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Highly sensitive investigation of initial growth process of sodium films by multiple-attenuated-total-reflection spectroscopy
Surface Science 78 (1978) 295-306 0 North-Holland Publishing Company
HIGHLY SENSITIVE ~VESTIGATION OF INITIAL GROWTH PROCESS OF SODI~ FILMS BY MULTIPLE-A~ENUATED-TOTAL-REFLECTION SPECTROSCOPY
T. YAMAGUCHI and H. TAKAHASHI Research Institute of Electronics, Shizuoka University,Hamamatsu 432, Japan Received 25 April 1978; manuscript received in final form 9 June 1978
Multiple-attenuated-total-reflection spectroscopy (MATRS) is applied to highly sensitive measurement of small amount of metal adsorbed on a fused quartz substrate. A sodium film thinner than 0.01 A in mean thickness or 1O-1o g/cm2 in weight could be detected. MATRS brings information also about shape and electronic properties of small island particles. Sodium islands were predicted to have cap-shape and the number of free electrons per unit volume in sodium islands was found to decrease from the bulk value as the mean thickness decreases to less than 1 A.
1. Introduction
In the experimental study of the initial growth process of a film formation, the most important thing is how small an amount of deposit we can detect sensitively. The sensitivity of the torsion balance of the top class [l] has been of the order of lo-’ g. A higher sensitivity is desirable for the investigation of the initial nucleation process of fdm growth. Holm and Palik [Z] showed recently that the multiple-attenuated-tots-resection spectroscopy (MATRS) ia s useful technique for the sensitive measurement of the low absorption coefficient of thin films. In the present paper, we apply the technique to a metal island film in the initial growth process, and show that the sensitivity becomes 0.01 A in mean thickness or 10-“” g/cm2 in weight for the case of sodium. MATRS in this paper uses the phenomenon of optical plasma-resonance absorption in small metal particles, so that material which is applicable to this technique will be restricted to a few kinds of metal, such as alkali metals and silver. Very thin films of these metals show plasma-resonance absorption peaks in some spectral region. The shape and the position of the resonance peak include information about size, shape and electronic properties of the small islands of which the fdm consists. The principle of MATRS and an example of measurement and analysis with sodium are given below. 295
296
T. Yamaguchi, H. Takahashi / Sensitive investigation of initial growth
2. Principle of MATRS In a previous paper [3] we derived equations for ATR of a metal. island film by integrating scattered waves. The reflectances for s- and p-polarized light are approximated as
2y1
R,
=
1-
2kodw
__...s(z;“x;; t&&x;), zT
+
(1)
z’,‘2
where d,
is the mean thickness of the deposit, ke = 2n/X, Xi = sin Bi/nj, Yj = and Zi = cos 8jl~j, nj and Bj being, respectively, index of and angle of refraction in the medium of substrate (j = 1) or vacuum Q = 3). The double prime expresses the imaginary part. x11,1is the effective polarizability of the island particles and includes all information about a metal island film. The subscripts II and 1 mean, respectively, the component for electric field of light parallel to and perpendicular to the substrate surfaces. We consider the case where kodW is so small that eq. (1) is approximated well by an exponential term. Then after N times total reflections the transmitted intensity through the substrate is given by cos t9j * nj
(2)
L,p = Io~,~ exd-~4,pdwlV , where 1, is the intensity A,=
[8nY,/‘(Y;
without metal deposit, and
+ Y;“)] x;; ,
A, = [Srr.Z,/(Z~ + Z;*)] (2;‘~;; Putting the logarithm
+ X,X,x;I)
.
(3)
of eq. (2) equal to --I&,
B SIP = loa 0 (~o~,p/~~,p~= ~~~,p(logl
od G/h
-
we have (4)
This will be measured in the experiments.
3. Experiments Fig. 1 shows a block diagram of the ex~~ent~ system. The substrate used was a fused quartz plate of 0.15 mm* X 30 mm’, whose edges were cut and polished at 45’. The substrate was put on a copper block of 10 X 30 X 50 mm3 in which a W-wire heater was embedded. The copper block was attached to a cold fmger mounted in a UHV system (V143N, NEVA). The whole substrate system was enclosed by a liquid-nitrogen temperature shroud that had windows for light beams and a sodium-atom flux.
T. Y4~4~ch~, UHV
H. T4k4h4sh~ / Sensitive investig4fion of initialgrowth
297
Chamber (V143N.
