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The effect of structure on surface plasmons
Surface Science 49 (1975) 275-292 0 North-Poland Publishing Company
THE EFFECT OF STRUCTURE ON SURFACE PLASMONS f A.D. BOARDMAN*,
B.V. PARANJAPE and R. TESHIMA ofAlberta, Edmonton, Alberta, Canada
Theoretical Physics Institute, University
Received 6 May 1974; revised manuscript received 26 November 1974
It is shown, for small wavenumbers, that the principal surface plasmon mode, on the surface of a semi-infinite metal has a frequency w = wp/~ which is independent of the nature of the surface density profile. A double step-functron model of a surface is proposed which leads to an analytical form of dispersion equation. This dispersion equation contains two parameters which are adjusted to obtain good agreement between theory and experiment. These parameters can be used to determine nature of the electron density surface profile. The model gives a simple analytic criterion for the existence of other frequency modes.
1. Introduction The precise form of the dispersion of surface plasmons which can be excited in plane films of metal is still a topic of debate. Specifically the question which has arisen for Mg and Al, the principal systems under observation [l-4] , is how the surface plasmon frequency, w, depends upon its wave vector R in the region where retardation is negligible. In practice, if the surface plasma frequency assumes the collisionless form _ _ ~2 kL pFk +@I +ia2)-t(fr3+iQ4)~+... I
wP oz . fork>kp = w,/c, where wp is the bulk plasma frequency, k is the wave number, c is the velocity of light in vacua and a 1, a2, a3 and a4 are real, then the principal interest centres upon the sign and magnitude of al and a3. Some early theoretical work by Kanazawa [S] using an approximate quantum mechanical random phase approach, for a system in which the surface is an infinite barrier, predicts that al = 0, a2 = 0 thus giving w a k2 for small k. This result contrasts sharply with the classical, hydrodynamic, calculation of Ritchie [6] which gives, for a jellium model in which a homogeneous semi-infinite electron gas moves against a uniform distribution of positive charge, aI > 0; thus making w, essentially, linear in upport from the N.R.C. is gratefully acknowledged. PPermanent address: Physics Department, University of Saiford, England.
Salford, Salford MS 4WT,
216 k.
A.D. Boardmarr et al/Effect
of structure on surface piasmorls
This, theoretical, situation is in conflict with the experiments of Kunz [l] who, from energy loss measurements using keV electrons passing through Mg films has shown that aI < 0, a3 > 0. Indeed the negative characteristic of al is quite pronounced and the q(w) diagram exhibits a clearly defined minimum. The same features have been observed in Al by Bagchi et al. [2] after a somewhat complicated analysis of ILEED observations. In spite of this the current experimental situation is still not settled quantitatively. Firstly, Kloos and Raether [3], from energy loss experiments on Mg and Al films, have concluded that there is, for k > k , at best, only a weak dependence of 32(w) upon k, in either material, and that CISf w) + w IGas k -+ k _Furthermore, although their results for Mg are similar to those obtaines by Kunz [ 1Jn, their results for Al are completely different from those of Bagchi et al. [2] ; thus implying that the latter are in error. Recent ILEED measurements by Porteus and Faith [4} and their subsequent analysis by Duke and Landman [7] corroborate this conclusion by also claiming that the work of Bagchi et al. [2] is in error. However, although the magnitude of the correction is similar, Porteus et al. [4,7] find that aI > 0 and a3 c 0 which, once again, conflicts with the results of Kloos and Raether [3] . The fact that the Porteus and Faith [4] experiments require an extremely complex analysis in order to produce the dispersion curve may be partly responsible for the discrepancy. Since the work of Kloos and Raether [3) embraces the work of Kunz (I], without serious conflict, we will assume, for the purposes of this paper, that their data together with those of Kunz [l] is correct. Several theoretical models of surface plasmons have been constructed both from quantum and classical microscopic random phase approximations and classical hydrodynamics. A variety of approximations, of necessity, have been employed, or are implicit, in the fabric of these models, thus rendering them, in a strict sense, inapplicable to real metals. Nevertheless, since a measure of success can be claimed for all of them it seems to be worth presenting a brief guide to the current theoretical position before introducing the details of the new model analysed in this paper. A lot of attention has been focussed recently on a microscopic approach which employs a random phase approximation (RPA). A useful model of this kind has been developed by Feibelman [8] which avoids the approximation, made by Fedders [9] , that the electrons are trapped in a well possessing infinitely high walls. Feibelman [8] has produced what amounts to a time Fourier transform of Laplace’s equation; a procedure which has been criticised by Newns [lo] who pointed out that its validity is strictly confined to density variations of wavelength greater than that of the plasmon excitation. In the Feibetman form~ation [S] the surface barrier is expressed in terms of the static density profile which, in turn, is assumed to be a step-function. The real coefficient of the linear term, i.e. a is found to vanish and it is proved that the result !’ of the structure of the surface. Newns 313(w) + w,/&as k -+ k, is quite independent [lo], however, has concluded using a time-dependent Hartree approximation applied to an infinite square barrier model that al f 0. This conclusion is also supported by Beck [ 1I] using a full quantum mechanical RPA.
AD. Bourdman et al.JEffect of structure on surface plasmons
277
Harris and Griffin [12] have developed an RPA model, based upon a Wigner distribution function, and conclude that it is important to use a realistic model of the surface structure. Indeed they show that, within the limits of the RPA, the linear term of the dispersion equation can only arise as a consequence of the surface structure. They also show that, in the high density RPA iimit, the use of a sharp density profile or an infinite barrier is not justifiable. Nevertheless, Beck and Celli [ 131 have attempted a more realistic RPA calculation by introducing the surface as a step-function potential barrier. Their calculations are performed, with the aid of a new variational principle [ 13, 141. However, although it is claimed that no adjustable parameters are involved they do introduce variational parameters and a good choice of trial function seems to be quite ~port~t. Their results show that aI # 0 and are in modest agreement with the Kunz Mg data. Unfortunately their work is also in good agreement with the data of Bagchi et al. [2] which, as we have already discussed, now seems to be in error. Feibelman [ 151 developing his earlier RPA theory, which gave al = 0 for an infinite barrier model, to include a smooth, finite, potential barrier has now found that the linear term of the dispersion equation is finite and that not only does it depend strongly upon the diffuseness (i.e. the distance over which the height of the barrier changes), but it is extremely sensitive to the barrier shape. However, no real attempt is made to contact experiment apart from the observation that al, being some three times larger than the value obtained by Beck and Celli [ 131 is commensurate with the recent data of Porteus and Faith [4]. This sensitivity of the linear term of the dispersion equation to the diffuseness agrees with the numerical conclusions of Bennett [ 161 which are obtained from a hydrodynamic model incorporating a linear surface density profile. Bennett on the other hand has not determined whether the dispersion equation is sensitive to the shape of surface density profile; but his results are in good agreement with the Mg data of Kunz [l] . The sensitivity calculated by Feibelman [ 151 may be a real feature or perhaps a function of the model or the degree of appro~mation. Therefore, since the features of the dispersion curve computed by Bennett agree so well with experimeni and are qualitatively supported by RPA approaches [ 171 it is worthwhile making a further examination of the hydrodynamic model to determine any such degree of sensitivity to shape. Before leaving the question of microscopic models, however, and passing on to a consideration of the hydrodynamic model it should be mentioned that Inglesfield and Wikborg [IS] using a step-function potential, coupled to an RPA Green function matching method, have calculated a value of al which is in disagreement with Beck and Celli [ 131 and closer to the experimental value of Porteus and Faith 141. They do not include diffuseness or damping and make no comments upon the influence of the barrier shape. There have now been several hydrodyn~ic fo~ulations of the surface plasmon problem [6,16,20,25]. The earliest of these is due to Ritchie [6] who has shown that aI > 0 for a step-function of electron density profile. Subsequently Bennett [ 161, including the influence of surface structure by allowing the surface electron density to
278
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of structure on surface plasmons
fall to zero over a distance of a few angstrom, has shown that it is possible to have aI < 0. In both of these hydrodynamic models collisional and collisionless (Landau) damping has been omitted even though a surface plasmon, unlike a bulk plasmon, can suffer Landau damping at all wave numbers. If a hydrodynamic model which omits any form of damping is to produce any meaningful results then surface plasmon ought, in practice, to possess a long enough lifetime for it to maintain its identity as a quasiparticle in the regime of experimental interest. Experiments involving surface plasmon excitation by passing electrons through thin films seem to produce results which fall into this category; since they are explained extremely well by the hydrodynamic model [ 161. Direct experimental evidence of surface plasmon lifetime is limited [2,4] but it does show that ILEED measurements in Al films reveal both collisional and Landau type damping. The former type is k independent and inversely proportional to the relaxation time r [20] ; this type of damping is obviously a function of the electronic state of the metal and can be minimised experimentally by making r + 00. The collisionless damping is proportional to k [2,4] . In order to be more precise let us examine the ratio r = 9(0)/32(w) i.e. the ratio of the imaginary part to the real part of the frequency. If this is sufficiently small then it can be assumed that the concept of quasi-particle or eigenmode is a good approximation. Since the collisional damping can be eliminated, as we discussed above, we will neglect it and discuss the principal Landau damping in terms of r = a.&(1
+ a14),
q =
kvF/wp.
