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Electronic band structure of silver low-index surfaces: a tight-binding study
Journal: Canadian Journal of Physics Manuscript ID cjp-2019-0218.R1
Manuscript Type: Article Date Submitted by the
Author: 14-Aug-2019
Complete List of Authors: Herrera–Suárez, H.J.; Universidad de Ibague, Facultad de Ciencias Naturales y Matemáticas
Rubio-Ponce, A.; Universidad Autónoma Metropolitana Azcapotzalco, Ciencias Básicas
Olguin, D.; Cinvestav Departamento de Física, Física
Keyword: Electronic structure of surfaces, Transition metals, The LCAO method, SIlver, Surface states Is the invited manuscript for
consideration in a Special
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Electronic band structure of silver low–index surfaces: a
tight–binding study
H. J. Herrera–Su´arez
Universidad de Ibagu´e, Facultad de Ciencias Naturales y Matem´aticas,
Carrera 22 Calle 67 Barrio Ambal´a, Ibagu´e, Tolima. Colombia.
A. Rubio–Ponce∗
Departamento de Ciencias B´asicas,
Universidad Aut´onoma Metropolitana–Azcapotzalco,
Av. San Pablo 180, Ciudad de M´exico, 02200, M´exico.
D. Olgu´ın
Departamento de F´ısica, Centro de Investigaci´on y de Estudios Avanzados
del Instituto Polit´ecnico Nacional, Ciudad de M´exico 07360, M´exico.
(Dated: September 20, 2019)
Abstract
We studied the electronic band structure and the corresponding local density of states of low– index fcc Ag surfaces (100), (110) and (111), by using the empirical tight–binding method in the framework of the Surface Green’s Function Matching formalism. The energy values for different surface and resonance states are reported, and comparison with the available experimental and theoretical data is also done.
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I. INTRODUCTIONFrom fundamental physics, the study of the ideal low index surfaces of different noble metals is of particular importance since they help us to test other theoretical and exper-imental investigations of the electronic and surface structure. Detailed knowledge of free surface properties are relevant to understand several phenomena, such as catalysis, sur-face reconstruction, sursur-face impurities (adsorption), which play a crucial role elucidating the chemistry, magnetism, surface reactivity, crystal growth, the creation of steps and kinks that can be used to construct nanosystems displaying desired optical or electronic properties [1–9].
Not long ago, the Ag surfaces have attracted the experimental attention once again, and early works have been reviewed [8–15]; at the same time and to help in the interpretation of the different experimental results, specialized and complicated calculations have been done [16–21]. However, a full explanation of the experimental data is yet to be completed. For this reason, we consider that for an initial interpretation of the results it could be useful to have a simple approach to help with it.
In practice, two primary calculation types are used: The first type includes empirical methods, from which tight–binding (TB) is one of the most transparent and widely used; where recent applications of the TB method can be found in the literature, the same for metal surfaces [5] and elemental semiconductors [22], as well as for more complex systems like the layered transition metal dichalcogenides [23], the layered III–IV compounds [24], quantum dots and wires [25–27]. A second type involves density functional theory (DFT) that are between the reliable and widely used methods nowadays. At present, the TB method is mainly used to model band structures, especially useful for large structures, while DFT is limited for this kind of systems. We can even do simulations with the TB method in real time; it is easy and simple to implement for the study of low dimensional systems, reducing greatly computational effort, without losing reasonable accuracy. By using the TB method, one can easily diagonalize a reasonable size matrix to obtain the electronic properties of a given material. This avoids the complexity of the DFT techniques, where considerable effort is needed to use them efficiently and reliably. Hence, the TB method is easily accesible for the analysis of experiments. There is an evident importance in this approach: its simplicity and transparency allow us to have a better understanding of the involved concepts. These
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are some reasons why the TB method is applicable to large and complex systems, where by using a minimum set of parameters, we are able to describe and compare our results with experimental data.
In this work and as a continuation of an extensive study of different noble and transition metal surfaces, the electronic band structures of ideal Ag (100), (110) and (111) surfaces are discussed. Here, we try to keep the numerical approach as simplest as it is possible.
The rest of the paper is organized as follows: In Section 2, we present the numerical approach. Results will be presented and discussed in Section 3. Finally, Section 4 presents our conclusions.
