- Email: [email protected]
Anomalous optical constants of thin films
Surface Science 56 (1976) 425-439 o Norm-Holland Publi~jng Company
ANOMALOUSOPTlCALCONSTANTSOFTHINFILMS E.F.I. ROBERTS and D. ROSS Dept. of Metallurgy and Materials, Sir John Cuss School of Science and Technology, Ciry of London Polytechnic, London E. I, England
The explanation of anomalous optical constants in thin chemically distinct layers on substrates offered by Plumb is re-examined and extended. The model invokes the concept of the space charged boundary layer and treats the charge carrier population as a freeelectron gas to derive the optical behaviour of thin surface films. The implication of the space charge means that the optical constants of a dielectric film on a metal will vary over a distance directly proportional to the dielectric constant of the film and inversly proportional to the concentration of the electrons at the metal/film interface. Similarly as the temperature increases this space charge region should extend to larger distances from the interface.
1. Introduction Many workers have recognised that thin films (generally less than 100 A) exhibit anomalous optical constants and numerous references are found,in reviews by Rouard and Bousquet [ 1 ] , Heavens [2] , Berning [3] , and Chopra [4] . The literature cites a large number of possible causes of this variation of optical properties: sizeeffects [S-7] ; anisotropy [8] ; film morphology [9-l I] ; surface (substrate) roughening [ 12,131 or of incorrectly assessed geometrical thickness. Drude [ 141 dealt theoretically with reflection at a plane boundary between two dielectrics and discussed the presence of a very thin transitional layer; the refractive index of which varies continuously between the refractive indices of the two bounding media. VaSicek [ 1S] has dealt further with reflection of light from glass with thin and thick transparent non-homogeneous films, he also acknowledged the work of Bauer [ 161 and Schroder [ 171, the former adopting a hyperbolic variation in refractive index of the film and both workers deriving expressions for the reflectivity and transmissivity of the glass. VaSiEek [ 1S] substituted one transparent non-homogeneous film by a number of thin homogeneous films the refractive index of each being considered constant. The refractive index of the film was assumed to change linearly with thickness. Abel& [Is] subdivided the transparent inhomogeneous layer into mu~tiiayers, each layer being exponential in form. He pointed out that the method could be extended to absorbing films but also that if homogeneous subfiims are used it is diffi425
426
E.F.I. Roberts, D. Ross /Anomalous
optical constants of thin films
cult to assess how many subdivisions should initially be made. In many chemically-formed films a high degree of non-stoichiometry will exist across the film [ 191. This and ageing or annealing effects will produce changes in optical properties [20] . Cathcart and Peterson [21] show that in thermally grown cuprous oxide, besides a variation in lattice parameter, a gradient in the refractive index exists across the film. Extreme changes in stoichiometry may lead to different phases and in such a system varaitions in optical constants are accepted ]11,22,23]. Luke: [24] gives surface contamination as an alternative explanation to changes observed on GaP (110) single crystals; Adams and Bashara [25] have re-emphasized the importance of back surface reflections for thin substrates in giving anomalous optical constants. This communication deals with the application of a model, suggested by Plumb [26], to a series of film/substrate systems. Plumb developed his model for ultra-thin dielectric films deposited on metal substrate (in particular barium stearate on gold). He invoked the concept of the electric double-layer arising at the interface between two distinct phases to account for the anomalous optical absorption that occurs in such a system. As indicated above and in the discussion of ref. [23] there is an obvious interest in such a model for application to studies in wet and dry corrosion, micro-electronics or any system in which charge transfer is important.
2. The Plumb model 2. I. The space-charge region Plumb pointed out that when two chemically distinct phases are placed in contact a charge transfer may be anticipated that will create an electric double-layer. The differences in mobility of the migrating charge carriers leads to charge accumulation and hence to the double-layer+ Phases with high dielectric constants would tend to have a diffuse double-layer extending some several hundred angstroms while materials with low dielectric constants would have a hsarp, localized electric double-layer. Since such a layer is characterized by a separation of electric charge, i.e. positive on one side and negative on the other, a difference in electrical conductivity and hence in optical absorption would be expected in such a region. It is abundantly clear that the electrical properties of thin films bear little resemblence to what is intrinsically expected of such a material from its bulk properties and that these anomalous properties are determined less by the intrinsic property of the material than by such factors as the substrate/film contact. Thus, while an insulator may normally be regarded as possessing perhaps only a few voI~lme-generated carriers per cm3 a suitable ohmic contact is capable of injecting additional carriers far in excess of the bulk-generated carriers [27-291. These carriers are in addition to those derived from the presence of impurity and trapping centres.
