Unusually weak electron-phonon coupling in the Shockley surface state on Pd ( 111 )

I. Yu. Sklyadneva,1,2,3 R. Heid,3 V. M. Silkin,1 A. Melzer,4 K. P. Bohnen,3 P. M. Echenique,1,5 Th. Fauster,4 and E. V. Chulkov1,5 1Donostia International Physics Center (DIPC), Paseo de Manuel Lardizabal, 4, 20018 San Sebastián/Donostia, Basque Country, Spain 2Institute of Strength Physics and Materials Science, pr. Academicheski 2/1, 634021 Tomsk, Russia 3Forschungszentrum Karlsruhe, Institut für Festkörperphysik, P.O. Box 3640, D-76021 Karlsruhe, Germany 4Lehrstuhl für Festkörperphysik, Universität Erlangen–Nürnberg, D-91058 Erlangen, Germany 5Departamento de Física de Materiales and Centro Mixto CSIC–UPV/EHU, Facultad de Ciencias Químicas, UPV/EHU, Apartado 1072, 20080 San Sebastián/Donostia, Basque Country, Spain Received 14 July 2009; published 31 July 2009


I. INTRODUCTION
Dynamics of excited electrons on metal surfaces are of paramount importance for molecular motion induced by femtosecond laser pulses, 1 for energy transfer in photochemical reactions, 2 for catalytical reactions, 3 and for charge and spin transfer across surfaces and interfaces. 4Excited electrons interact with other electrons ͑e-e͒, phonons ͑e-ph͒, and defects ͑e-d͒. 5The associated scattering processes contribute to the total linewidth ⌫ tot = ⌫ e-e + ⌫ e-ph + ⌫ e-d observed in spectroscopy.In time-resolved spectroscopy the decay of the population in a particular state can be followed directly and is described by the lifetime .The lifetime broadening ប / is included in the total linewidth and describes the scattering of the excited electron into other states associated with a change in energy or/and momentum.Electron-electron scattering is predominantly inelastic and ⌫ e-e is contained in ប / . 5In e-ph and e-d scattering the energy exchange is small compared to the experimental resolution.Therefore most scattering events leave the population of the observed state apparently unchanged.Contributions from ⌫ e-ph and ⌫ e-d to ប / are limited to scattering events with large momentum transfer, e. g., from surface to bulk bands. 6n estimate of ⌫ e-ph is normally performed by using a Debye model with the experimental e-ph coupling parameter measured at E F of bulk metal.For surfaces studied up to now including noble-metal surfaces, 5,[7][8][9][10] Mo͑110͒, 11,12 as well as simple and semimetal surfaces 13,14 this picture works quite well.Probably the only exception is beryllium whose surfaces show density of states ͑DOS͒ at E F much higher than that in bulk Be due to the appearance of strong surface electronic states at E F . 15 In this work we show an example when the e-ph coupling in a Shockley surface state is very distinct from that at E F of bulk metal despite that the bulk DOS at E F is not very different from the surface DOS.This distinction being important at low temperature essentially affects the linewidth at elevated temperatures when ⌫ e-ph can significantly exceed ប / .By using time-resolved two-photon photoemission Schäfer et al. 16 measured the lifetime broadening of the ⌫ ¯Shockley surface state on Pd͑111͒, ប / = 54 meV at T = 450 K and the total linewidth ⌫ tot = 100 meV.The difference between ⌫ e-e and ⌫ tot was attributed to the quasielastic scattering due to phonons.The theoretical estimation of the e-e inelastic scattering with a model that combines one-dimensional potential and many-body effects 16 gave ⌫ e-e = 37 meV which is slightly smaller than the experimental values for ប / .The Debye model with e-ph coupling parameter = 0.40 taken from ab initio calculations 17 of bulk Pd gives ⌫ e-ph = 60 meV at T = 450 K.This leads to theoretical ប / + ⌫ e-ph = 97 meV in good agreement with the measured linewidth ⌫ tot but not with the measured ប / .
Here we show by performing ab initio calculations that in the unoccupied ⌫ ¯surface state on Pd͑111͒ the e-ph coupling parameter is very small, = 0.08, i. e., by a factor of 5 smaller than obtained at the Fermi energy of bulk Pd. 17 This reduces the calculated ⌫ e-ph to 19 meV and cannot explain the large difference between the experimental values for ប / and ⌫ tot at T = 450 K.In order to clarify this discrepancy we performed temperature-dependent linewidth measurements by time-resolved two-photon photoemission.We also carried out an ab initio evaluation of the e-e contribution to the lifetime broadening of the ⌫ ¯surface state.