II. NUMERICAL APPROACH
The TB calculations were done using a minimal set of parameters in terms of two–center integrals together with an orthogonal basis within the Slater–Koster formalism [28]. Here, a set of nine s, p, and d atomic orbitals per atom in the unit cell were used, where in our approach first nearest– and next nearest–neighbors interactions were taken into account. The parameters of the model are those proposed by Papaconstantopoulos [29], where a
total of 24 parameters were used: four of them are the on–site s, p, and dx2−y2, d3z2−r2
parameters for the atomic orbital energies, ten parameters are used to describe the nearest neighbor interactions, and the last ten parameters for the next nearest neighbor interactions. As it is discussed in Ref. [29], the parameters are obtained by a least–squares fitting to self– consistent APW calculations that include the scalar relativistic effects (Darwin and mass– velocity terms), it is known that the obtained parameters reproduce the bulk electronic energies of Ag within an accuracy of 0.04 eV [29].
To study the electronic properties of the surfaces, we will use the Surface Green Func-tion Matching (GFM) method. Firstly, we introduce the infinite periodic medium Green’s function
G(ε, k) = (εI − H(k))−1, (1)
where k is the crystal momentum, ε is the quasiparticle energy, and H the Hamiltonian for the infinite periodic system with translational symmetry. Periodic boundary conditions are assumed. Then, for a semi–infinite medium we introduce the surface Green’s function
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as [30]: GS = G − GGB−1G + GG −1 B G −1 S G −1 B G, (2)where the first term on the right hand side is just the Green’s function for the infinite medium. The second and third terms describe the total effect of introducing a surface into an infinite medium. The second term describes the hard wall effect. The third term describes the new states, the new solutions due to the matching conditions, and gives rise to surface states that decay a few atomic layers away from the surface.
We have calculated the bulk–projected (GB) and the surface–projected (GS) Green’s
func-tions, given in Eq. 2. From GB we can obtain the effects on the band structure derived from
the hard wall effect (second term contribution above) and from GS we obtain the surface
band structure (third term contribution). To give continuity to the paper, here we just introduce the surface Green’s functions, extended formulae are given in the Appendix, a detailed derivation of the model can be found in Ref. [30]. The projected surface and bulk Green’s functions, in the language of the TB method are given as:
Gs−1(ε, k) = (εI − H0,0(k)) − H0,1(k)T, (3)
Gb−1(ε, k) = Gs−1(ε, k) − H1,0(k) ˜T . (4)
where ε is the energy, I the unit matrix, T and ˜T are transfer matrices that depend on
the energy through H0,0, H0,1 and H1,0, the intralayer and interlayer interaction
Hamiltoni-ans [31] (for complete formulae see Appendix).
In eq. 3, the first term gives the secular matrix for the surface layer. The nondiagonal
matrix element of the Hamiltonian, H0,1, introduces the coupling of the surface layer with
the rest of the crystal, and T transfers this information into the bulk. While in Eq. 4, ˜T
describes propagation in the opposite direction, and H1,0 brings the coupling of the semi–
infinite medium to the surface.
The SGFM method takes into account the perturbation caused for the surface exactly, the TB parameters used for the surface are those used for the bulk, where the difference between the bulk and surface TB parameters are taken into account by the matching of the surface Green’s functions [30].
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The presence of the surface modifies the boundary conditions, altering the energy spec-trum and generating new states. These states appear only in a few atomic layers nearby to the surface, their wave functions decay exponentially into the volume (surface states). Also, there are other states that do not decay and propagate into the bulk (resonant states). The states have two–dimensional character and can only be detected by photoemission. These effects related to the presence of the surface can be described correctly in the Green’s functions formalism.
From the knowledge of the Green’s function, the surface states (SS) and the resonance states (RS) can be calculated from the poles of the real part of the corresponding Green’s function. Similarly, from the imaginary part of the proper Green’s function, the correspond-ing density of states can be obtained by uscorrespond-ing the well known formula
N (ε, k) = −1
πIm[Tr G(ε, k)], (5)
then, the local density of states (LDOS) can be obtained from the surface projected Green’s function, integrating the density of states in the two–dimensional first Brillouin zone (2D–FBZ) using the Cunningham’s method [32].
The SGFM method, in conjunction with the TB approach, has been used successfully to study transition metals [33–39], and semiconductor surfaces [40–45]. For a complete formula of the TB method to the formalism of the SGFM see details in Refs. [30, 40–45]. A recent application of the method for the study of the electronic structure of different Pt surfaces has been done [47, 48].