Hence, knowing from experiment [ 1,3] that q lies in the range 0 d q < 0.35, we find that the recent theoretical microscopic RPA results of Feibelman [ 151 give r % 20% whilst those of Beck and Celli [ 131 give r = 7%. These results are an order of magnitude higher than those found by Ritchie and Marusak [ 191 using a classical BoltzmannVlazov equation. This theoretical discrepancy could be attributed to the different theoretical approaches but Heger and Wagner [21] have pointed out, quite strongly, that the microscopic theories discussed above depend upon a high frequency approximation of very doubtful validity. Furthermore they prove that if the RPA high density approximation is used consistently then the classical result is corroborated. The experimental results [2,4] (which it should be remembered, as far as the m(w) is concerned, are in conflict with those of Kloos and Raether [3]) give a Landau damping where r could be as high as 29% or as low as 10%. Thus the situation, with respect to collisionless damping, is not clear experimentally or theoretically. It would appear, however, that the concept of quasi particle is good enough to expect that any formalism which, in the first instance, neglects damping altogether is going to give meaning& ful description of the behaviour of 32 (w) with k. The numerical results of Bennett [16] showed that there is a principal mode, which is in excellent agreement with experiment, and a number of higher modes for which, ask-+kp,wp/fi
A.D. Boardman et aLlEffect
of structure on surface phmons
219
damping. This may, indeed, be the case; but Heinrich found a family of lower modes in the range 0 < w < wP/fi when collisional damping was omitted from his hydrodynamic scheme, whereas there is some possibility that higher modes really do exist on the grounds that they have been observed, by Dattner [22], in gaseous plasma columns. Apart from the omission of damping the application of hydrodynamics to metals can be criticised on the grounds that the true hydrodynamic condition, i.e. that the frequency with which the plasma changes is much less than the collision frequency, is not appropriate to the metallic state. A hydrodynamic condition in other words requires the carrier distribution to relax extremely rapidly, via collisions, to local equilibrium. It is possible to avoid these approximations by using the less restrictive quasi hydrodynamic approach [23] which involves the use of moments of the Boltzmann equation in a form which does not incorporate the condition w < V, where w is the frequency of the plasma excitation and u is the collision frequency. The resulting equations are very similar to the pure hydrodynamic equations. Harris [24] recently cast a strong shadow upon the hydrodynamic approach by emphasising that the pressure tensor P(r, t) is normally approximated to V. P(r, t) = ml3 VGp(r, t), where m is the electron mass, 6p(r, t) is the mass density fluctuation; and fl is taken to be (3/5)uc in a forcing move to make the dispersion e uation agree with the result 9 derived from micorscopic models. Actually, 0 would be UP/~ if the bulk plasmon was a true hydrodynamic mode. Now the numerical difference between (3/S@ and $3 is not such a serious objection but there is no particular reason for supposing that the bulk value of /3 is going to be valid for surface plasmons. Harris [24] maintains that the, high density, RF’A formalism can be reconciled with a hydrodynamic approach provided /3 is taken to be (3/5)ugwi/w2. For bulk plasmons, neglecting dispersion, this is (3/5)vi while for surface plasmons it is 42 larger. If this was the sole content of the criticism then it is, once again, not qualitatively important. However, Harris [24] also maintains that for surface plasmons a hydrodynamic solution does not exist; a viewpoint which has been shown recently, by Kleinman [25], to be in error. In fact Kleinman [25] using the form of 0 proposed by Harris [24] has shown that, for a uniform step-function density model, the results obtained by using fl= (3/5)$ and new form suggested by Harris [24] are both very similar to the original result of Ritchie [6] . In the above brief survey it is shown that surface plasmon phenomena, in the wave number range k > kp = wP/c, can be studied with the aid of RPA, microscopic, models or by hydrodynamics. It is important to emphasise that the RF’A model is, strictly, a high density infinite electron gas model and that there are valid criticisms of the hydrodynamic method. It is clear that no method is completely justified so the merits of any particular model must not only include its ability to account for the current experimental results but its capacity for extension to more complex situations such as those involving surface electric fields or external magnetic fields.
A.D. Boardman et al/Effect
280
of structure on surface plasmons
It is with these points in mind that this paper examines the collisionless, hydrodynamic model of Bennett whose results were obtained entirely numerically, for a piece-wise linear surface, density profile, using the simple form of 0, namely (3,‘S)vE. We present details of an analytical proof; derived from a hydrodynamic model, of the insensitivity of the k + k,, w -+ up/& result to surface structure. It is shown that the dispersion is insensitive to the shape of the surface density profile and we introduce a novel double-step-function surface structure which possesses the advantage of leading to an analytical form for the dispersion equation, The results obtained are in excellent agreement with the data of Kloos and Raether [3] . It is clear that this type of model will be extremely useful in assessing the dispersion characteristics of magnetoplasmons and the effect of sample geometry [26].
2. The basic equations The metal is assumed to consist of a collisionless electron gas, moving against a smoothed out positive background charge (jellium model), for which the basic Euler equation [27] is m@(z) h/at = e p(z) grad (9) t e&p grad (@) - grad (P),
(1)
where m, -e and v are, respectively, the mass, charge and average velocity of the electron. p(z) is the unperturbed electron density, G(z) is the electrostatic potential which exists inside the unperturbed solid, J/(r, t) is the perturbed part of the electrostatic potential, 6p(r, t> is the electron density perturbation and p(r, t) is the perturbation hydrostatic pressure which is a measure of the thermal velocity spread of the electrons [28] . The existence of $(z) is a consequence of the surface structure. The perturbation pressure is assumed to be [I 6,29,30] P(C t) = P MC 0,
(2)
Here, uF is the usual Fermi velocity. where /3 is a constant equal to (3/5)m$. The perturbation electric field is assumed to be E= -Vii/. This is the electrostatic approximation which is only valid for wave number greater than the critical wave numbe k , defined in the introduction. The linearised basic equations of the problem are [16] ) a$,uming a space-time dependence of the form eickxewt) h(z), --PYKJ~6p = div [grad @Sp)] - e div [p grad (J/) + 6~ grad (@)I,
(3)
V2J/=
(4)
and (s-k')
where eO is the permittivity -nrti2
&p =Q, -Q,,
$=$!,
of free space. Now eq. (3) can be written as (5)
A.D. Boardman et aLlEffect
281
of structure on surface plasmons
where
and
Q2=e
[“” ~Z+Pd
($4
+!cL(.$
42)
(7) If the equations are now made dimensionless by writing K = kR, Z = z/R, ho/(epoR2), \II= @O/(epOR2), s2 = w/tiP2afdl(Z) = pd_(Z), where R2 = $/(3w;5) is the screening radius of the electrons, w - e ~~/(~~~) and pu is the electron density in the homogeneous part of the me&l, then the equation for * becomes
where A, =9/S,
(94
A, = -d@/dZ,
WI
A, = -d2@/dZ2
- I8 K2/5 f R2 -f_(Z),
(9c)
A, = - {df_/dZ - K2 d@/dZ},
(9d)
A, = 9K4/5 + K2 d2@/dZ2 - f12K2 + f_(Z) K2,
(9e)
and, defining the positive charge density as p+(Z) d2WdZ2
pd;(Z),
=f_(Z) -f+(Z).
Eqs. (9) are the dimensionless
3. The boundary
=
(10) equivalent
of those derived by Bennett
[14]
conditions
Since the time-dependence of the perturbations in the system is emiwt the effective, perturbation, displacement vector in the metal plasma can be written as Dp = - j/iw f eOEP, where j is the conduction
current density and EP is the macroscopic
(11) electric field. This
282
A.D. Boardman
displacement tively , div@
vector, and D’,
et al./Effect
of strucfwe on surface plasmom
the displacement
vector in the vacuum, satisfy, respec-
div Dv = 0.
