- Email: [email protected]
Thickness dependence of the surface-polariton relaxation rates in a crystal slab
Solid State Communications, Vol. 44, No. 6, pp. 937-939, 1982. Printed in Great Britain.
0038-1098/82/420937--03 $03.00/0 Pergamon Press Ltd.
THICKNESS DEPENDENCE OF THE SURFACE-POLARITON RELAXATION RATES IN A CRYSTAL SLAB M.S. Toma§ Rudjer Bo§kovi6 Institute, 41001 Zagreb, Croatia, Yugoslavia and Z. Lenac* Pedagogical Faculty, Rijeka, Yugoslavia
(Received lO March 1982 by P.H. Dederichs) A few useful formulas are derived concerning the damping properties of long-wavelength surface polaritons (SP) in a crystal slab. For example, it is shown that the SP propagation length in a metal slab with thickness 2a satisfying 6ona/c < 1 (c is the velocity of light and con the electronic plasma frequency) is increased by the factor ~(clone) 4 in comparison with the corresponding value for SP in a semi-infinite metal.
THE PROPERTIES OF long-wavelength surface polaritons (SP) in crystals are usually investigated experimentally using optical methods, such as scattering of light from corrugated surface, attenuated total reflection, and Raman scattering [ 1], or by use of electron-loss spectroscopy [2]. In analyzing the spectra, it is usually assumed that SP relaxation rates (damping) and therefore the linewidths are properly described by the imaginary part of the dielectric function, which appears in any dielectric theory of crystal collective oscillations [2]. In most cases, SP damping was regarded simply as a constant essentially equal to the damping of bulk modes. For example, even in high resolution electrontransmission spectroscopy of metals, it was approximated by the deconvoluted width of the volumeplasmon line [3]. However, SP damping is always dependent on SP frequency. This is due not only to various frequency-dependent intrinsic damping processes, but also to the geometry of the solid. In fact, the SP field is localized at the surface between two media and decays exponentially on both sides of the surface. The penetration depth in each medium is a function of SP frequency, so that the amount of SP field in each medium varies with frequency. As a consequence, the contributions of the intrinsic damping mechanisms in the two media to the overall SP damping vary with SP frequency. Thus, even if we describe all intrinsic damping processes in the bulk by some constant damping parameter 7, the damping of surface modes is still frequency dependent. If one of the media is inert (does not provide a mechanism for damping), the damping of SP is always weaker * Also at the Rudjer Bo§kovi6 Institute, 41001 Zagreb, Croatia, Yugoslavia. 937
than that of bulk modes because only part of the SP field is damped. This fact has been proved theoretically for surface polaritons on a semi-inf'mite solid by Nkoma, Loudon, and Tilley [4]. In a crystal slab, the SP frequency and, therefore, the SP damping become dependent on the thickness of the slab. This behaviour of the SP relaxation rate has recently attracted considerable attention [5, 6] in the search for surface polaritons with long lifetimes and their possible application to nonlinear interactions [7, 8]. Fukui, So, and Normandin [5] investigated SP temporal damping theoretically by solving numerically the dispersion relation for surface polaritons in a thin silver film. As it is well known [9], in a thin crystal film, there exist two SP modes with different symmetry properties. Analyzing the variation of the imaginary part of their frequencies with film thickness (keeping the wave vector real), Fukui, So, and Normandin have found that the lifetime of the antisymmetric SP mode in unsupported films increases with decreasing film thickness, in contrast to the behaviour of the symmetric mode. They have also shown that, in a very thin film (~ 160 A), this mode can provide SP with a lifetime of the order of 10 -12 sec, which is much larger than the lifetime of bulk plasmons (~ 10 -14 see) and the lifetime of SP on a semi-infinite solid. Since the propagation length of any collective excitation is simply its group velocity times its lifetime, long-living SP obviously travel long distances before they decay. The value of their propagation length has been estimated to be a few centimeters [5]. SP spatial damping was first investigated, both theoretically and experimentally, by Schoenwald, Burstein, and Elson [10]. Very recently, Sarid [6] has investigated theoretically
