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A surface plasmon dispersion relation in the presence of spatial dispersion
Surface Science 81 (1979) 13-27 0 North-Holland Publishing Company
A SURFACE PLASMON DISPERSION RELATION
IN THE PRESENCE OF
SPATIAL DISPERSION ** A.A. MARADUDIN Department
of Physics, University of California, Irvine, California 92717,
USA
Received 16 May 1978; manuscript received in fmal form 25 September 1978
A dispersion relation for surface plasmons (retardation neglected) is obtained in the presence of spatial dispersion. The physical system considered is one of vacuum separated by a plane interface from a dielectric medium occupying the half-space xg > 0, and characterized by a nonlocal dielectric function e(kllw Ixsxh), assumed to be symmetric in xa and xi. Here $1 is a twodimensional wave vector whose components are parallel to the interface, and w is the frequency of the electromagnetic field in the medium. The dispersion relation has the form 1 + for obtaining the function kllx(kll~ Ix3 = 0, xj = 0) = 0, and an explicit prescription x(kllw Ixsx$) is presented. The use of the dispersion relation is illustrated by applying it to two examples: (1) a local dielectric constant; and (2) the nonlocal dielectric function used previously by Maradudin and Mills. In both cases previously obtained results are recovered.
1. Introduction Several years ago Ritchie and Marusak [l] published a derivation of a dispersion relation for surface plasmons [2] in the presence of spatial dispersion. Their result has the simple form
1
s
+k” R
s3
_m k%(k, w) =
0.
(1.1)
Here e(k, w) is the frequency and wave vector dependent dielectric constant of the material out of which the semi-infinite medium is composed, along whose planar interface with vacuum the surface plasmon propagates. In eq. (l.l), k~ is the magnitude of the projection of the three-dimensional wave vector k on the plane of the surface, and kB is the component of ‘k normal to the surface. A central assumption in the derivation of eq. (1 .l) is that the electrons in the * This research was supported N00014-76Cal21. ** Technical Report No. 78-16.
in part by the Office of Naval Research under Contract No.
13
14
A.A. Maradudin /Surface
plasmon despersion relation
semi-infinite dielectric medium are reflected specularly from the surface. It is this assumption which ultimately leads to the appearance of the bulk dielectric constant e(k, 0) in eq. (1.1). However, in any real solid it is unlikely that all of the electrons are reflected specularly from the surface. It is more likely that some fraction are scattered specularly and the rest diffusely, or that more general boundary conditions obtain. In any case, it would seem to be desirable to have a dispersion relation for surface plasmons which does not depend, in its derivation, on some particular assumption about the nature of the interaction of an electron with the boundary. Moreover, it would also seem to be desirable to have such a dispersion relation which is not based on the assumption that the dielectric properties of the medium supporting the surface plasmon are due to a specific collective excitation in the medium, viz. an electron plasma, so that it applies to systems in which these properties are associated with other electric dipole excitations such as IR active phonons and excitons, for example, or combinations of them. It seemed to be worthwhile, therefore, to try to obtain a dispersion relation for surface plasmons in the presence of spatial dispersion of a form similar to that of eq. (l.l), but without invoking the restrictive assumption of specular reflection of electrons at the boundary of the dielectric, in which the central role is played by the nonlocal dielectric constant of the medium, about which a minimum number of assumption are made. In this paper we present the derivation of such a dispersion relation. It is obtained in section 2, and its use illustrated by application to systems for which the dispersion relation is already known in section 3. A discussion of the results obtained is given in section 4.
2. The dispersion relation We assume a dielectric medium which occupies the semi-infinite region x3 > 0. The region x3 < 0 is occupied by vacuum. Because the system possesses infmitesimal translational invariance in directions parallel to the surface, the macroscopic electric field and the displacement in it can be expressed in the forms (2.la) D(x, t) = D(kllolx3)
(2.lb)
exp(iklt * x11- iwt),
where kll = z?rki t ~?&a, xl1 = z?rxi + &xa, and 2i and 2a are two mutually perpendicular unit vectors in the plane of the dielectric-vacuum interface. Within the dielectric medium the relation between D(kilw Cy3)and E(kttw lxa) is assumed to be D(kIlOlX3)
=
f 0
cl‘4
&WIX3X;)
wwI4),
x3
2
0,
(2.2)
A.A. ~aradud~n / Surf&e plasmon despersion retation
15
where E(IEIIW Ix&) is assumed to be symmetric in the variable xs and x(3. Thus, we assume the nonlocal dielectric tensor of the semi-infmite medium to be isotropic. This assumption is similar to and, because of the lower symmetry of the semi-infmite medium, may be as restrictive as the combined assumption by Ritchie and Marusak of an isotropic, nonlocal, bulk dielectric tensor and specular reflection of electrons from the boundary. The assumption of isotropy, however, can be removed from both the present theory and that of Ritchie and Marusak. The result in each case is a more complicated dispersion relation. The assumption of specular reflection from the boundary, on the other hand, is essential in the RitchieMarusak approach, while it is not in the present treatment. The Maxwell equation, v * D(x, t) = 0,
(2.3) is now combined with eq. (2.2) and with the equation of the electrostatic approximation
E(x, t) = -Q&X, t),
(2.4)
where tp(x, t) is a scalar potential, to yield an integro-differential equation for cp(ktlwlxs), the Fourier coefficient of cp(x,t):
--
d k3
= s
CL&e(k,,wlx&)
d --&~,,c&)
= 0.
h3
0
(2.5)
To solve eq. (2.5) we begin by formally expanding both e(kttwIxfl$) and (d/b,) e@tlwjx&) in double series of functions complete and orthonormal in the semi-infinite interval (0, -), and localized in the vicinity of the interface at x3 = 0: (2.6a)
r
d
$kllW 1x34)
=
3
2 m,n=Q
bmn(bQ)
pm(x3)
$%(x9.
