- Email: [email protected]
A simple dispersion relation for the surface plasmon: Dependence on electron density profile
Surface Science 48 (1975) 241-252 0 North-Holland Publishing Company
A SIMPLE DISPERSION RELATION
FOR THE SURFACE PLASMON:
DEPENDENCE ON ELECTRON DENSITY PROFILE
Basab B. DASGUPTA, Pradeep KUMAR and D.E. BECK Department of Physics and Laboratory for Surface Studies, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201,
USA
We know from the quantum-mechanical, random-phase-approximation (RPA) calculations for the surface plasmon that the electron density profile at the surface plays an important role in determining the dispersion relation. The full RPA calculations are very difficult to perform, and we present a simple expression for the dispersion relation which depends on the electron density profile at the surface. Our derivation is essentially semi-classical and parallels that of Ritchie and Marusak. The validity of our expression for the dispersion relation is assessed by computing the dispersion relations for different electron density profiles and comparing them with those obtained in the full RPA calculations.
1. Introduction
The surface plasmon (SPO) is an important feature in the electronic excitation spec. urn of metal and semiconductor surfaces [l] . Hence, one needs to incorporate its effects into surface studies such as low-energy electron diffraction, photoemission, surface energy and response of the surface to external charges. The full quantummechanical random-phase-approximation (RPA) calculations [2-61 of the surface plasmon excitation spectrum are very difficult, and numerous approximate semiclassical treatments [7-lo] have been employed to calculate the dispersion relation for the SPO. An early calculation of the SPO dispersion relation for a metal surface was performed by Ritchie and Marusak [8] (RM). These authors used the bulk RPA dielectric function in the metal and matched the solution of Poisson’s equation at the metal vacuum interface. This semi-classical calculation resulted in a simple integral expression for the SPO dispersion relation which did not depend on the electron density profile at the surface. Subsequently, the hydrodynamic calculation of Bennett [9] and the quantum-mechanical RPA calculations [3-.5,1 l] revealed a strong dependence of the dispersion on the electron density profile. In this paper, we derive a semi-classical, integral expression for the SPO dispersion relation which admits a dependence on the surface electron density profile. Our derivation which parallels that of RM is presented in section 2. In section 3, we de-
242
B.B. Dasgupta et al.lSimple dispersion relation for surface plasmon
velop a simple, density-dependent, surface dielectric function which reduces to the correct limiting forms in the vacuum and deep in the metal. Using this dielectric function and the hydrodynamic approximation for the bulk-dielectric function [I 21, we are able to perform most of the integrations in our calculations analytically. The calculation of the dispersion relation for different electron density profiles is carried out in section 4. These calculations demonstrate the application of our formulation and show that the corrections to the dispersion relation for the step-density profile have the correct magnitude. We test our surface dielectric function by comparing our dispersion relation to those obtained in calculations which treat the surface in a more complete manner; i.e., the hydrodynamic calculation of Bennett [9], and the quantum-mechanical RPA calculations [3,5,6].
2. Derivation of the dispersion relation The general expression relating the displacement
and electric fields in a medium
is D(r,r)=Sdr’P(r,r’,t).E(r’,t),
(1)
where i(r, Y’,t) is the dielectric tensor. In an infinite, homogeneous pression reduces to D(r, t) =Jd
P‘ E~(M’,
t) ECr’, t),
medium this ex-
(2)
where eB(r,t) is the bulk longitudinal dielectric function. For an infinite electron gas the Fourier transform of the bulk dielectric function has within the RPA, the limiting value
where wp is the bulk-plasmon
frequency
mi = 4rrnOe2fm.
(4)
Here e and m are the electron charge and mass, and n0 is the bulk electronic charge density. We consider a metal in the half-space z > 0 which consists of an electron gas and its compensating, homogeneous ionic background. The system is uniform in the x andy directions. For this system the Fourier transforms are taken with respect to the coordinates of the x and y planes, R, and the time, t, to give dt e-i(K-R-wt) ~K,w(z>=JdRJ-
F(R, z, t).
B.B. Dasgupta et nl./Simple dispersion relation for surface plasmon
The additional
transformation
FK,,(kZ)
243
with respect to z is given by
=J’dz e-ikzz&Jz),
and k = (I(, iiz) is the total momentum. For this model of the surface, we employ a dielectric function which is consistent with the requirement of translational invariance parallel to the surface, and eq. (1) reduces to (5)
=f^d z’ f&w(z? z’) &,,(z’).
&,,(z)
In their calculation, RM use the bulk RPA dielectric function for the metal in (5). This gives the correct response in the bulk region of the electron gas, but it is certainly not correct near the surface. We wish to retain the bulk response in our calculation and include an approximate treatment of the surface electron density profile. To facilitate this we write the dielectric function as eQJz*
z’) = &,,(z
-z’)
f Y&JZ,
z’)
(6a)
for z > 0 where r>(z, z’) vanishes far from the surface. In a vacuum the displacement and electric fields are equal, so for z < 0 we write the dielectric function as, elyJz,z‘)
= Nz-z’)
+ ?&Jz,
z’),
(6b)
where y<(z, z’) also vanishes far from the surface. The procedure adopted by RM in their derivation of the SPO dispersion relation was to solve Poisson’s equation for the fields in the metal and vacuum, and to match these fields at the surface. The dispersion relation for the SPO is obtained as the necessary condition for the existence of a solution for the surface. Our derivation parallels theirs’, and we obtain their relation by setting our y’s equal to zero. Consider the response of the system described by the dielectric function in eq. (6a) to an external charge disturbance. In the presence of an external charge, p(r, t), Poisson’s equation [ 131 is i k-I),,,@,)
= 4~ P~Jkz).