III. RESULTS AND DISCUSSION
A. Local density of states
As we have found, our calculated LDOS projected on the bulk compares very well with that reported by Papaconstantopoulos [29], showing that our method reproduces properly the bulk DOS. Table I shows the s, p, and d atomic orbital partial contribution to the surface and bulk LDOS.
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The numerical integration, in the 2D–FBZ, to calculate the LDOS for the different studied surfaces was realized by using the Cunningham’s method [32]. Where for the Ag(100) surface we used 528 Cunningham’s k–points, for the Ag(110) surface we used 256 Cunningham’s k–points, and for the Ag(111) surface we used 136 Cunningham’s k–points.
The calculated LDOS projected on the surface (full black line), and the LDOS projected on the bulk (broken red line) are shown in Fig. 1. As we can observe, the valence band bandwidth of the projected LDOS for the three oriented surfaces is very similar, the same for the bulk as well as for the surface. Our calculated bandwidth for the lower VB states is almost 3.5 eV for the three oriented surfaces. As it is also clear, significant differences between the projected surface and the bulk LDOS were obtained. From these differences, information about the surface– and resonance–states will be found. As we have commented in previous work [46–48]: “The electronic states at the surface of a single crystal are strictly two–dimensional and show a dispersion that significantly differs from the bulk states. It is convention to distinguish between surface states, which lie in the projected bandgap, and surface resonances, which lie within the bulk band structure projected on the surface [49]. These states are obtained in the projected bulk bands when the translational symmetry of the crystal is broken to create the surface”.
However, to get a better description of the different obtained SS and RS, a detailed discussion of the calculated projected bulk band structure (PBBS) of each oriented surface will be necessary.
B. Projected bulk bands structure, surface states, and resonance states
We have calculated the electronic band structure of ideal surfaces. The surface–states and the resonance–states are electronic states found at the surfaces of the solid. The surface states are characterized by energy bands that are not degenerate with the bulk energy bands, and only exist in the forbidden energy gaps. At energies for which the surface and bulk states are degenerate (i.e., where the surface states and the bulk states are mixed), a surface resonance forms. Such states can propagate into the bulk, similar to Bloch waves, and can retain enhanced amplitudes near the surface [47, 48]. The calculated SSs and RSs, as well as the projected bulk bands of the studied surfaces, are shown in Figs. 2–4.
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PBBS are represented by small black dots, red points represent the SSs, while blue points do it for the RSs. Some local energy gaps can be observed in the PBBS. The SSs are expected in these local gaps, whereas the RSs must be observed in the continuum of the PBBS.
1. Ag(100)–surface
In an early experimental work Kolb et al., by using electroreflectance in the infrared
frequency range, reported two SS for Ag(100) located in the ¯X high symmetry point of the
two dimensional Brillouin zone: the first SS was found at 3.1 eV above the Fermi level, while the second state was estimated to occur at 0.73 eV [50]. In the same work, it was found that both states were in good agreement with ab initio pseudopotential calculations. Where, in their calculations only the s and p valence electrons are used in the pseudopotential, and the d electrons were not included.
Then, Altmann et al., [51] by using angle–resolved Bremsstrahlung isochromat spec-troscopy corroborated the states found by Kolb et al., [50] and reported other high energy surface states.
More recently, Savio et al., [52] by using ab initio pseudopotential calculations, on the one hand, and the ultraviolet photoemission spectroscopy technique, on the other, refined the previous experimental values for the SS. Here, the ab initio calculations were done using the LDA approach for the exchange–correlation part of the total energy, and the authors used the known ultrasoft pseudopotentials, obtained in the Vanderbilt scheme, where the Ag–4d states were explicitly included.
Figure 2, shows our calculated projected band structure, the SS and RS were obtained from the poles of the real part of the Surface Green’s function and the bulk Green’s function,
respectively. As we found, there are two SS labeled Es1 and Es2 (solid red points), and three
resonance states marked Er1, Er2, and Er3 (blue circles). The shadow zones represent the
calculated bulk band structure projected on the (100) surface. We found that our calculated projected bulk bands are in good agreement with that reported in Refs. [50, 52].
Tables II and III show our calculated energies for the SS and RS, and compare them with
values reported in previous works. The states Es3 and Es4 reported in the literature are
listed in Table II although our calculations do not reproduce them.