NEVA) Quartz Substrate (015mm’ x 30mm’): x 5mm’)
long-path Ishort-path
Fig. 1. Block diagram of the experimental system.
of the copper block was measured by a c-c thermocouple and was assumed to be the same as the substrate temperature. A light beam from a 75 W Xe short-arc was introduced into the substrate through one of edges. The light beam traveled in the substrate by repeating total reflection about 80 times in this experiment to reach the opposite edge. The beam that came out from the edge was introduced into a double-monochrometer (H-lOD, Jovin Yvon) and was detected by a photomultiplier. The output signal of the photomultiplier was fed into a logarithmic amplifier and recorded by an X-Y recorder as a function of the wavelength of light. The polarization of the incident light was selected by rotating a quartz-crystal Rochon-prism mounted near the light source. No lens was inserted in the optical path. Sodium of 99.9% purity was cut into a cube of about (5 mm)’ in oil, rinced by trichrol ethylene, and dried by a rotary pump; and the oxide layer was cut off. Immediately the sodium cube was fed into a conical basket mounted at 126 mm apart from the substrate, and the vacuum chamber was evacuated as soon as possible. The vacuum pressure was ultimately 5 X iOe9 Torr and was maintained less than 1 X 10m8 Torr during deposition. In addition to depositing on the whole substrate length of 30 mm, deposition on a limited length of 5 mm was made by masking to lower the sensitivity. The former and the latter are called, respectively, long-path and short-path hereafter. A sensing head of quartz-crystal thickness monitor (EVM-32, NEVA) was set at a near distance of 40 mm from the evaporation source to monitor the incident flux with a one order higher sensitivity. Temperature
4. Results Figs. 2 and 3 show the observed B S,p spectra of the sodium films deposited at a substrate temperature of 77 K with an incident rate of 0.9 &‘min. The mean thick-
T. Yamaguchi, H. Takahashi / Sensitive investigation of initial growth
298
0.6
0.5
0.4
A (urn)
0.6
0.5
0.4
3
A(rrm)
(
3
4
hw(eV)
Fig. 2. f& and BP spectra measured by long-path. The values of d, (in A) are: (a) 1.05, (b) 0.75, (c) 0.57, td) 0.39, fe) 0.26, (f) 0.13.
ness d, corresponding to each curve was obtained by analysis as described below. The four curves in fig. 4 show &,,k) at 80 = 2.6 eV versus the incident thickness din (quartz monitor output) relations for long-path, which were directly
06
2
0.5
0.4
A(wm)
3 ‘lw(eV)
4
I
0.6
0.6
2
Fig. 3. Bs and BP spectra measured by short-path. The values of d, 3.79, (c) 2.91, (d) 2.10, (e) 1.41, ffj 0.72.
0.4
3
A(clm) 0
fw(eV)
4
(in A) are: (a) 4.78, (b)
T. Yamaguchi, H. Takahashi/Sensitive investigationof initialgrowth
0
2
1
4
299
5
Fig. 4. Bs(peak) versus din relations measured with four incident rates. Solid circles are Bs(peak) of fig. 2 plotted against dW taken on the horizontal axis. d, corresponding to &(peak) by the broken line is scaled on the right-hand side.
recorded using a X-Y recorder during deposition. Fiiw= 2.6 eV is the most probable value that gives the peak position of B, in several experiments.
5. Analysis
Multiplying
eq. (4) by X, we have
= CI x;;dw ,
W
B,h = C,x;;d,
+ Gx;Idw
(9
The factors C1, CZ and Cs include Xi, Yi, Zi and N, and are known quantities, although they depend on X because of the dispersion of nl. The unknown parameters are x,, ;’ and d,. Here, x,,,~ is given by [3] XII ,I
-
1 -~--nf t ns FII ,* + (Ei -v 2n:
(6)
Fll and FL are the effective depolarizing given by
y2 FII =~II - --
n: - n;
factors of the island particles and they are
F*=f,_y2+$
24~~ nt t ni ’
The first term is the real depolarizing
i27j3 nl
+n3
(7)
factor of the particles. As the shape of the
300
T. Yamaguchi, H. Takahashi /Sensitive
investigation
of initial
growth
island particles is assumed to be isotropic in the film plane (e.g., rotational ellipsoid or cap-shaped), the following relation holds for 41 and fi,
The second term of eq. (7), arises from the mirror image effect due to the substrate surface [4]. y is the axial ratio of the rotational ellipsoid which is optically equivalent to the island particles, and n is the height of the actual point dipole moment of a real island from the substrate surface normalized by the height of the center of the rotational ellipsoid. The term arising from the dipole-dipole interactuons between island particles [S] is omitted in eq. (7) because its contribution is small for very thin films as are considered in this paper. In eq. (6), 6 is the dielectric constant of sodium within the particles and it can be approximated well by the free-electron model: fi = 1 - W$(02 - iww,) ,
(9)
where or, and w7 are, respectively, the plasma and relaxation frequencies. Analysis of figs. 2 and 3, was made by a curve Gtting method using a computer as follows. First we obtained curves of x1,”d, and a”dw from the measured spectra of B,,, by the use of eq. (5), i.e., x’;dw = B,h/CI ,
x;Id, = (4$,X - B&XC1
GQ
YC, .
