Dynamical Processes in Open Quantum Systems from a TDDFT Perspective: Resonances and Electron Photoemission

  • Ask Hjorth Larsen
  • Umberto De Giovannini
  • Angel Rubio
Part of the Topics in Current Chemistry book series (TOPCURRCHEM, volume 368)


We present a review of different computational methods to describe time-dependent phenomena in open quantum systems and their extension to a density-functional framework. We focus the discussion on electron emission processes in atoms and molecules addressing excited-state lifetimes and dissipative processes. Initially we analyze the concept of an electronic resonance, a central concept in spectroscopy associated with a metastable state from which an electron eventually escapes (electronic lifetime). Resonances play a fundamental role in many time-dependent molecular phenomena but can be rationalized from a time-independent context in terms of scattering states. We introduce the method of complex scaling, which is used to capture resonant states as localized states in the spirit of usual bound-state methods, and work on its extension to static and time-dependent density-functional theory. In a time-dependent setting, complex scaling can be used to describe excitations in the continuum as well as wave packet dynamics leading to electron emission. This process can also be treated by using open boundary conditions which allow time-dependent simulations of emission processes without artificial reflections at the boundaries (i.e., borders of the simulation box). We compare in detail different schemes to implement open boundaries, namely transparent boundaries using Green functions, and absorbing boundaries in the form of complex absorbing potentials and mask functions. The last two are regularly used together with time-dependent density-functional theory to describe the electron emission dynamics of atoms and molecules. Finally, we discuss approaches to the calculation of energy and angle-resolved time-dependent pump–probe photoelectron spectroscopy of molecular systems.


Absorbing boundaries Complex scaling Photoemission Resonances 



We acknowledge financial support from the European Research Council Advanced Grant DYNamo (ERC-2010-AdG-267374), Ministerio de Economía y Competitividad or MINECO, Spanish Grant (FIS2013-46159-C3-1-P), Grupos Consolidados UPV/EHU del Gobierno Vasco (IT578-13), European Commission FP7 project CRONOS (Grant number 280879-2), COST Actions CM1204 (XLIC), and MP1306 (EUSpec).


  1. 1.
    Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev 136:B864–B871. doi: 10.1103/PhysRev.136.B864, Scholar
  2. 2.
    Runge E, Gross EKU (1984) Density-functional theory for time-dependent systems. Phys Rev Lett 52(12):997–1000CrossRefGoogle Scholar
  3. 3.
    Kohn W, Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Phys Rev 140:A1133–A1138. doi: 10.1103/PhysRev.140.A1133, Scholar
  4. 4.
    Burke K, Car R, Gebauer R (2005) Density functional theory of the electrical conductivity of molecular devices. Phys Rev Lett 94(14):146803CrossRefGoogle Scholar
  5. 5.
    Tempel DG, Aspuru-Guzik A (2011) Relaxation and dephasing in open quantum systems time-dependent density functional theory: properties of exact functionals from an exactly-solvable model system. Chem Phys 391(1):130–142CrossRefGoogle Scholar
  6. 6.
    Tempel DG, Watson MA, Olivares-Amaya R, Aspuru-Guzik A (2011) Time-dependent density functional theory of open quantum systems in the linear-response regime. J Chem Phys 134(7):074116CrossRefGoogle Scholar
  7. 7.
    Yuen-Zhou J, Rodríguez-Rosario C, Aspuru-Guzik A (2009) Time-dependent current-density functional theory for generalized open quantum systems. Phys Chem Chem Phys 11(22):4509–4522CrossRefGoogle Scholar
  8. 8.
    Yuen-Zhou J, Tempel DG, Rodríguez-Rosario CA, Aspuru-Guzik A (2010) Time-dependent density functional theory for open quantum systems with unitary propagation. Phys Rev Lett 104(4):043001CrossRefGoogle Scholar
  9. 9.
    Marques MAL, Maitra NT, Nogueira F, Gross EKU, Rubio A (2011) Fundamentals of time-dependent density functional theory. Springer, BerlinGoogle Scholar
  10. 10.
    Fano U (1961) Effects of configuration interaction on intensities and phase shifts. Phys Rev 124(6):1866–1878CrossRefGoogle Scholar
  11. 11.
