Surface plasmons and the extreme anomalous skin effect

Surface plasmons and the extreme anomalous skin effect

Surface Science 107 (1981) 165-175 North-Holland Publishing Company SURFACEPLASMONSANDTHEEXTREMEANOMALOUSSKINEFFECT We have derived a simple formula...

704KB Sizes 4 Downloads 84 Views

Surface Science 107 (1981) 165-175 North-Holland Publishing Company

SURFACEPLASMONSANDTHEEXTREMEANOMALOUSSKINEFFECT

We have derived a simple formula for the dispersion relation of surface plasmons on metals in the extreme anomalous skin effect region. Comparisons of this theory with the usual locai dielectric theory is made for Cu, Ag, Au, Sn, and Pb. The two theories can be more than two orders of magnitude different in their predictions of propagation lengths, depending on the temperature.

In a recent article [I ], Begley et al. (BAWMB) reported that surface pIasmons, at a wide variety of metal-air interfaces, at a frequency w/2nc = 84.2 cm-‘, and at room temperature, had attenuation lengths of only a few cm in the direction of propagation, whereas the local dielectric theory [2] predicts propagation lengths on the order of several meters. This is all the more surprising because at a frequency o/Zrrc = 1000 cm-l, theory and experiment are in fairly good agreement, with propagation lengths on the order of centimeters f3-51; at higher frequencies the propagation lengths are too small to be measured directly, but there is at least rough agreement between theory and experiment [2,6] (in, say, the dispersion curves), whereas one expects the agreement to continue to improve as one eontinuously lowers the frequency to zero, One possible explanation for the two orders of magnitude discrepancy in the far infrared that was casually considered, then rejected, by BAWMB was the anomalous skin effect [7] (ASE). Actually, it is extremely unlikely that any metal, at any frequency, is in the region of the anomalous skin effect at room temperature. Nonetheless, we felt it worthwhile to investigate the behavior of surface plasmons in that region because most metals do

* Present address: Sch~umbe~ger-Dam Research Center, R~dge~e~d, Connecticut 06877, USA.

166

D.L. Johnson,

W.H. McNeil1 / Surface plasmons and EASE

show anomalous behavior at low temperatures, and we wish to determine just what the effects would be. In this article, we have derived an algebraic expression for the dispersion relation of surface plasmons in the extreme anomalous skin effect (EASE) region, and we have compared our results against those of the local dielectric theory for Cu, Ag, Au, Sn, and Pb at several frequencies and temperatures; implicit in this article is that we are in the usual anomalous skin effect region o < wp. A similar result had been derived for the case where electrons are reflected specularly at the surface of the metal but our simple result, eq. (9b), is valid generally. If the response of the system can be adequately represented by a spatially local dielectric function, E&O), the surface plasmon/polariton dispersion relation is simply :

q2c2/w2= %,(~Y [flod~) + 11.

(1)

If the system’s non-local response is taken into account [i.e., if the bulk dielectric function has appreciable wave-vector (4) dependence, as well as a frequency dependence, E = e(q, w)] , then the dispersion relation is not simply obtained by substitution of e(q, o) into eq. (1). This is because the presence of a surface breaks the translational symmetry in the surface normal direction and a new approach must be taken. This is most conveniently done by means of the surface impedance Zn(w, sin 0) = E,.(O’)/H,(O’) for light polarized in the plane of incidence (0 is the angle of incidence). With the identification 4 = o/c sin 8 the dispersion relation for surface polaritons is then given by eqs. (1) and (7a) of ref. [ 1 l] : (q2 - ~~/c~)l’~ = i(w/c) Zn(w, qc/o)

,

(2)

and this is valid for the local as well as the non-local case; the former can be shown [ 1 l] to reduce to eq. (1). In a classic paper on the non-local response of the electrons in the ASE region, Reuter and Sondheimer [7] (RS) showed how one could calculate the electromagnetic properties of a vacuum-simple metal interface in terms of the normal incidence surface impedance [8] ZN = E(O)/H(O) = Z,(w, 8 = 0). Specific formulae for the surface impedance in the EASE region for specular 07 = 1) and diffuse (p = 0) scattering of the electrons at the inner surface of the metal were presented: Z,@ = 0) = (c/47r)(&rw21/c4a)“3(1 Z,(p=l)=$Z,(p=O),