= 0,
(12)
Hence, if the boundary between a semi-infinite metal and a vacuum is at z = 0, the continuity of the normal component of the displacement vector requires that en [E:(O) - E:(O)]
= [p(O) eiiw] u,(O) = o*,
(13)
where e* is a possible surface charge density. We shall now determine the conditions under which such a surface charge density will vanish, and thereby determine a set of boundary conditions which are valid at a plane metal-vacuum interface. The z-component of eq. (1) is (14) Thus eq. (13), in terms of potentials
wP
L
az
3$/v _-
wi(o)aGP
Gp and $I’, can be written as
e2
-z- +mw2e0
az = ,2
where 02(z) = e2p(z)/(eonz). Now, because the metal is electrically 0 at z = B so that the expression for o* reduces to
0% neutral, d@/dz =
(16) Several important cases can now be delineated. Firstly, for a cold homogeneous plasma the second term in eq. (16) vanishes and U;(O) f: 0. Such a plasma will, therefore, support a surface charge density
and 6p will vanish everywhere except at z = 0. Secondly, if the plasma is homogeneous, and warm, 6p will not vanish inside the plasma, but o* will be zero implying that in this case the surface charge density diffuses into the body of the plasma. This vanishing of o* leads to two conditions, at z = 0, namely
alliV_ wP az
(17)
a2 y
(18) Thirdly, if in addition to being warm, the plasma is i~omogeneous, 0, then eq. (18) in combination with eq. (4) is simply the boundary d3$’
k2dGP
d2
dz
--P=
o
I
such that e+,(O) = condition (19)
A.D.
Boardmanet al./Effect
of structure on
surface plasmons
283
The complete set of boundary conditions at a metal-vacuum interface, including the conditions that (i) the tangential electric fieId is continuous at z = 0, (ii) the plasma is warm, (iii) the electron density vanishes at z = 0 is, in dimensionless symbols (using a dash to denote differentiation with respect to Z), ,kP(0)
= *v(o),
(20)
*‘p(o)
= #V(O),
(20
\k”‘p(o)= 9”‘v(o),
(22)
where eq. (20) has been used to modify eq. (19) to the form given in eq. (22). In a model we use later, the surface is represented as follows. Suppose that the electron density is po over the region - m
(23)
‘k; (-A) = \Ir&h),
(24)
$(-A)
(25)
= !I$(-A),
9[\ki(-X)
- $&A)]
/5 + *;(-A)(1
-b) = 0,
(26)
*z(O) = *j (O),
(27)
$(O) = 9;(o),
(28)
9 [\kT (0) - \k; (O)] /5 f &Pi(O) = 0,
(29)
where the two density regions and the vacuum have been labelled 1,2 and 3 respectively.
4. Persistence of the Q2 = l/Z mode The electrostatic potential which exists in the unperturbed solid has the following properties (using a dash to denote a single differentiation) @(O)= 0,
@‘(O)= 0,
CI+=) = 0,
(30)
ca
.f -00
W’(Z)dz = 0.
(31)
284
A.D. Boardman et aLlEffect of structure on surface plasmons
Let us define the non-divergent solutions of eq. (8) for the metal, as 9 (Z) and \ky(Z), which for distances far removed from the surface have the form e& and eYZ , respectively, where ?2 = K2 + (1 - a21
(32)
.
915
The potential in the vacuum is of the form emKz. The application conditions (20), (21) and (22) yields the determinantal equation -1
*K
\I’7
%
*:,
*II; - K%;C
9;-K2’Py
where all the functions
Kl=O,
(33)
0
are evaluated at Z = 0. The determinant,
\Ir;qk;ll- K+P; + K
of the boundary
{UqPl;;
-
Qq;}
+
on expansion,
K3 {qK\k:, - 9$,}
= 0.
gives (34)
The limit of eq. (34) as K + 0 is
Since the electrostatic theory is valid only for k > k (i.e. K > K - 10m3) it is of interest to determine the limiting behaviour of R as K 5 K [3 l] . however, since K is so small we can, for all practical purposes, consider the likting process K + Kp top be the same as the process K -+ 0. For small K the function q(Z) can be written as ‘J’(Z) = A 1 [‘@o(Z) + Kx(Z)l + A,‘49>
(36)
where A 1 and A 2 are arbitrary constants and *u(Z) is lim,+, [*K(Z)] . This function satisfies eq. (8) so that in the limit K + 0 [and denoting q,,(Z), x(Z) or ‘I’JZ) as g(Z)] we have 9g”“(Z)/S - Q’(Z) g”‘(Z) + [CL2-f(Z)
- Q”(Z)] g”(Z) -f(Z)
g'(Z) =
O,i(37)
which can be re-written as $
(9 g”‘(Z)/5 - a’(z) g”(Z) + [!L? -f(Z)]
g’(Z)} = 0.
(38)
At large distances from the surface i.e. as Z -+ -m, 9 (Z) -+ eKz, x(Z) = Z, \E (Z) + eTz, $(Z) + K3 eKz, G;(Z) + K eKz, \ky -+y3 e~~,$r-+~e7Z.Thusas~+-m we can see that q”‘(Z) + 0, \Ir’ (Z) + 0, lim,+u [‘PK(Z)] = 1, lim,+,, [x(Z)] = Z and the limit as KY+ 0 of $$?), *i(Z) is zero. Now eq. (38) has a solution g(Z) = constant and it is obvious from the foregoing arguments that this solution is *o(Z) = 1. Thus eq. (35) reduces to Q”
- VJl
+ x’) = 0.
(39)
A.D. Boardman et aLlEffect
ofstructure on surface
285
plasmons
Eq. (38) also gives 9 x”‘(z)/s
- (a’(z) x”(z) t [a2 -f_(Z)
where B is a constant
(40)
which, since for Z + -00,
a’(Z)
G(Z) -+ 0,
x’(Z) = B,
-+ 0,
x”‘(Z)
+ 0,
x”(Z)
+ 0,
x’(Z)
f_(z) + 13
-+ I,
and (41)
is given by -B=CG
- 1.
(42)
Similarly, 9 V(Z)/5
- @‘(Z) *G(z) f [s22 -f_(Z)]
q;(z)
= 0.
(43)
Hence, at Z = 0 9 x”‘/S t R2$ = Q2 - 1,
(44)
9 *;I5
(45)
+ “““i,
The substitution *;(a2
= 0.
of eqs. (44) and (45) into eq. (39) gives
- l/2) = 0.
(46)
Therefore, if \k’ # 0, there always exists a mode with w = #~/~which is independent of the sur irace structure. This point has not previously been demonstrated for a hydrodynamic model. The other result of eq. (46) is that q’ = 0, a2 f l/2. This is the condition for the possible existence of other modes. Fe1‘6elman [8], using a manybody RPA approximation, with an unspecified f_(Z), also showed that o = tip/< but did not prove the possible existence of the other modes.
5. A model of the surface The experimental investigation of surface plasmons invariably employs thin films of thickness -100 A. For K > K,, the interaction between the surfaces of such films is negligible. It is therefore legitimate to base the theoretical interpretation on single surface models. At the surface of a metal the electron density decreases from its bulk value towards zero within the space of a few angstroms. This feature can be built into a hydrodynamic electrostatic surface plasmon theory by means of a model which employs step-functions The advantage of such a model is the convenient analytic form of the dispersion equation in which the basic parameters can be readily varied until a best fit to the experimental results is obtained. The solid is assumed to be semi-infinite with a double step-function electron number
286
AD. Boardman et aLleffect of structure on surface plasmons
density profile which, in equilibrium, is exactly equal to the positive charge density profile. The step-functions are defined as P@)=Pu
-m
Pm = bP()’
-h
P(Z)= 0,
O
(47)
If the three regions aiong the Z-axis are labelied 1,2 and 3, respectively, turbation electrostatic potentials are q,
then the per-
= P eKz t Q eyiz,
(48)
~~=ReKztSe-Kz+Te~2Z+Ue-~~Z, 9,
(49)
= VemKz,
GO)
where P, Q, R, S, T, U and V are arbitrary constants
and
y; = K2 t S(1 - L?2)/9,
(51)
y; = K2 t S(b - a2)/9.
(52)
The application of boundary conditions (23) to (29) leads to the condition (see p. 287). This determinant reduces to (T; - K2)(r;
+ Y$-
(53)
c) cash (y2h)
+ (yl +A”) {y2(yg - K2 + c) sinh (“i2h) + CK [eTKh - cash (y2h)] } (71 + WY; = &b i c
- r$3 ~
(y; - K2) cash (yZh) +
sinh (T2h)
72
t -[4K(yI + K) ema(^I; - K2)
(1 t e-2N)(ri
+2y,K
tK2)cosh(T2X)
\ (54)
where C= S(1 - b)/18.
(5%
If b -% I, then eq. (54) reduces to the particularly (?I + K) {(l/2 - f12h2
simple form
sinh (y2X) + (K/2) [eeKh - cash (rzA)] }
= !F12[5(1/2 - st2)/9 + ylK + K2] cash (y$).
