Empirical electronic band structure study of silver low-index surfaces
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Empirical electronic band structure study of silver low–indexsurfaces
H.J. Herrera–Su´arez
Universidad de Ibagu´e, Facultad de Ciencias Naturales y Matem´aticas,Colombia, Carrera 22 Calle 67 Barrio Ambal´a
A. Rubio–Ponce
Departamento de Ciencias B´asicas,Universidad Aut´onoma Metropolitana–Azcapotzalco,Av. San Pablo 180, M´exico, D.F. 02200, M´exico
D. Olgu´ın
Departamento de F´ısica, Centro de Investigaci´on y de Estudios Avanzadosdel Instituto Polit´ecnico Nacional, M´exico, D.F. 07360, M´exico
Abstract
We studied the electronic band structure of the low–index fcc Ag surfaces (001), (110) and (111),by using the empirical tight–binding method in addition with the surface Green function matchingmethod. We report the energy values for different surface and resonance states and compare withthe available experimental and theoretical data. . INTRODUCTION In order to predict several surface and bulk crystal properties such as mesoscopic equi-librium shape, surface and catalyst activity [1], growth, creation of rungs and kinks [2, 3],is essential to have a detailed analysis of its electronic band structure. Experimental data iscomplemented with calculation to obtain a deeply understand of this systems. Two primarytypes of calculations are used in practice. The first one includes empirical and semi-empiricaltreatments, from wich empirical tight-binding (ETB) is one of the most transparent andwidely used methods, several recent applications of the ETB method can be found in theliterature, both for noble metal surfaces [3] and elemental semiconductors [4], as more com-plex systems like the layered II-IV compunds [5]. The second type are the first-principlecalculations based on approximate methods such as density functional theory (DFT), theseare located on the most reliable and widely used methods nowadays. In this work the elec-tronic band structures of ideal Ag (001), (110) and (111) surfaces are discussed. This is acontinuation of an extensive study of different noble and transition metal surfaces [6].
II. NUMERICAL APPROACH
The ETB calculations were done using a minimal orthogonal basis set. Here, a set of nine“ s p d ” atomic orbitals per atom in the unit cell were used, and we have included the firstnearest and next nearest neighbors as proposed by Papaconstantopoulos [7]. The parametersof the model are those used by Papaconstantopoulos, it is known that these parametersproperly reproduce the bulk electronic properties of Ag, according to DFT calculations [7].To calculate the surface electronic band structure, the Surface Green Function Matching(SGFM) method was used, as suggested by Garc´ıa–Moliner and Velasco [8]. The SGFMmethod, in conjunction with the ETB approach, was used successfully to study transitionmetals [9] and semiconductor surfaces [10]. For a complete formulae of the ETB method tothe formalism of the SGFM see details in Refs. [8, 10]. A recent application of the methodto the study of the electronic structure of different Pt surfaces has been done. [11]From the knowledge of the Green function, the surface and the surface resonances statescan be calculated from the poles of the real part of the corresponding Green function. In asimilar way, from the imaginary part the local density of states (LDOS) can be obtained.2
II. RESULTS AND DISCUSSIONA. Local density of states
Figure 1 shows our calculated LDOS projected on the surface (broken line) and the LDOSprojected on the bulk (full line), see figure caption for details.For the Ag(001) surface a total of 528 k-points were used, here we used the Cunninghammethod to found the k-point set in the irreducible two dimentional segment of the firstBrillouin zone (2D-SBZ) (see inset of Fig. 2(a)) [12]. For Ag(110) surface we used 256Cunningham points in the 2D-SBZ (see inset of Fig. 2(b)). While for the Ag(111) surfacewe used 136 Cunningham points in the 2D-SBZ (see inset of Fig. 2(c)).Table I shows the s, p, d atomic orbital partial contribution to the surface and bulkLDOS. A comparison of our calculations with that reported by Papaconstantopoulos [7]shows that our method reproduces properly the bulk DOS.
