- Email: [email protected]
Enhanced transmission of the femtosecond laser pulse through metallic nanofilm
Physics Letters A 378 (2014) 975–977
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Enhanced transmission of the femtosecond laser pulse through metallic nanofilm S.G. Bezhanov a,b , A.P. Kanavin a , S.A. Uryupin a,b,∗ a b
P.N. Lebedev Physical Institute of the RAS, 119991, Leninskiy pr., 53, Moscow, Russia NRNU MEPhI, 115409, Kashirskoe sh., 31, Moscow, Russia
a r t i c l e
i n f o
Article history: Received 25 December 2013 Accepted 15 January 2014 Available online 31 January 2014 Communicated by V.M. Agranovich
a b s t r a c t We investigate the optical properties of metallic nanofilm heated by the femtosecond laser pulse. Equations for fields and temperatures were solved to calculate the transmission coefficient. It is found that fast electron heating upon absorption of the pulse leads to an increase in the intensity of radiation passing through the metallic nanofilm. © 2014 Elsevier B.V. All rights reserved.
Keywords: Femtosecond pulse Metallic nanofilm Electron temperature
1. Introduction Optical properties of metallic films heated in the absorption of femtosecond pulses are studied for quite long time (see e.g. [1–5]). Regularities of absorption and reflection by the films having thickness much greater than skin depth do not differ from those inherent to bulk metal [1,6]. These regularities permit adequate description within the framework of nonlinear electrodynamics by taking into account relatively fast strong electron heating and rather weak slow heating of the lattice. It is found that quantitative description of the peculiar optical properties of bulk metallic samples implies accounting of the inhomogeneity of the temperature distribution inside the skin layer [7–9]. For nanoscale films having thickness comparable to the skin depth decent accuracy could be obtained by using the uniform temperature approximation. Such possibility arises due to very high electron thermal conductivity which leads to electron temperature smoothing across the film thickness within time comparable to the femtosecond pulse duration [10]. In the present report we use the uniform temperature approximation to reveal the effect of enhanced transmission of the heating laser pulse through metallic nanofilm. Due to the heating of electrons and lattice in the absorption of a femtosecond pulse effective electron collision frequency ν is substantially increasing. When ν becomes comparable with fundamental frequency of the pulse ω the transition from high-frequency skin effect to the normal one takes place. For such conditions the skin layer depth increases, enhancing intensity of the pulse transmitted through the film. The
*
Corresponding author. E-mail address: [email protected] (S.A. Uryupin).
0375-9601/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2014.01.043
possibility of implementing this effect is quantitatively demonstrated by the example of the interaction of femtosecond pulse with nanoscale gold film. It is shown that the effect of enhanced transmission is greater for lower radiation fundamental frequencies and for higher incident radiation flux densities. 2. Basic equations Let us consider an interaction of femtosecond pulse of laser radiation with metallic film occupying space 0 z L, where film thickness L is comparable with skin layer thickness δ . We assume that fundamental frequency of radiation and effective collision frequency of the conduction electrons satisfy the inequality
|ω + i ν | v F /δ,
(1)
where v F is the Fermi velocity. The electric field of the pulse incident normally on the surface of the film can be expressed as
E L (t − kr/ω) = Een (t − kr/ω) sin(ωt − kr),
(2)
where wave vector k = (0, 0, k), k = ω/c, c is the speed of light, Een (t ) = (0, E en (t ), 0), and the envelope E en (t ) weakly changes over time ∼ 1/ω . Under the conditions of inequality (1) optical properties of the uniform film which parameters weakly change during the time ∼ 1/ω are approximately described by dielectric function
ε(ω) = ε0 (ω) − ω2p /ω(ω + i ν ),
(3)
where ε0 (ω) describes the contribution of bound electrons and lattice and ω p is the plasma frequency of the metal. For the aforementioned conditions the solution of the Maxwell equations in the
976
S.G. Bezhanov et al. / Physics Letters A 378 (2014) 975–977
film and in the vacuum, which takes into account the continuity of tangential components of electric and magnetic fields, for the field components along y-axis can be written in the following form (see [10] for details):
1
E ref ( z, t ) =
2i
E en (t + z/c )
× exp(i ωt + ikz) R (−ω) − c .c . , 1 E en (t ) exp(i ωt ) F ( z, −ω) − c .c . , E ( z, t ) = 2i 1
E tr ( z, t ) =
2i
E en t − ( z − L )/c
z < 0,
(4)
0 < z < L,
(5)
× exp i ωt − ik( z − L ) T (−ω) − c .c . ,
z > L,
(6)
where functions F ( z, ω), R (ω) and T (ω), that define the field in the media, reflected field and transmitted field, are given by
2ω
F ( z, ω) =
D
R (ω) = −1 − T (ω) = −
2ω
D
2ω
2 2
D
i κ c exp −i
D = κ c −ω 2
ω sinh(κ L ) + i κ c cosh(κ L ) , ω c
2
2
(7) (8)
(9)
L .
