https://doi.org/10.22463/0122820X.1324 https://doi.org/10.22463/0122820X.1324
Recibido: 23 de Julio de 2017 - Aprobado: 02 de Diciembre de 2017
The motivation of this work is based on the importance of Palladium in processes such as catalysis and hence the need to know its electronic properties. We present a detailed study of the electronic structure of Palladium bands in the crystallographic direction (111). First we verify that the local density of states, projected in the volume, agrees with the results obtained for the case of the infinite medium previously reported. Next, a detailed study is made of different surface states and characteristic resonant states of the Palladium in the crystallographic direction (111). It was found that the results obtained are compared with the values published in the literature, and the prediction of different states not yet reported is made.
Keywords:Surface states, resonant states, local density of states, bulk projected electronic band structure.
La motivación del presente trabajo se fundamenta en la importancia del Paladio en procesos de catálisis y sus propiedades electrónicas. En tal sentido, se presenta un estudio detallado de la estructura electrónica de bandas del Paladio en la dirección cristalográfica (111). De tal manera, se verificó que la densidad local de estados proyectada en el volumen, concordara con los resultados obtenidos para el caso del medio infinito previamente reportados, para ello, se realizó un estudio detallado de diferentes estados de superficie y estados resonantes característicos del Paladio en la dirección cristalográfica (111). Se halló que los resultados obtenidos se comparan con los valores publicados en la literatura, y se hizo la predicción de diferentes estados no reportados aún.
Palabras clave:Estados de superficie, estados resonantes, estructura electrónica de bandas.
Theoretically, the surface states can be calculated employing
different methods: empirical and principled [1]-[3]. In that
sense, a partial collection of the electronic structure of surfaces,
surface states and resonant states, are reported in the Landolt-
Börstein series [4].
Experimentally, the surface states occupied can be explored by
ultraviolet photoemission spectroscopy determined in angle
and the surface states unoccupied by inverse photoemission
spectroscopy determined in angle [4].
The palladium is an important catalyst for rusting processes
such as: methane catalytic rusting in gas turbines [5],
hydrocarbon rusting and carbon rusts in exhaust pipes [6],
which is particularly interesting due to high hydrogen solubility
in the metal [7]-[10]. In the same order, it is considered as one
of the best catalysts for the alkynes and dienes partial hydrogenation
[11].
Moreover, the relation between the surface structure of the
catalyst and the reaction activity or selectivity is an open
question of interest from the technological and fundamental
point of views.
For this reason, it is necessary a systematic study that explains
the electronic characteristics of the palladium surfaces. In a
recent research [12], the selective hydrogenation of acetylene
through Pd nanoparticles with different shapes, concluded the
catalyst that had Pd cube-shaped nanoparticles, which surface
consisted of Pd structure (100), exhibited a higher acetylene
conversion and ethylene selectivity than catalysts that had
spherical nanoparticles, having a significant quantity of Pd
nanoparticles (111).
However, the calculation of the functional density determined
that in the Pd surface (111) there would be more ethylene
activity and selectivity in contrast with the Pd surface (100) for
acetylene hydrogenation [13].
In this research, a detailed analysis is done about the electronic
structure of Pd bands (111), employing the strong link
approximation and Green´s surface function coupling method.
On the same hand, strong link Hamiltonians were built based
on Slater Koster´s formalism (SK), in which the model
parameters, in the approximation of two centers, were taken
from the manual charts of band electronic structures published
by Papaconstantopoulos [14].
For the surface analysis (111), it was used the Green´s surface
function coupling method and it was verified that the local
density in states, reflected in the volume, agrees with the
results obtained for the reference case [14]. Also, it was
determined the type of symmetry of the wave functions that
contributed to the surface states and resonant states of Pd
(111), the obtained results were compared with the values
published in the literature and the prediction is done about the
different states that are not reported yet.
The present research was organized in the following way: The first section describes Green´s surface function coupling method deducing the expressions for the Green´s functions projected in the surface and the volume. The second section presents the projected volume bands, the dispersion of surface states and resonant states for the Pd in the crystallographic direction (111), there, it is compared the obtained results with the findings on the experimental and theoretical literature related. Finally, the third section is focused on the conclusions.