(2.6b)
Although our final result will be independent of any particular choice for the {am), a convenient choice, for definiteness, is the set of Laguerre functions defmed by (p,@3)
= P2
ewi--+3)
Lz(px3) 7,
(2-V
16
A.A. Maradudin / Surface plasmon despersion relation
where &(x3) is the nth Laguerre polynomi~, and /_3is a real, positive parameter with the dimensions of an inverse length. Because of our assumption that e(kl+r 1xsx:) is symmetric in x3 and XL,it’is the case that the coefficients {a,,(k~~o)} are symmetric in the subscripts m and n. We next expand rp(k&~) and (d/~3)~(~11~~~) in terms of the &m(xa)):
(28a)
(2.8b) so that (2.9a)
(2.9b)
When we substitute eqs. (2.6) and (2.8) into eq. (2.5) and use the o~hono~~ity of the (~,&)], we find the latter takes the form (2.10) The coefficients ~~~(~,,~)~ and {B&t w )} are not independent, however. If we integrate by parts the integral in eq. (2.9b) we obtain the result that
where the notation cp(kllwIO+)is intended to emphasize that it is the potential on the medium side of the interface that is being considered. If we substitute eq. (2.8a) into the second term on the right hand side of eq. (2.1 l), we obtain fmally the relation &A&-IIW) = -cp,(O) cp(~ll~Io+) - ,co X where
%zp&&l~),
(2.12)
(2.13a)
17
A.A. Maradudin /Surface plasmon despersion relation
We note for future reference that enp can be expressed equivalently
as (2.13b)
0
0
On combining given by
B&w)
eqs. (2.10)
= -k$&~IO+)
and (2.12) we find that the coefficient
,$e
[Q
+ ea-’
B,#ctlo)
is
(2.14)
61;; cp,@),
where for convenience we have suppressed the argument (k,lw) of the matrices on the right hand side of this equation. It follows from this result and eqs. (2.2), (2.4), (2.6a) and (2.9b) that
a(kllw) and b(kllo)
5
~3(~llwlO+)=k~cp(k~~wlO+)
m.n=O
cp,(O) [kia-’
+ ea-lba-l]~n
cp,(O).
We now turn to the vacuum region x3 < 0. In this region the function is readily found to have the form CP(~IIW~X~)= A
x3
ev(W3),
where A is an arbitrary becomes D3(k~~olx3) = -&
<
constant,
exp(hx3),
cp(k~~w1x3)
(2.16)
0,
while the displacement
x3
(2.15)
<
0.
component
D3(kltwlx3)
(2.17)
The boundary conditions in the problem are the continuity of cp(x, t) and D3(x, t) across the surface x 3 = 0. The first of these yields the relation cp(kllo lot) = A; the use of this result together with eqs. (2.15) and (2.17) in the second boundary condition yields the dispersion relation for surface plasmons in the form
(2.18a) where xmn(kllw) = [kia-’
+ ea-‘ba-‘I&
.
(2.18b)
It is useful to point out that x~,&,o) is symmetric in m and n. For this purpose it suffices to show that the matrix k;fa-’ + ea-%a-’ is a symmetric matrix. Inasmuch as the matrix a-l IS ’ symmetric because the matrix a is, it is necessary to show only that the matrix X = ea-‘ba-’ is symmetric. To show this we first relate the coefficients {b,,(kltw)) to the coefficients
18
A.A. Maradudin /Surface
(a,&~)}.
plasmon despersion relation
We see from eq. (2.6b) that
(2.19)
An integration
by parts yields
bmn(klla)= -$ dx13 cPm(“) e(kllwlOxi) IPi2(Xt3) 0
(2.20) We now substitute (2.13a) to obtain
eq. (2.6a) into the right hand side of this equation,
and use eq.
(2.21) The second equality in eq. (2.21) follows from the use of eq. (2.13b). It follows from eq. (2.21) that the matrix X can be written equivalently X = ee-1 (&)e-’
= ea-ieT
= XT,
as (2.22)
where MT denotes the transpose of the matrix M. The matrix x(kllw) is therefore symmetric, and can be written alternatively as Xmn(kllw) = [k&s-’ + eadleT]&.
(2.18c)
Eq. (2.18) may have some interest of its own in connection with purely numerical studies of surface plasmons in spatially dispersive media. However, at this point we direct our efforts to re-expressing eq. (2.18) in a form in which the {qm(x3)} no longer appear explicitly or implicitly. We begin by introducing the function X(kllw lx3x;), which is defined by X(kIlcJJIx,&)
=
5 Prn(X3) m,n=O
In terms of this function
Xrnn(kll~)
Pp,(xk),
x3,x;
> 0.
the dispersion relation (2.18) becomes (2.24)
1 + kllx(kllw 100) = 0. To obtain the function x(kllw kg;) or more accurately, ing it, we return to eq. (2.18c), which we rewrite as
c
ki za%pxpn + prs empGksr ;Crn = 6mn . P
(2.23)
the equations
determin-
(2.25)
A.A. ~~~~dudi~ /Surface plasmon despersion relation
19
We now multiply both sides of this equation by ~~(x~)#~(x~) and sum on M and n from zero to infinity. We recall the orthonormality and completeness conditions 6 mn = s
&3
(Prn(X3)
(2.26%)
(Pry,
0
(2.26b) We next note that
(2.27) where ~-~(kl+~Ix~xj) is the inverse of e(kllolx,x;)
in the sense that
m
J
dx;
e-l
(k,,w
lx,xj’)
e&p
1x:x;)
0
= 6(x3
=
dx;
.I0
e(k,,wlx34)
E-‘(k,,WIX;X;)
-xi).
(2.28)
We also use the relations
5
Y&(X3)
emn%2fxj)=
m, n=O
fi
-
&6@3
-x’,)
+ ~(x3~so]
9
(2.29a)
[
Ipm(x3)
enm
Cpn=
$s(x3
-xi3)*
(2.29b)
m,n=O
The result of ail of this is to transform eq. (2.25) into
k;
r
dx’;
e-‘(k,,wIx3xlj)
X(krpMhS)
0
(2.30)
20
A.A. Maradudin /Surface
Finally, we see that the function tial equation
plasmon despersion relation
x(kIlw Ixsx;) which satisfies the integro-differen-
kf! dx~E-1(k,,wlX3XI;)X(k,,wIX);X;) 0 _-
d &3
J 0
dx; ~-~(~,,wlx,x';)~x(~,,wlx;x;)= ax;
together with the boundary
J
dx; E-l(k,,wIOxLJ
0
6(x3 -XL),
(2.31a)
condition
&&,,C&‘;Xj) dx;
(2.31b)
= 0,
is a solution of eq. (2.30). As we are seeking electric fields localized in the vicinity of the surface x3 = 0, the boundary condition which must be imposed on X(k,,ti b3x5) at infinity is that lim
x(k1lwIX3X;)=O.