- fii)

,

@a) (3b)

where u is the DC conductivity, I = ur is the mean free path of the Fermi surface electrons (u is the Fermi velocity, r is the lifetime). The parameter l/o = 4nv/wi is, therefore, independent of the damping parameter l/r as expected. The criterion for being in the EASE region is o(w)/ [ 1 + (,r)2]3i2

9 1,

(4)

D.L. Johnson, W.H.McNeil1/Surface plasmons and EASE

167

where (u(w) = oe2m2vs73/nc2t23

,

(5)

m being the effective mass of the otherwise free electrons. This criterion (essentially that the mean free path is much larger than the skin depth) precludes the EASE in DC conductivity and for frequencies in the range of the plasma frequency wP, or higher, in typical metals. To calculate the electromagnetic properties of the interface at a non-normal angle of incidence, 0 for frequencies w < wp in s or p polarization, Reuter and Sondheimer showed that the non-local surface impedance for light polarized in the plane of incidence 2, was approximately related to the nonlocal surface impedance for normal incidence by z, = Zi(Zl;’

.

- sir&)“2

(6)

A similar expression holds for light polarized perpendicular to the plane of incidence, Z,. This last assumption, which is equivalent to the assumption of an effective local dielectric function given by T(w) = zr;’ = (Z,ZJ’

)

(7)

and nor by the usual local theory, E&W)

= 1 - ,~Qu(u

t iy) ,

(8)

was critiqued in some detail by Fuchs and IUiewer [9] who showed, in their own theory of specular electron reflection, that when non-local effects are important, there is in general no E(O) which gives correct results for both polarizations of light. Instead, one must use the non-local expressions for Z,(w, sin O), Zs(o, sin 0) which in general are not simply related to Z,(w) via eq. (7). This is basically because longitudinal as well as transverse bulk excitations are important in p (but not s) polarization. However, ref. [9] did demonstrate that, in the frequency region of the usual anomalous skin effect (whether extreme or not), (1) their non-local theory is formally identical with that of RS for s-polarized light at an arbitrary angle of incidence, and (2) their theory is numerically identical with that of RS for p-polarized light, thus providing a justification of eq. (6) or equivalently eq. (7) in that region. This is the critical observation for our own work, for it allows us to immediately write the dispersion relation for the surface plasmon in the ASE region by comparison with that of the local theory [2]. Combining either eqs. (6) and (2) or eqs. (7) and (1) we are led to equivalent expressions for the dispersion relation of surface plasmons, valid in the frequency region of the anomalous skin effect, w < wp:

q2 = (w2/c2)F/(Es+1) = (w2/c2)(1

+zQ-’

(9a) .

(9b)

168

D.L. Johnson,

W.H. McNeilf / Surface plasmons

and EASE

In the EASE region, eq. (3)applies. Ref. [9] applied only to the case of specular electron reflection; for the diffuse case, there are problems of interpretation as to how to proceed for non-normally incident p-polarized light because of the nonvanishing of the normal component of current density at the surface [lo]. It is clear from ref. [lo] that our eq. (7) still holds in the ASE region, the point being that neither 2, nor 2, has any appreciable angular dependence in this frequency region. To see this, it is necessary to consult Keller, Fuchs and Kliewer [lo]. They state on p. 2022 of this reference, which considers the diffuse case, that there is no difference between p and s polar~ation as far as the surface impedance (diffuse) is concerned, i.e. Z, = 2, = ZN. Furthermore, the surface impedance is very small in the low frequency region, as indicated by their numerical results. That Z, is independent, for the diffuse case, of angle is shown by the numerical results of Kliewer and Fuchs [lo]. In this reference, which considers s polarization infrared absorbance for diffuse electron scattering in thick samples, numerical results show that the surface impedance Z, is independent of 19 in the low frequency region. Thus 2, = 2, = ZN for the EASE region. A physical argument for the independence of Z, and Z, on angle can be provided as follows: In the low frequency limit which we are considering, wr < 1, there are no longitudin~ effects. Charge fluctuations are damped out in a FermiThomas screening distance, so effectively V. E = 0, or iqE,+ aE&ik= 0. At the surface of the metal, 4 < (lJE=)~E~/~~ because of the rapid decay of the surface plasmon into the metal. Thus, the transverse field is much larger than the longitudinal field,E,~E,,andZ,=Z,=ZN. For the diffuse case, replacing Z, directly by ZN in eq. (2), we obtain q2 = w”/ c2(1 ----Z&), which is equivalent to eq. (9b) for ZN small as it is in the low frequency limit. We shall, then, consider eqs. (3a) and (9b) to be a valid theory of surface plasmons in the diffuse limit of the EASE region. Again, we emphasize that we mean the usual anomalous region, 0 e op. Fuchs and Khewer also calculated the surface plasmon [ 111 from their aforementioned non-local (specular) theory; they did not investigate the usual ASE region, but concentrated on the region w Z w,,L.f2, where non-local effects greatly affect the dispersion relation of the surface plasmon. It would be far preferable to carry out the calculation of ref. ]l I] directly into the EASE region; lacking that, considerable insight into the properties of surface plasmons in the EASE region can be gained from eq. (9b). We see that 4 x w/c for any reasonable theory because 121 4 1; the significant effects are in the imaginary part of 4 = qn + iqr which governs the propagation length of the surface plasmon L = l/(291). From eqs. (3a) and (9b) we have, for the diffuse EASE, qr(EASE) = ~rr[2(3)5’2(cZ/a)‘(o/2nc)7]

