Time-Domain ab Initio Modeling of Electron–Phonon Relaxation in

Letter pubs.acs.org/JPCL

Time-Domain ab Initio Modeling of Electron−Phonon Relaxation in High-Temperature Cuprate Superconductors Run Long*,†,‡ and Oleg V. Prezhdo*,§ †

College of Chemistry, Key Laboratory of Theoretical & Computational Photochemistry of Ministry of Education, Beijing Normal University, Beijing 100875, PR China ‡ School of Physics, Complex & Adaptive System Lab, University College Dublin, Belfield, Dublin 4, Ireland § Departments of Chemistry, Physics, and Astronomy, University of Southern California, Los Angeles, California 90089, United States ABSTRACT: Superconducting pairing due to electron−phonon coupling is investigated in recent pump−probe experiments. Combining time-dependent density functional theory and nonadiabatic molecular dynamics, we report the first direct modeling of such experiments and show how the electron−phonon relaxation depends on chemical bonding, electron−phonon coupling, and electronic state density. The relaxation rate is determined primarily by the nonadiabatic charge−phonon coupling strength, which in turn depends on the strength of chemical interactions between the key atoms, reflected in the wave function delocalization. The differences in the electronic density of states constitute the secondary factor. Having obtained good agreement with the experimental data on YBa2Cu3O6.5, we predict that the relaxation slows if Y is replaced with Sc or Ba with Sr, while the relaxation accelerates if O is replaced with S, indicating that YBa2Cu3S6.5 can exhibit improved superconducting performance.

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motivation for development of theoretical techniques capable of predicting electron−phonon relaxation times at a nonperturbative ab initio level, avoiding assumptions such as weak electron−phonon coupling and harmonic phonons. The ability to model the electron−phonon relaxation dynamics and to generate a mechanistic interpretation of the pump−probe measurements provides a route to search for novel high-Tc materials. In this Letter, we report the first time-domain ab initio study of the electron−phonon relaxation dynamics in high-Tc superconductors. The technique combines real-time timedependent density functional theory (DFT) for the evolution of the electronic subsystem18 with nonadiabatic (NA) molecular dynamics (MD) for description of the atomic motions.19,20 The ab initio atomistic methodology mimics the pump−probe optical experiments in the most direct way and has been applied successfully to a broad range of semiconducting and metallic systems.21−30 Focusing on the widely studied5,8,16 YBCO superconductor, we show that the relaxation time depends on the extent of chemical bonding, strength of NA electron−phonon coupling, and electronic density of states (DOS). Having established the relaxation mechanism and encouraged by excellent agreement with the experimental time scale,5 we consider substitutions of Y, Ba, and O in YBCO with Sc, Sr, and S, forming three new systems, ScBCO, YSrCO, and YBCS. Substitution of O with S

igh-temperature (high-Tc) superconductors have been attracting intense interest since the first discovery in 1986.1 With many experiments to follow, 2−9 multiple theoretical studies have explored the electronic structure and superconductivity mechanisms.10−13 In conventional superconductors, electrons form Cooper pairs bound via virtual phonon exchange. The original Bardeen−Cooper−Schrieffer theory14 was extended to the strong coupling regime by Eliashberg,15 who took into account that the electron−phonon interaction is local in space and retarded in time. Key theoretical quantity, the electron−phonon interaction, determines the material functional properties, such as the superconducting transition temperature. Recent breakthroughs in the field include the above room-temperature superconductivity in YBa2Cu3O6.5 (YBCO) lasting several picoseconds.8,16 In the past, the electron−phonon interaction strength in superconductors was determined experimentally from phonon line widths in Raman or neutron scattering spectra. These are often biased by selection rules and inhomogeneous broadening, giving rise to controversial results. Starting from the pioneering work of Gadermaier et al.,5 the electron−phonon interaction strength can be determined by measuring the electron−phonon relaxation time, τe−ph, using pump−probe optical spectroscopy. Focusing on the metallic, nonmagnetic ortho-II cuprate YBCO superconductor, the authors estimated the second moment of the Eliashberg function according to the nonequilibrium model.17 Faster electron−phonon relaxation increases the value, leading to a higher superconductivity transition temperature, Tc. The ability of pump−probe laser spectroscopies to characterize the superconducting properties provides a strong © XXXX American Chemical Society

Received: November 20, 2016 Accepted: December 16, 2016 Published: December 16, 2016 193