TABLE I: spd atomic partial contribution to the Local Density of States (LDOS) at the surface(surf) and bulk (bulk) at the Fermi Level (states/eV/atom), for the different oriented surfacesstudied in this work Surface s surf p surf d surf s bulk p bulk d bulk Ag(001) 0.600 0.344 9.901 0.671 0.448 9.879Ag(110) 0.674 0.454 9.869 0.650 0.350 10.010Ag(111) 0.670 0.440 9.870 0.650 0.350 10.010 B. Ag (001)
1. Projected bulk electronic band strucure, surface– and resonance–states
In an early experimental work Kolb et al. [13], by using electroreflectance in the infraredfrequency range, reported two surface states (SS) for Ag(001) located in the ¯ X high symme-try point of the two dimensional Brillouin zone: the first SS was reported at 3.1 eV abovethe Fermi level, while the second state was reported a few meV below the Fermi level (seeFig. 2 on Ref. [13]). In the same work it was found that both states were in good agreement3 L D O S ( s t a t e s / e V / a t o m ) Energy (eV) Ag (001) bulk surface -8 -6 -4 -2 0 2 4 6 8 1001020 L D O S ( s t a t e s / e V / a t o m ) Energy (eV) Ag(110) bulk surface -8 -6 -4 -2 0 2 4 6 8 10020406080 L D O S ( s t a t e s / e V / a t o m ) Energy (eV) Ag(111) bulk surface
FIG. 1: (Color online) Bulk local density of states (LDOS) (black line) and surface LDOS (redline) obtained from the SGFM method. (a) Ag(001)–, (b) Ag(110)–, and (c) Ag(111)–surface. Thezero of energies represents the Fermi level. with ab initio pseudopotential calculations.Then, Altmann et al. [14] by using angle-resolved Bremsstrahlung isochromat spec-troscopy corroborate the states found by Kolb et al. ,[13] and reported other high energysurface states.More recently, Savio et al. [15] by using an ab initio pseudopontential calculations com-bined with ultraviolet photoemission spectroscopy technique refined the previous experimen-tal values for the surface states.Figure 2 shows our calculated projected bulk band structure and the surface– andresonance–states (RS) obtained from the poles of the real part of the bulk Green functionand from the surface Green function, respectively. As we found, there are two SS labeled Es and Es (full points), and three resonance states labeled Er , Er , and Er (empty points).The shadow zones represent the calculated bulk band structure projected on the (001) sur-face. We found that our calculated projected bulk bands are in good agreement with thatreported in Refs. [13, 15]. Tables II and III list our calculated energies for the SS and RSfound in this work and compare them with other reports. The states E S and S S reportedin the literature, are listed in Table II although our calculations do not reproduce thesestates.
2. Detailed discussion of the found SS and RS
As we have found the SS Es is located at 4.43 eV at ¯Γ showing a parabolic dispersion.The state is located in the ¯Γ − gap which has a width of 5.03 eV, approximately, according4 -20246810 Wavevector ( k ) Er3 Er3Er2Er1Es3 Es1Es1 __ _X M Γ E F Ag(001) _ _ _ Σ Y ∆ ____ Γ MX Γ E ne r g y ( e V ) -10-8-6-4-20246810 Er5 X Er3 E F Er4 Er2Er1Er2 Es7Es6Es5Es3Es2Es1Er1 _ ___ SY X Γ X Ag (110) _____ Γ YS Γ E ne r g í a ( e V ) -10-8-6-4-20246810 _ __MK Γ Vector de onda )( → k Er5Er4Er3Er2Er1 Es9Es8Es3Es2 E F Ag(111) ____K KM Γ E ne r g í a ( e V ) FIG. 2: (Color on line) Projected bulk band structure found in our calculation (blue zone). The full(red) points represent the SS, while