Nonequilibrium electron–phonon coupling after ultrashort laser excitation of gold

Accepted Manuscript Title: Nonequilibrium Electron-Phonon Coupling after Ultrashort Laser Excitation of Gold Author: B.Y. Mueller B. Rethfeld PII: DOI: Reference:

S0169-4332(13)02345-3 http://dx.doi.org/doi:10.1016/j.apsusc.2013.12.074 APSUSC 26899

To appear in:

APSUSC

Received date: Revised date: Accepted date:

7-7-2013 12-12-2013 14-12-2013

Please cite this article as: B.Y. Mueller, B. Rethfeld, Nonequilibrium Electron-Phonon Coupling after Ultrashort Laser Excitation of Gold, Applied Surface Science (2013), http://dx.doi.org/10.1016/j.apsusc.2013.12.074 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Highlights (for review)

Dear Reviewer,

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Our highlights: It is well known that the electron phonon coupling plays a fundamental role in ultrafast laser-matter processing. A few years ago, this parameter has been calculated for several materials in a large range of temperatures. Here we show for the example of gold that the electronic temperature is not a proper variable, but rather the complete distribution of free electrons influences the coupling strongly.

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please excuse the size of the figures in the compiled text. All figures are also attached as pdf and you may refer to the figures at the end of the submission file.

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Thank you for consideration, sincerely yours, Baerbel Rethfeld

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*Manuscript

Nonequilibrium Electron-Phonon Coupling after Ultrashort Laser Excitation of Gold B. Y. Mueller, B. Rethfeld

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Fachbereich Physik und Forschungszentrum Optimas, Erwin-Schroedinger Str. 46 67663 Kaiserslautern, Germany

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Abstract

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Keywords: nonequilibrium, metal, electron-phonon coupling

1. Introduction

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The development of ultrashort laser pulses opened up a wide range of applications [1–4] including new research fields of ultrafast processes like laser ablation [5– 8], laser stimulated electron emission [8–15] and ultrafast changes in magnetization [16–19]. Theoretical efforts have been performed to find models which describe the behavior of the target after ultrashort laser excitation. One robust and easy to use model is the socalled two temperature model (TTM) [20] which was derived for picosecond laser pulses. Moreover, kinetic approaches like Monte Carlo methods [21–24] as well as Boltzmann scattering integrals were applied [7, 25– 28] to overcome the limitations of the TTM for femtosecond excitations. In the first application of the TTM to ultrashort laser irradiation of metals by Anisimov et al. [20] the material parameters were estimated as constants. Further investigations of these parameters have studied their dependence on electron and phonon temperature [29–33]. Moreover, Lin et al. [32] pointed out the differences arising through the particular density of states of different metals. As a further step, simulations for ultrashort laser excitations revealed strong

Email addresses: [email protected] (B. Y. Mueller), [email protected] (B. Rethfeld) Preprint submitted to Applied Surface Science

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Exciting a metal by an ultrashort laser pulse, the electrons are driven out of thermal equilibrium, while the phonon system remains almost unaffected. During and after the irradiation, the electrons thermalize and transfer energy to the phonons. In this work, we investigate the electron-phonon coupling in gold. The dependence of the coupling strength on the phonon properties as well as the nonequilibrium electrons have been taken into account. For the phonon system, we utilize several phonon temperatures. For the electrons we apply different excitation scenarios depending on the laser fluence and the photon energy. We observe that for gold at electron temperatures below 2000 K, the phonon distribution may affect the coupling slightly. However, the electron distribution, especially under nonequilibrium conditions, governs the electron-phonon coupling factor significantly.

deviations between the nonequilibrium case after ultrafast excitation and the equilibrium case of a thermalized electron system [25, 34, 35]. This indicates, that the “electron-phonon coupling constant” is no constant and also not simply a function of temperature, but for very short excitations a complex functional α[ f (t), g(t)] depending on the current distribution functions of electrons and phonons. In this work, we are interested in the influence of the phonon temperature and several situations of electron nonequilibrium on the electron-phonon coupling parameter. 2. Theory We consider a homogeneously heated metal film, which, for the case of gold, may be of thickness of up to ∼ 100 nm [36]. The energy balance between the primarily heated electrons and the lattice is governed by a temperature-dependent relaxation term leading to the time-dependent formulation of the TTM: d ue ∂T e = Ce (T e ) = −α · (T e − T p ) + S laser (t) , (1a) dt ∂t d up ∂T p = C p (T p ) = +α · (T e − T p ) . (1b) dt ∂t

