286
14 Principles: Bond-Band-Barrier Correlation
SPB parameters leads to finite numerical solutions that corresponds to reality and be
physically meaningful.
Most importantly, the variation in bond length and geometry and the behavior of
electrons, both in real space (represented by ImV(r)) and in energy space (described
by ImV(E) or DOS), depend on each other, as they are consequences of the surface
bond forming and the electrons connecting to the sites, sizes and the chemical states
of the atoms.
14.4.3 One-Dimensional SPB
Jones, Jennings and Jepsen formulated the ReV(z) [1] that approximated the results
of jellium and density functional theory calculations of the SPB. This model has
widely been used for the fitting both of the VLEED fine-structure features and of the
inverse-photoemission image states, and has the form [4]:
ReV(z) =
−V 0
1 + A exp[−B(z − z 0 )]
, z ≥ z 0 (a pseudo-Fermi-z function)
=
1 − exp[λ(z − z 0 )]
4(z − z 0 )
, z < z 0 (the image potential)
(14.4)
where A and B are constants given by B = V 0 /A and A = −1 + 4V 0 /λ. The z-axis
is directed into the crystal. V 0 is the muffin-tin inner potential constant of the crystal
and z 0 is the origin of the image potential. The degree of saturation is described by
the λ parameter.
The ReV(z) transforms at z = z 0 from the pseudo-Fermi z function to the 1/(z −
z 0 ) type classical image potential with a spatial decay term. The Poisson equation
correlates the ReV(z) and the charge density,
∇
2 ReV(z)
−ρ(z) (z > z 0 )
0
(z ≤ z 0 )
(14.5)
The origin of the image-plane, z 0 , acts as the boundary of the region occupied by
surface electrons. If we permit z 0 to vary with the surface coordinates, then z 0 (x, y)
provides a contour of spatial electron distribution being mimic of the STM image.
The SPB features are thus characterized by the z 0 and this effect allows us to choose
z 0 as the key character in the subsequent single-variable parameterization of the
nonuniform-SPB.
There are several models for the energy-dependence of the inelastic damping,
ImV(E), for pure metal. An E
1/3 dependence of the inelastic damping worked well in
the LEED calculations of Ni surfaces [51]. An alternative was proposed by McRae
and Caldwell [57] in 1976 from their VLEED investigations of Ni surfaces, which
was widely used in dealing with other metal surfaces. The damping varies with energy
14 Principles: Bond-Band-Barrier Correlation
SPB parameters leads to finite numerical solutions that corresponds to reality and be
physically meaningful.
Most importantly, the variation in bond length and geometry and the behavior of
electrons, both in real space (represented by ImV(r)) and in energy space (described
by ImV(E) or DOS), depend on each other, as they are consequences of the surface
bond forming and the electrons connecting to the sites, sizes and the chemical states
of the atoms.
14.4.3 One-Dimensional SPB
Jones, Jennings and Jepsen formulated the ReV(z) [1] that approximated the results
of jellium and density functional theory calculations of the SPB. This model has
widely been used for the fitting both of the VLEED fine-structure features and of the
inverse-photoemission image states, and has the form [4]:
ReV(z) =
−V 0
1 + A exp[−B(z − z 0 )]
, z ≥ z 0 (a pseudo-Fermi-z function)
=
1 − exp[λ(z − z 0 )]
4(z − z 0 )
, z < z 0 (the image potential)
(14.4)
where A and B are constants given by B = V 0 /A and A = −1 + 4V 0 /λ. The z-axis
is directed into the crystal. V 0 is the muffin-tin inner potential constant of the crystal
and z 0 is the origin of the image potential. The degree of saturation is described by
the λ parameter.
The ReV(z) transforms at z = z 0 from the pseudo-Fermi z function to the 1/(z −
z 0 ) type classical image potential with a spatial decay term. The Poisson equation
correlates the ReV(z) and the charge density,
∇
2 ReV(z)
−ρ(z) (z > z 0 )
0
(z ≤ z 0 )
(14.5)
The origin of the image-plane, z 0 , acts as the boundary of the region occupied by
surface electrons. If we permit z 0 to vary with the surface coordinates, then z 0 (x, y)
provides a contour of spatial electron distribution being mimic of the STM image.
The SPB features are thus characterized by the z 0 and this effect allows us to choose
z 0 as the key character in the subsequent single-variable parameterization of the
nonuniform-SPB.
There are several models for the energy-dependence of the inelastic damping,
ImV(E), for pure metal. An E
1/3 dependence of the inelastic damping worked well in
the LEED calculations of Ni surfaces [51]. An alternative was proposed by McRae
and Caldwell [57] in 1976 from their VLEED investigations of Ni surfaces, which
was widely used in dealing with other metal surfaces. The damping varies with energy