Vacuum-ultraviolet radiation damage of the KCl surface—application of combined spectroscopic ellipsometry and reflectometry

Surface Science 74 (1978) 568-595 0 North-Holland Publishing Company

VACUUM-ULTRAVIOLET RADIATION DAMAGE OF THE KC1 SURFACE APPLICATION OF COMBINED SPECTROSCOPIC ELLIPSOMETRY AND REFLECTOMETRY Victor M. BERMUDEZ Naval Research Laboratory,

Washington, DC 20375,

USA

Received 24 October 1977; manuscript received in final form 19 December 1977

A spectroscopic study (1.7 to 4.0 eV) of the surface damage layer, formed when KC1 is exposed to vacuum-ultraviolet radiation, is carried out using ellipsometry and reflectometry. The results indicate the formation of a thin layer (-30-300 A) with a high density of F-centers (-3 x 1019 cm-3 and of aggregate centers near the surface. The F-center absorption band in the damage layer appears at 0.04 f 0.02 eV to lower energy and with 30% larger width, relative to the bulk F-band. An approximate theoretical description of the defect-surface interaction indicates that this shift and broadening is of the correct magnitude and direction to be the result of such an interaction. However, the possibility that the apparent shift may also result from the presence of unresolved aggregate absorptions cannot be ruled out. It is also shown that a thin exponentially inhomogeneous film, such as that assumed to result from photochemical coloration of KCI, can be approximated as an “equivalent homogeneous film” for the purpose of analysis of ellipsometric data.

1. Introduction Although well developed [l] as a tool for measuring the thickness and optical constants of a surface film at futed wavelength, ellipsometry is not routinely used as a spectroscopic technique expect for “clean” homogeneous surfaces. The purpose of the work described in this paper is twofold. First, we wish to explore the techniques for measuring, as a continuous function of wavelength, the absorption and dispersion of a film of unknown thickness on a substrate of known (measurable) optical constants. As a model “three-phase” system (ambient-film-substrate) we have chosen a KC1 surface exposed to vacuum ultraviolet (VW) radiation. Henceforth, “film” refers to the surface damage layer and “substrate” to the undamaged bulk material. Second, we seek to discover whether evidence for interaction between the resulting F-centers and the crystal surface can be found in the absorption spectrum of the damaged surface. KC1 was selected largely for practical reasons. The F-band lies within the range of the apparatus (1.7-5.0 eV), and the intrinsic absorption overlaps the intense VUV line emission of the Ha glow dis568

V.M. Bermudez

/ VUV radiation damage of KCI surface

569

charge. The material is readily available, cleaves easily and can be handled in room air.

2. Ellipsometry

and reflectometry

The elhpsometry experiment measures the relative change, upon reflection at an interface, of the phase and amplitude of light polarized parallel (p) and perpendicular (s) to the plane of incidence. The ellipsometry parameters \Il and A are defined by

c,pl&p

R

(1)

-_L=~exp[i(6,-63]=tan$eiA, e;/cf R,

where Lr and e: are the incident (i) and reflected (r) field amplitudes of polarization v and Rgi6’ is.the Fresnel reflection coefficient. For the three-phase system, the coefficients are defined by [2] Gl t J;2 e-2i6

ei6v =

R,

(2)

1 t 4131;2 emzi6 . The (complex) reflection coefficients (42) interfaces are given by

r,j

P

;;i =

COS9i

-

z

?lj COS

9i

+

The (complex) 2n 6 = hd(T

Ei COS

TiiCOS9i

9j

iTiCOS ’

at the ambient-film

$j

=

Gj

COS

9j

COS

9j

9i

•t iij

and film-substrate

(3)

5

nj COS

9j

-

(&)

.

phase factor in eq. (2) is defined by

- H; sin29#12

,

where d is the film thickness, zi, = nl - iki the film optical constant and 90 and h the angle of incidence and wavelength in the ambient. The angles of incidence, 9k, which may be complex if one or more of the phases is absorbing, are obtained from cpovia Snell’s law no sin 90 = nk sin 9k .

(5)

For the problem at hand, the ambient is vacuum and the substrate is undamaged KCl, so that no = 1, %2 = n2 and cpoand 92 are real. In principle, if g2 is measured before film formation and if any one of the three parameters nl, kl and d is known, then $ and A measured after film formation can be used to derive the other two parameters. Substitution of eqs. (2)-(5) into (1) yields an expression that cannot be solved analytically for the two unknown

570

V.M. Bermudez / VUV radiation damage of KCl surface

film parameters in terms of J/ and A. Hence, some form of iterative technique is required. We have used the method of Cahan [3], which is based on Newton’s approximation in two or three dimensions. Here, successive approximations to the true film parameters are obtained from

C(X.j,m + 1 -

Xj,m)

i

W

vi)/axj,m

= -F(

where Xi = n 1, kl, d and Vi= A, $ or film formation). F( Vi) is defined by F(vi)

= Cvik

-

(vihvl

Vj), 6R/R (fractional change in reflectivity with

)

(7)

where (Vi)c is the value of the ith observable calculated with the present set of test values of Xi, and (Vi)M is the experimental value. The coupled linear equations were solved for Xi, m + 1 using Cramer’s rule. The necessary derivatives a Vi/ax, are given by Cahan [3]. Preliminary data reductions were also attempted using the “trial and error” method of McCrackin [4]. Although the method works well when kl = 0 is assumed and ni and d are to be obtained from $ and A, convergence is relatively slow for the case when d is fixed and nl and kl are to be computed. The Newton’s approximations, on the other hand, converge sufficiently rapidly that a large number of data points (-500) can be reduced in less than 10 set of computer time. In a practical situation, it is often found that measurement of J/ and A alone is not sufficient, even if d is known independently or if k, is known to be zero. In the former case, multiple solutions sometimes occur [3,5] - i.e., two different sets (nr, k,) for the same ($, A, d) set. In the latter case, of a transparent film, the phase factor [eq. (4)] is real, making eq. (1) invariant with respect to changes in d that change 6 by integral multiples of n. Therefore, in addition to measuring $ and A before and after film formation, we also measure 6RfR,with R given by R = Ri sin20 t

R$ cos'a = Rg(sin*a tan2Jl + cos2@-)

,

(8)

where OLis the azimuth of the incident polarization vector. In the present experiment, the precision in &R/R is much less than that in $ and A, and a full threeparameter solution of eq. (6) is not obtained. Nevertheless, it is still possible to avoid multiple solutions using even a crude measurement of &R/R. Aside from a reflectivity measurement, the only other applicable methods of obtaining more than two independent experimental quantities are multiple-angle-ofincidenceellipsometry [6,7] (MAIE), angle-of-incidence-derivative ellipsometry and reflectometry [8] (AIDER) and combined reflection and transmission ellipsometry [9] (CRTE). MAIE is not compatible with our experimental arrangement, since the placement of the ports on the ultrahigh-vacuum (III-IV) chamber permits angles of

V.M. Bermudez / VW radiation damage of KCl surface

571

incidence within a range of -4’ centered about cpe= 22”, 46”, 62”, or 77’. The sensitivity of the experiment (a$/&,, a$/%,, etc.) is found through direct calculation to be a maximum for cpo= 62” and to be smaller at the other available angles by a factor of two or more. Furthermore, because of correlation among nl, kl and d (to be discussed below), it appears that MAIE will not permit a. unique determination of all three terms in this case. The AIDER technique was not used because of practical difficulties in incorporating an angle-of-incidence modulation with the polarization-modulation ellipsometer. CRTE was not employed because of the rather stringent requirements imposed on the shape of the sample.