In Eqs. (7)–(9) the notations D and
ω sinh κ (z − L ) − i κ c cosh κ (z − L ) ,
κ are used:
sinh(κ L ) − 2i ωκ c cosh(κ L ),
(10) (11)
According to (4)–(10) the structure and magnitude of the fields are strongly dependent on parameter κ (11). From (3) and (11) we see that value of κ is determined by effective electron collision frequency. In pure normal metals this frequency can be approximated by the expression [6,11]
ν = νep
T lat T0
+a
2
T h¯ ε F
2
1+
h¯ ω
2
2πκ T
,
(12)
where νep is the electron–phonon collision frequency at temperature T 0 which is close to the room temperature, ε F is the Fermi energy, h¯ is the Plank constant, is the Boltzmann constant, a is the numerical value which depends on band structure, T lat and T are lattice and electron temperatures respectively. These temperatures change in time upon absorption of field inside the film. Strictly speaking due to field inhomogeneity the heating is also inhomogeneous. However, for L ∼ δ uniform temperature approximation provides good accuracy due to high electron thermal conductivity of typical metals [10]. In this approximation neglecting heat losses on the boundaries of the film for T and T lat we have the equations
Ce
∂T 1 = ∂t L
L dz Q ( z) − G ( T − T lat ),
(13)
0
∂ T lat = G ( T − T lat ), C lat ∂t
(14)
where C e and C lat are electron and lattice heat capacities, G is the factor defining the energy exchange between the electrons and the lattice. Before the heating we assume that T = T lat = T 0 . Function Q ( z) in (13) describes Joule heating in the absorption of the nonuniform field:
Q ( z) =
2 ω2p 2 ν
E (t ) F ( z, ω) . 8π en ω2 + ν 2
The heating of the film described by Eqs. (13)–(15) leads up to changes in effective electron collision frequency (12) and parameter κ (11) which determines field structure in the film. Due to the alterations in κ the optical properties of the film are also evolving in time. 3. Radiation transmission upon film heating
2
κ = −ω ε(ω)/c .
Fig. 1. The ratio of the energy flux density in the pulse transmitted through the gold film to the energy of the incident pulse, as a function of the maximum flux density in the incident pulse.