(2.31~)
x34=
Eq. (2.24), together with the prescription for obtaining x(kllw Jx3x;) given by eqs. (2.31), is the central result of this section: the surface plasmon dispersion relation. In the next section we illustrate its use by applying it to two cases for which the surface plasmon dispersion relation is known.
3. Examples In this section we solve eqs. (2.3 1) for two choices of e(kjl~ lxsxh), one local and the other nonlocal, for which the surface plasmon dispersion relation is already known, to illustrate the use of the dispersion relation, eq. (2.24). 3.1. A local dielectric tensor We first consider the case in which e(ICit~ (x3x;) is given by 4~llWIX3X;)=E(W)~(X3
where e(w) is independent e-l (k,,wlx,x;)
1 = --6(x,
-XL),
of k,, and of the coordinates
(3.1) x3 and x;. It follows that
-x$>
f(W)
The equation
(3.2)
satisfied by x(kllw 1x3x\) in this case becomes (3.3a)
A.A. Maradudin /Surface
together with the boundary
21
plasmon despersion relation
condition
-$x(q.4x,x;)lx3hg = 0.
(3.3b)
3
The solution
of eq. (3.3a) which vanishes as x3 + 00 is +A
exp(-Q3)
,
(3.4)
3
where the first term is the particular integral and the second is the complementary function. The coefficient A is determined with the aid of eq. (3.3b), with the result that e(u) x(~llwlx3x~)=--
2ks
bp(-hlx3
When eq. (3.5) is substituted dispersion relation
-xiI)+exp[-k11(x3
into
+&>I}.
eq. (2.24), we obtain
as the surface plasmon
1 te(o)=0, a well-known
(3.5)
(3.6) result for this case [3].
3.2. A nonlocal dielectric constant The second example we consider is based on the nonlocal dielectric function
E(~IIwIx~x)~) = eo6(x3 - xi) + i -2or
exp(irIx3
--&I>,
x3,x$>@
(3.7)
which was used by Birman and Sein [4], Maradudin and Mills [S], and Agarwal, Pattanayak, and Wolf [6], in studies of various optical optical properties of semiinfinite media in the presence of spatial dispersion. There are some unphysical features of the model underlying this form for e(kllo lxaxk), as will be discussed in the following section. However, because results based on its use are available it serves as a useful example to illustrate eq. (2.24), which is not restricted by these unphysical features. In eq. (3.7) ee is the optical frequency dielectric constant, which is assumed to be real and frequency independent; on is a plasma frequency; D is a positive constant which defines the curvature of the exciton or transverse optical phonon dispersion curve at k = 0; and r is given by
(3 J) so that Re P > 0, Im r > 0. The frequency w. in eq. (3.8) is that of the exciton or transverse optical phonon at k = 0. The equation for the inverse dielectric function eM1(kp/x3xk), eq. (2.28), in
22
A.A. Maradudin /Surface
plasmon despersion relation
the present case takes the form io’p e-l(k,,WIXaX$) + __ 2e($r =
m dxg exp(iP(xs .I-
- x’; I) e-l(kljW 1x:x;)
0
Ely S(x3 - x’,).
(3.9)
We apply the operator (d2/dxi) into
d’ t a2
(
2
)
=A-J- +r*
1
d.4
+ I’2 to both sides of this equation to transform
E-l(k,,w,x3x;)
co
(
d-4
6(x3 -
xi),
it
(3.10)
where tY=
W”-Wf_
(
iwy II2 -ki+y
D
1
(3.1 la)
9
0: = w;+(w2p/eo),
(3.1 lb)
so that Re OL> 0, Im CY > 0. We now use the fact that
(-$+02)
[6(xaLx;)tI$$exp(i~lx,
-xkiq=
($+F2)
6(x3-x$), (3.12)
to write the solution of eq. (3.10) in the form E-r(k,,WIXsX:)
wi exp(icrlxs -xhl) 1 = 0 6(x3 - xk) +2 2iQ eoD
t A exp(icrxs)
(3.13)
The coefficient A is obtained by requiring that the expression (3.13) also satisfy the original integral equation (3.9). In this way we obtain finally the result that e-l &,w 1x,x;) = ;
1
6(x3 - x$)
a-r
4 +=A 0
(
=;
-x)3) +f(k,,wIX3X:).
6(x,
exp(iculx3 -x’,l)+--(y +
r
I
expMx3 + &)I
(3.14)
We note the result that (3.15)
23
A.A. Maradudin /Surface plasmon despersion reletion
The integral equation (2.3 la) for the function x(kt+~ 1x3x;) now takes the form 1 d2 -~eo ( dx:
kj
= -Ei(Xj -
1
x(k,$.Q,x,x’,) + $
p
&x(3) f&lo
(x3x;)
30
x;>.
(3.16)
We now apply the operator (d~~~~) + o2 to both sides of this equation, and use eq. (3.15), to obtain
($- k~)(~ -B2) xVq,olxsx;)=-eo (-$+a2) %x3 -xi,,
(3.17)
where p=
(wgiw2+k?_~~‘2=-ir,
(3.18)
so that Re $> 0, Im 0 < 0. The solution of eq. (3.17) which vanishes at infmity is given by
X(kll~l~3X~)= -
p2 “” k’:
[(n2+ exp(-k~~~ - xf3 1)
- (a2 + 02) exp(-Plx3 - 4 I) Izp
k;)
I+
Cl exp(-knx,)
+
C2 exp(-8x,).
(3.19)
The two coefficients Cr and C2 in this solution are determined from the two equations obtained by requiring that the solution (3.19) satisfy the integro-differential equation (3.16) on the one hand, and the boundary condition (2.3 lb) on the other. After some tedious analysis the following result is obtained:
+
44 2il?