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located at 4.44 eV, and shows a parabolic dispersion. The state is placed in the ¯Γ–gap which
has a width of 5.03 eV approximately, as was obtained in our calculations, the state is the
hybridization of the s and pz atomic orbitals. This state was measured at 4.1(2) eV, by
Altmann et al., [51]. The SS Es2 is located at 0.97 eV, showing also a parabolic dispersion,
the state is inside the ¯X–gap which has a width of 3.87 eV. This SS has the symmetry of
the px, and py wave functions. The state was measured and calculated by Kolb et al., [50]
Ershbaumer et al., [53] and recently by Savio et al., [52].
For the Ag–(100) surface, we report here three calculated RS’s: Er1, Er2, and Er3. Where
Er1 was found at −4.07 eV in the
¯X ¯M
direction and shows almost zero dispersion, and we
found that the state has the hybridization of the dz2 atomic orbitals.
The RS Er2 begins at −3.18 eV in the ¯M point showing a small dispersion, as we have
found the character of this state is dxy. The RS Er3 is located at −4.41 eV at ¯Γ, the
symmetry found for this RS is dz2.
2. Ag(110)–surface
In an early ab initio calculation Ho et al., reported the projected bulk band structure and SS of Ag(110), and predicted a SS on the upper band energy region of the 2D–BZ (see Fig. 1 in Ref. [55]). After that, Reihl et al., using the experimental technique of k–resolved inverse
photoemission spectroscopy found, in the energy gap around the ¯X point of the 2D–BZ (see
Fig. 2 in Ref. [56]) an unoccupied SS at 1.65 eV for Ag(110), in good agreement with the state predicted by Ho et al., [55].
Figure 3 depicts our calculated projected bulk band structure for the Ag(110) surface, which shows nine SS (solid red points) and eight RS (blue circles). The details of the calculated surface and resonance states are shown in Tables IV and V.
As we found, the state Es1 that appears in the upper energy gap located at ¯X, begins
at 5.21 eV and ends at 4.19 eV in the interval ¯Γ − ¯X − ¯S, crossing the ¯X point at 2.90 eV,
according to our calculations the state has px symmetry of the atomic orbital. Although
there is no experimental evidence for this SS, from a theoretical point of view, by using the empirical tight–binding method, the state was calculated by Tjeng et al., [57].
Moreover, we found another SS in the same gap energy (Es2). The state shows few
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for this state has the s, pz symmetry of the atomic orbitals. From the experimental point
of view this SS was reported by Altmann et al., [51] and was found around 5.0(2) eV, while the one calculated by Ho et al., [55] was at 4.25 eV, and the calculated value by Tjeng et al., [57] was at 5.1 eV.
Then in the ¯X − ¯S interval at energies that range from 7.0 − 10.0 eV, we found the
state Es3 which shows a significant slope. The wave function for this state has the py+ pz
symmetry of the atomic orbitals. We do not find experimental evidence for this SS, although the state was reported theoretically, in the already mentioned work, by Tjeng et al., [57].
The SS Es4 is placed at 6.44 eV in the ¯S point and shows a high slope, the state was
found in an energy gap of approximately 0.54 eV. We found the state is a hybridization
of the s and pz atomic orbitals. This SS was reported by Tjeng et al., [57] and was found
around 6.45 eV.
In the energy gap at ¯Y point, we found two SS showing a parabolic shape. The lower
state (Es5) is located at 1.47 eV. Our wave function for the Es5 state has the py symmetry.
This SS was found experimentally in Refs. [12, 51, 56], and theoretically in Refs. [55, 57].
The upper state (Es6) is placed at 3.10 eV, and the state has the symmetries s, pz. The
Es7 starts from 4.66 eV in 0.35
¯X0− ¯S0
to 3.03 eV in ¯S0, located in an energy gap whose
width is 1.76 eV in ¯S0, the state character is mainly of py composition. As well, we identify
in the same energy gap the SS Es8 which starts from 4.87 eV in 0.35
¯X0− ¯S0
to 3.85 eV in
¯
S0, this state has a mixed orbital character s, px+ pz. Finally, the SS Es9 starts from −5.13
eV in 0.25 ¯X − ¯S to 0.55 ¯X − ¯S
with slight dispersion. This state has a mixed orbital
character dxy+ dx2−y2 + dz2.