E.F.I. Roberts, D. Ross /Anomalous optical constants of thin films
421
If we consider the space-charge region associated with a dielectric then three types of effect may be manifest depending upon the height and form of the potential barrier extending from the substrate Fermi level to the bottom of the dielectric conduction band. The bands in the two sides must be aligned so that in thermal equilibrium the vacuum and Fermi levels must be the same throughout the system. Furthermore the shape of the potential barrier just within the dielectric depends on the number and type of bulk carriers i.e. whether it is intrinsic or extrinsic, and upon the relative magnitudes of the work function of the metal, $r,, , and insluator, pi. If pi < G, an ohmic or Mott-Gurney contact arises at the interface and the substrate #readily able to inject electrons into the dielectric films conduction band. Such an “accumulation region” extends to a distance X, into the film which has been shown to be given, for pi - $,,, > 4kT, by
(1) where k is Boltzmann’s constant, T is the absolute temperature, K is the dielectric constant, e,, is permittivity of free space, e is the electronic charge, N, the effective density of states in the insulator at the interface and x is the insulator affinity. This equation, along with others defining the conditions within the space region, is derived from solutions of the Poisson-Boltzmann equation [26,27,30] . Using the fact that the density N, of electrons in the insulator at the interface is determined at the substrate (here assumed to be metal) interface by the same Fermi distribution functions as for the electrons in the metal Mott [27] shows
(2) for the number of electrons per unit volume at a distance x from the metal substrate and N C
exp I-
(Gi - X)/kTI
,
where m is the mass of the electron, h is Planck’s constant. It is possible that the substrate and the film form a neutral contact. In this case there is no reservoir of charge at the contact, implying $,,, = pi and the conduction band is flat up to the interface and no band bending occurs. In such a situation no space-charge would imply no variations in either electrical conductivity or optical absorption. Thirdly, when $, > Gi electrons flow from the film dielectric into the metal substrate to establish thermal-equilibrium conditions. A space-charge region of positive charge, the‘depletion region, is created in the insulator and an equal negative
E.F.I. Roberts, D. Ross /Anomalous
428
optical constants of thin films
change resides on the metal electrode. Normally such a situation is unlikely and the insulator would have to be very thick to provide sufficient electrons to satisfy the space charge. First studied by Schottky [3 1,321 for doped semiconductor-metal contact it is possible to apply a method of deriving quantitative information about this depletion region. The insulator is assumed to contain a large density, I%~~,of donors, these are fully ionized and uniformly distributed in the depletion region to a depth h,. Again a solution to Poisson’s equation is sought leading to
2(tim - ILiKE
11’2
x, = ___- ..-.--_ [ e2ND i
(4)
Thus an ohmic contact would produce a space-charge potential with distance, x, as [27]
V(x) which varies
V(x) = 2kT log ((~/~*) + 1) .
W4
A neutral contact does not produce a space charge and hence no potential i.e.
gradient
V(x) = 0 *
(5b)
,A blocking contact or Schottky
barrier would produce a space charge potential
V(x)
[301
Plumb’s expression
1261 assumes the form
V(x) = const. X exp(-x/h*)
CW
.
In the simplest model we assume the space charge to be composed essentially of free charge carriers, e.g. electrons whereas it is also likely to form as localized trapped charges as in defect ionic lattices and the exact frequency dependence and absorption characteristics would only be possible if an exact and detailed knowledge of the defect structure of the phases was known. 2.2. The optical characteristics * Defining the optical characteristics ‘2, =n(l
of a material thus (331
+iK)=12+%,
* The convention used to define the complex optical constants used in elIipsometry. See R.H. Muller, Surface Sci. 16 (1969)
(6)
is different 15 (Ed.).
from that commonly
E. F.I. Roberts, D. Ross /Anomalous
optical constants of thin films
429
where K = k/n is the absorption index; k = extinction index; n = refractive index; and using derivations from the free-electron theory of solids we have: 2rtNe2/3
n2K =
mw(w2
+ p2)
(74
or 2rNe2/3
k=
mno(w2
+ fi2) ’
(7b)
and 2rrNe2/3
n=
mkw(w2
+ f12) ’
(7c)
where N = concentration of charge carriers; w = frequency of incident radiation; (3= l/r = damping constant for the motion of charges under the influence of the applied field, T is the decay time. From eqs. (7) we may adduce something of the shape and turning points of the K versus w curve since
Firstly there should be three turning points from eqs. (7) which is a cubic in w. In seeking the solution of eq. (8) when dK/dw = 0 we may note that the condition utilized by Plumb [26] that is dn/dw = 0 is restricted. (N.B. the same result is obtained if n is assumed to vary and dk/dw = 0.) Then (9)
The condition for the turning point is that dK/dw = 0; two of the required solutions are thus found; the first for t p2)2 = m )
(104
+ 02) = 0.
(lob)
,2(,2 the second for -(3w2
The first solution corresponds to the observed behaviour at high frequency where The second solution gives a critical frequency w,rit where
K + 0 i.e. a minimum. Wc,it =
i(PIdB.
This latter criticality
invites further comment.
(1Oc) If the expression in (1 Oc) is regarded
430
E.F.I. Roberts, D. Ross /Anomalous
Fig. 1. Schematic
as a resultant
diagram
showing
the variation
optical constants of thin films
of n, k, and K as a function
of form Oc,it = p(A t W) then to fulfill the identity
of frequency
w.
A must be zero
valued and B = l/d.
Such a solution would indicate that /I is $rr out of phase with the incident radiation. Since 0 = l/r eq. (10~) would place this critical frequency at a frequency lower than l/r. Now Abel&s [34] presents a schematic outline of the variation of n and k for a system of free carriers as a function of o. The behaviour is characterized by three regions (fig. 1): (i) the absorbing region (0 < w, r < 1) in which Hagen-Rubens [35] apply n2 = k2 ‘c w;r{2w,
(11)
whence K = k/n = const = 1 and dK/dw = 0 this represents the Oc,it = i(pld3) tion above; (ii) the reflecting region (i-r < o < oP) in which
solu-
whence K = k/n = 207; (iii) the transparent region (o > wP) in which n =Jl
- (op/w)2
K = wP/2w2r
= 0 ,
C,61 ) (13)
and so lim w > wpK = lim k/n = 0, again this represents one of the solutions found above, i.e. where w -+ 00.
E.F.I. Roberts, D. Ross /Anomalous
In the above equations c.$ = hNe2/m,,,,
optical constants of thin films
wP is the plasma frequency
431
and is defined by
,
(14)
where N is the concentration of free carriers and rnopt is the optical mass; at wp, n=kandK=l. Thus, if K is also plotted on Abel&s figure, we observe that it should pass through a maximum and possess a value of K = 1 at the plasma frequency wP. Consequently if a dielectric on a metal substrate possesses a space-charge region at the interface it may possibly exhibit free carrier behaviour. In particular it should show a maximum in the value of K as a function of frequency w. Considering the relationships for n and k for the reflecting region we may determine dK/dw and the solution for the critical turning point, we, where dK/dw = 0. Thus and
-1
WC = w&/2.