Eo
exp(iUxs - x; t >,
(3 -20)
where e(w) = co t o~/(o$ - w2 - ioy)
(3.21)
is the bulk dielectric constant at k = 0, and where eq. (3.18) has been used. Substitution of eq. (3.20) into eq. (2.24) yields as the dispersion relation for sur-
24
A.A. Maradudin /Surface
plasmon despersion relation
ikjj -r +2 [E(W) -EJ=0. -P+; -aikll tr‘1 [ ik11
face plasmons 1
+
E(O)
iklj t (Y
This equation
can be rearranged into the form
1
(Y--r + e(w) - fo 1 t E(W) = ikll E(O) (k,, t icu)(klj - ir) 2r [ Inasmuch obtaining it suffices dispersion
(3.22)
’
(3.23)
as the right hand side of this equation is explicitly proportional to kli, in the frequency of the surface plasmon to lowest nonvanishing order in kll to set kll = 0 in the expression in braces. We thus obtain the approximate relation
1 te(o)riktl
[e(~)(~-~)+c(wz)~80]~,l;o.
At this point we neglect the intrinsic damping in the dielectric medium tend to zero through positive values. In this limit we find that r(kr = O)= a-1/2(aZg = D-“*(J o(kl, = 0) = a-r’*
- o~*)l’*
o
_ (.Qi)
w>oo;
((& - ,*)I’*
= o-l/2(w2
_
&l/2
O
(3.24) and let y
(3.25)
(3.26)
W>f&,.
We next note that the equation E(W) t 1 = 0 is the surface plasmon dispersion relation in the absence of spatial dispersion. Consequently the frequency of the surface plasmon in the absence of spatial dispersion, w,, is the solution of the equation (3.27)
f(W& + 1 = 0. We thus write the solution of eq. (3.24) in the form
(3.28)
a=W,+Aw,,
where Aw, is of first order in k,,. Substituting eq. (3.28) into eq. (3.24) and linearizing the resulting equation in Aw, with the aid of eq. (3.27) we obtain (3.29) If we denote the static (w = 0) value of the bulk dielectric constant eS, we have the relation E, = Ee + (w”,/&. With the substitution
at k = 0 by (3.30)
of eq. (3.21) (with y = 0) into eq. (3.27) we find that the
A.A. Maradudin /Surface
frequency
phmon
25
despersion relation
of the surface plasmon in the absence of spatial dispersion is given by (3.31a)
0; = w; + w2p/(l + Ee), or equivalently 0, = [(l + E,)/(l + ee)]“Z It is readily determined
(3.31b)
we.
that (3.32)
WO
These inequalities, together with eqs. (3.25) and (3.26), enable us to express Aw, given by eq. (3.29) in the form
Aw, =p E’(WS) (c.$, - 1w;)r’, D”‘k,,
1 i(eo + 3) _p_ (wi - wi)l’2 I 2 +
-i----_ 60 2
1’
3
(3.33)
which agrees completely with the result for Aw, obtained by Maradudin and Mills [5] on letting the speed of light tend to infinity in the surface polariton dispersion relation they obtained on the basis of the dielectric constant (3.7).
4. Discussion In this paper we have obtained an explicit dispersion relation for surface plasmons at the interface between vacuum and a semi-infinite dielectric medium occupying the half-space x3 > 0 and characterized by a nonlocal dielectric constant &w b3x;). It should be remarked that the derivation of eqs. (2.24) and (2.31) presented here is somewhat artificial. It should be possible to obtain these results more directly, but we have not attempted to do so. It also appears as if the use of eqs. (2.24) and (2.3 1) to obtain the surface plasmon dispersion relation may involve lengthier calculations than are required in a more conventional approach based on a direct solution of the equations of electrostatics. Nevertheless, it still seems useful to have an explicit dispersion relation, eq. (2.24), with a definite prescription for obtaining the function x(kllw Ix&) entering it, eq. (3.31). For example, in the case of dielectric constants of more complicated form than those considered in the preceding section eq. (2.3 1) can serve as the basis for approximate determinations of x(/cllw lxaxj), e.g. by variational methods. The fact that the method developed here yields the same surface plasmon frequency, eqs. (3.28), (3.31), and (3.33), as was obtained by a rather different method by Maradudin and Mills [5] is of some independent interest for the following reason. The dielectric contant (3.7) is obtained by partially Fourier transform-
A.A. ~~r~d~d~~ /Surface
26
plasmon despersion relation
ing the bulk dielectric constant
E(k, 0) = Eo
+
4 w$+Dk’-w2-iwy’
(4.1)
according to
(4.2) and restricting x3 and x; to be positive. Recently Horing [73 used e(k, w) as given by eq. (4.1) in the Ritchie-Marusak dispersion relation, eq. (l.l), and obtained an expression for the frequency of a surface plasmon which differs from the Maradudin-Mills result, eqs. (3.28), (3.31), and (3.33). This difference is not very surprising in fact. As we have noted in the Introduction, underlying the RitchieMarusak dispersion relation is the assumption that electrons are reflected specularly from the surface of the solid. The dielectric constant (3.7), on the other hand, represents the so-called dielectric approximation, in which the bulk dielectric constant is partially Fourier transformed according to eq. (4.2), after which x3 and .xj are restricted to be positive, so that the resulting e(kllo1xax;) is no longer a function of x3 -xk, but depends on x3 and xi separately. It is known that the dielectric approximation does not conserve particle number [8], and causes the surface to act as a source or sink of energy [9], neither of which is the case when specular reflection is assumed. Thus a different physical situation is being considered in Horing’s work from that assumed in the work of Maradudin and Mills. Consequently, it is not surprising that different surface plasmon frequencies are obtained in these two calculations. However, this example points up the desirability of having a dispersion relation for surface plasmons which is valid for more general physical situations than is represented by the assumption of specular reflection at the surface. Eqs. (2.24) and (2.31) provide such a dispersion relation.
I am grateful to Professor N.J.M. Horing for informing me of his work and for a discussion of it. The present paper is a direct result of that work.
References [I] R.H.
Ritchie and A.L. Marusak, Surface Sci. 4 (1966) 243. 121 In this paper we use the term “surface plasmon” to denote a surface localized electromagnetic wave in the limit that the effects of retardation are neglected. 131 See, for example, the discussion by D.L. Mills and E. Burstein, Rept. Progr. Phys. 37 (1974) 817.
