonal elements with n>4 to zero, which substantially reduces the computational costs. For the numerical simulations we choose the BCS ground state at zero temperature as the initial state. In what follows, we consider two model systems, one with weak attractive V12, relevant for MgB2, and the other with strong repulsive V12, relevant for iron-based superconductors.
We start by discussing the model appropriate for the multiband superconductor MgB2. The electronic structure of MgB2 consists of two sets of bands: σ-bands forming quasicylindrical hole pockets and π-bands giving rise to tubular networks of electronlike Fermi surface sheets. Superconductivity in MgB2 arises from couplings between the electrons and the E2g-phonon mode. These couplings lead to attractive intraband interactions, which are stronger in the σ-bands than in the π-bands. Since the interband pair scattering is weak, this gives rise to two gaps with different magnitudes on the σ- and π-bands.
We describe the band structure of MgB2 in a somewhat oversimplified manner by two parabolic bands  , i.e, a hole band, with Fermi velocity vσF=0.273×106 m/s and effective mass mσ=−3m0, and an electron band, with vπF=1.0×106 m/s and mπ=m0. Here, m0 denotes the free electron mass. The intraband pairing interactions on the σ- and π-bands are chosen to be V11=24.8 meV and V22=11.6 meV, respectively, whereas the attractive interband coupling is taken to be much smaller than either V11 or V22, i.e., we set V12=0.2V11. With these material parameters, the superconducting gaps in the initial state Δl(ti) can be computed from the self-consistency equation (4). Assuming that the cut-off energy ℏωc is approximately equal to the E2g-phonon energy, i.e, ℏωc=50 meV, we find that in the initial state the gaps on the σ- and π-bands are given by Δσ(ti)=7 meV and Δπ(ti)=3 meV, respectively. The central energy of the optical pump pulse ℏω0 is chosen to be either in between 2Δπ(ti) and 2Δσ(ti), or equal to twice the gap on the σ-band, i.e., ℏω0=14 meV.
where bll′ and ϕll′ are constants that depend on the initial conditions. In other words, the superconducting gaps oscillate with an amplitude decaying as 1/√t and two frequencies, which are determined by the long-time asymptotic gap values Δ∞l<|Δl(ti)| [see Figs. 1(a) and 1(b)]. Interestingly, the relative strength of the two frequency components ωΔσ=2Δ∞σ/ℏ and ωΔπ=2Δ∞π/ℏ depends sensitively on the magnitude of V12. That is, the amplitudes b12 and b21 increase with increasing interband pair scattering V12. For V12=0, on the other hand, the amplitudes b12 and b21 vanish, and hence Eq. (10) reduces to the well-known result for single band models, where the gap oscillates with a single frequency [12, 17, 14, 13, 18] .
Figure 2: (Color online) Asymptotic values of the order parameters Δ∞σ and Δ∞π as a function of integrated pump pulse intensity for three different pump pulse widths τ. The central energy of the pump pulse is taken to be ℏω0=8 meV in panels (a) and (b), and ℏω0=14 meV in panels (c) and (d). The left and right panels show the results for V12=0.2V11 and V12=0, respectively. The symbols represent calculated values, while the solid and dashed lines are a guide to the eye. Here, the arbitrary units in A20τ have to be multiplied by the constant d≈1.06×10−27J2s3C2m2 to get the physical units.
