Influence of hydrogen absorption on low-energy electronic collective excitations in palladium

V. M. Silkin,1 I. P. Chernov,2 P. M. Echenique,1,3 Yu. M. Koroteev,4 and E. V. Chulkov1,3 1Donostia International Physics Center (DIPC), Paseo de Manuel Lardizabal 4, 20018 San Sebastián, Basque Country, Spain 2Tomsk Polytechnical University, Prospekt Lenina 30, 634050 Tomsk, Russia 3Departamento de Física de Materiales and Centro Mixto CSIC-UPV/EHU, Facultad de Ciencias Químicas, Universidad del País Vasco, Apartado 1072, 20080 San Sebastián, Basque Country, Spain 4Institute of Strength Physics and Materials Science, Russian Academy of Sciences, Prospekt Academicheskii 2/1, 634021 Tomsk, Russia Received 26 June 2007; revised manuscript received 22 October 2007; published 6 December 2007


I. INTRODUCTION
The physical properties of metal-hydrogen systems have extensively been studied over many years. 1-3 These systems are interesting both from practical and theoretical points of view. They are important in energy-storage systems, in sensor devices, in catalysis, as well as in other fields of modern technology. The presence of hydrogen in metals significantly influences their transport, mechanical, electrical, magnetic, and other properties. The remarkable example of this is the palladium-hydrogen systems which attracted attention for many decades due to peculiar properties. 4 The change of physical properties of metals upon hydrogen absorption is closely related to strong modification of electronic structure of the materials. [5][6][7][8][9] In the case of palladium, the hydrogen absorption leads to three major effects in PdH x : ͑i͒ appearance of hydrogen induced electron states below valence d band, ͑ii͒ reduction of the width of valence d band, and ͑iii͒ sinking of this d band relative to the Fermi level with increase of H concentration that results in drastic reduction of density of states at the Fermi level at x ജ 0.6. The modification of electronic structure is reflected in excitation spectra which contain information on single-particle and collective excitations in the system. Extensive work on collective excitations in palladium has been done with the use of electronenergy-loss spectroscopy 5,10-13 ͑EELS͒ ͑see also data on optical spectra of palladium in Refs. 7 and 14͒.
The interpretation of experimental electron-energy-loss spectra is a rather difficult task, as the data can strongly be influenced by the surface excitations. Therefore, there is a noticeable uncertainty in the interpretation of experimentally observed peaks. 13 Nevertheless, the majority of experimental conclusions [11][12][13] and first-principles calculations [15][16][17] agree: The dominant peak found in the energy range from 7 to 8 eV is the bulk plasmon peak and the others are due to interband transitions ͑here, we refer to the valence electron energy range only, i.e., below the core electron excitations͒.
Momentum dependence of electron-energy-loss spectra in Pd has been studied by Netzer and El Gomani in EELS measurements. 18 They observed four features in the energy interval up to 10 eV. The dominant peak at ϳ6.6 eV was accounted for by the overlap of the surface and bulk plasmons. This peak shows weak dispersion to higher energy with increasing momentum. A shoulder at ϳ4 eV ͑interpreted as associated with interband transition͒ disappears already at small momenta. As to other broad features, no conclusion about their dispersion has been done.
In EELS measurements of collective excitations in PdH 0.8 , a shift of the dominant plasmon peak from around 7 eV in Pd to around 5 eV in PdH 0.8 has been found. 5 The significant change contradicts a simple picture based on a free-electron gas model where the only parameter determining the system property is an average electron density n o . This density is slightly higher in PdH 0.8 than in clean Pd. Following the free-electron gas model, this should lead to slight increase of the plasmon energy instead of almost 30% reduction observed experimentally. The electronic structure of pure Pd around the Fermi level is dominated by 4d states which form a strong peak in density of states at the Fermi level. The modification of electronic structure upon hydrogen absorption has been studied in a number of first-principles calculations; 8,9,[19][20][21][22] however, we are not aware of the corresponding ab initio calculations of the excitation spectra in palladium hydrides, although it was done for some other metal hydrides. [23][24][25] To our knowledge, theoretical investigations of anisotropic properties of the excitation spectra have not been reported either.