(56)
As K -+ Kp the terms of O(K), and above, become negligible. Hence, in this limit eqs.
A.D. &ar&nan et al. fEffect of structure on surface plasmons
d II
0
x
F
“i
e
P
i0
0
0
0
7
287
A.D. Boardman et al/Effect
288
of structure on surface plasmons
(51) to (52) become y; = 172= 5(1 - s22)/9,
(57)
y; = -g2 = 5(b - n2>/9,
(58)
(a2 - l/2) [i cos (@) + 1) sin (@)I = 0,
(59)
where it is assumed that, for practical applications, b G R2. Eq. (59) gives the principal mode as L? = l/fland the condition for existence of higher modes as (0)
cot (@) = --77&
(60)
which can be re-written as x cot (x) = -@
- x2),
(61)
where D* = 5(1 -b)h2/9. The condition for the existence of higher modes requires the circles of radius D to intersect the family of curves x cot (x) and is in fact a resonance condition. Now x cot (x) vanishes at x = mn/2, m being an integer. Thus higher modes cannot exist unless
‘>:
112 9 5(1 -b) ’ ( 1
and the total number of modes of the system, including the principal mode, is tz= 1[3[;(l-b)]1’2++j,
(62)
where Ka denotes the integer part of the quantity inside the brackets. Fig. 1 shows, for a selection of (h, b) values, the principal mode solutions of eq. (54). These can be compared to the results obtained from the Ritchie mode [4], which can be obtained from eq. (56) by setting h = 0, b = 0. The results obtained from the double step-function model show a weak, initial, dependence upon k; the flattest curve being for a small value of b with X in the region of 2.5. Physically, the model developed here simulates the fall in the electron density at a real surface by a step-function profile. The distance over which the electron density in a real surface would become zero, is governed by the width of the step. In fig. 1 it can, also, be seen that (X, 0) curves possess strong curvature for large values of h, but that this curvature diminishes as b increases. There are in fact two independent parameters in this model, and they can be fitted to the experimental data. The parameters required to give the best fit to the data give a physically realistic description of the surface. Fig. 2 shows, for Mg, the experimental data of Kloos and Raether [3] together with a good theoretical fit. Although it has been claimed, for Mg, that no variation of a with K can be observed the error bar on the experimental points can sustain the
A.D. Boardman et al./Effect of structure on surface plasmons
289
085 t 0.80 -
I
I
I
I
I
I
0.05
0.10
0.15
0.20
0.25
0.30
K Fig. 1. Theoretical dependence of frequency upon wave number for different values of the parameters A and b. Each curve is tabelled (h, b). The result Iabelled Ritchie is the case (0,O).
theoretical, weak, dependence on K shown here. The values of X and b required to fit the experimental data are not unique. On the other hand, all pairs of (h, b) which lead to the same curve can be interpreted in such a way as to give the same order of magnitude for the surface transition region. Thus for the curve shown in fig. 2 the surface transition region is in dimensionless units -2h which, since the screening radius is -0.5 A, is roughly 2 to 3 A.
0.80 -
Mg
0.75 -
0.65~
I
0.05
I
040
I
I
0.15
0.20
I
I
O-23
0.30
K Fig. 2. The circles are the Kloos-Raether experimental results for Mg. The theoretical curve labelled (2.5, 0.1) is obtained from the step-function model. The theoretical curve labelled Gaussian is obtained after numerically integrating eq. (8).
AD. Boardman et aLlEffect of structure on surface plasmons
290
0 650
0.05
0.10
K
0.15
0.20
Fig. 3. The circles are the Kunz data for Mg. The theoretical curve is labelled (3.0,O.l).
Fig. 2 also includes a curve calculated by a numerical integration of eq. (8) for a Gaussian electron distribution compensated by a rectangular positive charge distribution. It is found that a half-width of 2.5 is required which implies the existence of 2.5 A surface region. Such an entirely numerical approach was employed by Bennett [ 161 using a piece-wise linear profile for which analytical solutions cannot be obtained. It is clear from fig. 2 that the much simpler, analytic, step-function model gives results which have exactly the same quantitative reliance. Fig. 3 shows, for Mg, some alternative data produced by Kunz [ 1] in which the K-dependence is more pronounced. It has, however, been suggested [3] that the quality of sample surfaces used by Kunz is open to doubt. Nevertheless, the data in fig. 3 are fitted, for example, by the parameters (3,0.1) showing that in this case a somewhat larger transition region exists.
080-
Al
0.75 L?
0.651)
I 0.05
I 0.10
I 0.15
I
0.20
K Fig. 4. The circles are the Kloos-Raetber
data for Al. The theoretical curve is iabelled (2.5,O.l).
A.D. Boardman
et aLlEffect
of structure
291
on surface plasmons
Fig. 4 shows the recent data on Al [3]. Here, although the K-dependence is particularly weak the data are fitted quite well by the curve with parameters (2.5,O. 1). These are the same as the parameters found for Mg; but, since they are dimensionless parameters, the ratio of the thicknesses of the surface regions of Al and Mg is simply the ratio of their respective screening lengths. It now remains to discuss the possibility of the occurrence of other frequency modes. As we have seen earlier, as K + Kn, an extra mode cannot appear unless
A>!!
l/2
( 1 L
2 5(1 -b)
which, for b < 1, becomes h > 2.22. Thus the requirement that h must be -2.5 in order to fit the experimental results means that the condition for the existence of one higher mode is just fulfilled. For (2.5,O.l) the second mode occurs at 52 = 0.987 which is very close to R = 1 the bulk oscillation frequency. For (3.0,O.l) the higher mode occurs at 52 = 0.932. It is interesting to note as b + 1 the condition for the existence of other modes requires h to be very large.
References [ll C. Kunz, Z. Physik 196 (1966) 311. 121 A. Bagchi, C.B. Duke, P.J. Feibelman
and J.O. Porteus, Phys. Rev. Letters [31 T. Kloos and H. Raether, Phys. Letters 44A (1973) 157. 141 LO. Porteus and W.N. Faith, Phys. Rev. B8 (1973) 491. 151 H. Kanazawa, Progr. Theoret. Phys. 26 (1961) 851. 161 R.H. Ritchie, Progr. Theoret. Phys. 29 (1963) 607. [71 C.B. Duke and U. Landman, Phys. Rev. 8 (1973) 505. 181 P.J. Feibelman, Phys. Rev. 176 (1968) 551; Phys. Rev. B3 (1971) 220.
27 (1971)
998.
191 P.A. Fedders, Phys. Rev. 153 (1967) 438. I101 D.M. Newns, Phys. Rev. Bl (1970) 3304. 1111 D.E. Beck, Phys. Rev. B4 (1971) 1555. 1121 J. Harris and A. Griffin, Can. J. Phys. 48 (1970) 2592; Phys. Rev. B3 (1971) 749; Phys. Letters 34A (1971) 51; Phys. Letters 37A (1971) 387. D.E. Beck and V. Celli, Phys. Rev. Letters 28 (1972) 1124; Surface Sci. 37 (1973) 48. P.J. Feibelman, Surface Sci. 27 (1971) 438. P.J. Feibelman, Phys. Rev. Letters 30 (1973) 975. A.J. Bennet, Phys. Rev. Bl (1970) 203. R.H. Ritchie, Surface Sci. 34 (1973) 1. 1181 J.E. Inglesfield and E. Wikborg, J. Phys. C6 (1973) L158. 1191 R.H. Ritchie and A.L. Marusak, Surface Sci. 4 (1966) 234. 1201 J. Heinrichs, Phys. Rev. B7 (1973) 3487. 1211 Ch. Heger and D. Wagner, Z. Physik 244 (1971) 449. 1221 A. Dattner, Phys. Rev. Letter 10 (1963) 205. of Electromagnetic Waves in Plasmas (Pergamon, Oxford, ~31 V.L. Ginzburg, The Propagation 1964). ~41 J. Harris, Phys. Rev. B4 (1971) 1022. 1251 L. Kleinman, Phys. Rev. B7 (1973) 2288. 1261 J.C. Ashley and L.C. Emerson, Surface Sci. 41 (1974) 615. [I31 (141 [I51 [I61 [I71
292 [27]
A.D. Boardman et al/Effect B.V. Paranjape
and W.J. Heaney, Phys. Rev. B6 (1972) 1743. and A.D. Boardman, Plasma Effects in Semiconductors (Taylor and Francis, Edinburgh, 1971). J.V. Parker, J.C. Nickel and R.W. Gould, Phys. Fluids 7 (1964) 1489. F.C. Hoh, Phys. Rev. 133 (1964) 1016. A.D. Boardman, B.V. Paranjape and R. Teshima, Phys. Letters 48A (1974) 327.
