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THE INTERACTION OF PHOTONS AND PHONONS WITH CRYSTAL SURFACES
A. Grimm, A. Maradudin, S. Tong
To cite this version:
A. Grimm, A. Maradudin, S. Tong. THE INTERACTION OF PHOTONS AND PHONONS WITH CRYSTAL SURFACES. Journal de Physique Colloques, 1970, 31 (C1), pp.C1-9-C1-19.
�10.1051/jphyscol:1970102�. �jpa-00213733�
JOURNAL DE PHYSIQUE Colloque C 1, supplément au no 4, Tome 31, Avril 1970, page C 1 - 9
THE INTERACTION OP PHOTONS AND PHONONS WITH CRY STAL SURFACES
A. GRIMM (*), A. A. MARADUDIN (**) AND S. Y. TONG (**) Department of Physics, University of California, Irvine, California 92664
Résumé. - Dans la première partie de l'article, on détermine le tenseur diélectrique c,,(o) pour une couche mince d'un cristal ionique ayant la structure NaCl et limitée par deux surfaces paral- lèles (001) le résultat sert de base à un calcul de coefficient d'absorption de la couche et une discus- sion sur la possibilité d'étudier les modes de surfaces optiques actifs dans l'infrarouge par la tech- nique de l'absorption infrarouge. Dans la seconde partie, on établit une expression formelle du temps de relaxation inverse pour la diffusion des phonons par les bords d'un cristal. Cette expression est calculée pour deux cristaux simples et l'on compare brièvement avec les formules utilisées dans les calculs de conductibilité thermique de cristaux isolants.
Abstract. - In the first part of this paper the dielectric tensor c,,(w) is determined for a thin slab of an ionic crystal of the rocksalt structure, bounded by a pair of parallel [O011 surfaces. The result serves as the basis for a calculation of the absorption coefficient of the slab, and a discussion of the feasibility of studying infrared active optical surface modes by the technique of infrared absorption.
In the second part a derivation is presented of a forma1 expression for the inverse relaxation time for the scattering of phonons by the boundaries of a crystal. This expression is evaluated for two simple crystals, and the results are compared briefly with expressions used in calculations of the lattice thermal conductivity of insulating crystals.
1. Introduction. - The normal modes of vibration of a crystal bounded by free surfaces differ from those of a hypothetical, ideal crystal of the same substance, containing the same number of atoms, but in which the atomic displacements obey cyclic boundary condi.
tions. The frequencies of the modes of a finite crystal are shifted with respect to those of a cyclic crystal, by amounts proportional to the ratio of its surface area to its volume, and exceptional modes can occur, whose frequencies lie outside the range allowed the modes of the cyclic crystal. These exceptional modes, which have no counterpart in the cyclic crystal, are characterized by displacement patterns which are wavelike in directions parallel to the free surfaces, but which decay exponentially with increasing dis- tance into the crystal from the surfaces. Such modes, localized spatially in the vicinity of crystal surfaces, are called surface modes. These effects of surfaces on the phonon spectrum of a crystal are reflected to a greater or lesser extent in al1 of the physical properties of the crystal which have their oiigins in the atomic vibrations.
In this paper we study the consequences of free surfaces for two functions, each of which plays an essential role in determining important physical pro-
(*) Studienstiftung des Deutschen Volkes.
(**) This research was supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, U. S. A. F., under AFOSR Grant No. 68-1448.
Technical Report No. 69-28.
perties of a crystal, viz. its optical properties and its lattice thermal conductivity. In section 2 we determine the dielectric tensor of a slab of an ionic crystal of the rocksalt structure, bounded by a pair of parallel [O011 surfaces. This result is then used to obtain expressions for the transmission, reflection, and absorp- tion coefficients of the slab. The results of this analysis furnish some insight into the conditions which must be satisfied in order that infrared active surface modes of the slab can be observed in its absorption spectrum.
In section 3 we obtain a forma1 result for the inverse relaxation time for the scattering of phonons by crystal surfaces, and evaluate it for two simple crystal models. The inverse relaxation time for boundary scattering governs the lattice thermal conductivity of an insulator at very low temperatures, when al1 other resistive mechanisms become ineffective. The results obtained in section 3 show that the usual phenomenolo- gical expressions used for this function in calculations of lattice thermal conductivities are oversimplified.