3. Experimental

details

The principles of operation of the wavelength-scanning polarization-modulation ellipsometer have been discussed extensively (refs. [ 1 JO-1 21 and references cited therein) and will not be reviewed here. The ellipsometry apparatus permits measurement of three quantities, iV, S, and C, defined by N = cos(2$)

)

S = sin(2$)

sin A ,

C = sin(2$)

cos A .

(9)

Since $ and A are both small angles, optimum precision is obtained by calculating $ from sin(2$), via S and C, and A from sin A, via $J and S. N was not measured. The angle of incidence was cpo= 61.8 f 0.2” which is close to the KC1 Brewster angle (0, = 56.5” at h = 400 nm). In general, angles too close to 19~are to be avoided [7] because of the resulting extreme sensitivity of the data reduction to error in measurement of cpo.Evaluation of the derivatives [3] a$/&,, a$/ak,, etc. for the available angles of incidence verified that this is the best choice for cpo. Since the changes in $ and A upon irradiation are small, separate lock-in amplifiers were used to measure the signals at the first (S) and second (C) harmonic of the modulation frequency (50 kHz). This eliminates error resulting from readjustment of the electronics during the experiment. S and Care measured with the same set of azimuths for polarizer, modulator and analyzer (Configuration II of refs. [lo] and [ 121) and are obtained by the two-zone-averaging technique with correction for sample imperfection [12]. A small signal-averaging computer [lo] is used to digitize the lock-in output (12 bit resolution), storing one point every -0.8 nm. This device also permits averaging of repeated scans to suppress noise. The punched paper-tape output is processed off-line, with numerical smoothing to remove “highfrequency” noise [lo], to yield a continuous plot of nl - n2 and kl. The strain birefringence of the entrance window of the vacuum chamber, which is the only major systematic error not eliminated in first-order by two-zone averaging [ 121, is measured in a separate experiment. The resulting small (-0.4”) h-dependent correction to A is catalogued on the main computer disc-file and accessed by the datareduction program.

Vhf. Bermudez

/ VW

r------------i

ARIZET L_______

ARC

radiation damage of KCl surface

~MODULATOR

/ A

----?Y LiF UHV WINDOW

LIGHT -TIGHT 1 ENCLOSURE : l/4 -METER DOUBLE MONOCHROMATOR

VACUUM

n

vuv

r-

-

SOURCE

ANALYZER-

Fig. 1. Schematic diagram of the experimental arrangement.

Several modifications of the basic system [lo] were made to accomodate the present experiment. A schematic diagram of the optical arrangement is shown in fig. 1. In an effort to reduce the effects of structural imperfections of the surface, steps were taken to obtain a small and stable source image on the sample. A 1 kW xenon arc, having about 3; times the brightness of the original 150 W arc, was used as the source, with a 15 cm distilled-water cell as an infrared filter. The light emerging from the monochromator was focused onto a 250 pm pinhole, and the illuminated area of the sample was about 1 X 2 mm. Also, a more sensitive detector was used for this work, an EM1 9785QB, with S-20 spectral response. The range of the apparatus, determined by the source, monochromator and detector characteristics, is 250-700 nm. The ellipsometry vacuum chamber [lo] was equipped with a LiF UHV window. A Tanaka-type hydrogen VUV source [13] was mounted on this port, with an ordinary rubber O-ring seal around the laboratory side of the UHV LiF window. A thin expendable piece of LiF was mounted directly in front of the source to reduce coloration of the UHV window. The resulting coloration of the LiF plate will attenuate somewhat [ 141 the VUV intensity reaching the sample and will also strongly absorb most of the near-UV emission. The source was operated at 15 kV, 50 mA and an Ha pressure of about 9.5 Torr; no attempt was made to measure the light flux at the sample. Alignment was checked by placing a sample coated with sodium salicylate in the beam. Under the present operating conditions, the emission consists [ 131 of a line spectrum between -165 nm and the LiF cut-off, -115 nm, with a continuum extending into the visible. At room temperature, the intrinsic absorption of KC1 begins at about 168 nm, with the first exciton peak appearing at 161 nm [15].

V.M. Bemudez / VW radiation damage of KC1 surface

573

The relative reflectivity change, SR/R, was measured by the double-beam method described by Treu et al. [ 161. The incident polarization was at 45” with respect to the plane of incidence [o = 45’ in eq. (S)]. A mirror and two beam-splitter cubes, one before the polarizer and one in front of the phototube, were used to direct the reference beam around the outside of the UHV chamber. The analyzer is moved from the “reflected” arm to a position on the “incident” arm in front of the modulator. In our experiment the light intensity in the sample channel is modulated at the first harmonic, rather than at the second [ 161, by inserting a servo-controlled Babinet-Soleil compensator [lo] between the modulator and analyzer. This eliminates complications resulting from small changes in the phase modulation amplitude with wavelength. The lock-m signal obtained in the double-beam mode is [ 161 X= Rcrc,(V)l]cr(N

+ R4VI

(11)

7

where cr is a constant. Rc,(h) and c,(h) are the wavelength-dependent fractions of the total detected intensity in the sample and reference beams and include, implicity, all the losses associated with lenses, windows, etc. The lock-in output is processed by an analog computation circuit, the output voltage of which is I’= logro](cr

- x)/Xl

= logro]cr(V/c,(Vl

The change in Vupon irradiation, -6 V= loglo(R’/R) = logro(l

+ logro(l/R)

.

(12)

6 V= V’ - I’, gives

+ 6R/R).