[ 281 A.C. Baynham [29] [30] [31]
of structure on surface plasmons
THE EFFECT OF STRUCTURE ON SURFACE PLASMONS f A.D. BOARDMAN*,
B.V. PARANJAPE and R. TESHIMA ofAlberta, Edmonton, Alberta, Canada
Theoretical Physics Institute, University
Received 6 May 1974; revised manuscript received 26 November 1974
It is shown, for small wavenumbers, that the principal surface plasmon mode, on the surface of a semi-infinite metal has a frequency w = wp/~ which is independent of the nature of the surface density profile. A double step-functron model of a surface is proposed which leads to an analytical form of dispersion equation. This dispersion equation contains two parameters which are adjusted to obtain good agreement between theory and experiment. These parameters can be used to determine nature of the electron density surface profile. The model gives a simple analytic criterion for the existence of other frequency modes.
1. Introduction The precise form of the dispersion of surface plasmons which can be excited in plane films of metal is still a topic of debate. Specifically the question which has arisen for Mg and Al, the principal systems under observation [l-4] , is how the surface plasmon frequency, w, depends upon its wave vector R in the region where retardation is negligible. In practice, if the surface plasma frequency assumes the collisionless form _ _ ~2 kL pFk +@I +ia2)-t(fr3+iQ4)~+... I
wP oz . fork>kp = w,/c, where wp is the bulk plasma frequency, k is the wave number, c is the velocity of light in vacua and a 1, a2, a3 and a4 are real, then the principal interest centres upon the sign and magnitude of al and a3. Some early theoretical work by Kanazawa [S] using an approximate quantum mechanical random phase approach, for a system in which the surface is an infinite barrier, predicts that al = 0, a2 = 0 thus giving w a k2 for small k. This result contrasts sharply with the classical, hydrodynamic, calculation of Ritchie [6] which gives, for a jellium model in which a homogeneous semi-infinite electron gas moves against a uniform distribution of positive charge, aI > 0; thus making w, essentially, linear in upport from the N.R.C. is gratefully acknowledged. PPermanent address: Physics Department, University of Saiford, England.
Salford, Salford MS 4WT,
216 k.
A.D. Boardmarr et al/Effect
of structure on surface piasmorls
This, theoretical, situation is in conflict with the experiments of Kunz [l] who, from energy loss measurements using keV electrons passing through Mg films has shown that aI < 0, a3 > 0. Indeed the negative characteristic of al is quite pronounced and the q(w) diagram exhibits a clearly defined minimum. The same features have been observed in Al by Bagchi et al. [2] after a somewhat complicated analysis of ILEED observations. In spite of this the current experimental situation is still not settled quantitatively. Firstly, Kloos and Raether [3], from energy loss experiments on Mg and Al films, have concluded that there is, for k > k , at best, only a weak dependence of 32(w) upon k, in either material, and that CISf w) + w IGas k -+ k _Furthermore, although their results for Mg are similar to those obtaines by Kunz [ 1Jn, their results for Al are completely different from those of Bagchi et al. [2] ; thus implying that the latter are in error. Recent ILEED measurements by Porteus and Faith [4} and their subsequent analysis by Duke and Landman [7] corroborate this conclusion by also claiming that the work of Bagchi et al. [2] is in error. However, although the magnitude of the correction is similar, Porteus et al. [4,7] find that aI > 0 and a3 c 0 which, once again, conflicts with the results of Kloos and Raether [3] . The fact that the Porteus and Faith [4] experiments require an extremely complex analysis in order to produce the dispersion curve may be partly responsible for the discrepancy. Since the work of Kloos and Raether [3) embraces the work of Kunz (I], without serious conflict, we will assume, for the purposes of this paper, that their data together with those of Kunz [l] is correct. Several theoretical models of surface plasmons have been constructed both from quantum and classical microscopic random phase approximations and classical hydrodynamics. A variety of approximations, of necessity, have been employed, or are implicit, in the fabric of these models, thus rendering them, in a strict sense, inapplicable to real metals. Nevertheless, since a measure of success can be claimed for all of them it seems to be worth presenting a brief guide to the current theoretical position before introducing the details of the new model analysed in this paper. A lot of attention has been focussed recently on a microscopic approach which employs a random phase approximation (RPA). A useful model of this kind has been developed by Feibelman [8] which avoids the approximation, made by Fedders [9] , that the electrons are trapped in a well possessing infinitely high walls. Feibelman [8] has produced what amounts to a time Fourier transform of Laplace’s equation; a procedure which has been criticised by Newns [lo] who pointed out that its validity is strictly confined to density variations of wavelength greater than that of the plasmon excitation. In the Feibetman form~ation [S] the surface barrier is expressed in terms of the static density profile which, in turn, is assumed to be a step-function. The real coefficient of the linear term, i.e. a is found to vanish and it is proved that the result !’ of the structure of the surface. Newns 313(w) + w,/&as k -+ k, is quite independent [lo], however, has concluded using a time-dependent Hartree approximation applied to an infinite square barrier model that al f 0. This conclusion is also supported by Beck [ 1I] using a full quantum mechanical RPA.
AD. Bourdman et al.JEffect of structure on surface plasmons
277
Harris and Griffin [12] have developed an RPA model, based upon a Wigner distribution function, and conclude that it is important to use a realistic model of the surface structure. Indeed they show that, within the limits of the RPA, the linear term of the dispersion equation can only arise as a consequence of the surface structure. They also show that, in the high density RPA iimit, the use of a sharp density profile or an infinite barrier is not justifiable. Nevertheless, Beck and Celli [ 131 have attempted a more realistic RPA calculation by introducing the surface as a step-function potential barrier. Their calculations are performed, with the aid of a new variational principle [ 13, 141. However, although it is claimed that no adjustable parameters are involved they do introduce variational parameters and a good choice of trial function seems to be quite ~port~t. Their results show that aI # 0 and are in modest agreement with the Kunz Mg data. Unfortunately their work is also in good agreement with the data of Bagchi et al. [2] which, as we have already discussed, now seems to be in error. Feibelman [ 151 developing his earlier RPA theory, which gave al = 0 for an infinite barrier model, to include a smooth, finite, potential barrier has now found that the linear term of the dispersion equation is finite and that not only does it depend strongly upon the diffuseness (i.e. the distance over which the height of the barrier changes), but it is extremely sensitive to the barrier shape. However, no real attempt is made to contact experiment apart from the observation that al, being some three times larger than the value obtained by Beck and Celli [ 131 is commensurate with the recent data of Porteus and Faith [4]. This sensitivity of the linear term of the dispersion equation to the diffuseness agrees with the numerical conclusions of Bennett [ 161 which are obtained from a hydrodynamic model incorporating a linear surface density profile. Bennett on the other hand has not determined whether the dispersion equation is sensitive to the shape of surface density profile; but his results are in good agreement with the Mg data of Kunz [l] . The sensitivity calculated by Feibelman [ 151 may be a real feature or perhaps a function of the model or the degree of appro~mation. Therefore, since the features of the dispersion curve computed by Bennett agree so well with experimeni and are qualitatively supported by RPA approaches [ 171 it is worthwhile making a further examination of the hydrodynamic model to determine any such degree of sensitivity to shape. Before leaving the question of microscopic models, however, and passing on to a consideration of the hydrodynamic model it should be mentioned that Inglesfield and Wikborg [IS] using a step-function potential, coupled to an RPA Green function matching method, have calculated a value of al which is in disagreement with Beck and Celli [ 131 and closer to the experimental value of Porteus and Faith 141. They do not include diffuseness or damping and make no comments upon the influence of the barrier shape. There have now been several hydrodyn~ic fo~ulations of the surface plasmon problem [6,16,20,25]. The earliest of these is due to Ritchie [6] who has shown that aI > 0 for a step-function of electron density profile. Subsequently Bennett [ 161, including the influence of surface structure by allowing the surface electron density to
278
A.D. Boardman et al/Effect
of structure on surface plasmons
fall to zero over a distance of a few angstrom, has shown that it is possible to have aI < 0. In both of these hydrodynamic models collisional and collisionless (Landau) damping has been omitted even though a surface plasmon, unlike a bulk plasmon, can suffer Landau damping at all wave numbers. If a hydrodynamic model which omits any form of damping is to produce any meaningful results then surface plasmon ought, in practice, to possess a long enough lifetime for it to maintain its identity as a quasiparticle in the regime of experimental interest. Experiments involving surface plasmon excitation by passing electrons through thin films seem to produce results which fall into this category; since they are explained extremely well by the hydrodynamic model [ 161. Direct experimental evidence of surface plasmon lifetime is limited [2,4] but it does show that ILEED measurements in Al films reveal both collisional and Landau type damping. The former type is k independent and inversely proportional to the relaxation time r [20] ; this type of damping is obviously a function of the electronic state of the metal and can be minimised experimentally by making r + 00. The collisionless damping is proportional to k [2,4] . In order to be more precise let us examine the ratio r = 9(0)/32(w) i.e. the ratio of the imaginary part to the real part of the frequency. If this is sufficiently small then it can be assumed that the concept of quasi-particle or eigenmode is a good approximation. Since the collisional damping can be eliminated, as we discussed above, we will neglect it and discuss the principal Landau damping in terms of r = a.&(1
+ a14),
q =
kvF/wp.
