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Laser-excitation of electrons and nonequilibrium energy transfer to phonons in copper
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Laser-excitation of electrons and nonequilibrium energy transfer to phonons in copper S.T. Weber, B. Rethfeld ∗ Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, 67663 Kaiserslautern, Germany
a r t i c l e
i n f o
Article history: Received 28 October 2016 Received in revised form 9 February 2017 Accepted 20 February 2017 Available online xxx Keywords: Nonequilibrium Metal Electron–phonon coupling Ultrafast laser-excitation
a b s t r a c t After the irradiation of a copper sample with an ultrashort laser pulse, electrons do not follow a Fermi distribution anymore but instead are in a nonequilibrium state. In contrast, the lattice cannot be excited directly by the laser pulse, due to the frequency mismatch. The energy increase in the phononic system only happens due to electron–phonon scattering. We investigate the initial electron dynamics using full Boltzmann-type collision integrals, including material-dependent characteristics by implementing a realistic density of states. We show results on the absorbed energy, details of the electronic nonequilibrium and the resulting electron–phonon coupling parameter in dependence on the photon energy. Our results show a counteracting dependence on the photon energy, which, on the one hand, enables the d-band electrons to absorb high-energy photons and on the other hand, increases the probability of multi-photon absorption. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Since the invention of the laser, laser-matter interaction has become an important topic of fundamental as well as applied research. With help of ultrashort laser pulses in the femtosecond regime, scattering processes in solid materials can be induced and studied on their intrinsic timescales. Particularly, the electron–phonon coupling strength is of broad interest for a large variety of phenomena ranging from superconductivity [1,2] over plasmonics [3,4] and spintronics [5–9] to the observation of new transient states of matter [10,11]. Dynamical processes, like ultrafast laser-induced demagnetization or nanostructuring of solids, are strongly influenced by the interplay of electron–electron and electron–phonon scattering processes [8,9,12,13]. The electron–phonon coupling parameter, describing the strength of the energy transfer from the initially heated electrons to the crystal lattice, is often assumed to be constant. However, different publications have shown, that it depends on macroscopic parameters like the electron and phonon temperature [14,1,15,16] or the specific density of states (DOS) [16–18,15,19,20]. In this work, we investigate the influence of different photon energies and fluences on the energy gain of the electrons and on the
∗ Corresponding author. E-mail addresses: [email protected] (S.T. Weber), [email protected] (B. Rethfeld).
electron–phonon coupling parameter after an ultrashort laser excitation. We focus our calculations on the effects of nonequilibrium electron distributions. In the next section, we introduce the basic equations and assumptions, necessary to understand the analysis of the results presented thereafter. These are, on the one hand, considerations of the energy deposition in dependence on the laser fluence and photon energies applied, and on the other hand, peculiarities in the simulations of the nonequilibrium electron distribution for the excitation with different photon energies. Finally, we present the influence of this nonequilibrium on the electron–phonon coupling strength. The latter again depends on the photon energy, as well as on the energy absorbed by the electrons. The results are presented for copper. They are compared to the case of gold, which was discussed in Ref. [15], revealing commonalities of the behavior of nobel metals. 2. Theory One of the basic models to describe the electron and phonon dynamics during and after the laser excitation is the twotemperature model (TTM) [21]
∂ue = −˛(Te − Tp ) + S(t), ∂t
(1a)
∂up = +˛(Te − Tp ). ∂t
(1b)
http://dx.doi.org/10.1016/j.apsusc.2017.02.183 0169-4332/© 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: S.T. Weber, B. Rethfeld, Laser-excitation of electrons and nonequilibrium energy transfer to phonons in copper, Appl. Surf. Sci. (2017), http://dx.doi.org/10.1016/j.apsusc.2017.02.183
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It traces the time development of the internal energies ue and up of the electrons and the phonons, respectively. The electron energy is increased due to laser excitation, represented by the source term S(t). This energy increase leads to an elevated electron temperature Te . The energy transfer from the electrons to the phonons is described by a term proportional to the difference of their respective temperatures, Te and Tp , and an electron–phonon coupling parameter ˛. The source term, S(t), describes the energy deposition into the material in dependence on time and is proportional to the intensity in the material. The intensity I = I0 (1 − R)
(2)
the collision term of the inverse Bremsstrahlung, but also the collision term of the electron–phonon interaction. This reflects the fact that photon absorption can also take place during electron–phonon collisions [35]. We indicate this in the notation el–ph(-phot) for this process. The major part of the absorption is, however, determined by the inverse Bremsstrahlung absorp [33]. The probability of single- or multi-photon processes increases with decreasing photon energy ωL and increasing intensity I, which can be connected to the later used fluence of the laser pulse F, which is given by
∞
F=
I(t)dt.
