Optical excitations in small particles and thin films
Ronald Fuchs
Ames Laboratory, U.S. Department of Energy, and the Department of Physics, Iowa State University, Ames,Iowa 50011
Received May 30, 1980
The optical absorption in small cubic particles and thin films, composed of an ionic crystal or a free-electron metal,is calculated by using both a local dielectric constant and more-exact microscopic methods. It is found that a localtheory gives a qualitatively correct description of the absorption in cubes but not in thin films. The electric fieldis calculated in thin metallic films, and the results are applied to the theory of surface-enhanced Raman scat-tering.
1. INTRODUCTION
The method of local optics can be used for calculating ab-sorption and scattering of light by a small particle or a thinfilm. The particle or film is imagined to consist of the samelocal, frequency-dependent dielectric constant Edco) thatcharacterizes the bulk material. One writes the electric dis-placement as D(r,w) = E(c))E(r,co) and solves Maxwell'sequations using standard boundary conditions at the sur-face.
A more exact approach is to use a nonlocal dielectric con-stant e(r - r', w), which is the same as that of the bulk mate-rial, in the expression
D(r,co) = S E(r - r',w)E(r',w)d 3 r'.
scopic-will be compared using calculations for simple modelsystems.
2. ELECTRICAL POLARIZABILITY OF SMALLPARTICLES
We consider particles much smaller than the wavelength oflight, so that scattering is negligible compared with absorption.If E(r,t) = Eoe-iwt is the (uniform) electric field of the light,the induced dipole moment of the particle or collection ofparticles is of the form1
M, = X, aE, = -p E 2 F.I 47r j,, Wj - (W + jy)(1)
In such a theory the modification of the dielectric constantnear the surface of the material is neglected. The surface istaken into account approximately by introducing appropriateadditional boundary conditions. For a surface lying in thez = 0 plane, with a vacuum in the region z < 0, the specular-scattering boundary condition is the easiest to use. Thematerial is imagined to extend into the vacuum region, andthe symmetry conditions E,(z) = E,(-z), EY(z) = EY(-z)Ez(Z ) = -Ez (-z) are imposed. This is equivalent to the ad-ditional boundary condition that the normal component ofthe current or polarization must vanish at the surface. Justas in an infinite medium, the Fourier transform of Eq. (1) isD(q,cw) = E(q,w)E(q,w). The bulk dielectric constant E(q,co)is, in general, a tensor, so D and E are not necessarily in thesame direction. Also, one can express the components ofE(q,w) in terms of the longitudinal and transverse dielectricconstants Ej(q,w) and Et(q,w).
A still more microscopic or exact method, applicable to ametal, is to write Eq. (1) using a dielectric constant c(r,r',w)that depends on r and r' separately, not only on the differencer - r'. This dielectric constant contains information aboutthe modification of the response of the material near thesurface and includes effects of surface states. Another mi-croscopic method, applicable to infrared properties of ioniccrystals, relates the optical properties directly to the nor-mal-mode eigenvectors and eigenvalues.
These three approaches-local, nonlocal, and micro-
(2)
Here p and v are Cartesian indices, wj are excitationfrequencies, Cj1AP are dipole strengths of the excitations, -y isa damping factor, and wo = (4ir2iNjq 2/M,)1/2 is a generalizedplasma frequency involving the masses mi, charges qj, andnumbers No of all electrons and nuclei. For a symmetricallyshaped particle or collection of randomly oriented particles,M and E are parallel, so Cj- = CJ86 is diagonal. Equation(2) applies to an arbitrary collection of particles. For example,if it is applied to a unit volume of a composite medium fillinga thin slab, it gives the average dielectric constant of thecomposite,
e = 1+ 47rM/E = 1 + 2 E p_j 92-w((W +i4) (3)
The dipole strengths satisfy the sum rule 1jCjgw = are,equivalent to f-sum rules for the imaginary parts of the po-larizability all, and average dielectric constant e:
cva 2l"(co)dw =I CO 265v,
o cWE2 (c4)dco = 2P .
(4a)
(4b)
These sum rules state that the integrated optical absorption(conversion of electromagnetic energy to heat) depends on theamount and type of matter present but is independent ofparticle shape.2
If the particle or particles have total volume v and are
composed of a material described by the local dielectric con-stant E(C), Eq. (2) reduces to 2
M P = E -= (e-1)1 + nj (5)
where the n. are depolarization factors. All information aboutthe particle shape and relative location and orientation of theparticles is contained in the quantities CjQv and ni, which areindependent of the material dielectric constant E(c). Forexample, if M and E are in a given direction (take A = v = x),for an isolated sphere there is only one dipole mode with unitstrength Cxx = 1 and dipolarization factor n = 1/3, and for anisolated cube there are six important modes with strengthsand depolarization factors shown in Fig. 1. The depolariza-tion factors nj obey the sum rule3 Yj,,Cj1 inj = 1. If theparticle has a symmetry such that the x, y, and z directionsare equivalent, this means that the centroid of the nj for agiven direction is 1/3, the same as for a sphere. It also meansroughly that the centroid of the absorption spectrum for anyparticle averaged over orientation is the same as that of aspherical particle composed of the same material.