II. DETAILS OF THE CALCULATION
The e-ph calculations were performed using densityfunctional theory and the local-density approximation for the exchange-correlation functional.Phonon frequencies and polarization vectors were obtained within the mixed-basis density-functional perturbation theory 18,19 using a normconserving scalar-relativistic pseudopotential. 20The planewave energy cutoff was 20 Ry and, additionally, one d-type localized wave function at each atomic site of Pd was employed.Integrations over the surface Brillouin zone ͑SBZ͒ were performed with a uniform mesh of 576 special points and a Gaussian energy smearing scheme with a width of 0.2 eV.The surface is presented by nine-layer Pd slabs separated by five layers of vacuum.This thickness is quite sufficient to avoid a strong interaction between the two surfaces of a slab: the splitting of the ⌫ ¯surface state consists of only 0.25 eV.For comparison with experimental data we average the energies of these two states.Atomic positions inside the Pd͑111͒ planes are fixed at the theoretical lattice parameter, a = 7.34 a.u., which is close to the experimental value. 21The calculated relaxation of the Pd͑111͒ layers shows a very small contraction of the outermost interlayer spacing ͑about 0.15%͒ relative to the bulk distance, that is, consistent with the experimental expansion of 0.9Ϯ 1.3%. 22he phonon-induced lifetime broadening of an electron state with momentum k and energy ⑀ k is related to the Eliashberg spectral function ␣ 2 F through the integral over all the scattering events that conserve energy and momentum 23 ͑we use atomic units, i.e., Here, f and n are the Fermi and Bose distribution functions, and m is the maximum phonon frequency.␣ 2 F E͑A͒ ͑k ; ͒ is the electron state-dependent Eliashberg spectral function corresponding to phonon emission ͑E͒ and absorption ͑A͒ processes.The state-dependent e-ph coupling parameter is just the first reciprocal moment of the Eliashberg function The evaluation of the e-e scattering contribution to the linewidth, ⌫ e-e , was performed within the first-principles GW self-energy formalism.Using the calculated ground-state electronic structure we evaluate ⌫ e-e of an electron ͑hole͒ in the state k ͑r͒ as Here G and GЈ are the reciprocal-lattice vectors, W G,G Ј ͑q , ͒ is the Fourier transform of the screened interaction W͑r , rЈ , ͒, and B k ,k ͑q + G͒ are coefficients More details of the GW calculation can be found in Ref. 5.