the empty points are for the RS. (a) Ag(001)–, (b) Ag(110)–,and (c) Ag(111)–surface. The Fermi level is the zero of the energies.TABLE II: Surface states for Ag(001). The first column is the found SS, the second column listthe k -point where the SS was found, the third column shows the experimental energy reported forthe related state, the fourth list the theoretical energy value reported in the literature for the SS,the next column shows our calculated energy value for the found SS, finally the last column showsthe symmetry of the atomic orbitals that form the SS, according to our calculation. The k − vectoris given in units of (cid:2) πa (cid:3) , while the energies are in eV.SS ~k E exp E theo E our SFO our Es (0,0) 4.00[14] 4.43 s, p z Es (1 ,
1) 0.07[15] -0.13[13] 0.96 p x , p y -0.15[15]3.80[17]Es (1,1) 3.50[14] 3.50[13]3.03[15] 2.99[15]Es (1,1) − . − . to our calculations this state is the hibridization of the s, p z atomic orbitals. This SS wasmeasured at 4.0 eV by Altmann et al. [14] 5 ABLE III: Resonance states for the Ag(001) surface. The first column shows the labeled resonancestate, the second one shows the wavevector of the state in units of [ πa ], the next column shows therelated energy in eV, finally the last column shows the wave symmetry found for the differentstates. E r ~k E our SFO our Er (2,0) –4.06 d z − r Er (2,0) –3.18 d xy Er (0,0) –4.40 d z − r We found that the SS E S is located at 0.96 eV at ¯ X showing a parabolic dispersion,the state is located in the ¯ X − gap which width is 3.87 eV. We found that this state has thesymmetry of the p x , p y wavefunctions. The state was theoretically reported by Kolb et al. [13], and was recently mesured and calculated by Savio et al. [15]The RS Er was found at − .
06 eV in the (cid:12)(cid:12) ¯X ¯M (cid:12)(cid:12) direction and shows almost zero disper-sion. We conclude that the state is the hibridization of the d z − r atomic orbitals.The RS Er begins at − .
18 eV in the ¯ M point showing a small dispersion, as we havefound that the character of this state is d xy .The RS Er is located at –4.40 eV at ¯Γ, the symmetry found for this RS is d z − r .The SS’s found in this work for Ag(001) Es and Es were reported previously in theliterature, and we have found that our calculated energy values for these states show goodagreement with published values, as we show in Table II. While the new states found in thiswork are the resonant states E r , E r , and E r . On the other hand, we do not found the SSE S and E S listed in Table II and reported in the Refs. [14, 15, 17]. C. Ag(110)
1. Projected bulk band structure, surface– and resonance–states
In an early ab initio calculation Ho et al. [16] reported the projected bulk band structureand SS of Ag(110), and predicted a SS on the upper band energy region of the 2D BZ (seeFig. 1(b) in Ref. [16]). After that, Reihl et al. [18] using the experimental technique of k − resolved inverse photoemission spectroscopy found an unoccupied SS at the energy of6.65 eV for Ag(110), that matches in good agreement with the state predicted by Ho et al. ,in the energy gap around the ¯ X point of the 2D-BZ (see Fig. 2 in Ref.[18]).Figure 2(b) shows our calculated projected bulk band structure for the Ag(110) surface,wich shows eight SS (full points) and six RS (empty points). The details of the calculatedsurface– and resonance–states are showed in Tables IV and V.