(5)
−∞
In our work, we assume a laser pulse that is rectangular in time. Thus the intensity is constant over the whole pulse duration and the fluence F = I L within the material is proportional to the intensity and the pulse duration L . This assumption was made to better distinguish between excitation and relaxation processes. Furthermore, we only fix the intensity inside the material, I, in order to enable a better comparison between results obtained for different photon energies. The electrons are described by an effective oneband model [18]. Moreover, only longitudinal phonon modes are considered and described within the Debye model. For the comparison with the two-temperature model, the internal energies of electrons and phonons,
inside the material itself depends on the external intensity I0 and the reflectivity R of the material. The TTM has been widely and successfully applied to the description of a large variety of experimental measurements [22–26]. Moreover, different expansions facilitate the study of heat transport [14,27,28], excitation of semiconductors or distinct phonon modes [17,29,30] and magnetization dynamics [5,31,32]. However, during and directly after the laser excitation, the electrons are driven out of equilibrium and do not follow a Fermi distribution anymore. Thus, no temperature is defined and the application of temperature-dependent calculations is questionable [33,18,15,34]. In order to follow this nonequilibrium dynamics, we apply full Boltzmann-type collision integrals, described in Ref. [18]. The Boltzmann equation
ue =
t) df (k, = el–el + el–ph(-phot) + absorp dt
(3a)
up =
, t) dg(q = ph–el(-phot) dt
(3b)
can be determined from the corresponding distributions f and g and density of states De/p (E) for electrons and phonons, respectively. The electron density of states is taken from Lin et al. [16] and is assumed to be constant during the calculations. To further compare our results with the input parameters of the TTM, we calculate an electron–phonon coupling parameter
does not trace a macroscopic parameter such as the internal energy, but the microscopic distribution function f of the electrons and g of the phonons. The internal energies can be calculated with help of these distribution functions as will be described below. The change of the distribution function is calculated by applying collision terms for electron–electron el–el , electron–phonon el–ph(-phot) and phonon–electron ph–el(-phot) scattering. The energy deposition of the laser pulse in the material is achieved either by the electron–phonon(-photon) collision or by inverse Bremsstrahlung. The latter is represented by the absorption term absorp . All collision integrals, , depend on the density of states D(E) of the participating particles, Pauli-factors ensuring Pauli-blocking within the electron system, matrix elements representing the collision probability. For further details of the collisions integrals see Eqs. (10), (15), (17) and (19) in Ref. [18]. In particular, electron–phonon–photon collisions and inverse Bremsstrahlung require a Bessel function J¯2 (, ω), which represents the absorption of a certain number of photons [35,36]. The parameter = (eE0 /me )p depends on the electrical laser field E0 , the effective band mass of the electrons me and the exchanged momentum during the process p. For low intensities ( ωL2 ) it can be approximated as J¯2 (, ω) ≈
1 2
(!)
2ωL2
2||
∝
I ωL4
|| (4)
for | |≥1 [35,36]. Without an external laser-field, the Bessel function for =0 equals unity, while all other orders vanish, thus no photons are absorbed. If an external laser field is applied, the zeroorder Bessel function decreases, while all other orders start to increase and absorption processes take place. For the intensities analyzed here, the absorption of higher photon orders is much less probable then of lower orders, which is reflected in the fact that J¯ 2 +1
is always much smaller then J¯2 . The Bessel function not only enters
dE f (E)De (E)E,
(6a)
dE g(E)Dp (E)E,
(6b)
˛[f (t), g(t)] =
dup [g(t)]/dt , Te [f (t)] − Tp [g(t)]
(7)
based on Eq. (1) [15], with help of the phonon–electron collision term. Either laser-excited (nonequilibrium) electrons or hot, thermalized (equilibrium) electrons interact with cold phonons. For the nonequilibrium case, we calculate “corresponding temperatures” Tp [g(t)] and Te [f(t)] by comparing the internal energy and number of particles of the nonequilibrium distribution function to a Fermi–Dirac or Bose–Einstein distributed system, respectively. This yields well-defined “corresponding temperatures”, chemical potentials and equilibrium distribution functions. The equilibrium distribution, on the other hand, can be used to calculate an energy change of the equilibrated system using our Boltzmann approach, to extract an equilibrium coupling parameter. 3. Energy deposition There are two important factors that influence the total internal energy gain of the electrons at the end of laser excitation. These are the fluence and the photon energy of the laser pulse applied to the sample [35,18]. In our work, we do not consider the reflectivity explicitly. Thus, all fluences shown here are inside the material. The probability for the absorption of photons increases with increasing fluence and decreasing photon energy [35], see Eq. (4). Moreover, single-photon processes are more probable than multiphoton processes. In our simulations, we assume a 10 fs-laser pulse with rectangular pulse shape, to better distinguish between excitation and
Please cite this article in press as: S.T. Weber, B. Rethfeld, Laser-excitation of electrons and nonequilibrium energy transfer to phonons in copper, Appl. Surf. Sci. (2017), http://dx.doi.org/10.1016/j.apsusc.2017.02.183
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Fig. 1. Energy gain ue of the electronic system after the end of the laser pulse for different fluences F plotted over the photon energy ωL .