On the other hand, we do not locate the states labeled Es10 and Es11, and listed in
Table IV. Experimentally the state Es10 was reported by Altmann et al., at 4.1(2) eV on
the ¯Γ point [51]. While the state Es11was found experimentally in Ref. [58], and theoretically
in Ref. [57]. More recently both SS have attracted the attention, and have been studied the same experimentally as well as theoretically by Pascual et al., [12] where these authors reported an occupied SS at −0.05 eV, and the unoccupied SS at 1.7 eV (see Table IV).
In the VB region we found five RS’s, and according to our calculations the states are: a)
The RS Er1 is located at ¯Γ at −5.13 eV, and its wave function has the full symmetry of the
d atomic orbitals. b) The RS Er2 is situated at ¯Γ at −4.18 eV, and its wave function also
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point and its wave function has the symmetry of the dx2−y2 atomic orbitals. d) The RS Er4
is found at −3.56 eV at ¯Y point, and its wave function is a hybridization of the dxy and
dzx atomic orbitals. From the theoretical point of view, these four states were calculated in
Ref. [55]. e) The RS Er7 is located at ¯Γ at −5.47 eV, and its wave function has a mixed
orbital character dyz+ dx2−y2.
Then, in the CB we found the RS Er5, the state is located at 8.20 eV in the ¯S point,
and it wave function has the py symmetry. The RS Er6 is situated at ¯X0 at 6.16, and its
wave function has the hybridization of the s, px atomic orbitals. These two states were also
reported in Ref. [57]. The RS Er8 is located at ¯Γ at 7.32 eV, and has the symmetry of the
pz atomic orbitals.
3. Ag(111)–surface
In an early photoemission measurement, Roloff and Neddermeyer reported a SS for
Ag(111) just below the Fermi level at the ¯Γ point [59]. Afterward, Bertoni et al., [60]
using the linear combination of atomic orbitals (LCAO) method, described the bulk band structure, SS and RS for Ag(111).
Figure 4 shows our calculated projected bulk band structure for Ag(111). There we
identify four SS’s labeled Es1, Es2, Es3, and Es4, and five RS’s labeled Er1, Er2, Er3, Er4,
and Er5. In general, from the figure we observe that the calculated projected bulk bands
structure are in agreement with that reported in Refs. [61, 62]. The characteristics of the different SS and RS found are shown in Tables VI and VII.
The SS Es1 is located in the lower gap at −6.08 eV at ¯Γ. The state shows a parabolic
shape and its wave function is the hybridization of the s, dz2 atomic orbitals.
Our SS Es2 was calculated at 2.29 eV at ¯Γ, showing an important dispersion, its wave
function has the symmetry of the s, pz atomic orbitals. From the experimental point of view
a similar SS was reported in Refs. [51, 62–64] as an unoccupied state. Theoretically in Ref. [60] the state was identified at 2.04 eV.
There are a couple of SS, Es3 and Es4, located in the upper gap at ¯K. Es3 was found
at 6.64 eV and its wave function has the symmetry of the px, py atomic orbitals. Es4 was
calculated at 6.98 eV, and its wave function has the symmetry of the s, and pz atomic
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On the other hand, we do not locate the SS Es5 listed in Table VI and reported in the
Refs. [59, 61–63, 65]. While Reihl et al., [62] described this SS as an unoccupied state, in
Refs. [11, 59, 61, 63, 65, 66] was reported as an occupied state. Even more, B¨urgi et al.,
[66] compare their measurements with a simple tight–binding description, where they use a single s–band, and found that this simple approach describes the dispersion of this SS
properly. This means that the orbital composition of the SS Es5 comes mainly from the s
orbitals, as was noted by Nicolay et al., [11].
The RS Er1 was calculated at −6.42 eV at ¯Γ, and we found that the state has the
symmetry of the dz2 atomic orbitals. Although there is no experimental evidence for this
RS, theoretically the state was predicted in Ref. [61] (see Table VII for energy comparison).
In the energy bulk bands above the lower gap at ¯Γ, we found the state Er2 at −4.79 eV,
the calculated wave function for this RS was the hybridization of the dxy and dx2−y2 atomic
orbitals. The state Er3 was calculated at −6.35 eV at ¯M point, and has the character of
the d orbitals. The state Er4 was calculated at −3.36 eV at ¯M point, and its wave function
has the symmetry of the dxy, dyz, dzx and dx2−y2 atomic orbitals. Theoretically, these three
states were reported in the Refs. [61, 62].
Finally, the state Er5 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, px, and py
atomic orbitals.