The purpose of the present paper is the ab initio study of the excitation spectra in pure Pd and PdH. The energy-loss spectra for both systems are investigated for three symmetry directions in the low-energy domain. The role played by crystal local-field effects is analyzed as well. The paper is organized as follows. In Sec. II, the calculation procedure is described. In Sec. III, we present the main results which include the loss spectra for both pure Pd and PdH systems. Finally, the conclusions are drawn in Sec. IV.

II. CALCULATION DETAILS
Within the first Born approximation, the inelastic scattering cross section of x rays and electrons for momentum transfer q and energy ប transferred to the system is proportional to the dynamical structure factor S͑q , ͒ of manyelectron systems. S͑q , ͒ is connected by the fluctuationdissipation theorem 26 to the dielectric function ⑀͑r , rЈ , ͒ which, in general, is a nonlocal energy-dependent function. For a periodic solid, S͑q , ͒ is where ⍀ represents the normalization volume, G's are the reciprocal lattice vectors, and Im͓⑀ G=0,G Ј =0 −1 ͑q , ͔͒ is the electron energy-loss function related to the Fourier coefficients of the density-response function G,G Ј ͑q , ͒ through

͑5͒
The Fourier coefficients G,G Ј 0 ͑q , ͒ have the form where the factor of 2 accounts for spin, the sums over n and nЈ run over the band structure for wave vectors k in the first Brillouin zone ͑BZ͒, f nk is the Fermi distribution function, nk and nk ͑r͒ are Bloch eigenvalues and eigenfunctions of the Kohn-Sham Hamiltonian, and is positive infinitesimal.
The numerical evaluation of G,G Ј 0 ͑q , ͒ matrix is the most time-demanding part of the calculations, and one should put some finite value for which introduces an additional broadening in calculated spectra. In practice, instead of direct use of Eq. ͑6͒, the matrix G,G Ј 0 ͑q , ͒ can be evaluated by performing calculations at imaginary 29 or complex 30 frequencies with subsequent analytical continuation to the real axis. Here, we use an alternative route calculating at the first step the spectral function matrix S G,G Ј 0 ͑q , ͒ as 31,32 with subsequent evaluation of the imaginary part of Here, sgn͑͒ =1 ͑−1͒ for Ͼ 0 ͑ Ͻ 0͒. The real part of the G,G Ј 0 ͑q , ͒ matrix elements is obtained from the corresponding imaginary part by the Hilbert transform.
The present calculations have been performed for fcc Pd and the stoichiometric palladium hydride PdH in rocksalt structure where the palladium atoms form a fcc lattice and hydrogen atoms occupy all the octahedral sites. For Pd, we use the experimental lattice parameters a Pd = 7.3512 a.u. Lattice parameter for PdH, a PdH , has been taken as 7.7921 a.u., similar to Ref. 8 and close to the experimental value. 33 This corresponds to the lattice expansion of 6% for PdH with respect to the pure Pd. The one-particle energies nk and wave functions nk ͑r͒ were obtained as a self-consistent solution of the Kohn-Sham equations with the use of exchangecorrelation potential in the form of Ref. 34. The electron-ion interaction was described by a nonlocal norm-conserving ionic pseudopotential for Pd ͑Ref. 35͒ and the hydrogen local ionic pseudopotential taken from Ref. 8. Wave functions nk ͑r͒ were expanded in plane waves up to a kinetic-energy cutoff of 70 Ry. Up to 40 reciprocal vectors G were included in the Fourier expansions of 0 , , and ⑀ matrices. Hence, the crystal local-field effects were incorporated in the evaluation of Im͓⑀ G=0,G Ј =0 −1 ͑q , ͔͒ through the inclusion of nondiagonal matrix elements of G,G Ј 0 ͑q , ͒ and matrix equation ͑5͒. In Eq. ͑7͒, the sum over k includes a 96ϫ 96ϫ 96 sampling that corresponds to 442 368 points in the BZ. The sums over n and nЈ included all energy bands up to energy of 50 eV above the Fermi level. The S G,G Ј 0 ͑q , ͒ matrices were calculated at the discrete mesh of energies ranging from 0 to 50 eV with a step of 0.05 eV. In the numerical calculations, the ␦ function in Eq. ͑7͒ was replaced by a Gaussian with a broadening of 0.1 eV. We solve Eq. ͑5͒ using a random-phase approximation, i.e., taking K xc = 0. It was demonstrated in Ref. 36 that for transition metals, this is a rather good approximation for low-energy electronic excitations, an energy domain of interest here.