Here, ue and u p are the internal energies of the electrons and phonons, respectively, α is the electron-phonon December 12, 2013

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10000 K), as can be seen by temperature-dependent DFT calculations [42–45]

coupling parameter and S laser (t) the source term for electron heating by the laser pulse. Equation (1) assumes that the electrons as well as the phonons can be described each by a respective temperature, T e and T p . For nonequilibrium conditions we apply Boltzmann collision integrals for metals [25, 35] df = Γel−el + Γel−ph + Γlaser , dt dg = Γph−el , dt

In this work we consider electron-phonon coupling in gold. We implement the density of states of the electrons for gold within an effective one band model as described in detail in Ref. [35]. Investigations considering in particular interband transitions between dand s-electrons were made in Ref. [28]. Here, we neglect the influence of different bands but assume the effective mass of electrons according to the combined density of states of d- and s-bands. The resulting dispersion relation is shown in Figure 2 as a dashed line. For the phonons, we assume the Debye model with E D = 32 meV as the Debye energy which leads to a Debye temperature of T D = 375 K. The volume of unit cell is 1.695 × 10−29 m3 [46] and the sound velocity of the longitudinal mode was taken as 3240 m/s [46].

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(2a) (2b)

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where f and g are the transient distribution functions for the electrons and the phonons, respectively. The collision integrals Γindex describe collisions among electrons, between electrons and phonons, and inverse Bremsstrahlung leading to absorption. We find that both electron-electron collisions and electron-phonon collisions depend significantly on the screening of the electrons [35]. Therefore, we determine the transient screening in each time step during our simulation. Note, that in our model umklapp processes have been neglected. An improvement might be taken into account by applying the matrix elements from Ref. [37, 38]. However, we believe, that umklapp processes are most influential during the excitation of the material [37]. For the coupling between electrons and phonons and during electron-electron collision, the required momentum transfer can be achieved without an additional umklapp vector. For the phonon system we only trace the longitudinal mode within the Debye model, since only those phonons can contribute to the electron-phonon coupling [25, 35, 39, 40]. Details of the collision integrals and their implementation are given in Refs. [25, 35, 39–41]. In a rough comparison of Eq. (1) with Eq. (2), both systems of equations describe electron heating by the laser as well as electron-phonon interaction, while the process of electron-electron thermalization enters Eq. (2a) additionally. All considered processes are sketched in Fig. 1. In an isotropic material, the internal energy of the electrons and of the phonons can be determined by the moments of the distribution functions: Z ue = dE f (E)De (E)E , (3a) Z up = dE g(E)D p (E)E , (3b)

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With the DOS of the electrons and the phonons, respectively, we are thus able to calculate the transient distribution functions of electrons and phonons, f and g, respectively.

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To compare our results to the TTM, we, in a first step, determine the ”corresponding temperature”, T e [ f (t)] for a given transient nonequilibrium electron distribution f (t). It can be obtained by comparing the internal energy ue of the electrons (3a) with the internal energy of a Fermi-distributed electron gas in gold, fFermi (T e ). If both energies are equal, the temperature of the Fermi distribution determines uniquely the corresponding temperature of the nonequilibrium distribution, thus T e [ f (t)] = T e [ fFermi ] when ue [ f (t)] = ue [ fFermi (T e )]. The temperature for a nonequilibrium phonon system can be determined in the same manner. Finally, in the second step, we determine the corre-


 






 

 



 

 









  


 



 

Figure 1: Sketch of the considered interaction processes: laser excitation, electron thermalization as well as relaxation between the electron and the phonon subsystem. The latter process is driven by the equilibration of the temperature of the electrons T e , and the temperature of the phonons T p .

where De (E) is the density of states (DOS) of the electrons and D p (E) that of the phonons. We assume that both DOS do not change due to excitation. This is justified for our considered excitation strength (∆T . 2

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effective one-band dispersion relation

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Figure 3: Electron-phonon coupling strength in dependence on the phonon and electron temperature. The simulations were performed without a laser excitation. The Boltzmann equation was solved by initially assuming two different temperatures for the phonons and electrons.

sponding electron-phonon coupling parameter as [35]

tion. Consequences of a nonequilibrium phonon distribution are currently under study in our group. For the electron-phonon coupling we expect no large deviations for high electron temperatures and a phonon specific heat near 3nkB. For low material temperatures, however, a nonequilibrium phonon distribution may affect the electron-phonon coupling slightly.