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The Journal of Physical Chemistry Letters accelerates the energy relaxation, which is counterintuitive, because heavier S should slow the electron−phonon energy exchange. Replacement of Y with Ba and Sc with Sr make the relaxation slower. The accelerated energy losses in YBCS are attributed to the following effects. First, 3p orbitals of sulfurs are more delocalized than 2p orbitals of oxygens, leading to a strong S−Cu interaction that facilitates charge delocalization and enhances the NA coupling. Second, the substitution increases DOS in the relevant energy range. Third, replacement of O with S increases anharmonicity and hence the electron− phonon coupling. In contrast, substitution of Sc with Sr and Y with Ba decreases the NA coupling, reduces DOS, and lowers the active phonon frequencies, slowing the relaxation. The simulations suggest that the Tc of YBCS should be higher than that of YBCO, ScBCO, and YSrCO. The simulations are performed using the fewest switching surface hopping technique31 implemented within the timedependent Kohn−Sham DFT,32 as explained in detail in refs 19, 20, and 33. The technique can be viewed as a nonMarkovian quantum master equation, in which transition rates vary with time and are computed nonperturbatively at the ab initio, atomistic level. An in-plane 2 × 1 × 1 Y2Ba4Cu6O13 simulation cell is used, containing 25 atoms. Subsequently, the cell is modified to model the ScBCO, YSrCO, and YBCS systems. The VASP package34,35 is used for electronic structure calculations, geometry optimization, and adiabatic MD. The simulations are carried out with the Perdew−Burke−Ernzerhof exchange-correlation functional,36 projector-augmented wave pseudopotentials,37,38 and 420 eV plane-wave basis energy cutoff. The use of more advanced functionals, e.g., rangeseparated hybrids accounting for the self-interaction error of semilocal DFT, or the GW theory, as well as explicit consideration of the spin−orbit interactions39,40 involving the heavy elements is desirable. Unfortunately, it is not yet feasible because of excessive computational expense associated with NAMD involving thousands of electronic structure calculations. A 6 × 12 × 6 Γ-centered k-point Monkhorst−Pack mesh is employed during geometry optimization and MD.41 Both lattice constants and atomic coordinates are allowed to relax during the optimization, which is stopped when the residual forces reach below 0.01 eV/Å. To obtain accurate DOS, a denser 12 × 24 × 12 Monkhorst−Pack Γ-centered mesh is used.41 The default electronic smearing parameter is employed. The relaxed systems are heated to 300 K by repeated velocity rescaling. Then, 5 ps microcanonical MD trajectories are generated with a 1 fs atomic time-step. To simulate electron− phonon relaxation, 1000 geometries are randomly selected from these adiabatic trajectories and are used as initial conditions for NAMD. A one attosecond electronic time-step is used. A more detailed description can be found in refs 20 and 42. Figure 1 shows the optimized YBCO geometry in the Pmmm space group. The optimized lattice constants summarized in Table 1 show excellent agreement with the experimental data.43 YBCO is characterized by a periodic alternation of empty and filled Cu−O b-axis chains, doubling the size of the unit cell of YBa2Cu3O6.5 in the a direction. Two conducting CuO2 planes are separated by Y atoms and form a bilayer unit. Ba atoms between the CuO4 ribbons in the bc-plane separate the bilayers units.43 The presence of oxygen in the chains gives rise to an attractive potential for the holes in the planes. When holes are doped into the planes, high-temperature superconductivity is induced. Replacement of Y, Ba, and O by Sc, Sr, and S gives rise

Figure 1. Geometric structure of YBa2Cu3O6.5 (YBCO). Y is light blue, Ba green, Cu dark blue, and O red. Two conducting Cu(1)O2 planes are separated by Y atoms and form a bilayer unit. Ba atoms and Cu(2)O4 ribbons in the bc-plane separate the bilayer units. The other three systems are obtained by replacing Y, Ba, and O with Sc, Sr, and S, respectively. The optimized lattice constants are listed in Table 1. Replacement of Y and Ba by Sc and Sr induces little change in the lattice constants because of similar ionic sizes of the initial and final atoms. Substitution of O with S expands the lattice because the S ion is much larger than the O ion.