2. Optical Properties of a Thin Ionic Crystal Slab.
- In several recent papers [l-51 it has been shown that among the normal modes of vibration of an ionic crystal slab are several whose amplitudes, while wavelike in directions parallel to the surfaces of the slab, decay exponentially with increasing distance into the slab from the surface. Such modes, which are localized in the vicinity of the crystal surfaces, are called surface vibration modes. They can be subdivided
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970102
C l - 1 0 A. GRIMM, A. A. MARADUDIN A N D S. Y. TONG into acoustic and optical surface modes, depending
on whether their frequencies tend to zero or to nonzero values in the limit as the two-dimensional wavevector characterizing these modes tends to zero. In the long wavelength acoustic surface modes the sublattices comprising the crystal slab move in parallel, with nearly equal displacement amplitudes. In the long wavelength optical surface modes the sublattices vibrate essentially rigidly against each other. Because the sublattices are composed of ions of different char- ges, the surface optical modes have dipole moments associated with them, and consequently can couple to electromagnetic radiation to extract energy from it.
This fact means that lattice vibration absorption of infrared radiation provides a means in principle of observing surface optical modes. In this section we present a theory of the optical properties of a thin slab of an ionic crystal of the rocksalt structure, which, it is hoped, will make clear the conditions under which optical surface modes can in fact be observed experi- mentally.
Central to this discussion is the dielectric tensor for the slab, to whose calculation we now turn. We begin by determining the dielectric response susceptibility tensor of the slab, oriented as in Figure 1, which
FIG. 1. - The orientation of the slab of an ionic crystal studied in this paper.
relates the induced polarization to an external electro- magnetic field. This tensor can be expressed as [6]
where !2 is the volume of the crystal slab, o is the fre- quency of the incident light, assumed to be spatially homogeneous and to have a time dependence propor- tional to exp(- iot), and M , ( t ) is the dipole moment operator of the crystal in the Heisenberg representa- tion,
where H is the vibrational Hamiltonian. The angular brackets in eq. (2.1) denote an average over the cano- nical ensemble defined by the Hamiltonian H.
In the approximation that the ions are represented by point charges, the dipole moment is given by
= C eK u , ( W ,
IK (2.3)
where u,(lrc) is the p Cartesian component of the dis- placement of the rcth ion in the lth unit cell, and the ionic charges are given by
The displacement of the ion (lx) from its equilibrium position can be expanded in terms of the normal modes of the slab,
eil~rpi + i l z v z (bVj + bTVj). (2.5) In this expansion 2 N2 is the number of ions ( N 2 of each parity) in each layer of the slab parallel to the free surfaces, Mx is the mass of the ion of type IC, and cp = (cp,, cp,) is the two-dimensional wavenumber describing the wavelike propagation of the modes parallel to the free surfaces. The allowed values of cp are determined by cyclic boundary conditions to be the N 2 values given by
where
and pl and p, are integers. The integers (II, 12, 1,) give the components of the position vector ~(IIc) of the equilibrium position of the rcth ion in the Ith primitive unit cell according to
x(l +) = ro(lI, 12, 1,) 1, + 1, + l3 even
1 -) = r 0 ( , 1 , l ) 1 + 1 + 1 O (2.7) where ro is the distance between nearest neighbor ions, and we have chosen the origin of coordinates at a positive ion site. oj(cp) is the frequency of the normal mode described by the wave vector <p and the « branch)) index j (= 1,2, ..., 6 L), where L is the number of layers in the slab, and ~ d ( c p j ; 13) is the corresponding unit eigenvector. b:j and bqj are creation and destruc- tion operators for phonons in the mode (cpj).
When eqs. (2.2)-(2.5) are substituted into eq. (2. l), we obtain the result
e
;,v<o> = - C f,(j)fv(j)
, (2.8) Lu, j oS(0) - o2 - 2 ieo
where v, = 2 r i is the volume of a primitive unit ce11
of the crystal slab, and
THE INTERACTION OF PHOTONS AND PHONONS C l - 1 1 We see from eqs. (2.8) and (2.9) that because the
crystal slab retains its periodicity in directions parallel to the free surfaces, and because we have assumed a spatially uniform external electromagnetic field, only the normal modes of the slab corresponding to cp = O contribute to the response susceptibility.