A.A. Maradudin /Surface phsmon despersion relation [4] [5] [6] [7]
21
J.L. Birman and J.J. Sein, Phys. Rev. B6 (1972) 2482. A.A. Maradudin and D.L. MilJs, Phys. Rev. B7 (1973) 2787. G.S. Agarwal, D.N. Pattanayak and E. Wolf, Phys. Rev. B10 (1974) 1447. N.J.M. Horing, in: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Eds. R. Dobrozemsky, F. Rtidenauer, F.P. Viebock and A. Breth (Berger, Vienna, 1977) p. 419. [8] J.T. Foley and A.J. Devaney, Phys. Rev. B12 (1975) 3104. [9] M.F. Bishop and A.A. Maradudin, Phys. Rev. B14 (1976) 3384.
A SURFACE PLASMON DISPERSION RELATION
IN THE PRESENCE OF
SPATIAL DISPERSION ** A.A. MARADUDIN Department
of Physics, University of California, Irvine, California 92717,
USA
Received 16 May 1978; manuscript received in fmal form 25 September 1978
A dispersion relation for surface plasmons (retardation neglected) is obtained in the presence of spatial dispersion. The physical system considered is one of vacuum separated by a plane interface from a dielectric medium occupying the half-space xg > 0, and characterized by a nonlocal dielectric function e(kllw Ixsxh), assumed to be symmetric in xa and xi. Here $1 is a twodimensional wave vector whose components are parallel to the interface, and w is the frequency of the electromagnetic field in the medium. The dispersion relation has the form 1 + for obtaining the function kllx(kll~ Ix3 = 0, xj = 0) = 0, and an explicit prescription x(kllw Ixsx$) is presented. The use of the dispersion relation is illustrated by applying it to two examples: (1) a local dielectric constant; and (2) the nonlocal dielectric function used previously by Maradudin and Mills. In both cases previously obtained results are recovered.
1. Introduction Several years ago Ritchie and Marusak [l] published a derivation of a dispersion relation for surface plasmons [2] in the presence of spatial dispersion. Their result has the simple form
1
s
+k” R
s3
_m k%(k, w) =
0.
(1.1)
Here e(k, w) is the frequency and wave vector dependent dielectric constant of the material out of which the semi-infinite medium is composed, along whose planar interface with vacuum the surface plasmon propagates. In eq. (l.l), k~ is the magnitude of the projection of the three-dimensional wave vector k on the plane of the surface, and kB is the component of ‘k normal to the surface. A central assumption in the derivation of eq. (1 .l) is that the electrons in the * This research was supported N00014-76Cal21. ** Technical Report No. 78-16.
in part by the Office of Naval Research under Contract No.
13
14
A.A. Maradudin /Surface
plasmon despersion relation
semi-infinite dielectric medium are reflected specularly from the surface. It is this assumption which ultimately leads to the appearance of the bulk dielectric constant e(k, 0) in eq. (1.1). However, in any real solid it is unlikely that all of the electrons are reflected specularly from the surface. It is more likely that some fraction are scattered specularly and the rest diffusely, or that more general boundary conditions obtain. In any case, it would seem to be desirable to have a dispersion relation for surface plasmons which does not depend, in its derivation, on some particular assumption about the nature of the interaction of an electron with the boundary. Moreover, it would also seem to be desirable to have such a dispersion relation which is not based on the assumption that the dielectric properties of the medium supporting the surface plasmon are due to a specific collective excitation in the medium, viz. an electron plasma, so that it applies to systems in which these properties are associated with other electric dipole excitations such as IR active phonons and excitons, for example, or combinations of them. It seemed to be worthwhile, therefore, to try to obtain a dispersion relation for surface plasmons in the presence of spatial dispersion of a form similar to that of eq. (l.l), but without invoking the restrictive assumption of specular reflection of electrons at the boundary of the dielectric, in which the central role is played by the nonlocal dielectric constant of the medium, about which a minimum number of assumption are made. In this paper we present the derivation of such a dispersion relation. It is obtained in section 2, and its use illustrated by application to systems for which the dispersion relation is already known in section 3. A discussion of the results obtained is given in section 4.
2. The dispersion relation We assume a dielectric medium which occupies the semi-infinite region x3 > 0. The region x3 < 0 is occupied by vacuum. Because the system possesses infmitesimal translational invariance in directions parallel to the surface, the macroscopic electric field and the displacement in it can be expressed in the forms (2.la) D(x, t) = D(kllolx3)
(2.lb)
exp(iklt * x11- iwt),
where kll = z?rki t ~?&a, xl1 = z?rxi + &xa, and 2i and 2a are two mutually perpendicular unit vectors in the plane of the dielectric-vacuum interface. Within the dielectric medium the relation between D(kilw Cy3)and E(kttw lxa) is assumed to be D(kIlOlX3)
=
f 0
cl‘4
&WIX3X;)
wwI4),
x3
2
0,
(2.2)
A.A. ~aradud~n / Surf&e plasmon despersion retation
15
where E(IEIIW Ix&) is assumed to be symmetric in the variable xs and x(3. Thus, we assume the nonlocal dielectric tensor of the semi-infmite medium to be isotropic. This assumption is similar to and, because of the lower symmetry of the semi-infmite medium, may be as restrictive as the combined assumption by Ritchie and Marusak of an isotropic, nonlocal, bulk dielectric tensor and specular reflection of electrons from the boundary. The assumption of isotropy, however, can be removed from both the present theory and that of Ritchie and Marusak. The result in each case is a more complicated dispersion relation. The assumption of specular reflection from the boundary, on the other hand, is essential in the RitchieMarusak approach, while it is not in the present treatment. The Maxwell equation, v * D(x, t) = 0,
(2.3) is now combined with eq. (2.2) and with the equation of the electrostatic approximation
E(x, t) = -Q&X, t),
(2.4)
where tp(x, t) is a scalar potential, to yield an integro-differential equation for cp(ktlwlxs), the Fourier coefficient of cp(x,t):
--
d k3
= s
CL&e(k,,wlx&)
d --&~,,c&)
= 0.
h3
0
(2.5)
To solve eq. (2.5) we begin by formally expanding both e(kttwIxfl$) and (d/b,) e@tlwjx&) in double series of functions complete and orthonormal in the semi-infinite interval (0, -), and localized in the vicinity of the interface at x3 = 0: (2.6a)
r
d
$kllW 1x34)
=
3
2 m,n=Q
bmn(bQ)
pm(x3)
$%(x9.