(15)
I Using the definition
(14) for wp gives .
we = (2nNe2/m,pt)1/2 Further,
(16)
since K = k/n = 2~7, then for we and K, pertaining
to it we have
7 =K,/2w,.
(17)
Finally since the k varies with the concentration (2) in eq. (7). Thus
of carriers we may substitute
eq.
2 n
K+
2ne2NoP + f12) [ xo X0+x n 2 moptw(w2
wP2
P
[
h0
=_
2n2 o(w2
1
1 2
(18)
’
+ p2> x,
should indicate the behaviour of K through an insulating film on a metal as a function of thickness x at a frequency w when an ohmic contact is formed. Further since 2n2(K) = 2nk = Im e(x) = e2(x) then E2(X) = w;
w(w2
2
X0
P + 02)
[ho+x
1
.
(19)
432
E. F.J. Roberts, D. Ross / Anomoious optical consrunts
From eq. (19) we see that when x > Xu, Ed as a bulk dielectric.
3. Observations
of thin fihzs
+ 0 and the dielectric film behaves
on some thin-film systems
3.1. Gold substrate/barium
stearate film [26]
We have examined Plumb’s results in which he shows variations of the absorption index K at three wavelengths: h = 4358 a, 5461 8, 5780 A as a function of thickness, x, of the film. He also shows the experimental (tan J/, a) plots along with several calculated loci to illustrate the faifure to achieve a correspondence using a uniform film complex refractive index. He points out that the variation with frequency goes through a maximum, contrary to his expectation that dK/dw c 1/a_ However, as we observe from eqs. (7)-( 15) the situation is more complicated than this and furthermore the variation should exhibit a maximum (i.e. we = w&/2). Assuming the maximum to occur at h, = 5500 A (we = 2nv/h, where u is the velocity of light in vacua, w, = 3.43 X lOI set-*), then wn = o,t/Z = 4.85 X lOI
Fig. 2. The variation of K as a function of film thickness, x, and space charge “accumulation region” ho for different wavelengths, for baruim stearate films on gofd substrate (Plumb [ 261).
433
E.F.I. Roberts, D. Ross /Anomalous optical constants of thin films
set-l , X = 3880 A. The maximum in K, occurring as it seems about 5461 A > X, < 5780 1, tempts one to associate it with the similar wavelength region in pure gold, where the changes have been associated with the d + s interband transitions. Using the o, above and assuming K, N 1 the relaxation time, r = 1.5 X lo-l6 sec. Greenaway and Harbeke [36] quote the relaxation time for Ge, I&b, Si. GaAs, GaP as a group as lying in the range 1.4 X 1O- l6 - 1.8 X lo-l6 sec. This invites the speculation that the dielectric barium stearate in these thicknesses on the gold is behaving more like a semiconductor than an insulator, the extra carriers being injected from the gold and these carriers being the same that one responsible for the above transitions. Finally by plotting log K versus log@&, + x)~ it is possible to test eq. (18) and the model upon which it rests since the insulator/metal system should be ideally ap-
Fig. 3. The variation thicknesses.
of K versus w for Au/Se
system
(Weitzenkamp
[37])
with various
film
434
E.F.I. Roberts. D. Ross /Anomalous
Fig. 4. The variation of K versus w for Glass/Au films of different thickness.
optical constunts of thin films
system
(Krautkramer
[2,39]
) for evaporated
plicable (fig. 2). Here we assume X0 = 50 A and 100 A although only three points are availabie from Plumb’s results they do show that eq. (15) is followed for the longer wavelengths the change iI1slope for the shorter wavelength would imply a wavelength dependent X0. 3.2. Gold substrate/selenium firm [37] Here we use the results obtained for amorphous selenium on substrates of thick evaporated gold using ellipsometry by Weitzenkamp [37]. In fig. 3 we have shown the dispersion of the K values for the system and have included our own bulk Au values. As the gold is covered by increasing thicknesses of selenium we observe that the film acquires a distinct turning point in K which (a) shifts to longer wavelengths
E.F.I. Roberts, D. Ross /Anomalous
optical constants
of thin films
435
Fig. 5. Plot of lag K versus log(ho/ho +x)’ for glass/Au system for films of different thicknesses.
and (b) decreases until it finally merges into the bulk Se K values (fig, 3). There were insufficient data points in the very thin film region to fit an eq. (15). However, calculations of the plasma frequencies associated with the films of increasing thickness showed a decrease. This is as one would expect [38] if the carrier concentration decreases with distance from the gold/film interface: Thickness (A)
56
137
316
tip X lOi 7 X lo-l7
4.76 8.9
4.37 4.0
4.10 1.0
set-l set
436
E. F. f. Roberts, D. Ross
/A nomolous opticaf
constants
of thin films
Figa 6. The variation of K versus w for bulk and evaporated Gg films on glass.
Using the results of Krautkr~er [2,39] similar dispersion curves were prepared for the glass/gold system (fig. 4). These results show the opposite trends to those observed in the gold~selenium system namely as the gold film thickens K increases and the maximum, possibly associated with the gold interband transition, observed in the thin film (17 a) merges into the bulk gold curve for the thicker film (281 A). The maximum A, also initially moves towards longer wavelengths probabiy demonstrative the influence of the insulating substrate, however, the max~um returns as the film thickens and exceeds, presumably, the X, value.
E.F.I. Roberts, D. Ross /Anomalous
Fig. 7. Plot of K versus log(hg/h-, +x)’
optical constants of thin firms
437
for bulk and evaporated Ag films on glass.
Again a plot of log K versus log@,&, t x)* gives a reasonable straight line and the curves for different h, values converge to the same value at log(Xu/Xu + x)~ = 0 (fig. 5).
When this system is examined (fig. 6) both bulk silver and thick evaporated films have high values of K which peak at h, = 5500 A for Kc = 83 (wc = 3.42 X 1015 set-l) which gives T(= K,/wz) CSI1.21 X lo-l4 set a value reasonable by order of magnitude. Similarly using results by Clegg f40] a straight line is obtained for log K, versus 10~~~1~~ + x)~ using ho = 300 A (fig. 7).
