Electron–phonon coupling in 122 Fe pnictides analyzed by femtosecond time-resolved photoemission

Although an enormous amount of research has been conducted in the last few years on the Fe pnictide high-temperature superconductors (HTSCs) [1, 2], the search for the superconducting pairing mechanism in these materials is still ongoing. Besides other excitations such as spin fluctuations [3], the electron–phonon (e–ph) coupling responsible for Cooper pairing in conventional superconductors in the Bardeen–Cooper–Schrieffer (BCS) theory is considered a potential candidate. Therefore the quantitative determination of the e–ph coupling constant λ in the Fe pnictides is of particular interest. Several methods to determine λ have been established in the literature and successfully used in various studies. In the energy domain, angle-resolved photoemission spectroscopy (ARPES) analyzes the renormalization of the single-particle spectral function by the electronic self-energy Σ, which includes the e–ph coupling. However, such type of analysis requires very high sample and data quality, which is difficult to obtain in the Fe pnictides, and only a few ARPES studies of FeAs compounds were able to determine the effects of Σ [4–6]. In addition, in thermal equilibrium, contributions from other degrees of freedom to Σ are often difficult to disentangle from e–ph coupling.

In contrast, femtosecond (fs) time-resolved spectroscopies allow separating of e–ph scattering from other relaxation channels such as e.g. electron–electron (e–e) scattering or heat diffusion due to their different intrinsic timescales out of thermal equilibrium [7–9]. Here, time-resolved ARPES (trARPES) is used, which combines the energy and momentum resolution of ARPES with fs time resolution into a powerful tool to directly investigate the dynamics of the electronic structure in a non-equilibrium state [10, 11].

The e–ph relaxation in metals after photoexcitation has been successfully described using the two-temperature model (2TM) [7–9, 12–14]. This model describes a system of two coupled heat baths for the conduction electrons and the ion lattice with temperatures T_e and T_l by coupled rate equations for T_e and T_l, respectively. One key assumption of this model is that e–e and phonon–phonon (ph–ph) scattering occurs on a much faster timescale than the e–ph scattering, leading to a thermal distribution within each subsystem. However, in correlated electron systems such as the cuprate or Fe pnictide HTSCs, e–ph scattering might occur on similar timescales [15], which poses some questions on the applicability of the 2TM for these materials and care has to be taken. Therefore we analyze our data in terms of a suitable version of the 2TM and compare the results with two complementary methods for the determination of the e–ph coupling strength. We find that all three methods to analyze the e–ph coupling strength from time-resolved photoemission data yield results for the second moment of the Eliashberg coupling function $\lambda \langle \omega ^2 \rangle$, which agree well within error bars and reproduce the trends for three different 122 FeAs compounds consistently.

For trARPES experiments, single crystals of EuFe₂As₂ and BaFe₂As₂ parent compounds and optimally doped BaFe_1.85Co_0.15As₂ (T_c = 23 K) were cleaved in ultrahigh vacuum (p < 10⁻¹⁰ mbar) at T = 100 K, where most measurements were carried out. The experimental setup is sketched in figure 1(a): the output of a commercial regenerative Ti:sapphire amplifier (Coherent RegA 9050) delivering ultrashort laser pulses at hν₁ = 1.5 eV photon energy with a pulse duration of 55 fs, and operating at 300 kHz is used to optically excite the samples (pump pulse). A time-delayed frequency-quadrupled probe pulse at hν₂ = 6.0 eV photon energy with a pulse duration of 80 fs leads to the emission of photoelectrons, which are detected using an electron time-of-flight (TOF) spectrometer with an acceptance angle of ±3°. The energy resolution of 50 meV is mainly determined by the spectral width of the probe pulses, and the overall temporal resolution is <100 fs. For details see [9].