Although the presence of a pair of parallel free sur- faces normal to the z-direction destroys the equiva- lence of the z-axis to the x- and y-axes, the crystal slab possesses tetragonal symmetry, with the z-axis the fourfold rotation axis. Consequently &,(cc>) is diagonal in p and v, and the only nonzero elements of this tensor are
What is required for a discussion of the optical properties of a crystal slab is not the dielectric response susceptibility, which relates the induced polarization to an external electromagnetic field, but the dielectric susceptibility ~,,(w), which relates the induced polari- zation to the macroscopic field inside the slab. To obtain the relation between xPv(w) and ~py(w), we note that for light incident normally on the slab, so that the electric field vector is parallel to the plane of the free surfaces, the macroscopic field inside the slab is equal to the external field. There is no depolarizing field in this configuration, because the sources of the depolarizing field are at infinity. Consequently, the xx and yy components of the susceptibility tensor are equal to the xx and yy components of the response susceptibility tensor, respectively
x,(o) = Xxx(w) x y y ( d = - (2.11)
To obtain the zz component of the susceptibility tensor of the slab, we note that the z-component of the induced polarization is related to the external electric field and the macroscopic field by the equations
A
Pz = xzZ(o) EFt = xzZ(w) E: . (2.12)
The macroscopic field is related to the external field by
because the depolarizing field for a slab with the elec- tric vector normal to it is - 4 nP. Combining eqs. (2.12) and (2.13) we obtain for the relation between x,,(w) and ji,,(w)
xzz(w> = x Z z ( 4 (2.14) 1 - 4 &,(O)
'Since we have treated the ions as rigid point charges, the electronic (or optical frequency) contribution to the susceptibility vanishes, and the elements of the dielectric constant tensor for the crystal slab are given by
It is the case that at cp = O the three eigenvectors 5(")(0, j ; 1,) for a = x, y, z are mutually perpendicular, with 5(K)(0, j ; 1,) parallel to the z-axis. The two eigen- vectors parallel to the xy-plane with no loss of gene- rality can be taken to be parallel to the x- and y-axes, and satisfy the condition
The frequencies associated with the latter two eigen- vectors are degenerate. Consequently, we find that only transverse modes (i. e., those corresponding to ion motions in the x-direction) contribute to &,(CO), while only longitudinal modes (i. e., those in which the ion motions are in the z-direction) contribute to E,,(o). AS a result, we see from eqs. (2.8), (2.9), and (2.15), the poles of Re &,,(w) occur at the cp = O transverse mode frequencies, while the zeroes of Re &,,(O) occur at the cp = O longitudinal mode fre- quencies.
The normal mode frequencies ( wj(0) ) and the corresponding eigenvectors { SC'(0 j ; 1,) ] have been determined for a 15 layer slab of NaCl in reference [4].
The values of wj(0) and off f ( j ) and f Z(j) obtained from these results are presented in Table 1. In the case off fÿ), we see that there are two modes for which the values of f f(j) far exceed those associated with al1 other transverse modes. One is the bulk transverse optical (TO) mode, whose frequency in the slab is 2.491 x 1013 s-l, and the other is the transverse opti- cal surface mode, whose frequency is 2.41 8 x 1013 s- l .
The strengths f f(j) associated with these two modes have qualitatively different characters, however. The former is associated with a mode which is not spatially localized near the crystal surface, so that al1 planes contribute to it. As the thickness of the slab increases, so does the value off f(j) for the bulk TO mode, in such a way that its contribution to E~,(O) becomes independent of L for a thick crystal. On the other hand, the principal contribution to f&(j) for the trans- verse optical surface mode arises from the vibrations of atoms in layers close to the surface, so that the value of f&(j) for this mode is independent of L as the thick- ness of the slab increases. The contribution to &,,(CO) from the transverse optical surface mode therefore decreases with increasing L as const./L. An extrapo- lation of the results obtained for a 15 layer slab to a slab of 100 layers indicates that a very good approxi- mation to &,(O) for such a slab is given by
26.65 00
+ --, (2.16a)
(2.491 w , ) ~ - w2 - iyo
where w, = 1013 S- '. In writing eq. (2.16) we have
replaced 2 8 by y, and have kept y finite rather than
passing to the limit y -+ O +. The damping constant y
is intended to represent the anharmonic damping of
C l - 1 2 A. GRIMM, A. A. MARADUDIN A N D S. Y. TONG each of the modes which contributes t o &,(a). A
different value of y could be used for each mode, but we have made the simplest choice. Values of y of the order of 0.02 o, are not unreasonable for NaCl at room temperature [7].
We see from Table 1 that only one mode at cp = O contributes significantly to ?,,(o), even for a 15 layer slab, namely the bulk longitudinal optical mode, whose frequency is 5.837 x IOi3 s-' in the slab.
Consequently, a very good approximation to ~,,(o) for a 100 layer slab is
28.79
&,,(O) = 1 + . (2.16b)
(2.298 O,)' - 02 - iyo
It must be kept in mind that Our assumption of rigid point ions forces the background dielectric constant
E, to equal unity.
