ReviewHot electron lifetimes in metals probed by time-resolved two-photon photoemission
Introduction
A large variety of phenomena in modern research topics such as photoinduced phase transition, femtosecond-demagnetization, and ultrafast nano-optics are governed by the ultrafast dissipation of energy directly after the absorption of a driving (heating) laser pulse. The photons are absorbed by the creation of excited carriers, i.e., through the production of electron–hole pairs, within the characteristic penetration depth of the laser. Initially, the thermal equilibrium between electronic system and lattice is destroyed and a non-thermal electron distribution is generated. On a finite timescale the ‘nascent’ electron distribution will equilibrate internally establishing a Fermi–Dirac distribution of elevated temperature. A primary goal of many studies on hot electron dynamics in solids is the development of a fundamental understanding of the elementary processes of energy dissipation involved in such light-matter interaction. A profound knowledge of the relevant relaxation processes of excited carriers is of crucial importance for the development of new materials with exceptional properties (such as superconductivity, graphene-like characteristics, or non-trivial topology) as well as the design of new devices for ultrafast applications (such as magnetic storage systems, magnetic sensors, or electronic and opto-electronic devices).
The present review on the inelastic decay of optically excited volume electrons in different types of metals is concerned with the most elementary process determining the dynamics of the transition from the nascent electron distribution towards a Fermi–Dirac distribution: the energy relaxation of a single electron populating an excited state. The relevant timescale for this process is referred to as the inelastic lifetime of the electronic excitation. It is governed by the number of available decay channels generally offered by the interaction with the (cold) electron gas and the phonon system. Defect and impurity scattering affect the phase of the electronic excitation (dephasing time) but do not contribute to the energy relaxation.
Next to screening and electronic dephasing, the inelastic decay of photo-excited carriers is the fastest quasiparticle process in a metal (see Fig. 1.1). More important, the inelastic decay determines the primary energy loss of the excited carrier right after photoabsorption. Hence, it is the relevant process limiting the (ballistic) ultrafast transport of energy out of the photo-excitation area.
As it will be shown in this review the fundamental timescale associated with the excitation of electronic volume states in metals is always short, typically on the order of a few femtoseconds (fs), only. The systematic and comprehensive study of related relaxation phenomena in the time domain [2], [3] became possible in the 1990s following the establishment of Kerr lens mode locked Ti:Sapphire femtosecond lasers with time-resolved two-photon photoemission (TR-2PPE) providing the most direct access to the inelastic lifetime [4], [5].
These investigations have shown that excitation energy, (spin-dependent) electronic band structure and density of states as well as carrier screening are the most relevant parameters determining the inelastic lifetime of the excited carriers (see Fig. 1.2). As a consequence, e.g., the inelastic decay of optically excited electrons in transition metals happens on a timescale which typically is one order of magnitude faster than what is observed for noble metals [6]. Additionally, the inelastic lifetime of ferromagnetic 3d-transition metals shows a spin-dependence due to the exchange splitting of the d-bands [7].
The inelastic lifetime of photo-excited (hot) electrons in a metal is linked to the single electron picture, which occupy some well-defined energy levels in the unoccupied energy regime [8]. For a correct description of the energy levels, many-particle correlations must be taken into account. One can remain in a single-particle picture by introducing the quasiparticle concept that considers many-body effects in terms of electronic level shifts and broadening and furthermore allows for introducing effects associated with collective bosonic excitations such as plasmons. In metals the inelastic relaxation process is almost exclusively attributed to the scattering with ‘cold’ electrons below the Fermi level (e–e scattering) and with phonons (e–ph scattering). These scattering processes were implicitly considered already in the first microscopic theories of electrical resistance (Drude theory [9]) and have been studied extensively under quasi-equilibrium conditions, e.g., within transport measurements. Note that due to the extremely efficient screening of the Coulomb interaction in metals, radiative recombination of the excited electron–hole pairs is extremely unlikely.
The first theoretical works on hot electron relaxation processes were based on the work by Quinn and Ferrell from the late 1950s [10] and assumed a free-electron like behavior. This approach yields a quadratic dependency of the inelastic scattering rate on the excitation energy, which is often referred to as Fermi-liquid behavior for e–e scattering. The simplicity of this model is extremely appealing, however, already Quinn pointed out that this basic approach can only be considered as a first approximation for the description of the hot electron dynamics in real metals, since band structure effects are not incorporated at all. In general, inelastic lifetimes in real metals might therefore show significant deviations from the Fermi-liquid behavior. As an example, also localized d- and f-bands in the vicinity of the Fermi energy may significantly affect the relaxation of the hot electrons.
In the past two decades different theoretical approaches have been developed and applied to describe hot electron dynamics beyond the Fermi-liquid theory. The most powerful of them is the self-energy formalism of the many-body theory, and first calculations of the e–e scattering rate in solids which take band structure effects into account were carried out in 1999. An overview of different theoretical approaches will be given in Chapter 2 of the review. It will provide the necessary background for the discussion and interpretation of experimental and theoretical lifetime results.
The lifetime of electronic excitations can be measured either directly in the time domain or, via homogeneous linewidth analysis, in the frequency domain. The latter approach, which is commonly used in angle-resolved photoemission (ARPES), inverse photoemission (IPES), and scanning tunneling spectroscopy (STS), relies, however, on the presence of sharp spectral features, which in general are not provided by bulk bands in metals. In a TR-2PPE experiment the lifetime of an optically induced electronic excitation is measured in the time domain by detection of the signal decrease as a function of the time delay between a pump and a probe laser pulse.