938
SURFACE-POLARITON RELAXATION RATES IN A CRYSTAL SLAB
the SP spatial damping in a thin metal slab. He studied numerically the variation of the imaginary part of the SP wave vector with slab thickness (keeping the SP frequency real) and concluded that SP modes which propagate over macroscopic distances can exist in a thin silver film. He called them long-range SP modes. Physically, the possibility for the existence of longliving or long-range surface polaritons in unsupported metal films is due to the fact that, for an antisymmetric SP mode, the electromagnetic field is expelled from the crystal as film thickness decreases [5]. As we have already said, if the surrounding medium is inert, the damping mechanisms in the crystal become less effective and SP damping should decrease. This may not be true for metals only. The same features should be expected for surfacephonon polaritons in dielectrics, for example [ 10]. As we have seen, previous theoretical investigations of SP damping properties in crystal slabs were [5, 6] performed for SP in metals and by use of numerical methods. The transparency of the results was, therefore, somewhat lost. In this Communication we report a few interesting results concerning the dependence of SP relaxation rates in metals on slab thickness. We have derived them from an analytical expression for SP damping in a crystal slab (not necessarily a metallic one) in which its dependence on SP frequency and, particularly, on slab thickness is more transparent. The expression for SP damping in crystal slabs was originally obtained by Nkoma [ 11 ] as a by-product in the theory of Raman scattering by surface polaritons in a thin film. We have rederived it starting from a quantized form for the surface polariton field [12] and using a perturbative approach. Our theory applies to crystals having a single kind of polar optical excitation and is therefore suitable for dielectrics such as alkali halides, for I I I - V compounds, and for free-electron-like metals. Among various damping mechanisms, we took into account those which are due to friction forces between ions in dielectrics or Drude processes in metals, and described them by a small phenomenological constant % Then, to the first order in % the frequency-dependent SP relaxation rate is given by [4] ,/(co) = 27((K)/(H)), where K and H are the SP kinetic (mechanical) and total energy, respectively, in the absence of damping and ( ) denote the cycle average in classical physics or the diagonal matrix element in quantum mechanics. Having the quantized form for the SP field, as in [12], it is a matter of simple algebra to show that surface polaritons in an unsupported crystal slab lose their energy at the rate:
where the SP frequency co, the two-dimensional SP wave vector k, parallel to the slab surfaces, and the slab thickness 2a are intercorrelated through the surface-polariton dispersion relation [9]
e(w+_) = -- a/ao(th aa) +-1.
(2)
Here the real quantities a = (k 2 - - e ( x ) 2 / c 2 ) 1/2 and a0 = (k 2 - co2/c2)V2 are the penetration depths of the SP field in the crystal and in the vacuum, respectively, c is the velocity of light, and e(co) the long-wavelength real dielectric function of the crystal. The subscripts + and denote, respectively, the antisymmetric and symmetric SP modes in the slab. [We have omitted them in the curly brackets of equation (1).] In equation (1), k and, therefore, a and a0 are regarded as functions of co through the solution of the dispersion relations (2). We apply equation (1) to a metallic slab which we describe by the free-electron dielectric function e(co) = 1 - co,/co 2, where cop is the electronic plasma frequency. This is a good approximation for most metals, especially noble ones, in the low-frequency region co ¢ cop. For example [13], it holds for silver when hco < 2.0 eV (hcop = 9.08 eV), for copper when boo < 1.85 eV (hWp = 9.15 eV), etc. In any case, small damped SP are expected at frequencies co~cop ~- 10-1-10 -2, i.e. they can be excited by light with wavelengths from the infrared, X ~ 1-10t.tm [5, 10]. In this frequency region our Drude model for the dielectric function of metals should be satisfactory. Regarding w/cop as a small quantity, we can make the approximations e(co) "" -¢o2/co 2, a ~-- cop/c = kp, and obtain solutions of the SP dispersion relations (2) -
-
k~ ~-- co2/c2(1 + co2/cop2 th -+2 kpa).