Defining the potential ik+~,,(k,)
(7)
by
= -&,Jk,),
(8)
and using (6a) in (5), we obtain &,(k,)
=
-ie3 x,w(k,)#~,,(k,)k-in-li3~yj;,,(k,,kl)~~,~(k:,)k1,(9)
where k’ = (K, kl). Substituting potential,
this into (7), we obtain an integral equation for the
244
B.B. Dasgupta et al./Sitnplt dispersion relo tion for surf&e piastnnn
These fields are to be matched at the surface, z = 0, to those found using the dielectric function given in (6b). Hence, we employ an external charge density which is confined to the surface, PK,&?
(11)
= 0K.w 6(z),
The system is then completed by requiring a symmetric continuation of the metal to the half-space z < 0 [ 141. For the system described by the dielectric function in C6b) the displacement field and potential are given by
and
The external charge here is f&(z) In order to (9), (lo), (12) on the electric the tangential tinuous at the
= CJ;,, 6(z). obtain the dispersion refatiun For the SPO, we seek soIut~ons ofeqs. and (13) dhich are consistent with the boundary conditions imposed and displacement fields by’Maxwell’s equations. Thus we require that component of the electric field, or equivalently the potential be consurface,
#O(z = o- ) = @(z = o+), and that the normal component face
(14) of the displacement
[DO@ = o--)] z = [D(z = O”)],.
field be continuous
at the sur-
(15)
[In these and the following expressions we suppress the subscripts Kand o which are common to all functions.] Using the symmetry of the y>, y’(k,, ki) = -y’(-k,, -ki), and the potential, Cp(k,) = 4(-k,), and evaluating the discontinuity of the displacement field across the surface charge at z = 0, we find
B.5. Dasgupta et aLlSimple dispersion relation for surface plasmon
,ik,O+
~r>(k,&) [k,g -k;] z
z-p i?
GK,,Jk;)
=
245
0.
This argument also holds for Do, and we have
Hence the boundary o. c
condition
eikzo- kz = ~ c k2
kz
on the displacement
field, (15), yields
eik,O+ k
k,
’
k2
or o” =-u and the dispersion relation for the SPO follows from (14). To obtain a simple integral expression for the dispersion relation we keep only the first order term in an iterative solution of the integral equation for tik,), (lo), hence we use the approximation, $(k,)
= = k2eB(k
1 - Q-113 Z
)
k;)k*k’
k12 eB(k’)Z
‘4
An iterative solution of $O(k,) 4n00 @‘(k,) = F
y’(k,,
c
1
leads to a similar approximation
1 _ Q_1,3 CY’(kz> kz
k:) k-k k’2
(16)
’
to (13), namely
1.
(17)
The dispersion relation for the SPO follows from (14) as a-113
eikz O+ ___
c
=
k,
[ k2 eB(k,)
a-213
c
I&;
,ik,O+p
fi k2kr2
k2
1
eikzo+y’(kz,k~)
_
+ elkzo y<(k,, eB(k,)
EB(k;)
ki)
1
(18)
.
Using the symbolic identity,
lim ~
k-OK2
2K -t k;
= alI3 6(k,),
we find that in the limit of K + 0 the dispersion relation becomes E! ,(O) = -1. Inserting the expression for the dielectric function in this limit, (3), we have the usual limiting value for the SPO frequency, wso = wP/d2.
A.B. Dnsgupta et al./Simple dispersion relation for surfhce plasmon
246
To proceed further we must make some assumption concerning the form of E(z,z’). In the next section we develop a form for the dielectric function and, in section 4, present calculations using a simple form for the bulk dielectric function. This allows us to present a concrete illustration of the application of our formalism.
3. Model dielectric function We know from the quantum-mechanical RPA calculations of the SPO dispersion relation that the linear response function which is proportional to the polarizability, 4nx(z,z’) = E(Z, z’) - 6(z -z’), is not a simple function of the electronic density. However, one can show that m
lim
K-t0
s
I@ “2 n(z)
dz’ L(z, z’) = __-.!--
-co
47wblo
’
where n(z) is the static electron density and the density response of the system is given by &z(z) =Jdz’L(z,z’)
I’(.+
Here V(z’) is the total electric potential. We also see from (3) and (4) that ~~,~(k, = 0) is proportional to the electronic density no. Hence, we shall use a dielectric function of the form E(Z,z’) 7 6(z - 2’) + 4n n(z) xB(z ~ z’)/no.
(19)
This expression has the desired features of a dielectric function for a metal surface. It goes to the bulk dielectric function deep in the metal, and in the vacuum, z < 0, where the electron density is zero it gives D(z) = E(Z). To understand the type of system implied by this model of the dielectric function, we consider the linear Boltzmann-Vlasov equation for a system of electrons in the presence of an external electric field, E,
Here the total distribution function for the electrons at a position, I, with a velocity u, isf(r,u, t) tfo(u,z) and the distribution function for the undistributed electrons isf,(o,z). This is the undisturbed distribution function and is related to the electronic charge density by n(z) = ,fduf&z>. The potential
U 115 ] arises from kinetic, exchange and correlation
effects near the
B.B. Dasgupta et al./Simple dispersion relation for surface plasmon
241
surface. If there is a dipole field due to the electronic and ionic charge density this field will also contribute to U. The induced current can be related to the total field by the polarizability. For a system where the undistributed electron distribution is independent of position and given by the usual Fermi distribution,fF(u), the calculation of the dielectric function is straightforward [ 161. If we make the ansatz, (20)
f,(U> z) = n(z)fI:(u)>
and neglect the term V,U* V, f in the Boltzmann-Vlasov equation, we can carry through the calculation of the dielectric function and obtain (19). Since it is the total electric field that appears in the expression for the dielectric function we need a system which has no dipole field at the surface, or the electronic and ionic charge distributions must coincide. In addition, neglecting the extra term requires that the variation of n(z) with z is small which is clearly not the case for a metal surface. Comparing eqs. (6) and (19), we have y;,,(z,
z’) = -An(z)
where An(z) = 1 - n(z)/nu, &(z,
[e;,,(z
-z’)
- F(z -z’)]
,
(214
and
z’) = 12(z) [e&(z
- z’) - 6(z -z’)]
.