    Brandbyge M, Mozos JL, Ordejón P, Taylor J, Stokbro K (2002) Density-functional method for nonequilibrium electron transport. Phys Rev B 65:165401. doi: 10.1103/PhysRevB.65.165401 CrossRefGoogle Scholar
  12. 12.
    Chen J, Thygesen KS, Jacobsen KW (2012) Ab initio. Phys Rev B 85:155140. doi: 10.1103/PhysRevB.85.155140, Scholar
  13. 13.
    Larsen AH, Vanin M, Mortensen JJ, Thygesen KS, Jacobsen KW (2009) Localized atomic basis set in the projector augmented wave method. Phys Rev B 80:195112. doi: 10.1103/PhysRevB.80.195112, Scholar
  14. 14.
    Soler JM, Artacho E, Gale JD, García A, Junquera J, Ordejón P, Sánchez-Portal D (2002) The SIESTA method for ab initio order-N materials simulation. J Phys Condens Matter 14:2745–2779. doi: 10.1088/0953-8984/14/11/302, Scholar
  15. 15.
    Newns DM (1969) Self-consistent model of hydrogen chemisorption. Phys Rev 178:1123–1135. doi: 10.1103/PhysRev.178.1123, Scholar
  16. 16.
    Gellene GI (1995) Resonant states of a one-dimensional piecewise constant potential. J Chem Educ 72(11):1015. doi: 10.1021/ed072p1015, Scholar
  17. 17.
    Siegert AJF (1939) On the derivation of the dispersion formula for nuclear reactions. Phys Rev 56:750–752. doi: 10.1103/PhysRev.56.750, Scholar
  18. 18.
    Hatano N, Sasada K, Nakamura H, Petrosky T (2008) Some properties of the resonant state in quantum mechanics and its computation. Prog Theor Phys 119(2):187–222. doi: 10.1143/PTP.119.187, Scholar
  19. 19.
    Aguilar J, Combes J (1971) A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun Math Phys 22:269–279. doi: 10.1007/BF01877510, Scholar
  20. 20.
    Balslev E, Combes JM (1971) Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions. Commun Math Phys 22(4):280–294CrossRefGoogle Scholar
  21. 21.
    Simon B (1973) Resonances in n-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory. Ann Math 97:247–274CrossRefGoogle Scholar
  22. 22.
    Simon B (1979) The definition of molecular resonance curves by the method of exterior complex scaling. Phys Lett A 71(2):211–214CrossRefGoogle Scholar
  23. 23.
    Ho Y (1983) The method of complex coordinate rotation and its applications to atomic collision processes. Phys Rep 99(1):1–68. doi: 10.1016/0370-1573(83)90112-6, CrossRefGoogle Scholar
  24. 24.
    McCurdy CW, Baertschy M, Rescigno TN (2004) Solving the three-body Coulomb breakup problem using exterior complex scaling. J Phys B At Mol Opt 37(17):R137, = 17/a = R01 CrossRefGoogle Scholar
  25. 25.
    Moiseyev N (1998) Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling. Phys Rep 302(5–6):212–293. doi: 10.1016/S0370-1573(98)00002-7, CrossRefGoogle Scholar
  26. 26.
    Reinhardt WP (1982) Complex coordinates in the theory of atomic and molecular structure and dynamics. Annu Rev Phys Chem 33(1):223–255. doi: 10.1146/annurev.pc.33.100182.001255, CrossRefGoogle Scholar
  27. 27.
    Simon B (1978) Resonances and complex scaling: a rigorous overview. Int J Quantum Chem 14(4):529–542. doi: 10.1002/qua.560140415, Scholar
  28. 28.
    Cerjan C, Hedges R, Holt C, Reinhardt WP, Scheibner K, Wendoloski JJ (1978) Complex coordinates and the Stark effect. Int J Quantum Chem 14(4):393–418. doi: 10.1002/qua.560140408, Scholar
  29. 29.
    Herbst IW (1979) Dilation analyticity in constant electric field. Commun Math Phys 64(3):279–298. doi: 10.1007/BF01221735 CrossRefGoogle Scholar
  30. 30.