0.8
0.6
U
0.4
0.2
0 0.2 0.4 0.6 0.8 1.0
Fig. 1. Dipole strengths Cj as a function of depolarization factorsnj for the six most prominent excitations of a cube.
0.:1
0.
C
O.:
CfY
0.
C X
0
- I I IMgO CUBELATTI CE
DYNAMICS
2
.1 -
I & 11|.1 , 111 .JI .II I8 10 12 14 16
FREQUENCY (1013 rod-i 1)
Fig. 2. Dipole strengths Cj", Cjyy, Cj21 as functions of the nor-mal-mode frequencies wj for a MgO cube containing 900 atoms cal-culated using lattice dynamics.
0.6
0.4
Cl
0.2
08 10 12 3 14 16
FREQUENCY (10 (ad r sod )
Fig. 3. Dipole strengths Cj as a function of normal-mode frequencieswj for MgO calculated using a local dielectric constant.
8 10I I 14 16
6-
4 -4
0B IC 12 14 16
FREQUENCY (10i3 rod/si
Fig. 4. Lattice-dynamical (solid curve) and local (dashed curve)optical absorption of a MgO cube as functions of frequency.
The infrared polarizability a, defined by Eq. (2), has beencalculated for a MgO cube contained 10 X 10 X 9 ions usinga point-ion model in the harmonic approximations It is ofinterest to compare this microscopic calculation with a localcalculation for a cube using a bulk dielectric constant E(W)
consistent with the point-ion model. In Eq. (2), the wj are thenormal-mode eigenfrequencies, and the dipole strengths canbe obtained from the eigenvectors 4i(s,gp), where s labels theatoms in the solid:
Ci WP= 2 S s, sM~)l2(J)js~) 6
Figure 2 shows the dipole strengths Cjxx, CjYy, and Cjzz asfunctions of the normal-mode frequencies wj. Because themicrocrystal is not quite a cube and because of the choice ofaxes, the x, y, and z directions are not equivalent. Onlymodes with strengths greater than 5 X 10-3 are shown. In Fig.3 the dipole strengths Cj are shown as a function of normal-mode frequencies wj, according to a local description. Thesesix frequencies wj, which correspond to the depolarizationfactors nj, are those for which the denominators of Eq. (5)vanish: e(ccj) = 1 - 1/nj.
The microscopic and local theories are best compared bycalculating the optical absorption, which is proportional toIm(Zi), where -a = 1/32Msoy is the orientationally averagedpolarizability defined by Eqs. (2) and (5). In order to smoothout the detailed mode structure of the microscopic theory, adamping factor y = 0.3 X 1013 sec- 1 has been introduced intoEq. (2). The same value of -y in the bulk dielectric constant
I I I I I I i i l i i _
MgO CUBELOCAL0
F THEORY
4 I, I , 1 1, I,1,1
I .1 L I .
. . . . . . . .
-
;11 ' ' ' - -
Ronald Fuchs
Vol. 71, No. 4/April 1981/J. Opt. Soc. Am. 381
E(w) in Eq. (5) causes a similar smoothing of the local modestructure. Figure 4 compares the microscopic and local cal-culations. Although there is some difference, the main fea-tures of the absorption agree. This shows that a local theoryis adequate even for an extremely small ionic-crystal cube.5
3. ABSORPTION OF A THIN FILM
Although the polarizability of a thin film can be expressed inthe form of Eq. (2), it is more useful to calculate the absorp-tance for obliquely incident p-polarized light. All threetheoretical approaches mentioned above can be compared fora 15-layer point-ion model of a NaCl film: (1) a lattice-dy-namical calculation, (2) a nonlocal calculation using bulk di-electric constants El (q,w) and Et (q,w), and (3) a local calcula-tion using a bulk dielectric constant consistent with thepoint-ion model. 6
The local calculation gives absorptance peaks at COT and COL.