Hence, knowing from experiment [ 1,3] that q lies in the range 0 d q < 0.35, we find that the recent theoretical microscopic RPA results of Feibelman [ 151 give r % 20% whilst those of Beck and Celli [ 131 give r = 7%. These results are an order of magnitude higher than those found by Ritchie and Marusak [ 191 using a classical BoltzmannVlazov equation. This theoretical discrepancy could be attributed to the different theoretical approaches but Heger and Wagner [21] have pointed out, quite strongly, that the microscopic theories discussed above depend upon a high frequency approximation of very doubtful validity. Furthermore they prove that if the RPA high density approximation is used consistently then the classical result is corroborated. The experimental results [2,4] (which it should be remembered, as far as the m(w) is concerned, are in conflict with those of Kloos and Raether [3]) give a Landau damping where r could be as high as 29% or as low as 10%. Thus the situation, with respect to collisionless damping, is not clear experimentally or theoretically. It would appear, however, that the concept of quasi particle is good enough to expect that any formalism which, in the first instance, neglects damping altogether is going to give meaning& ful description of the behaviour of 32 (w) with k. The numerical results of Bennett [16] showed that there is a principal mode, which is in excellent agreement with experiment, and a number of higher modes for which, ask-+kp,wp/fi
A.D. Boardman et aLlEffect
of structure on surface phmons
219
damping. This may, indeed, be the case; but Heinrich found a family of lower modes in the range 0 < w < wP/fi when collisional damping was omitted from his hydrodynamic scheme, whereas there is some possibility that higher modes really do exist on the grounds that they have been observed, by Dattner [22], in gaseous plasma columns. Apart from the omission of damping the application of hydrodynamics to metals can be criticised on the grounds that the true hydrodynamic condition, i.e. that the frequency with which the plasma changes is much less than the collision frequency, is not appropriate to the metallic state. A hydrodynamic condition in other words requires the carrier distribution to relax extremely rapidly, via collisions, to local equilibrium. It is possible to avoid these approximations by using the less restrictive quasi hydrodynamic approach [23] which involves the use of moments of the Boltzmann equation in a form which does not incorporate the condition w < V, where w is the frequency of the plasma excitation and u is the collision frequency. The resulting equations are very similar to the pure hydrodynamic equations. Harris [24] recently cast a strong shadow upon the hydrodynamic approach by emphasising that the pressure tensor P(r, t) is normally approximated to V. P(r, t) = ml3 VGp(r, t), where m is the electron mass, 6p(r, t) is the mass density fluctuation; and fl is taken to be (3/5)uc in a forcing move to make the dispersion e uation agree with the result 9 derived from micorscopic models. Actually, 0 would be UP/~ if the bulk plasmon was a true hydrodynamic mode. Now the numerical difference between (3/S@ and $3 is not such a serious objection but there is no particular reason for supposing that the bulk value of /3 is going to be valid for surface plasmons. Harris [24] maintains that the, high density, RF’A formalism can be reconciled with a hydrodynamic approach provided /3 is taken to be (3/5)ugwi/w2. For bulk plasmons, neglecting dispersion, this is (3/5)vi while for surface plasmons it is 42 larger. If this was the sole content of the criticism then it is, once again, not qualitatively important. However, Harris [24] also maintains that for surface plasmons a hydrodynamic solution does not exist; a viewpoint which has been shown recently, by Kleinman [25], to be in error. In fact Kleinman [25] using the form of 0 proposed by Harris [24] has shown that, for a uniform step-function density model, the results obtained by using fl= (3/5)$ and new form suggested by Harris [24] are both very similar to the original result of Ritchie [6] . In the above brief survey it is shown that surface plasmon phenomena, in the wave number range k > kp = wP/c, can be studied with the aid of RPA, microscopic, models or by hydrodynamics. It is important to emphasise that the RF’A model is, strictly, a high density infinite electron gas model and that there are valid criticisms of the hydrodynamic method. It is clear that no method is completely justified so the merits of any particular model must not only include its ability to account for the current experimental results but its capacity for extension to more complex situations such as those involving surface electric fields or external magnetic fields.
A.D. Boardman et al/Effect
280
of structure on surface plasmons
It is with these points in mind that this paper examines the collisionless, hydrodynamic model of Bennett whose results were obtained entirely numerically, for a piece-wise linear surface, density profile, using the simple form of 0, namely (3,‘S)vE. We present details of an analytical proof; derived from a hydrodynamic model, of the insensitivity of the k + k,, w -+ up/& result to surface structure. It is shown that the dispersion is insensitive to the shape of the surface density profile and we introduce a novel double-step-function surface structure which possesses the advantage of leading to an analytical form for the dispersion equation, The results obtained are in excellent agreement with the data of Kloos and Raether [3] . It is clear that this type of model will be extremely useful in assessing the dispersion characteristics of magnetoplasmons and the effect of sample geometry [26].
2. The basic equations The metal is assumed to consist of a collisionless electron gas, moving against a smoothed out positive background charge (jellium model), for which the basic Euler equation [27] is m@(z) h/at = e p(z) grad (9) t e&p grad (@) - grad (P),
(1)
where m, -e and v are, respectively, the mass, charge and average velocity of the electron. p(z) is the unperturbed electron density, G(z) is the electrostatic potential which exists inside the unperturbed solid, J/(r, t) is the perturbed part of the electrostatic potential, 6p(r, t> is the electron density perturbation and p(r, t) is the perturbation hydrostatic pressure which is a measure of the thermal velocity spread of the electrons [28] . The existence of $(z) is a consequence of the surface structure. The perturbation pressure is assumed to be [I 6,29,30] P(C t) = P MC 0,
(2)
Here, uF is the usual Fermi velocity. where /3 is a constant equal to (3/5)m$. The perturbation electric field is assumed to be E= -Vii/. This is the electrostatic approximation which is only valid for wave number greater than the critical wave numbe k , defined in the introduction. The linearised basic equations of the problem are [16] ) a$,uming a space-time dependence of the form eickxewt) h(z), --PYKJ~6p = div [grad @Sp)] - e div [p grad (J/) + 6~ grad (@)I,
(3)
V2J/=
(4)
and (s-k')
where eO is the permittivity -nrti2
&p =Q, -Q,,
$=$!,
of free space. Now eq. (3) can be written as (5)
A.D. Boardman et aLlEffect
281
of structure on surface plasmons
where
and
Q2=e
[“” ~Z+Pd
($4
+!cL(.$
42)
(7) If the equations are now made dimensionless by writing K = kR, Z = z/R, ho/(epoR2), \II= @O/(epOR2), s2 = w/tiP2afdl(Z) = pd_(Z), where R2 = $/(3w;5) is the screening radius of the electrons, w - e ~~/(~~~) and pu is the electron density in the homogeneous part of the me&l, then the equation for * becomes
where A, =9/S,
(94
A, = -d@/dZ,
WI
A, = -d2@/dZ2
- I8 K2/5 f R2 -f_(Z),
(9c)
A, = - {df_/dZ - K2 d@/dZ},
(9d)
A, = 9K4/5 + K2 d2@/dZ2 - f12K2 + f_(Z) K2,
(9e)
and, defining the positive charge density as p+(Z) d2WdZ2
pd;(Z),
=f_(Z) -f+(Z).
Eqs. (9) are the dimensionless
3. The boundary
=
(10) equivalent
of those derived by Bennett
[14]
conditions
Since the time-dependence of the perturbations in the system is emiwt the effective, perturbation, displacement vector in the metal plasma can be written as Dp = - j/iw f eOEP, where j is the conduction
current density and EP is the macroscopic
(11) electric field. This
282
A.D. Boardman
displacement tively , div@
vector, and D’,
et al./Effect
of strucfwe on surface plasmom
the displacement
vector in the vacuum, satisfy, respec-
div Dv = 0.