(3)
This gives or0 = kpco2/co~ th -+1 kpa for the SP penetration depth in the vacuum, and instead of equation (1) we have
7+_(co, a) ~ V(co/cop)2F+(kpa), _
+
2X
F+(x) = th-+2x(1-s-ff--~- ) . Equations (4) show explicitly the dependence of SP damping on slab thickness. The functions F+_(x) are rather smooth for x > 1 and approach 1 for large x. However, for x < 1, F._(x) changes rapidly and the 2 4 values of the leading terms are F÷(x) ~ ~x and F_(x) ~ 2x -2. Therefore, from equations (4) we obtain -1
s~_~o~(k~l~ _ ~Io~) -+ (k~l~,~ - I) 1
7±(co, a) = 7 l l + m co de sh 2aa -(1 + k ~ / @ ) +- ( k ~ l a ~ - - 1) 2 dco 2ota
Vol. 44, No. 6
(1)
(4)
Vol. 44, No. 6
SURFACE-POLARITON RELAXATION RATES IN A CRYSTAL SLAB
( z3(kna)4 ]
~+_(co,a) ~- ~(co,~) 2(~na)-U'
kna < 1,
(5)
where 3'(co, ~ ) = 7(co[con) 2 is the SP damping in a semiinfinite metal. For metals, kn is of the order of 10 -3 A -1 (for example, for silver (copper) with [13] con = 1.38 (1.39) x 1016 sec -1 , we have k;, ~ 4.6 x 10 -3 A-l), so that for a reasonably thin slab (2a = 200 A), we have kna = 0.5 and we can employ equation (5) with a relatively small error. From equation (5) it is perfectly clear that, owing to the slab thickness, the damping of the antisymmetric (+) SP mode can be order of magnitude smaller than that of the symmetric (--) mode. One should expect such behaviour because the antisymmetric SP mode is more "photon-like" with the electromagnetic field extending far outside the slab, while the symmetric SP mode has a more mechanical character, with the electromagnetic field strongly attenuated outside the slab. As SP damping decreases with increasing ratio of the electromagnetic to mechanical energy, it follows that the antisymmetric mode must be less damped than the symmetric one. Furthermore, the decrease of slab thickness makes this difference between the two modes more pronounced. Therefore, in thin t'rims, the antisymmetric mode can provide SP with very long lifetime, T÷ = 7+1(co, a). The propagation length L+(co, a) of these long-living SP is, according to equation (3) given by
L+(co, a) = ldcol 1--~['/+-1 ( co, a) "" cl,+l(co, a),
(6)
or, if we use equation (5), it is given by
L+(co, a) = 3 (kna)-4L+(co, oo), kna < 1,
we can, in principle, allow for SP of almost arbitrarily long lifetimes or propagation lengths. For example, Schoenwald, Burstein, and Elson [ 10] estimated that for SP in semi-infinite copper excited by laser light at a wavelength of 10.6 #m, the propagation length should be 1.9 cm. The value they obtained by measurement was 1.6 cm. However, if they had used a thin (~ 200 A) copper ftlm, they would have obtained a 24 times larger value, as follows directly from equation (5) with kpa 0.5 (copper case). This also agrees with the result of Sarid [6] for SP with frequencies corresponding to those of visible light. Of course, SP with such long propagation lengths can hardly be observed in a real metal where the effects associated with various inhomogeneities in solid, surface roughness, or the existence of an oxide layer on metal also influence SP damping. In conclusion, we have derived expressions which can be easily used to estimate the damping properties (lifetime, propagation length) of long-wavelength surface polaritons in an unsupported slab. We have analytically discussed their dependence on surface-polariton frequency and slab thickness for metals. The theory can be easily applied to surface-phonon polaritons in dielectrics and extended to more realistic geometries, e.g. supported slab, sandwiches, etc. [ 14]. We have also shown that a suitable choice of material and geometrical parameters offers the possibility for the existence of surface polaritons with very long lifetimes and large propagation lengths. When excited, they can be used as a tool for further investigations, e.g. surface roughness, various nonlinear effects, etc.