(2.6b)
Although our final result will be independent of any particular choice for the {am), a convenient choice, for definiteness, is the set of Laguerre functions defmed by (p,@3)
= P2
ewi--+3)
Lz(px3) 7,
(2-V
16
A.A. Maradudin / Surface plasmon despersion relation
where &(x3) is the nth Laguerre polynomi~, and /_3is a real, positive parameter with the dimensions of an inverse length. Because of our assumption that e(kl+r 1xsx:) is symmetric in x3 and XL,it’is the case that the coefficients {a,,(k~~o)} are symmetric in the subscripts m and n. We next expand rp(k&~) and (d/~3)~(~11~~~) in terms of the &m(xa)):
(28a)
(2.8b) so that (2.9a)
(2.9b)
When we substitute eqs. (2.6) and (2.8) into eq. (2.5) and use the o~hono~~ity of the (~,&)], we find the latter takes the form (2.10) The coefficients ~~~(~,,~)~ and {B&t w )} are not independent, however. If we integrate by parts the integral in eq. (2.9b) we obtain the result that
where the notation cp(kllwIO+)is intended to emphasize that it is the potential on the medium side of the interface that is being considered. If we substitute eq. (2.8a) into the second term on the right hand side of eq. (2.1 l), we obtain fmally the relation &A&-IIW) = -cp,(O) cp(~ll~Io+) - ,co X where
%zp&&l~),
(2.12)
(2.13a)
17
A.A. Maradudin /Surface plasmon despersion relation
We note for future reference that enp can be expressed equivalently
as (2.13b)
0
0
On combining given by
B&w)
eqs. (2.10)
= -k$&~IO+)
and (2.12) we find that the coefficient
,$e
[Q
+ ea-’
B,#ctlo)
is
(2.14)
61;; cp,@),
where for convenience we have suppressed the argument (k,lw) of the matrices on the right hand side of this equation. It follows from this result and eqs. (2.2), (2.4), (2.6a) and (2.9b) that
a(kllw) and b(kllo)
5
~3(~llwlO+)=k~cp(k~~wlO+)
m.n=O
cp,(O) [kia-’
+ ea-lba-l]~n
cp,(O).
We now turn to the vacuum region x3 < 0. In this region the function is readily found to have the form CP(~IIW~X~)= A
x3
ev(W3),
where A is an arbitrary becomes D3(k~~olx3) = -&
<
constant,
exp(hx3),
cp(k~~w1x3)
(2.16)
0,
while the displacement
x3
(2.15)
<
0.
component
D3(kltwlx3)
(2.17)
The boundary conditions in the problem are the continuity of cp(x, t) and D3(x, t) across the surface x 3 = 0. The first of these yields the relation cp(kllo lot) = A; the use of this result together with eqs. (2.15) and (2.17) in the second boundary condition yields the dispersion relation for surface plasmons in the form
(2.18a) where xmn(kllw) = [kia-’
+ ea-‘ba-‘I&
.
(2.18b)
It is useful to point out that x~,&,o) is symmetric in m and n. For this purpose it suffices to show that the matrix k;fa-’ + ea-%a-’ is a symmetric matrix. Inasmuch as the matrix a-l IS ’ symmetric because the matrix a is, it is necessary to show only that the matrix X = ea-‘ba-’ is symmetric. To show this we first relate the coefficients {b,,(kltw)) to the coefficients
18
A.A. Maradudin /Surface
(a,&~)}.
plasmon despersion relation
We see from eq. (2.6b) that
(2.19)
An integration
by parts yields
bmn(klla)= -$ dx13 cPm(“) e(kllwlOxi) IPi2(Xt3) 0
(2.20) We now substitute (2.13a) to obtain
eq. (2.6a) into the right hand side of this equation,
and use eq.
(2.21) The second equality in eq. (2.21) follows from the use of eq. (2.13b). It follows from eq. (2.21) that the matrix X can be written equivalently X = ee-1 (&)e-’
= ea-ieT
= XT,
as (2.22)
where MT denotes the transpose of the matrix M. The matrix x(kllw) is therefore symmetric, and can be written alternatively as Xmn(kllw) = [k&s-’ + eadleT]&.
(2.18c)
Eq. (2.18) may have some interest of its own in connection with purely numerical studies of surface plasmons in spatially dispersive media. However, at this point we direct our efforts to re-expressing eq. (2.18) in a form in which the {qm(x3)} no longer appear explicitly or implicitly. We begin by introducing the function X(kllw lx3x;), which is defined by X(kIlcJJIx,&)
=
5 Prn(X3) m,n=O
In terms of this function
Xrnn(kll~)
Pp,(xk),
x3,x;
> 0.
the dispersion relation (2.18) becomes (2.24)
1 + kllx(kllw 100) = 0. To obtain the function x(kllw kg;) or more accurately, ing it, we return to eq. (2.18c), which we rewrite as
c
ki za%pxpn + prs empGksr ;Crn = 6mn . P
(2.23)
the equations
determin-
(2.25)
A.A. ~~~~dudi~ /Surface plasmon despersion relation
19
We now multiply both sides of this equation by ~~(x~)#~(x~) and sum on M and n from zero to infinity. We recall the orthonormality and completeness conditions 6 mn = s
&3
(Prn(X3)
(2.26%)
(Pry,
0
(2.26b) We next note that
(2.27) where ~-~(kl+~Ix~xj) is the inverse of e(kllolx,x;)
in the sense that
m
J
dx;
e-l
(k,,w
lx,xj’)
e&p
1x:x;)
0
= 6(x3
=
dx;
.I0
e(k,,wlx34)
E-‘(k,,WIX;X;)
-xi).
(2.28)
We also use the relations
5
Y&(X3)
emn%2fxj)=
m, n=O
fi
-
&6@3
-x’,)
+ ~(x3~so]
9
(2.29a)
[
Ipm(x3)
enm
Cpn
$s(x3
-xi3)*
(2.29b)
m,n=O
The result of ail of this is to transform eq. (2.25) into
k;
r
dx’;
e-‘(k,,wIx3xlj)
X(krpMhS)
0
(2.30)
20
A.A. Maradudin /Surface
Finally, we see that the function tial equation
plasmon despersion relation
x(kIlw Ixsx;) which satisfies the integro-differen-
kf! dx~E-1(k,,wlX3XI;)X(k,,wIX);X;) 0 _-
d &3
J 0
dx; ~-~(~,,wlx,x';)~x(~,,wlx;x;)= ax;
together with the boundary
J
dx; E-l(k,,wIOxLJ
0
6(x3 -XL),
(2.31a)
condition
&&,,C&‘;Xj) dx;
(2.31b)
= 0,
is a solution of eq. (2.30). As we are seeking electric fields localized in the vicinity of the surface x3 = 0, the boundary condition which must be imposed on X(k,,ti b3x5) at infinity is that lim
x(k1lwIX3X;)=O.