(21b)
These expressions for y> and y< are small except near the surface, since An(z) is small in the metal, and n(z) is small outside the metal. Making use of the symmetric continuation of the metal in our derivations, the Fourier transform of the y’s are y>(k,,
k;) = -2 i
dz An(z) cos (kz - k;) z [en@;) - l] ,
(22a)
0
and 7’(K;,k;)=2Pdz,*(z)cos(k,-~~~z _m
[eB(k;)-I].
(22b)
These expressions in combination with (18) give us an integral expression for the dispersion relation of the SF’0 which depends explicitly on the electron density profile at the surface. To assess our expressions for the dispersion relation and surface dielectric function, we compute the dispersion relations for different density profiles and compare them with those obtained in other calculations. In the next section we carry out these calculations using the hydrodynamic expression for the bulk dielectric function. With this dielectric function we can perform most of the integrals for the dispersion relation analytically which greatly simplifies our comparison.
B.3. Dasgupta et al./Simple dispersion relation fbr surjbce plasmon
248
4. Hydrodynamic
calculation
The hydrodynamic +Jk,)
expression for the bulk dielectric function
= 1 -- w;/(wZ
[ 121 is
- fi2k’),
(23)
where [7] p2 = $u$ and uF is the Fermi velocity. This value of fi is chosen so that the bulk plasmon dispersion relation eE w(k,) = 0 agrees with the RPA expression (171 for small k2. Inserting this dielec&c function into (22) we obtain
Y’@,t k;)
cos (kz -k;)
= -2 2
j
eB(k;)
z
(24a)
dz An(z) kt2 + .2 i
0
’
and cos (k, - k:) z
r’(rr.h:)=2~~dz”(‘)--k~~~i.
(24b) z
0
where a2 = K 2 + (ws - 02)/p2 and r2 = m2/f12 - K2. For a uniform background of positive charge the sums over k, in the dispersion relation (18) go over to integrals in the usual way R-‘j3
t) + (2n)-1 kZ
m dkZ
s -Cc
and the integrals are easily performed by contour integration
(1-$)=-!$(w/ dzAn(z)[?(l
1 -y
,2 +--II.
to give
-w)e-z~z
O
s wz -_m
,
dzn(z) )li
where w = (1 - u2/w$l. For our comparisons we will only compute the term in the dispersion relation which is hnear in K, w2 = w; (1 +AK)/2. Substituting have
(26)
this expression into (25) and retaining only the terms linear in K, we
B.B. Dasgupta
A
+
et al/Simple
.I-dz An(z)
_
[2 -
dispersion
249
(27)
0
s _m
for surface plasmon
3 eeacz + 2 e-2aez]
0
t
relation
dzn(z)
[2 - coscuoz]
where o. = w,/flfi. In this relation An(z) and n(z) are positive quantities, hence we find that the linear coefficient, A, is a sensitive function of the location of the matching point, z = 0. This feature of the model is demonstrated by computing A for three density profiles at two matching points. One matching point, case (i), is taken at the position where a step positive background would have to be placed to insure the charge neutrality of the system; i.e., 0 [
n(z) dz = s
An(z) dz.
0
-ca
The other matching point, case (ii), is taken at the position where the electron charge density goes to zero, hence the second integral on the right-hand side of (27) vanishes. Linear density profile: Bennett in his hydrodynamic calculations [9] used an electron density profile, n(z) =
no
forz>b,
nOi%
fora
For this model of the metal surface the integrals in (27) can be performed analytically and we present a comparison with Bennett’s results in table 1. Infinite barrier model: A model of the metal surface which has been employed in many theoretical studies [3] is that of a semi-infinite electron gas confined by an infinite potential barrier. For this model the density is given by [ 181 n(z)=no
[
x2 x31,
3 cosx 1 t----
3 sinx __
where x = 2k,(z -a). Here k, is the Fermi momentum, and a is the location of the infinite barrier. The edge of the step positive background for this model is at x = 3n/4. Our results for this density profile are also presented in table 1 and compared with other calculations. Self-consistent density profile: The best theoretical density profile for a metal is that computed by Lang and Kohn [19] using the inhomogeneous electron gas theory. Their paper contains tables and graphs of the electron density profile. To perform
250
B.B. Dasgupta et ai./Sinrple
dispersim
relatiorl for surface plnsr~otr
Table 1 Values of the linear coefficient, A, in the SI’O dispersion relation, ccl. (26). (The density profiles are described in the text: the matching points are chosen at the edge of the step-ion backpound, case (i), and at the edge of the electron density profile, cast (ii): the step density result for ,.1 is %-I = I. 10 Ul;/W* Metal
Density profile
Other calculations ----__--_ Source -
MK (k~=1.3711-‘)
Linear b =211. b=3# b=4tY
Bennett
[9
This calculation __~ j1 fq/wp) fi)Af(uF/w$ -___~___I^
(ii)~(u~/~~~
0.01 -1.15 -1.72
-0.01 -0.75 ml.63
] 1.13 1.19 1.30
__I__. Infinite barrier
Al (X-I:=
Infinite
0.68
1191
Eeibelman
1.11
0.12
I .08
0.38
0.10
Beck [3) Heger and Wagner 161
1.75A-’ ) barrier
Self-consistent
~ Beck (31 lleger and Wagner [6J
0.73 0.30 [S 1
-0.55
l.I8
-~ 1.49
the calculation for the matching point at the edge of the electron density profile, case (ii), we choose a point a distance z = ?r/lcl: from the edge of the positive background where the density dropped to 0.01 nO. Since the density falls exponentially to zero the choice of the cutoff point is arbitrary and affects the value of A. Our results for this density profile are compared with Feibelman’s quantum-mechanical RPA calculation [S] in table 1. It is clear that our iteration procedure should give the best answer for a matching point chosen at the edge of the step-ion background, since the corrections. y> and -Y<. are smallest for this case. Hence, for our dielectric function, (19), we find that the linear coefficient,n, in the SPO dispersion relation is not very different from that found for a step-density profile using the bum-hydrodynamic dielectric function; i.e., A = ~20~. Examining table 1, we see that the best agreement with other calculations is obtained with the matching point chosen to be the edge of the electron density, case (ii). This seems to be a coincidence resulting from the strong dependence of the linear coefficient, A, on the matching point. This strong dependence is a feature which is introduced into the calculation by the iteration scheme. It means that the neglected terms in the iterative solutions may be inlportant in calculating the exact value of the linear coefficient of the dispersion. Nevertheless, our dispersion relatjon provides a means of including the effects due to the electronic density profile in the calculation.