(13)

In actual practice, the properties of the beam-splitter cubes limited the usefulness of this technique to h > 500 nm. For a single sweep before and after irradiation, the minimum experimentally-significant value of 16R/R I is found to be 0.02 for X > 550 nm. We have also investigated a single-beam method, in which the light intensity entering the monochromator is maintained constant by the technique of Katzir and Rosmann [ 171. Here a fiber-optic light guide, with one end behind the entrance slit, passes a small fraction of the incoming flux to a phototube. The amplified phototube output is used in a feedback loop to regulate a shunt [18] in parallel with the source. Changes in flux through the entrance slit, caused by “arc wandering”, are thereby compensated by small changes in arc current. The detected intensity is given by 0)

= IO F(X) R(X) ,

(14)

where IO is the total intensity entering the monochromator, and flX> expresses the wavelength-dependent source spectrum, grating efficiency, detector response, etc. Recording I(X) before and after irradiation gives the desired result if IO remains constant during both scans. The advantage of this method is that ellipsometry and reflectometry can be carried out simultaneously with the same arrangement of optical components and without the need for beam-splitters. However, the stability of

574

V.M. Bemudez

/ VlJV radiation damage of KCI surface

the input flux was such that the minimum significant value of 16R/R 1 was -0.08. Nevertheless, in the visible and infrared, where the sources used are intrinsically more stable than the xenon arc, the single beam technique may be feasible. The samples used were cleaved from a block of KC1 obtained from the Harshaw Chemical Co. The material was checked for impurity-induced absorption in the 700-170 nm range under conditions where the minimum detectable absorption coefficient was (Y= 7 X 10B4 cm-‘. Bands were found at 201 nm ((u = 0.88 cm-‘) and at 303 nm (o = 0.018 cm-‘) the former being due to hydroxide impurity and the latter to an unknown contaminant. These are of no consequence in the present work. One experiment was also performed using high-purity KCl, grown at NRL, having no detectable impurity absorptions. Prior to the final cleaving, which exposed the (100) face to be studied, the samples were annealed at 650°C for 3: h and cooled at lO’C/h. Since the sample is transparent, steps must be taken to insure that light reflected from the back surface does not enter the collection optics [ 191. Surprisingly, roughening of the back surface with abrasive paper, with or without subsequent blackening with a colloidal suspension of graphite in alcohol, was ineffective in eliminating reflections at the back surface. Therefore, the samples used were -14 mm thick, which results in a displacement of -7 mm between the beams [19] reflected from the front and back surfaces. The crystals were cleaved in room air and mounted in vacuum within 15 min. All experiments were conducted at chamber pressures of
4. Precison and accuracy The precision of the experiment, i.e., the minimum significant change in G, A and R, was found to be 0.02’, 0.05” and 0.02, respectively. These values are comparable to the scatter or “noise level” in the spectra of J/, A and 6R/R. Averaging of several sweeps (4 for 6 $ and 6A and 10 for &R/R) gives a small improvement in precision,yielding0.01°,0.040 and0.008for lS$l, l&Al and l&R/RI. With regard to accuracy, we first note the results for the substrate index, ;a = n2 - ik2, obtained from JI and A before irradiation. The values of n2 differ from

V.M. Bermudez

/ VUV radiation damage of KG’ surface

575

the best literature values [26] by less than 0.002 for X > 350 nm. Farther into the UV, the deviation increases to 50.004. The value of k2 is no more than 0.001 for h > 3.50 nm and ~50.002 at shorter wavelength. These values of k, correspond to values of A between 0.15 and 0.40”, which are within the estimated limit of the absolute accuracy of A (see below). These results, together with the independent results given above, suggest that the surface is “clean” prior to WV irradiation. An error analysis has been carried out by the procedure in ref. 11, revised slightly to correspond to the present mode of operation of the ellipsometer. The estimates of the upper limits on the error in J/ and A are kO.5’ for both quantities, using values of $ and A appropriate to KCl. The actual error is probably somewhat less than this “worst-case” estimate. The most significant systematic errors involve the azimuths of the various optical components, including the windows.

5. Experimental

results

The change in 9 and A, resulting from a 20 min irradiation of a cleaved KCl surface at room temperature, is shown in fig. 2. A total of eight experiments, each

700

600

I””

-0.3oot

500

NANOMETERS 400 I

300 I

’

,

/

2.0

3.0 ELECTRON

, 4.0 VOLTS

Fig. 2. 6 ti = @after - tiLefore ad &A = Aafter - Abefore (in degrees) obtained from ellipsometric measurements before and after vacuum-ultraviolet irradiation of a KCl sample. The angle of incidence is 61.8”, and the spectral slitwidth is indicated by the vertical bars. All procedures were carried out at room temperature.

576

V.M. Bermudez

/ VUVradiation

damage of KCI surface

with a fresh sample, were carried out. Qualitatively and, in most respects quantitatively, the results were all as shown in fig. 2. Irradiation for 10 min, instead of 20, produced similar Sll, and &A, indicating that saturation of the damage, for a given VUV flux, occurs within a few minutes at most. One sample was baked for 10 h at 200°C in vacuum (5 X lo-’ Torr) prior to the run, yielding results similar to those in fig. 2. Another experiment, performed using the high-purity KC1 described above, yielded similar results. However, the magnitudes of Sri, and 66 at a given X, for different samples run under nominally identical conditions, were found.to differ somewhat. For four such runs, the peak value of 6 + varied between -0.27’ and -0.36”. The extrema in 6A varied between -0.49” and -0.69” and between +1.31” and t1.75”. A similar variation, of about O.l”, was encountered in the values of S$ in the 280-450 nm region, and two of the eight samples showed small positive values of S$ in this region. These differences, which will be discussed below, are well above the experimentally-determined precision limits of the apparatus and are, therefore, to be considered real. For reasons discussed previously, reliable values of 6R/R can be obtained with the present apparatus only in the 500-700 nm region. Fortunately, this range includes the peak of the KC1 F-band, where 6R/R is at a maximum. In measuring &R/R, we find that a distinct band can be seen in the F-band region if a sufficient number of scans is averaged. However, &R/R is found to be significantly different from zero both above and below the F-band region, in contrast to predictions based on eqs. (l)-(S), (8) an d reasonable trial values of nl and /cr. Fig. 3a shows &R/R obtained in the same run that yielded fig. 2. The anomalous values of 6R/R observed above and below the band are found to be sample-dependent in both sign and magnitude. A probable explanation is that - whereas $ and A are measured in such a way that the effects of small sample imperfections are reduced [ 121 - the absolute reflectivity is sensitive to the physical condition of the surface. This is a fundamental difficulty in combining ellipsometric and reflectometric data [3]. Therefore, in addition to the formation of an absorbing color-center layer, changes in the surface roughness caused by VUV irradiation will affect &R/R. That such changes do, in fact, occur through sputtering has been amply documented through electron microscopy (refs. [27] and [28] and references quoted therein). Hence, in using the reflectivity data, we have subtracted a “baseline” representing the non-zero 6R/R on either side of the band, which we assume to result from a sputtering-induced change in surface roughness. The resulting corrected values of 6R/R are then believed to represent the effect of the adsorbing surface layer. As will be shown below, even this crude measurement of 6R/R is of considerable use in reducing the ellipsometric data. The initial problem in analysis of the data is to obtain a value of the damagelayer thickness, d. Three semi-quantitative arguments can be used to establish an upper limit for d. First, following Aspnes (ref. [ 11, fig. 3.3), we note that for d - X