Fig. 2. Energy gain ue in the electronic system after the end of the laser pulse for different fluences F. For the photon energy of 2.48 eV, the absorbed energy increases linearly with the fluence. For the photon energy of 1.55 eV, the energy gain at low fluences is smaller than for the photon energy of 2.48 eV. At higher fluences, multiphoton absorption becomes more probable for the case of ωL = 1.55 eV, so the energy gain increases stronger than linearly with the fluence and exceeds that for ωL = 2.48 eV.
relaxation processes. Fig. 1 shows the energy gain, ue , of the electronic system after the end of the laser pulse plotted over the photon energy ωL for different fluences F. The energy gain decreases with increasing photon energy as described by Eq. (4). However, above a photon energy of about 1.8 eV, the energy gain increases slightly, which is similar to results obtained for gold [15]. This increase is a consequence of the energy distance between the d-bands and the Fermi-edge [15]. In our work, we use the density of states (DOS) for copper obtained by of Lin et al., where this energy distance is 1.65 eV [16]. Photons with a larger energy can excite electrons directly from the region of high DOS associated with the d-bands, which leads to a higher absorption probability. However, the increase in the electron energy gain does not become visible below a photon energy of about 1.8 eV. Prior to that, the increase due to direct excitation cannot compensate the global decrease in absorption probability due to the higher photon energy. Note that the minimum of the energy gain slightly depends on the fluence. For low excitation, strengths the minimum in the energy gain is visible for photon energies higher of about 1.7 eV, while it appears at about ωL = 1.8 eV for the highest fluence, due to the higher probability for multi-photon processes. Fig. 2 depicts the electron energy gain over fluence for photon energies of 1.55 eV and 2.48 eV. For a photon energy of 2.48 eV, the energy gain increases almost linearly with the fluence, because only one-photon absorption is favorable. For ωL = 1.55 eV, the energy gain is slightly smaller than for ωL = 2.48 eV at low fluences, which
Fig. 3. Nonequilibrium electron distribution for an excitation with photon energies of (a) ωL = 1.55 eV and (b) ωL = 2.48 eV directly after the end of the laser pulse compared to the equilibrium distribution at 300 K before excitation. The high peak in the DOS (DOS-peak) is reproduced in the excited distributions and marked with DOS + 1pt. For the smaller photon energy in (a), also a second replication (DOS + 2pt) is visible. The Fermi edge is reproduced as well due to excitation; the positions of the replica are marked with Fermi + 1pt and Fermi + 2pt, respectively. Two-photon replications are only visible for the lower photon energy of 1.55 eV.
seems contrary to the basic assumptions based on Eq. (4). Here, direct excitation of d-band electrons leads to a higher energy gain for a high photon energy as compared to an excitation with a low photon energy. At higher fluences, two-photon processes come into account and, as described by Eq. (4), these are more probable for the excitation with lower photon energy. Thus, as shown in Fig. 2, the energy gain for a photon energy of 1.55 eV increases stronger than linearly and exceeds the energy gain associated with the excitation with 2.48 eV at the same fluence. Note that the photons, which are not absorbed, are transmitted through the considered volume. 4. Transient electron distribution Directly after the end of the laser pulse, the electronic system is in a strong nonequilibrium. As the energy gain, the shape of the nonequilibrium distribution also depends on the photon energy applied. Fig. 3 compares two nonequilibrium distributions directly after laser excitation with two different photon energies. We have chosen distributions with almost the same internal energy gain of ue ≈ 6.3e9 Jm−3 , which are generated by pulses of two different fluences. For a photon energy of 1.55 eV, the distribution after an excitation with a fluence F = 0.33 mJ/cm2 is shown whereas for 2.48 eV the distribution after an excitation with F = 0.48 mJ/cm2 was chosen. When neglecting energy dissipation, both nonequilibrium distributions would thermalize toward a temperature of Te ≈ 7700 K. Additionally, the distribution function at Te = 300 K, i.e., the distribution before laser excitation, is depicted. Fig. 3 shows reproductions of distinct features of the density of states (DOS) as well as the initial distribution. For both nonequilibrium distributions, the high peak in the DOS at about 1.65 eV below Fermi energy, EF , is reproduced in the distribution function after
Please cite this article in press as: S.T. Weber, B. Rethfeld, Laser-excitation of electrons and nonequilibrium energy transfer to phonons in copper, Appl. Surf. Sci. (2017), http://dx.doi.org/10.1016/j.apsusc.2017.02.183
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Fig. 4. Electron–phonon coupling parameter ˛ in equilibrium and nonequilibrium depicted over the corresponding equilibrium electron temperature Te [f]. The equilibrium coupling parameter increases due to the shape of the density of states. The nonequilibrium coupling parameter depends on photon energy and fluence, as well as the density of states.