III. CALCULATION RESULTS AND DISCUSSION
The calculated energy of the unoccupied surface state at ⌫ īs 1.21 eV above E F ͑Fig. 1͒.Our measured value at 450 K is 1.31 eV with a temperature-dependent shift of 0.12 meV/K ͑see also Fig. 4͒.This yields at T = 0 a value of 1.26 eV in agreement with the calculation and previous results. 16First, we calculated exactly the electron-phonon matrix elements using 576 wave vectors q in the SBZ.Then, to check the convergence the summation was carried out on a denser mesh of 2304 q points.For the additional q points, the matrix elements were calculated using a Fourier interpolation scheme for the change in the self-consistent potential with respect to atomic displacements as well as for the dynamical matrices.
Figure 2 shows ␣ 2 F͑k ; ͒ for the unoccupied ⌫ ¯surface state.Since the contributions from phonon emission and adsorption processes nearly coincide, only the average  ␣ 2 F͑k ; ͒ is shown.We calculated for both split surface states at ⌫ ¯and obtained ⌫ ¯= 0.07 for the lower state and ⌫ ¯= 0.09 for the upper state.Extrapolating these results to the thick films ͑semi-infinite crystal͒ we take the average value of ⌫ ¯= 0.08.The obtained ⌫ ¯is very small and differs drastically from the e-ph coupling parameter averaged over momenta at the Fermi energy of the nine-layer film of Pd͑111͒ and of bulk Pd, E F = 0.4. 17s a rule, the strength of the e-ph interaction in a Shockley surface state is close to at the Fermi energy of the corresponding bulk material ͑see Table I͒.On the ͑111͒ surface of noble metals, the Shockley states lie just below E F and well inside the surface-projected band gap.As a result, their wave functions are mostly localized near the surface and the contribution to the e-ph coupling coming from surface-phonon modes is important. 26The Shockley state at Pd͑111͒ resembles much the ⌫ ¯surface state on Al͑001͒. 25oth states lying very close to the bottom of the band gap at ⌫ ¯are characterized by a deep penetration into the bulk.In both cases, the e-ph coupling is determined by bulk phonons and the contribution coming from the surface Raleigh modes is very small.It does not exceed 6% for the surface state on Pd͑111͒ ͑see Fig. 2͒.However, like in the case of noblemetal ͑111͒ surfaces, ⌫ ¯for the Al͑001͒ surface state is close to that at E F of bulk Al though the state is much deeper in energy. 25The strength of the e-ph coupling for the Shockley state on Pd͑111͒ is closer to that of image-potential states, especially, when the latter lies close to the band edge, e.g., on the Cu͑111͒ surface. 28Such negligible values of for the image states were attributed to a small penetration of the states into the bulk, e.g., for Pd͑111͒ the penetration of the first image-potential state is 4.6%. 16However, the situation with the unoccupied surface state on Pd͑111͒ is quite different.It is located almost completely ͑90%͒ inside the bulk.A plausible reason could be an abrupt decrease in the density of states above E F and, correspondingly, the number of electronic states which may participate in the e-ph coupling ͑see Fig. 1͒.Though such a relation is not evident.The values of DOS at the energies of the Pd͑111͒ surface state and the ⌫ surface electronic states on Al͑001͒ and Mg͑0001͒ do not differ substantially ͑ϳ2, 2.4, and 3.4 states/Ry/atom/spin, re-spectively͒ unlike the values of : 0.08, 0.51, and 0.30, respectively. 24,25On the other hand, the calculations for the Al͑001͒ surface showed that there was no direct relationship between the value of DOS and , e.g., the density of states at the energy of the surface state at the X ¯point is half as large than that at the ⌫ ¯point while X ¯= 0.78.
The evaluation of the e-e scattering contribution to the linewidth gives ⌫ e-e = 29 meV for the surface state.That is smaller than 37 meV obtained from the one-dimensional calculation. 16This difference can be attributed to use of ab initio electronic structure that includes d states just above E F in the electron transitions to final states and as well as in the screening.
Figure 3 shows the calculated linewidth of the unoccupied surface state as a function of temperature by a solid line.The temperature-independent linewidth, ⌫ e-e = 29 meV, has been added and is indicated by the dashed line.We also show in Fig. 3 two data sets for the intrinsic linewidth derived from two-photon photoemission.The data were obtained by exciting the unoccupied surface state by a photon with energy h = 1.55 eV from occupied states below the Fermi energy E F ͑see inset of Fig. 4and Ref. 16͒.A second photon with energy 3h = 4.65 eV lifts the energy above the vacuum level E vac .Electrons emitted along the surface normal are detected by a hemispherical energy analyzer. 16The kineticenergy scale in Fig. 4 refers to the analyzer and the work function can be obtained by adding the analyzer work function of 4.32 eV to the low-energy cutoff of the spectra.The decrease in −0.11 meV/ K is similar to values for other surfaces. 29The spectra in Fig. 4 show an intensity decrease and linewidth increase with temperature for the peak assigned to the Shockley surface state. 16Spectra were fitted using a Lorentzian for the intrinsic linewidth convoluted with a Gaussian accounting for the experimental resolution  of the analyzer and the bandwidth of the laser pulses.The experimental data in Fig. 3 can be fitted using the hightemperature approximation ⌫ e-ph =2kT ͑Refs.23 and 30͒ with = 0.38Ϯ 0.05.This value is much larger than the calculated one and would extrapolate the linewidth to negative values for T = 0.These variances can be resolved by including the electron scattering due to thermally activated defects following Ref. 30.This contribution is modeled by an added term ⌫ e-d = C exp͑−E a / kT͒ to the expression for the temperature-dependent linewidth.The activation energy E a and cannot be fitted simultaneously in the available temperature range, so we used the calculated value = 0.08.The resulting fits are shown as dotted and dash-dotted lines in Fig. 3.The obtained activation energy, 115Ϯ 13 meV is in the range of the values for Au͑111͒ ͑81 meV͒ and Al͑100͒ ͑170 meV͒. 30The value of the prefactor C = 390Ϯ 20 meV compares similarly.When the exponential factor is interpreted as defect concentration the obtained value corresponds to 4 meV linewidth increase for 1% defects.This value is similar to the linewidth increase due to adatoms for image-potential states on Cu͑100͒. 31easurements of the linewidth for the image-potential state on Pd͑111͒ as a function of temperature using a straight-line fit give also a rather high value for = 0.14.For the image-potential state on Cu͑111͒, = 0.06 has been measured which is attributed to a rather strong coupling to bulk bands at an energy near the band edge 28 due to the significant overlap of the image-potential state and bulk states wave functions.For Ag͑100͒ and Cu͑100͒ which have band gaps similar to Pd͑111͒ = 0.01 has been calculated 32 and no temperature dependence of the linewidth has been observed, 33 respectively.A consistent fit of the data for the imagepotential state on Pd͑111͒ assuming = 0 can be obtained using the thermally induced defect model with an activation energy E a =74Ϯ 15 meV and a prefactor C = 135Ϯ 12 meV.Because the image-potential state is measured with the available photon energy close to the Fermi edge the linewidths might be underestimated somewhat.This would lead in turn to a larger value for E a and explains the deviation to the value obtained for the Shockley surface state.
The linewidth extrapolated to T = 0 for the best sample preparations is 39Ϯ 5 meV which has to be compared to the calculated contribution from electron-electron scattering, ⌫ e-e = 29 meV.Adding the contribution from the electronphonon scattering, ⌫ e-ph = 4 meV at T = 0, to the calculated ⌫ e-e yields a reasonable agreement between the experiment and calculation.For the image-potential state a similar agreement is reached with a calculated value of 27 meV ͑Ref.16͒ compared to the experimental value of 24Ϯ 6 meV for the linewidth extrapolated to T = 0. Time-resolved measurements at 450 K ͑Ref.16͒ gave for the surface state ប / =54Ϯ 13 meV which includes besides ⌫ e-e also contributions from phonon-and possible defect-induced interband scattering from the surface state to bulk bands.The measured value is in good agreement with the calculated ⌫ e-e + ⌫ e-ph = 29+ 19 meV result.The measured lifetime broadening ប / for the surface state increases with increasing temperature.This observation as well as the comparison of the different linewidth contributions at 450 K indicate that phonon-and defect-induced scattering to bulk bands is about a factor of 5 smaller than the corresponding elastic intraband scattering.