V.M. Bermudez/

~~V~ud~a~onda~ageofK~s~rfa~e

577

NANOMETERS 700

600

-

+.02-

cc \ Ir m

o-

-.02-

Fig. 3. The fractional change in reflectivity, 6R/R, induced by irradiation. The incident polarization is at 45” with respect to the plane of incidence. In (a), the solid curve is the experimental result, obtained in the same run as for fig. 2, and the dashed line is the “background” caused by rad~tion-induced structural changes in the surface (see text). In (b), the curve is the calculated peak value of fiR/Rversus damage-layer thickness d, computed using (nr, kr) versus assumed value of d,calculated from SJ, and SA. Note scale change at d = 100 a.

an oscillatory behavior in Sll, and 6A would be expected in the region away from the F-band, with 6A displaying maxima of alternating positive and negative sign. This effect results from thin-film interference in the surface layer; the fringe spacing in eV, AI!?, can be estimated by rewriting eq. (4) as [l] AE= 12397.7 eV * 8/[2d(nf

- sin2p,-,)1/2] = 5107 eV - a/d,

(15)

with n, = 1 S, p. = 61.@ and d in a. The fact that no indication of such an interference effect is found in 6 J, and &A in 2-4 eV range suggests that AE > 10 eV, so that d < 500 A.

578

V.M. Bermudez

/ VUV radiation damage of KCl surface

Second, the intrinsic absorption coefficient of KC1 [15] for X < 165 nm is 23 X lo5 cm-‘. Hence, 90% of the VUV intensity is attenuated in a surface layer of G33OA. The argument that the damage-layer thickness is of similar magnitude is based on the assumption that the damage results from WV absorption at energies at or above the 1s exciton. This is supported by measurements of the F-center creationefficiency spectrum in KC1 by Bichevin et al. [29] and by the observation noted above that S$ and 6A are independent of impurity concentration. It should be pointed out that the resulting damage-layer is then expected to be exponentially inhomogeneous, with a complex index of the form

Tl(z) = Z2 + Fi’ e-z’r , where { is a characteristic length, z is the depth beneath the surface and n, and G’ (= n’ - ik’) are implicit functions of wavelength. k’ (and therefore n’) is proportional to the total number of absorbing centers. In analyzing such a system with the formalism appropriate to the three-phase model, we assume that the exponentiallyinhomogeneous tilm can be replaced by an equivalent homogeneous layer. This will be justified in the Appendix, in which the relationship between the homogeneous (n 1, kl, d) and inhomogeneous (n’, k’, c) film parameters will be presented. A third indication of the value of d is obtained by reducing the data in fig. 2 for various assumed thicknesses, using eqs. (6), (7). It is then found that divergence occurs throughout all or most of the spectrum for d > 300 A; the divergence results regardless of the initial choice of ni and k, in the iteration. In other words, the data for some or all X are inconsistent with the three-phase model if a film thickness much in excess of 300 A is assumed. These three observations suggest 300 A as a reasonable upper limit for d. A lower limit can be obtained from the reflectivity data. From SJ, and 6A versus X, nr and k, versus X are obtained for trial values of d S 300 A; from nl and kl, &R/R versus X is computed. Fig. 3b shows the calculated 6R/R,at the F-band peak, versus d. From 30 to 300 A, the calculated 6R/R is nearly constant at tO.0175, versus the observed value of &R/R = +0.023IV0.008.Below 30 A, the computed (r,R/R falls rapidly to zero. These arguments lead to the conclusion that the damage-layer thickness is between 30 and 300 A. It is not possible, with the present apparatus and techniques, to obtain a more precise estimate of this quantity. Fig. 4 shows nr - n2 and k, obtained from the data in fig. 2, under the assumption of d = 75 A.The uncertainty of a factor of four in the value of d results in an uncertainty of a factor of -3-4 in the absolute values of n, - n2 and k,. However, the qualitative appearance of the spectra, particularly the locations of the extrema in n, - n2 and k, and the full-width at half-maximum (FWHM) of kl, are independent (within experimental error) of the choice of d in the 30-300 A range. Hence, some conclusions may be drawn without a more precise value of d. First, we note that the anomalous dispersion in nl - n2 is asymmetric, being displaced toward negative values. This suggests the presence of one or more absorption

V.M. Bermudez

/ VUV radiation damage of KC1 surface

579

NANOMETERS 000

700

I

1.5

600

1

2.0

500

I

I

2.5 3.0 ELECTRON VOLTS

/

3.5

I

4.0

O

Fig. 4. The damage-layer optical constants nl and kl obtained from 6J/, SA and n2 for an assumed film thickness of 75 A. The arrow marks the F-band maximum in transmission spectra of bulk centers.

bands to lower energy, relative to the F-band. The contribution of these bands to nr in the vicinity of the F-band will be negative. The lower-energy absorption is most likely due to an F-center aggregate [30], such as the M-center, which absorbs at -1.5 eV, or the R-center, with bands between 1.5 and 1.9 eV. The possibility of aggregation is further suggested by the peak value of kl, 0.05, which corresponds to an absorption coefficient of OL= 4rrkJh = 1.14 X lo4 cm-‘. This in turn corresponds [31] to an F-center concentration of -3.2 X 1019 cme3, which is accurate to within a factor of four. This is a rather high density of centers, but is comparable to those obtainable in electron bombardment of KC1 [31]. As a check on the selfconsistency of the results for nl and k,, the Kramers-Kronig transform, nl(/j)

= 1 +

s

2p =E’EF;“‘2-y’ n

0

F-band

E’ kl(E’) dE’ El2 _ E2 ’

(17)

was computed, with k,(E) represented by a gaussian. The results are consistent with the observed extrema in nr - n2. A second interesting point is the appearance of the peak in k, at 2.19 f 0.01 eV (56.5 * 2nm); whereas, transmission measurements [32] on X-irradiated or additively colored KC1 samples give an F-band peak at 2.23 + 0.01 eV (555 f 2 nm) at

580

V.M. Bennudez

/ VUV radiation damage of KC1 surface

room temperature. The shift of 0.04 * 0.02 eV is found in all eight of the different sets of data, as is the asymmetry in n1 - n2 disc u sse d ab ove. It is also found that the FWHM of k, is about 0.43 eV versus a value of 0.34 eV for a bulk center [32] at room temperature. There are two possible explanations for the shift and broadening of the F-band. The first is that unresolved aggregate absorption bands are falling on the low-energy side of the F-band. Specifically, the so-called [30] Mz and Mb bands of the M-center occur at 2.27 and 2.30 eV, respectively, at 77 K, with FWIXM of 0.15 and 0.19 eV. The R, and Rz bands of the R-center are found at 1.89 and 1.70 eV, respectively, with FWHM of 0.08 and 0.05 eV. For the F-center, the corresponding quantities at 77 K are [32] 2.30 and 0.19 eV. In view of the high density of F-centers and the asymmetry of n1 - n2 versus h, it seems likely that aggregate absorption bands make some contribution to k,. A significant aggregate contribution might then also account for the small sample-to-sample variations in the magnitudes of 6 $ and 6A noted previously. Aggregation of F-centers can be induced by illumination in the F-band [30], which is not expected to be identical from one experiment to the next. For this reason, care was taken to eliminate light leaks in the vacuum chamber. In one experiment, A versus h (after WV irradiation) was measured before and after turning on the ionization gauge for 20 min. The characteristic F-band dispersion (fig. 2) decreased in amplitude and became more asymmetric t i.e., displaced more toward negative n 1 - n2). Both observations suggest F + M-center conversion induced by the white light emission of the gauge. A second possibility, which will be examined in the following section, is that the shift and broadening of the band is caused by interaction of the F-center with the crystal surface.