2. Theoretical ModelTo do the calculation of the surface electronic properties is used the Green´s surface function coupling method (GFCM) [15], [16]. For the construction of Green´s surface function, it proceeds like this: With the Hamiltonian H of strong link it is defined a Green´s function associated like G, such that it satisfies the equation: (eq. 1)
In which H is the Hamiltonian of strong link, I is the unitary matrix and ω is the own energy value. A main layer is defined as the minimum group of atomic layers that reproduce the translational periodicity of the system carried out. (Fig 1)
The main layer conventions are labeled with positive numbers
and the atomic layers are labeled with negative numbers, and
number zero is assigned to the surface atomic layer (figure 1).
From this definition about the main layer, it is introduced the
approximation of the GFCM method, which considers
interactions in first neighbors among main layers. The
quantity of atomic interactions will be determined by the
number of atomic layers that define each main layer.
Being |𝑛⟩ the wave function associated with the n – th main
layer. This wave function is a lineal combination of a certain
number of atomic orbitals per atom in the unit cell, in our case
the orbitals (s, p, d). Considering the space generated by the
system of complete functions |𝑛⟩, it can be calculated the
matrix elements of the equation (1). That space obtains: (eq. 2)
In the approximation of first neighbors among main layers, the identity operator is given as: (eq. 3)
Replacing the equation (3) in which the equation (2) obtains: (eq. 4)
Where in this approximation and for the surface, n=0, we have
H_(0,m)=0 for m≥2 (as in the strong link model the
interactions are dominated by first neighbors interactions [17].
Then, the expression for Green´s function of surface projection
will be: (eq. 5)
Meanwhile Green´s function for the volume projected in the first main layer will be [15], [16].[15, 16]
Here, H0,0 and H0,1 are the interaction hamiltonians in
intralayers and among main layers, respectively. To calculate
these matrixes, it is needed to know the strong link matrixes of
interaction among atomic layers (see details in references [18],
[19]).
Then, T and T are the transfer matrixes, and they have all the
data on surface electronic properties. These matrixes are
coupled, this is why they cannot be evaluated exactly, so that
they are developed in many iterative algorithms in order to
obtain them numerically [19].
Due to the existence of a surface, the energetic states can differ
from the states of inner parts of solids or their volume; if the
surface states which energy value and wave vector coincide
with the inner states of the solid, they are nominated resonant
states [18]. The surface states which energy value and wave vector
do not coincide with any inner state of the solid are
nominated surface states [20]. These have wave functions that
buffer themselves exponentially at the same time that they
enter the crystal.
The surface states are located in the energetic breaches of the
volume bands structure and emerge due to the break of the
crystal symmetry. Finally, once we know the differences of
Green´s function mentioned beforehand, it can be calculated
the surface states and the resonant states directly from the poles
from the real part of Green´s function, whereas from the
imaginary part we can find the density of local states [15], [16].
Density of States Figure number 2 shows volume states density of the Pd applying the reference parameters [14], that are related to the model through energy integrals for the crystal in terms of integrals of two centers [21]. Figure 2b exhibits the local density of states projected on the surface (dotted red line) and the local states density projected on the volume (solid black line) for the Pd (111).
In figures 2a and 2b, the zero of the energy axis represents the Fermi level (EF). For calculating figure 2b there were used 136 points in the irreducible segment of the first Brillouin zone [22] (the number of the extracted points reproduces the most significant features of the local density of volume states and optimize the time of calculation). (Fig 2)
A direct comparison on the local density of states projected on
the volume, with the states density reported in the reference
[14], allows to observe that the calculation properly reproduces
the main features of the density of states. In this respect, the
local density of volume states that is calculated using the
MAFG, is adequately compared for energies below the Fermi
level, where are localized three spikes in the energies of: -4.66,
-2.46 and -0.16 eV, which differ of the reported ones in the
reference [14], in 0.0748, 0.0162 and 0.01034 eV.
On the basis of these facts, it is shown that the surface
electronic band structure coincides with the reported results
[14].