4. Discussion and conclusions It appears on the basis of the foregoing examinations that it is at least worth reconsidering, perhaps with theoretical refinements, the model proposed by Plumb for the variation of optical properties as a function of thickness for very thin films.
E.F.I. Roberts, D. Ross /Anomalous
438
optical constants of thin films
In the case of a dielectric on metal substrate the extent of the space-charge region, and hence the region over which the variation in optical properties may be expected to vary, is directly proportional to the dielectric constant of the film and inversly proportional to the concentration of electrons at the metal/film interface (which in turn is dependent upon the electron concentration in the metal). Likewise as the temperature increases one would anticipate the extension of this space charge to larger h, values. If the dielectric is considered to possess a fairly high population of free carriers then the appearance of a critical maximum in the spectral dispersion of the film’s optical constants is not unexpected. Furthermore this maximum appears to be related to the bulk plasma frequency, wP. It is understood that wP/fi is also equal to the surface plasma frequency for an approximately free electron gas [41] . The concentration of free carriers being lower than that of the bulk metal substrate, will tend to give a plasma frequency lower than that for the bulk (i.e. longer wavelength). The work of Daude et al. [ 121 has shown that surface roughening can be correlated with the appearance of surface plasma photon resonance and it may very well be that the variations we have considered are complicated by this effect. At present we favour the idea of a carrier population present in the film which possesses many features similar to the substrate band structure.
Acknowledgements One of the authors (D.R.) would like to thank the ILEA for financial support received during the period of the work. Both authors would like to thank colleagues who contributed during discussions particularly Mr. K. Clarke.
References [l] [2] [3] [4] (51 [6] [ 71 (81 [9] [lo]
P. Rouard and P. Bousquet, in: Progress in Optics, Vol. 4, Ed. E. Wolf (North-Holland, Amsterdam, 1965) p. 147. O.S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965). P.H. Berning, in: Physics of Thin Films, Ed. G. Hass (Academic Press, New York, 1963) p. 69. K.L. Chopra, Thin Film Phenomena (McGraw-Hill, New York, 1969). J.C. Maxwell-Garnett, Phil. Trans. Roy. Sot. (London) A203 (1904) 385. D. Male’, Compt. Rend. (Paris) 235 (1952) 1630. U. Kreibig, J. Phys. F: Metal Phys. 4 (1974) 999. M.S. Tomar and U.K. Srivastava, Thin Solid Films 15 (1973) 207. J.B. Bateman, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington, D.C., 1964) p. 297. R-J. Archer, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington, D.C., 1964) p.255.
E.F.I. Roberts, D. Ross /Anomalous
optical constants of thin films
439
[ 1 l] P.C.S. Hayfield and G.W.T. White, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington, D.C., 1964) p. 157. [ 121 A. Daude, A. Savary and S. Robin, Thin Solid Film 13 (1972) 255. [13] J.O’M. Bock& M. Gershaw and V. Brusic, Symp. Faraday Sot. 4 (1979) 177. [ 141 P. Drude, Lehrbuch der Optik, 3rd Ed. (Leipzig, 1912) p. 273. [15] A. VaSicek, Optics of Thin Films (North-Holland, Amsterdam, 1960) pp. 150, 188. [ 161 G. Bauer, Ann. Physik 19 (1934) 434. [17] H. SchrBder, Ann. Physik 39 (1941) 55. [ 181 F. Abel&s, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington D.C., 1964) p. 41.
[ 191 E.F.I. Roberts, [20] [21] [22] [23] [24] [26] [27] [28] [29] [30] [31] [32] (331 [34] [35] [36] [37] [38] [39] [40] [41]
C.J.L. Booker, P. Osborne and M. Salim, Corrosion Sci. 14 (1974) 307. J.R. Adams and K.K. Rao, Surface Sci. 16 (1969) 382. J.V. Cathcart and G.F. Peterson, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington, D.C., 1964) p. 201. M.A. Barrett, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington, D.C., 1964) p. 213. J. Kruger, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington, D.C., 1964) p. 131. F. Luke:, Surface Sci. 49 (1975) 344. R.C. Plumb, J. Phys. Radium 25 (1964) 69. N.F. Mott and R.W. Gurney, Electronic Processes in Ionic Crystals, 2nd ed. (Oxford University Press, Oxford, 1950) Ch. V. A. Rose, Phys. Rev. 97 (1955) 1538. M.A. Lampert, Rept. Prog. Phys. 27 (1964) 329. J.G. Simmons, in: Handbook of Thin Film Technology, Eds. L.I. Maissel and R. Glang, Ch. 14. W. Schottky, Z. Physik 113 (1939) 367. H.K. Henisch, Rectifying Semi Conductor Contacts (Oxford University Press, Oxford, 1957). M. Born and E. Wolfe, Principles of Optics, (Pergamon Press, London, 1959) p. 611 et seq. F. Abel&, in: Optical Properties of Solids, Ed. F. Abel&s (North-Holland, Amsterdam, 1972) Ch. 3. E. Hagen and H. Rubens, Ann. Physik 11 (1903) 873. D.L. Greenaway and G. Harbeke, Optical Properties and Band Structure of Semi Conductors (Pergamon Press, London, 1968) p. 143. L.A. Weitzenkamp, Surf. Sci. 16 (1969) 353. J.I. Pankove, Optical Processes in Semi Conductors (Prentice-Hall, Englewood Cliffs, N.J., 1971) p. 92. J. Krautkramer, Ann. Physik 32 (1938) 537. P.L. Clegg, Proc. Phys. Sot. 65 (1952) 774. E.T. Arakawa, R.N. Hamm, W.F. Hanson and T.M. Jelinek, in: Optical Properties and Electronic Structure of Metals and Alloys, Ed. F. Abel&s (North-Holland, Amsterdam, 1966) p. 374.