In the nonlocal calculation Et (q,c) is approximated by thestandard point-ion local bulk dielectric constant E(O), andel(q,w) is constructed by requiring that the solution of el(qv)= 0 reproduce the longitudinal-optic (LO) phonon-dispersionrelation for q normal to the film. One finds that the absorp-tance peak at WiL breaks into a series of peaks associated withstanding-wave LO modes for which an odd-integer numberof half-waves appears across the film. The frequencies ofthese peaks are given accurately by the solution of fI(qn,co) =
0, where qn = n7r/a, with n = 1, 3, 5 .... ,15 and a is the filmthickness. The lattice-dynamical calculation also gives anabsorptance peak just below Tm. This is associated with alocalized surface mode that arises from a softening of inter-atomic forces near the surfaces. The nonlocal and lattice-dynamical results for the absorptance are compared in Fig.5. In these calculations the p-polarized light is incident ata 750 angle, and a damping factor y = 10-38 T gives a finitewidth to the peaks.
Calculations for a free-electron thin film give results similarto those found for infrared absorption in an ionic-crystal film,keeping in mind that plasmons in the metal are analogous toLO phonons in the ionic crystal.
nFig. 5. Lattice-dynamical (points) and nonlocal (solid curve) opticalabsorption of a NaCl film, as functions of frequency Q = c/CT.
z 10L
a 3x103 NON LOCAL /
O 10 04 06 08 10 12 14 16 18 20x 04 LOCAL. . t . 14 16 . .
Fig. 6. Nonlocal (solid curve) and local (dashed curve) optical ab-sorptance of a free-electron film as functions of frequency for p-polarized light incident at 75°. The electron density is that of po-tassium, and the film thickness is 46 A.
zL-1
0.0InCo
10-1
10-2
0.2 0.4 0.6 0.8if/Up
1.0 1.2 1.4 1.6
Fig. 7. Microscopic (solid curve) and local (dashed curve) opticalabsorptance of a free-electron film, as functions of frequency, usingthe same parameters as in Fig. 6.
The local theory shows a single absorptance peak at theelectron-plasma frequency ap. In the nonlocal theory, theabsorptance is found in terms of the bulk Lindhard-Mermindielectric constants Et (qw) and El (qc) .7 Figure 6 shows howthe absorptance peak at cop breaks into a series of peaks as-sociated with standing-wave plasmon modes for which anodd-integer number of half-waves appears across the film.The electron-hole pair excitation region in El (q,cv) gives ad-ditional absorptance at frequencies both lower and higherthan cop and produces damping of high-q plasmon excita-tions.
In the microscopic theory the electrons are assumed to befree and noninteracting within the film, which is terminatedby infinite potential barriers. The one-electron wave func-tions are simply sine waves in the z direction (normal to thefilm); from these wave functions the random-phase-approx-imation nonlocal dielectric tensor e(r,r',co) can be calculated. 8
Maxwell's equations are solved for the fields, keeping in mindthat the internal electric field that acts on the electrons differsfrom the external field because of screening by charges in-duced in the film. The absorptance for a 46-A film is shownin Fig. 7. The prominent peaks are again associated withstanding-wave plasmons, but the higher-frequency peaks,which were broadened by electron-hole pair excitations in the
; I I . I I I . I , I . I - I ,
1, i>
Ronald Fuchs
o10-4
382 J. Opt. Soc. Am./Vol. 71, No. 4/April 1981
0.2 0.6 1.0
Fig. 8. Normal component of the electric field in a free-electron film
as a function of position. The solid curve and the dot-dashed curve
show the microscopic and nonlocal results, respectively. The fre-quency is 1/2 wp.
nonlocal calculation, are now split. Except for a few small
peaks in the region X < up, there is no additional absorptance
as a result of the electron-hole pair excitations.
In Fig. 8 the normal component of the electric field is shown
as a function of position across the film. In the local theory
(not shown), this field component is discontinuous at thesurface. The nonlocal theory gives a more physically correct
result: the field is continuous everywhere, but its slope is still
discontinuous at the surfaces. In the microscopic theory both
the field and its derivative are continuous everywhere.There is a significant difference between the microscopic
theories for absorption in an ionic crystal and a metal. In an
ionic crystal, the coulomb interaction is included from the
beginning, and it enters into the normal-mode frequencies.Therefore the absorption-spectrum peaks coincide with the
normal-mode frequencies. In a metal, one starts with free
electrons, with excitation frequencies corresponding to elec-
tron-energy differences. Coulomb interaction between
electrons is not included at this stage but is included later bysolving Maxwell's equations using a dynamically screened
field. The final absorptance peaks therefore do not coincide
with the one-electron excitation frequencies, and there is, infact, no apparent similarity between them. (This statement
does not apply to interband absorption.)