    Herbst IW, Simon B (1978) Stark effect revisited. Phys Rev Lett 41:67–69. doi: 10.1103/PhysRevLett.41.67, Scholar
  31. 31.
    Scrinzi A, Piraux B (1998) Two-electron atoms in short intense laser pulses. Phys Rev A 58:1310–1321. doi: 10.1103/PhysRevA.58.1310, Scholar
  32. 32.
    Junker BR (1983) Complex virial theorem and complex scaling. Phys Rev A 27:2785–2789. doi: 10.1103/PhysRevA.27.2785, Scholar
  33. 33.
    Moiseyev N, Friedland S, Certain PR (1981) Cusps, θ trajectories, and the complex virial theorem. J Chem Phys 74(8):4739–4740. doi: 10.1063/1.441624, Scholar
  34. 34.
    McCurdy CW (1980) Complex-coordinate calculation of matrix elements of the resolvent of the Born–Oppenheimer Hamiltonian. Phys Rev A 21:464–470. doi: 10.1103/PhysRevA.21.464, Scholar
  35. 35.
    McCurdy CW, Rescigno TN (1978) Extension of the method of complex basis functions to molecular resonances. Phys Rev Lett 41:1364–1368. doi: 10.1103/PhysRevLett.41.1364, Scholar
  36. 36.
    Moiseyev N, Corcoran C (1979) Autoionizing states of H2 and H2 using the complex-scaling method. Phys Rev A 20:814–817. doi: 10.1103/PhysRevA.20.814, Scholar
  37. 37.
    Morgan JD, Simon B (1981) The calculation of molecular resonances by complex scaling. J Phys B At Mol Opt 14(5):L167CrossRefGoogle Scholar
  38. 38.
    Scrinzi A (2010) Infinite-range exterior complex scaling as a perfect absorber in time-dependent problems. Phys Rev A 81(5):053845CrossRefGoogle Scholar
  39. 39.
    Moiseyev N (1999) Derivations of universal exact complex absorption potentials by the generalized complex coordinate method. J Phys B At Mol Opt 31(7):1431–1441CrossRefGoogle Scholar
  40. 40.
    Krylstedt P, Carlsund C, Elander N (1989) On the calculation of electron–atom collision properties using exterior complex dilatated s-matrix expansions. J Phys B At Mol Opt 22(7):1051, = 7/a = 014 CrossRefGoogle Scholar
  41. 41.
    Rescigno TN, Baertschy M, Byrum D, McCurdy CW (1997) Making complex scaling work for long-range potentials. Phys Rev A 55:4253–4262. doi: 10.1103/PhysRevA.55.4253, CrossRefGoogle Scholar
  42. 42.
    Scrinzi A, Elander N (1993) A finite element implementation of exterior complex scaling for the accurate determination of resonance energies. J Chem Phys 98(5):3866–3875. doi: 10.1063/1.464014, Scholar
  43. 43.
    Simons J (1980) The complex coordinate rotation method and exterior scaling: a simple example. Int J Quantum Chem 18(S14):113–121. doi: 10.1002/qua.560180814, Scholar
  44. 44.
    Kar S, Ho YK (2009) Isotope shift for the 1De autodetaching resonance in H and D. J Phys B At Mol Opt 42(5):055001, = 5/a = 055001CrossRefGoogle Scholar
  45. 45.
    Scrinzi A, Geissler M, Brabec T (1999) Ionization above the Coulomb barrier. Phys Rev Lett 83:706–709. doi: 10.1103/PhysRevLett.83.706, Scholar
  46. 46.
    McCurdy CW, Rescigno TN, Davidson ER, Lauderdale JG (1980) Applicability of self-consistent field techniques based on the complex coordinate method to metastable electronic states. J Chem Phys 73(7):3268–3273. doi: 10.1063/1.440522, Scholar
  47. 47.
    Samanta K, Yeager DL (2008) Investigation of 2P Be shape resonances using a quadratically convergent complex multiconfigurational self-consistent field method. J Phys Chem B 112(50):16214–16219. doi: 10.1021/jp806998n, Scholar
  48. 48.
    Zdánská PR, Moiseyev N (2005) Hartree–Fock orbitals for complex-scaled configuration interaction calculation of highly excited Feshbach resonances. J Chem Phys 123(19):194105. doi: 10.1063/1.2110169, Scholar
  49. 49.