1’3 ,

to be compared with the local result in the same frequency eqs. (8) and (9a), qI(loc) = w2/2crw;

= w2/8n5c.

(lo? region which is, from

01)

i3.L. Johnson,

W.H. McNeiII / Surface plasmons and EASE

Because of the existence of an additional qr(EASE) > qr(loc). In fact, one has ~~(EASE)~~~(loc) = 0.403%~“~

absorption

mechanism,

,

169

one might expect

cm

so that the surface mode is more attenuated than the local theory predicts if one is indeed in the EASE region a2 % [ 1 t (oT)~]~, although eq. (12) says that fairly large values of a (>15) are needed in order to really be in the EASE region. We have focussed on the diffuse limit because of the data of Chambers [ 121 in the early fifties, which showed that the diffuse theory was applicable for his samples at frequencies f= 1.2 X 10’ and 3.6 X 10s Hz. He was able, by varying the temperature from -2 to -90 K, to continuously vary his samples from the extreme anomalous limit to the classical (local) limit. At every temperature, the agreement with the diffuse theory was no less than astounding once appropriate values of u/j (considered as a single parameter) were deduced. ~~0~~ the data are quite old, ref. [12] seems to be the only really exhaustive study of the anomalous skin effect in metals in that a given sample was made to span the region from classical to EASE. The metals considered are not really simple free-electron-like ,metals, but this is not a serious consideration as discussed by Chambers (significant Fermi surface information can be derived from the EASE in metals [13]); we are not so much interested in the “true” values of u/l for ideal perfect crystals of a pure metal as we are in the fact that there are metals which really can be made to span the range from EASE to classical, and these metals are describable by the diffuse theory of Reuter and Sondheimer. The local theory has two independent parameters (e.g. wp and T); the non-local theory has two more (e.g. p and u). We have focussed on the p = 0 theory here. Moreover, if the inertial term can be neglected or < I, then the two material parameters o/l and o are all that are needed when one is in the usual ASE region (whether extreme or not), and eqs. (10) and (11) bear this out. From Chambers’ values of LY(1200 MHz) (temperature dependent through r) it is possible to deduce the value of the DC conductivity from (I = [(U/~2(~CZ~3~~~] 1’3 . We wish approximate use

values of r only in order to test inequality

(4); hence we shall

r* = (~2e2ffzf3~2~2)(~/~~3 in order to deduce approximate lifetimes from Chambers’ data [ 121. It is possible to measure m using infrared techniques, but resultant values are appreciably sample dependent, sometimes varying by a factor of -1.5. Note that it is completely invalid to extract r values directly from the literature, as they are very much sample and temperature dependent. Unfortunately, the temperature was not monitored, but Chambers has tabulated the largest values of (Y(1200 MHz) reached by his samples - presumably at 2 K;

102 102 10’ 100

5x 1x 1x 1x

102 102 10’ 100

w/2nc (cm-l)

cm-’