It is also worth pointing out that as the thickness of the slab increases, L + CD, and the contribution to E,,(w) from the transverse optical surface mode goes to zero leaving only one pole (at o = o,,) and one zero ( o = CO,,), while the pole of ~,,(o) moves to
O = WTO, making &,(CO) = ~,,(o).
Having obtained the dielectric constant for a thin slab of an ionic crystal, we now proceed on the basis of this result to obtain the transmission, reflection, and absorption coefficients for the slab. We follow here, with some obvious modifications, the treatment of Fuchs, Kliewer, and Pardee [SI. Our procedure will be to solve Maxwell's equations to obtain the electromagnetic normal modes of the system consisting of the dielectric slab and the surrounding vacuum, and to superpose these solutions to obtain the solution describing an electromagnetic field incident on the slab from above and the accompanying reflected and transmitted fields. From these results expressions for the transmission, reflection, and absorption coefficieiits of the slab follow directly.
Maxwell's equations in the absence of net charge and
The displacement vector D and the macroscopic field E are related through the dielectric tensor E,,(w) by
We assume that the magnetic permeability of the crystal slab and the surrounding vacuum equals unity, so that B = H inside and outside the crystal.
Let us write the E field and the H field for the system of slab plus surrounding vacuum in the forms E,(x, t) = E,(z) eikxx-iwt (2.19a)
where we have assumed that the only nonzero compo- nent of the wave vector k is the x-component. The pair of Maxwell's equations given by eqs. ( 2 . 1 7 ~ ) and (2.17d) separate into the following set of six equations :
d i o
- Ex(z) = ik, E,(z) + - Hy(z) (2.20b)
dz c
i o
ik, Hy(z) = - - C E,,(o) E,(z) . (2.20f)
current are
V.D = O V.B = O
If we adopt the geometry depicted in figure 1, then (2. 17a) inside the slab (1 z 1 < a) the dielectric tensor e,,(o) is given by eqs. (2.16) ; outside the slab (1 z 1 > a) it is (2.17b) given simply by E,,(w) = . a,,
TABLE 1
Values of oj(0) and of j:(j) and f:(j) jor the normal modes a t <p = O for a 15 layer slab of NaCl
Transverse Modes Longitudinal Modes
~ ~ ( 0 ) (i0l3 S-II f Z ( j ) (10" gm-l)
- -
THE INTERACTION OF PHOTONS A N D PHONONS C l - 1 3 The boundary conditions at the surfaces of the slab
are obtained from eqs. (2.17a) and (2.17b), and can be written
A
n.(D, - D,) = O (2.21a)
A
n x (El - E,) = O (2.21b)
In these equations g i s a unit vector normal to a given surface, while the subscripts 1 and 2 indicate values of the associated quantities inside and outside the slab, respectively.
When we eliminate Hy(z) and dldz Hy(z) from eqs. (2.20b), (2.20d), and (2.20f ), we obtain a pair of equations for Ex(z) and E,(z) :
Because the system of slab plus surrounding vacuum has reflection symmetry in the midplane of the slab, each component of the electromagnetic field can be chosen to have either even or odd parity with respect to the midplane of the slab, and it is convenient to make this choice. However, in an optical experiment, in which light is incident on the slab from one side only, there is no definite parity. Consequently, a description of the optical properties of a crystal slab in terms of the eigensolutions of the system consisting of the slab and the surrounding vacuum must involve the mixing of solutions of opposite parity.
In what follows we will be interested only on the case that the electric vector is in the xz-plane (P-pola- rization), so that E,(z) = Hx(z) = Hz@) = O. Thus, in the region outside the slab the z-component of the electric field is obtained from eq. (2.22a) in the form
for the solutions of even parity, and
for the solutions of odd parity. Here
and P+ and P- are two arbitrary complex constants which permit us an arbitrary choice of the amplitude and phase of the waves leaving the slab. Here, and in what follows, we use the indices + and - to denote the even and odd parity solutions, respectively. We
can add the fields E,(+)(z) and E,(-)(z) to obtain the asymmetric field encountered in an optical experiment, Ez(z) = 2 e-iPOz + ( P + + P-) eiso" z > a (2.25a)
= (P+ - P-) eëiBoz , z < a (2.25b) in which the field is incident on the slab froin the
+ z-direction only. From eqs. (2.22b) and (2.20f) we obtain for the remaining nonzero components of the electromagnetic field outside the slab
To obtain expressions for the reflection, transmission and absorption coefficients for the crystal slab we must determine the energy fluxes in the incident, reflec- ted, and transmitted electromagnetic fields. The time- averaged flux of energy is given by the real part of the complex Poynting vector [9].