= 0,
(12)
Hence, if the boundary between a semi-infinite metal and a vacuum is at z = 0, the continuity of the normal component of the displacement vector requires that en [E:(O) - E:(O)]
= [p(O) eiiw] u,(O) = o*,
(13)
where e* is a possible surface charge density. We shall now determine the conditions under which such a surface charge density will vanish, and thereby determine a set of boundary conditions which are valid at a plane metal-vacuum interface. The z-component of eq. (1) is (14) Thus eq. (13), in terms of potentials
wP
L
az
3$/v _-
wi(o)aGP
Gp and $I’, can be written as
e2
-z- +mw2e0
az = ,2
where 02(z) = e2p(z)/(eonz). Now, because the metal is electrically 0 at z = B so that the expression for o* reduces to
0% neutral, d@/dz =
(16) Several important cases can now be delineated. Firstly, for a cold homogeneous plasma the second term in eq. (16) vanishes and U;(O) f: 0. Such a plasma will, therefore, support a surface charge density
and 6p will vanish everywhere except at z = 0. Secondly, if the plasma is homogeneous, and warm, 6p will not vanish inside the plasma, but o* will be zero implying that in this case the surface charge density diffuses into the body of the plasma. This vanishing of o* leads to two conditions, at z = 0, namely
alliV_ wP az
(17)
a2 y
(18) Thirdly, if in addition to being warm, the plasma is i~omogeneous, 0, then eq. (18) in combination with eq. (4) is simply the boundary d3$’
k2dGP
d2
dz
--P=
o
I
such that e+,(O) = condition (19)
A.D.
Boardmanet al./Effect
of structure on
surface plasmons
283
The complete set of boundary conditions at a metal-vacuum interface, including the conditions that (i) the tangential electric fieId is continuous at z = 0, (ii) the plasma is warm, (iii) the electron density vanishes at z = 0 is, in dimensionless symbols (using a dash to denote differentiation with respect to Z), ,kP(0)
= *v(o),
(20)
*‘p(o)
= #V(O),
(20
\k”‘p(o)= 9”‘v(o),
(22)
where eq. (20) has been used to modify eq. (19) to the form given in eq. (22). In a model we use later, the surface is represented as follows. Suppose that the electron density is po over the region - m
(23)
‘k; (-A) = \Ir&h),
(24)
$(-A)
(25)
= !I$(-A),
9[\ki(-X)
- $&A)]
/5 + *;(-A)(1
-b) = 0,
(26)
*z(O) = *j (O),
(27)
$(O) = 9;(o),
(28)
9 [\kT (0) - \k; (O)] /5 f &Pi(O) = 0,
(29)
where the two density regions and the vacuum have been labelled 1,2 and 3 respectively.
4. Persistence of the Q2 = l/Z mode The electrostatic potential which exists in the unperturbed solid has the following properties (using a dash to denote a single differentiation) @(O)= 0,
@‘(O)= 0,
CI+=) = 0,
(30)
ca
.f -00
W’(Z)dz = 0.
(31)
284
A.D. Boardman et aLlEffect of structure on surface plasmons
Let us define the non-divergent solutions of eq. (8) for the metal, as 9 (Z) and \ky(Z), which for distances far removed from the surface have the form e& and eYZ , respectively, where ?2 = K2 + (1 - a21
(32)
.
915
The potential in the vacuum is of the form emKz. The application conditions (20), (21) and (22) yields the determinantal equation -1
*K
\I’7
%
*:,
*II; - K%;C
9;-K2’Py
where all the functions
Kl=O,
(33)
0
are evaluated at Z = 0. The determinant,
\Ir;qk;ll- K+P; + K
of the boundary
{UqPl;;
-
Qq;}
+
on expansion,
K3 {qK\k:, - 9$,}
= 0.
gives (34)
The limit of eq. (34) as K + 0 is
Since the electrostatic theory is valid only for k > k (i.e. K > K - 10m3) it is of interest to determine the limiting behaviour of R as K 5 K [3 l] . however, since K is so small we can, for all practical purposes, consider the likting process K + Kp top be the same as the process K -+ 0. For small K the function q(Z) can be written as ‘J’(Z) = A 1 [‘@o(Z) + Kx(Z)l + A,‘49>
(36)
where A 1 and A 2 are arbitrary constants and *u(Z) is lim,+, [*K(Z)] . This function satisfies eq. (8) so that in the limit K + 0 [and denoting q,,(Z), x(Z) or ‘I’JZ) as g(Z)] we have 9g”“(Z)/S - Q’(Z) g”‘(Z) + [CL2-f(Z)
- Q”(Z)] g”(Z) -f(Z)
g'(Z) =
O,i(37)
which can be re-written as $
(9 g”‘(Z)/5 - a’(z) g”(Z) + [!L? -f(Z)]
g’(Z)} = 0.
(38)
At large distances from the surface i.e. as Z -+ -m, 9 (Z) -+ eKz, x(Z) = Z, \E (Z) + eTz, $(Z) + K3 eKz, G;(Z) + K eKz, \ky -+y3 e~~,$r-+~e7Z.Thusas~+-m we can see that q”‘(Z) + 0, \Ir’ (Z) + 0, lim,+u [‘PK(Z)] = 1, lim,+,, [x(Z)] = Z and the limit as KY+ 0 of $$?), *i(Z) is zero. Now eq. (38) has a solution g(Z) = constant and it is obvious from the foregoing arguments that this solution is *o(Z) = 1. Thus eq. (35) reduces to Q”
- VJl
+ x’) = 0.
(39)
A.D. Boardman et aLlEffect
ofstructure on surface
285
plasmons
Eq. (38) also gives 9 x”‘(z)/s
- (a’(z) x”(z) t [a2 -f_(Z)
where B is a constant
(40)
which, since for Z + -00,
a’(Z)
G(Z) -+ 0,
x’(Z) = B,
-+ 0,
x”‘(Z)
+ 0,
x”(Z)
+ 0,
x’(Z)
f_(z) + 13
-+ I,
and (41)
is given by -B=CG
- 1.
(42)
Similarly, 9 V(Z)/5
- @‘(Z) *G(z) f [s22 -f_(Z)]
q;(z)
= 0.
(43)
Hence, at Z = 0 9 x”‘/S t R2$ = Q2 - 1,
(44)
9 *;I5
(45)
+ “““i,
The substitution *;(a2
= 0.
of eqs. (44) and (45) into eq. (39) gives
- l/2) = 0.
(46)
Therefore, if \k’ # 0, there always exists a mode with w = #~/~which is independent of the sur irace structure. This point has not previously been demonstrated for a hydrodynamic model. The other result of eq. (46) is that q’ = 0, a2 f l/2. This is the condition for the possible existence of other modes. Fe1‘6elman [8], using a manybody RPA approximation, with an unspecified f_(Z), also showed that o = tip/< but did not prove the possible existence of the other modes.
5. A model of the surface The experimental investigation of surface plasmons invariably employs thin films of thickness -100 A. For K > K,, the interaction between the surfaces of such films is negligible. It is therefore legitimate to base the theoretical interpretation on single surface models. At the surface of a metal the electron density decreases from its bulk value towards zero within the space of a few angstroms. This feature can be built into a hydrodynamic electrostatic surface plasmon theory by means of a model which employs step-functions The advantage of such a model is the convenient analytic form of the dispersion equation in which the basic parameters can be readily varied until a best fit to the experimental results is obtained. The solid is assumed to be semi-infinite with a double step-function electron number
286
AD. Boardman et aLleffect of structure on surface plasmons
density profile which, in equilibrium, is exactly equal to the positive charge density profile. The step-functions are defined as P@)=Pu
-m
Pm = bP()’
-h
P(Z)= 0,
O
(47)
If the three regions aiong the Z-axis are labelied 1,2 and 3, respectively, turbation electrostatic potentials are q,
then the per-
= P eKz t Q eyiz,
(48)
~~=ReKztSe-Kz+Te~2Z+Ue-~~Z, 9,
(49)
= VemKz,
GO)
where P, Q, R, S, T, U and V are arbitrary constants
and
y; = K2 t S(1 - L?2)/9,
(51)
y; = K2 t S(b - a2)/9.
(52)
The application of boundary conditions (23) to (29) leads to the condition (see p. 287). This determinant reduces to (T; - K2)(r;
+ Y$-
(53)
c) cash (y2h)
+ (yl +A”) {y2(yg - K2 + c) sinh (“i2h) + CK [eTKh - cash (y2h)] } (71 + WY; = &b i c
- r$3 ~
(y; - K2) cash (yZh) +
sinh (T2h)
72
t -[4K(yI + K) ema(^I; - K2)
(1 t e-2N)(ri
+2y,K
tK2)cosh(T2X)
\ (54)
where C= S(1 - b)/18.
(5%
If b -% I, then eq. (54) reduces to the particularly (?I + K) {(l/2 - f12h2
simple form
sinh (y2X) + (K/2) [eeKh - cash (rzA)] }
= !F12[5(1/2 - st2)/9 + ylK + K2] cash (y$).
(56)
As K -+ Kp the terms of O(K), and above, become negligible. Hence, in this limit eqs.