B.B. Dasgupta et al./Simple dispersion relation for surface plasmon
251
5. Conclusions
We have developed a formalism to calculate the dispersion relation for the SPO that allows for the incorporation of the effects of a realistic electron density profile. Previous calculations of the SPO which have included a dependence of the density profile [3-6,9] require extensive calculations to obtain the dispersion. Our dispersion relation is given by a simple integral expression and recovers the expression obtained by RM [8] when the density profile is represented by a step function. The existence of the extensive hydrodynamic [9] and quantum-mechanical RPA [3-61 calculations provide a means of evaluating our dispersion relation, and we find qualitative agreement with the previous calculations for an infinite barrier model and a linear density profile. In order to obtain a simple expression for the coefficient of the linear K term in the dispersion relation, we have made several approximations. Firstly, the inhomogeneous part of the dielectric function has been assumed to be proportional only to the density. The linear density dependence can be derived from the Boltzmann-Vlasov equation only after ignoring the terms that are proportional to the gradient of the unperturbed potential. The unperturbed potential has contributions not only from the dipole field near the surface, but also from exchange.correlation and kinetic energy terms that vary as a function of z. In a local equilibrium approximation these terms give rise to a term in the dielectric response proportional to the density gradient. The density gradient is large near the surface and such a term would have to be included in a more realistic representation of the surface dielectric function. A second approximation is the use of an iteration procedure to solve for the potential in the metal and in the vacuum. The choice of matching point for the potentials changes y> and y< and makes suspect the accuracy of the truncated results for either one of them. It is thus essential that the matching point be properly chosen so that y> and y< are optimum. However with the choice of a proper matching point, we believe that our formalism yields the essential dependence of the dispersion relation on the density profile. Lastly, we perform the calculation using the hydrodynamic form of the bulk dielectric function. This is consistent with the assumption of linear density dependence but it also requires the total momentum to be small. However, the localization of the SPO near the surface requires the use of the full range of k, dependence in the calculation. This approximation is in fact unnecessary and has only been used to simplify the calculations. The choice of a surface dielectric function is a critical step in obtaining the dispersion of the SPO. The main requirements are that it be simple enough to permit easy calculation and faithfully reproduce the response of the real surface. We have chosen an extremely simple density-dependent expression for the surface dielectric function and carried through the calculation to illustrate the application of the formalism. That this approximation is not sufficient is illustrated by the small corrections we obtain to the step-function density profile within the hydrodynamic approximation. To
3.3. Dusguptu et ai./Simple dispersion relation fbr surface plasrn~~l
252
obtain a better estimate of the linear term we will have to go beyond the linear density dependence of 7> and r<.
References [ 1] R.N. Ritchie, Surface Sci. 34 (1973) 1. [2] P.J. Feibelman, Phys. Rev. 176 (1968) 551, [3] D.E. Beck, Phys. Rev. B4 (1971) 1555. 141 D.E. Beck and V. Celli, Phys. Rev. Letters 28 (1972) 1124; Surface Sci. 37 (1973) 48. JSJ P.J. Feibelman, Phys. Rev. Letters 30 (1973) 975; Phys. Rev. B 9 (1974) 50. 161 Ch. Heger and D. Wagner, Phys. Letters 34 A (1971) 448; Z. Physik 244 (1971) 499. 17 J R.H. Ritchie, Progr. Theoret. Phys. 29 (1963) 607. [8 J R.H. Ritchie and A.L. Marusak, Surface Sci. 4 (1966) 234. [9] A.J. Bennett, Phys. Rev. B 1 (1970) 203. [lo] D. Wagner, Z. Naturforsch. 21a (1966) 634. [ 111 P.J. Feibelman, Phys. Rev. B 3 (1971) 220. [ 12 J 0. Pines and P. Nozieres, The Theory of Quantum Liquids (Benjamin, New York, 1966). [ 13 J Retardation effects are neglected in this calculation so the results are only applicable for K > tip/c = 0.005 kF. [ 141 Ritchie and Marusak [8] argue that this procedure corresponds to requiring the specular reflection of the electrons at the surface for a step density profile. IIS] The potential U and the undisturbed distribution of electrons satisfy the zero-order equation. afo(u, 2) i au(r) afbru.z) % -------=---_-~ a,_ m a2 au,
4161 .J. Lindhard,
Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd. 28 (1954) No. 8. (171 Objections have been raised against this value for p [J. Harris, Phys. Rev. B 4 (1971) 10221 since bulk plasmons are not the same type of excitations as SPO. On the other hand the catculation by Kleinman [L. Kleinman, Phys. Rev. B7 (1973) 2288 J shows 13to be w depenp = 0.53 11::; :Lsmdl difference. dent but for w = tip/J2 [18] D.M. Newns, Phys. Rev. B 1 (1970) 3304. [ 191 N.D. Lang and W. Kohn, Phys. Rev. B 1 (1970) 4555.