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due [ f (t)]/dt . T p [g(t)] − T e [ f (t)]

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α[ f (t), g(t)] =

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Figure 2: The density of states of gold (left axis) and the corresponding averaged dispersion relation (right axis) in the framework of the effective one-band model. The Fermi energy of gold is at EF = 9.2 eV. Below Fermi energy, three different photon energies, as applied in this work, are marked.

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3. Influence of the phonon temperature

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This is the coupling parameter α entering the TTM in Eq. (1), in case that the electrons and phonons are described with the approximation of corresponding temperatures.

4. Influence of the laser wavelength In order to evaluate the influence of different nonequilibrium situations in the electronic system on the electron-phonon coupling, we study different irradiation conditions by varying the absorbed laser fluence F and the photon energy. The excitation was assumed to occur under constant intensity within a short time of τ = 10 fs to avoid a thermalization during the irradiation process. For instance, in Ref. [35] we determined the influence of the photon energy on the absorption strength. We extended these simulations and Fig. 4 shows the gained electron energy after different fluences and at different photon energies. On the right y-axis the corresponding electron temperatures after the thermalization process are depicted. Since the absorption probability is expected to decrease with increasing photon energy [35], we expect a decreasing curve also for the gained electron energy. This is observed in Fig. 4 for low photon energies. However, it is observed, that increasing the photon energy over a critical energy (∼ 2 eV) increases the absorption in gold significantly. This is attributed to the d-electrons which are located ∼ 2 eV below the

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In this section we investigate the dependence of the electron-phonon coupling on the phonons. As a first step we consider undisturbed electrons and phonons both with equilibrium distributions, however, different temperatures T p , T e . Directly at T p = T e , we observe that Eq. (4) has a removable singularity α(T e = T p ) = a(T e 7→ T p ) within the accuracy of our numerical simulations. Figure 3 depicts the electron-phonon coupling depending on the electron temperature for five different phonon temperatures. By changing the phonon temperature, almost no changes are observed for high electron temperatures. For low electron temperatures, the deviations are moderate. With phonon temperatures below the Debye temperature, the coupling is reduced, since not all phonon modes are thermally excited. In this case, less phonons couple to the electron system. Here, thermal equilibrium was assumed for electrons as well as for phonons. Nonequilibrium situations for electrons will be discussed in the next sec3

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final electron temperature [K]

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corresponding electron temperature [K] Figure 5: Electron-Phonon coupling in dependence on the photon energy of the laser pulse. The same seven laser fluences were applied as in Fig. 4. For ~ω = 1.55 eV only the lowest six fluences are shown. Note, that for the same fluence but different laser wavelength the absorption differs.

Fermi energy. With a sufficient photon energy, these electrons can be directly excited. In Fig. 2 such photon energies are marked with the green and black arrows, here ~ω is 2.48 eV and 2.14 eV, respectively. Excitations of electrons across the Fermi edge with only one photon are of considerably higher probability than the two-photon absorption processes, as for instance with a photon energy of ~ω = 1.55 eV marked with a red arrow in Fig. 2. The contribution of the excited electrons to the electron-phonon coupling strength is studied in the present work. Here, we determine the electron-phonon coupling under nonequilibrium conditions for both, varying the wavelength and the fluence. The fluence F was tuned between 0.01 mJ/cm2 and 0.65 mJ/cm2 and no reflection was taken into account, i.e., the values refer to the absorbed fluence. We applied three different wavelengths 500 nm, 580 nm and 800 nm that correspond to photon energies ~ω of 2.48 eV, 2.14 eV and 1.55 eV, respectively. Figure 5 shows the electron-phonon coupling for these three different laser wavelengths and the comparison to the equilibrium case. For better comparison the coupling parameter calculated through Eq. (4) is plotted in dependence on the corresponding electronic temperature T e [ f (t = τ)] calculated directly after the excitation. This corresponding electron temperature can be considered as the temperature, the electrons would have after the electron thermalization process without any other effects, in particular without cooling of the electrons due to electron-phonon interaction. As sketched in Fig. 2, the low photon energy of

