14.4 Energy-Dependent 3D-SPB
291
In order to reduce numerical efforts and ensure the uniqueness of solutions, one
can define the SPB parameters as functional dependents on z 0 in the following form
[42]:
z 1 (z 0 ) = z 0 × exp
−
z 0 − z 0M
τ 1
2
,
α(z 0 ) = 1/λ(z 0 ) × exp
−
z 0 − z 0m
τ 2
2
,
(14.11)
where the constants τ 1 and τ 2 are the full width at half maximum (FWHM) of the
Gaussian functions and optimized to be 0.75 and 1.50, respectively, by minimizing
the z 0 in the calculations. The z 0m is estimated to be −1.75 Bohr radii.
The λ in Eq. (14.11) increases monotonically with the outward shift of z 0 [42]:
λ(z 0 ) = λ 0M
x + (1 − x) × exp
−
z 0 − z 0M
λ z
2
, (x = 0.4732) (14.12)
where the λ 0M = 1.275 is the maximum of λ corresponding to z 0M = −3.425 Bohr
radii and λ z = 0.8965. Constants for O–Cu(001) surface were obtained by leastsquare simulation of a z 0 (E) − λ(E) curve.
Equations (14.10) and (14.11) represent not only correlation among SPB parameters but also hypotheses that, at the dipole site, z 1M ≈ z 0M , α ≈ λ
−1 , while in the
missing-row or ion position, z 1m << z 0m and that the ImV(z) is much less saturated than is ReV(z) in the depressed sites of STM images. The SPB increases its
degree of saturation with the outward shift of the image plane z 0 by polarization.
The z 0 -dependent of the SPB parameters is illustrated in Fig. 14.10.
14.4.5.3 Physical Indication of the SPB
Figure 14.11 shows an interface between the bulk and the vacuum for typical O-metal
surfaces to illustrate the coordinate-resolved z 0 and z 1 . The z-axis originating from
the top layer (z 0L ) directs into the bulk. The two typical broken curves, at the sites
of the dipole and the missing-row vacancy, are ReV(z) of Fig. 14.9a to illustrate the
difference in z 0 and λ from site to site on the surface. That z 0 (x, y) is usually farther
away from the surface than the z 1 (x, y), which results from the contribution of the
surrounding electrons to the image potential. It is also to be noted that:
ρ(z 1 ) = 0.5ρ(bulk, z ≥ z SL ) > ρ(z < z 0 ) ≈ 0,
as defined by the Fermi-z function and the correlation between the ReV(z) and the
ImV(z). In the vacant position, the smallest z 1m is much lower than z 0m because it is
291
In order to reduce numerical efforts and ensure the uniqueness of solutions, one
can define the SPB parameters as functional dependents on z 0 in the following form
[42]:
z 1 (z 0 ) = z 0 × exp
−
z 0 − z 0M
τ 1
2
,
α(z 0 ) = 1/λ(z 0 ) × exp
−
z 0 − z 0m
τ 2
2
,
(14.11)
where the constants τ 1 and τ 2 are the full width at half maximum (FWHM) of the
Gaussian functions and optimized to be 0.75 and 1.50, respectively, by minimizing
the z 0 in the calculations. The z 0m is estimated to be −1.75 Bohr radii.
The λ in Eq. (14.11) increases monotonically with the outward shift of z 0 [42]:
λ(z 0 ) = λ 0M
x + (1 − x) × exp
−
z 0 − z 0M
λ z
2
, (x = 0.4732) (14.12)
where the λ 0M = 1.275 is the maximum of λ corresponding to z 0M = −3.425 Bohr
radii and λ z = 0.8965. Constants for O–Cu(001) surface were obtained by leastsquare simulation of a z 0 (E) − λ(E) curve.
Equations (14.10) and (14.11) represent not only correlation among SPB parameters but also hypotheses that, at the dipole site, z 1M ≈ z 0M , α ≈ λ
−1 , while in the
missing-row or ion position, z 1m << z 0m and that the ImV(z) is much less saturated than is ReV(z) in the depressed sites of STM images. The SPB increases its
degree of saturation with the outward shift of the image plane z 0 by polarization.
The z 0 -dependent of the SPB parameters is illustrated in Fig. 14.10.
14.4.5.3 Physical Indication of the SPB
Figure 14.11 shows an interface between the bulk and the vacuum for typical O-metal
surfaces to illustrate the coordinate-resolved z 0 and z 1 . The z-axis originating from
the top layer (z 0L ) directs into the bulk. The two typical broken curves, at the sites
of the dipole and the missing-row vacancy, are ReV(z) of Fig. 14.9a to illustrate the
difference in z 0 and λ from site to site on the surface. That z 0 (x, y) is usually farther
away from the surface than the z 1 (x, y), which results from the contribution of the
surrounding electrons to the image potential. It is also to be noted that:
ρ(z 1 ) = 0.5ρ(bulk, z ≥ z SL ) > ρ(z < z 0 ) ≈ 0,
as defined by the Fermi-z function and the correlation between the ReV(z) and the
ImV(z). In the vacant position, the smallest z 1m is much lower than z 0m because it is