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2.1. Three-temperature model

A dispersion of the hole pocket in EuFe₂As₂ without the pump pulse is shown in figure 1(b). Exemplary time-dependent trARPES spectra near Γ are shown in figure 1(c) for various pump–probe delays after excitation on a logarithmic intensity scale. After excitation, a pronounced distribution of excited charge carriers is formed at E − E_F > 0.1 eV, which deviates from the Fermi–Dirac distribution of the electronic system before excitation and which originates from hot, non-thermalized electrons. Subsequently, e–e scattering quickly leads to thermalization of these non-thermal electrons and to the formation of a hot thermalized electron distribution [9]. At t > 250 fs, the non-thermal contribution has decayed to less than 1% of the electron population.

The temperature of the thermalized part of the electronic system T_e can be extracted from the trARPES data by fitting a Fermi–Dirac distribution function to the high-energy cutoff of the transient spectra, multiplied by a phenomenological density-of-states function and convoluted with an instrumental resolution function [9, 14], as shown in figure 1(c) (solid lines). T_e determined by the fitting is shown in figure 1(d) as a function of pump–probe delay. After the steep rise of the electronic temperature at zero pump–probe delay, we find a relaxation on two distinct timescales τ_α and τ_β. Such a behavior indicates the selective coupling of hot electrons to a subset of strongly coupled phonon modes on the faster timescale τ_α and the subsequent energy transfer to the rest of the phonons by anharmonic ph–ph scattering, determined by τ_β. Thus, the system is described by an extended version of the 2TM, which had been developed for cuprate HTSCs [14], where a similar relaxation dynamics was found, and which has been also used recently for the Fe pnictides [16, 17]. This three-temperature model (3TM) consists of a system of three coupled differential equations describing the temperature of the electrons, T_e, of a hot phonon distribution T_p and of the rest of the lattice modes, T_l:

$$\begin{equation} \frac{\mathrm{d} T_{\mathrm{e}}}{\mathrm{d} t} = -H(T_{\mathrm{e}}, T_{\mathrm{p}}) + \frac{S}{C_{\mathrm{e}}}, \end{equation} \tag{ 1 }$$

$$\begin{equation} \frac{\mathrm{d} T_{\mathrm{p}}}{\mathrm{d} t} = +\frac{C_{\mathrm{e}}}{C_{\mathrm{p}}}H(T_{\mathrm{e}}, T_{\mathrm{p}}) - \frac{T_{\mathrm{p}}-T_{\mathrm{l}}}{\tau_\beta}, \end{equation} \tag{ 2 }$$

$$\begin{equation} \frac{\mathrm{d} T_{\mathrm{l}}}{\mathrm{d} t} = +\frac{C_{\mathrm{p}}}{C_{\mathrm{l}}}\frac{T_{\mathrm{p}}-T_{\mathrm{l}}}{\tau_\beta}. \end{equation} \tag{ 3 }$$

The source term S describes the optical excitation; C_e, C_p and C_l are the specific heat capacities of electrons, strongly and weakly coupled phonon modes, respectively. The anharmonic decay of phonons is described by τ_β. For the energy transfer from the electrons to the more strongly coupled phonons, the formula derived by Allen [7] is used:

$$\begin{equation} H(T_{\mathrm{e}}, T_{\mathrm{p}})=\gamma_{\mathrm{T}} (T_{\mathrm{e}}-T_{\mathrm{p}})=\frac{3\hbar\lambda\langle\omega^2\rangle}{\pi k_{\mathrm{B}}}\frac{T_{\mathrm{e}}-T_{\mathrm{p}}}{T_{\mathrm{e}}}, \end{equation} \tag{ 4 }$$

where k_B is the Boltzmann constant. This relation allows the determination of the second moment of the Eliashberg e–ph coupling function $\lambda \langle \omega ^2 \rangle$.

A fit of a numerical solution of the 3TM to T_e is shown in figure 1(d) and yields a good agreement to the experimental data. Owing to an ambiguity of the model parameters such as the electron and lattice specific heat capacities reported in the literature, a range of values for $\lambda \langle \omega ^2 \rangle$ is retrieved, where the fits show similar good agreement to the data. Details of this analysis can be found in [17]; here, only the main results are discussed. From the 3TM, we find values of $\lambda \langle \omega ^2\rangle =56\mbox {--}65\,\mathrm {meV^2}$ for EuFe₂As₂, whereas in BaFe_1.85Co_0.15As₂ it is slightly smaller ($\lambda \langle \omega ^2 \rangle =46\mbox {--}55\,\mathrm {meV^2}$) and even smaller in undoped BaFe₂As₂ ($\lambda \langle \omega ^2 \rangle =30\mbox {--}46\,\mathrm {meV^2}$).