A. Pd
As the basic ingredient of the dielectric matrix evaluation is the band structure, we plot it for Pd in Fig. 1. It agrees well with previous calculations, 8,9,37,38 although some small quantitative differences in energy positions owing to the use of different calculation methods can be noted.
In Fig. 2, we present the calculated imaginary part of the inverse dielectric matrix, Im͓⑀ G=0,G Ј =0 −1 ͑q , ͔͒, for pure Pd as a function of q and along the ͑100͒, ͑110͒, and ͑111͒ symmetry directions and for energies up to 15 eV. In order to enhance the fine structures in the low region, we scale Im ⑀ −1 by −1 . Whereas this kind of presentation distorts to some extent the shape of the features, nevertheless, the positions of peak structures along the axis are not affected and it gives a general idea about the Im ⑀ −1 behavior in the ͑q , ͒-phase space. Figure 3 summarizes the peak positions in the calculated loss spectra of Pd along the three symmetry directions. From the figures, one can appreciate that the excitation spectra in Pd along all the symmetry directions are dominated by the main broad feature centered at ϳ7.2 eV for the smallest momenta and at ϳ8 eV for q Ӎ 0.3 a.u. −1 The position of this peak at small q is in good agreement with other ab initio calculations 16,39 and with the majority of values obtained in numerous optical experiments 7,14 which, despite a rather large spread, place this peak around 7.3 eV. Our value of ϳ8 eV for larger wave vector is close to 8.2 eV of Ref. 40 and 8.4 eV of Ref. 15, where a significantly more scarce k point mesh was used in the 0 evaluation. Also, we find a fairly good agreement with electron-energy-loss spectra measurements which place a dominating feature around 7 eV. 13 For small q, this peak is almost nondispersing in ͑q , ͔͒ / ͑in eV −1 ͒ for pure Pd as a function of q and along the ͑100͒, ͑110͒, and ͑111͒ directions at top, middle, and bottom panels, respectively. Crystal local-field effects are included. The low-energy peaks were cut at 0.05 eV −1 . agreement with electron-energy-loss investigations, 10,12 but with increasing q, its absorption strength is gradually redistributed to the higher-energy peak. Thus, along the ͑100͒ direction, this nondispersing peak is dying at q Ӎ 0.45 a.u. −1 , whereas another feature appears at 7.5 eV for q Ӎ 0.15 dispersing almost linearly with q up to Ӎ9 eV at q Ӎ 0.4 a.u. −1 Along the ͑110͒ direction, the nondispersing part disappears at much smaller q ͑Ӎ0.3 a.u.͒, and the dominating part consists of a linear-dispersing feature which is clearly visible up to Ӎ 9 eV at q Ӎ 0.45 a.u. −1 Along the ͑111͒ direction, similar to the ͑110͒ the nondispersing peak disappears at q Ӎ 0.25 a.u. −1 , and the dominating feature disperses linearly up to almost 10 eV at q Ӎ 0.5 a.u. −1 . Whereas behavior of the lower nondispersing peak at Ӎ7.2 eV is in agreement with the energy-loss experiments, 10-12 the upper dispersing part was apparently not revealed in the experiments.
The dominant feature at ϳ7 eV in loss spectra is normally attributed to a bulk plasmon, although, taking into account the number of valence electrons in palladium, N v = 10, the bulk plasmon energy calculated within a free-electron gas model should be significantly larger. A close inspection of Fig. 1 reveals that three bands cross the Fermi level, making possible a great number of interband transitions with energies close to the bulk plasmon energy. Therefore, the freeelectron gas model is clearly not adequate in the case of Pd.