Next we calculated the theoretical curves of xi,,; by substituting the appropriate
0
2.0
*
3.0
hw(eV)
* 4.0
*.
2.0
3.0
4.0
%I (eV)
Fig. 5. An example of curve fitting for curve c of fig. 2. Solid circles are sampled points from the observed spectra and full lines are fitted theoretical curves.
T. Yamaguchi, H. Takahashi / Sensitive investigation ofinitial growth
301
values of F/Q, or, and w, in eqs. (6) and (9). Then we normalized both the experimental and the theoretical curves so as to make the peak height unity and compared them to each other to find the values of FI~,I, on and w, that make the best fitting between both the normalized curves by the least-square method. It was found that excellent fitting was realized. An example of the fitting is shown in fig. 5 for the third curve (c) of fig. 2. The solid circles are sampled points from the observed curves and the full lines represent the fitted theoretical curves. By comparing the normalization factors of the observed and theoretical curves, we can determine the values of d,, which are plotted in fig. 4 against &(p&) for the case of long-path. As is seen in the figure, those points are well approximated by a straight line. Therefore, &peak) is understood to be a good measure of & in this thickness region. d, corresponding to &(p&) is scaled on the right side of the figure. The
0.3
. oooc’ 0
l
0.2 : 00
000
O 0
l
l
.
‘
l
.
.
*
l
Fl F,,
*
:
~
O.ll (a)
1.6 1 .4
0 0 000 !
-d 0
l l
l
l
1
0
Cc)
*
2
l
4
d,
5
(Al3
Fig. 6. FII and FL (a), 7 (b) and q (c) versus dw relations obtained by long-path (open circle) and by short-path (solid circle). h is the height of the actual point-dipole moment of a real island particle.
302
T. Yamaguchi, H. Takahashi /Sensitive investigation of initial growth
figure tells us that we can detect & of 0.01 A which corresponds to 10T’Og/cm2 for sodium. The results of Fll and FL are plotted against d, in fig. 6a. The open and solid circles are determined from, respectively, long-path and short-path data. From eqs. (7) and (8), the fo~ow~g relations hold for F, and FL 2Fll-FL=2fw-fi=4fu-1,
(11) Thus fil is obtained from the determined values of Fll and FL using the first equation of (11). The axial ratio y of the rotational eliipsoid corresponding tofil is plotted in fig. 6b. By the use of the second equation of (1 l}, rl is determined and plotted in fig. 6c. The resulting values for fiw, are given in fig. 7a. They are much larger than the bulk value of 0.02 eV. If they are attributed to the classical size effect of a spherical
’
(b)’
’
2 d,
3
4
5
(A>
Fig. 7. llwz (a) and r (b) versus d, relations. r was determined by the ciassical size effect; w7 = vF/r with v~ = 10’ 5 A/s.
T. Ya~a~chi, If. Ta~ha~h~ f Sensitive tnve~tigatio~ ofinitiaf growth
303
5.9
_________ bulk l 0g l
5.8
hap (eV) l
l
l
0
o*
57 0 0
5.6
O (a)
I
I
1
I
’
2
3
4
7.00
5
I
0.9
0.9
0.9
’
(b)
’
2 d,
4
5
Cli, 3
Fig. 8. Bw, (a) and N (b) versus c&,, relations. N (=o@&bul@ trons per atom, the bulk value being assumed to be unity.
is the number of free elec-
particle:
where uF (=lO” &see) is the Fermi velocity of sodium and r is the readius of the spherical particle, then r is determined and plotted in fig. 7b. The results of hw, are given in fig. 8a. It can be seen that fiw, decreases from the bulk value when dw decreases to less than 1 8. If they are attributed to the number of free electrons per stom, it also decreases with decrease of thickness as shown in fig. 8b.