single-photon excitation (marked “DOS + 1pt”). For an excitation with a photon energy of 1.55 eV, two-photon processes create a second replication above Fermi edge, marked DOS + 2pt. For 2.48 eV, on the other hand, only one replication is visible. In this case, electrons are excited from the DOS peak to energies above Fermi edge directly. As a consequence, parts of the peak structure are clearly reproduced in the electron distribution above Fermi edge. Both distributions shown here have the same internal energy, however, the energy is distributed differently among the electrons. With a photon energy of 2.48 eV less electrons are excited, but every excited electron has gained more energy as compared to the excitation with a photon energy of 1.55 eV. This leads also to differences in the electron–phonon coupling, which will be discussed below. 5. Electron–phonon coupling With help of Eq. (7), we calculate electron–phonon coupling parameters ˛ of the laser-excited nonequilibrium distribution directly after irradiation. We compare them to coupling parameters obtained for the corresponding equilibrium distributions of the same internal energy. Fig. 4 depicts the coupling parameter for equilibrium and nonequilibrium for different photon energies in dependence on the corresponding electron temperature. The values are calculated directly after the end of the rectangular laser pulse, where we expect the largest differences [15]. First, we take a closer look at the equilibrium curve. At low temperatures, the electrons have a steep Fermi edge. As a consequence, at room temperature, only scattering at the Fermi edge is possible. For elevated temperatures, the Fermi edge broadens and scattering becomes possible on a broader energy range. Thus, the coupling factor increases. When the temperature reaches about 4000 K, the d-band peaks are no longer fully occupied and scattering in this energy range is possible as well. Due to the much higher electron density in this region, the slope of the coupling parameter is stronger. When the whole d-band region is excited at about 9000 K the slope decreases again. The nonequilibrium case is more complicated and depends on the photon energy. High photon energies such as 2.48 eV suffice to excite electrons directly from the d-bands. Thus, the scattering rate and therefore the electron–phonon coupling parameter is higher than the corresponding equilibrium value. For low photon energies, here for 1.55 eV, and for low fluences, the d-band peaks cannot be excited directly and the electron distribution still almost looks like a Fermi distribution at 300 K. Thus, scattering is
less probable than in the equilibrium case [33]. For higher fluences, two-photon processes become important. Therefore, the coupling strength of the nonequilibrium distribution exceeds the equilibrium value also for lower photon energies. It even exceeds the coupling for the higher photon energy, since the multi-photon absorption probability depends on the laser frequency, cf. Eq. (4). A larger amount of d-band electrons can be excited for ωL = 1.55 eV than for ωL = 2.48 eV at these fluences. This leads to a larger scattering rate, even when the same amount of energy was absorbed by the electrons. Therefore, the coupling parameter after an excitation with a photon energy of 1.55 eV exceeds the values of both other cases at high fluences. Note that this observation depends on the specific DOS of the material and the excitation parameters. In Ref. [15], where nonequilibrium electron-phonon coupling for gold has been studied, no intersection of these curves has been shown. However, it is likely that such conditions can be found in each noble metal depending on the laser parameters applied. As has been shown in Ref. [15] for gold, also in copper the influence of the laser-excitation on the electron–phonon coupling parameter decreases on the timescale of electron thermalization which depends on excitation strength [15].
6. Conclusion In this work, we investigated the influence of nonequilibrium effects on electron dynamics after ultrashort laser excitation. We have shown that the energy gain of the electrons depends on the photon energy as well as the fluence applied during excitation. For high photon energies, direct excitation of d-band electrons is possible and the energy gain increases linearly with the fluence. For low photon energies and low fluences, the d-band electrons cannot be excited directly and the energy gain is lower. However, the photon energy influences the probability for multi-photon processes, which at sufficiently high fluences enables direct excitation of important d-band peaks particularly for low photon energies. This process counteracts the effect of the density of states and leads to a higher energy gain for the lower photon energy considered in this work. These different excitation characteristics also have an impact on further material behavior such as electron–phonon coupling. The electron distribution after the end of the laser pulse also depends on the photon energy used for excitation. For larger photon energies, electrons in the d-band peaks of the density of states can be excited directly to energies Fermi edge. For smaller photon energies, on the other hand, two-photon processes become more likely and a second representation of the high DOS peak can be found in the distribution function. As a consequence of the excitation and the particular electron distribution after the excitation, electron–phonon coupling differs for equilibrium and nonequilibrium and for different photon energies. Moreover, the coupling parameter also depends on the excitation strength.
Acknowledgments Financial support of the Deutsche Forschungsgemeinschaft through the Heisenberg Program (Grant No. RE 1141/15), through the Carl-Zeiss Stiftung, and through the SFB/TRR-173 “Spin+X” is gratefully acknowledged. Additionally, the authors appreciate the Allianz für Hochleistungsrechnen Rheinland-Pfalz for providing computing resources through project LAINEL on the Elwetritsch high performance computing cluster. We thank B.Y. Mueller for his intellectual and numerical heritage.