To address the role that intra-and interband transitions play in the formation of this plasmon peak, we have performed calculations taking into account only the intraband contributions in 0 . The calculation gives a Drude plasmon peak at 7 eV for small q = 2 a Pd ͑ 1 48 , 1 48 ,0 ͒ , which is in good agreement with the value of 7.1 eV obtained in optical calculations. 16,41 The excitation spectrum of Pd evaluated with the intraband transitions only is significantly modified by numerous interband transitions from the occupied d bands to the unoccupied bands. The most efficient transitions from the band of d symmetry should be to the states of p and f symmetries. The p states are presented in the low-energy unoccupied part of the palladium electron-energy spectra, whereas the f states are situated at significantly higher energies. This picture of transitions is corroborated by Fig. 4, where we place the imaginary part of ⑀ G=0,G Ј =0 ͑q , ͒ for q = 2 a Pd ͑ 1 8 ,0,0 ͒ ͑solid line͒ which is very similar to those reported previously. 16,42 This function reflects all one-particle transitions available in the electronic system and their strength. One can see that it does not contain sharp structures in the energy range from ϳ7 to ϳ 20 eV, whereas for higher energies, some peak features arise. Nevertheless, the enhanced values of Im ⑀ in this and upper energy regions do not change significantly the position around = 7 eV where the real part of ⑀ crosses the = 0 line. 16 This fact, together with the relatively low value of Im ⑀ at ϳ7 eV, leads to the appearance of a pronounced peak in the loss function.
Additionally, we have decomposed the total Im ⑀ spectra into the partial contributions from the initial bands i into the final bands f in the same fashion as it was done by Lässer and Smith, 42 and some of them are presented in Fig. 5. Comparing with the figures of Ref. 42, where detailed analysis of all transitions for these energies was performed, we observe rather good agreement with the data of Ref. 42. In Fig. 5, one can see that interband transitions from all the occupied bands ͑mainly of d symmetry͒ to the unoccupied part of band 6 ͑mainly of p symmetry͒ dominate contributions in the energy range from ϳ2 to ϳ 7 eV. The contributions from bands 3, 4, and 5 to band 7 ͑which has a large contribution of p character͒ determine Im ⑀ in the energy range from ϳ7 up to ϳ13 eV. Hence, the peak in the loss function at ϳ7 eV is influenced by charge fluctuations between the occupied d orbitals and the lowest unoccupied p orbital.
In addition to the main peak, in Figs. 2 and 3, a clear peak is seen for large momenta and → 0 along all three symme- try directions. It occupies large phase space along the ͑111͒ direction being less pronounced in the ͑110͒ direction. Indeed, in all the directions for → 0, Im ⑀ −1 also goes to zero, but due to the chosen representation in Fig. 2, ͑Im ⑀ −1 ͒ / diverges, signaling that Im ⑀ −1 tends to zero slower than . Some features corresponding to interband transitions can also be observed for large q at Ӎ4.3 and Ӎ5.5 eV for the ͑100͒ and ͑111͒ directions, at Ӎ2.5 eV along the ͑110͒ direction, and below 2 eV for all the directions. Also, in all the directions, some narrow peaks are present at 0.5 eV for small momenta dispersing linearly with increasing q. A closer inspection of this feature shows that it is more pronounced along the ͑111͒ direction ͑where another small peak appears even for smaller energy͒ and is less visible along the ͑110͒ direction. In general, all these low-energy features reflect a complicated electronic structure of Pd around the Fermi level.