IV. SUMMARY
We have presented the results of an ab initio study and time-resolved two-photon photoemission measurements of the electron linewidths for the Shockley surface state on Pd͑111͒.The calculations show that the e-ph interaction is mostly determined by coupling of bulk electronic states to bulk-phonon modes.This interaction results in an unusually small e-ph coupling parameter at the ⌫ ¯surface state due to a low density of bulk electron states at the surface-state energy.Time-resolved two-photon photoemission measurements taking into account thermally created defects confirm the calculated results.The kinetic energy is relative to the vacuum level of the analyzer.
The inset shows the energy diagram of the two-photon photoemission process.
FIG. ͑Upper panel͒ The Shockley surface electronic state ͑black circles͒ on Pd͑111͒.The gray background represents bulk Pd bands projected onto the two-dimensional Brillouin zone.͑Lower panel͒ The calculated total density of electronic states, DOS ͑solid line͒ and DOS of d states only ͑dotted line͒ for a nine-layer Pd slab.

FIG. 2 .
FIG. 2. Electron-phonon Eliashberg spectral function ␣ 2 F͑͒ for the ⌫ ¯unoccupied surface state.The contribution from the Rayleigh surface mode is indicated by hatched area.Phonon density of states for a nine-layer Pd slab, F͑͒, is shown by dashed line.

FIG. 3 .
FIG. 3. Temperature dependence of the lifetime broadening for the unoccupied surface state at ⌫ ¯.The solid curve shows the calculated electron-phonon contribution plus the electron-electron contribution ͑dashed line͒.The symbols present two experimental data sets with the corresponding fits including the contribution by thermally generated defects.

TABLE I .
Electron-phonon coupling parameters for surface electronic states at ⌫ ¯͑Refs.24-27͒.The surface-state energies E are shown with respect to the Fermi level.Also averaged over momentum at E F of bulk Al, Mg, Pd ͑Ref.17͒, and noble metals ͑Ref.23͒ are presented.
Two-photon photoemission spectra for the unoccupied surface state at the ⌫ ¯symmetry point for different temperatures.