A SIMPLE DISPERSION RELATION
FOR THE SURFACE PLASMON:
DEPENDENCE ON ELECTRON DENSITY PROFILE
Basab B. DASGUPTA, Pradeep KUMAR and D.E. BECK Department of Physics and Laboratory for Surface Studies, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201,
USA
We know from the quantum-mechanical, random-phase-approximation (RPA) calculations for the surface plasmon that the electron density profile at the surface plays an important role in determining the dispersion relation. The full RPA calculations are very difficult to perform, and we present a simple expression for the dispersion relation which depends on the electron density profile at the surface. Our derivation is essentially semi-classical and parallels that of Ritchie and Marusak. The validity of our expression for the dispersion relation is assessed by computing the dispersion relations for different electron density profiles and comparing them with those obtained in the full RPA calculations.
1. Introduction
The surface plasmon (SPO) is an important feature in the electronic excitation spec. urn of metal and semiconductor surfaces [l] . Hence, one needs to incorporate its effects into surface studies such as low-energy electron diffraction, photoemission, surface energy and response of the surface to external charges. The full quantummechanical random-phase-approximation (RPA) calculations [2-61 of the surface plasmon excitation spectrum are very difficult, and numerous approximate semiclassical treatments [7-lo] have been employed to calculate the dispersion relation for the SPO. An early calculation of the SPO dispersion relation for a metal surface was performed by Ritchie and Marusak [8] (RM). These authors used the bulk RPA dielectric function in the metal and matched the solution of Poisson’s equation at the metal vacuum interface. This semi-classical calculation resulted in a simple integral expression for the SPO dispersion relation which did not depend on the electron density profile at the surface. Subsequently, the hydrodynamic calculation of Bennett [9] and the quantum-mechanical RPA calculations [3-.5,1 l] revealed a strong dependence of the dispersion on the electron density profile. In this paper, we derive a semi-classical, integral expression for the SPO dispersion relation which admits a dependence on the surface electron density profile. Our derivation which parallels that of RM is presented in section 2. In section 3, we de-
242
B.B. Dasgupta et al.lSimple dispersion relation for surface plasmon
velop a simple, density-dependent, surface dielectric function which reduces to the correct limiting forms in the vacuum and deep in the metal. Using this dielectric function and the hydrodynamic approximation for the bulk-dielectric function [I 21, we are able to perform most of the integrations in our calculations analytically. The calculation of the dispersion relation for different electron density profiles is carried out in section 4. These calculations demonstrate the application of our formulation and show that the corrections to the dispersion relation for the step-density profile have the correct magnitude. We test our surface dielectric function by comparing our dispersion relation to those obtained in calculations which treat the surface in a more complete manner; i.e., the hydrodynamic calculation of Bennett [9], and the quantum-mechanical RPA calculations [3,5,6].
2. Derivation of the dispersion relation The general expression relating the displacement
and electric fields in a medium
is D(r,r)=Sdr’P(r,r’,t).E(r’,t),
(1)
where i(r, Y’,t) is the dielectric tensor. In an infinite, homogeneous pression reduces to D(r, t) =Jd
P‘ E~(M’,
t) ECr’, t),
medium this ex-
(2)
where eB(r,t) is the bulk longitudinal dielectric function. For an infinite electron gas the Fourier transform of the bulk dielectric function has within the RPA, the limiting value
where wp is the bulk-plasmon
frequency
mi = 4rrnOe2fm.
(4)
Here e and m are the electron charge and mass, and n0 is the bulk electronic charge density. We consider a metal in the half-space z > 0 which consists of an electron gas and its compensating, homogeneous ionic background. The system is uniform in the x andy directions. For this system the Fourier transforms are taken with respect to the coordinates of the x and y planes, R, and the time, t, to give dt e-i(K-R-wt) ~K,w(z>=JdRJ-
F(R, z, t).
B.B. Dasgupta et nl./Simple dispersion relation for surface plasmon
The additional
transformation
FK,,(kZ)
243
with respect to z is given by
=J’dz e-ikzz&Jz),
and k = (I(, iiz) is the total momentum. For this model of the surface, we employ a dielectric function which is consistent with the requirement of translational invariance parallel to the surface, and eq. (1) reduces to (5)
=f^d z’ f&w(z? z’) &,,(z’).
&,,(z)
In their calculation, RM use the bulk RPA dielectric function for the metal in (5). This gives the correct response in the bulk region of the electron gas, but it is certainly not correct near the surface. We wish to retain the bulk response in our calculation and include an approximate treatment of the surface electron density profile. To facilitate this we write the dielectric function as eQJz*
z’) = &,,(z
-z’)
f Y&JZ,
z’)
(6a)
for z > 0 where r>(z, z’) vanishes far from the surface. In a vacuum the displacement and electric fields are equal, so for z < 0 we write the dielectric function as, elyJz,z‘)
= Nz-z’)
+ ?&Jz,
z’),
(6b)
where y<(z, z’) also vanishes far from the surface. The procedure adopted by RM in their derivation of the SPO dispersion relation was to solve Poisson’s equation for the fields in the metal and vacuum, and to match these fields at the surface. The dispersion relation for the SPO is obtained as the necessary condition for the existence of a solution for the surface. Our derivation parallels theirs’, and we obtain their relation by setting our y’s equal to zero. Consider the response of the system described by the dielectric function in eq. (6a) to an external charge disturbance. In the presence of an external charge, p(r, t), Poisson’s equation [ 131 is i k-I),,,@,)
= 4~ P~Jkz).