In order to investigate the role played by crystal localfield effects in Pd, we plot in Fig. 6 the loss function evaluated without inclusion of crystal local fields, i.e., Im ͓⑀ G=0,G Ј =0 ͑q , ͔͒ −1 / . Its comparison with Fig. 2 shows that the main effect of the inclusion of crystal local fields in Pd is the smoothing of sharp interband features for large q without notable effect on the number and position of the peaks. This situation is similar to that observed in the 3d metal series. 36,43

B. PdH
Absorption of hydrogen causes significant modifications of the Pd band structure. 8,[44][45][46][47][48] In particular, hydrogen ͑1͒ lifts the Fermi level, ͑2͒ results in the formation of a mixed Pd-H band at the bottom of valence band, and ͑3͒ gives rise to new bands above E F . In Fig. 7, we show the evaluated band structure of PdH which is in good agreement with other calculations. 8,48 As seen from the figure, all palladium 4d bands in PdH are occupied and the Fermi level is crossed by the s-p band only. This results in a much simpler Fermi surface in PdH compared to pure Pd. Another visible change of electronic structure is the position of the lowest occupied band which is about 2 -3 eV lower than the bottom of the pure Pd spectrum. This change is more pronounced at the ⌫ point where the state has predominantly s character and less pronounced at the BZ borders, where it is mainly of d sym-  6. ͑Color online͒ Normalized energy-loss spectrum Im͓⑀ G=0,G Ј =0 ͑q , ͔͒ −1 / ͑in eV −1 ͒ for pure Pd as a function of q and along the ͑100͒, ͑110͒, and ͑111͒ directions at top, middle, and bottom panels, respectively. Crystal local-field effects are not included. The low-energy peaks were cut at 0.05 eV −1 . metry. Also an unoccupied band appears at ϳ10 eV at the ⌫ point which disperses downward toward the BZ borders. This leads to a significant reduction of a gap at the BZ borders between the top of 4d band and band 6 which has predominantly p symmetry. In pure Pd, this band is situated at ϳ24 eV at the ⌫ point. At the same time, comparison with Fig. 1 shows that the bands which have predominantly d symmetry are affected by hydrogen at much less scale.
The H modifications of band structure of clean Pd lead to dramatic changes in the response function. This can be seen in Fig. 8 where the imaginary part of the inverse dielectric function for PdH is presented for the three symmetry directions of BZ ͓again, as in the pure Pd case, we plot Im ⑀ G=0,G Ј =0 −1 ͑q , ͒ scaled by −1 to emphasize to the low-domain͔. The corresponding plot for the peak positions is shown in Fig. 9. The most prominent feature in the case of PdH is a dominant peak structure observed at 4.2 eV for small momenta in all the directions. This peak is almost nondispersing and has a large amplitude up to q Ӎ 0.35 a.u. −1 from where it starts to gradually lose its strength. Nevertheless, along the ͑100͒ direction, it is clearly distinguished up to q = 0.7 a.u. −1 . In Fig. 8, a well defined peak in ͓Im ⑀ −1 ͑q , ͔͒ / arises at ϳ3 eV for small q, although the corresponding amplitude of Im ⑀ −1 ͑q , ͒ is significantly lower than the amplitude of the main peak at 4.2 eV. This feature disperses upward almost linearly with q in all three directions and disappears at q Ӎ 0.2 a.u. −1 Even for lower energies, a weak peak is visible in the ͑100͒ direction. For small q, it is located at Ӎ1 eV and disperses upward linearly with increasing q. The downward shift of the main peak structure to 3 -6 eV region correlates very well with the experimental loss peak at ϳ5 eV. 5 As all the experiments have been performed for substoichiometric PdH x with x Ͻ 1, it would be desirable to perform EELS measurements for stoichiometric PdH samples for direct comparisons of ab initio and experimental results.
A low density of states at the Fermi level in PdH leads to dramatic reduction of the intensity of Im ⑀ −1 for small energies. Thus, instead of a large value of ͓Im ⑀ −1 ͑q , ͔͒ / for → 0, a significant depression is visible in PdH for all mo-menta and the threshold for interband transitions from occupied d bands at large momenta is observed at energies of 0.8, 0.6, and 0.5 eV for the ͑100͒, ͑110͒, and ͑111͒ directions, respectively. Additionally, two less pronounced nondispersing peaks appear at Ӎ 5 eV for q ഛ 0.3 a.u. −1 and Ӎ 6 eV for q ഛ 0.5 a.u. −1 . Along the ͑100͒ direction, a slightly dispersing upward feature appears at Ӎ 7 eV for q ജ 0.4 a.u. −1 . In the higher-energy region, a noticeable peak arises at Ӎ8.2 eV for q Ͻ 0.25 a.u. −1 in all the directions. A strong broad almost flat feature is located at Ӎ 10.5 eV. It is more pronounced along the ͑110͒ and ͑111͒ directions and has the maximum intensity for 0.2 a.u. −1 ഛ q ഛ 0.5 a.u. −1 . The strong structure appears for energies higher than 12 eV.