3.05 6.1 x 10-l 6.1 X lo-* 6.1 x 1O-3

CY

x X X X

10-l 10-l lo-* 1O-3

4.16 8.33 8.33 8.33

01

x x x x

100 100 10”” 10-2

1.47 1.89 8.12 8.32

x X X X

10-l 10’ 10-l lo-*

[ 1 + (w7)*]y*

CY

90 K a(1200 MHz) = 3.33 x 1O-3 o = 6.32 X 10” s-l 7 = 6.89 X lo-l4 s

cm-*

4.14 5.2 6.09 6.1

[ 1 + (w7)*]s*

CY

90 K a(1200 MHz) = 2.44 x lo4 D = 3.9 x 10” s-1 7 = 1.77 x 10-14 s

Table 2 Silver, o/l = 8.6 X 1O’O ohm-’

5x 1x 1x 1x

(cm-‘)

W/22%

Table 1 Copper, o/Z = 15.4 X 10’ o ohm-’

x x x x

10” 102 104 106 X x X x

2.68 6.70 6.70 6.70

(cm)

L(loc)

x X X x

lo* lo* lo4 lo6

(1.92 (8.19 1.76 3.80

(cm)

L(ease)

(2.83 (1.21 2.6 5.61 10’) 103) 10’ 10’

x x X x

IO’) 102) 10’ 10’

x X X x

103 10” 106) 10’)

2.12 5.44 5.44 5.44

a!

x x x x

2.16 6.90 (6.90 (6.90

L(loc)

x x x x

IO4 10’ 10’) 109)

4.54 9.08 9.08 9.08

a

x X x x

10’0 lo9 10’ IO7

10’ 106 105 104

2K (~(1200 MHz) = 3.63 x lo6 CJ= 6.5 x lO*O s-l 7=7.09x10-11s

3.43 8.58 (8.58 (8.58

(cm)

1.656 4.14 4.14 4.14

L(loc)

(cm)

2K ~~(1200 MHz) = 2.18 x lo3 D = 8.08 X lOi9 s-l 7 = 3.68 x 10-12 s

L(loc)

(cm)

L (ease)

10-l 10’ 103 lo4

01

X x x x

1.52 3.80 3.80 3.77

x x x x

10-l 100 102 104

[ 1 + (w7)2]“2

6.53 1.63 1.58 3.02

[l + (W7)*]w*

CY

102

1.58 1.58 X IO-’ 1.58 x lo-*

7.91

cy

cm-*

7.97 x 10-l 1.57 x 10-l 1.58 X IO-*

x 10-l

1.61 x 10’ 3.22 X 10-l 3.22 x 1O-2

3.22 x 1O-3

1 x 100

a

7 = 2.15 x lo-l4

a

3.22 x 1O-3

1.40 x 10-l 2.57 X 10-l 3.22 X lo-*

[ 1 + (w7)*]w*

s

2.42 x lo6

4.06 X lo7

(5.66 X 10’)

2.26 X lo3 5.66 x lo4 (5.66 x 106)

(cm)

x 109 x 108 x 107

4.10 x 104

2.05 X lo7 4.10 x 106 4.10 x 10s

01

D = 5.33 x 1019 s-1 7 = 5.01 x 10-12 s L(loc)

9.69 X 10’ 2.42 X lo* 2.42 x lo4

2.5 2.5 2.5

1.25 x 10’0

a

2K a(1200 MHz) = 1.64 x lo3

4.42 X 10’ (4.42 X 107) (4.42 X 109)

1.77 x 104

(cm)

L(loc)

2K a(1200 MHz) = 1.0 X lo6 D = 4.16 x lO*O s-l 7 = 4.7 x lo-” s

(cm) (2.05 x 10’) (8.75 X lo*) 1.89 x 10’

(cm)

L(ease)

(8.06 x 102) 1.74 x 10s 3.74 x 107

(1.89 X 10’)

(cm)

L(ease)

L(loc)

3.79 x 102 3.79 x 104 3.79 x 106

1.52 x 10’

(cm)

[ 1 + (w7)*]=

1.3

L(loc)

01

90 K a(1200 MHz) = 1.29 x lo4 D = 2.28 x 10’ 7 s-l

5 x 102 1 x 102 1x 10’

w/2lrc (cm-‘)

cm-’