A nonvanishing interband pair scattering V12 not only affects the relative strength of the oscillation frequencies but also strongly influences the long-time asymptotic gap values Δ∞l and hence the frequencies of the order parameter oscillations. This is illustrated in Figs. 2(a) and 2(b), which show the dependence of Δ∞l on the integrated pump pulse intensity A20τ for ℏω0=8 meV and both for zero and nonzero V12. Comparing Figs. 2(a) and 2(b), we observe that in the presence of nonzero interband pair scattering the time evolutions of the two gaps are coupled together. The change in Δ∞l due to interband interactions seems to be roughly proportional to V12Δ∞l′, with l′≠l [cf. Eq. (9)]. That is, the larger gap Δσ(t) influences the smaller gap Δπ(t) more strongly than vice versa. With increasing integrated laser intensity both gaps are suppressed simultaneously, until for sufficiently large A20τ they vanish at the same value of A20τ. For zero V12, in contrast, the order parameters can be suppressed independently. For example, we find that in the nonadiabatic regime the larger gap Δσ(t) vanishes at a smaller integrated intensity than the smaller gap Δπ(t) [solid green curve in Fig. 2(b)]. We observe that in general the asymptotic gap values Δ∞l decrease linearly at small A20τ, but deviate from this behavior at larger integrated intensities. With increasing A20τ, the curves corresponding to short pump pulses (τ=0.4 ps) show a downward bend, while those with longer pump pulses (τ=1.0 and 5.0 ps) flatten due to Pauli blocking.
Let us also briefly discuss the case where the pump pulse energy ℏω0 is equal to 2Δσ(ti)=14 meV. In general, the behavior of the gaps in this case is rather similar to the previously discussed case, where ℏω0=8 meV. As before, we find that for pump pulse widths τ≫τΔl both gap amplitudes |Δl(t)| decrease monotonically from their initial equilibrium values to their final values Δ∞l, whereas for τ≪τΔl the gaps approach their asymptotic values in an oscillatory fashion according to Eq. (10). However, as can be seen from Figs. 2(c) and 2(d), the dependence of Δ∞l on the integrated intensity A20τ is quite different than in Figs. 2(a) and 2(b). Interestingly, we find that a pump pulse with central energy ℏω0=2Δσ(ti) depletes the superconducting condensate more efficiently on the σ-band than on the π-band. In particular, for V12=0 the gap on the π-band is almost unaffected by the pump pulse, even for large A20τ, whereas the asymptotic gap value on the σ-band decreases steadily with increasing A20τ. For nonzero V12 the behavior is similar, although here the gap on the π-band is slightly suppressed, which is due to the coupling with the gap on the σ-band.
Figure 3: (Color online) (a) Calculated time evolution of |Δh(t)| and |Δe(t)| for the case of iron-based superconductors. The inset shows the spectral distribution of the gap oscillations for the same parameters as in the main panel. (b) Asymptotic values of the order parameters Δ∞h and Δ∞e versus integrated pump pulse intensity for three different pump pulse widths τ. In this figure we choose V12=−10V11 and ℏω0=14 meV. In panel (a), we set τ=0.4 ps and A20τ≈1.44×10−26Js/Cm [corresponding to the point A20τ=13 arb. units in panel (b)]. The symbols in panel (b) represent calculated values, while the lines are a guide to the eye. Here, the arbitrary units in A20τ have to be multiplied by the constant d≈1.11×10−27J2s3C2m2 to get the physical units.
Secondly, we consider a model appropriate for the iron pnictide superconductors  . Many features of iron-based superconductors can be captured within a simple two-band model with quasinested holelike and electronlike pockets centered at the Γ and M points in the Brillouin zone, respectively  . Due to the quasinesting of the Fermi surfaces, the repulsive interband pair scattering in such a description is much larger than the intraband pairing potentials. The latter are assumed to be overscreened due to the nesting effects and become attractive for low energies relevant for superconductivity. Therefore, in what follows, we model the band structure of iron-based superconductors by two parabolic bands, a hole band with effective mass mh=−3m0 and Fermi velocity vhF=0.168×106 m/s, and an electron band with me=m0 and veF=1.453×106 m/s  . For convenience we shift the location of the electron Fermi surface from the M point to the Γ point, i.e., we set ε2k+Q=ε2k, where Q=(π,π). The attractive intraband pairing interactions on the electron and hole Fermi surfaces are chosen to be V11=2.466 meV and V22=2.463 meV, while the repulsive interband pair scattering is set to V12=−24.42 meV. With these material parameters and assuming that the cut-off energy ℏωc is determined by the energy of the spin fluctuations in the paramagnetic state, ℏωc=30 meV  , we find that the equilibrium gaps are given by Δh(ti)=−Δe(ti)=6.5 meV. That is, the gaps on the hole and electron bands are equal in magnitude but opposite in phase, which is roughly in line with experimental findings in some of the hole-doped iron-based superconductors  . The two-band superconductor is driven out of equilibrium by a pump pulse with energy ℏω0=14 meV, which is of the same order but slightly larger than twice the gap amplitudes in the initial state.