Defining the potential ik+~,,(k,)
(7)
by
= -&,Jk,),
(8)
and using (6a) in (5), we obtain &,(k,)
=
-ie3 x,w(k,)#~,,(k,)k-in-li3~yj;,,(k,,kl)~~,~(k:,)k1,(9)
where k’ = (K, kl). Substituting potential,
this into (7), we obtain an integral equation for the
244
B.B. Dasgupta et al./Sitnplt dispersion relo tion for surf&e piastnnn
These fields are to be matched at the surface, z = 0, to those found using the dielectric function given in (6b). Hence, we employ an external charge density which is confined to the surface, PK,&?
(11)
= 0K.w 6(z),
The system is then completed by requiring a symmetric continuation of the metal to the half-space z < 0 [ 141. For the system described by the dielectric function in C6b) the displacement field and potential are given by
and
The external charge here is f&(z) In order to (9), (lo), (12) on the electric the tangential tinuous at the
= CJ;,, 6(z). obtain the dispersion refatiun For the SPO, we seek soIut~ons ofeqs. and (13) dhich are consistent with the boundary conditions imposed and displacement fields by’Maxwell’s equations. Thus we require that component of the electric field, or equivalently the potential be consurface,
#O(z = o- ) = @(z = o+), and that the normal component face
(14) of the displacement
[DO@ = o--)] z = [D(z = O”)],.
field be continuous
at the sur-
(15)
[In these and the following expressions we suppress the subscripts Kand o which are common to all functions.] Using the symmetry of the y>, y’(k,, ki) = -y’(-k,, -ki), and the potential, Cp(k,) = 4(-k,), and evaluating the discontinuity of the displacement field across the surface charge at z = 0, we find
B.5. Dasgupta et aLlSimple dispersion relation for surface plasmon
,ik,O+
~r>(k,&) [k,g -k;] z
z-p i?
GK,,Jk;)
=
245
0.
This argument also holds for Do, and we have
Hence the boundary o. c
condition
eikzo- kz = ~ c k2
kz
on the displacement
field, (15), yields
eik,O+ k
k,
’
k2
or o” =-u and the dispersion relation for the SPO follows from (14). To obtain a simple integral expression for the dispersion relation we keep only the first order term in an iterative solution of the integral equation for tik,), (lo), hence we use the approximation, $(k,)
= = k2eB(k
1 - Q-113 Z
)
k;)k*k’
k12 eB(k’)Z
‘4
An iterative solution of $O(k,) 4n00 @‘(k,) = F
y’(k,,
c
1
leads to a similar approximation
1 _ Q_1,3 CY’(kz> kz
k:) k-k k’2
(16)
’
to (13), namely
1.
(17)
The dispersion relation for the SPO follows from (14) as a-113
eikz O+ ___
c
=
k,
[ k2 eB(k,)
a-213
c
I&;
,ik,O+p
fi k2kr2
k2
1
eikzo+y’(kz,k~)
_
+ elkzo y<(k,, eB(k,)
EB(k;)
ki)
1
(18)
.
Using the symbolic identity,
lim ~
k-OK2
2K -t k;
= alI3 6(k,),
we find that in the limit of K + 0 the dispersion relation becomes E! ,(O) = -1. Inserting the expression for the dielectric function in this limit, (3), we have the usual limiting value for the SPO frequency, wso = wP/d2.
A.B. Dnsgupta et al./Simple dispersion relation for surfhce plasmon
246
To proceed further we must make some assumption concerning the form of E(z,z’). In the next section we develop a form for the dielectric function and, in section 4, present calculations using a simple form for the bulk dielectric function. This allows us to present a concrete illustration of the application of our formalism.
3. Model dielectric function We know from the quantum-mechanical RPA calculations of the SPO dispersion relation that the linear response function which is proportional to the polarizability, 4nx(z,z’) = E(Z, z’) - 6(z -z’), is not a simple function of the electronic density. However, one can show that m
lim
K-t0
s
I@ “2 n(z)
dz’ L(z, z’) = __-.!--
-co
47wblo
’
where n(z) is the static electron density and the density response of the system is given by &z(z) =Jdz’L(z,z’)
I’(.+
Here V(z’) is the total electric potential. We also see from (3) and (4) that ~~,~(k, = 0) is proportional to the electronic density no. Hence, we shall use a dielectric function of the form E(Z,z’) 7 6(z - 2’) + 4n n(z) xB(z ~ z’)/no.
(19)
This expression has the desired features of a dielectric function for a metal surface. It goes to the bulk dielectric function deep in the metal, and in the vacuum, z < 0, where the electron density is zero it gives D(z) = E(Z). To understand the type of system implied by this model of the dielectric function, we consider the linear Boltzmann-Vlasov equation for a system of electrons in the presence of an external electric field, E,
Here the total distribution function for the electrons at a position, I, with a velocity u, isf(r,u, t) tfo(u,z) and the distribution function for the undistributed electrons isf,(o,z). This is the undisturbed distribution function and is related to the electronic charge density by n(z) = ,fduf&z>. The potential
U 115 ] arises from kinetic, exchange and correlation
effects near the
B.B. Dasgupta et al./Simple dispersion relation for surface plasmon
241
surface. If there is a dipole field due to the electronic and ionic charge density this field will also contribute to U. The induced current can be related to the total field by the polarizability. For a system where the undistributed electron distribution is independent of position and given by the usual Fermi distribution,fF(u), the calculation of the dielectric function is straightforward [ 161. If we make the ansatz, (20)
f,(U> z) = n(z)fI:(u)>
and neglect the term V,U* V, f in the Boltzmann-Vlasov equation, we can carry through the calculation of the dielectric function and obtain (19). Since it is the total electric field that appears in the expression for the dielectric function we need a system which has no dipole field at the surface, or the electronic and ionic charge distributions must coincide. In addition, neglecting the extra term requires that the variation of n(z) with z is small which is clearly not the case for a metal surface. Comparing eqs. (6) and (19), we have y;,,(z,
z’) = -An(z)
where An(z) = 1 - n(z)/nu, &(z,
[e;,,(z
-z’)
- F(z -z’)]
,
(214
and
z’) = 12(z) [e&(z
- z’) - 6(z -z’)]
.