IV. CONCLUSIONS
In this work, we have obtained the local density of states and the electronic band structure of the low–index fcc Ag surfaces (100), (110) and (111) by using an empirical tight–binding Hamiltonian within the Surface Green’s Function Matching formalism. We report the energy values found for the different surface and resonance states in the selected directions.
The surface states found in this work for Ag(100) Es1 and Es2 were reported previously
in the literature, our calculated energy values for these states show good agreement with published values, as we show in Table II. While the new states found in this work are the
resonant states Er1, Er2 and Er3.
As we found our calculated SS’s and RS’s for Ag(110) Es1, Es2, Es3, Es4, Es5, Er1, Er2,
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precision of the method, show good agreement with them, as we can see in Tables IV and
V. The new states found in this work are Es6, Es7, Es8, Es9, Er7 and Er8.
Finally, the SS’s and RS’s for Ag(111) are presented in Tables VI and VII. The states
Es2, Er1, Er2, Er3 and Er4 have been reported previously, and we found that our calculations
agree with the reported values. The new states found in this work are Es1, Es3, Es4 and
Er5.
As a general feature, we get that our LDOS, SS’s and RS’s agree with those reported in the literature. Our results for ideal surfaces demonstrate the predictive power of the empirical method. Therefore, we can conclude that in general, the calculations with TB– SGFM are in agreement with theoretical and experimental results. However, to improve our results more work will be necessary. As we have stated above, our calculations could be improved if we include some effects that have not been taken into account in the present work, such as, surface relaxation and reconstruction, nonorthogonal basis approach, among others.
Acknowledgements
This work was partially supported by the project UAM-A-CB001-14 M´exico. We
grate-fully acknowledge the SNI-CONACYT-M´exico for the distinction of our membership and
the stipend received.
Appendix: Surface Green’s Function Method[30]
In order to describe the electronic band structure, we use the tight–binding method in the Slater and Koster formalism [28]. We define a hamiltonian for the surface, where we assume the ideal truncation. With this hamiltonian, the associated Green’s function satisfies the Eq. 1. Let |ni be the principal layer wave function, if we take matrix elements of Eq. 1 we get
hn|(εI − H(k))G(ε, k)|mi = δm,n, (6)
where I is the unit matrix of the Hilbert space in which H and G are defined, and ε is the energy eigenvalue. Since interactions will be included up to second nearest neighbors only,
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the principal layers will consist of two atomic planes and, there are only nearest neighbour interactions between principal layers, therefore, the identity operator can be rewritten as
I = |n − 1ihn − 1| + |nihn| + |n + 1ihn + 1|. (7)
By using Eq. 6 and Eq. 7 we get
(ε − Hn,n)Gn,m− Hn,n−1Gn−1,m− Hn,n+1Gn+1,m= δm,n, (8)
where Hm,m+i = 0 for i ≥ 2. The hamiltonian, Hn,m is a 2 × 2 supermatrix where each of
the elements is a 9×9 matrix since we are using a wave function with one s–like, three p–like, and five d–like atomic functions for each atom as a basis. In terms of principal layers, we label them with positive numbers and zero for the surface atomic layer. Now, we assume the
simplification of an ideal surface in equation 8, i.e., H0,0 = H1,1 = . . . and H0,1 = H1,2 = . . ..
Using Eq. 8 for m = n it is straightforward to get the surface Green’s function
G−1s = εI − H0,0− H1,0T, (9)
and the principal–layer–projected bulk Green’s function
G−1b = G−1s − H0,1† T ,˜ (10) where H0,0 = h0,0 h0,−1 h−1,0 h−1,−1 , H0,1 = h0,−2 h0,−3 h−1,−2 h−1,−3 . (11)
We label the main layers with positive numbers and the atomic planes with negative
numbers. Note that H0,0 contains atomic layers 0 and −1 while, for H0,1, atomic layers 0,
−1, −2 and −3. The surface is labeled with zero, this means that H0,0 = H1,1 = . . . = Hn,n
for any n. Also h0,−1 = h−1,−2 = . . ., h0,−2 = h−1,−3 = . . . and for h0,−3 = h−1,−4 = . . . = 0.
Now, we need to know only H0,0 and H0,1 and, therefore, h0,0, h0,−1 and h0,−2 which are
9×9 matrices. These matrices are written in a tight–binding language and can be calculated with the bulk parameters mentioned above. They depend on the energy, ε, and on the wave vector k.