(7)
where L+(co, oo) = cT-l(con/co)2 is the SP propagation length in a semi-infinite metal. To check our results, we compared them with those of Fukui, So, and Normandin [5], because their results are expected to be correct to all orders in 7- They estimated that in a thin ( " 160 A) silver film the propagation length is L+ "" 3 cm for SP with frequency corresponding to the frequency of light with the wavelength X = 1.06 /~m. Using the same parameters as those employed in [5], i.e. cop = 1.48 × 1016 sec -1 and 7 = 3.22 x 1013 sec -1, we obtained rather large values for co/con and kna, namely, co/con ~ 0.12 and kna ~--0.4, and found from equations (4) and (6) that L+ ~- 3.4 cm. When we used equation (5) instead of equation (4), we obtained a slightly larger value, L+ = 3.8 cm. Therefore, we concluded that an estimate of the damping properties of surface polaritons in a very thin metal film, based on the use of equations (1), (4), or (5), can be even quantitatively correct for the case of small intrinsic damping and for low-frequency SP. Equations (5) and (7) contain interesting results. They show that simply by changing slab thickness,
939
REFERENCES 1. 2.
See, for example, A. Otto, FestkOrperprobleme XIV, 1 (1974). A recent review on this field is H. Raether, Exci-
tation of Plasmons and lnterband Transitions by Electrons. Springer Tracts in Modem Physics 88 3. 4. 5.
6. 7. 8. 9. 10. 11. 12. 13. 14.
(1980). R.B. Petit, J. Silcox & R. Vincent, Phys. Rev. B11, 3116 (1975). J. Nkoma, R. Loudon & D. TiUey, J. Phys. C7, 3547 (1974). M. Fukui, V.C.Y. So & R. Normandin, Phys. Status Solidi 91, K61 (1979). D. Sarid, Phys. Rev. Lett. 47, 1927 (1981). D.L. Mills, Solid State Commun. 24, 669 (1977). M. Fukui & G.I. Stegeman, Solid State Commun. 26,239 (1978). K.L. Kiiewer & R. Fuchs, Phys. Rev. 144,495 (1966). J. Schoenwald, E. Burstein & J.M. Elson, Solid State Commun. 12, 185 (1973). J.S. Nkoma, J. Phys. C8, 3919 (1975). M.S. Toma§, M. Sunji6 & Z. Lenac, Fizika 14, 77 (1982). T. Hollstein, U. Krebig & F. Leis, Phys. Status Solidi (b) 82, 545 (1977). M.S. Toma~ & Z. Lenac (in preparation).
0038-1098/82/420937--03 $03.00/0 Pergamon Press Ltd.
THICKNESS DEPENDENCE OF THE SURFACE-POLARITON RELAXATION RATES IN A CRYSTAL SLAB M.S. Toma§ Rudjer Bo§kovi6 Institute, 41001 Zagreb, Croatia, Yugoslavia and Z. Lenac* Pedagogical Faculty, Rijeka, Yugoslavia
(Received lO March 1982 by P.H. Dederichs) A few useful formulas are derived concerning the damping properties of long-wavelength surface polaritons (SP) in a crystal slab. For example, it is shown that the SP propagation length in a metal slab with thickness 2a satisfying 6ona/c < 1 (c is the velocity of light and con the electronic plasma frequency) is increased by the factor ~(clone) 4 in comparison with the corresponding value for SP in a semi-infinite metal.
