6
M. I. Stockman
1
2
3
20
40
60
80
100
1
2
3
5
10
15
20
Q
Q
(eV)
(eV)
(a)
(b)
Fig. 1.2 a Quality factor Q for silver and b for gold calculated according to Eq. (1.5) (red) and
Eq. (1.6) (blue) as a function of frequency ω
The SP quality factors Q calculated according to Eqs. (1.5)–(1.6) for gold and
silver using the permittivity data of Ref. [32] are shown in Fig. 1.2. The Q-factors
found from these two definitions agree reasonably well in the red to near-infrared
(near-ir) region but not in the yellow to blue region of the visible spectrum. The
reason is that these two definitions would be equivalent if metals’ permittivity were
precisely described by a Drude-type formula Re ε m (ω) = −ω 2
p /ω 2 , where ω p is the
bulk plasma frequency; ω p ≈ 9 eV for one-electron metals such as silver, copper,
gold, and alkaline metals. This formula is reasonably well applicable in the red and
longer wavelength part of the spectrum, but not in the yellow to blue part where the
D-band transitions are important. Note that silver is a much better plasmonic metal
than gold: its Q-factor is several-fold of that of gold.
The finite skin depth of real metals leads to an effect related to nanoplasmonic
confinement: a phase shift Δϕ for light reflected from a metal mirror deviates from
a value of Δϕ = π characteristic of an ideal metal. As suggested in Ref. [33], this
allows for ultrasmall cavities whose length L ∪ λ. While generally this is a valid
idea, there two problems with Ref. [33] that affect the validity of its specific results.
First, the Fresnel reflection formulas used in this article to calculate Δϕ are only valid
for infinite surfaces but not for the “nanomirrors” in a nanocavity. Second, Eq. (1.1)
of this article expressing Q is incorrect: it contains in the denominator a quantity
∂ [ωImε m (ω)]/∂ω instead of 2Im ε m (ω) as in Eq. (1.5). The correct expression [30]
for Ohmic losses defining the Q-factor, which we reproduce as Eq. (1.108), is proportional to Im ε m (ω) as in Eq. (1.5) and not to ∂ [ωImε m (ω)]/∂ω, which constitutes
a significant difference.
The lifetime τ of the SPs is related to the spectral width as
τ =
1
2γ
.
(1.7)
Note that the SP spectral width γ, quality factor Q, and lifetime τ depend explicitly only on frequency ω and the type of the metal (permittivity ε m ) but not on
the nanosystem’s geometry or surrounding dielectric. However, this geometry and
M. I. Stockman
1
2
3
20
40
60
80
100
1
2
3
5
10
15
20
Q
Q
(eV)
(eV)
(a)
(b)
Fig. 1.2 a Quality factor Q for silver and b for gold calculated according to Eq. (1.5) (red) and
Eq. (1.6) (blue) as a function of frequency ω
The SP quality factors Q calculated according to Eqs. (1.5)–(1.6) for gold and
silver using the permittivity data of Ref. [32] are shown in Fig. 1.2. The Q-factors
found from these two definitions agree reasonably well in the red to near-infrared
(near-ir) region but not in the yellow to blue region of the visible spectrum. The
reason is that these two definitions would be equivalent if metals’ permittivity were
precisely described by a Drude-type formula Re ε m (ω) = −ω 2
p /ω 2 , where ω p is the
bulk plasma frequency; ω p ≈ 9 eV for one-electron metals such as silver, copper,
gold, and alkaline metals. This formula is reasonably well applicable in the red and
longer wavelength part of the spectrum, but not in the yellow to blue part where the
D-band transitions are important. Note that silver is a much better plasmonic metal
than gold: its Q-factor is several-fold of that of gold.
The finite skin depth of real metals leads to an effect related to nanoplasmonic
confinement: a phase shift Δϕ for light reflected from a metal mirror deviates from
a value of Δϕ = π characteristic of an ideal metal. As suggested in Ref. [33], this
allows for ultrasmall cavities whose length L ∪ λ. While generally this is a valid
idea, there two problems with Ref. [33] that affect the validity of its specific results.
First, the Fresnel reflection formulas used in this article to calculate Δϕ are only valid
for infinite surfaces but not for the “nanomirrors” in a nanocavity. Second, Eq. (1.1)
of this article expressing Q is incorrect: it contains in the denominator a quantity
∂ [ωImε m (ω)]/∂ω instead of 2Im ε m (ω) as in Eq. (1.5). The correct expression [30]
for Ohmic losses defining the Q-factor, which we reproduce as Eq. (1.108), is proportional to Im ε m (ω) as in Eq. (1.5) and not to ∂ [ωImε m (ω)]/∂ω, which constitutes
a significant difference.
The lifetime τ of the SPs is related to the spectral width as
τ =
1
2γ
.
(1.7)
Note that the SP spectral width γ, quality factor Q, and lifetime τ depend explicitly only on frequency ω and the type of the metal (permittivity ε m ) but not on
the nanosystem’s geometry or surrounding dielectric. However, this geometry and