Based on this model for pnictide superconductors, we compute the temporal evolution of the superconducting order parameters on the hole and electron bands. In Fig. 3(a), the time dependence of |Δl(t)| is shown for a pump pulse with length τ=0.4 ps, i.e., for the nonadiabatic regime. We observe that both gaps exhibit almost the same behavior: they first decrease monotonically and then start to oscillate with different frequencies producing a beatinglike pattern in the amplitudes. Due to the smaller Fermi velocity and the larger effective mass of the hole band, the gap on the hole band Δh(t) is slightly less suppressed than the gap on the electron band Δe(t). Because of the nonzero interband coupling V12, both gaps oscillate with two frequencies, ωΔe=2Δ∞e% /ℏ and ωΔh=2Δ∞h% /ℏ, which differ very little, i.e., ∣∣ωΔh−ωΔe∣∣≪ωΔe. This in turn leads to a pronounced beating phenomenon, as can be seen in Fig. 3(a).
The dependence of the asymptotic gap values Δ∞l on the integrated pump pulse intensity A20τ is plotted in Fig. 3(b). Due to the large interband coupling V12 the gaps on both bands decrease almost synchronously with increasing A20τ. However, the asymptotic value of the gap on the electron band Δ∞e is always slightly smaller than the asymptotic value of the gap on the hole band Δ∞h. As discussed above, this is because of the larger effective mass and the smaller Fermi velocity on the hole band. In general, we find that the relative difference between Δ∞h and Δ∞e gradually increases with increasing pump pulse intensity. Correspondingly, the difference between the two oscillation frequencies ωΔh and ωΔh increases, and hence the beating phenomenon becomes less pronounced, as the integrated pump pulse intensity is increased.
In this work we analyzed the nonequilibrium dynamics of two-band superconductors after excitation by a short opitcal pump pulse. We considered two model systems: one with dominant intraband pairing and weak attractive interband pair scattering, and one where the repuslive intrerband interactions are much larger than the intraband pairing potentials. The former model is relevant for MgB2 superconductors, whereas the latter one is appropriate for iron-based superconductors. For both of these model systems we numerically computed the time evolution of the order parameters as a function of pump pulse duration and integrated pump pulse intensity. Our main observation is that the ratio between the gaps for asymptotically large times depends sensitively on the interband Cooper-pairing strength and differs from its equilibrium value. This allows in principle to use pump-probe experiments for direct identification of the interband Cooper-pair scattering strength. In addition, sufficiently short pump pulses create fast oscillations in the gap amplitudes and the quasiparticle densities of these two-band superconductors. We found that for nonzero interband pair scattering V12 these oscillations are characterized by two frequencies which are determined by the long-time asymptotic values of the gaps on the two bands (see Fig. 1). We showed that the relative strength of the two frequency components sensitively depends on the magnitude of the interband interaction V12. When the gaps on the two bands are of similar magnitude (which is the case, for example, in iron-based superconductors) the relative difference between the two oscillation frequencies is small, and hence the gaps oscillate with a beatinglike pattern in the amplitudes.
We thank A. Avella, U. Bovensiepen, B. Kamble, N. Hasselmann, and G. Uhrig for useful discussions. The work of A.A. and I.E. is supported by the Merkur Foundation.
where we have introduced the short-hand notation M±lk,k′=uklu∗k′l±vklv∗k′l and L±lk,k′=uklvk′l±vkluk′l, with the coherence factors ukl and vkl. The functions η1kl and η2kl are defined in the main text. In deriving Eqs. (0.2), we used the relation A∗q=A−q.
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