We calculated the bare plasmon energy ͑Drude peak͒ in PdH taking into account in the evaluation of 0 the intraband contributions corresponding to band 6 only. The Drude plasmon energy of 6.8 eV calculated for q = 2 a PdH ͑ 1 24 , 1 24 ,0 ͒ is only 0.2 eV smaller than the value obtained for Pd. Hence, it cannot explain the observed shift of plasmon peak from 7.2 eV in Pd to 4.2 eV in PdH. In order to investigate the origin of this significant downward shift of the low-energy dominant peak structure in PdH in comparison with pure Pd, we plot in Fig. 4 the Im ⑀͑q , ͒ for q = 2 a PdH ͑ 1 8 ,0,0 ͒ . One can see that in PdH in contrast to pure Pd, an additional sharp two-peak structure in Im ⑀ arises in the energy range from 7 to 14 eV. Due to interrelation between imaginary and real parts of the dielectric function via the Hilbert transform, this additional feature pushes the real part of the dielectric function to cross  11. ͑Color online͒ Normalized energy-loss spectrum Im͓⑀ G=0,G Ј =0 ͑q , ͔͒ −1 / ͑in eV −1 ͒ for PdH as a function of q and along the ͑100͒, ͑110͒, and ͑111͒ directions at top, middle, and bottom panels, respectively. Crystal local-field effects are not included. the = 0 axis at the much lower energy. In Fig. 10, we plot the contributions to Im ⑀ in PdH which are relevant for this energy range. Comparing Figs. 10 and 5, one can see the shift to higher energies of the low edge of all transitions to band 6. One can also see that the contribution due to the 1 → 6 transitions maintains, in general, its shape and strength, while the contributions from bands 4 and 5 are significantly reduced. These effects produce, in PdH, the drastic decrease of Im ⑀ for energies below 6 eV with respect to that in Pd, Fig. 4. The contributions from the i → 7 transitions are significantly stronger in PdH than in Pd, and they are shifted to lower energies by 2 -3 eV. These contributions strongly enhance Im ⑀ at 7 -14 eV, although some minor contributions result from the i → 8 and i → 9 transitions. The 1 → 7 transitions contribute for Ͼ 10 eV and therefore have less impact on Re ⑀ in comparison with the other i → 7 transitions. Therefore, one can account for the shift of the main peak in the PdH loss function to 4.2 eV as a result of fluctuating density oscillations between the occupied Pd 4d bands ͑only slightly modified by H 8 ͒ and the strongly modified unoccupied p band, although transitions from the lowest mixed Pd-H band to the unoccupied part of band 6 ͑of p character͒ have some impact. Figure 11 shows loss function for PdH without the inclusion of crystal local-field effects. The comparison of Fig. 11 with Fig. 8 reveals that, as in the case of pure Pd, the inclusion of these effects results in some changes of the amplitude of Im ⑀ −1 especially for large q but does not produce a significant effect on the peak positions. Nevertheless, the comparison of Figs. 2 and 6 and of Figs. 8 and 11 reveal that the crystal local fields modify the loss function shape in PdH stronger than in Pd.

IV. CONCLUSIONS
We have presented a theoretical study of electronic excitation spectra in Pd and PdH based on ab initio calculations of band structure of these materials. It has been shown that complicated band structures in both systems are responsible for the rich structure observed in the excitation spectra. Good agreement with available experimental optical and energyloss experiments and other theoretical studies is obtained for small momenta in Pd. The various peak features are obtained in Pd for momenta up to q = 0.7 a.u. −1 . All the calculated features show slight momentum anisotropy.
We have demonstrated that energy-loss spectra in PdH is strongly modified with respect to that in Pd for the entire q region studied. The main plasmon peak in the PdH loss function is situated at Ӎ4.2 eV and has almost flat dispersion in all three symmetry directions up to q = 0.7 a.u. −1 with gradual reduction of amplitude for q Ͼ 0.4 a.u. −1 . We have studied the origin of the pronounced shift of the main plasmon peak from Ӎ7.2 eV in Pd to Ӎ4.2 eV in PdH and found that it is mainly due to interband transitions in PdH from occupied Pd d bands to hydrogen-modified unoccupied p band, although some effect is due to transitions from the lowest occupied Pd-H band to the lowest p unoccupied band.
In both systems studied, the crystal local-field effects are shown to play only a minor role in the formation of the loss function.