7 Ultrafast and Nonlinear Plasmon Dynamics
261
at the surface varies over approximately the Thomas-Fermi screening length, which
leads to the spatial confinement of the induced nonlinear current to a subnanometer
region. Therefore, a classical electromagnetic description to model surface SHG fails
and a quantum mechanical treatment is necessary in order to accurately incorporate
the surface charge density and screening effects.
The nonlinear surface polarization is described by susceptibility tensor components, with χ
(2)
zzz,s describing the surface normal current, which is expected to be
the largest contribution to surface SHG as it is the most sensitive to the structural
and electric field change across the interface. The other components for the in-plane
surface current are χ
(2)
x xz,s , or equivalently χ
(2)
yyz,s , χ
(2)
xzx,s , etc., and χ
(2)
zx x,s = χ
(2)
zyy,s ,
due to the symmetries of the tensor.
Rudnick and Stern [26] parametrized three contributions to SHG in terms of the
phenomenological constants a(ω) ◦ χ
(2)
∞,s , b(ω) ◦ χ
(2)
∗,s , and d(ω) ◦ χ
(2)
B . In the
Drude model, b(ω) = −1, d(ω) = 1, and a(ω) was initially assumed to be close
to 1. Several models for calculating the spatial distribution of the electron density
close to the surface and deriving a(ω) were developed, chiefly using hydrodynamic
arguments to derive the surface potential within a jellium framework, which treats
the metal surface as a homogeneous gas of interacting free electrons in a background
of uniform positive charge. These models provided an intuitive description of the
system, but underestimated the magnitude of the SHG by an order of magnitude
[27]. Subsequent models used density functional theory to describe the electron–
electron interactions at the surface, which incorporates the screening of the external
electric field [28]. While these models typically agree qualitatively with experimental
observations, particularly in the long wavelength limit, other effects can also become
significant and change the relative contributions of the different polarization terms.
Additional susceptibility components may also appear when the lattice is considered.
For example, close to resonances the bound electrons may contribute more strongly
to the nonlinear polarization, producing a bulk response larger than the surface, even
in centrosymmetric materials. Consistent with this, a strong enhancement in SHG
has been observed in noble metals close to the interband transition, in addition to the
usual off-resonant nonlinear signal [29, 30].
For noble metals, the k-dependence of the electronic structure is typically neglected in modeling the SHG response. The high density of states and overlapping
d-bands allow for a continuum of transitions with different symmetries, as shown
in Fig. 7.10, producing broad SHG peaks. When the excitation frequency is such
that the band gap E g is less than 2ω, the SHG response is generally dominated by
transitions where both the initial and the intermediate states are in the d-band. This
sensitivity of SHG to the d-band can provide spectroscopic material specificity.
The literature disagrees on quantitative measurements of the magnitude of the
SHG signal and its components, due to the high sensitivity of SHG to surface structure
and contamination. In particular for Ag and Al, accurate measurements require ultra
high vacuum to ensure clean surfaces. The second-harmonic responses from Ag(111)
and Au(111) surfaces, far off-resonant at ω = 0.81 eV, were found to be dominated
by the surface normal susceptibility, as expected [31]. Significant contributions were
261
at the surface varies over approximately the Thomas-Fermi screening length, which
leads to the spatial confinement of the induced nonlinear current to a subnanometer
region. Therefore, a classical electromagnetic description to model surface SHG fails
and a quantum mechanical treatment is necessary in order to accurately incorporate
the surface charge density and screening effects.
The nonlinear surface polarization is described by susceptibility tensor components, with χ
(2)
zzz,s describing the surface normal current, which is expected to be
the largest contribution to surface SHG as it is the most sensitive to the structural
and electric field change across the interface. The other components for the in-plane
surface current are χ
(2)
x xz,s , or equivalently χ
(2)
yyz,s , χ
(2)
xzx,s , etc., and χ
(2)
zx x,s = χ
(2)
zyy,s ,
due to the symmetries of the tensor.
Rudnick and Stern [26] parametrized three contributions to SHG in terms of the
phenomenological constants a(ω) ◦ χ
(2)
∞,s , b(ω) ◦ χ
(2)
∗,s , and d(ω) ◦ χ
(2)
B . In the
Drude model, b(ω) = −1, d(ω) = 1, and a(ω) was initially assumed to be close
to 1. Several models for calculating the spatial distribution of the electron density
close to the surface and deriving a(ω) were developed, chiefly using hydrodynamic
arguments to derive the surface potential within a jellium framework, which treats
the metal surface as a homogeneous gas of interacting free electrons in a background
of uniform positive charge. These models provided an intuitive description of the
system, but underestimated the magnitude of the SHG by an order of magnitude
[27]. Subsequent models used density functional theory to describe the electron–
electron interactions at the surface, which incorporates the screening of the external
electric field [28]. While these models typically agree qualitatively with experimental
observations, particularly in the long wavelength limit, other effects can also become
significant and change the relative contributions of the different polarization terms.
Additional susceptibility components may also appear when the lattice is considered.
For example, close to resonances the bound electrons may contribute more strongly
to the nonlinear polarization, producing a bulk response larger than the surface, even
in centrosymmetric materials. Consistent with this, a strong enhancement in SHG
has been observed in noble metals close to the interband transition, in addition to the
usual off-resonant nonlinear signal [29, 30].
For noble metals, the k-dependence of the electronic structure is typically neglected in modeling the SHG response. The high density of states and overlapping
d-bands allow for a continuum of transitions with different symmetries, as shown
in Fig. 7.10, producing broad SHG peaks. When the excitation frequency is such
that the band gap E g is less than 2ω, the SHG response is generally dominated by
transitions where both the initial and the intermediate states are in the d-band. This
sensitivity of SHG to the d-band can provide spectroscopic material specificity.
The literature disagrees on quantitative measurements of the magnitude of the
SHG signal and its components, due to the high sensitivity of SHG to surface structure
and contamination. In particular for Ag and Al, accurate measurements require ultra
high vacuum to ensure clean surfaces. The second-harmonic responses from Ag(111)
and Au(111) surfaces, far off-resonant at ω = 0.81 eV, were found to be dominated
by the surface normal susceptibility, as expected [31]. Significant contributions were