6. The 1s + 2p transition

of an F-center near a surface

In this section, we will use the semi-continuum model of the F-center, described by Fowler [33,34], to compute the effect of a nearby (100) surface on the KC1 F-band. The approach employs first- and second-order perturbation theory applied to the variational 1s and 2p wave functions that describe the bulk center. The coordinates are defined in fig. 5; R, is the cavity radius and r, is the distance to the surface along the normal. The potential in the z-direction is also shown, with the zero of energy at the bottom of the conduction band. Inside the well, the unperturbed hamiltonian is given by BIo =p2/2m,

+

V, .

(18)

Outside the well, but inside the crystal, the hamiltonian 81, = p2/2m* - e2/Kr

,

where m* is the electron

is (19)

effective mass and K is an effective dielectric constant.

V.M. Bermudez / VUV radiation damage of KCl surface

581

VACU

CRYS

CONDUCTION BAND ------------

(2p), (We

Fig. 5. Coordinates and potential function used in computing the effect of a nearby ideal surface on an F-center. The unperturbed energy levels, (Is)0 and (Zp),, are shown. The energy scale is appropriate to the parameters in table 1, and the zero of ernergy is at the bottom of the conduction band.

Outside the crystal, we assume the electron to be free, with energy equal to the electron affinity x, where x > 0 means that the bottom of the conduction band lies below the vacuum level. If, in addition, we consider the effect of a small change, ue, in the well depth, Vo, due to the change in the Madelung energy near the surface, we can express the perturbation operator, %‘, as 91’=Uo

inside well ;

SC=0

outside well, but inside crystal ;

%‘=$p*(l/m,-

I/m*)te*/Krtx

outside crystal .

(20)

582

V.M. Bermudez

/ VUV radiation damage of KCl surface

Thus, we assume that the crystal-vacuum interface is infinitely sharp and that the bulk dielectric properties are constant up to the interface. We also neglect contributions to the surface potential arising from surface charge, which may result from photoemission during VW irradiation. 91’ is then defined so that 9(, + 91’ = p2/2m, + x outside the crystal. Recently, several authors 135-391 have computed the electrostatic potential for ion sites at or near crystal surfaces. For the (100) plane of the rocksalt lattice, the Madelung energy on the surface plane is smaller than in the bulk by -4-S%. At one lattice unit beneath the surface, the difference is less than 0.1% (see table 5 of ref. [37]). Hence, we may take u. = 0 for r, L 5 8, and u. equal to 4% of the bulk Madelung energy (and positive in sign) for r, < 5 8. Consistent with the findings of Kassim and Matthew [39], we neglect the effects of net and differential (“rumpling”) surface relaxation on the surface Madelung energy. The dipolar contribution to the surface potential is believed to be small [37], and is therefore neglected, although this term will contribute to the splitting of the 2p level of a surface center [39]. Qualitatively, since the 1s state is more localized, we expect the shifts due to u. > 0 to vary as Ah’,, B AWzp > 0. Outside the crystal, 81’ will produce splitting between the 2p, and 2p, Y levels as well as second-order interaction between s and p levels. Qualitatively, we expect the shifts in energy resulting from this perturbation to vary as Aw(2pZ) 2 AlU( AW(2p,,,) Z 0, based on the spatial distributions of charge density. The unperturbed wave functions are given by [34] (1 t or) e-&r ,

$rS = (03/7n)“*

rl,

2po

=

eePr cos 0 = I)*~, ,

(p5/n)‘4

G2n*r = (/P/2n)‘/*r

eeP’ sin 8 ekiv = 2-‘/*($~~~~ + i $aPY) ,

where (Yand 0 are variational parameters. The desired energy shifts reduce to the following evaluated numerically: Ro

4a31Jo

Al+‘;, = -

7

X (1 - rs/r)

r*( 1 t or)* e-lorr drt$

j

0

j

radial integrals,

(21) which are

r2 e-2&r

Ts

(1 + ar)*(e*/Kr

+ x) - a2A2 -zj- (l/ m, - l/m*) (cu*r* - kr

- 3)

dr , (22)

Ah’;,,

4s5uo = __ j 3 0

X [l - (rs/r)3]

Ro

r4 e-?!fir dr + 9 3

r(e*/Kr+

x)-$(1/m,

s rs

m r3 e-2/.+

- l/m*) @r - 4)

(23)

V.M. Bermudez / WV radiation damage of KCI surface

4P5ue AW&r = __ s 3

0

RO

r4 ee2@ dr t -PS 3 s rs

583

r3 e-2Pr

X {2 + (rS/r)[(rS/r)2 - 37) r(e2/Kr + x) - $

(l/m, - l/m*) (@ - 4) dr , (24)

and

-Ah’:, = +AW&,, = l(Gr,l 9f’I~2po)l*/Ws

+ 2P) 5

where

X [ 1 - @Jr)*]

r(e*/Kr + x) - ~T2 (l/m,-

l/m*>(/3r-4)

dr, 1

(25)

and FV(Ts+ 2p) is the bulk F-band energy. The superscripts 1 and 2 refer to firstand second-order. The terms in (r,fr) arise from integration over 0 outside the crystal, which is from 0 to cos-‘(r,/r). The variational parameters QIand 0 are obtained by minimizing the calculated energies of the Is and 2p levels in the bulk, using the analytic expressions of Fowler [34]. The parameters appearing in the Ham~tonian are given in tabel 1, together Table I Parameters and calculated results for bulk KC1 F-center in the semi-continuum model Effective dielectric constant Effective electron mass Electron affinity Cavity radius Well depth

K = 2.13 a m* = 0.6 me = 0.6 au b x = 0.5 eV = 0.018 au c Ro = 2.28 A = 4.30 au -V,= 5.85 eV = 0.215 au

Variational constants (au)-i

CY = 0.546

Ground-state energy relative to vacuum, -Wr,(eV) F-band energy, W(1s -+ 2p) (eV) F-band oscillator strength f(ls + 2~)

fl = 0.178

Calculated

Observed

3.32 2.33 0.11

2.8 f 0.1 d 2.23 e 0.85 f

a Equal to the KC1 optical dielectric constant, Ko, as in ref. 1341. b Same value as for NaCl, ref. [ 341. c Poole et al. [40]. d Photoemission results of Kashkai et al. 1411 (room temp.). e Ref. 1321 (room temp.). f Ref. [ 331,Appendix B.