(15)
Features of the nonuniform heating of the film and the absorption of field in it are considered in [10]. Unlike [10] further discussion will be devoted to peculiarities of the transmission of relatively strong short pulse through quickly heated homogeneous nanoscale film. For definiteness sake we assume that the energy flux density in the pulse described by the Gaussian distribution
I (t ) =
c 8π
2 E en (t ) =
c 8π
2 E en exp −t 2 /t 2p = I exp −t 2 /t 2p ,
√
(16)
where time t p defines FWHM duration τ = 2 ln 2t p . Since physical properties of gold are known quite well we will present the results of calculation of energy flux density in the transmitted pulse I tr (t ) for the gold film. For numerical solution of Eqs. (13), 2 (14) and calculation of the function I tr (t ) = (c /8π ) E tr (t ), where E tr (t ) ≡ E tr ( z = L , t ), following data are used: T 0 = 300 K, ε F = 5.5 eV, ω p = 1.37 · 1016 s−1 , νep = 1.2 · 1014 s−1 [12], C lat = 2.5 · 107 erg cm−3 K−1 , a = 1 [13]. For electron heat capacity and parameter G the following approximations [13], based on the data of paper [14], are used: C e = C Au T [1 + 3.37(10−4 T ) − 1.28(10−4 T )2 ], G = G Au [1 + 5(10−4 T )2 − 0.79(10−4 T )4 ], where C Au 5.25 · 102 erg cm−3 K−2 and G Au 2.7 · 1018 erg s−1 K−1 cm−3 , and T is measured in kelvins. ∞ Fig. 1 shows the dependence of function Tr( I ) = −∞ dt I tr (t )/ √ π It p from I – maximum energy flux density in the incident pulse. Calculations are performed for L = 2c /ω p , t p = 90 fs when τ 150 fs, ω = 2 · 1015 s−1 , ε0 (ω) 9.8 + 0.06i [12] and for intensities in the range from 1011 W/cm2 to 8 · 1012 W/cm2 . For these I maximum electron temperature does not exceed 2 eV. Temperature T lat calculated at t = 2t p (when the pulse effect becomes almost unimportant) is never above 600 K which is less than the melting temperature T m . From Fig. 1 it is clear that heating of electrons results in the relative increase of the transmission effectiveness by approximately twenty percent. For the action of heating pulse with lower fundamental frequency the effect of enhanced transmission is significantly greater. It can be seen from Fig. 2 which shows the same dependence that in Fig. 1 for the carbon-dioxide laser, when ω = 2 · 1014 s−1 , t p = 200 fs, τ 330 fs, and energy flux density is in the range
S.G. Bezhanov et al. / Physics Letters A 378 (2014) 975–977
977
causes the significant enhancement of energy flux density of the transmitted radiation. 4. Conclusion A significant increase in the efficiency of the radiation transmission through the rapidly heated nanoscale gold film has been predicted. Although calculations were demonstrated only for gold sample, it is clear that such effect occurs for films of other metals where the electron collision frequency increases upon the heating of the electrons. Also we note that the uniform temperature approximation which allowed us to relatively easy uncover the effect is not strictly necessary. The effect persists even if this approximation is lifted, but the description will require additional numerical solution of the field equations and a more complicated equation for the electron temperature. Fig. 2. The same dependence as in Fig. 1, but for the infrared radiation.
from 1011 W/cm2 to 6 · 1012 W/cm2 . For such low frequency the contribution of ε0 (ω) in ε (ω) can be neglected. As can be seen from Fig. 2 for IR radiation the function Tr( I ) increases with I by more than an order of magnitude. In these calculations maximum value of T also does not exceed 2 eV, and T lat at t = 2t p does not exceed 900 K which is again less than T m . The reason of significant increase in transmission can be easily seen from expressions (3), (6), (9)–(11). When inequalities ν ω , ω2p ων · max{1, ε0 (ω)} are satisfied, from (3) and (11) we have
κ 2 c 2 = −i ω2p ω/ν . Let us also assume that |κ c | 2ω| coth(κ L )|. For parameters chosen for calculations the latter inequality is safely fulfilled. Under these conditions from (3), (6), (9)–(11) we obtain
−1