A.D. &ar&nan et al. fEffect of structure on surface plasmons
d II
0
x
F
“i
e
P
i0
0
0
0
7
287
A.D. Boardman et al/Effect
288
of structure on surface plasmons
(51) to (52) become y; = 172= 5(1 - s22)/9,
(57)
y; = -g2 = 5(b - n2>/9,
(58)
(a2 - l/2) [i cos (@) + 1) sin (@)I = 0,
(59)
where it is assumed that, for practical applications, b G R2. Eq. (59) gives the principal mode as L? = l/fland the condition for existence of higher modes as (0)
cot (@) = --77&
(60)
which can be re-written as x cot (x) = -@
- x2),
(61)
where D* = 5(1 -b)h2/9. The condition for the existence of higher modes requires the circles of radius D to intersect the family of curves x cot (x) and is in fact a resonance condition. Now x cot (x) vanishes at x = mn/2, m being an integer. Thus higher modes cannot exist unless
‘>:
112 9 5(1 -b) ’ ( 1
and the total number of modes of the system, including the principal mode, is tz= 1[3[;(l-b)]1’2++j,
(62)
where Ka denotes the integer part of the quantity inside the brackets. Fig. 1 shows, for a selection of (h, b) values, the principal mode solutions of eq. (54). These can be compared to the results obtained from the Ritchie mode [4], which can be obtained from eq. (56) by setting h = 0, b = 0. The results obtained from the double step-function model show a weak, initial, dependence upon k; the flattest curve being for a small value of b with X in the region of 2.5. Physically, the model developed here simulates the fall in the electron density at a real surface by a step-function profile. The distance over which the electron density in a real surface would become zero, is governed by the width of the step. In fig. 1 it can, also, be seen that (X, 0) curves possess strong curvature for large values of h, but that this curvature diminishes as b increases. There are in fact two independent parameters in this model, and they can be fitted to the experimental data. The parameters required to give the best fit to the data give a physically realistic description of the surface. Fig. 2 shows, for Mg, the experimental data of Kloos and Raether [3] together with a good theoretical fit. Although it has been claimed, for Mg, that no variation of a with K can be observed the error bar on the experimental points can sustain the
A.D. Boardman et al./Effect of structure on surface plasmons
289
085 t 0.80 -
I
I
I
I
I
I
0.05
0.10
0.15
0.20
0.25
0.30
K Fig. 1. Theoretical dependence of frequency upon wave number for different values of the parameters A and b. Each curve is tabelled (h, b). The result Iabelled Ritchie is the case (0,O).
theoretical, weak, dependence on K shown here. The values of X and b required to fit the experimental data are not unique. On the other hand, all pairs of (h, b) which lead to the same curve can be interpreted in such a way as to give the same order of magnitude for the surface transition region. Thus for the curve shown in fig. 2 the surface transition region is in dimensionless units -2h which, since the screening radius is -0.5 A, is roughly 2 to 3 A.
0.80 -
Mg
0.75 -
0.65~
I
0.05
I
040
I
I
0.15
0.20
I
I
O-23
0.30
K Fig. 2. The circles are the Kloos-Raether experimental results for Mg. The theoretical curve labelled (2.5, 0.1) is obtained from the step-function model. The theoretical curve labelled Gaussian is obtained after numerically integrating eq. (8).
AD. Boardman et aLlEffect of structure on surface plasmons
290
0 650
0.05
0.10
K
0.15
0.20
Fig. 3. The circles are the Kunz data for Mg. The theoretical curve is labelled (3.0,O.l).
Fig. 2 also includes a curve calculated by a numerical integration of eq. (8) for a Gaussian electron distribution compensated by a rectangular positive charge distribution. It is found that a half-width of 2.5 is required which implies the existence of 2.5 A surface region. Such an entirely numerical approach was employed by Bennett [ 161 using a piece-wise linear profile for which analytical solutions cannot be obtained. It is clear from fig. 2 that the much simpler, analytic, step-function model gives results which have exactly the same quantitative reliance. Fig. 3 shows, for Mg, some alternative data produced by Kunz [ 1] in which the K-dependence is more pronounced. It has, however, been suggested [3] that the quality of sample surfaces used by Kunz is open to doubt. Nevertheless, the data in fig. 3 are fitted, for example, by the parameters (3,0.1) showing that in this case a somewhat larger transition region exists.
080-
Al
0.75 L?
0.651)
I 0.05
I 0.10
I 0.15
I
0.20
K Fig. 4. The circles are the Kloos-Raetber
data for Al. The theoretical curve is iabelled (2.5,O.l).
A.D. Boardman
et aLlEffect
of structure
291
on surface plasmons
Fig. 4 shows the recent data on Al [3]. Here, although the K-dependence is particularly weak the data are fitted quite well by the curve with parameters (2.5,O. 1). These are the same as the parameters found for Mg; but, since they are dimensionless parameters, the ratio of the thicknesses of the surface regions of Al and Mg is simply the ratio of their respective screening lengths. It now remains to discuss the possibility of the occurrence of other frequency modes. As we have seen earlier, as K + Kn, an extra mode cannot appear unless
A>!!
l/2
( 1 L
2 5(1 -b)
which, for b < 1, becomes h > 2.22. Thus the requirement that h must be -2.5 in order to fit the experimental results means that the condition for the existence of one higher mode is just fulfilled. For (2.5,O.l) the second mode occurs at 52 = 0.987 which is very close to R = 1 the bulk oscillation frequency. For (3.0,O.l) the higher mode occurs at 52 = 0.932. It is interesting to note as b + 1 the condition for the existence of other modes requires h to be very large.
References [ll C. Kunz, Z. Physik 196 (1966) 311. 121 A. Bagchi, C.B. Duke, P.J. Feibelman
and J.O. Porteus, Phys. Rev. Letters [31 T. Kloos and H. Raether, Phys. Letters 44A (1973) 157. 141 LO. Porteus and W.N. Faith, Phys. Rev. B8 (1973) 491. 151 H. Kanazawa, Progr. Theoret. Phys. 26 (1961) 851. 161 R.H. Ritchie, Progr. Theoret. Phys. 29 (1963) 607. [71 C.B. Duke and U. Landman, Phys. Rev. 8 (1973) 505. 181 P.J. Feibelman, Phys. Rev. 176 (1968) 551; Phys. Rev. B3 (1971) 220.
27 (1971)
998.
191 P.A. Fedders, Phys. Rev. 153 (1967) 438. I101 D.M. Newns, Phys. Rev. Bl (1970) 3304. 1111 D.E. Beck, Phys. Rev. B4 (1971) 1555. 1121 J. Harris and A. Griffin, Can. J. Phys. 48 (1970) 2592; Phys. Rev. B3 (1971) 749; Phys. Letters 34A (1971) 51; Phys. Letters 37A (1971) 387. D.E. Beck and V. Celli, Phys. Rev. Letters 28 (1972) 1124; Surface Sci. 37 (1973) 48. P.J. Feibelman, Surface Sci. 27 (1971) 438. P.J. Feibelman, Phys. Rev. Letters 30 (1973) 975. A.J. Bennet, Phys. Rev. Bl (1970) 203. R.H. Ritchie, Surface Sci. 34 (1973) 1. 1181 J.E. Inglesfield and E. Wikborg, J. Phys. C6 (1973) L158. 1191 R.H. Ritchie and A.L. Marusak, Surface Sci. 4 (1966) 234. 1201 J. Heinrichs, Phys. Rev. B7 (1973) 3487. 1211 Ch. Heger and D. Wagner, Z. Physik 244 (1971) 449. 1221 A. Dattner, Phys. Rev. Letter 10 (1963) 205. of Electromagnetic Waves in Plasmas (Pergamon, Oxford, ~31 V.L. Ginzburg, The Propagation 1964). ~41 J. Harris, Phys. Rev. B4 (1971) 1022. 1251 L. Kleinman, Phys. Rev. B7 (1973) 2288. 1261 J.C. Ashley and L.C. Emerson, Surface Sci. 41 (1974) 615. [I31 (141 [I51 [I61 [I71
292 [27]
A.D. Boardman et al/Effect B.V. Paranjape
and W.J. Heaney, Phys. Rev. B6 (1972) 1743. and A.D. Boardman, Plasma Effects in Semiconductors (Taylor and Francis, Edinburgh, 1971). J.V. Parker, J.C. Nickel and R.W. Gould, Phys. Fluids 7 (1964) 1489. F.C. Hoh, Phys. Rev. 133 (1964) 1016. A.D. Boardman, B.V. Paranjape and R. Teshima, Phys. Letters 48A (1974) 327.
[ 281 A.C. Baynham [29] [30] [31]
of structure on surface plasmons