Table 1. Optimized Lattice Constants (Å) of YBCO, ScBCO, YSrCO, and YBCS a b c

YBCO

ScBCO

YSrCO

YBCS

7.69 3.90 11.81

7.58 3.84 11.77

7.53 3.84 11.53

9.01 4.62 13.89

to the other three materials under consideration, ScBCO, YSrCO, and YBCS, respectively. The symmetry remains the same in all systems after geometry optimization. The optimized lattice constants of the three additional systems are also presented in Table 1. Table 1 shows that the lattices in ScBCO and YSrCO shrink slightly compared to YBCO because of a smaller ionic size of Sc2+ versus Y2+ and Sr2+ versus Ba2+. Compared to those of YBCO, the lattice constants of YBCS expand by 17%, 18%, and 18% in the a, b, and c directions, respectively, because of the larger radius of S2− compared to that of O2−. The substantial geometry changes in YBCS lead to significant changes in the electronic structure (Figure 2). The DOS of ScBCO and YSrCO are similar to the YBCO DOS, because the lattice constants are nearly the same in these three materials (Table 1). Consider the asymmetry between the conduction band (CB) and valence band (VB), the large peak just below the Fermi energy, and the energy gap in the CB. The DOS at the Fermi energy is slightly greater in YBCO than in ScBCO and YSrCO, as represented in Figure 2 by the numbers in parentheses. Generally, higher DOS results in faster electron−phonon energy relaxation. The DOS of YBCS differs 194

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Figure 2. DOS of the optimized (a) YBCO, (b) ScBCO, (c) YSrCO, and (d) YBCS. The dashed line represents the Fermi level. The values in parentheses represent DOS at the Fermi level. Larger DOS favors faster electron−phonon relaxation.

contrast, substitution of O with S significantly increases the S− Cu(1) bonding interaction, making the wave function more delocalized, bottom right panel of Figure 3, and enhancing the NA coupling. The change arises because the S2− ion is much larger than the O2− ion, and the 3p orbitals of the S atom are much more extended than the 2p orbitals of O. The analysis of the wave function localization (Figure 3) and DOS (Figure 2) suggests that the electron−phonon relaxation in YBCO should be slower than in YBCS and faster than in YSrCO and ScBCO. Fourier transforms (FTs) of the phonon-induced fluctuations in the energies of the photoexcited states characterize the phonon modes that couple to the electronic subsystem (Figure 4). The FT magnitude reflects the strength of the electron− phonon coupling. The magnitude is largest for YBCS, while YBCO shows the highest frequencies. The 475 cm−1 mode observed in YBCO is the high-frequency longitudinal Josephson plasma mode,8 which quantifies the strength of the Josephson coupling between pairs of Cu(1)O2 layers (cf. Figure 1). Only YBCO exhibits strong coupling to this mode. The lowest-energy peak agrees with the other longitudinal Josephson plasma mode at 30 cm−1.8 The slow modes modulate the interlayer interaction and separation, thereby also contributing to the electron−phonon coupling. YBCS exhibits the largest FT magnitude, suggesting that this material should exhibit the fastest electron−phonon relaxation. Figure 5 represents relaxation of the energy of the electronic subsystem due to coupling to phonons. The relaxation time scales reported in the figure are obtained by Gaussian fitting. The 35 fs energy relaxation time computed for YBCO agrees well with the sub-100 fs experimental observation.5 Comparing the four systems, we observe that the relaxation time decreases in the following order: ScBCO ≈ YSrCO > YBCO > YBCS. This sequence corresponds to the increasing absolute values of the NA coupling, computed by averaging over the NA dynamics: 9.5 ≈ 9.9 < 10.5 < 15.5 meV. According to Fermi’s golden rule, the relaxation rate is proportional to the coupling