6. Discusions
The sensitivity for detecting adsorbed sodium in this experiment exceeds that of the usual torsion balance by more than two orders. Further, enhancement of one order can be made with relative ease, by thinning the substrate plate and taking the path length in the substrate longer. The validity of the obtained high sensitivity
304
T. Ymmg~chi,
H. Takahash~ / Sensitive i~ves~~g~~ion of initialgrowth
can be confirmed by fig. 4. If the sensitivity is low, i.e., the slope of the broken line is small, then the observed growth curve with the highest incident rate of 7.2 A/min will intersect the broken line. Since the broken line is considered to be a growth curve with sticking coefficient of unity, the intersection should not occur. It must be noted that the weight sensitivity of lo-” g/cm’ can apply only to alkali metals. The sensitivity for silver will be by one order lower because of its high density. It can be seen in fig. 4 that the sticking coefficient of sodium on a fused quartz substrate is still very small at the low temperature of 77 K in this small-thickness range. It was also found in another experiment that sodium films are not stable and re-evaporate completely in a few hours even at 77 K. As the re-evaporation was accelerated by a slight increase of the substrate temperature, a number of depositions could be made in a short time without changing the substrate. Although the decay of the optical absorption can occur by the film oxidation, the rate of the decay did not change by increasing the residual gas pressure by more than one order. It is understood from fig. 6b that island particles are optically equivalent to the rotational ellipsoid with axial ratio of 1.4-l S. Since n is less than unity as shown in fig. 6c, the particle shape may be rather cap-shape like. Decrease of 77 with decrease of & may be attributed to the decrease of contact angle of the cap-shaped island. The discontinuity of hw, in fig. 7a between long-path and short-path may be due to a small difference in the condition of deposition. It seems curious that the change in particle size with d, is small, as is seen in fig. 7b. Other causes might exist for the reason of the very large value of AU,. So far the statistical distribution in fil [6,7] or in r [8] has been taken into account for the explanation of the broad optical absorption peak metal island films much thicker than the present films. However, the fitting between observed and theoretical curves in fig. 5 seems too good to take the statistics distribution of particle shapes into consideration, because the theory used is based on a single particle shape. Possibly, the theory of the classical size effect, that is based on the limit of the mean free path of electrons by the particle surface, can not be applied to such extremely small particles. No matter what is the reason, a small change in fiiw, with d,., is a convenient thing for the purpose of using it as a detector of deposit, because the linear relation between &(p&) and d, in fig. 4 is due to this fact. The decrease of the number of free electrons per atom with an decrease of dw as shown in fig 8b seems reasonable, for the fraction of surface atoms increases with decrease of particle size. It is interesting that we can detect the change in the electronic properties of a small metal particle. In the case of silver, we must take into account the contribution not only from the free electrons but also from the bound electrons. We found that the contribution from the bound electrons must also differ from the bulk value for the explanation of the results of MATRS.
T. Yamaguchi, H. Takahashi/Sensitive investigationof initialgrowth
30.5
Short path --__ _ __
0 0
0
0
1
0
0
o
2
O3
4
5
d,(A)
Fig. 9. X~(peak)~X~i(peak) versus C& relations for spectra observed (crosses), calculated with the fictitious plane-panel f&n model (solid circle) and calculated by integration method (open circles).
In the previous paper [3] we obtained the ATR equation using a fictitious planeparallel ftim model as well as integrating waves scattered by island particles. The factor of XT in the second equation of (1) becomes X$ in the former calculation instead of X,X, in the latter. In order to check the validity of both calculations, ratios between peak values of xl” and xi; are compared in fig. 9. The observed values (crosses) are between values calculated using a fictitious plane-parallel model (solid circle) and those calculated by integrating scattered waves (open circles). For films thinner than 1 A, the calculation by the integration method looks better than the other.
7. Summary It was shown that MATRS is a sensitive and useful tech~que for the investigation of the initial growth process of metal ftims. The sensitivity for detecting adsorbed sodium was found to exceed that of the usual torsion balance by more than two orders. It was also shown that MATRS brings information about shape and electronic properties of small metal islands. Other merits of MATRS are, e.g., in mechanical stability, easy temperature control and continuous in situ study with quick responce.