The partial contributions of atomic orbitals s, p y d to the local
density of Pd states (111), are shown in figure 3. Particularly,
figure 3c shows that the approximate energy rate is -5.42 eV a
0.44 eV, the main contribution of local density is obtained from
electrons Pd - 5d, in contrast with the contributions of the
orbitals s y p (see figures 3a and 3b). (Fig 3)
Projected bands of volume, surface states and resonant states of Pd (111).The measured values of the surface states (solid black circles) and the measured resonant states (white circles), as well as the projected bands of surface volume (111), are shown in figure 4.
The electronic features of these states are respectively shown in
tables I and II. According to the calculations, there have been
obtained ten surface states and four resonant states. Regarding
the electronic and symmetry characteristics found for the
different surface states and resonant states, it is had:
The state Es1 does not present dispersion and it is located in
3.48 eV below Fermi level as it is observed in figure 3, in 0.05
|K Γ|, being located in an energetic breach, and its width in the
K point from the two-dimensional Brillouin zone is 0.47 eV, in
which the kind of symmetry of the wave function that
contributes to this state is d 3z2 - r2 .
Theoretically, this state has been identified in the references [10] and [23], results that differ from the current research in 0.01 and 0.54 eV, respectively. (Fig 4)
The surface state is Es in approximately -3.21 eV in 0.45 |K Γ| and it scatters until -3.14 eV in 0.55 |K Γ|, being located in an energetic breach which width is 0.20 eV in 0.50 |K Γ|, being the result of the hybridization of atomic orbitals dxy. Experimentally and theoretically, this state has not been identified.
The surface state Es starts from -2.12 eV in 0.80 |MK|, crosses
point K in -2.39 eV, approximately, and disperses to -2.25 eV
en 0.25 |K Γ|, being in an energetic breach whose width is 1.08
eV in K , and whose orbital character is d.
Theoretically, this state has been identified in the references
[10] and [23], results that differ from our result in 0.22 eV.
The surface state Es4 starts from -0.62 eV in 0.70 |MK|, crosses
point K in -0.69 eV, approximately, and disperses to -0.76 eV
en 0.10 |K Γ|, being in an energetic breach whose width is 0.68
eV in K, in which the kind of symmetry of the wave function
that contributes to this state is d3z 2 - r2 . Theoretically, this state
has been identified in the references [10] and [23], results that
differ in 0.62 y 0.38 eV with our results. (table 1)
Theoretically, this state has been identified in the references
[10] and [23], results that differs in 0.62 and 0.38 eV with our
results.
The surface state Es₅ starts from -4.16 eV in 0.60 |K Γ|,
crossing the Γ point in -5.11 eV, approximately, and it is
scattered in a parabolic form until -4.33 eV in 0.35 |Γ M|, being
located in an energetic breach which width is 2.10 eV in Γ ,
being the result of hybridization of atomic orbitals s, d 3z2 - r2.
Theoretically, this state has been identified in the references
[10] and [23], result that differs in 0,06 and 1.15 eV,
respectively with the results of the current article.
The surface state Es starts from 8.42 eV in 3.35 |K Γ|, and disperses to -7.74 eV en 0.70 |ΓM|, being in an energetic breach whose width is 6.59 eV in Γ , and whose orbital character is s,p_z. This state has been identified experimentally in the reference [24] and theoretically in the reference [23], results that differ in 2.91 eV and 2.81 eV, respectively. A possible explanation for the error found in the energies of the states Es , Es and Es , may be the fact that our calculation has considered an ideal surface. A correction to this approach should include the effects of reconstruction and relaxation of the surface, effects that remain pending.
The surface state Es₇ starts from -2.25 eV in 0.40 |K Γ| and it is
scattered until -2.19 eV in 0.50 |K Γ|, in which the kind of
symmetry of the wave functions that contribute to this state are
dxy, yz, zx, finding themselves in the same energetic breach
than the state Es₃.
The surface state Es₈ starts from 3.93 eV in M and it is
scattered until 4.88 eV in 0.40 |MK|, finding itself in an energetic
breach which width is 3.40 eV in M, being the result of
hybridization of atomic orbitals s, px,y.
The surface state Es₉ starts from 7.74 eV in 0.85 |M K| and it is
scattered until 7.80 eV in 0.10 |K Γ|, being located in an
energetic breach which width is 1.49 eV in K , and its orbital
character is px, y.