Please cite this article in press as: S.T. Weber, B. Rethfeld, Laser-excitation of electrons and nonequilibrium energy transfer to phonons in copper, Appl. Surf. Sci. (2017), http://dx.doi.org/10.1016/j.apsusc.2017.02.183
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Please cite this article in press as: S.T. Weber, B. Rethfeld, Laser-excitation of electrons and nonequilibrium energy transfer to phonons in copper, Appl. Surf. Sci. (2017), http://dx.doi.org/10.1016/j.apsusc.2017.02.183
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Laser-excitation of electrons and nonequilibrium energy transfer to phonons in copper S.T. Weber, B. Rethfeld ∗ Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, 67663 Kaiserslautern, Germany
a r t i c l e
i n f o
Article history: Received 28 October 2016 Received in revised form 9 February 2017 Accepted 20 February 2017 Available online xxx Keywords: Nonequilibrium Metal Electron–phonon coupling Ultrafast laser-excitation
a b s t r a c t After the irradiation of a copper sample with an ultrashort laser pulse, electrons do not follow a Fermi distribution anymore but instead are in a nonequilibrium state. In contrast, the lattice cannot be excited directly by the laser pulse, due to the frequency mismatch. The energy increase in the phononic system only happens due to electron–phonon scattering. We investigate the initial electron dynamics using full Boltzmann-type collision integrals, including material-dependent characteristics by implementing a realistic density of states. We show results on the absorbed energy, details of the electronic nonequilibrium and the resulting electron–phonon coupling parameter in dependence on the photon energy. Our results show a counteracting dependence on the photon energy, which, on the one hand, enables the d-band electrons to absorb high-energy photons and on the other hand, increases the probability of multi-photon absorption. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Since the invention of the laser, laser-matter interaction has become an important topic of fundamental as well as applied research. With help of ultrashort laser pulses in the femtosecond regime, scattering processes in solid materials can be induced and studied on their intrinsic timescales. Particularly, the electron–phonon coupling strength is of broad interest for a large variety of phenomena ranging from superconductivity [1,2] over plasmonics [3,4] and spintronics [5–9] to the observation of new transient states of matter [10,11]. Dynamical processes, like ultrafast laser-induced demagnetization or nanostructuring of solids, are strongly influenced by the interplay of electron–electron and electron–phonon scattering processes [8,9,12,13]. The electron–phonon coupling parameter, describing the strength of the energy transfer from the initially heated electrons to the crystal lattice, is often assumed to be constant. However, different publications have shown, that it depends on macroscopic parameters like the electron and phonon temperature [14,1,15,16] or the specific density of states (DOS) [16–18,15,19,20]. In this work, we investigate the influence of different photon energies and fluences on the energy gain of the electrons and on the
∗ Corresponding author. E-mail addresses: [email protected] (S.T. Weber), [email protected] (B. Rethfeld).
electron–phonon coupling parameter after an ultrashort laser excitation. We focus our calculations on the effects of nonequilibrium electron distributions. In the next section, we introduce the basic equations and assumptions, necessary to understand the analysis of the results presented thereafter. These are, on the one hand, considerations of the energy deposition in dependence on the laser fluence and photon energies applied, and on the other hand, peculiarities in the simulations of the nonequilibrium electron distribution for the excitation with different photon energies. Finally, we present the influence of this nonequilibrium on the electron–phonon coupling strength. The latter again depends on the photon energy, as well as on the energy absorbed by the electrons. The results are presented for copper. They are compared to the case of gold, which was discussed in Ref. [15], revealing commonalities of the behavior of nobel metals. 2. Theory One of the basic models to describe the electron and phonon dynamics during and after the laser excitation is the twotemperature model (TTM) [21]
∂ue = −˛(Te − Tp ) + S(t), ∂t
(1a)
∂up = +˛(Te − Tp ). ∂t
(1b)
http://dx.doi.org/10.1016/j.apsusc.2017.02.183 0169-4332/© 2017 Elsevier B.V. All rights reserved.
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It traces the time development of the internal energies ue and up of the electrons and the phonons, respectively. The electron energy is increased due to laser excitation, represented by the source term S(t). This energy increase leads to an elevated electron temperature Te . The energy transfer from the electrons to the phonons is described by a term proportional to the difference of their respective temperatures, Te and Tp , and an electron–phonon coupling parameter ˛. The source term, S(t), describes the energy deposition into the material in dependence on time and is proportional to the intensity in the material. The intensity I = I0 (1 − R)
(2)
the collision term of the inverse Bremsstrahlung, but also the collision term of the electron–phonon interaction. This reflects the fact that photon absorption can also take place during electron–phonon collisions [35]. We indicate this in the notation el–ph(-phot) for this process. The major part of the absorption is, however, determined by the inverse Bremsstrahlung absorp [33]. The probability of single- or multi-photon processes increases with decreasing photon energy ωL and increasing intensity I, which can be connected to the later used fluence of the laser pulse F, which is given by
∞
F=
I(t)dt.