1.55 eV (800 nm), indicated by a red arrow, is not sufficient to excite the d-electrons. Since the density of states around the Fermi edge is very smooth, the electrons behave like a free electron gas and the electronphonon coupling is reduced in comparison with the equilibrium case [25]: Due to the reduced phase space of the nonequilibrium electrons, the coupling is decreased as compared to the equilibrium case. Such behavior can be seen in the red curve in Fig. 5 for low fluences. By increasing the fluence, the photon absorption is increased as well and the d-electrons are excited above the Fermi edge by two-photon processes. This leads to a higher electron-phonon coupling parameter. For the higher photon energies considered here (black and green arrow in Fig. 2 and respective curves in Fig. 5), d-electrons are excited directly above the Fermi level and contribute to the electron-phonon coupling, which is larger than the equilibrium coupling for all considered fluences. As Fig. 5 indicates, there is no distinct value for the electron phonon-coupling strength under nonequilibrium conditions. Depending on the strength of the nonequilibrium the corresponding electron-phonon coupling parameter according to Eq. (4) can deviate by a factor of two or even more from the equilibrium case. For larger photon energies, this deviation is larger than for lower photon energies. Thus, the coupling under nonequilibrium conditions strongly depends on the particular excitation, thus on details of the particular nonequilibrium. The coupling parameter entering the TTM, Eq. (1), has therefore to be calculated explicitly by the electron distribution.

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Figure 4: Gained electron energy after laser excitation depending the photon energy and laser fluence. Since the internal energy ue (T e ) of the electrons is not linear in temperature, see Eq. (3a), the right y-axis is also not linear.

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where τ is the thermalization time [35] and αNEQ [F, λ] is the electron-phonon coupling in nonequilibrium as given in Fig. 5. The temperature-dependent coupling factor α(T el , T ph ) can be taken from Fig. 3. However, the definition of temperature for very short times is still crucial after an ultrashort laser excitation. Therefore, a kinetic model should be applied to get reliable results for times shorter than the thermalization time.

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the electron and phonon distribution. In particular we showed the influence of the fluence, wavelength and thermalization state on the electron-phonon coupling. We conclude, that for ultrashort laser excitation, the two temperature model which relies on the coupling factor α is questionable. One might think about implementing a transient excitation-dependent electron-phonon coupling parameter into the two-temperature model (1), like h i α = α(T el , T ph ) + αNEQ (F, λ) − α(T el , T ph ) e−t/τ

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Figure 6: Electron-phonon coupling under equilibrium conditions compared with the static as well as the transient nonequilibrium case. Here, a laser fluence of 0.21 mJ/cm2 and a photon energy of ~ω = 1.55 eV are applied.

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5. Transient effects

Acknowledgments

To stress even more the deviations of the electronphonon coupling strength under nonequilibrium conditions, Fig. 6 shows the transient coupling α[ f (t), g(t)], derived by our model. Here, the combined effect of nonequilibrium electron-phonon coupling and thermalization is visible. As shown in Ref. [35], the electronic system in gold thermalizes on a timescale of about 50 fs. Though this value also depends on excitation parameters as fluence and wavelength, we conclude that it is neither correct to assume a “temperature-dependent” equilibrium parameter nor a nonequilibrium expression for the electron-phonon coupling. During the relaxation and thermalization of the electrons there is a transition between both cases. Note, that also the equilibrium curve (blue solid line in Fig. 6) changes slightly in time (barely seen), since the electron temperature changes in time as well.

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Financial support of the Deutsche Forschungsgemeinschaft through the Heisenberg Programm (grant No. RE 1141/15) and the AHRP for providing the computational facilities is gratefully acknowledged.

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References

6. Conclusion

In conclusion, a model to determine the electronphonon coupling strength under equilibrium and nonequilibrium conditions has been applied. We showed that the influence of the phonon system has only slight effects for low electron and phonon temperatures. However, the electron distribution influences the coupling significantly. This means that the electron-phonon coupling parameter α is neither approximately a constant nor approximately a function on the electron and phonon temperatures but a functional depending on

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