THE PROPERTIES OF long-wavelength surface polaritons (SP) in crystals are usually investigated experimentally using optical methods, such as scattering of light from corrugated surface, attenuated total reflection, and Raman scattering [ 1], or by use of electron-loss spectroscopy [2]. In analyzing the spectra, it is usually assumed that SP relaxation rates (damping) and therefore the linewidths are properly described by the imaginary part of the dielectric function, which appears in any dielectric theory of crystal collective oscillations [2]. In most cases, SP damping was regarded simply as a constant essentially equal to the damping of bulk modes. For example, even in high resolution electrontransmission spectroscopy of metals, it was approximated by the deconvoluted width of the volumeplasmon line [3]. However, SP damping is always dependent on SP frequency. This is due not only to various frequency-dependent intrinsic damping processes, but also to the geometry of the solid. In fact, the SP field is localized at the surface between two media and decays exponentially on both sides of the surface. The penetration depth in each medium is a function of SP frequency, so that the amount of SP field in each medium varies with frequency. As a consequence, the contributions of the intrinsic damping mechanisms in the two media to the overall SP damping vary with SP frequency. Thus, even if we describe all intrinsic damping processes in the bulk by some constant damping parameter 7, the damping of surface modes is still frequency dependent. If one of the media is inert (does not provide a mechanism for damping), the damping of SP is always weaker * Also at the Rudjer Bo§kovi6 Institute, 41001 Zagreb, Croatia, Yugoslavia. 937
than that of bulk modes because only part of the SP field is damped. This fact has been proved theoretically for surface polaritons on a semi-inf'mite solid by Nkoma, Loudon, and Tilley [4]. In a crystal slab, the SP frequency and, therefore, the SP damping become dependent on the thickness of the slab. This behaviour of the SP relaxation rate has recently attracted considerable attention [5, 6] in the search for surface polaritons with long lifetimes and their possible application to nonlinear interactions [7, 8]. Fukui, So, and Normandin [5] investigated SP temporal damping theoretically by solving numerically the dispersion relation for surface polaritons in a thin silver film. As it is well known [9], in a thin crystal film, there exist two SP modes with different symmetry properties. Analyzing the variation of the imaginary part of their frequencies with film thickness (keeping the wave vector real), Fukui, So, and Normandin have found that the lifetime of the antisymmetric SP mode in unsupported films increases with decreasing film thickness, in contrast to the behaviour of the symmetric mode. They have also shown that, in a very thin film (~ 160 A), this mode can provide SP with a lifetime of the order of 10 -12 sec, which is much larger than the lifetime of bulk plasmons (~ 10 -14 see) and the lifetime of SP on a semi-infinite solid. Since the propagation length of any collective excitation is simply its group velocity times its lifetime, long-living SP obviously travel long distances before they decay. The value of their propagation length has been estimated to be a few centimeters [5]. SP spatial damping was first investigated, both theoretically and experimentally, by Schoenwald, Burstein, and Elson [10]. Very recently, Sarid [6] has investigated theoretically
938
SURFACE-POLARITON RELAXATION RATES IN A CRYSTAL SLAB
the SP spatial damping in a thin metal slab. He studied numerically the variation of the imaginary part of the SP wave vector with slab thickness (keeping the SP frequency real) and concluded that SP modes which propagate over macroscopic distances can exist in a thin silver film. He called them long-range SP modes. Physically, the possibility for the existence of longliving or long-range surface polaritons in unsupported metal films is due to the fact that, for an antisymmetric SP mode, the electromagnetic field is expelled from the crystal as film thickness decreases [5]. As we have already said, if the surrounding medium is inert, the damping mechanisms in the crystal become less effective and SP damping should decrease. This may not be true for metals only. The same features should be expected for surfacephonon polaritons in dielectrics, for example [ 10]. As we have seen, previous theoretical investigations of SP damping properties in crystal slabs were [5, 6] performed for SP in metals and by use of numerical methods. The transparency of the results was, therefore, somewhat lost. In this Communication we report a few interesting results concerning the dependence of SP relaxation rates in metals on slab thickness. We have derived them from an analytical expression for SP damping in a crystal slab (not necessarily a metallic one) in which its dependence on SP frequency and, particularly, on slab thickness is more transparent. The expression for SP damping in crystal slabs was originally obtained by Nkoma [ 11 ] as a by-product in the theory of Raman scattering by surface polaritons in a thin film. We have rederived it starting from a quantized form for the surface polariton field [12] and using a perturbative approach. Our theory applies to crystals having a single kind of polar optical excitation and is therefore suitable for dielectrics such as alkali halides, for I I I - V compounds, and for free-electron-like metals. Among various damping mechanisms, we took into account those which are due to friction forces between ions in dielectrics or Drude processes in metals, and described them by a small phenomenological constant % Then, to the first order in % the frequency-dependent SP relaxation rate is given by [4] ,/(co) = 27((K)/(H)), where K and H are the SP kinetic (mechanical) and total energy, respectively, in the absence of damping and ( ) denote the cycle average in classical physics or the diagonal matrix element in quantum mechanics. Having the quantized form for the SP field, as in [12], it is a matter of simple algebra to show that surface polaritons in an unsupported crystal slab lose their energy at the rate:
where the SP frequency co, the two-dimensional SP wave vector k, parallel to the slab surfaces, and the slab thickness 2a are intercorrelated through the surface-polariton dispersion relation [9]
e(w+_) = -- a/ao(th aa) +-1.