(5)
−∞
In our work, we assume a laser pulse that is rectangular in time. Thus the intensity is constant over the whole pulse duration and the fluence F = I L within the material is proportional to the intensity and the pulse duration L . This assumption was made to better distinguish between excitation and relaxation processes. Furthermore, we only fix the intensity inside the material, I, in order to enable a better comparison between results obtained for different photon energies. The electrons are described by an effective oneband model [18]. Moreover, only longitudinal phonon modes are considered and described within the Debye model. For the comparison with the two-temperature model, the internal energies of electrons and phonons,
inside the material itself depends on the external intensity I0 and the reflectivity R of the material. The TTM has been widely and successfully applied to the description of a large variety of experimental measurements [22–26]. Moreover, different expansions facilitate the study of heat transport [14,27,28], excitation of semiconductors or distinct phonon modes [17,29,30] and magnetization dynamics [5,31,32]. However, during and directly after the laser excitation, the electrons are driven out of equilibrium and do not follow a Fermi distribution anymore. Thus, no temperature is defined and the application of temperature-dependent calculations is questionable [33,18,15,34]. In order to follow this nonequilibrium dynamics, we apply full Boltzmann-type collision integrals, described in Ref. [18]. The Boltzmann equation
ue =
t) df (k, = el–el + el–ph(-phot) + absorp dt
(3a)
up =
, t) dg(q = ph–el(-phot) dt
(3b)
can be determined from the corresponding distributions f and g and density of states De/p (E) for electrons and phonons, respectively. The electron density of states is taken from Lin et al. [16] and is assumed to be constant during the calculations. To further compare our results with the input parameters of the TTM, we calculate an electron–phonon coupling parameter
does not trace a macroscopic parameter such as the internal energy, but the microscopic distribution function f of the electrons and g of the phonons. The internal energies can be calculated with help of these distribution functions as will be described below. The change of the distribution function is calculated by applying collision terms for electron–electron el–el , electron–phonon el–ph(-phot) and phonon–electron ph–el(-phot) scattering. The energy deposition of the laser pulse in the material is achieved either by the electron–phonon(-photon) collision or by inverse Bremsstrahlung. The latter is represented by the absorption term absorp . All collision integrals, , depend on the density of states D(E) of the participating particles, Pauli-factors ensuring Pauli-blocking within the electron system, matrix elements representing the collision probability. For further details of the collisions integrals see Eqs. (10), (15), (17) and (19) in Ref. [18]. In particular, electron–phonon–photon collisions and inverse Bremsstrahlung require a Bessel function J¯2 (, ω), which represents the absorption of a certain number of photons [35,36]. The parameter = (eE0 /me )p depends on the electrical laser field E0 , the effective band mass of the electrons me and the exchanged momentum during the process p. For low intensities ( ωL2 ) it can be approximated as J¯2 (, ω) ≈
1 2
(!)
2ωL2
2||
∝
I ωL4
|| (4)
for | |≥1 [35,36]. Without an external laser-field, the Bessel function for =0 equals unity, while all other orders vanish, thus no photons are absorbed. If an external laser field is applied, the zeroorder Bessel function decreases, while all other orders start to increase and absorption processes take place. For the intensities analyzed here, the absorption of higher photon orders is much less probable then of lower orders, which is reflected in the fact that J¯ 2 +1
is always much smaller then J¯2 . The Bessel function not only enters
dE f (E)De (E)E,
(6a)
dE g(E)Dp (E)E,
(6b)
˛[f (t), g(t)] =
dup [g(t)]/dt , Te [f (t)] − Tp [g(t)]
(7)
based on Eq. (1) [15], with help of the phonon–electron collision term. Either laser-excited (nonequilibrium) electrons or hot, thermalized (equilibrium) electrons interact with cold phonons. For the nonequilibrium case, we calculate “corresponding temperatures” Tp [g(t)] and Te [f(t)] by comparing the internal energy and number of particles of the nonequilibrium distribution function to a Fermi–Dirac or Bose–Einstein distributed system, respectively. This yields well-defined “corresponding temperatures”, chemical potentials and equilibrium distribution functions. The equilibrium distribution, on the other hand, can be used to calculate an energy change of the equilibrated system using our Boltzmann approach, to extract an equilibrium coupling parameter. 3. Energy deposition There are two important factors that influence the total internal energy gain of the electrons at the end of laser excitation. These are the fluence and the photon energy of the laser pulse applied to the sample [35,18]. In our work, we do not consider the reflectivity explicitly. Thus, all fluences shown here are inside the material. The probability for the absorption of photons increases with increasing fluence and decreasing photon energy [35], see Eq. (4). Moreover, single-photon processes are more probable than multiphoton processes. In our simulations, we assume a 10 fs-laser pulse with rectangular pulse shape, to better distinguish between excitation and
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Fig. 1. Energy gain ue of the electronic system after the end of the laser pulse for different fluences F plotted over the photon energy ωL .