    Larsen AH, Whitenack DL, De Giovannini U, Wasserman A, Rubio A (2013) Stark ionization of atoms and molecules within density functional resonance theory. J Phys Chem Lett 4:2734–2738CrossRefGoogle Scholar
  50. 50.
    Whitenack DL, Wasserman A (2011) Density functional resonance theory of unbound electronic systems. Phys Rev Lett 107(16):163002CrossRefGoogle Scholar
  51. 51.
    Wasserman A, Moiseyev N (2007) Hohenberg-Kohn theorem for the lowest-energy resonance of unbound systems. Phys Rev Lett 98:093003. doi: 10.1103/PhysRevLett.98.093003, Scholar
  52. 52.
    Perdew JP, Wang Y (1992) Accurate and simple analytic representation of the electron-gas correlation energy. Phys Rev B 45:13244–13249. doi: 10.1103/PhysRevB.45.13244, Scholar
  53. 53.
    Telnov DA, Sosnova KE, Rozenbaum E, Chu SI (2013) Exterior complex scaling method in time-dependent density-functional theory: multiphoton ionization and high-order-harmonic generation of Ar atoms. Phys Rev A 87(5):053406CrossRefGoogle Scholar
  54. 54.
    van Leeuwen R, Baerends EJ (1994) Exchange–correlation potential with correct asymptotic behavior. Phys Rev A 49:2421–2431. doi: 10.1103/PhysRevA.49.2421, Scholar
  55. 55.
    Ammosov MV, Delone NB, Krainov VP (1986) Tunnel ionization of complex atoms and atomic ions in a varying electromagnetic-field. Zh Éksp Teor Fiz 91:2008–2013Google Scholar
  56. 56.
    Parker SD, McCurdy C (1989) Propagation of wave packets using the complex basis function method. Chem Phys Lett 156(5):483–488. doi: 10.1016/S0009-2614(89)87316-6, CrossRefGoogle Scholar
  57. 57.
    Bengtsson J, Lindroth E, Selstø S (2008) Solution of the time-dependent Schrödinger equation using uniform complex scaling. Phys Rev A 78:032502. doi: 10.1103/PhysRevA.78.032502, Scholar
  58. 58.
    Bengtsson J, Lindroth E, Selstø S (2012) Wave functions associated with time-dependent, complex-scaled Hamiltonians evaluated on a complex time grid. Phys Rev A 85:013419. doi: 10.1103/PhysRevA.85.013419, Scholar
  59. 59.
    Gilary I, Fleischer A, Moiseyev N (2005) Calculations of time-dependent observables in non-Hermitian quantum mechanics: the problem and a possible solution. Phys Rev A 72:012,117. doi: 10.1103/PhysRevA.72.012117,
  60. 60.
    García-Moliner F, Flores F (2009) Introduction to the theory of solid surfaces. Cambridge University Press, CambridgeGoogle Scholar
  61. 61.
    Kudrnovský J, Drchal V, Turek I, Weinberger P (1994) Magnetic coupling of interfaces: a surface-Green’s-function approach. Phys Rev B 50:16105–16108. doi: 10.1103/PhysRevB.50.16105, Scholar
  62. 62.
    Boucke K, Schmitz H, Kull HJ (1997) Radiation conditions for the time-dependent Schrödinger equation: application to strong-field photoionization. Phys Rev A 56(1):763–771CrossRefGoogle Scholar
  63. 63.
    Ermolaev A, Puzynin I, Selin A, Vinitsky S (1999) Integral boundary conditions for the time-dependent Schrödinger equation: atom in a laser field. Phys Rev A 60(6):4831–4845CrossRefGoogle Scholar
  64. 64.
    Hellums J, Frensley W (1994) Non-Markovian open-system boundary conditions for the time-dependent Schrödinger equation. Phys Rev B 49(4):2904–2906CrossRefGoogle Scholar
  65. 65.
    Kurth S, Stefanucci G, Almbladh CO, Rubio A, Gross EKU (2005) Time-dependent quantum transport: a practical scheme using density functional theory. Phys Rev B 72(3):035308CrossRefGoogle Scholar
  66. 66.