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Now we are in a position to discuss the transfer matrix. The principal–layer–projected Green’s funtion on the layers next down from the surface is given by
Gn,n = Gb+ Tn(Gs− Gb) ˜Sn. (12)
The transfer matrices are defined as
Gk+1,p = T Gk,p, Gk+1,p = Gk,pS, k ≥ p ≥ 0,
Gi,j+1 = ˜T Gi,j, Gi,j+1= Gi,jS,˜ j ≥ i ≥ 0.
These matrices can be calculated by the quick algorithm of L´opez–Sancho el al., [31].
They get T = t0+ ˜t0t1+ · · · + ˜t0˜t1· · · ˜ti−1ti+ · · · (13) ˜ T = ˜t0+ t0t˜1+ · · · + t0t1· · · ti−1˜ti+ · · · (14) S = s0+ s1s˜0+ · · · + sis˜i−1· · · ˜s1˜s0+ · · · (15) ˜ S = ˜s0+ ˜s1s0+ · · · + ˜si˜si−1· · · ˜s1s0+ · · · (16) where t0 ≡ (ε − H0,0)−1H † 0,1, ˜t0 ≡ (ε − H0,0)−1H0,1,
ti ≡ Mi−1t2i−1, t˜i ≡ Mi−1t˜2i−1,
with Mi−1 = (1 − ti−1t˜i−1− ˜ti−1ti−1)−1,
s0 ≡ H
†
0,1(ε − H0,0)−1, ˜s0 ≡ H0,1(ε − H0,0)−1,
si ≡ s2i−1Ni−1, s˜i ≡ ˜s2i−1Ni−1,
with Ni−1 = (1 − si−1s˜i−1− ˜si−1si−1)−1.
The transfer matrices S and T describe the propagation in a perpendicular direction from
de surface, while the transfer matrices ˜S and ˜T describe the porpogation in the opposite
direction [30]
The i–th term in (11) is of order of 2i+1− 1 in H0,1 and it vanishes rapidly. Thus, a good
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the Gs, Gb, and Gn,n in an straightforward way from the formulae given above.
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TABLE I: s, p, and d atomic partial contribution to the Local Density of States (LDOS) at the surface(s) and bulk(b) at the Fermi Level (states/eV/atom), for the different oriented surfaces studied in this work. The last line lists the values reported by Papaconstantopoulos.
Surface ss ps ds sb pb db Ag(100) 0.091 0.095 0.024 0.074 0.139 0.031 Ag(110) 0.104 0.083 0.020 0.074 0.135 0.030 Ag(111) 0.102 0.128 0.030 0.071 0.138 0.030 Agbulk 0.078a 0.126a 0.060a aRef. [29].
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TABLE II: Surface states for Ag(100). The first column labels the found SS, the second column lists the k–point where the SS was found, the third column shows the experimental energy reported for the related state, the fourth column 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 shows the symmetry of the atomic orbitals (SAO) that form the SS, obtained from our calculations. The k–vector are given in units ofπ
a, while the energies are in eV.
SS ~k Eexp Etheo Eour SAO
Es1 (0, 0) 4.1(2)a 4.44 s, pz Es2 (1, 1) 0.73b 0.07c 0.97 px, py 0.15d Es3 (1, 1) 3.1b 2.99d 3.5(2)a 3.03c 3.8(4)e 3.3e Es4 (1, 1) −0.4c −0.5e −0.53c
aRef. [51];bRef. [50]; cRef. [52]; dRef. [53];eRef. [54].
TABLE III: Resonance states for the Ag(100) surface. The first column shows the labeled resonance state, the second one shows the wave vector of the state in units of [πa], the next column shows the related energy in eV, finally the last column shows the wave symmetry found for the different states.
RS ~k Eour SAO
Er1 (2, 0) −4.07 dz2 Er2 (2, 0) −3.18 dxy
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TABLE IV: Calculated energy values for SS on the Ag(110) surface. The labels of the different columns follow the format given in Table II.
SS ~k Eexp Etheo Eour SAO
Es1 ( √ 2, 0) 2.04a 2.90 px Es2 ( √ 2, 0) 5.0(2)b 4.25c 6.16 s, pz 5.1a Es3 ( √ 2, 0.4) 8.06a 9.22 py,z Es4 ( √ 2, 1) 6.45a 6.44 s, pz Es5 (0, 1) 1.6(2)b 1.28c 1.47 py 1.65d 1.95a 1.7e Es6 (0, 1) 3.10 s, pz Es7 (√12, 1) 3.03 py Es8 (√12, 1) 3.85 s, px,z Es9 ( √ 2, 0.5) −5.13 dxy,x2−y2,z2 Es10 (0, 0) 4.1(2)b Es11 (0, 1) −0.05e 0.00a −0.10(10)f
aRef. [57];bRef. [51];cRef. [55]; dRef. [56]; eRef. [12]; fRef. [58].