(21b)
These expressions for y> and y< are small except near the surface, since An(z) is small in the metal, and n(z) is small outside the metal. Making use of the symmetric continuation of the metal in our derivations, the Fourier transform of the y’s are y>(k,,
k;) = -2 i
dz An(z) cos (kz - k;) z [en@;) - l] ,
(22a)
0
and 7’(K;,k;)=2Pdz,*(z)cos(k,-~~~z _m
[eB(k;)-I].
(22b)
These expressions in combination with (18) give us an integral expression for the dispersion relation of the SF’0 which depends explicitly on the electron density profile at the surface. To assess our expressions for the dispersion relation and surface dielectric function, we compute the dispersion relations for different density profiles and compare them with those obtained in other calculations. In the next section we carry out these calculations using the hydrodynamic expression for the bulk dielectric function. With this dielectric function we can perform most of the integrals for the dispersion relation analytically which greatly simplifies our comparison.
B.3. Dasgupta et al./Simple dispersion relation fbr surjbce plasmon
248
4. Hydrodynamic
calculation
The hydrodynamic +Jk,)
expression for the bulk dielectric function
= 1 -- w;/(wZ
[ 121 is
- fi2k’),
(23)
where [7] p2 = $u$ and uF is the Fermi velocity. This value of fi is chosen so that the bulk plasmon dispersion relation eE w(k,) = 0 agrees with the RPA expression (171 for small k2. Inserting this dielec&c function into (22) we obtain
Y’@,t k;)
cos (kz -k;)
= -2 2
j
eB(k;)
z
(24a)
dz An(z) kt2 + .2 i
0
’
and cos (k, - k:) z
r’(rr.h:)=2~~dz”(‘)--k~~~i.
(24b) z
0
where a2 = K 2 + (ws - 02)/p2 and r2 = m2/f12 - K2. For a uniform background of positive charge the sums over k, in the dispersion relation (18) go over to integrals in the usual way R-‘j3
t) + (2n)-1 kZ
m dkZ
s -Cc
and the integrals are easily performed by contour integration
(1-$)=-!$(w/ dzAn(z)[?(l
1 -y
,2 +--II.
to give
-w)e-z~z
O
s wz -_m
,
dzn(z) )li
where w = (1 - u2/w$l. For our comparisons we will only compute the term in the dispersion relation which is hnear in K, w2 = w; (1 +AK)/2. Substituting have
(26)
this expression into (25) and retaining only the terms linear in K, we
B.B. Dasgupta
A
+
et al/Simple
.I-dz An(z)
_
[2 -
dispersion
249
(27)
0
s _m
for surface plasmon
3 eeacz + 2 e-2aez]
0
t
relation
dzn(z)
[2 - coscuoz]
where o. = w,/flfi. In this relation An(z) and n(z) are positive quantities, hence we find that the linear coefficient, A, is a sensitive function of the location of the matching point, z = 0. This feature of the model is demonstrated by computing A for three density profiles at two matching points. One matching point, case (i), is taken at the position where a step positive background would have to be placed to insure the charge neutrality of the system; i.e., 0 [
n(z) dz = s
An(z) dz.
0
-ca
The other matching point, case (ii), is taken at the position where the electron charge density goes to zero, hence the second integral on the right-hand side of (27) vanishes. Linear density profile: Bennett in his hydrodynamic calculations [9] used an electron density profile, n(z) =
no
forz>b,
nOi%
fora
For this model of the metal surface the integrals in (27) can be performed analytically and we present a comparison with Bennett’s results in table 1. Infinite barrier model: A model of the metal surface which has been employed in many theoretical studies [3] is that of a semi-infinite electron gas confined by an infinite potential barrier. For this model the density is given by [ 181 n(z)=no
[
x2 x31,
3 cosx 1 t----
3 sinx __
where x = 2k,(z -a). Here k, is the Fermi momentum, and a is the location of the infinite barrier. The edge of the step positive background for this model is at x = 3n/4. Our results for this density profile are also presented in table 1 and compared with other calculations. Self-consistent density profile: The best theoretical density profile for a metal is that computed by Lang and Kohn [19] using the inhomogeneous electron gas theory. Their paper contains tables and graphs of the electron density profile. To perform
250
B.B. Dasgupta et ai./Sinrple
dispersim
relatiorl for surface plnsr~otr
Table 1 Values of the linear coefficient, A, in the SI’O dispersion relation, ccl. (26). (The density profiles are described in the text: the matching points are chosen at the edge of the step-ion backpound, case (i), and at the edge of the electron density profile, cast (ii): the step density result for ,.1 is %-I = I. 10 Ul;/W* Metal
Density profile
Other calculations ----__--_ Source -
MK (k~=1.3711-‘)
Linear b =211. b=3# b=4tY
Bennett
[9
This calculation __~ j1 fq/wp) fi)Af(uF/w$ -___~___I^
(ii)~(u~/~~~
0.01 -1.15 -1.72
-0.01 -0.75 ml.63
] 1.13 1.19 1.30
__I__. Infinite barrier
Al (X-I:=
Infinite
0.68
1191
Eeibelman
1.11
0.12
I .08
0.38
0.10
Beck [3) Heger and Wagner 161
1.75A-’ ) barrier
Self-consistent
~ Beck (31 lleger and Wagner [6J
0.73 0.30 [S 1
-0.55
l.I8
-~ 1.49
the calculation for the matching point at the edge of the electron density profile, case (ii), we choose a point a distance z = ?r/lcl: from the edge of the positive background where the density dropped to 0.01 nO. Since the density falls exponentially to zero the choice of the cutoff point is arbitrary and affects the value of A. Our results for this density profile are compared with Feibelman’s quantum-mechanical RPA calculation [S] in table 1. It is clear that our iteration procedure should give the best answer for a matching point chosen at the edge of the step-ion background, since the corrections. y> and -Y<. are smallest for this case. Hence, for our dielectric function, (19), we find that the linear coefficient,n, in the SPO dispersion relation is not very different from that found for a step-density profile using the bum-hydrodynamic dielectric function; i.e., A = ~20~. Examining table 1, we see that the best agreement with other calculations is obtained with the matching point chosen to be the edge of the electron density, case (ii). This seems to be a coincidence resulting from the strong dependence of the linear coefficient, A, on the matching point. This strong dependence is a feature which is introduced into the calculation by the iteration scheme. It means that the neglected terms in the iterative solutions may be inlportant in calculating the exact value of the linear coefficient of the dispersion. Nevertheless, our dispersion relatjon provides a means of including the effects due to the electronic density profile in the calculation.