ANOMALOUSOPTlCALCONSTANTSOFTHINFILMS E.F.I. ROBERTS and D. ROSS Dept. of Metallurgy and Materials, Sir John Cuss School of Science and Technology, Ciry of London Polytechnic, London E. I, England
The explanation of anomalous optical constants in thin chemically distinct layers on substrates offered by Plumb is re-examined and extended. The model invokes the concept of the space charged boundary layer and treats the charge carrier population as a freeelectron gas to derive the optical behaviour of thin surface films. The implication of the space charge means that the optical constants of a dielectric film on a metal will vary over a distance directly proportional to the dielectric constant of the film and inversly proportional to the concentration of the electrons at the metal/film interface. Similarly as the temperature increases this space charge region should extend to larger distances from the interface.
1. Introduction Many workers have recognised that thin films (generally less than 100 A) exhibit anomalous optical constants and numerous references are found,in reviews by Rouard and Bousquet [ 1 ] , Heavens [2] , Berning [3] , and Chopra [4] . The literature cites a large number of possible causes of this variation of optical properties: sizeeffects [S-7] ; anisotropy [8] ; film morphology [9-l I] ; surface (substrate) roughening [ 12,131 or of incorrectly assessed geometrical thickness. Drude [ 141 dealt theoretically with reflection at a plane boundary between two dielectrics and discussed the presence of a very thin transitional layer; the refractive index of which varies continuously between the refractive indices of the two bounding media. VaSicek [ 1S] has dealt further with reflection of light from glass with thin and thick transparent non-homogeneous films, he also acknowledged the work of Bauer [ 161 and Schroder [ 171, the former adopting a hyperbolic variation in refractive index of the film and both workers deriving expressions for the reflectivity and transmissivity of the glass. VaSiEek [ 1S] substituted one transparent non-homogeneous film by a number of thin homogeneous films the refractive index of each being considered constant. The refractive index of the film was assumed to change linearly with thickness. Abel& [Is] subdivided the transparent inhomogeneous layer into mu~tiiayers, each layer being exponential in form. He pointed out that the method could be extended to absorbing films but also that if homogeneous subfiims are used it is diffi425
426
E.F.I. Roberts, D. Ross /Anomalous
optical constants of thin films
cult to assess how many subdivisions should initially be made. In many chemically-formed films a high degree of non-stoichiometry will exist across the film [ 191. This and ageing or annealing effects will produce changes in optical properties [20] . Cathcart and Peterson [21] show that in thermally grown cuprous oxide, besides a variation in lattice parameter, a gradient in the refractive index exists across the film. Extreme changes in stoichiometry may lead to different phases and in such a system varaitions in optical constants are accepted ]11,22,23]. Luke: [24] gives surface contamination as an alternative explanation to changes observed on GaP (110) single crystals; Adams and Bashara [25] have re-emphasized the importance of back surface reflections for thin substrates in giving anomalous optical constants. This communication deals with the application of a model, suggested by Plumb [26], to a series of film/substrate systems. Plumb developed his model for ultra-thin dielectric films deposited on metal substrate (in particular barium stearate on gold). He invoked the concept of the electric double-layer arising at the interface between two distinct phases to account for the anomalous optical absorption that occurs in such a system. As indicated above and in the discussion of ref. [23] there is an obvious interest in such a model for application to studies in wet and dry corrosion, micro-electronics or any system in which charge transfer is important.
2. The Plumb model 2. I. The space-charge region Plumb pointed out that when two chemically distinct phases are placed in contact a charge transfer may be anticipated that will create an electric double-layer. The differences in mobility of the migrating charge carriers leads to charge accumulation and hence to the double-layer+ Phases with high dielectric constants would tend to have a diffuse double-layer extending some several hundred angstroms while materials with low dielectric constants would have a hsarp, localized electric double-layer. Since such a layer is characterized by a separation of electric charge, i.e. positive on one side and negative on the other, a difference in electrical conductivity and hence in optical absorption would be expected in such a region. It is abundantly clear that the electrical properties of thin films bear little resemblence to what is intrinsically expected of such a material from its bulk properties and that these anomalous properties are determined less by the intrinsic property of the material than by such factors as the substrate/film contact. Thus, while an insulator may normally be regarded as possessing perhaps only a few voI~lme-generated carriers per cm3 a suitable ohmic contact is capable of injecting additional carriers far in excess of the bulk-generated carriers [27-291. These carriers are in addition to those derived from the presence of impurity and trapping centres.
E.F.I. Roberts, D. Ross /Anomalous optical constants of thin films
421
If we consider the space-charge region associated with a dielectric then three types of effect may be manifest depending upon the height and form of the potential barrier extending from the substrate Fermi level to the bottom of the dielectric conduction band. The bands in the two sides must be aligned so that in thermal equilibrium the vacuum and Fermi levels must be the same throughout the system. Furthermore the shape of the potential barrier just within the dielectric depends on the number and type of bulk carriers i.e. whether it is intrinsic or extrinsic, and upon the relative magnitudes of the work function of the metal, $r,, , and insluator, pi. If pi < G, an ohmic or Mott-Gurney contact arises at the interface and the substrate #readily able to inject electrons into the dielectric films conduction band. Such an “accumulation region” extends to a distance X, into the film which has been shown to be given, for pi - $,,, > 4kT, by
(1) where k is Boltzmann’s constant, T is the absolute temperature, K is the dielectric constant, e,, is permittivity of free space, e is the electronic charge, N, the effective density of states in the insulator at the interface and x is the insulator affinity. This equation, along with others defining the conditions within the space region, is derived from solutions of the Poisson-Boltzmann equation [26,27,30] . Using the fact that the density N, of electrons in the insulator at the interface is determined at the substrate (here assumed to be metal) interface by the same Fermi distribution functions as for the electrons in the metal Mott [27] shows
(2) for the number of electrons per unit volume at a distance x from the metal substrate and N C
exp I-
(Gi - X)/kTI
,
where m is the mass of the electron, h is Planck’s constant. It is possible that the substrate and the film form a neutral contact. In this case there is no reservoir of charge at the contact, implying $,,, = pi and the conduction band is flat up to the interface and no band bending occurs. In such a situation no space-charge would imply no variations in either electrical conductivity or optical absorption. Thirdly, when $, > Gi electrons flow from the film dielectric into the metal substrate to establish thermal-equilibrium conditions. A space-charge region of positive charge, the‘depletion region, is created in the insulator and an equal negative
E.F.I. Roberts, D. Ross /Anomalous
428
optical constants of thin films
change resides on the metal electrode. Normally such a situation is unlikely and the insulator would have to be very thick to provide sufficient electrons to satisfy the space charge. First studied by Schottky [3 1,321 for doped semiconductor-metal contact it is possible to apply a method of deriving quantitative information about this depletion region. The insulator is assumed to contain a large density, I%~~,of donors, these are fully ionized and uniformly distributed in the depletion region to a depth h,. Again a solution to Poisson’s equation is sought leading to
2(tim - ILiKE
11’2
x, = ___- ..-.--_ [ e2ND i
(4)
Thus an ohmic contact would produce a space-charge potential with distance, x, as [27]
V(x) which varies
V(x) = 2kT log ((~/~*) + 1) .