I tr (t ) 2I (t )κ 2 δ 2 cosh(2L /δ) − cos(2L /δ) , (17) √ where δ = (c /ω p ) 2ν /ω is the skin layer depth for normal skin effect. Strong rapid heating of electrons leads to an increase of electron effective collision frequency√ν (12). This rise of ν results in increase of the skin depth δ ∼ ν which, according to (17),
Acknowledgements This work was supported by RFBR (N13-02-01377) and RAS Presidium Program N24. References [1] H.M. Milchberg, R.M. Freeman, S.C. Davey, R.M. More, Phys. Rev. Lett. 61 (1998) 2364. [2] A. Kubo, K. Onda, H. Petek, Z. Sun, Y.S. Jung, H.K. Kim, Nano Lett. 5 (2005) 1123. [3] J. Wang, C. Guo, Phys. Rev. B 75 (2007) 184304. [4] J. Krüger, D. Dufft, R. Koter, A. Hertwig, Appl. Surf. Sci. 253 (2007) 7815. [5] A.M. Chen, H.F. Xu, Y.F. Jiang, L.Z. Sui, D.J. Ding, H. Liu, M.X. Jin, Appl. Surf. Sci. 257 (2010) 1678. [6] S.G. Bezhanov, A.P. Kanavin, S.A. Uryupin, Opt. Spectrosc. 114 (2013) 384. [7] V.A. Isakov, A.P. Kanavin, S.A. Uryupin, Quantum Electron. 36 (2006) 928. [8] A.P. Kanavin, K.N. Mishchik, S.A. Uryupin, J. Russ. Laser Res. 29 (2008) 123. [9] S.G. Bezhanov, A.P. Kanavin, S.A. Uryupin, Quantum Electron. 41 (2011) 447. [10] S.G. Bezhanov, A.P. Kanavin, S.A. Uryupin, Quantum Electron. (2014). [11] R.N. Gurzhi, Zh. Eksp. Teor. Fiz. 35 (1958) 965, Sov. Phys. JETP 8 (1959) 673. [12] P.B. Johnson, R.W. Christy, Phys. Rev. B 6 (1972) 4370. [13] S.G. Bezhanov, A.P. Kanavin, S.A. Uryupin, Quantum Electron. 42 (2012) 447. [14] Zh. Lin, L.V. Zhigilei, V. Celli, Phys. Rev. B 77 (2008) 075113.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Enhanced transmission of the femtosecond laser pulse through metallic nanofilm S.G. Bezhanov a,b , A.P. Kanavin a , S.A. Uryupin a,b,∗ a b
P.N. Lebedev Physical Institute of the RAS, 119991, Leninskiy pr., 53, Moscow, Russia NRNU MEPhI, 115409, Kashirskoe sh., 31, Moscow, Russia
a r t i c l e
i n f o
Article history: Received 25 December 2013 Accepted 15 January 2014 Available online 31 January 2014 Communicated by V.M. Agranovich
a b s t r a c t We investigate the optical properties of metallic nanofilm heated by the femtosecond laser pulse. Equations for fields and temperatures were solved to calculate the transmission coefficient. It is found that fast electron heating upon absorption of the pulse leads to an increase in the intensity of radiation passing through the metallic nanofilm. © 2014 Elsevier B.V. All rights reserved.
Keywords: Femtosecond pulse Metallic nanofilm Electron temperature
1. Introduction Optical properties of metallic films heated in the absorption of femtosecond pulses are studied for quite long time (see e.g. [1–5]). Regularities of absorption and reflection by the films having thickness much greater than skin depth do not differ from those inherent to bulk metal [1,6]. These regularities permit adequate description within the framework of nonlinear electrodynamics by taking into account relatively fast strong electron heating and rather weak slow heating of the lattice. It is found that quantitative description of the peculiar optical properties of bulk metallic samples implies accounting of the inhomogeneity of the temperature distribution inside the skin layer [7–9]. For nanoscale films having thickness comparable to the skin depth decent accuracy could be obtained by using the uniform temperature approximation. Such possibility arises due to very high electron thermal conductivity which leads to electron temperature smoothing across the film thickness within time comparable to the femtosecond pulse duration [10]. In the present report we use the uniform temperature approximation to reveal the effect of enhanced transmission of the heating laser pulse through metallic nanofilm. Due to the heating of electrons and lattice in the absorption of a femtosecond pulse effective electron collision frequency ν is substantially increasing. When ν becomes comparable with fundamental frequency of the pulse ω the transition from high-frequency skin effect to the normal one takes place. For such conditions the skin layer depth increases, enhancing intensity of the pulse transmitted through the film. The
*
Corresponding author. E-mail address: [email protected] (S.A. Uryupin).