2.2. Electronic excess energy

The analysis within the 3TM was based on the energy relaxation within the thermalized part of the transient electronic distribution function and thus neglected the non-thermal electrons present at early delay times (see figure 1(c)). However, the analysis of the energy relaxation of these non-thermal, excited electrons can also provide information about the strength of e–ph coupling.

The scattering rate of electrons excited at energy  = E − E_F above the Fermi level is in the self-energy formulation of many-body theory determined by Γ = ℏτ⁻¹ = 2Im Σ(), where Im Σ is the imaginary part of the electronic self-energy [18]. Important energy-dependent contributions to Γ arise from e–e and e–ph scattering, where e–e scattering is considered to follow the quadratic energy scaling of the Fermi liquid theory, $\Gamma _{\mathrm {e}\mbox {--}\,\mathrm {e}}=2\beta [(\pi k_{\mathrm {B}} T)^2+ \epsilon ^2 ]$ (figure 2(a)). As the proportionality coefficient β is rather small in typical metals (of the order of 10⁻²–10⁻¹ eV⁻¹) [19, 20], this contribution is negligible compared to the e–ph scattering at low-enough excitation energies. The latter increases up to the maximal phonon energy ℏω_max and is constant above (figure 2(a)). For  > ℏω_max and T = 0 K, Γ_e– ph results for an Einstein mode ω₀ to [21]

$$\begin{equation} \Gamma_{\mathrm{e}\mbox{--}\,\mathrm{ph}}=\pi\hbar\lambda\omega_0. \end{equation} \tag{ 5 }$$

Within the energy window between ℏω_max and the crossover regime, where Γ_e– e becomes dominant (shaded area in figure 2(a)), the rate of energy dissipation of an electron due to the emission of a phonon with energy ℏω₀ is given by

$$\begin{equation} \frac{\mathrm{d} E}{\mathrm{d} t} = \frac{\hbar\omega_0}{\tau}=\pi\hbar\lambda\omega_0^2, \end{equation} \tag{ 6 }$$

which leads to a linear relaxation of the electron energy [22].

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The rate of energy relaxation of excited electrons can be extracted from the experimental trARPES intensity I(,t) by analyzing the mean excess energy

$$\begin{equation} \left\langle\Delta E(t)\right\rangle = \frac{\int^{\epsilon_1}_{\epsilon_0}{\epsilon \Delta I(\epsilon,t)\textrm{d}\epsilon}}{\int^{\epsilon_1}_{\epsilon_0}{\Delta I(\epsilon,t)\textrm{d}\epsilon}}. \end{equation} \tag{ 7 }$$

However, this integral represents the mean excess energy within a selected partition of a transient distribution function and not the energy relaxation of single individual electrons. Thus, special care has to be taken with the determination of the integration boundaries, ₀ and ₁. A careful investigation of the influence of the integration boundaries and the excitation fluence reveals that a reasonable choice of ₀ = 70 meV and ₁ = 300 meV allows us to give a good estimate of the e–ph coupling strength, as detailed in [23].

Experimental data of EuFe₂As₂, BaFe_1.85Co_0.15As₂ and BaFe₂As₂ near k_F and at T = 100 K are shown in figure 2(b). To minimize lattice heating and the influence of the hot thermalized electron distribution, low excitation fluences of F ∼ 50 μJ cm⁻² have been used. The linear fits to equation (6) within the first 100 fs yield values of $\lambda \langle \omega ^2\rangle ={65(5)}\,\mathrm {meV^2}$ for EuFe₂As₂, 50(3) meV² for BaFe_1.85Co_0.15As₂ and 34(6) meV² for BaFe₂As₂. Here, error bars represent the numerical uncertainties of the fits, whereas the overall errors have to be considered larger due to the uncertainty of the integration boundaries discussed above and of the elevated lattice temperature. Nevertheless, we find a good agreement with the values obtained in the 3TM simulations and a considerably higher e–ph coupling in EuFe₂As₂ than in BaFe_1.85Co_0.15As₂ and even lower $\lambda \langle \omega ^2\rangle$ in BaFe₂As₂.