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TABLE V: Resonant states found for Ag(110) surface. The labels for the different columns follow the format given in III.
RS ~k Etheo Eour SAO
Er1 (0, 0) −4.95a −5.13 d Er2 (0, 0) −4.05a −4.18 d Er3 ( √ 2, 1) −3.75a −3.50 d x2−y2 Er4 (0, 1) −3.86a −3.56 dxy,zx Er5 ( √ 2, 1) 7.67b 8.20 py Er6 (√12, 0) 6.45b 6.16 s, px Er7 (0, 0) −5.47 dyz,x2−y2 Er8 (0, 0) 7.32 pz aRef. [55]; bRef. [57]
TABLE VI: Surface states for the Ag(111). The labels for the different columns follow the format given in II.
SS ~k Eexp Etheo Eour SAO
Es1 (0, 0) −6.08 s, dz2 Es2 (0, 0) 3.75a 2.04b 2.29 s, pz 3.90c 4.00d 4.2(2)e Es3 43 √ 2, 0 6.64 px,y Es4 43 √ 2, 0 6.98 s, pz Es5 (0, 0) 0.33d −0.23f −0.04a −0.12g,h
aRef. [63]; bRef. [60]; cRef. [64];dRef. [62];eRef. [51]; fRef. [61]; gRef. [65];hRef. [59].
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TABLE VII: Resonant states found for Ag(111) surface. The labels for the different columns follow the format given in III.
RS ~k Etheo Eour SAO
Er1 (0, 0) −6.27a −6.42 dz2 Er2 (0, 0) −4.94a −4.79 dxy,x2−y2 −5.00b Er3 ( √ 2, √ 6 3 ) −6.11a −6.35 d −6.25b Er4 ( √ 2, √ 6 3 ) −3.70a −3.36 dxy,yz,zx,x2−y2 −3.20b Er5 (43 √ 2, 0) 2.08 s, px, py aRef. [61];bRef. [62]
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0 2 4 6 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 L D O S (st a te s/ e V /ato m ) Energy (eV) 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 L D O S (st a te s/ e V /a to m ) Energy (eV) s b s b (c) (b) E F (a) 0 2 4 6 L D O S ( st a t e s/ e V / a t o m ) -8 -6 -4 -2 0 2 4 6 0 2 4 6 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 L D O S (st a te s/ e V /a to m ) Energy (eV) s b Energy (eV)FIG. 1: (Color online) Surface local density of states (LDOS) (full black line) and bulk LDOS (broken red line) obtained from the SGFM method. (a) Ag(100)–, (b) Ag(110)–, and (c) Ag(111)– surface. The zero of energies represents the Fermi level.
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-10 -8 -6 -4 -2 0 2 4 6 8 10 W ave vector Er3 Er3 Er2 Er1 Es2 Es1 Es1 E F Ag(100) _ _ _ _ M X E n e r g y ( e V )FIG. 2: (Color online) Projected bulk band structure found in our calculation (black zone). The solid red points represent the SS, while the blue circles are for the RS. The Fermi level is the zero of the energies.
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-10 -8 -6 -4 -2 0 2 4 6 8 10 Er7 Er5 Es9 Er8 Er4 Er3 Er2 Er1 Es3 Es2 Es1 Es4 Es6 Es5 S Ag (110) E n e r g y ( e V ) _ W ave vector Er6 Es7 Es8 E F _ _ _ _ _ _ S ' X ' S X YFIG. 3: (Color online) Projected bulk band structure found in our calculation (black zone). The solid red points represent the SS, while the blue circles are for the RS. The Fermi level is the zero of the energies.
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-10 -8 -6 -4 -2 0 2 4 6 8 10 W ave vector Er4 Er3 Er2 Er1 Es4 Es3 Er5 Es2 Es1 E F Ag(111) _ _ _ _ K M K E n e r g y ( e V )FIG. 4: (Color online) Projected bulk band structure found in our calculation (black zone). The solid red points represent the SS, while the blue circles are for the RS. The Fermi level is the zero of the energies.