W4
A neutral contact does not produce a space charge and hence no potential i.e.
gradient
V(x) = 0 *
(5b)
,A blocking contact or Schottky
barrier would produce a space charge potential
V(x)
[301
Plumb’s expression
1261 assumes the form
V(x) = const. X exp(-x/h*)
CW
.
In the simplest model we assume the space charge to be composed essentially of free charge carriers, e.g. electrons whereas it is also likely to form as localized trapped charges as in defect ionic lattices and the exact frequency dependence and absorption characteristics would only be possible if an exact and detailed knowledge of the defect structure of the phases was known. 2.2. The optical characteristics * Defining the optical characteristics ‘2, =n(l
of a material thus (331
+iK)=12+%,
* The convention used to define the complex optical constants used in elIipsometry. See R.H. Muller, Surface Sci. 16 (1969)
(6)
is different 15 (Ed.).
from that commonly
E. F.I. Roberts, D. Ross /Anomalous
optical constants of thin films
429
where K = k/n is the absorption index; k = extinction index; n = refractive index; and using derivations from the free-electron theory of solids we have: 2rtNe2/3
n2K =
mw(w2
+ p2)
(74
or 2rNe2/3
k=
mno(w2
+ fi2) ’
(7b)
and 2rrNe2/3
n=
mkw(w2
+ f12) ’
(7c)
where N = concentration of charge carriers; w = frequency of incident radiation; (3= l/r = damping constant for the motion of charges under the influence of the applied field, T is the decay time. From eqs. (7) we may adduce something of the shape and turning points of the K versus w curve since
Firstly there should be three turning points from eqs. (7) which is a cubic in w. In seeking the solution of eq. (8) when dK/dw = 0 we may note that the condition utilized by Plumb [26] that is dn/dw = 0 is restricted. (N.B. the same result is obtained if n is assumed to vary and dk/dw = 0.) Then (9)
The condition for the turning point is that dK/dw = 0; two of the required solutions are thus found; the first for t p2)2 = m )
(104
+ 02) = 0.
(lob)
,2(,2 the second for -(3w2
The first solution corresponds to the observed behaviour at high frequency where The second solution gives a critical frequency w,rit where
K + 0 i.e. a minimum. Wc,it =
i(PIdB.
This latter criticality
invites further comment.
(1Oc) If the expression in (1 Oc) is regarded
430
E.F.I. Roberts, D. Ross /Anomalous
Fig. 1. Schematic
as a resultant
diagram
showing
the variation
optical constants of thin films
of n, k, and K as a function
of form Oc,it = p(A t W) then to fulfill the identity
of frequency
w.
A must be zero
valued and B = l/d.
Such a solution would indicate that /I is $rr out of phase with the incident radiation. Since 0 = l/r eq. (10~) would place this critical frequency at a frequency lower than l/r. Now Abel&s [34] presents a schematic outline of the variation of n and k for a system of free carriers as a function of o. The behaviour is characterized by three regions (fig. 1): (i) the absorbing region (0 < w, r < 1) in which Hagen-Rubens [35] apply n2 = k2 ‘c w;r{2w,
(11)
whence K = k/n = const = 1 and dK/dw = 0 this represents the Oc,it = i(pld3) tion above; (ii) the reflecting region (i-r < o < oP) in which
solu-
whence K = k/n = 207; (iii) the transparent region (o > wP) in which n =Jl
- (op/w)2
K = wP/2w2r
= 0 ,
C,61 ) (13)
and so lim w > wpK = lim k/n = 0, again this represents one of the solutions found above, i.e. where w -+ 00.
E.F.I. Roberts, D. Ross /Anomalous
In the above equations c.$ = hNe2/m,,,,
optical constants of thin films
wP is the plasma frequency
431
and is defined by
,
(14)
where N is the concentration of free carriers and rnopt is the optical mass; at wp, n=kandK=l. Thus, if K is also plotted on Abel&s figure, we observe that it should pass through a maximum and possess a value of K = 1 at the plasma frequency wP. Consequently if a dielectric on a metal substrate possesses a space-charge region at the interface it may possibly exhibit free carrier behaviour. In particular it should show a maximum in the value of K as a function of frequency w. Considering the relationships for n and k for the reflecting region we may determine dK/dw and the solution for the critical turning point, we, where dK/dw = 0. Thus and
-1
WC = w&/2.
(15)
I Using the definition
(14) for wp gives .
we = (2nNe2/m,pt)1/2 Further,
(16)
since K = k/n = 2~7, then for we and K, pertaining
to it we have
7 =K,/2w,.