2. Detailed discussion of the found SS and RS
As we found the state Es , that appears in the upper energy gap located at ¯ X , beginsat 5.21 eV and ends at 4.19 eV in the interval ¯Γ − ¯ X − ¯ S , crossing the ¯ X point at 2.89 eV,according to our calculations the state has p x symmetry of the atomic orbitals. Althoughthere is not experimental evidence for this SS, from a theoretical point of view this state hasbeen reported in the Refs. [13, 16, 19]. We found that the calculated energy for this stateis within the reported values (see Table IV).Moreover, we found another SS in the same gap energy (E S ). The state shows fewdispersion, and it is located around 6.2 eV on the ¯ X point. Our calculated wavefunction forthis state has the s, p z symmetry of the atomic orbitals. From the experimental point ofview this SS was reported by Altmann et al. [14] and was found around 5.0 ± et al. , [16] was located at 4.25 eV .Also, we found that the state Es shows a great slope, and it is located in the ¯ X − ¯ S interval at energies that range from 8.0–10.0 eV. The wavefunction for this state has the p y − p x symmetry of the atomic orbitals. We do not find experimental evidence for this SS,although the state was reported in a theoretical work by Tjeng et al. [19]The SS Es is located at 6.05 eV in the ¯ S point and shows a great slope, the state wasfound in a energy gap of approximately 0.54 eV. As we found the state is an hybridizationof the s, p z atomic orbitals.In the energy gap at ¯ Y point, we found two SS showing a parabolic shape. The lower state(E S ) is located at 1.46 eV. Our wavefunction for the E S state has the p y symmetry. This SSwas found experimentally in Refs.[14, 19, 20], and theoretically in Refs. [13, 16]. The upperstate (E S ) is located at 3.1 eV, and the state has the symmetries s, p z . Experimentally thisstate was reported in Refs.[14, 18, 19], and theoretically in Ref.[16]. The energy differencefound betwen our calculation and the reported values can bee seen in Table IV.7 ABLE IV: Calculated energy values for SS on the Ag(110) surface. The first column is thefound SS, the second column list the k -point where the SS was found, the third column showsthe experimental energy reported for the related state, the fourth list the theoretical energy valuereported in the literature for the SS, the next column shows our calculated energy value for thefound SS, finally the last column shows the symmetry of atomic orbitals that form the SS, accordingto our calculation. The k − vector are given in units of (cid:2) πa (cid:3) , while the energies are in eV.surface state ~k E exp E theo E our SFO our Es ( √ − p x ( √ ,
0) 4.86[14] 5.26[19] 6.16 s, p z Es ( √ , .
4) – 8.06[19] 9.22 p y,z Es ( p (2),1) – – 6.43 s, p z (0,1) –0.10[20] 1.15[13] 1.46 p y s, p z Es (0,1) 1.65[18]1.95[19] In the VB region we found four RS, and according to our calculations the states are: 1)The RS Er is located at ¯Γ at -5.12 eV, and its wavefunction has the full symmetry of the d atomic orbitals. From the theoretical point of view, the state was reported in Ref. [16].2) The RS Er is located at ¯Γ at –4.17 eV, and its wavefunction has also the full symmetryof the d atomic orbitals. The state was calculated in Ref. [16]. 3) The RS Er is located at-3.40 at the ¯ S point, and its wavefunction has the symmetry of the d x − y atomic orbitals.Theoretically the state was reported in Ref. [16]. 4) The RS Er is found at -3.50 eV at ¯ Y point, and its wavefunction is an hybridization of the d xy , d zx atomic orbitals.Then, in the CB we found the RS Er , the state is located at 8.20 eV in the ¯ S point, andit wavefunction has the p y symmetry. The state was calculated in Ref. [16].8 ABLE V: Calculated energy values for the RS of the Ag(110) surface. The first column showsthe labeled resonance state, the second one shows the wavevector of the state in units of [ πa ], thenext column shows the related energy in eV, finally the last column shows the wave symmetryfound for the different states. E r ~k E theo E our SFO our Er (0,0) –4.74[16] -5.12 d Er (0,0) –3.76[16] -4.17 d Er ( √ d x − y Er (0,1) –3.48[16] -3.50 d xy,zx Er ( √ p y Er ( √ ,0) 6.77[19] 6.16 s, p x Er (0,0) -5.46 d yz,x − y Er (0,0) 7.31 p z The RS Er is located at ¯Γ at -5.46, and its wavefunction has the hybridization of the s, p x atomic orbitals. This state was also calculated in Ref. [16].The RS Er is located at ¯Γ at 7.31 eV, and has the symmetry of the p z atomic orbitals.As we found our calculated SS and RS for Ag(110) Es , Es , Es , Es , Es , Er , Er , ..., Er , have been reported previously in the literature, and in the usual precision of themethod, show good agreement with them, as we can see in Tables IV and V. The new statesfound in this work are Es , Es , Es , Es , Es , Er y Er . D. Ag(111)
1. Projected bulk band structure, surface states, and resonance states
In an early calculation Ho et al. [21] by using the ab initio pseudopotential method,reported the bulk band structure, SS and RS for Ag(111). They predicted a SS just abovethe Fermi level in the ¯Γ point. Then in a photoemission measurement Kevan and Gaylord[22] corroborated the existence of this state at ¯Γ and present a discusson of it (see Fig. 5 inRef. [22]). 9igure 2(c) shows our calculated projected bulk band struture for Ag(111). There weidentify four SS labeled Es , Es , Es , and Es , and five RS labeled Er , . . . , Er . Ingeneral, from the figure we observe that the calculated projected bulk bands structure arein agreement with that reported in Ref. [21]. The characteristics for the different found SSand RS are shown in Tables VI and VII.