significantly from the other three systems because of a large geometry change (Table 1). The energy gap in the CB disappears; the peak just below the Fermi energy becomes lower and broadens, eliminating the deep trough at −0.4 eV; and the DOS at the Fermi energy and below increases, compared to YBCO, ScBCO, and YSrCO. We focus on the pump−probe experiments5 performed with the 530 nm pump wavelength, corresponding to 2.34 eV. The computed oscillator strengths for excitations between occupied and vacant Kohn−Sham orbitals, with the orbital energy gaps between 2 and 2.5 eV, have the same order of magnitude. Because the VB DOS is significantly higher than the CB DOS (Figure 2), and the DOS generally decreases with increasing energy in the −2 to 2 eV window, absorption of a photon creates an asymmetric excitation, with more of the photon energy deposited into the hole. The significantly higher VB DOS compared to the CB DOS also guarantees that the hole relaxes much faster than the electron; therefore, the hole relaxation determines the experimental time scale. Figure 3 shows the charge densities of the photogenerated holes in the four systems. The NA electron−phonon coupling arises because of changes in the excited-state densities in response to nuclear motions. States responsible for chemical bonding with densities delocalized between atoms are much more sensitive to atomic displacements than localized states. More significant delocalization implies stronger NA electron−phonon coupling and involvement of a broader spectrum of phonon modes, favoring faster energy relaxation. The top left panel of Figure 3 shows that the interaction between in-plane O and Cu(1) in YBCO contributes to the bonding interaction and creates the NA coupling. It is known that the coupling to the O−Cu(1) phonon is responsible for the Cooper pair formation, making YBCO superconductive.44 Figure 3 show that replacement of Y and Ba by Sc and Sr in YBCO, respectively, make the wave function slightly more localized, decreasing the NA coupling. In 195

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Figure 5. Electron−phonon energy relaxation in the four systems.

squared. This relationship holds for the current systems, e.g. (15.5 meV/9.5 meV)2 ≈ 51 fs/20 fs. Thus, the value of the NA electron−phonon coupling is the main factor that determines the electron−phonon relaxation time in the current systems. The coupling is closely related to the bonding interaction and wave function delocalization (Figure 3). The differences in the DOS (Figure 2) constitute a secondary contribution to the relative relaxation rates. For example, the VB DOS is highest for YBCS, which exhibits the fastest decay. The relationship between the electron−phonon relaxation rate and the second moment of the Eliashberg function5,17 suggests that ScBCO and YSrCO should have lower superconducting transition temperatures than YBCO, while YBCS should have a higher transition temperature. In summary, we have reported the first time-domain atomistic study of electron−phonon relaxation dynamics in superconducting materials. The used ab initio technique mimics the experimental systems in the most direct way, avoids perturbative and harmonic approximations, and is easily transferrable between materials. The relaxation time obtained for YBCO agrees well with the experimental data. The calculations show that the relaxation in YBCO is promoted by the longitudinal Josephson plasma modes, also in agreement with the previous results. The ab initio simulations provide a detailed mechanism of the relaxation process and allow for a straightforward prediction of the electron−phonon relaxation dynamics in other materials yet to be studied experimentally. The study shows that the relaxation rate is determined primarily by the NA electron−phonon coupling strength, which in turn depends on the strength of chemical interactions between the key atoms, reflected in the wave function delocalization. The differences in the electronic DOS constitute the secondary factor. The insights into the relaxation mechanism provided by the simulations and the good agreement with the available experimental data have allowed us to consider, in a systematic manner, several derivatives of the original YBCO superconductor and to predict that YBCS should exhibit stronger electron−phonon coupling, faster relaxation, and likely, higher superconducting transition temperature. This prediction is counterintuitive a priori, because one expects that substitution of the lighter oxygen for the heavier sulfur should decrease the electron−phonon interaction. However, the calculations show that the extent of chemical bonding is more important than the atomic mass. The

Figure 3. Charge density of the initially excited state. Density delocalization implies coupling to a broader spectrum of phonon modes and stronger NA electron−phonon coupling. The delocalization changes in the sequence YBCS > YBCO > ScBCO > YSrCO. The isosurface value is set to 0.003 e/Å3 in all cases.

Figure 4. Fourier transfroms of the phonon-induced fluctuations in the energies of the initially excited states. They characterize the phonon modes involved in the electron−phonon relaxation. Higher frequency modes dominate in YBCO, while low-frequency phonons contribute to the relaxation in the other systems. The signal amplitude reflects the electron−phonon coupling strength. The amplitude is strongest in YBCS.

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reported simulations generate a unique and comprehensive description of the electron−phonon dynamics in superconductors and provide a novel computational approach to search for high-Tc superconducting materials.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Oleg V. Prezhdo: 0000-0002-5140-7500 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS R.L. is grateful to the National Science Foundation of China, Grant No. 21573022; the Recruitment Program of Global Youth Experts of China; the Beijing Normal University Startup Package; and the Science Foundation Ireland, Grant No. 11/ SIRG/E2172. O.V.P. acknowledges the U.S. National Science Foundation, Grant No. CHE-1565704, and is grateful to the High-End Foreign Experts Recruitment Program for support during the visit to Beijing Normal University.

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