(2)
Here the real quantities a = (k 2 - - e ( x ) 2 / c 2 ) 1/2 and a0 = (k 2 - co2/c2)V2 are the penetration depths of the SP field in the crystal and in the vacuum, respectively, c is the velocity of light, and e(co) the long-wavelength real dielectric function of the crystal. The subscripts + and denote, respectively, the antisymmetric and symmetric SP modes in the slab. [We have omitted them in the curly brackets of equation (1).] In equation (1), k and, therefore, a and a0 are regarded as functions of co through the solution of the dispersion relations (2). We apply equation (1) to a metallic slab which we describe by the free-electron dielectric function e(co) = 1 - co,/co 2, where cop is the electronic plasma frequency. This is a good approximation for most metals, especially noble ones, in the low-frequency region co ¢ cop. For example [13], it holds for silver when hco < 2.0 eV (hcop = 9.08 eV), for copper when boo < 1.85 eV (hWp = 9.15 eV), etc. In any case, small damped SP are expected at frequencies co~cop ~- 10-1-10 -2, i.e. they can be excited by light with wavelengths from the infrared, X ~ 1-10t.tm [5, 10]. In this frequency region our Drude model for the dielectric function of metals should be satisfactory. Regarding w/cop as a small quantity, we can make the approximations e(co) "" -¢o2/co 2, a ~-- cop/c = kp, and obtain solutions of the SP dispersion relations (2) -
-
k~ ~-- co2/c2(1 + co2/cop2 th -+2 kpa).
(3)
This gives or0 = kpco2/co~ th -+1 kpa for the SP penetration depth in the vacuum, and instead of equation (1) we have
7+_(co, a) ~ V(co/cop)2F+(kpa), _
+
2X
F+(x) = th-+2x(1-s-ff--~- ) . Equations (4) show explicitly the dependence of SP damping on slab thickness. The functions F+_(x) are rather smooth for x > 1 and approach 1 for large x. However, for x < 1, F._(x) changes rapidly and the 2 4 values of the leading terms are F÷(x) ~ ~x and F_(x) ~ 2x -2. Therefore, from equations (4) we obtain -1
s~_~o~(k~l~ _ ~Io~) -+ (k~l~,~ - I) 1
7±(co, a) = 7 l l + m co de sh 2aa -(1 + k ~ / @ ) +- ( k ~ l a ~ - - 1) 2 dco 2ota
Vol. 44, No. 6
(1)
(4)
Vol. 44, No. 6
SURFACE-POLARITON RELAXATION RATES IN A CRYSTAL SLAB
( z3(kna)4 ]
~+_(co,a) ~- ~(co,~) 2(~na)-U'
kna < 1,
(5)
where 3'(co, ~ ) = 7(co[con) 2 is the SP damping in a semiinfinite metal. For metals, kn is of the order of 10 -3 A -1 (for example, for silver (copper) with [13] con = 1.38 (1.39) x 1016 sec -1 , we have k;, ~ 4.6 x 10 -3 A-l), so that for a reasonably thin slab (2a = 200 A), we have kna = 0.5 and we can employ equation (5) with a relatively small error. From equation (5) it is perfectly clear that, owing to the slab thickness, the damping of the antisymmetric (+) SP mode can be order of magnitude smaller than that of the symmetric (--) mode. One should expect such behaviour because the antisymmetric SP mode is more "photon-like" with the electromagnetic field extending far outside the slab, while the symmetric SP mode has a more mechanical character, with the electromagnetic field strongly attenuated outside the slab. As SP damping decreases with increasing ratio of the electromagnetic to mechanical energy, it follows that the antisymmetric mode must be less damped than the symmetric one. Furthermore, the decrease of slab thickness makes this difference between the two modes more pronounced. Therefore, in thin t'rims, the antisymmetric mode can provide SP with very long lifetime, T÷ = 7+1(co, a). The propagation length L+(co, a) of these long-living SP is, according to equation (3) given by
L+(co, a) = ldcol 1--~['/+-1 ( co, a) "" cl,+l(co, a),
(6)
or, if we use equation (5), it is given by
L+(co, a) = 3 (kna)-4L+(co, oo), kna < 1,