2.3. Temperature-dependent hole relaxation rates

Finally, another estimate of the e–ph coupling strength can be gained from the temperature dependence of quasiparticle (QP) relaxation rates [7, 15, 24, 25]. A recent theoretical investigation of the Boltzmann equation for e–e and e–ph interaction found an analytic solution for the temperature dependence of the QP relaxation rate τ [24]. In the limit of an e–ph scattering rate comparable to or exceeding the rate for e–e scattering and at elevated temperatures, τ depends linearly on the lattice temperature T [15, 24, 25]:

$$\begin{equation} \tau=\frac{2\pi k_{\mathrm{B}} T}{3\hbar\lambda\langle\omega^2\rangle}. \end{equation} \tag{ 8 }$$

This relation allows for an independent determination of the e–ph coupling strength $\lambda \langle \omega ^2 \rangle$.

Experimentally, the temperature-dependent relaxation of holes in the hole pocket near Γ in EuFe₂As₂ is used, which are found to be independent from the antiferromagnetic transition [26]. The respective hole relaxation time τ_holes is determined from the temperature-dependent trARPES intensity by fitting exponential decay curves (figure 3(a)) and is shown in figure 3(b) as a function of temperature. The fit to equation (8) for T > 100 K reveals a good agreement to the linear behavior and yields a value of $\lambda \langle \omega ^2 \rangle ={90(30)}\,\mathrm {{meV}^2}$, where the error represents a confidence interval of 80%. This value, albeit considerably higher than the values found in the 3TM and in the evaluation of the excess energy, is in agreement with the other evaluations within error bars.

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The values of $\lambda \langle \omega ^2 \rangle$ determined by the three methods discussed above are compared in table 1. Despite the limitations and drawbacks of the various methods, we find a perfect agreement of all three methods within error bars. In particular, the trend for larger e–ph coupling in EuFe₂As₂ than in BaFe_1.85Co_0.15As₂ and in BaFe₂As₂ is well represented. Remarkably, all methods produce a small $\lambda \langle \omega ^2\rangle <100\,\mathrm {meV^2}$, which already indicates weak e–ph coupling.

Table 1. Values of $\lambda \langle \omega ^2 \rangle$ determined by the three methods.

Compound	Te(t)	$\left \langle \Delta E(t)\right \rangle$	τholes(t)
EuFe2As2	56–65	65(5)	90(30)
BaFe1.85Co0.15As2	46–55	50(3)	–
BaFe2As2	30–46	34(6)	–

Our values of $\lambda \langle \omega ^2 \rangle$ can be compared to the recently published values determined from optical pump–probe experiments. Mansart et al [16] report a comparable value of $\lambda \langle \omega ^2 \rangle \approx {64}\,\mathrm {meV^2}$ for Co-doped BaFe₂As₂, which was determined using the 3TM. Stojchevska et al [25] derived a somewhat higher value of $\lambda \langle \omega ^2 \rangle ={110(10)}\,\mathrm {meV^2}$ in SrFe₂As₂ using temperature-dependent QP relaxation times, and for SmFeAsO_1−xF_x an even larger value of $\lambda \langle \omega ^2 \rangle ={135(10)}\,\mathrm {meV^2}$ is reported [27].

Based on our results of $\lambda \langle \omega ^2 \rangle$ we can estimate the value of the e–ph coupling constant λ for a particular value of ω. Considering the Raman active A_1g mode at 23 meV, which can be coherently excited [17, 28, 29] and which therefore shows an enhanced e–ph coupling in the system, we find λ < 0.2 for all compounds. This estimate is in agreement with the calculations of various Fe-pnictide compounds, which report the values of λ < 0.35 [30, 31]. Taking the mean of the phonon spectrum as reference, λ gets even smaller, in agreement with other publications [16, 25]. Even if we consider the lowest coupled modes around 12 meV to be most important for e–ph coupling, λ does not exceed a value of 0.5. Similarly small values for λ have been found in the cuprate HTSCs [14, 15], which suggests limited importance of e–ph coupling for the pairing mechanism in both classes of materials.