The surface state Es₁₀ starts from 9.64 eV in 0.90 |M K| and it
is scattered until 9.57 eV in K , finding itselt in an energetic
breach which width is 1.49 eV in K, in which the kind of
symmetry of wave functions that contribute to this state is px,y.
The resonant state Er₁ starts from -4.57 eV in 0.65 |K Γ| and it
is scattered until -5.31 eV in 0.15 |Γ M|, in which the kind of
symmetry of wave functions that contribute to this state is
d 3z2-r2 .Theoretically, it has been identified in the reference [23],
result that differs from ours in 1.14 eV. (Table 2)
The resonant state Er₂ starts from -3.27 eV in 0.85 |K Γ| and it
is scattered until -3.14 eV in 0.05 |Γ M|, being the result of
hybridization of atomic orbitals d. Theoretically this state has
not been identified. The resonant state Er₃ starts from -1.17 eV
in 0.95 |K Γ | and it is scattered until -1.17 eV in 1.05 |Γ M|, and
its character is dyz,zx. Theoretically, this state has been identified
in the reference [23], result that differs from ours in 0.59 eV.
The resonant state Er₄ starts from -0.14 eV in 0.95 |Γ M| and it
is scattered until -0.21 eV in 0.25 |M K|, in which the kind of
symmetry of wave functions that contribute to this state is
dxy,x²-y². Theoretically, this state has been identified in the
reference [23], result that differs from ours in 1.02 eV.
The states calculated in the current research for the Pd (111)
Es₁, Es₃, Es₄, Es₅, Es₆, Er₁ and Er₃ have been reported in the
literature and the results obtained agree acceptably with the
values published, with the exception of state Es₆, as it is
observed in the charts I and II. The new states found are the
surface states Es₂, Es₈, Es9 and Es10 and the resonant state Er₄.
It has been studied the electronic structure of palladium in the
crystallographic direction (111). The present research used
Slater- Koster hamiltonian, with parameters that duplicate the
electronic volume structure adequately.
To describe the surface, it was exterted Green´s function
coupling method. In general terms, it was found that the calculation
duplicates qualitatively the surface and resonant states,
previously reported, as well as it preditcs the existence of no
reported surface and resonant states for the Pd surface (111).
Although there are quantitative problems to duplicate the
energy reported for the surface states Es4 and Es6, the
calculation made is for the case of an ideal surface, a correction
for this approximation would have to include recons truction
and relaxation effects of the surface, effects that are pending.
To the Natural Science and Mathematics Department from the University of Ibague.
6. Reference[1] K. M. Ho, C. L. Fu, S. H. Liu, D. M. Kolb y G. Pi, «Observation of surface states at the metal- electrolyte interface by electroreflectance spectroscopy,» Journal of Electroanalytical Chemistry, vol. 150, nºs 1-2, pp. 235-240, 1983.
[2] H. J. Herrera Suárez, A. Rubio Ponce y D. Olguín, «Electronic band structure of platinum low–index surfaces: an ab initio and tight–binding study. II,» Revista mexicana de física, vol. 58, pp. 46–54, 2012.
[3] H. J. Herrera Suárez, A. Rubio Ponce y D. Olguín, «Electronic band structure of the Pt (1 1 1) surface: An ab initio and tight-binding study – I,» Computational Materials Science, vol. 56, pp. 141-146, 2012.
[4] C. Calandra y G. Chiarotti, Landolt-Börnstein - Group III Condensed Matter Electronic and Vibrational Properties, vol. 24b, New York: Springer-Verlag, 1994, pp. 29-303.
[5] R. B. Anderson, K. C. Stein y J. J. Feenan, «Catalytic Oxidation of Methane,» Ind. Eng. Chem, vol. 53, pp. 809-812, 1961.
[6] V. A. Bondziea, P. H. Klebanb y D. J. Dwyer, «Kinetics of PdO formation and CO reduction on Pd (110),» Surface Science, vol. 465, p. 266–276, 2000.
[7] H. Wipf, Diffusion of hydrogen in metals. In Hydrogen in metals III., Berlin Heidelberg: Springer, 1997, pp. 215-278.