References [ 1] S. Fujiwara and H. Terajima, Phil. Mag. 27 (1973) 853. [2] R.T. Holm and E.D. Palik, Appl. Opt. 17 (1978) 394.
306
T. Yamaguchi, H. Takahashi j Sensitive investigation of initial growth
[3] T. Yamaguchi, H. Takahashi and A. Sudoh, J. Opt. Sot. Am., to be published.
[4] [S] [6] [7] [8]
T. Yamaguchi, S. Yoshida and A. Kinbara, Thin Solid Films 21 (1974) 173. T. Yamaguchi, S. Yoshida and A. Kinbara, Thin Solid Films 18 (1973) 63. E. David, Z. Physik 114 (1939) 389. H. Schopper, Z. Physik 130 (1951) 565. S. Yoshida, T. Yamaguchi and A. Kinbara, J. Opt. Sot. Am. 61 (1971) 62,463.
HIGHLY SENSITIVE ~VESTIGATION OF INITIAL GROWTH PROCESS OF SODI~ FILMS BY MULTIPLE-A~ENUATED-TOTAL-REFLECTION SPECTROSCOPY
T. YAMAGUCHI and H. TAKAHASHI Research Institute of Electronics, Shizuoka University,Hamamatsu 432, Japan Received 25 April 1978; manuscript received in final form 9 June 1978
Multiple-attenuated-total-reflection spectroscopy (MATRS) is applied to highly sensitive measurement of small amount of metal adsorbed on a fused quartz substrate. A sodium film thinner than 0.01 A in mean thickness or 1O-1o g/cm2 in weight could be detected. MATRS brings information also about shape and electronic properties of small island particles. Sodium islands were predicted to have cap-shape and the number of free electrons per unit volume in sodium islands was found to decrease from the bulk value as the mean thickness decreases to less than 1 A.
1. Introduction
In the experimental study of the initial growth process of a film formation, the most important thing is how small an amount of deposit we can detect sensitively. The sensitivity of the torsion balance of the top class [l] has been of the order of lo-’ g. A higher sensitivity is desirable for the investigation of the initial nucleation process of fdm growth. Holm and Palik [Z] showed recently that the multiple-attenuated-tots-resection spectroscopy (MATRS) ia s useful technique for the sensitive measurement of the low absorption coefficient of thin films. In the present paper, we apply the technique to a metal island film in the initial growth process, and show that the sensitivity becomes 0.01 A in mean thickness or 10-“” g/cm2 in weight for the case of sodium. MATRS in this paper uses the phenomenon of optical plasma-resonance absorption in small metal particles, so that material which is applicable to this technique will be restricted to a few kinds of metal, such as alkali metals and silver. Very thin films of these metals show plasma-resonance absorption peaks in some spectral region. The shape and the position of the resonance peak include information about size, shape and electronic properties of the small islands of which the fdm consists. The principle of MATRS and an example of measurement and analysis with sodium are given below. 295
296
T. Yamaguchi, H. Takahashi / Sensitive investigation of initial growth
2. Principle of MATRS In a previous paper [3] we derived equations for ATR of a metal. island film by integrating scattered waves. The reflectances for s- and p-polarized light are approximated as
2y1
R,
=
1-
2kodw
__...s(z;“x;; t&&x;), zT
+
(1)
z’,‘2
where d,
is the mean thickness of the deposit, ke = 2n/X, Xi = sin Bi/nj, Yj = and Zi = cos 8jl~j, nj and Bj being, respectively, index of and angle of refraction in the medium of substrate (j = 1) or vacuum Q = 3). The double prime expresses the imaginary part. x11,1is the effective polarizability of the island particles and includes all information about a metal island film. The subscripts II and 1 mean, respectively, the component for electric field of light parallel to and perpendicular to the substrate surfaces. We consider the case where kodW is so small that eq. (1) is approximated well by an exponential term. Then after N times total reflections the transmitted intensity through the substrate is given by cos t9j * nj
(2)
L,p = Io~,~ exd-~4,pdwlV , where 1, is the intensity A,=
[8nY,/‘(Y;
without metal deposit, and
+ Y;“)] x;; ,
A, = [Srr.Z,/(Z~ + Z;*)] (2;‘~;; Putting the logarithm
+ X,X,x;I)
.
(3)
of eq. (2) equal to --I&,
B SIP = loa 0 (~o~,p/~~,p~= ~~~,p(logl
od G/h
-
we have (4)
This will be measured in the experiments.