90 K ~~(1200 MHz) = 6.33 x lo4 o = 3.6 x 10” s-l 7=4 * .04 x lo-‘78

ohm-’

Table 4 Tin, o/l = 9.5 X 1O1O ohm-’

1 x 102 1 x IO’ 1 x 100

5x

w/2rrc (cm-‘)

Table 3 Gold, o/l = 15.4 X 10”

1.58 x lo4

1.95 x 10-l 4.88 X 10’ 4.80 X lo*

[l + (wr)*p*

a

3.59 x 100 3.58 X lo* 3.52 x lo4

1.43 x 10-l

[ 1 + (W7)*]3’*

a

$ +

5

$ 3 8 : ?A. P

S z

S

:

% ;; -3

6

P P

5x 1x 1x 1x

102 102 10’ 100

w/2?tc (cm-’ )

3.62 7.25 7.25 7.25

a

X X X X

lo2 10’ 10’ lo-’

1.88 3.79 6.63 1.24

x x X x

10-l 100 10’ 10-l

(1 + (w7)2p2

a

90 K ~~(1200 MHz) = 2.9 X lo-’ u= 1.38 x 10”s s-1 7 = 1.32 X lo-l3 !

Table 5 Lead, o/l = 9.4 X 1O1* ohm-” cm-’

X X x X

10’ IO3 10’ 10’

(2.03 i8.69 1.87 4.03

X 10’) X 102j x 105 x 10’

6.64 1.66 (1.66 (1.66

(cm)

5.85 1.46 1.46 1.46

L(loc)

(cm) x x x x

lo3 105 10’) 109)

5.26 1.05 1.05 1.05

01

X x x x

2K (~(1200 MHz) = 4.21 X lo4 (I = 1.56 x 102* s-* 7=1.49x 10-l* s

L(loc)

(cm)

L(ease)

10’ 108 10’ 106

a!

1.9 4.75 4.74 3.97

x x x x

10-l 100 102 104

[ 1 -t (w*)2]3’2

D.L. Johnson, W.H.~~~~ill/ Surface plasrn~~satrdEASE

173

from his graphs we have picked off the smallest values of (Y- presumably at 90 K, and only for the five metals Cu, Ag, Au, Sn, and Pb. Accordingly, we have calculated their attenuation lengths, using eq. (10) (temperature independent) and eq. (11) (temperature dependent), and the results are presented in tables l-5 for several different frequencies. The column L(EASE) applies only if cr/ [ 1 + (oT)‘] 3’2 < 15 and it is temperature independent. Quantities which are never really applicable are placed in parentheses. Several points emerge: (1) None of these metals are in the EASE region at 90 K (or higher temperatures) at any frequency. That is, the criterion a/[ 1 t (~r)*]~‘* 9 15 is not satisfied at any frequency for any metal at 90 K. However, there is an indication that the propagation length of the local and the EASE theories are comparable at certain frequencies. For example, from table 2, which pertains to silver, it can be seen that at a frequency of 100 cm-’ , L(loc) is 670 cm, while L(EASE) is 819 cm. Presumably a complete nonlocal theory accounting for both normal and anomalous losses would predict a propagation length approximately one half of L(loc). (2) All of the metals are in the EASE region over a very large frequency region at 2 K; the high frequency limit is -100 cm- ‘; the low frequency limit is -10’ Hz. (3) When the metal is definitely in the EASE region, the attenuation length L(EASE) can be more than two orders of magnitude less than that calculated from the local formula L(loc) at the given temperature. The biggest differences between the two formulae typically occur around 0.1 to 10 cm-‘. (4) All propagation lengths involved are enormously large, especially at low frequencies, thus making a directly experimental confirmation of this theory difficult. We do feel, however, that when a theory predicts macroscopitally large propagation lengths (10-100 cm), it ought to be subjected to a rigorous experimental verification; experiments that measure properties of materials in macroscopically small regions are insufficient. For example, the far infrared reflectivities of metals are in accord with the predictions of the local dielectric function, eq. (5), but the same theory predicts propagation lengths much larger than seen in ref. [I]. It would be desirable to perform propagation length experiments on large samples (tens of centimeters) at about o/&z + 100 cm-’ using a different technique from that of ref. [ 11, (5) The signature of the EASE will be, as usual, that as the temperature is lowered, the propagation length L = $qr remains constant even as the DC conductivity continues to increase. At higher frequencies, where inequality (2) has failed and one is no longer in the extreme anomalous region, the non-local effects are still large but must be taken into account presumably by means of the more general method of ref. [ 1I] or its equivalent for the diffuse case. One can only conjecture whether ~r(non-lot) will smoot~y decrease to qr(loc) as the frequency is increased above l/7 (as does the absorptivity for the specular case f9]> or whether it remains essentially constant up to the region of or, (as does the absorption for the diffuse case) [lo]. Finally, we feel that as a consequence of our quantitative analysis of propagation length in the EASE region, we have ruled out the possibility of the ASE as an explanation of the two orders of magnitude discrepancy (theoretical versus experi-