0375-9601/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2014.01.043
possibility of implementing this effect is quantitatively demonstrated by the example of the interaction of femtosecond pulse with nanoscale gold film. It is shown that the effect of enhanced transmission is greater for lower radiation fundamental frequencies and for higher incident radiation flux densities. 2. Basic equations Let us consider an interaction of femtosecond pulse of laser radiation with metallic film occupying space 0 z L, where film thickness L is comparable with skin layer thickness δ . We assume that fundamental frequency of radiation and effective collision frequency of the conduction electrons satisfy the inequality
|ω + i ν | v F /δ,
(1)
where v F is the Fermi velocity. The electric field of the pulse incident normally on the surface of the film can be expressed as
E L (t − kr/ω) = Een (t − kr/ω) sin(ωt − kr),
(2)
where wave vector k = (0, 0, k), k = ω/c, c is the speed of light, Een (t ) = (0, E en (t ), 0), and the envelope E en (t ) weakly changes over time ∼ 1/ω . Under the conditions of inequality (1) optical properties of the uniform film which parameters weakly change during the time ∼ 1/ω are approximately described by dielectric function
ε(ω) = ε0 (ω) − ω2p /ω(ω + i ν ),
(3)
where ε0 (ω) describes the contribution of bound electrons and lattice and ω p is the plasma frequency of the metal. For the aforementioned conditions the solution of the Maxwell equations in the
976
S.G. Bezhanov et al. / Physics Letters A 378 (2014) 975–977
film and in the vacuum, which takes into account the continuity of tangential components of electric and magnetic fields, for the field components along y-axis can be written in the following form (see [10] for details):
1
E ref ( z, t ) =
2i
E en (t + z/c )
× exp(i ωt + ikz) R (−ω) − c .c . , 1 E en (t ) exp(i ωt ) F ( z, −ω) − c .c . , E ( z, t ) = 2i 1
E tr ( z, t ) =
2i
E en t − ( z − L )/c
z < 0,
(4)
0 < z < L,
(5)
× exp i ωt − ik( z − L ) T (−ω) − c .c . ,
z > L,
(6)
where functions F ( z, ω), R (ω) and T (ω), that define the field in the media, reflected field and transmitted field, are given by
2ω
F ( z, ω) =
D
R (ω) = −1 − T (ω) = −
2ω
D
2ω
2 2
D
i κ c exp −i
D = κ c −ω 2
ω sinh(κ L ) + i κ c cosh(κ L ) , ω c
2
2
(7) (8)
(9)
L .