we can, in principle, allow for SP of almost arbitrarily long lifetimes or propagation lengths. For example, Schoenwald, Burstein, and Elson [ 10] estimated that for SP in semi-infinite copper excited by laser light at a wavelength of 10.6 #m, the propagation length should be 1.9 cm. The value they obtained by measurement was 1.6 cm. However, if they had used a thin (~ 200 A) copper ftlm, they would have obtained a 24 times larger value, as follows directly from equation (5) with kpa 0.5 (copper case). This also agrees with the result of Sarid [6] for SP with frequencies corresponding to those of visible light. Of course, SP with such long propagation lengths can hardly be observed in a real metal where the effects associated with various inhomogeneities in solid, surface roughness, or the existence of an oxide layer on metal also influence SP damping. In conclusion, we have derived expressions which can be easily used to estimate the damping properties (lifetime, propagation length) of long-wavelength surface polaritons in an unsupported slab. We have analytically discussed their dependence on surface-polariton frequency and slab thickness for metals. The theory can be easily applied to surface-phonon polaritons in dielectrics and extended to more realistic geometries, e.g. supported slab, sandwiches, etc. [ 14]. We have also shown that a suitable choice of material and geometrical parameters offers the possibility for the existence of surface polaritons with very long lifetimes and large propagation lengths. When excited, they can be used as a tool for further investigations, e.g. surface roughness, various nonlinear effects, etc.
(7)
where L+(co, oo) = cT-l(con/co)2 is the SP propagation length in a semi-infinite metal. To check our results, we compared them with those of Fukui, So, and Normandin [5], because their results are expected to be correct to all orders in 7- They estimated that in a thin ( " 160 A) silver film the propagation length is L+ "" 3 cm for SP with frequency corresponding to the frequency of light with the wavelength X = 1.06 /~m. Using the same parameters as those employed in [5], i.e. cop = 1.48 × 1016 sec -1 and 7 = 3.22 x 1013 sec -1, we obtained rather large values for co/con and kna, namely, co/con ~ 0.12 and kna ~--0.4, and found from equations (4) and (6) that L+ ~- 3.4 cm. When we used equation (5) instead of equation (4), we obtained a slightly larger value, L+ = 3.8 cm. Therefore, we concluded that an estimate of the damping properties of surface polaritons in a very thin metal film, based on the use of equations (1), (4), or (5), can be even quantitatively correct for the case of small intrinsic damping and for low-frequency SP. Equations (5) and (7) contain interesting results. They show that simply by changing slab thickness,
939
REFERENCES 1. 2.
See, for example, A. Otto, FestkOrperprobleme XIV, 1 (1974). A recent review on this field is H. Raether, Exci-
tation of Plasmons and lnterband Transitions by Electrons. Springer Tracts in Modem Physics 88 3. 4. 5.
6. 7. 8. 9. 10. 11. 12. 13. 14.
(1980). R.B. Petit, J. Silcox & R. Vincent, Phys. Rev. B11, 3116 (1975). J. Nkoma, R. Loudon & D. TiUey, J. Phys. C7, 3547 (1974). M. Fukui, V.C.Y. So & R. Normandin, Phys. Status Solidi 91, K61 (1979). D. Sarid, Phys. Rev. Lett. 47, 1927 (1981). D.L. Mills, Solid State Commun. 24, 669 (1977). M. Fukui & G.I. Stegeman, Solid State Commun. 26,239 (1978). K.L. Kiiewer & R. Fuchs, Phys. Rev. 144,495 (1966). J. Schoenwald, E. Burstein & J.M. Elson, Solid State Commun. 12, 185 (1973). J.S. Nkoma, J. Phys. C8, 3919 (1975). M.S. Toma§, M. Sunji6 & Z. Lenac, Fizika 14, 77 (1982). T. Hollstein, U. Krebig & F. Leis, Phys. Status Solidi (b) 82, 545 (1977). M.S. Toma~ & Z. Lenac (in preparation).