(17)
Finally since the k varies with the concentration (2) in eq. (7). Thus
of carriers we may substitute
eq.
2 n
K+
2ne2NoP + f12) [ xo X0+x n 2 moptw(w2
wP2
P
[
h0
=_
2n2 o(w2
1
1 2
(18)
’
+ p2> x,
should indicate the behaviour of K through an insulating film on a metal as a function of thickness x at a frequency w when an ohmic contact is formed. Further since 2n2(K) = 2nk = Im e(x) = e2(x) then E2(X) = w;
w(w2
2
X0
P + 02)
[ho+x
1
.
(19)
432
E. F.J. Roberts, D. Ross / Anomoious optical consrunts
From eq. (19) we see that when x > Xu, Ed as a bulk dielectric.
3. Observations
of thin fihzs
+ 0 and the dielectric film behaves
on some thin-film systems
3.1. Gold substrate/barium
stearate film [26]
We have examined Plumb’s results in which he shows variations of the absorption index K at three wavelengths: h = 4358 a, 5461 8, 5780 A as a function of thickness, x, of the film. He also shows the experimental (tan J/, a) plots along with several calculated loci to illustrate the faifure to achieve a correspondence using a uniform film complex refractive index. He points out that the variation with frequency goes through a maximum, contrary to his expectation that dK/dw c 1/a_ However, as we observe from eqs. (7)-( 15) the situation is more complicated than this and furthermore the variation should exhibit a maximum (i.e. we = w&/2). Assuming the maximum to occur at h, = 5500 A (we = 2nv/h, where u is the velocity of light in vacua, w, = 3.43 X lOI set-*), then wn = o,t/Z = 4.85 X lOI
Fig. 2. The variation of K as a function of film thickness, x, and space charge “accumulation region” ho for different wavelengths, for baruim stearate films on gofd substrate (Plumb [ 261).
433
E.F.I. Roberts, D. Ross /Anomalous optical constants of thin films
set-l , X = 3880 A. The maximum in K, occurring as it seems about 5461 A > X, < 5780 1, tempts one to associate it with the similar wavelength region in pure gold, where the changes have been associated with the d + s interband transitions. Using the o, above and assuming K, N 1 the relaxation time, r = 1.5 X lo-l6 sec. Greenaway and Harbeke [36] quote the relaxation time for Ge, I&b, Si. GaAs, GaP as a group as lying in the range 1.4 X 1O- l6 - 1.8 X lo-l6 sec. This invites the speculation that the dielectric barium stearate in these thicknesses on the gold is behaving more like a semiconductor than an insulator, the extra carriers being injected from the gold and these carriers being the same that one responsible for the above transitions. Finally by plotting log K versus log@&, + x)~ it is possible to test eq. (18) and the model upon which it rests since the insulator/metal system should be ideally ap-
Fig. 3. The variation thicknesses.
of K versus w for Au/Se
system
(Weitzenkamp
[37])
with various
film
434
E.F.I. Roberts. D. Ross /Anomalous
Fig. 4. The variation of K versus w for Glass/Au films of different thickness.
optical constunts of thin films
system
(Krautkramer
[2,39]
) for evaporated
plicable (fig. 2). Here we assume X0 = 50 A and 100 A although only three points are availabie from Plumb’s results they do show that eq. (15) is followed for the longer wavelengths the change iI1slope for the shorter wavelength would imply a wavelength dependent X0. 3.2. Gold substrate/selenium firm [37] Here we use the results obtained for amorphous selenium on substrates of thick evaporated gold using ellipsometry by Weitzenkamp [37]. In fig. 3 we have shown the dispersion of the K values for the system and have included our own bulk Au values. As the gold is covered by increasing thicknesses of selenium we observe that the film acquires a distinct turning point in K which (a) shifts to longer wavelengths
E.F.I. Roberts, D. Ross /Anomalous
optical constants
of thin films
435
Fig. 5. Plot of lag K versus log(ho/ho +x)’ for glass/Au system for films of different thicknesses.
and (b) decreases until it finally merges into the bulk Se K values (fig, 3). There were insufficient data points in the very thin film region to fit an eq. (15). However, calculations of the plasma frequencies associated with the films of increasing thickness showed a decrease. This is as one would expect [38] if the carrier concentration decreases with distance from the gold/film interface: Thickness (A)
56
137
316
tip X lOi 7 X lo-l7
4.76 8.9
4.37 4.0
4.10 1.0
set-l set
436
E. F. f. Roberts, D. Ross
/A nomolous opticaf
constants
of thin films
Figa 6. The variation of K versus w for bulk and evaporated Gg films on glass.
Using the results of Krautkr~er [2,39] similar dispersion curves were prepared for the glass/gold system (fig. 4). These results show the opposite trends to those observed in the gold~selenium system namely as the gold film thickens K increases and the maximum, possibly associated with the gold interband transition, observed in the thin film (17 a) merges into the bulk gold curve for the thicker film (281 A). The maximum A, also initially moves towards longer wavelengths probabiy demonstrative the influence of the insulating substrate, however, the max~um returns as the film thickens and exceeds, presumably, the X, value.
E.F.I. Roberts, D. Ross /Anomalous
Fig. 7. Plot of K versus log(hg/h-, +x)’
optical constants of thin firms
437
for bulk and evaporated Ag films on glass.
Again a plot of log K versus log@,&, t x)* gives a reasonable straight line and the curves for different h, values converge to the same value at log(Xu/Xu + x)~ = 0 (fig. 5).
When this system is examined (fig. 6) both bulk silver and thick evaporated films have high values of K which peak at h, = 5500 A for Kc = 83 (wc = 3.42 X 1015 set-l) which gives T(= K,/wz) CSI1.21 X lo-l4 set a value reasonable by order of magnitude. Similarly using results by Clegg f40] a straight line is obtained for log K, versus 10~~~1~~ + x)~ using ho = 300 A (fig. 7).