2. Detailed discussion of the found SS and RS
The SS Es is located in the lower gap at –6.08 eV at ¯Γ. The state shows a parabolicshape and its wavefunction is the hybridization of the s, d z − r atomic orbitals.Our found SS E S was calculated at 2.28 eV at ¯Γ, showing an important dispersion,its wavefunction has the symmetry of the s, p z atomic orbitals. From the experimentalpoint of view a similar SS was reported by Altmann et al. [14] and Reihl et al. [23] as anunoccupied state, while Kevan and Gaylord et al. [22] reported the state as an occupiedstate. Theoretically in Ref. [21] the state was identified as an occupied state.There are a couple of SS E S and E S located in the upper gap at ¯ K . E S was foundat 6.63 eV and its wavefunction has the symmetry of the p x , p y atomic orbitals. E S wascalculated at 6.97 eV, and its wavefunction has the symmetry of the s, p z atomic orbitals.The RS Er was calculated at -6.42 eV at ¯Γ, and we found that the state has the symmetryof the d z − r atomic orbitals. Although there is no experimental evidence for this RS,theoretically the state was predicted in Ref. [21] (see Table VII for energy comparison).In the energy bulk bands above the lower gap at ¯Γ, we found the state Er at -5.04 eV, thecalculated wavefunction for this RS was the hybridization of the d xy,x − y atomic orbitals.The state was reported in Ref. [21].The state Er was calculated at -6.23 eV at ¯ M point, and has the character of the d orbitals. Theoretically, this state was predicted in Ref. [21].The state Er was calculated at –3.82 eV at ¯ M point, and its wavefunction has thesymmetry of the d xy,yz,zx,x − y atomic orbitals. Theoretically, the state was reported inRef. [21].The state Er was calculated in the lower border of the upper gap located in the ¯ M point.The state was located at 2.08 eV, and has the symmetry of the s, /p x , p y atomic orbitals.The states Es , Er , Er , Er y Er have been reported previously, and we found that our10alculations agree with the reported values, with the exception of the energy values for E S .The new states found in this work are E S , and Er . TABLE VI: Surface states for the Ag(111). The first column is the found SS, the second columnlist the k -point where the SS was found, the third column shows the experimental energy reportedfor the related state, the fourth list the theoretical energy value reported in the literature for theSS, the next column shows our calculated energy value for the found SS, finally the last columnshows the symmetry of atomic orbitals that form the SS, according to our calculation.Es ~k E exp E theo E our SFO our Es (0 ,
0) -6.08 s, d z − r (0,0) 0.33[23] -0.31[22] 2.28 s, p z -0.12[22]Es (cid:0) √ , (cid:1) . p x,y Es (cid:0) √ , (cid:1) . s, p z TABLE VII: Resonant states found for Ag(111) surface. The first column shows the labeledresonance state, the second one shows the wavevector of the state in units of [ πa ], the next columnshows the related energy in eV, finally the last column shows the wave symmetry found for thedifferent states. Er ~k E te´orico E our SFO our Er (0 , − .
26 [21] − . d z − r Er (0 , − .
04 [21] − . d xy,x − y Er ( √ , √ ) − . − . d Er ( √ , √ ) − . − . d xy,yz,zx,x − y Er ( √ ,
0) 2 . s, p x , p y
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