If one uses the expressions for E and H given by eqs.
(2.25)-(2.27) in eq. (2.28), we find that the magnitudes of the energy flow in the incident, reflected, and trans- mitted fields are
Consequently, the reflection, transmission, and absorp- tion coefficients are given by
R = 1 Ireflected = - 1 1 p+ + p- 1 v 2 . 3 2 )
1 Re lincident
T = I R e S ltransmitte2 = 1 1 p+ - p 12 (2.33)
1 Re lincident 4
We now determine P+ and P- in terms of the quanti- ties characterizing the slab.
The boundary conditions on the fields at the surface of the slab are obtained from eq. (2.21), and can be written in the form
E,,(~)I E l z W = &Zz(CO)z Ez~(z) (2.35)
Elx(z) = EzX(z) (2.36)
C l - 1 4 A. GRIMM, A. A. MARADUDIN AND S. Y. TONG
where, as before, the subscripts 1 and 2 denote values of the associated quantities inside and outside the slab, respectively. Thus the function
is continuous across the surface of the slab. Therefore we can use the values of the components E,(z) and Ez(z) outside the slab, where ~,,(o) = 1 to determine g(z) for the even and odd parity modes. We find that
while
Consequently, we find that
The fields inside the slab obtained from eqs. (2.22) are
E:+)(z) = A cos pz
E ~ - ) ( z ) = B s i n ~ z J z I < a (2.41) and
(+) AB ~ ~ sin ( 4
Ex (z) = -
lkx EX,(@)
= - " - Gzz(W) COS Pz 1 z 1 < a (2.42)
"k,
~ x x ( 4
where A and B are complex constants, and
The expressions for g(+)(a) and g(-)(a) obtained from the substitution of these results into eq. (2.37) are
$+)(a) - = 1 - - ' tan Ba (2.43a)
kx & x x ( ~ )
P 1
g(-)(a) = i - - CO^ Ba . (2.43b)
kx ~ x x ( 4 Using eqs. (2.43) in eqs. (2.40), we obtain
P+ = 1 + i ( B l & x x ( ~ ) BO) tan Ba - ~ i p o a (2 44a) 1 - i(PI&,(4 B O ) tan Ba e
results for films as thin as those considered here. The vacuum wavelenght of light is 2, = 2 n c l o . In the limit that this is much larger than the thickness of the slab, 2 a, we can expand the expressions for the reflection, transmission, and absorption coefficients in powers of the ratio 2 a/A0 and retain only the lea- ding contribution. Since 2, is of the order of 10-2 - 1OP3 cm for light of frequencies comparable to the frequencies of optical vibration modes in alkali- halide crystals, while the thickness of a slab a typical alkali-halide crystal of 100 layers is of the order of 3 x cm, 2 a/Ao is a very small parameter under these conditions, and comparatively simple expressions for the optical coefficients of the slab can be obtained.
Thus, by expanding tan (Ba) and cot (Ba) to first order in 6 = 2 a(w/c) = 2 ~ ( 2 alA,), we obtain from eqs.
(2.44) and (2.32)-(2.34) the results that
where &$)(O) and cg)(@) are the real and imaginary parts of E~,(o). In obtaining this result we have used the fact that k, = ( o l c ) sin O and Po = ( o l c ) cos O, where û is the angle of incidence of the light impinging on the slab (see Fig. 1).
We see from eq. (2.46) that the absorption coefficient for a thin crystal slab in general has peaks at the peaks of ~(2,)(o) and at the zeros of e(2)(0), provided that O # O. The former occur at the frequencies of the
<p = O transverse optical modes, while the latter occur at the frequencies of the cp = O longitudinal optical modes.
In figure 2 is plotted the absorption coefficient for a slab of NaCl of 100 atomic planes with y = 0.02 oO, corresponding to an angle of incidence of 300. It is
- W
the problem of obtaining the reflection, transmission, wo 1 - i ( B l & x x ( ~ ) B O )
Ba - 2 i B y ( 2 . 44b)
P- = e
1 + i(Pl~,,(o> BO) cet Ba
0and absorption coefficie~ts for an ionic crystal slab, FIG. 2. - The absorption coefficient for a slab of NaCl of 100 layers thickness is plotted as a function of (w/wo), where oo
The for obtained in is 1 0 1 3
s-i.me value of the damping in fhis
this ïlîanner are very complicated, more complicated calculation is 0.02 wo, and the angle of incidence is 0 = 300.