Fig. 2. Energy gain ue in the electronic system after the end of the laser pulse for different fluences F. For the photon energy of 2.48 eV, the absorbed energy increases linearly with the fluence. For the photon energy of 1.55 eV, the energy gain at low fluences is smaller than for the photon energy of 2.48 eV. At higher fluences, multiphoton absorption becomes more probable for the case of ωL = 1.55 eV, so the energy gain increases stronger than linearly with the fluence and exceeds that for ωL = 2.48 eV.
relaxation processes. Fig. 1 shows the energy gain, ue , of the electronic system after the end of the laser pulse plotted over the photon energy ωL for different fluences F. The energy gain decreases with increasing photon energy as described by Eq. (4). However, above a photon energy of about 1.8 eV, the energy gain increases slightly, which is similar to results obtained for gold [15]. This increase is a consequence of the energy distance between the d-bands and the Fermi-edge [15]. In our work, we use the density of states (DOS) for copper obtained by of Lin et al., where this energy distance is 1.65 eV [16]. Photons with a larger energy can excite electrons directly from the region of high DOS associated with the d-bands, which leads to a higher absorption probability. However, the increase in the electron energy gain does not become visible below a photon energy of about 1.8 eV. Prior to that, the increase due to direct excitation cannot compensate the global decrease in absorption probability due to the higher photon energy. Note that the minimum of the energy gain slightly depends on the fluence. For low excitation, strengths the minimum in the energy gain is visible for photon energies higher of about 1.7 eV, while it appears at about ωL = 1.8 eV for the highest fluence, due to the higher probability for multi-photon processes. Fig. 2 depicts the electron energy gain over fluence for photon energies of 1.55 eV and 2.48 eV. For a photon energy of 2.48 eV, the energy gain increases almost linearly with the fluence, because only one-photon absorption is favorable. For ωL = 1.55 eV, the energy gain is slightly smaller than for ωL = 2.48 eV at low fluences, which
Fig. 3. Nonequilibrium electron distribution for an excitation with photon energies of (a) ωL = 1.55 eV and (b) ωL = 2.48 eV directly after the end of the laser pulse compared to the equilibrium distribution at 300 K before excitation. The high peak in the DOS (DOS-peak) is reproduced in the excited distributions and marked with DOS + 1pt. For the smaller photon energy in (a), also a second replication (DOS + 2pt) is visible. The Fermi edge is reproduced as well due to excitation; the positions of the replica are marked with Fermi + 1pt and Fermi + 2pt, respectively. Two-photon replications are only visible for the lower photon energy of 1.55 eV.
seems contrary to the basic assumptions based on Eq. (4). Here, direct excitation of d-band electrons leads to a higher energy gain for a high photon energy as compared to an excitation with a low photon energy. At higher fluences, two-photon processes come into account and, as described by Eq. (4), these are more probable for the excitation with lower photon energy. Thus, as shown in Fig. 2, the energy gain for a photon energy of 1.55 eV increases stronger than linearly and exceeds the energy gain associated with the excitation with 2.48 eV at the same fluence. Note that the photons, which are not absorbed, are transmitted through the considered volume. 4. Transient electron distribution Directly after the end of the laser pulse, the electronic system is in a strong nonequilibrium. As the energy gain, the shape of the nonequilibrium distribution also depends on the photon energy applied. Fig. 3 compares two nonequilibrium distributions directly after laser excitation with two different photon energies. We have chosen distributions with almost the same internal energy gain of ue ≈ 6.3e9 Jm−3 , which are generated by pulses of two different fluences. For a photon energy of 1.55 eV, the distribution after an excitation with a fluence F = 0.33 mJ/cm2 is shown whereas for 2.48 eV the distribution after an excitation with F = 0.48 mJ/cm2 was chosen. When neglecting energy dissipation, both nonequilibrium distributions would thermalize toward a temperature of Te ≈ 7700 K. Additionally, the distribution function at Te = 300 K, i.e., the distribution before laser excitation, is depicted. Fig. 3 shows reproductions of distinct features of the density of states (DOS) as well as the initial distribution. For both nonequilibrium distributions, the high peak in the DOS at about 1.65 eV below Fermi energy, EF , is reproduced in the distribution function after
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Fig. 4. Electron–phonon coupling parameter ˛ in equilibrium and nonequilibrium depicted over the corresponding equilibrium electron temperature Te [f]. The equilibrium coupling parameter increases due to the shape of the density of states. The nonequilibrium coupling parameter depends on photon energy and fluence, as well as the density of states.