4. Discussion and conclusions It appears on the basis of the foregoing examinations that it is at least worth reconsidering, perhaps with theoretical refinements, the model proposed by Plumb for the variation of optical properties as a function of thickness for very thin films.
E.F.I. Roberts, D. Ross /Anomalous
438
optical constants of thin films
In the case of a dielectric on metal substrate the extent of the space-charge region, and hence the region over which the variation in optical properties may be expected to vary, is directly proportional to the dielectric constant of the film and inversly proportional to the concentration of electrons at the metal/film interface (which in turn is dependent upon the electron concentration in the metal). Likewise as the temperature increases one would anticipate the extension of this space charge to larger h, values. If the dielectric is considered to possess a fairly high population of free carriers then the appearance of a critical maximum in the spectral dispersion of the film’s optical constants is not unexpected. Furthermore this maximum appears to be related to the bulk plasma frequency, wP. It is understood that wP/fi is also equal to the surface plasma frequency for an approximately free electron gas [41] . The concentration of free carriers being lower than that of the bulk metal substrate, will tend to give a plasma frequency lower than that for the bulk (i.e. longer wavelength). The work of Daude et al. [ 121 has shown that surface roughening can be correlated with the appearance of surface plasma photon resonance and it may very well be that the variations we have considered are complicated by this effect. At present we favour the idea of a carrier population present in the film which possesses many features similar to the substrate band structure.
Acknowledgements One of the authors (D.R.) would like to thank the ILEA for financial support received during the period of the work. Both authors would like to thank colleagues who contributed during discussions particularly Mr. K. Clarke.
References [l] [2] [3] [4] (51 [6] [ 71 (81 [9] [lo]
P. Rouard and P. Bousquet, in: Progress in Optics, Vol. 4, Ed. E. Wolf (North-Holland, Amsterdam, 1965) p. 147. O.S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965). P.H. Berning, in: Physics of Thin Films, Ed. G. Hass (Academic Press, New York, 1963) p. 69. K.L. Chopra, Thin Film Phenomena (McGraw-Hill, New York, 1969). J.C. Maxwell-Garnett, Phil. Trans. Roy. Sot. (London) A203 (1904) 385. D. Male’, Compt. Rend. (Paris) 235 (1952) 1630. U. Kreibig, J. Phys. F: Metal Phys. 4 (1974) 999. M.S. Tomar and U.K. Srivastava, Thin Solid Films 15 (1973) 207. J.B. Bateman, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington, D.C., 1964) p. 297. R-J. Archer, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington, D.C., 1964) p.255.
E.F.I. Roberts, D. Ross /Anomalous
optical constants of thin films
439
[ 1 l] P.C.S. Hayfield and G.W.T. White, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington, D.C., 1964) p. 157. [ 121 A. Daude, A. Savary and S. Robin, Thin Solid Film 13 (1972) 255. [13] J.O’M. Bock& M. Gershaw and V. Brusic, Symp. Faraday Sot. 4 (1979) 177. [ 141 P. Drude, Lehrbuch der Optik, 3rd Ed. (Leipzig, 1912) p. 273. [15] A. VaSicek, Optics of Thin Films (North-Holland, Amsterdam, 1960) pp. 150, 188. [ 161 G. Bauer, Ann. Physik 19 (1934) 434. [17] H. SchrBder, Ann. Physik 39 (1941) 55. [ 181 F. Abel&s, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington D.C., 1964) p. 41.
[ 191 E.F.I. Roberts, [20] [21] [22] [23] [24] [26] [27] [28] [29] [30] [31] [32] (331 [34] [35] [36] [37] [38] [39] [40] [41]
C.J.L. Booker, P. Osborne and M. Salim, Corrosion Sci. 14 (1974) 307. J.R. Adams and K.K. Rao, Surface Sci. 16 (1969) 382. J.V. Cathcart and G.F. Peterson, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington, D.C., 1964) p. 201. M.A. Barrett, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington, D.C., 1964) p. 213. J. Kruger, in: Ellipsometry in the Measurement of Surfaces and Thin Films, Symp. Proc. Natl. Bur. Std. publ. 256 (Washington, D.C., 1964) p. 131. F. Luke:, Surface Sci. 49 (1975) 344. R.C. Plumb, J. Phys. Radium 25 (1964) 69. N.F. Mott and R.W. Gurney, Electronic Processes in Ionic Crystals, 2nd ed. (Oxford University Press, Oxford, 1950) Ch. V. A. Rose, Phys. Rev. 97 (1955) 1538. M.A. Lampert, Rept. Prog. Phys. 27 (1964) 329. J.G. Simmons, in: Handbook of Thin Film Technology, Eds. L.I. Maissel and R. Glang, Ch. 14. W. Schottky, Z. Physik 113 (1939) 367. H.K. Henisch, Rectifying Semi Conductor Contacts (Oxford University Press, Oxford, 1957). M. Born and E. Wolfe, Principles of Optics, (Pergamon Press, London, 1959) p. 611 et seq. F. Abel&, in: Optical Properties of Solids, Ed. F. Abel&s (North-Holland, Amsterdam, 1972) Ch. 3. E. Hagen and H. Rubens, Ann. Physik 11 (1903) 873. D.L. Greenaway and G. Harbeke, Optical Properties and Band Structure of Semi Conductors (Pergamon Press, London, 1968) p. 143. L.A. Weitzenkamp, Surf. Sci. 16 (1969) 353. J.I. Pankove, Optical Processes in Semi Conductors (Prentice-Hall, Englewood Cliffs, N.J., 1971) p. 92. J. Krautkramer, Ann. Physik 32 (1938) 537. P.L. Clegg, Proc. Phys. Sot. 65 (1952) 774. E.T. Arakawa, R.N. Hamm, W.F. Hanson and T.M. Jelinek, in: Optical Properties and Electronic Structure of Metals and Alloys, Ed. F. Abel&s (North-Holland, Amsterdam, 1966) p. 374.