In Eqs. (7)–(9) the notations D and
ω sinh κ (z − L ) − i κ c cosh κ (z − L ) ,
κ are used:
sinh(κ L ) − 2i ωκ c cosh(κ L ),
(10) (11)
According to (4)–(10) the structure and magnitude of the fields are strongly dependent on parameter κ (11). From (3) and (11) we see that value of κ is determined by effective electron collision frequency. In pure normal metals this frequency can be approximated by the expression [6,11]
ν = νep
T lat T0
+a
2
T h¯ ε F
2
1+
h¯ ω
2
2πκ T
,
(12)
where νep is the electron–phonon collision frequency at temperature T 0 which is close to the room temperature, ε F is the Fermi energy, h¯ is the Plank constant, is the Boltzmann constant, a is the numerical value which depends on band structure, T lat and T are lattice and electron temperatures respectively. These temperatures change in time upon absorption of field inside the film. Strictly speaking due to field inhomogeneity the heating is also inhomogeneous. However, for L ∼ δ uniform temperature approximation provides good accuracy due to high electron thermal conductivity of typical metals [10]. In this approximation neglecting heat losses on the boundaries of the film for T and T lat we have the equations
Ce
∂T 1 = ∂t L
L dz Q ( z) − G ( T − T lat ),
(13)
0
∂ T lat = G ( T − T lat ), C lat ∂t
(14)
where C e and C lat are electron and lattice heat capacities, G is the factor defining the energy exchange between the electrons and the lattice. Before the heating we assume that T = T lat = T 0 . Function Q ( z) in (13) describes Joule heating in the absorption of the nonuniform field:
Q ( z) =
2 ω2p 2 ν
E (t ) F ( z, ω) . 8π en ω2 + ν 2
The heating of the film described by Eqs. (13)–(15) leads up to changes in effective electron collision frequency (12) and parameter κ (11) which determines field structure in the film. Due to the alterations in κ the optical properties of the film are also evolving in time. 3. Radiation transmission upon film heating
2
κ = −ω ε(ω)/c .
Fig. 1. The ratio of the energy flux density in the pulse transmitted through the gold film to the energy of the incident pulse, as a function of the maximum flux density in the incident pulse.
(15)
Features of the nonuniform heating of the film and the absorption of field in it are considered in [10]. Unlike [10] further discussion will be devoted to peculiarities of the transmission of relatively strong short pulse through quickly heated homogeneous nanoscale film. For definiteness sake we assume that the energy flux density in the pulse described by the Gaussian distribution
I (t ) =
c 8π
2 E en (t ) =
c 8π
2 E en exp −t 2 /t 2p = I exp −t 2 /t 2p ,
√
(16)
where time t p defines FWHM duration τ = 2 ln 2t p . Since physical properties of gold are known quite well we will present the results of calculation of energy flux density in the transmitted pulse I tr (t ) for the gold film. For numerical solution of Eqs. (13), 2 (14) and calculation of the function I tr (t ) = (c /8π ) E tr (t ), where E tr (t ) ≡ E tr ( z = L , t ), following data are used: T 0 = 300 K, ε F = 5.5 eV, ω p = 1.37 · 1016 s−1 , νep = 1.2 · 1014 s−1 [12], C lat = 2.5 · 107 erg cm−3 K−1 , a = 1 [13]. For electron heat capacity and parameter G the following approximations [13], based on the data of paper [14], are used: C e = C Au T [1 + 3.37(10−4 T ) − 1.28(10−4 T )2 ], G = G Au [1 + 5(10−4 T )2 − 0.79(10−4 T )4 ], where C Au 5.25 · 102 erg cm−3 K−2 and G Au 2.7 · 1018 erg s−1 K−1 cm−3 , and T is measured in kelvins. ∞ Fig. 1 shows the dependence of function Tr( I ) = −∞ dt I tr (t )/ √ π It p from I – maximum energy flux density in the incident pulse. Calculations are performed for L = 2c /ω p , t p = 90 fs when τ 150 fs, ω = 2 · 1015 s−1 , ε0 (ω) 9.8 + 0.06i [12] and for intensities in the range from 1011 W/cm2 to 8 · 1012 W/cm2 . For these I maximum electron temperature does not exceed 2 eV. Temperature T lat calculated at t = 2t p (when the pulse effect becomes almost unimportant) is never above 600 K which is less than the melting temperature T m . From Fig. 1 it is clear that heating of electrons results in the relative increase of the transmission effectiveness by approximately twenty percent. For the action of heating pulse with lower fundamental frequency the effect of enhanced transmission is significantly greater. It can be seen from Fig. 2 which shows the same dependence that in Fig. 1 for the carbon-dioxide laser, when ω = 2 · 1014 s−1 , t p = 200 fs, τ 330 fs, and energy flux density is in the range
S.G. Bezhanov et al. / Physics Letters A 378 (2014) 975–977
977
causes the significant enhancement of energy flux density of the transmitted radiation. 4. Conclusion A significant increase in the efficiency of the radiation transmission through the rapidly heated nanoscale gold film has been predicted. Although calculations were demonstrated only for gold sample, it is clear that such effect occurs for films of other metals where the electron collision frequency increases upon the heating of the electrons. Also we note that the uniform temperature approximation which allowed us to relatively easy uncover the effect is not strictly necessary. The effect persists even if this approximation is lifted, but the description will require additional numerical solution of the field equations and a more complicated equation for the electron temperature. Fig. 2. The same dependence as in Fig. 1, but for the infrared radiation.