These boundaries for the e–ph coupling constant λ allow us to estimate the superconducting critical temperature T_c, assuming a conventional BCS-type pairing based on e–ph coupling. Here, T_c in isotropic systems and in a strong-coupling regime is given by McMillan's formula [32], modified by Allen and Dynes [33],

$$\begin{equation} T_{\mathrm{c}}=\frac{\omega_{\mathrm{log}}}{1.20}\exp\left(\!-\frac{1.04(1+\lambda)}{\lambda-\mu^*-0.62\lambda\mu^*}\!\right), \end{equation} \tag{ 9 }$$

where μ* is the effective Coulomb repulsion and ω_log is the logarithmic average of the phonon spectrum. Taking μ* = 0 and ω_log = 205 K [30], which provides an upper limit for T_c, we find T_c = 0.33 and 7.5 K for λ = 0.2 and 0.5, respectively. Comparing these results with the experimentally found T_c = 24 K in optimally doped BaFe_1.85Co_0.15As₂ [34] and even up to T_c = 38 K in K-doped BaFe₂As₂ [35] demonstrates that e–ph coupling in a conventional BCS pairing scenario cannot explain the high critical temperatures found in the pnictide HTSCs. In addition, albeit the stronger e–ph coupling, the critical temperatures found in EuFe₂As₂ upon Co [36] and K [37] substitution are considerably smaller compared to BaFe₂As₂.

However, e–ph coupling might still play a significant role in the Cooper pair formation in the Fe pnictides. For instance, a very strong sensitivity of the Fe magnetic moment on the As height in the FeAs tetrahedra was found in density functional theory (DFT) band structure calculations [38], with a rate of 6.8 μ_B Å⁻¹. Inelastic x-ray scattering experiments on CaFe₂As₂ concluded on strong coupling of phonons to magnetic excitations even in the high-temperature paramagnetic phase [39]. Furthermore, a strong sensitivity of the maximum critical temperature on the pnictogen height was found [1]. On the basis of these findings a strong magnetophonon coupling in these compounds was proposed. In particular, the strongly coupled A_1g mode perpendicular to the Fe layers, which modulates the pnictogen height, could mediate superconductivity in the Fe-pnictides [40]. Such a scenario is supported by the strong coupling of this particular mode to electronic states directly at the Fermi level [17] evidenced by the coherent excitation of this phonon mode [17, 28, 29].

In summary, we demonstrated the quantitative analysis of the e–ph coupling strength from femtosecond trARPES. In detail, we compared three different methods to determine the e–ph coupling strength in three 122 FeAs compounds from time- and angle-resolved photoemission experiments. The transient temperature of the thermalized electronic distribution is analyzed in EuFe₂As₂, BaFe_1.85Co_0.15As₂ and BaFe₂As₂ by a 3TM; the rate of energy relaxation of the non-thermal electrons yields direct information on the second moment of the Eliashberg coupling function $\lambda \langle \omega ^2 \rangle$. Finally, the temperature dependence of hole relaxation rates in EuFe₂As₂ also allows us to determine $\lambda \langle \omega ^2 \rangle$. All three methods consistently yield a small $\lambda \langle \omega ^2 \rangle <100\,\mathrm {meV^2}$, which results in λ < 0.2 using a reasonable choice of the phonon spectrum. This value is discussed to be too small to explain superconductivity in the Fe pnictides by a conventional BCS-type pairing.

Experiments were carried out at the Freie Universität Berlin. We acknowledge discussions with Martin Wolf and Hermann Dürr in the early stage of the project and experimental support by Setti Thirupathaiah. This work was supported by the Mercator Research Center Ruhr and the Deutsche Forschungsgemeinschaft through BO 1823/2 and SPP 1458. RC acknowledges the Alexander von Humboldt Foundation.