584

V.M. Bermudez / VUVradiation

damage of KC2 surface

with a comparison of the observed and calculated bulk quantities. Ve was obtained from the assumed value of R, via the approximate expression [34] v, = -(Y&r + (K, - 1)/2K&

+x )

(26)

where (YM= 1.74756 and a = 3.14 a define the Madelung energy and Ke is the optical dielectric constant. Ro is obtained by trial and error to give a good tit to the observed quantities, and it is seen that a reasonably good description of the bulk center is obtained. The results of the perturbation calculation are summarized in fig. 6. We have taken rs to be a continuous variable, since, in practice, the presence of surface microstructure means that r, will not be confined to discrete values. Bearing in mind that the full-width-at-half maximum of the F-band at room temperature is -0.34 eV, we see that a center that is only one unit cell away from the surface is indistinguishable from a bulk center in this experiment. This conclusion is in accord with the results of a point-ion calculation by Smart and Jennings [42] for near-surface F-centers in NaF and NaCl. As the distance to the surface approaches the well radius, a small splitting is predicted (-0.04 eV), with a net shift of the F-band to lower energy (by -0.25 eV). The estimated band shift is very sensitive to the exact value of uo, as shown in fig. 6. As expected on the basis of qualitative arguments, the splitting arises from the truncation of the semi-continuum potential at the surface and the shift mainly from the small decrease in the Madelung well-depth. Even for the extreme case of rs I?,, the convergence of the perturbation calculation is reasonably good, with the second-order term being smaller than the corresponding first-order terms by a fac-

BULK r,>10a.u

rs = 10 v,=o

rs =5 v. = 0

r* = 5 0.“. v. q+0.012a.u

Fig. 6. F-center 1s and 2p levels as a function of distance from an ideal surface, computed by the perturbation technique described in the text. rs is the distance from the surface in atomic units (1 au = 0.5292 A), and u. is the shift on the Madelung well-depth in atomic units (1 au = 27.21 eV). The calculated transition energies are in electron volts. The energy scale on which the shifts of either state are displayed is indicated to the left.

VA

Bermudez

/ VW

radiation damage of KCl surface

585

tor of eight or more. However, the calculation neglects the higherenergy ns and np states (n > 2), the latter of which form the K-band appearing -0.4 eV above the F-band [33]. The omitted second-order interactions will be significant for r, -R, and will probably have the effect of further reducing the value of W(ls + 2p) near the surface. A limited investigation was carried out of the dependence of the calculation on the values assumed for the parameters in Bl,. A change of + 100% in x had a negligible effect. Repeating the entire computation for m* = 0.9, instead of 0.6, leads to a more localized 2p state [34] ( i.e., 0 = 0.306 instead of fl= 0.178). AWzp induced by u. > 0 is, therefore, larger but is still less than the corresponding AW,,. Hence, the shift in the F-band energy is ---0.1 eV instead of --0.25 eV. The calculated splitting is somewhat larger for m * = 0.9, being -0.10 eV instead of 0.04 eV. Point-ion and finite-ion calculations for centers directly on the surface (r, = 0) have been carried out for the F-center in NaCl by Smart and Jennings [42] and by Kassim et al. [43]. The results are consistent with the trend shown in fig. 6, although AWzpo and AW,, are, of course, larger. The object of the preceding discussion is to show that perturbation of the F-center by the crystal surface can be expected to produce a net shift of the F-band to lower energy and to introduce a splitting. It is useful to estimate what fraction of the F-centers would have to interact with the surface in order to account for the observed shift of 0.04 f 0.02 eV. The center of gravity (or first moment) of the absorption band is given by (27) where the integral is over the whole band. If u is the fraction of “surface centers” with absorption spectrum klQsurf, the remainder being “bulk centers” characterized by k,(E),,,k, the observed spectrum is given by kl(‘?obs

= akl@%urf

+ (1 -

#l@%ulk

3

so that

(28)

The experiment gives 6E - - 0.04 eV and the perturbation calculation Esurf -0.2 eV, so that u - 20%. If we assume a damage-layer thickness of d 100 A and an exponentially-decreasing F-center concentration proportional to e-2r’d (see Appendix), we find that 20% of the centers are located within -10 A of the surface. The above estimate, while too rough to establish unambiguously a surface interaction, does show that such an effect is physically reasonable.

Ebul k -

586

V.M. Bermudez

/ VW

radiation damage of KC7 surface

7. Summary A combination of spectroscopic ellipsometry and reflectometry has been used to study the surface damage-layer produced when a KC1 cleavage face in UHV is exposed to VW radiation. The changes in the ellipsometric parameters J/ and A are readily detected and can be reduced to yield the optical constants of the damage layer, using a Newton’s method interative solution of the Fresnel equations for a three-phase (ambient-film-substrate) model of the system. This requires that the damage-layer thickness be known via some independent measurement. We have measured &R/R, the fractional change in reflectivity upon irradiation. Since the precision in &R/R is not as good as in S$ and &A, a full three-parameter iterative solution of the Fresnel equations is not feasible. Furthermore, it appears that VW irradiation may cause changes in the surface texture, which affect the reflectivity independently of the damage-layer formation. Another difficulty encountered was the insensitivity of the computed value of &R/R, to the value of d assumed in reducing 6 rl, and ISA, for some values of d. Nevertheless, it is possible to establish a lower limit of 30 A for d, on the basis of the observed and calculated &R/R at the F-band peak. An upper limit of 300 A is determined through semi-quantitative arguments based on: (1) the absence of interference fringes in the data, (2) the high fundamental absorption coefficient of KC1 and (3) the failure of the Newton’s method iteration to coverge for d > 300 A. A value of d = 75 A was used in reducing the 6 $ and 6A data, and the resulting mag nitudes of n, - nz and kI are uncertain by a factor of -3-4. However, the qualitative appearance of the spectra of n, - n2 and k, are found to be essentially independent of the assumed value of d, as well as of the sample purity, total irradiation time and prior bakeout of the sample. A fundamental assumption of the experiment is that the exponentially inhomogeneous damage-layer arising from optical absorption in the VUV can be represented as a equivalent homogeneous film. This has been justified, for the present problem, in the Appendix. The peak value of k, corresponds to a peak F-band absorption coefficient of -1 X lo4 cm-‘, which implies a density of centers of -3 X 1019 crnw3. This result suggests the possibility of aggregation of the F-centers. The displacement of the anomalous dispersion curve, n 1 - n2, toward negative values further implies the presence of aggregate-center absorption in the near infrared. The kl band is found to be shifted slightly (0.04 + 0.02 eV) to lower energy and to be slightly broader (FWHM of 0.43 versus 0.34 eV) relative to the F-band observed by transmission in additively colored or X-irradiated KC1 at room temperature. Two contributions to the shift and broadening of the k, band are considered, both of which may be important. The first involves the possible presence of unresolved aggregate-center absorption bands on the low-energy side of the F-band. The second involves interaction of a fraction of the centers with the surface. To show that such an interaction could, in fact, account for the observed small shift and broadening of the F-band, a perturbation calculation was carried out within the