During the roughly 25 years of femtosecond time-resolved photoemission a huge number of successful TR-2PPE studies were published which proves this technique as the method of choice for the investigation of optically excited electronic states in metals [4], at pristine metal surfaces (e.g. extensive work on image potential states, see e.g. [11], [12], [13]), and of adsorbate–surface hybrid systems [14], [15], [16], [17]. A first overview on the study of femtosecond dynamics of electrons in metals was given in 1997 in [5] by H. Petek and S. Ogawa. There are also a number of reviews on different theoretical approaches to this problem, see e.g. [18], [19], [20], [21]. A very general review on current developments within this field can furthermore be found in [22].
There has been parallel work on the study of carrier dynamics in semiconductors by means of time-resolved photoemission. As a pioneering TR-2PPE study on GaAs see e.g. [23]. Further examples are, for instance, the work by S. Ramakrishna et al. [24], who modeled TR-2PPE data using the Heisenberg equations of motion, and very recent works on the ultrafast carrier dynamics at the ZnO(1 0 0) surface [25] and within the conduction band of reduced rutile TiO2(1 1 0) [26]. However, most of the time-resolved photoemission experiments addressing the carrier dynamics in semiconductors focused not on single electron excitations as typically addressed in TR-2PPE experiments but on the ensemble dynamics after very intense optical excitation (on this point see, i.e., pioneering work by R. Height [3], [27] and J. Bokor [28]).
All these studies demonstrate that in this field a substantial level of maturity has already been achieved. Still, the interpretation of the experimental results was subject to many debates in the past. A particular challenge in these real-time studies is the separation of the different mechanisms involved in the hot electron generation and decay. One of the efforts of this review is to clarify in this context the limitations of the TR-2PPE approach. In Chapter 3 the TR-2PPE method is described in detail and secondary processes potentially affecting the measured lifetime signal are discussed. This part gives also a brief description of a TR-2PPE setup used for lifetime measurements of volume excitations and briefly summarizes the analysis method applied to the experimental data presented for the first time in this work. The actual lifetime results are reviewed in Chapters 4–7. A thorough overview of TR-2PPE lifetime data for simple metals, noble metals, transition metals, and rare earth metals will be given with a particular emphasis on the comparison to available calculations.
Section snippets
Theory of hot electron relaxation processes
Hot electrons relax to lower energy levels as a result of the interaction with the surrounding environment. At sufficiently low excitation densities the relaxation process is governed by inelastic interaction with ‘cold’ electrons, with collective excitations of the interacting electron gas (i.e., electron-plasmon scattering), and with phonons. Due to the extremely efficient screening of the Coulomb potential in metals, the transition rate for e-h recombination is strongly suppressed. Hence,
Time-resolved two-photon photoemission method
Photoelectron spectroscopy techniques probe the electronic structure of solids and provide therefore a very direct access to the ultrafast dynamics of excited carriers once combined with optical pump–probe schemes. In general, one distinguishes between two time-domain photoemission approaches. In a time-resolved photoelectron spectroscopy (TR-PES) experiment a pump-pulse of an intensity in the mJ/cm2 – range generates a high density of excited carrier (≈1 electron/unit cell) which efficiently
Aluminum – a Fermi liquid model system?
Among all metals aluminum was already quoted in the very early theoretical works on excited state lifetimes as one of the most appropriate model systems in order to experimentally validate the (E − EF)2-type behavior in the inelastic decay rate as predicted by the Fermi-liquid theory for a homogeneous electron gas (FLT behavior) [10].
The electronic properties of Al indeed resemble the behavior of a degenerate three-dimensional free electron gas fairly well. First to mention are the electronic
Noble metals
The majority of volume state investigations employing TR-2PPE have focused on noble metals. The energy bands of noble metals near the Fermi energy are derived from s- and p-orbitals and exhibit an overall parabolic dispersion character as expected for a free electron gas (see Fig. 5.1) [159]. However, as for aluminum, the interaction with the periodic lattice potential results in band structure distortions particularly at the boundary of the Brillouin zone giving for instance rise to a
Transition metals
Contrary to the noble metals, the d-orbitals of transition metals contribute also to the electron density of states at and above EF. Next to enhanced screening, this peculiarity opens up new channels for e–e scattering processes potentially causing an overall decrease in the inelastic lifetime. Advanced theoretical approaches [62], [70], [122] and TR-2PPE experiments [195] confirm this general trend. More specifically, details such as occupation, width and shape of the d-bands dominantly affect
Rare earth metals
In the lanthanide rare-earth metals shallow 4f core levels are energetically degenerated with the 5d and 6s valence states and add, therefore, another degree of complexity of potential relevance for hot electron lifetimes.
The electronic structure of the different lanthanides differs mainly in the occupancy of the 4f-levels so that a systematic study may unravel the relevance of these orbitals in the context of electron–hole pair excitation, spin-flip scattering, or magnon emission. For
Future directions and concluding remarks
Theoretical and experimental efforts over the last 20 years as reviewed in this article provided considerable insights into relevant mechanisms governing the inelastic decay of optically excited volume electrons in different types of metals. Yet, many fundamental issues still remain to be explored with problems regarding the spin-degree of freedom such as spin–spin (singlet vs. triplet scattering, exchange scattering) or spin–lattice scattering processes (as a first example, see [236]) being a
Acknowledgments
This research was carried out under the funding of the German Science Foundation (DFG) within AE 19/13-1. We are particularly grateful to P.M. Echenique, E.V. Chulkov, V.P. Zhukov, H.C. Schneider, B. Rethfeld, M. Cinchetti, and S. Mathias for many discussions over the years.
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