Contribution of inter- and intraband transitions into electron–phonon coupling in metals

Eur. Phys. J. D (2021) 75 :212 https://doi.org/10.1140/epjd/s10053-021-00200-w THE EUROPEAN PHYSICAL JOURNAL D Regular Article - Plasma Physics Contribution of inter- and intraband transitions into electron–phonon coupling in metals Nikita Medvedev1,2,a and Igor Milov3 1 Institute of Physics, Czech Academy of Sciences, Na Slovance 2, 18221 Prague 8, Czech Republic Institute of Plasma Physics, Czech Academy of Sciences, Za Slovankou 3, 182 00 Prague 8, Czech Republic 3 Industrial Focus Group XUV Optics, MESA+ Institute for Nanotechnology, University of Twente, Drienerlolaan 5, 7522 NB Enschede, The Netherlands 2 Received 16 March 2021 / Accepted 14 June 2021 / Published online 23 July 2021 © The Author(s) 2021 Abstract. We recently developed an approach for calculation of the electron–phonon (electron–ion in a more general case) coupling in materials based on tight-binding molecular dynamics simulations. In the present work, we utilize this approach to study partial contributions of inter- and intraband electron scattering events into total electron–phonon coupling in Al, Au, and Cu elemental metals and in AlCu alloy. We demonstrate that the interband scattering plays an important role in the electron–ion energy exchange process in Al and AlCu, whereas intraband d–d transitions are dominant in Au and Cu. Moreover, interand intraband transitions exhibit qualitatively different dependencies on the electron temperature. Our findings should be taken into account for the interpretation of experimental results on the electron–phonon coupling parameter. 1 Introduction Since the advent of powerful femtosecond lasers, the field of material response to irradiation has been developing fast [1]. It is driven by a wide range of applications in materials surface and bulk processing and nanostructuring for photonics [2], catalysis [3] and biomedicine [4]. Elemental metals and alloys are a class of materials that is widely used in the ultrafast community for its relative simplicity and versatile functionality [5–7]. An ultrafast transfer of the absorbed laser energy from an electronic system of metal to the lattice is a core process that defines the nature and dynamics of irradiated target evolution. Understanding and quantifying such processes is important to keep advancing the field of ultrafast light-matter interaction. Most often, the response of metals to ultrafast laser pulses is modeled with the two-temperature model This work benefited from networking activities carried out within the EU funded COST Action CA17126 (TUMIEE) and represents a contribution to it. The authors gratefully acknowledge financial support from the Czech Ministry of Education, Youth and Sports (Grants Nos. LTT17015, EF16_013/0001552, and LM2015083). I. Milov gratefully acknowledges support from the Industrial Focus Group XUV Optics of the MESA+ Institute for Nanotechnology of the University of Twente; the industrial partners ASML, Carl Zeiss SMT GmbH, and Malvern PANalytical, and the Netherlands Organisation for Scientific Research (NWO). a e-mail: (corresponding author) (TTM)—a set of coupled differential equations for the electronic and atomic/phononic heat conduction and exchange [8,9]. The latter is controlled with an electron–phonon coupling parameter, which, in a general case, is a function of many variables defining a material transient state, such as electron and ion temperature, density, etc. [10]. Despite shortcomings of the TTM approach (see, e.g., discussions in Refs. [11,12]), it remains one of the most widely used models in the community. Further refinements of the model are being developed and applied, resulting in multi-temperature approaches, treating different electronic bands and/or different phonon modes separately, each with its own temperature [13,14], and hence with different energy exchanges among them. Decoupling the contributions into the total coupling from the different electronic bands and interband transitions can help in the further development of advanced models that trace different bands separately, such as, e.g., in Refs. [15–17]. Such models require reliable calculations of various contributions to the coupling parameter. Here, we use the recently developed method of calculating the electron coupling to the ionic motion [10] and derive contributions of various interband (between different electronic bands) and intraband (within one band) electronic transitions to the total coupling parameter in aluminum, copper, gold, and AlCu alloy. We focus on a dependence of these partial couplings on the electronic temperature. 123 212 Page 2 of 6 Eur. Phys. J. D (2021) 75 :212 2 Model Electronic coupling to atomic/ionic motion is a process in which an electron transition from one energy level to another occurs, while the energy difference is transferred to or from the atoms. Each atomic displacement induces a change in the Hamiltonian, and correspondingly in its eigenfunctions and eigenstates. These sudden changes from one time step to another trigger electron transfers between the energy levels [18], known as nonadiabatic coupling between atomic displacements and the electronic wave function. In the solid-state community, it is known as the electron–phonon coupling when the atomic motion is harmonic within an ideal crystal lattice. We use XTANT-3 method described in Ref. [10] to calculate the electron–ion coupling parameter of selected materials. We use the term “electron–ion” instead of a more common “electron–phonon” since our model works beyond the harmonic approximation of the atomic system. (Hence, it is also capable of calculations of the coupling parameter in the disordered matter.) The model is based on tight-binding molecular dynamics simulations to evaluate the evolution of the Hamiltonian, which is dependent on transient positions of all atoms in the simulation box. A solution of the secular equation provides electron wave functions and eigenstates at each molecular dynamics time step, together with the interatomic forces [19]. Knowledge of the transient wave functions allows calculating the matrix elements of electrons coupling to ionic displacements [10]. Using the linear combination of atomic orbitals (LCAO) basis set (ci,α ) within the tight-binding  Hamiltonian, a wave function is presented as ψi = α ci,α φα , and the electron transition rate between the eigenstates i and j can be written in the following manner [10]: wij =  α,β αβ wij = 4e  2 |ci,α (t)cj,β (t0 )Sα,β | (1) δt2 α,β where e is the electron charge,  is the Planck’s constant, Sα,β is the tight binding overlap matrix, and the wave functions are calculated at two sequential molecular dynamics time steps t0 and t = t0 + δt, where δt is the molecular dynamics time step. The evaluated matrix elements (1) are then used in the Boltzmann collision integral to calculate the energy exchange rate between electrons and ions: e−a Iij = wij  f (Ej )[2−f (Ei )]−f (Ei )[2−f (Ej )]e−Eij /Ta for i < j f (Ej (...truncated)