(2.31~)
x34=
Eq. (2.24), together with the prescription for obtaining x(kllw Jx3x;) given by eqs. (2.31), is the central result of this section: the surface plasmon dispersion relation. In the next section we illustrate its use by applying it to two cases for which the surface plasmon dispersion relation is known.
3. Examples In this section we solve eqs. (2.3 1) for two choices of e(kjl~ lxsxh), one local and the other nonlocal, for which the surface plasmon dispersion relation is already known, to illustrate the use of the dispersion relation, eq. (2.24). 3.1. A local dielectric tensor We first consider the case in which e(ICit~ (x3x;) is given by 4~llWIX3X;)=E(W)~(X3
where e(w) is independent e-l (k,,wlx,x;)
1 = --6(x,
-XL),
of k,, and of the coordinates
(3.1) x3 and x;. It follows that
-x$>
f(W)
The equation
(3.2)
satisfied by x(kllw 1x3x\) in this case becomes (3.3a)
A.A. Maradudin /Surface
together with the boundary
21
plasmon despersion relation
condition
-$x(q.4x,x;)lx3hg = 0.
(3.3b)
3
The solution
of eq. (3.3a) which vanishes as x3 + 00 is +A
exp(-Q3)
,
(3.4)
3
where the first term is the particular integral and the second is the complementary function. The coefficient A is determined with the aid of eq. (3.3b), with the result that e(u) x(~llwlx3x~)=--
2ks
bp(-hlx3
When eq. (3.5) is substituted dispersion relation
-xiI)+exp[-k11(x3
into
+&>I}.
eq. (2.24), we obtain
as the surface plasmon
1 te(o)=0, a well-known
(3.5)
(3.6) result for this case [3].
3.2. A nonlocal dielectric constant The second example we consider is based on the nonlocal dielectric function
E(~IIwIx~x)~) = eo6(x3 - xi) + i -2or
exp(irIx3
--&I>,
x3,x$>@
(3.7)
which was used by Birman and Sein [4], Maradudin and Mills [S], and Agarwal, Pattanayak, and Wolf [6], in studies of various optical optical properties of semiinfinite media in the presence of spatial dispersion. There are some unphysical features of the model underlying this form for e(kllo lxaxk), as will be discussed in the following section. However, because results based on its use are available it serves as a useful example to illustrate eq. (2.24), which is not restricted by these unphysical features. In eq. (3.7) ee is the optical frequency dielectric constant, which is assumed to be real and frequency independent; on is a plasma frequency; D is a positive constant which defines the curvature of the exciton or transverse optical phonon dispersion curve at k = 0; and r is given by
(3 J) so that Re P > 0, Im r > 0. The frequency w. in eq. (3.8) is that of the exciton or transverse optical phonon at k = 0. The equation for the inverse dielectric function eM1(kp/x3xk), eq. (2.28), in
22
A.A. Maradudin /Surface
plasmon despersion relation
the present case takes the form io’p e-l(k,,WIXaX$) + __ 2e($r =
m dxg exp(iP(xs .I-
- x’; I) e-l(kljW 1x:x;)
0
Ely S(x3 - x’,).
(3.9)
We apply the operator (d2/dxi) into
d’ t a2
(
2
)
=A-J- +r*
1
d.4
+ I’2 to both sides of this equation to transform
E-l(k,,w,x3x;)
co
(
d-4
6(x3 -
xi),
it
(3.10)
where tY=
W”-Wf_
(
iwy II2 -ki+y
D
1
(3.1 la)
9
0: = w;+(w2p/eo),
(3.1 lb)
so that Re OL> 0, Im CY > 0. We now use the fact that
(-$+02)
[6(xaLx;)tI$$exp(i~lx,
-xkiq=
($+F2)
6(x3-x$), (3.12)
to write the solution of eq. (3.10) in the form E-r(k,,WIXsX:)
wi exp(icrlxs -xhl) 1 = 0 6(x3 - xk) +2 2iQ eoD
t A exp(icrxs)
(3.13)
The coefficient A is obtained by requiring that the expression (3.13) also satisfy the original integral equation (3.9). In this way we obtain finally the result that e-l &,w 1x,x;) = ;
1
6(x3 - x$)
a-r
4 +=A 0
(
=;
-x)3) +f(k,,wIX3X:).
6(x,
exp(iculx3 -x’,l)+--(y +
r
I
expMx3 + &)I
(3.14)
We note the result that (3.15)
23
A.A. Maradudin /Surface plasmon despersion reletion
The integral equation (2.3 la) for the function x(kt+~ 1x3x;) now takes the form 1 d2 -~eo ( dx:
kj
= -Ei(Xj -
1
x(k,$.Q,x,x’,) + $
p
&x(3) f&lo
(x3x;)
30
x;>.
(3.16)
We now apply the operator (d~~~~) + o2 to both sides of this equation, and use eq. (3.15), to obtain
($- k~)(~ -B2) xVq,olxsx;)=-eo (-$+a2) %x3 -xi,,
(3.17)
where p=
(wgiw2+k?_~~‘2=-ir,
(3.18)
so that Re $> 0, Im 0 < 0. The solution of eq. (3.17) which vanishes at infmity is given by
X(kll~l~3X~)= -
p2 “” k’:
[(n2+ exp(-k~~~ - xf3 1)
- (a2 + 02) exp(-Plx3 - 4 I) Izp
k;)
I+
Cl exp(-knx,)
+
C2 exp(-8x,).
(3.19)
The two coefficients Cr and C2 in this solution are determined from the two equations obtained by requiring that the solution (3.19) satisfy the integro-differential equation (3.16) on the one hand, and the boundary condition (2.3 lb) on the other. After some tedious analysis the following result is obtained:
+
44 2il?