    Inglesfield JE (2001) Embedding at surfaces. Comput Phys Commun 137(1):89–107CrossRefGoogle Scholar
  67. 67.
    Inglesfield JE (2011) A time-dependent embedding calculation of surface electron emission. J Phys Condens Matter 23(30):305004CrossRefGoogle Scholar
  68. 68.
    Antoine X, Arnold A, Besse C, Ehrhardt M, Schädle A (2008) A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Commun Comput Phys 4:729–796Google Scholar
  69. 69.
    Inglesfield JE (1981) A method of embedding. J Phys C Solid State 14(26):3795–3806CrossRefGoogle Scholar
  70. 70.
    Inglesfield J (2008) Time-dependent embedding. J Phys Condens Matter 20:095215CrossRefGoogle Scholar
  71. 71.
    Ehrhardt M (1999) Discrete transparent boundary conditions for general Schrödinger-type equations. VLSI Des 9(4):325–338CrossRefGoogle Scholar
  72. 72.
    Szmytkowski R, Bielski S (2004) Dirichlet-to-Neumann and Neumann-to-Dirichlet embedding methods for bound states of the Schrödinger equation. Phys Rev A 70(4):042103CrossRefGoogle Scholar
  73. 73.
    Frensley W (1990) Boundary conditions for open quantum systems driven far from equilibrium. Rev Mod Phys 62(3):745–791CrossRefGoogle Scholar
  74. 74.
    De Giovannini U, Larsen AH, Rubio A (2015) Modeling electron dynamics coupled to continuum states in finite volumes. Eur Phys J B 88(3):56. doi: 10.1140/epjb/e2015-50808-0
  75. 75.
    Neuhauser D, Baer M (1989) The application of wave packets to reactive atom–diatom systems: a new approach. J Chem Phys 91(8):4651–4657CrossRefGoogle Scholar
  76. 76.
    Neuhauser D, Baer M (1989) The time-dependent Schrödinger equation: application of absorbing boundary conditions. J Chem Phys 90(8):4351CrossRefGoogle Scholar
  77. 77.
    Berenger JP (1994) A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 114(2):185–200CrossRefGoogle Scholar
  78. 78.
    Elenewski JE, Chen H (2014) Real-time transport in open quantum systems from PT-symmetric quantum mechanics. Phys Rev B 90(8):085104CrossRefGoogle Scholar
  79. 79.
    Varga K, Pantelides S (2007) Quantum transport in molecules and nanotube devices. Phys Rev Lett 98(7):076804CrossRefGoogle Scholar
  80. 80.
    Wibking BD, Varga K (2012) Quantum mechanics with complex injecting potentials. Phys Lett A 376(4):365–369CrossRefGoogle Scholar
  81. 81.
    Muga J, Palao JP, Navarro B, Egusquiza IL (2004) Complex absorbing potentials. Phys Rep 395(6):357–426CrossRefGoogle Scholar
  82. 82.
    Andrade X, Aspuru-Guzik A (2011) Prediction of the derivative discontinuity in density functional theory from an electrostatic description of the exchange and correlation potential. Phys Rev Lett 107(18):183002CrossRefGoogle Scholar
  83. 83.
    Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made simple. Phys Rev Lett 77(18):3865–3868CrossRefGoogle Scholar
  84. 84.
    Crawford-Uranga A, De Giovannini U, Räsänen E, Oliveira MJT, Mowbray DJ, Nikolopoulos GM, Karamatskos ET, Markellos D, Lambropoulos P, Kurth S, Rubio A (2014) Time-dependent density-functional theory of strong-field ionization of atoms by soft X-rays. Phys Rev A 90(3):033412CrossRefGoogle Scholar
  85. 85.
    Krause J, Schafer K, Kulander K (1992) Calculation of photoemission from atoms subject to intense laser fields. Phys Rev A 45(7):4998–5010CrossRefGoogle Scholar
  86. 86.
    Kulander K, Mies F, Schafer K (1996) Model for studies of laser-induced nonlinear processes in molecules. Phys Rev A 53(4):2562–2570CrossRefGoogle Scholar
  87. 87.
    Lein M, Marangos J, Knight P (2002) Electron diffraction in above-threshold ionization of molecules. Phys Rev A 66(5):051404RCrossRefGoogle Scholar
  88. 88.