174

D.L. Johnson, W.H. ~c~ei~l/

Surface plasmons and EASE

mental) reported for the room teInperature data of ref. [7]); two of our metals, copper and gold, were explicitly considered in ref. [l] t and they had theoretical (local) room temperature propagation lengths of 250 and 546 cm, which is compatible with our tables 1 and 3. (We did, of course, repeat the c~culat~or~ of L(loc) at 84.2 cm-‘, using literature values for the dielectric parameters at room temperature, and were able to reproduce table 1 of ref. [I].) Finally, it should be mentioned that at 84.2 cm-’ the penetration depth of the surface plasmon into the vacuum side of the interface is several centimeters, according to the local theory, which may have made it difficult to eliminate problems of scattered light and couple into the surface plasmon only.

Note added in proof Hartmann and Luttinger [14] obtain a formula for surface impedance trary electron scattering as follows: .&&I) =2,&J

= 0) [I - cos $(cos-‘p)/(l

for arbi-

- p)] .

Substitution into eq. (9b) results in an ima~nary component of the propagation constant, for arbitrary electron reflection from the surface, as follows: q&EASE, p) = ~~(EASE) [ 1 - cos $(cos-r&/(1

- P)]~ ,

where qr(EASE) is given by eq. (10).

Acknowledgments We should like to acknowledge several very fruitful discussions with E. Burstein, D.L. Mills, and especially R. Fuchs and C.A. Shiffman.

References [ l] D.L. Begley, R.W. Alexander, CA. Ward, R. Miller and R.J. Bell, Surface Sci. 81 (1979) 245. [2] E.N. Economou and K.L. Ngai, in: Advances in Chemical Physics, Vol. 27, Eds. I. Prigogine and S.A. Rice (Wiley, New York, 1974). [3] D.A. Bryan, D.L. Begley, K. Bhasin, R.W. Alexander, R.3. Beil and R. Gerson, Surface Sci. 57 (1976) 53. 141 D.L. Begley, D.A. Bryan, R.W. Alexander, R.J. Bell and C.A. Goben, Surface Sci. 60 (1976) 99. [S] J. Schoenwald, E. Burnstein and J.M. Elson, Solid State Commun. 12 (1973) 185. [6] P.E. Ferguson, F.R. Wallis and G. Chauvet, Surface Sci. 82 (1979) 255. [7] G.E.H. Reuter and E.H, Sondheimer, Proc. Roy. Sot. (London) Al95 (1948) 336.

D.L. Johnson,

W.H. McNeil1 /Surface

plasmons and EASE

[8] Our definitions of Z differs from that of ref. [7] by the factor c/4a. In addition, assume a time dependence exp(-iwt). [9] K.L. Kliewer and R. Fuchs, Phys. Rev. 172 (1968) 607; 185 (1969) 905. [lo] K.L. Kliewer and R. Fuchs, Phys. Rev. B2 (1970) 2923; J.M. Keller, R. Fuchs and K.L. Kliewer, Phys. Rev. B12 (1975) 2012. [ 111 R. Fuchs and K.L. Kliewer, Phys. Rev. B3 (1971) 2270. [12] R.G. Chambers, Nature 165 (1950) 239; R.G. Chambers, Proc. Roy. Sot. (London) A215 (1952) 481. [ 131 P.H. Haberland, J.F. Cochran and C.A. Shiffman, Phys. Rev. 184 (1969) 655. [14] L.E. Hartmann and J.M. Luttinger, Phys. Rev. 151 (1964) 430.

115 we