than they need to be for the discussion of experimental There is a scale change in the frequency scale at (w/wo) = 3.
l l
I
lL
2 O 12 5 3 O 40 5 0
'
6 0Equations (2.44) together with eqs. (2.32)-(2.34) solve 2
WLGTHE INTERACTION OF PHOTONS AND PHONONS C l - 1 5
clear from this figure that the transverse optical surface modes should be observable as peaks in the absorption spectrum of a crystal slab of this thickness. However, in as much as the strenght of the absorption by transverse optical surface modes varies inversely with the thickness of the crystal slab, it is clear that an increase in the slab thickness by a factor of three or more, resulting in a thickness of approximately 0.1 micron, would make the transverse optical surface mode peak in the absorption spectrum very difficult, if not impossible, to observe. In addition because the transverse optical surface mode frequency is very close to that of the bulk transverse optical mode in the slab, unless experiments are done at sufficiently low temperatures, it is possible that the anharmonic broadening of each of these modes can be large enough that the two peaks in the absorption spectrum corresponding to these modes overlap so strongly that the weaker surface mode peak can not be separated from the much more intense bulk TO mode peak. In the present example of NaCl, if the estimate of y = 0.02 o, is a reliable one for the transverse optical surface mode, « low enough » means room temperature, or lower.
The extreme thinness of a crystal slab that is required in order that optical surface modes be observable in absorption makes it appear very likely that in practice such slabs will be thin films deposited on a substrate, rather than self-supporting films. The extension of the analysis presented in this section to the determination of the optical properties of supported films will be reported elsewhere.
3 . Scattering of Phonons by Crystal Boundaries. -
At very low temperatures, when al1 other possible scattering mechanisms become ineffective, and the mean free path for phonons in an insulating crystal becomes comparable to the linear dimensions of the crystal, the principal phonon scattering mechanism giving rise to thermal resistance is the scattering of phonons by the crystal boundaries. An important ingredient, therefore, in calculations of lattice thermal conductivities at very low temperatures is the inverse relaxation time zkjl for the scattering of phonons in the mode (itj) from crystal surfaces. This quantity is defined as the time required for the deviation of the occupation number Nkj for the mode (kj) from its equilibrium value NL;) at a given temperature to relax to Ile of its initial value when al1 of the remaining modes have the thermal equilibrium values for their occupation numbers. This quantity is ordinarily represented by zkjl = CIE, where C, called the Casimir lenght [IO], is roughly a typical dimension of the crystal, while c is an average speed of sound, and is usually regarded as being independent of the wave vector k , and sometimes as independent of the branch index j. This result can be understood physically.
Since zkj = Clc is the time for a phonon to traverse the crystal, if scattering from the boundaries is the
mechanism driving the phonon system to equilibrium, this is the only characteristic time for this relaxation process.
In this section we present a rnicroscopic lattice dynamical derivation of an expression for the inverse relaxation time for the scattering of phonons by crystal surfaces in a slab shaped sample, in which the interaction of the lattice waves with the free surfaces is treated realistically. For a description of the manner in which the results of the present discussion enter into a calculation of the lattice thermal conductivity of a crystal, the reader is referred to the review article by Carruthers [Il].
To calculate the inverse relaxation time for the scattering of phonons by crystal surfaces, we must determine the time rate of change of the number of phonons in the mode (kj) due to this scattering mechanism. Let us denote by W(k'jl + kj) the rate at which a phonon in the mode (k'j') is scattered by a crystal surface into a phonon in the mode (kj). Then, if we denote by Nkj the number of phonons in the mode (kj), the rate of change of Nkj is given by
x ( Nktjr W(kl j' -+ kj) - Nkj W(kj + k' j') ) . ( 3 . 1 ) For the crystals with which we will be concerned here, it is the case that
W(k'j'
-tkj) = W(kj -, k' j') . (3 2) We have been unable as yet to determine how general this result is for boundary scattering. Using eq. (3.2) in eq. (3.1) we obtain
dNkj
-- - C ~ ( k ' j'
-*kj) (IVkpj. - N k j ) . ( 3 . 3 ) dt k ' j r ( f k j )
The relaxation time for a mode (kj) has been defined as the time in which the deviation of its occupation number Nkj from its equilibrium value N L ~ ) falls to
lle of its initial value when al1 other modes have the thermal equilibrium values for their occupation numbers. Consequently, if we write Nkj as the surn of its equilibrium value and the departure from this value,
Nkj = Nkj (0) + nkj , ( 3 4)
and set
Nkrjt = ~ k ? ] , (kt j' # kj) , ( 3 . 5 ) Eq. (3.3) takes the form
- -
d n k j - ~ ( k ' j' + kj) (N$: - N;) - nkj)
dt k'j'(+ k j )
The equilibrium value of the occupation number for
C l - 1 6 A. GRIMM, A. A. MARADUDIN A N D S. Y. TONG the mode (kj) depends on k and j only through its
dependence on the normal mode frequency oj(k),
We will see shortly that W ( k ' j f -, k j ) vanishes unless the frequencies of the modes (kj) and ( k ' j ' ) are equal.