    Chelkowski S, Foisy C, Bandrauk AD (1998) Electron–nuclear dynamics of multiphoton H2 + dissociative ionization in intense laser fields. Phys Rev A 57(2):1176–1185CrossRefGoogle Scholar
  89. 89.
    Grobe R, Haan S, Eberly J (1999) A split-domain algorithm for time-dependent multi-electron wave functions. Comput Phys Commun 117(3):200–210CrossRefGoogle Scholar
  90. 90.
    De Giovannini U, Varsano D, Marques MAL, Appel H, Gross EKU, Rubio A (2012) Ab initio angle- and energy-resolved photoelectron spectroscopy with time-dependent density-functional theory. Phys Rev A 85:062515CrossRefGoogle Scholar
  91. 91.
    Shemer O, Brisker D, Moiseyev N (2005) Optimal reflection-free complex absorbing potentials for quantum propagation of wave packets. Phys Rev A 71(3):032716CrossRefGoogle Scholar
  92. 92.
    McCurdy CW, Stroud C, Wisinski M (1991) Solving the time-dependent Schrödinger equation using complex-coordinate contours. Phys Rev A 43(11):5980–5990CrossRefGoogle Scholar
  93. 93.
    Riss UV, Meyer HD (1995) Reflection-free complex absorbing potentials. J Phys B At Mol Opt 28(8):1475–1493CrossRefGoogle Scholar
  94. 94.
    Sosnova KE, Telnov DA, Rozenbaum EB, Chu SI (2014) Exterior complex scaling method in TDDFT: HHG of Ar atoms in intense laser fields. J Phys Conf Ser 488(1):012022CrossRefGoogle Scholar
  95. 95.
    Pohl A, Reinhard PG, Suraud E (2000) Towards single-particle spectroscopy of small metal clusters. Phys Rev Lett 84(22):5090–5093CrossRefGoogle Scholar
  96. 96.
    Dinh PM, Romaniello P, Reinhard PG, Suraud E (2013) Calculation of photoelectron spectra: a mean-field-based scheme. Phys Rev A 87(3):032514CrossRefGoogle Scholar
  97. 97.
    Caillat J, Zanghellini J, Kitzler M, Koch O, Kreuzer W, Scrinzi A (2005) Correlated multielectron systems in strong laser fields: a multiconfiguration time-dependent Hartree–Fock approach. Phys Rev A 71(1):012712CrossRefGoogle Scholar
  98. 98.
    Scrinzi A (2012) t-SURFF: fully differential two-electron photo-emission spectra. New J Phys 14(8):085008CrossRefGoogle Scholar
  99. 99.
    Tao L, Scrinzi A (2012) Photo-electron momentum spectra from minimal volumes: the time-dependent surface flux method. New J Phys 14(1):013021CrossRefGoogle Scholar
  100. 100.
    Crawford-Uranga A, De Giovannini U, Mowbray DJ, Kurth S, Rubio A (2014) Modelling the effect of nuclear motion on the attosecond time-resolved photoelectron spectra of ethylene. J Phys B At Mol Phys 47(12):124018CrossRefGoogle Scholar
  101. 101.
    De Giovannini U, Brunetto G, Castro A, Walkenhorst J, Rubio A (2013) Simulating pump-probe photoelectron and absorption spectroscopy on the attosecond timescale with time-dependent density functional theory. Chemphyschem 14(7):1363–1376CrossRefGoogle Scholar
  102. 102.
    Gazibegović-Busuladžić A, Hasović E, Busuladžić M, Milosevic D, Kelkensberg F, Siu W, Vrakking M, Lepine F, Sansone G, Nisoli M, Znakovskaya I, Kling M (2011) Above-threshold ionization of diatomic molecules by few-cycle laser pulses. Phys Rev A 84(4):043426CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Nano-bio Spectroscopy Group and European Theoretical Spectroscopy Facility (ETSF), Centro de Física de Materiales CSIC-UPV and DIPCUniversidad del País Vasco UPV/EHUDonostia–San SebastiánSpain
  2. 2.Max Planck Institute for the Structure and Dynamics of MatterHamburgGermany

Personalised recommendations