single-photon excitation (marked “DOS + 1pt”). For an excitation with a photon energy of 1.55 eV, two-photon processes create a second replication above Fermi edge, marked DOS + 2pt. For 2.48 eV, on the other hand, only one replication is visible. In this case, electrons are excited from the DOS peak to energies above Fermi edge directly. As a consequence, parts of the peak structure are clearly reproduced in the electron distribution above Fermi edge. Both distributions shown here have the same internal energy, however, the energy is distributed differently among the electrons. With a photon energy of 2.48 eV less electrons are excited, but every excited electron has gained more energy as compared to the excitation with a photon energy of 1.55 eV. This leads also to differences in the electron–phonon coupling, which will be discussed below. 5. Electron–phonon coupling With help of Eq. (7), we calculate electron–phonon coupling parameters ˛ of the laser-excited nonequilibrium distribution directly after irradiation. We compare them to coupling parameters obtained for the corresponding equilibrium distributions of the same internal energy. Fig. 4 depicts the coupling parameter for equilibrium and nonequilibrium for different photon energies in dependence on the corresponding electron temperature. The values are calculated directly after the end of the rectangular laser pulse, where we expect the largest differences [15]. First, we take a closer look at the equilibrium curve. At low temperatures, the electrons have a steep Fermi edge. As a consequence, at room temperature, only scattering at the Fermi edge is possible. For elevated temperatures, the Fermi edge broadens and scattering becomes possible on a broader energy range. Thus, the coupling factor increases. When the temperature reaches about 4000 K, the d-band peaks are no longer fully occupied and scattering in this energy range is possible as well. Due to the much higher electron density in this region, the slope of the coupling parameter is stronger. When the whole d-band region is excited at about 9000 K the slope decreases again. The nonequilibrium case is more complicated and depends on the photon energy. High photon energies such as 2.48 eV suffice to excite electrons directly from the d-bands. Thus, the scattering rate and therefore the electron–phonon coupling parameter is higher than the corresponding equilibrium value. For low photon energies, here for 1.55 eV, and for low fluences, the d-band peaks cannot be excited directly and the electron distribution still almost looks like a Fermi distribution at 300 K. Thus, scattering is
less probable than in the equilibrium case [33]. For higher fluences, two-photon processes become important. Therefore, the coupling strength of the nonequilibrium distribution exceeds the equilibrium value also for lower photon energies. It even exceeds the coupling for the higher photon energy, since the multi-photon absorption probability depends on the laser frequency, cf. Eq. (4). A larger amount of d-band electrons can be excited for ωL = 1.55 eV than for ωL = 2.48 eV at these fluences. This leads to a larger scattering rate, even when the same amount of energy was absorbed by the electrons. Therefore, the coupling parameter after an excitation with a photon energy of 1.55 eV exceeds the values of both other cases at high fluences. Note that this observation depends on the specific DOS of the material and the excitation parameters. In Ref. [15], where nonequilibrium electron-phonon coupling for gold has been studied, no intersection of these curves has been shown. However, it is likely that such conditions can be found in each noble metal depending on the laser parameters applied. As has been shown in Ref. [15] for gold, also in copper the influence of the laser-excitation on the electron–phonon coupling parameter decreases on the timescale of electron thermalization which depends on excitation strength [15].
6. Conclusion In this work, we investigated the influence of nonequilibrium effects on electron dynamics after ultrashort laser excitation. We have shown that the energy gain of the electrons depends on the photon energy as well as the fluence applied during excitation. For high photon energies, direct excitation of d-band electrons is possible and the energy gain increases linearly with the fluence. For low photon energies and low fluences, the d-band electrons cannot be excited directly and the energy gain is lower. However, the photon energy influences the probability for multi-photon processes, which at sufficiently high fluences enables direct excitation of important d-band peaks particularly for low photon energies. This process counteracts the effect of the density of states and leads to a higher energy gain for the lower photon energy considered in this work. These different excitation characteristics also have an impact on further material behavior such as electron–phonon coupling. The electron distribution after the end of the laser pulse also depends on the photon energy used for excitation. For larger photon energies, electrons in the d-band peaks of the density of states can be excited directly to energies Fermi edge. For smaller photon energies, on the other hand, two-photon processes become more likely and a second representation of the high DOS peak can be found in the distribution function. As a consequence of the excitation and the particular electron distribution after the excitation, electron–phonon coupling differs for equilibrium and nonequilibrium and for different photon energies. Moreover, the coupling parameter also depends on the excitation strength.
Acknowledgments Financial support of the Deutsche Forschungsgemeinschaft through the Heisenberg Program (Grant No. RE 1141/15), through the Carl-Zeiss Stiftung, and through the SFB/TRR-173 “Spin+X” is gratefully acknowledged. Additionally, the authors appreciate the Allianz für Hochleistungsrechnen Rheinland-Pfalz for providing computing resources through project LAINEL on the Elwetritsch high performance computing cluster. We thank B.Y. Mueller for his intellectual and numerical heritage.
Please cite this article in press as: S.T. Weber, B. Rethfeld, Laser-excitation of electrons and nonequilibrium energy transfer to phonons in copper, Appl. Surf. Sci. (2017), http://dx.doi.org/10.1016/j.apsusc.2017.02.183
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ARTICLE IN PRESS S.T. Weber, B. Rethfeld / Applied Surface Science xxx (2017) xxx–xxx
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Please cite this article in press as: S.T. Weber, B. Rethfeld, Laser-excitation of electrons and nonequilibrium energy transfer to phonons in copper, Appl. Surf. Sci. (2017), http://dx.doi.org/10.1016/j.apsusc.2017.02.183