It follows, therefore, that eq. (3.6) can be rewritten as dnkj
- -
dt - - n k j C ~ ( k ' j ' k j ) , (3.8)
k'j'(+ k j )
and we are led the result that the inverse relaxation time ~ k j l is given by
7;: = c W ( k r j' + k j ) .
krj'(# k j )
According to the forma1 theory of scattering [12], the rate W(kr j' + k j ) at which a lattice wave u(0)(kl j') is scattered by a crystal surface into the wave d o ) ( k j ) is given in terms of the scattering matrix
for boundary scattering by W ( k r j' -+ k j ) =
( O ) k' j'j, ~ ( o : ( k ) + io) d O ) ( k j ) ) 1' x
= x J ( u (
The lattice wave displacement field d O ) ( k j j can be chosen to be [13]
ur)(lrc ; k j ) = h ) " 1 k j ) e i k . ~ ( i ~ ) .
(3 . I l ) This expression is recognized as the amplitude multiplying the phonon field operator for the mode (kj), Akj = bkj + b f k j , in the expansion of the displacement field ua(Zrc) in terms of normal coordi- nates :
(' 7)" W ~ ( K 1 k j ) e i k . x ( L ~ ) A k j . (3.12)
u,(lTC) = --
2 N M , k j ( C O ~ ( ~ ) ) "
It is the amplitude for the destruction (or creation, when conjugated) of a phonon in the mode (kj) at the site (Zrc).
It is convenient to expand the scattering matrix in a double Fourier series [14]
When this expansion, together with eq. (3.1 l), is substituted into eq. (3. IO), the expression for
W(k' j'
4k j )
takes the form
n 1 t(kt j' ; - k j ; o:(k) + io) 1'
W ( k ' j l -+ k j ) =
-2 wj(k) w,,(k')
Combining eqs. (3.9) and (3.14), we obtain finally for the inverse relaxation time the expression
7c 1
C x
2 COS(k) k ' j 9 ( + k j )
x 1 t(kr j'; - k j ; wS(k) + io) l2
x 6 ( o j ( k ) - w f ( k ' ) ) . (3 .15)
The scattering matrix Tap(Zlc ; Z' a' ; 0 2 ) is the solution of the equation [14]
where
describes the perturbation of the time-independent equations of motion of a crystal due to the presence of a defect, in the present case a pair of free surfaces.
The Green's function for the unperturbed crystal has the expansion [15]
When we substitute this expansion, together with eq. (3.13), into eq. (3.16), we obtain as the equation for the Fourier coeacient t(kj ; k' j' ; w2)
t ( k j ; k ' j ' ; 0') = V ( k j ; k' j ' ; w2) f
where
~ ( k j ; k ' j ' ; w ' )
= @1 wz(rcl k j ) x
KU L ' K ' ~
6 L u f l ( 1 ~ ; 1' K '; o z )
x - W;(TC' ; k' j')
( M , M,,)"
Fairly simple expressions for t(kj ; k'j' ; w2) have been obtained for two lattice dynamical models of a simple cubic lattice bounded by two [O011 free surfaces.
The first is the nearest neighbor, central and noncentral
THE INTERACTION OF PHOTONS AND PHONONS C 1 - 1 7
force model, with equal central and noncentral force constants [16]. The second is the nearest and next nearest neighbor central force model 1141. We consider the calculation of zkjl for each of these two models in turn.
The primitive translation vectors of the simple cubic crystal will be chosen to be
a, = ao(l, O, O) a, = ao(O, 1, O) a, = ao(O, O, 1) , (3.21) where a, is the lattice parameter. The position vector to the equilibrium position of an atom is given by
where Il, 12, I, are three integers, positive, negative, or zero, to which we refer collectively as 1. We assume that the atomic displacements satisfy cyclic boundary conditions, where the periodicity volume is a cube with edges La,, La,, La,, so that the total number of atoms in the periodicity volume is N = L3.