3. Experiments Fig. 1 shows a block diagram of the ex~~ent~ system. The substrate used was a fused quartz plate of 0.15 mm* X 30 mm’, whose edges were cut and polished at 45’. The substrate was put on a copper block of 10 X 30 X 50 mm3 in which a W-wire heater was embedded. The copper block was attached to a cold fmger mounted in a UHV system (V143N, NEVA). The whole substrate system was enclosed by a liquid-nitrogen temperature shroud that had windows for light beams and a sodium-atom flux.
T. Y4~4~ch~, UHV
H. T4k4h4sh~ / Sensitive investig4fion of initialgrowth
297
Chamber (V143N.
NEVA) Quartz Substrate (015mm’ x 30mm’): x 5mm’)
long-path Ishort-path
Fig. 1. Block diagram of the experimental system.
of the copper block was measured by a c-c thermocouple and was assumed to be the same as the substrate temperature. A light beam from a 75 W Xe short-arc was introduced into the substrate through one of edges. The light beam traveled in the substrate by repeating total reflection about 80 times in this experiment to reach the opposite edge. The beam that came out from the edge was introduced into a double-monochrometer (H-lOD, Jovin Yvon) and was detected by a photomultiplier. The output signal of the photomultiplier was fed into a logarithmic amplifier and recorded by an X-Y recorder as a function of the wavelength of light. The polarization of the incident light was selected by rotating a quartz-crystal Rochon-prism mounted near the light source. No lens was inserted in the optical path. Sodium of 99.9% purity was cut into a cube of about (5 mm)’ in oil, rinced by trichrol ethylene, and dried by a rotary pump; and the oxide layer was cut off. Immediately the sodium cube was fed into a conical basket mounted at 126 mm apart from the substrate, and the vacuum chamber was evacuated as soon as possible. The vacuum pressure was ultimately 5 X iOe9 Torr and was maintained less than 1 X 10m8 Torr during deposition. In addition to depositing on the whole substrate length of 30 mm, deposition on a limited length of 5 mm was made by masking to lower the sensitivity. The former and the latter are called, respectively, long-path and short-path hereafter. A sensing head of quartz-crystal thickness monitor (EVM-32, NEVA) was set at a near distance of 40 mm from the evaporation source to monitor the incident flux with a one order higher sensitivity. Temperature
4. Results Figs. 2 and 3 show the observed B S,p spectra of the sodium films deposited at a substrate temperature of 77 K with an incident rate of 0.9 &‘min. The mean thick-
T. Yamaguchi, H. Takahashi / Sensitive investigation of initial growth
298
0.6
0.5
0.4
A (urn)
0.6
0.5
0.4
3
A(rrm)
(
3
4
hw(eV)
Fig. 2. f& and BP spectra measured by long-path. The values of d, (in A) are: (a) 1.05, (b) 0.75, (c) 0.57, td) 0.39, fe) 0.26, (f) 0.13.
ness d, corresponding to each curve was obtained by analysis as described below. The four curves in fig. 4 show &,,k) at 80 = 2.6 eV versus the incident thickness din (quartz monitor output) relations for long-path, which were directly
06
2
0.5
0.4
A(wm)
3 ‘lw(eV)
4
I
0.6
0.6
2
Fig. 3. Bs and BP spectra measured by short-path. The values of d, 3.79, (c) 2.91, (d) 2.10, (e) 1.41, ffj 0.72.
0.4
3
A(clm) 0
fw(eV)
4
(in A) are: (a) 4.78, (b)
T. Yamaguchi, H. Takahashi/Sensitive investigationof initialgrowth
0
2
1
4
299
5
Fig. 4. Bs(peak) versus din relations measured with four incident rates. Solid circles are Bs(peak) of fig. 2 plotted against dW taken on the horizontal axis. d, corresponding to &(peak) by the broken line is scaled on the right-hand side.
recorded using a X-Y recorder during deposition. Fiiw= 2.6 eV is the most probable value that gives the peak position of B, in several experiments.