V.M. Bermudez

/ VW

radiation damage of KCl surface

587

framework of the semi-continuum model. Although the representation of the surface potential is highly simplified, the description of the bulk (unperturbed) center is reasonably accurate, and the calculated effect of bringing the center close to the surface is consistent with independent and more detailed calculations [42,43]. It should be noted that the slow-electron absorption experiments of Fredericks and Cook [44] have indicated the presence of a thin layer of F-centers on the vacuum-cleaved KC1 surface, even in the absence of prior irradiation. The present results are not in agreement with this observation; however, cleavage in this case was in air, not in vacuum. From the precision limits stated above, it is estimated that such a preexisting layer would have to be well under -1.5 A thick to escape detection. Several improvements in the present experiment are required in order to obtain definitive results. First, carrying out the spectroscopic measurements at 77 K or below would sharpen the features considerably and would reduce aggregation of the F-centers induced by irradiation in the F-band. Second, isolation of the 160.8 nm hydrogen line would permit irradiation at the peak of the 1s exciton, where the absorption coefficient [IS] is 1.3 X lo6 cm-‘. The anticipated -4-fold reduction in damage-layer thickness might render surface effects more apparent. Third, extending the spectroscopic measurements down to 1000 nm would permit observation of both F- and F-aggregate absorptions. Also, the photoemission quantum efficiency of thin KC1 films on a conducting substrate is known [45] to approach 20% at the LiF cut-off. It is not known what effect, if any, the resulting positive surface charge may have on the structure of near-surface F-centers. It might therefore be desirable to discharge the sample, after VW irradiation, using a low-energy electron flood gun. Another useful experiment would be the observation of the effects on the spectrum of an in situ chemical treatment [42] to destroy selectively the surface color centers, such as exposure to gaseous NO, a good electron acceptor. However, as discussed by Smart [42], nitric oxide may also destroy centers not on the immediate surface, which would complicate interpretation of the results. Since it is evident that reflectivity is of only limited use in determining the damage-layer thickness, some other technique must be employed to obtain a third independent datum. Of the possible methods mentioned previously, it appears that in principle AIDER [8] would be the most promising. It is worth considering briefly whether MAIE would be useful in this particular problem. As discussed by Ibrahim and Bashara [7] it is necessary that the unknowns, nr, k, and d not be correlated if they are to be determined uniquely from a series of MAIE measurements. Correlation between unknowns x and y occurs if the following two conditions are satisfied at all angles of incidence ++,: (1)

(agax)sx

< -0.1’

(a rllar)sy

-c -0. 1’ for 6y/y > -0.3 ;

for

6x/x

>

aalax

(2) -e

waY

constant,

independent

-0.3

,

of cpo.

kl = 0.048, “2 = 1.499, k2 = 0.002. d = 75 A in all calculations.

isin Angstroms. For h= 350.2 nm;nl

______~ = 1.512, kl = 0.009;q

.~~__~ = 1.517, k2 = 0.002.

-2424

-5.625

-2.698

-4.914

-2187 -2337 -2385 -2410

-4.217 -5.221 -5.525 -5.622

-0.8736

66

x 1O-3

-2.789 -2.888 -2.872 -2.804

-5.351 -5.372 -5.287 -5.132

-1.269 -1.029 -0.9570 -0.9127

58 60 62 64

-1.038

-5.713

-5.509

-8.129

-1.423

66

-1.019 -1.028 -1.034

-5.217 -5.476 -5.620

-5.817 -5.815 -5.703

-8.847 -8.716 -8.472

-‘1.696 -1.591 -1.508

60 62 64

-0.9906

x lOA

-4.445

-5.542

-8.826

-1.986

58

a $ and A are indegreesandd

550.4

350.2

Table 2 Parameter correlation test for MAIE on KC1 substrate/damage-layer a

For h = 550.4

x lo4

nm: nl = 1.481,

518.7 447.6 431.8 428.5 430.9

1954 1887 1840 1817

2229

$

k?

3

B

5

B 3

3 a 9 g 0’

i; .

4 R

V.M. Bemudez

/ VW

radiation damage of KC1 surface

589

Table 2 gives the quantities necessary to apply the correlation test to the present data. The angles chosen cover the range available at a single exit port of the UHV chamber. Two points are considered, one well above the F-band and one near the peak. Although n,, ki and d all pass the second test, only k, and d pass the first and are, therefore, correlated. Hence, it appears that an independent estimate of d would be required, even in MAIE. Similar conclusions are reached if a larger variation in cpo(46”) 62” and 77’) is considered.

Acknowledgements Several of my colleagues helped in carrying out this research. V.H. Ritz designed the UHV system and aided in the design of the heatable sample stage. R.J. Ginther annealed the numerous samples used, and P.H. Klein provided the high-purity KCl. M.E. Gingerich constructed the analog computation circuit used in the reflectivity measurements. I am grateful to V.H. Ritz, I. Schneider and R.T. Williams for providing equipment and advice and to M.N. Kabler for suggesting this problem and for a critical reading of the manuscript. I also thank Professor O.S. Heavens for a copy of ref. [31] prior to publication and Professor J.A.D. Matthew for a preprint of ref. [43].

Appendix The purpose of this Appendix is to consider the validity of using the formalism appropriate to the three-phase model to analyze data obtained for VUV-irradiated KCl. The subject of reflection by inhomogeneous films has been reviewed by Jacobsson [46]. The particular example of interest here is an exponentially-inhomogeneous film of index nr(z) given by

Z,(z) = Z2 + 2 eezis ,

(A-1)

where zT, is the substrate index and E” (= n’ - ik’) and f are characteristic parameters of the film. 2 is the depth into the film, with z = 0 defining the ambient-film interface. Abel&s [47] has presented analytic expressions for the s- and p-polarized reflection and transmission coefficients for a non-absorbing film, with an index given by n(z) = nOeLI’,on a transparent substrate. Numerical computation of J/ and A using the resulting expressions for R, and R, shows that, in general, a thick exponentially&homogeneous film cannot be represented by an equivalent homogeneous film. This is seen [47] in the plots of ($, A) versus film thickness, which do not resemble those for a homogeneous fdm. However, McCrackin and Colson [48] have carried out a numerical study of thin non-absorbing films with linear, exponential or gaussian inhomogeneity. $ and A are computed by replacing the inhomogeneous film with a large number of very

590

V.M. Bermudez

/ VUV radiation damage of KCl surface

thin homogeneous films. Ic/ and A thus obtained are used to solve eqs. (l)-(S), appropriate to the three-phase system, giving (nr) and (d), which are the index and thickness of the equivalent homogeneous film. It is then found empirically that (nr) and (d) are related to the parameters of the inhomogeneous film via

s

ndz)

(n1)=-

s

[n1(z> - n21dz

bl@-n21

0

J

(d)=~ h(z)

-

d.z

(A.3

(nl) - n2 -’

~1 dz

0

With nr(z) = n2 + n’e-z/s, this reduces to

(nr ) = n2 + n’/2 ,

C&=25.