from 1011 W/cm2 to 6 · 1012 W/cm2 . For such low frequency the contribution of ε0 (ω) in ε (ω) can be neglected. As can be seen from Fig. 2 for IR radiation the function Tr( I ) increases with I by more than an order of magnitude. In these calculations maximum value of T also does not exceed 2 eV, and T lat at t = 2t p does not exceed 900 K which is again less than T m . The reason of significant increase in transmission can be easily seen from expressions (3), (6), (9)–(11). When inequalities ν ω , ω2p ων · max{1, ε0 (ω)} are satisfied, from (3) and (11) we have
κ 2 c 2 = −i ω2p ω/ν . Let us also assume that |κ c | 2ω| coth(κ L )|. For parameters chosen for calculations the latter inequality is safely fulfilled. Under these conditions from (3), (6), (9)–(11) we obtain
−1
I tr (t ) 2I (t )κ 2 δ 2 cosh(2L /δ) − cos(2L /δ) , (17) √ where δ = (c /ω p ) 2ν /ω is the skin layer depth for normal skin effect. Strong rapid heating of electrons leads to an increase of electron effective collision frequency√ν (12). This rise of ν results in increase of the skin depth δ ∼ ν which, according to (17),
Acknowledgements This work was supported by RFBR (N13-02-01377) and RAS Presidium Program N24. References [1] H.M. Milchberg, R.M. Freeman, S.C. Davey, R.M. More, Phys. Rev. Lett. 61 (1998) 2364. [2] A. Kubo, K. Onda, H. Petek, Z. Sun, Y.S. Jung, H.K. Kim, Nano Lett. 5 (2005) 1123. [3] J. Wang, C. Guo, Phys. Rev. B 75 (2007) 184304. [4] J. Krüger, D. Dufft, R. Koter, A. Hertwig, Appl. Surf. Sci. 253 (2007) 7815. [5] A.M. Chen, H.F. Xu, Y.F. Jiang, L.Z. Sui, D.J. Ding, H. Liu, M.X. Jin, Appl. Surf. Sci. 257 (2010) 1678. [6] S.G. Bezhanov, A.P. Kanavin, S.A. Uryupin, Opt. Spectrosc. 114 (2013) 384. [7] V.A. Isakov, A.P. Kanavin, S.A. Uryupin, Quantum Electron. 36 (2006) 928. [8] A.P. Kanavin, K.N. Mishchik, S.A. Uryupin, J. Russ. Laser Res. 29 (2008) 123. [9] S.G. Bezhanov, A.P. Kanavin, S.A. Uryupin, Quantum Electron. 41 (2011) 447. [10] S.G. Bezhanov, A.P. Kanavin, S.A. Uryupin, Quantum Electron. (2014). [11] R.N. Gurzhi, Zh. Eksp. Teor. Fiz. 35 (1958) 965, Sov. Phys. JETP 8 (1959) 673. [12] P.B. Johnson, R.W. Christy, Phys. Rev. B 6 (1972) 4370. [13] S.G. Bezhanov, A.P. Kanavin, S.A. Uryupin, Quantum Electron. 42 (2012) 447. [14] Zh. Lin, L.V. Zhigilei, V. Celli, Phys. Rev. B 77 (2008) 075113.