For each crystal model we create a pair of adjacent [O011 free surfaces by equating to zero al1 interactions between atoms in the planes I, = 0 and l, = 1.
Because of Our assumption of periodic boundary conditions on the atomic displacements in the unperturbed crystal, the result of this operation is a crystal slab of L atomic planes bounded by two [O011 free surfaces, in which the atomic displacements obey periodic boundary conditions on the faces of the periodicity volume normal to the x- and y-axes.
In the case of the nearest neighbor, central and noncentral force model the elements of the perturbed matrix are given by
where y is the nearest neighbor, central and noncentral force constant. If we keep in mind that for this model
and
independent of j, where oi = 12 y/M, and M is the atomic mass, we obtain from eqs. (3.19) and (3.20) the following result for t(kj ; k'j' ; o z )
a0 a0 '
sin- k, sin - k,
0, 2
X - 2
3 1 + M(k, k y ; 0 2 )
where
2 a0 sin - k,
~ ( k , k,,; o z ) = - a; 2
, (3.26b) 3 L kz o2 - w2(k)
and A(k, + k:) equals unity if kx + k: = 0, and vanishes otherwise.
If we replace summation over k, by integration according to
we find that
M(kx ky ; o; (k) + io) =
a
=/a0 sin2 2 2 klz
dk,, a 2 ao
I o sin2 3 k, - sin - k,, + i o
2 2
r n l a o
4J dklz cos a, klZ - k s a, k, + io
Consequently, we obtain the result that
1 t(kr j'; - k j ; o;(k) + io) 1' =
= ajj, A(k, - kJ A(k, - k;) x o; 2ao 2 a0
x-cos -kzsin -k:. (3.29)
9 L~ 2 2
When we substitute eq. (3.29) into eq. (3.15), we obtain for the inverse relaxation time
x 6(02(kx ky k,) - 02(kx ky k:))
where c = a, 0,/(2 J3) is the speed of sound for this crystal model. If we note that
a a
a ~ ( k ) 2 sin 2 k, cos 2 k, -- O, 2 2
dk, - 12 o(k)
y(3.31)
we can express equivalently as
C l - 1 8 A. GRIMM, A. A. MARADUDIN AND S. Y. TONG The result given by eq. (3.30) takes a particularly
simple form for k = (0, O, k,),
C a0
= - cos - k, . (3.30a)
i c = ( ~ , ~ , k z ) Lao 2
We see from eqs. (3.30)-(3.32) that for k , , =k 0, where k l l is the two-dimensional vector k l l = (k,, k,)
- 1
parallel to the free surfaces, z~~ vanishes when k, = O and when k, = nia,. The former result is readily understood : phonons with k, = O and kll =k O propagate parallel to the free surfaces, and consequently are not scattered by them. In the latter case ~k~~ vanishes because the z-component of the phonon group velocity vanishes for k, on the boundary of the Brillouin zone, irrespective of the values of k, and k,. Phonons for which the component of their group velocity normal to the free surfaces vanishes will not be scattered by them.
The Fourier coefficient t&j ; kt j' ; 0 2 ) has been determined by Maradudin and Wallis [14] for the simple cubic crystal with nearest and nextnearest neigh- bor, central force interactions. The result, which has a structure similar to that given by eq. (3.26), is
t(kj; k'j' ; co2) = A(k, + k;) A(k, + k;) x
4 a
x - ML exp [- i 2 (k, + k:)]
1 1
x sin - k . x(1) sin - kt . x(lt) [1 - M(k, k, ; 0 2 ) ] ,
2 2
(3.33a) where
sin 2 k . x(1) (x(lt) . e(kj)) sin .2 k . x(1')
x (x(1). e(kj)) ---
kzj o2 - CO:(k)
a, (- 1, O, - 1), a, (O, - 1, - 1), a, (1, O,
-1), and a, (O, 1, - 1). These are the five bonds joining an atom in the plane l3 = 1 to its five nearest and next nearest neighbors in the plane I3 = O, which are cut in creating the pair of free surfaces in this crystal model.
The inverse relaxation time z c h a s been calculated numerically as a function of k and j on the basis of eqs. (3.15) and (3.33), in the special case qr'(ao) = q"(&,). In this case the crystal is elastically isotropic in the long wavelength limit. Some of the results are displayed in figure 3, and are qualitatively similar to that given by eq. (3.30a). From these results it would appear that the approximation of
- 1
zkj by const./La,, is a poor one, even if a different constant is used for each phonon branch. The consequences of using a realistic result for the inverse
cl ' 3 k
z 7 T Z