Eo
exp(iUxs - x; t >,
(3 -20)
where e(w) = co t o~/(o$ - w2 - ioy)
(3.21)
is the bulk dielectric constant at k = 0, and where eq. (3.18) has been used. Substitution of eq. (3.20) into eq. (2.24) yields as the dispersion relation for sur-
24
A.A. Maradudin /Surface
plasmon despersion relation
ikjj -r +2 [E(W) -EJ=0. -P+; -aikll tr‘1 [ ik11
face plasmons 1
+
E(O)
iklj t (Y
This equation
can be rearranged into the form
1
(Y--r + e(w) - fo 1 t E(W) = ikll E(O) (k,, t icu)(klj - ir) 2r [ Inasmuch obtaining it suffices dispersion
(3.22)
’
(3.23)
as the right hand side of this equation is explicitly proportional to kli, in the frequency of the surface plasmon to lowest nonvanishing order in kll to set kll = 0 in the expression in braces. We thus obtain the approximate relation
1 te(o)riktl
[e(~)(~-~)+c(wz)~80]~,l;o.
At this point we neglect the intrinsic damping in the dielectric medium tend to zero through positive values. In this limit we find that r(kr = O)= a-1/2(aZg = D-“*(J o(kl, = 0) = a-r’*
- o~*)l’*
o
_ (.Qi)
w>oo;
((& - ,*)I’*
= o-l/2(w2
_
&l/2
O
(3.24) and let y
(3.25)
(3.26)
W>f&,.
We next note that the equation E(W) t 1 = 0 is the surface plasmon dispersion relation in the absence of spatial dispersion. Consequently the frequency of the surface plasmon in the absence of spatial dispersion, w,, is the solution of the equation (3.27)
f(W& + 1 = 0. We thus write the solution of eq. (3.24) in the form
(3.28)
a=W,+Aw,,
where Aw, is of first order in k,,. Substituting eq. (3.28) into eq. (3.24) and linearizing the resulting equation in Aw, with the aid of eq. (3.27) we obtain (3.29) If we denote the static (w = 0) value of the bulk dielectric constant eS, we have the relation E, = Ee + (w”,/&. With the substitution
at k = 0 by (3.30)
of eq. (3.21) (with y = 0) into eq. (3.27) we find that the
A.A. Maradudin /Surface
frequency
phmon
25
despersion relation
of the surface plasmon in the absence of spatial dispersion is given by (3.31a)
0; = w; + w2p/(l + Ee), or equivalently 0, = [(l + E,)/(l + ee)]“Z It is readily determined
(3.31b)
we.
that (3.32)
WO
These inequalities, together with eqs. (3.25) and (3.26), enable us to express Aw, given by eq. (3.29) in the form
Aw, =p E’(WS) (c.$, - 1w;)r’, D”‘k,,
1 i(eo + 3) _p_ (wi - wi)l’2 I 2 +
-i----_ 60 2
1’
3
(3.33)
which agrees completely with the result for Aw, obtained by Maradudin and Mills [5] on letting the speed of light tend to infinity in the surface polariton dispersion relation they obtained on the basis of the dielectric constant (3.7).
4. Discussion In this paper we have obtained an explicit dispersion relation for surface plasmons at the interface between vacuum and a semi-infinite dielectric medium occupying the half-space x3 > 0 and characterized by a nonlocal dielectric constant &w b3x;). It should be remarked that the derivation of eqs. (2.24) and (2.31) presented here is somewhat artificial. It should be possible to obtain these results more directly, but we have not attempted to do so. It also appears as if the use of eqs. (2.24) and (2.3 1) to obtain the surface plasmon dispersion relation may involve lengthier calculations than are required in a more conventional approach based on a direct solution of the equations of electrostatics. Nevertheless, it still seems useful to have an explicit dispersion relation, eq. (2.24), with a definite prescription for obtaining the function x(kllw Ix&) entering it, eq. (3.31). For example, in the case of dielectric constants of more complicated form than those considered in the preceding section eq. (2.3 1) can serve as the basis for approximate determinations of x(/cllw lxaxj), e.g. by variational methods. The fact that the method developed here yields the same surface plasmon frequency, eqs. (3.28), (3.31), and (3.33), as was obtained by a rather different method by Maradudin and Mills [5] is of some independent interest for the following reason. The dielectric contant (3.7) is obtained by partially Fourier transform-
A.A. ~~r~d~d~~ /Surface
26
plasmon despersion relation
ing the bulk dielectric constant
E(k, 0) = Eo
+
4 w$+Dk’-w2-iwy’
(4.1)
according to
(4.2) and restricting x3 and x; to be positive. Recently Horing [73 used e(k, w) as given by eq. (4.1) in the Ritchie-Marusak dispersion relation, eq. (l.l), and obtained an expression for the frequency of a surface plasmon which differs from the Maradudin-Mills result, eqs. (3.28), (3.31), and (3.33). This difference is not very surprising in fact. As we have noted in the Introduction, underlying the RitchieMarusak dispersion relation is the assumption that electrons are reflected specularly from the surface of the solid. The dielectric constant (3.7), on the other hand, represents the so-called dielectric approximation, in which the bulk dielectric constant is partially Fourier transformed according to eq. (4.2), after which x3 and .xj are restricted to be positive, so that the resulting e(kllo1xax;) is no longer a function of x3 -xk, but depends on x3 and xi separately. It is known that the dielectric approximation does not conserve particle number [8], and causes the surface to act as a source or sink of energy [9], neither of which is the case when specular reflection is assumed. Thus a different physical situation is being considered in Horing’s work from that assumed in the work of Maradudin and Mills. Consequently, it is not surprising that different surface plasmon frequencies are obtained in these two calculations. However, this example points up the desirability of having a dispersion relation for surface plasmons which is valid for more general physical situations than is represented by the assumption of specular reflection at the surface. Eqs. (2.24) and (2.31) provide such a dispersion relation.
I am grateful to Professor N.J.M. Horing for informing me of his work and for a discussion of it. The present paper is a direct result of that work.
References [I] R.H.
Ritchie and A.L. Marusak, Surface Sci. 4 (1966) 243. 121 In this paper we use the term “surface plasmon” to denote a surface localized electromagnetic wave in the limit that the effects of retardation are neglected. 131 See, for example, the discussion by D.L. Mills and E. Burstein, Rept. Progr. Phys. 37 (1974) 817.
A.A. Maradudin /Surface phsmon despersion relation [4] [5] [6] [7]
21
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