[8] H. Conrad, G. Ertl y E. E. Latta, «Adsorption of hydrogen on palladium single crystal surfaces. Surf. Sci., vol. 41, pp. 435-446., 1974.
[9] H. Kuipers y T. Engel, «A molecular-beam investigation of the scattering, adsorption and absorption of H2 and D2 from/on/in Pd (111).,» Surf. Sci, vol. 90, pp. 162-180, 1979.
[10] W. Dong y J. Hafner, «H2 dissociative adsorption on Pd (111),» Phys. Rev. B, vol. 56, pp. 15396–15403., 1997.
[11] J. S. Zhu y Y.-S. Shon, «Mechanistic interpretation of selective catalytic hydrogenation and isomerization of alkenes and dienes by ligand deactivated Pd nanoparticles,» Nanoscale, vol. 7, pp. 17786-17790, 2015.
[12] S. K. Kim, C. Kim y J. H. Lee, «Performance of shape-controlled Pd nanoparticles in the selective,» Journal of Catalysis, vol. 306, pp. 146-154, 2013.
[13] B. Yang, R. Burch y C. Hardacre, «Influence of surface structures, subsurface carbon and hydrogen, and surface alloying on the activity and selectivity of acetylene hydrogenation on Pd surfaces: A density functional theory study,» Journal of Catalysis, vol. 305, p. 264–276, 2013.
[14] D. A. Papaconstantopoulos, Handbook of the band structure of elemental solids, New York: Plenum Press, 1986., New York: Plenum Press, 1986, pp. 234-238.
[15] F. Garcia- Moliner y V. R. Velasco, Theory of Single and Multiple Interfaces, London: World Scientific, 1992, pp. 323-336.
[16] F. Garcia-Moliner y V. R. Velasco, «Theory of incomplete crystals, surfaces, defects, interfaces and layered structures, » Progress in Surface Science, vol. 21, nº 2, pp. 93-162, 1986.
[17] C.-Z. Wang, W.-C. Lu y Y.-X. Yao, «Tight-binding Hamiltonian from first-principles,» Sci Model Simul, vol. 15, p. 81–95, (2008).
[18] M. M. Lannoo, Atomic and electronic structure of surfaces: theoretical foundations, Berlin, 1991: (Springer-Verlag Vol 16, 1991, pp. 10-13.
[19] M. P. Lopez Sancho, J. M. Lopez Sancho y J. Ru, «Quick iterative scheme for the calculation of transfer matrices: application to Mo (100),» Phys. F: Met. Phys, vol. 14, pp. 1205-1215, 1984.
[20] P. V. Pavlov y A. F. Jojlov, Física del estado sólido, Moscú: MIR, 1987, p. p. 270.
[21] J. C. SLATER y G. F. KOSTER, «Simplified LCAO Method for the Periodic Potential Problem,» Physical Review, vol. 1524, p. 1458, 1954.
[22] D. l. Chadi y M. L. Cohen, «Special Points in the Brillouin Zone,» D. l. Chadi~ and Marvin L. Cohen, vol. 8, pp. 5747-5753, 1973.
[23] S. G. Louie, «Electronic states and adsorbate induced photoemission structure on the Pd (111) surface.,» Phys. Rev. Lett, vol. 40, pp. 1525-1528, 1978.
[24] P. D. Johnson y N. V. N. V. Smith, «Inverse photoemission observation of an unoccupied surface state on Pd (111),» Phys. Rev. Lett, vol. 49, pp. 290-292, 1982.
How to cite:
H.J. Herrera-Suárez, A. Rubio-Ponce and R.D. Olguín-Melo, "Surface States and Resonant States of Pd (111)", Respuestas, vol. 23, no. 1, pp.13-18, 2018.
a Universidad de Ibagué, Facultad de Ciencias
Naturales y Matemáticas, Colombia, Carrera 22 Calle 67 Barrio Ambalá. Correo: hernan.herrera@unibague.edu.co
b Dep. de Ciencias Básicas, Universidad Autónoma Metropolitana–Azcapotzalco, Av. San Pablo
180, México.
c Dep. de Física, Centro de Inv. y de Estudios Avanzados del Instituto Politécnico Nal,
México, D.F. 07300, México.