4. APPLICATION TO THE GIANT RAMANEFFECT
The Raman-scattering cross section of certain molecules ad-
sorbed on the surface of metals can be enhanced by more than
106, compared with the cross section of the same molecules in
solution or in vacuum. Although many explanations for the
enhancement mechanism have been suggested, the one that
appears most sound uses electron-hole pair excitations in the
metal as intermediate states for the scattering process.9
We take the metal to be a thin film, which permits us to use
the microscopic theory described in the last section. Theinitial state I consists of an incoming photon a', the moleculein a vibrational state v, and the electrons in the ground state0; the final state F has an outgoing photon c', the molecule in
a vibrational state v', and electrons again in the ground state0. The scattering cross section is determined by the squareof a matrix element' 0
M (FIHePIi ) ( tIhvibII) (iIHepIIY (7)
ij (ES -Ej )(E] - Ei )
0.B
0.6
OA
0.2wON
I I I I I I - I -
T r
:4
0.0
-0.2
-rn-4-1.0 -0.6 -0.2
Z/A
1.0 - W -. r . . . . . - . .I
Ronald Fuchs
Here the electron-photon matrix element on the right-handside destroys the incoming photon and produces an inter-mediate electronic excited state j (an electron-hole pair ex-citation in the metal), the matrix element in the middle si-multaneously changes the vibrational state of the moleculefrom v to v' and the electronic state from j to i by means of theelectron-vibration interaction, and the electron-photon ma-trix element on the left-hand side returns the electrons to theground state while creating the final photon. Since both ofthe intermediate electronic states essentially form an energycontinuum, the energy denominators can vanish, and one has,
resonant Raman scattering.The electron-photon matrix elements can be calculated
exactly; one finds that the rapid variation of the normalcomponent of the electric field just inside the film is importanthere. In order to estimate the electron-vibrational matrixelement, it is assumed that the excited electron can leave thefilm and scatter off the molecule."1 This matrix element canbe determined either from independent measurements ofinelastic electron-scattering cross sections or from simplemolecular models.
At a photon frequency of about one half of the plasma fre-quency, the Raman-scattering cross section is about 10-27
cm2/sr, which is an enhancement by a factor of only 102 to 103.
It is believed that surface roughness can cause an additionalenhancement of 103 to 104, giving an overall enhancementconsistent with the observed value.
Ames Laboratory is operated for the U.S. Department ofEnergy by Iowa State University under contract no. W-7405-Eng-82. This research was supported by the Directorfor Energy Research, Office of Basic Energy Sciences,WPAS-KC-02-02-03.
REFERENCES
1. The symbol up stands for two somewhat different quantities. Ifwp is related to the total number of charges, as in Eqs. (2) and (4a),it has units cm 3 12 sec'. If it is related to the number of chargesper unit volume, as in Eqs. (3) and (4b), it has units sec-1, and onlyin this case is it actually a frequency.
2. R. Fuchs, "Theory of the optical properties of ionic crystal cubes,"Phys. Rev. B 11, 1732-1740 (1975); "Theory of the optical prop-erties of small cubes," Phys. Lett. 48A, 353-354 (1974).
3. R. Fuchs and S. H. Liu, "Sum rule for the polarizability of smallparticles," Phys. Rev. B 14, 5521-5522 (1976).
4. T. S. Chen, F. S. de Wette, and L. Kleinman, "Infrared absorptionin MgO microcrystals," Phys. Rev. B 18, 958-962 (1978).
5. R. Fuchs, "Infrared absorption in MgO microcrystals," Phys. Rev.B 18, 7160-7162 (1978).
6. W. E. Jones and R. Fuchs, "Surface modes of vibration and opticalproperties of an ionic crystal slab," Phys. Rev. B 4, 3581-3603(1971); R. Fuchs, "Nonlocal optical properties of an ionic crystalfilm," Phys. Lett. 43A, 42-44 (1973).
7. W. E. Jones, K. L. Kliewer, and R. Fuchs, "A nonlocal theory ofthe optical properties of thin metallic films," Phys. Rev. 178,1201-1203 (1969).
8. G. Mukhopadhay and S. Lundqvist, "The electromagnetic fieldnear a metal surface," Phys. Scr. 17, 69-81 (1979).
9. C. Y. Chen, E. Burstein, and S. Lundqvist, "Giant Raman scat-tering by pyridine and CN absorbed on silver," Solid StateCommun. 32, 63-66 (1979).
10. R. London, "Theory of the first-order Raman effect in crystals,"Proc. R. Soc. (London) A 275, 218-232 (1963).
11. Although the idea that electrons can leave the film is not consis-tent with the infinite potential barriers at the surfaces, we believethat our calculation of Eq. (7) is not greatly in error.