(A.3)

The computation of McCrackin and Colson was carried out for { = 100 A and n’/na = 0.231 or n’/n2 = 0.035. According to eq. (A.3), the quantity (nr) - n2, plotted in fig. 4, is one-half the difference between the film index at the surface and the substrate index. The film thickness represents the depth at which the difference between film and substrate indicies has diminished to l/e2 (= 13.5%) of the value at the surface. To investigate this point further, we follow the thin inhomogeneous film approximation described by Jacobsson [46]. The reflection coefficient of the film-covered substrate is approximated as (A.4) where k = 21r/h < 1. If we keep only terms first-order in k, we find that

64.5) where R is the reflection coefficient of the clean substrate. hand, the terms in eq. (AS) may be defined by [46]

For the problem

at

s-polarization Pa= (1 - sin2po)-1/2 d =j

,

0, = (g - sin2qo)-1/2

[&(z) - sin2qo] dz ,

0

,

,=p,=,;

(‘4.6)

0

ppolarization 0, = (1 - sin2po)-1/2

,

& = F&3 - sin2&-1/2

,

d sc=j[l0

sin2qo/Yl(z)]

dz ,

93 =

s 0

F~tz> dz ;

(A-7)

V.M. Bermudez / VW radiation damage of KCl surface

where F = (;i)*. For a homogeneous to

R,

=

= Fr, and eqs. (AS)-(A.7)

(g - G;>[gg - <&+Fl’,> sin* 901

R

P

film, or

P

rr(F2 - 1)[F2 - (F2 + 1) sin2qo]

.

591

reduce

(A4

These can be shown to be identical to the expressions derived by McIntyre and Aspnes [ref. [49], eqs. (19) and (21)], with the appropriate changes in notation and an ambient vacuum. For the inhomogeneous film specified by eq. (l), we have Fr(.a) = (g2)’ t 2z2T

e-z/r t (;;1)* e-2z/r e-z/r ,

z F2 t 2(F2F)l/*

(A.9)

where the last term is dropped because we are assuming 11’Q n2 and k’ < 1, consistent with fig. 4. Eqs. (A.6), (A.7) then yield s-polarization PQ= Q(g

- sin2qo) + 2{(&Z+)1’2 ,

9 = Cg ;

(A.lO)

p-polarization 54 = Q(1 - sin2po&)

t {(sin2~e/~2) log,[ 1 + 2(?&)“*]

,

(A.ll)

.

cx3 = :*2’2, + 2&?)“2

Here CDis a dummy variable, representing the upper limit in the integrations over z, such that e -cg/t - 0. (2, cancels in the final expressions for R6 and Rb. Combining eqs. (AS)-(A.7) (A.lO) and (A.1 l), we obtain R’

=R

s

s

+~~~YSY'O

(2~~‘f’~“)].

(A.12)

To evaluate Rb, we expand the second term in 94 in eq. (A.1 1) to obtain

0 1~2~~*%[(~)‘I’_~], 7

sin2po

112

2z-

CEz

(A.13)

E2

which gives 4n cos cpe x x

2f(&F’)“*

(

g2 - 2 sin2qo + (?/G)“’

G2 - 1) [F2 - (F2 + I) sin2po]

sin2qo

)I-

(A. 14)

V.M. Bermudez

592

/ VW

radiation damage of KC7 surface

We then note that eqs. (A.12) and (A.14) can be obtained from eq. (A.8) if the substitutions d = 2{ and Fr = rZ + (FZ?)1’2 are made in eq. (A.8). For the case at hand, k2 = 0 and ‘?Z is real. We then obtain Re(Fr) = IZ: - k! = rzz + n2n’ ,

Im(FI) = 2nr k, = n,k’

,

(A. 15)

which yield, finally, nl =

n2 + n’/2 ,

2k, = k’(1 - n’/2n,)

= k’ ,

d = 2[ ,

(A.16)

correct to first order in rz’/n2. Eq. (A.16) expresses the relationship between the parameters of the actual inhomogeneous film and those of the “equivalent homogeneous film”. The derivation of eq. (A.16) is based on the first-order approximation for thin films, which is expected to be valid for thickness of a few tens of Angstroms, at most [49]. However, the numerical treatment of McCrackin and Colson [48] suggests that eq. (A.16) itself may be valid for films of r = 100 A and for large, as well as small, values of n’/n,. A second topic to consider is whether the whole damage-layer can be represented by a single layer, inhomogeneous or otherwise. To be specific, we have argued that F-centers within 5 A of the surface have optical properties somewhat different from those lying farther away. Hence, it is necessary to consider a -5 a thick film on the surface of the -100 A thick damage-layer. The 5 A thick film has an index %r characteristic of surface-perturbed F-centers, while the thicker damagelayer is characterized by eq. (A.l), with g’ representative of the bulk F-center. We treat this question qualitatively by invoking arguments related to those of Meyer et al. [50]. Namely the contributions to S$ and 6A made by a surface film this thin will simply add to that of the thicker damage-layer, so that

where “f” and “dl” refer to “film” and “damage layer”. If in addition we assume that S$f (&A,) is small compared to 6$dl @Ad,), then nl and kl derived from will be superpositions of the contributions from the film and the 6tiobs and h&b, damage-layer.

Note added in proof A few other studies of surface F-centers have been published recently. Fryburg and Lad [51] report the EPR spectrum and Fassler and Graness [52] the diffuse reflectance spectrum of electron-irradiated LiF and NaCl powders, respectively. In both cases, the results differ very slightly, if at all, from those for bulk F-centers. Lord and Gallon [53] tentatively identify bands at 4.59 and 5.64 eV in the transmission spectrum of LiF, after 900 eV electron-irradiation in UHV at 540 K, with the Is -+ 2p,, and 1s + 2p, transitions of surface F-centers.

V.M. Bennudez

/ VW

radiation damage of KCl surface

593

References [I] D.E. Aspnes, in: Optical Properties of Solids: New Developments,

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