B.B. Dasgupta et al./Simple dispersion relation for surface plasmon
251
5. Conclusions
We have developed a formalism to calculate the dispersion relation for the SPO that allows for the incorporation of the effects of a realistic electron density profile. Previous calculations of the SPO which have included a dependence of the density profile [3-6,9] require extensive calculations to obtain the dispersion. Our dispersion relation is given by a simple integral expression and recovers the expression obtained by RM [8] when the density profile is represented by a step function. The existence of the extensive hydrodynamic [9] and quantum-mechanical RPA [3-61 calculations provide a means of evaluating our dispersion relation, and we find qualitative agreement with the previous calculations for an infinite barrier model and a linear density profile. In order to obtain a simple expression for the coefficient of the linear K term in the dispersion relation, we have made several approximations. Firstly, the inhomogeneous part of the dielectric function has been assumed to be proportional only to the density. The linear density dependence can be derived from the Boltzmann-Vlasov equation only after ignoring the terms that are proportional to the gradient of the unperturbed potential. The unperturbed potential has contributions not only from the dipole field near the surface, but also from exchange.correlation and kinetic energy terms that vary as a function of z. In a local equilibrium approximation these terms give rise to a term in the dielectric response proportional to the density gradient. The density gradient is large near the surface and such a term would have to be included in a more realistic representation of the surface dielectric function. A second approximation is the use of an iteration procedure to solve for the potential in the metal and in the vacuum. The choice of matching point for the potentials changes y> and y< and makes suspect the accuracy of the truncated results for either one of them. It is thus essential that the matching point be properly chosen so that y> and y< are optimum. However with the choice of a proper matching point, we believe that our formalism yields the essential dependence of the dispersion relation on the density profile. Lastly, we perform the calculation using the hydrodynamic form of the bulk dielectric function. This is consistent with the assumption of linear density dependence but it also requires the total momentum to be small. However, the localization of the SPO near the surface requires the use of the full range of k, dependence in the calculation. This approximation is in fact unnecessary and has only been used to simplify the calculations. The choice of a surface dielectric function is a critical step in obtaining the dispersion of the SPO. The main requirements are that it be simple enough to permit easy calculation and faithfully reproduce the response of the real surface. We have chosen an extremely simple density-dependent expression for the surface dielectric function and carried through the calculation to illustrate the application of the formalism. That this approximation is not sufficient is illustrated by the small corrections we obtain to the step-function density profile within the hydrodynamic approximation. To
3.3. Dusguptu et ai./Simple dispersion relation fbr surface plasrn~~l
252
obtain a better estimate of the linear term we will have to go beyond the linear density dependence of 7> and r<.
References [ 1] R.N. Ritchie, Surface Sci. 34 (1973) 1. [2] P.J. Feibelman, Phys. Rev. 176 (1968) 551, [3] D.E. Beck, Phys. Rev. B4 (1971) 1555. 141 D.E. Beck and V. Celli, Phys. Rev. Letters 28 (1972) 1124; Surface Sci. 37 (1973) 48. JSJ P.J. Feibelman, Phys. Rev. Letters 30 (1973) 975; Phys. Rev. B 9 (1974) 50. 161 Ch. Heger and D. Wagner, Phys. Letters 34 A (1971) 448; Z. Physik 244 (1971) 499. 17 J R.H. Ritchie, Progr. Theoret. Phys. 29 (1963) 607. [8 J R.H. Ritchie and A.L. Marusak, Surface Sci. 4 (1966) 234. [9] A.J. Bennett, Phys. Rev. B 1 (1970) 203. [lo] D. Wagner, Z. Naturforsch. 21a (1966) 634. [ 111 P.J. Feibelman, Phys. Rev. B 3 (1971) 220. [ 12 J 0. Pines and P. Nozieres, The Theory of Quantum Liquids (Benjamin, New York, 1966). [ 13 J Retardation effects are neglected in this calculation so the results are only applicable for K > tip/c = 0.005 kF. [ 141 Ritchie and Marusak [8] argue that this procedure corresponds to requiring the specular reflection of the electrons at the surface for a step density profile. IIS] The potential U and the undisturbed distribution of electrons satisfy the zero-order equation. afo(u, 2) i au(r) afbru.z) % -------=---_-~ a,_ m a2 au,
4161 .J. Lindhard,
Kgl. Danske Videnskab. Selskab, Mat.-Fys. Medd. 28 (1954) No. 8. (171 Objections have been raised against this value for p [J. Harris, Phys. Rev. B 4 (1971) 10221 since bulk plasmons are not the same type of excitations as SPO. On the other hand the catculation by Kleinman [L. Kleinman, Phys. Rev. B7 (1973) 2288 J shows 13to be w depenp = 0.53 11::; :Lsmdl difference. dent but for w = tip/J2 [18] D.M. Newns, Phys. Rev. B 1 (1970) 3304. [ 191 N.D. Lang and W. Kohn, Phys. Rev. B 1 (1970) 4555.