5. Analysis
Multiplying
eq. (4) by X, we have
= CI x;;dw ,
W
B,h = C,x;;d,
+ Gx;Idw
(9
The factors C1, CZ and Cs include Xi, Yi, Zi and N, and are known quantities, although they depend on X because of the dispersion of nl. The unknown parameters are x,, ;’ and d,. Here, x,,,~ is given by [3] XII ,I
-
1 -~--nf t ns FII ,* + (Ei -v 2n:
(6)
Fll and FL are the effective depolarizing given by
y2 FII =~II - --
n: - n;
factors of the island particles and they are
F*=f,_y2+$
24~~ nt t ni ’
The first term is the real depolarizing
i27j3 nl
+n3
(7)
factor of the particles. As the shape of the
300
T. Yamaguchi, H. Takahashi /Sensitive
investigation
of initial
growth
island particles is assumed to be isotropic in the film plane (e.g., rotational ellipsoid or cap-shaped), the following relation holds for 41 and fi,
The second term of eq. (7), arises from the mirror image effect due to the substrate surface [4]. y is the axial ratio of the rotational ellipsoid which is optically equivalent to the island particles, and n is the height of the actual point dipole moment of a real island from the substrate surface normalized by the height of the center of the rotational ellipsoid. The term arising from the dipole-dipole interactuons between island particles [S] is omitted in eq. (7) because its contribution is small for very thin films as are considered in this paper. In eq. (6), 6 is the dielectric constant of sodium within the particles and it can be approximated well by the free-electron model: fi = 1 - W$(02 - iww,) ,
(9)
where or, and w7 are, respectively, the plasma and relaxation frequencies. Analysis of figs. 2 and 3, was made by a curve Gtting method using a computer as follows. First we obtained curves of x1,”d, and a”dw from the measured spectra of B,,, by the use of eq. (5), i.e., x’;dw = B,h/CI ,
x;Id, = (4$,X - B&XC1
GQ
YC, .
Next we calculated the theoretical curves of xi,,; by substituting the appropriate
0
2.0
*
3.0
hw(eV)
* 4.0
*.
2.0
3.0
4.0
%I (eV)
Fig. 5. An example of curve fitting for curve c of fig. 2. Solid circles are sampled points from the observed spectra and full lines are fitted theoretical curves.
T. Yamaguchi, H. Takahashi / Sensitive investigation ofinitial growth
301
values of F/Q, or, and w, in eqs. (6) and (9). Then we normalized both the experimental and the theoretical curves so as to make the peak height unity and compared them to each other to find the values of FI~,I, on and w, that make the best fitting between both the normalized curves by the least-square method. It was found that excellent fitting was realized. An example of the fitting is shown in fig. 5 for the third curve (c) of fig. 2. The solid circles are sampled points from the observed curves and the full lines represent the fitted theoretical curves. By comparing the normalization factors of the observed and theoretical curves, we can determine the values of d,, which are plotted in fig. 4 against &(p&) for the case of long-path. As is seen in the figure, those points are well approximated by a straight line. Therefore, &peak) is understood to be a good measure of & in this thickness region. d, corresponding to &(p&) is scaled on the right side of the figure. The
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Fig. 6. FII and FL (a), 7 (b) and q (c) versus dw relations obtained by long-path (open circle) and by short-path (solid circle). h is the height of the actual point-dipole moment of a real island particle.
302
T. Yamaguchi, H. Takahashi /Sensitive investigation of initial growth
figure tells us that we can detect & of 0.01 A which corresponds to 10T’Og/cm2 for sodium. The results of Fll and FL are plotted against d, in fig. 6a. The open and solid circles are determined from, respectively, long-path and short-path data. From eqs. (7) and (8), the fo~ow~g relations hold for F, and FL 2Fll-FL=2fw-fi=4fu-1,
(11) Thus fil is obtained from the determined values of Fll and FL using the first equation of (11). The axial ratio y of the rotational eliipsoid corresponding tofil is plotted in fig. 6b. By the use of the second equation of (1 l}, rl is determined and plotted in fig. 6c. The resulting values for fiw, are given in fig. 7a. They are much larger than the bulk value of 0.02 eV. If they are attributed to the classical size effect of a spherical
’
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3
4
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Fig. 7. llwz (a) and r (b) versus d, relations. r was determined by the ciassical size effect; w7 = vF/r with v~ = 10’ 5 A/s.
T. Ya~a~chi, If. Ta~ha~h~ f Sensitive tnve~tigatio~ ofinitiaf growth
303
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Fig. 8. Bw, (a) and N (b) versus c&,, relations. N (=o@&bul@ trons per atom, the bulk value being assumed to be unity.
is the number of free elec-
particle:
where uF (=lO” &see) is the Fermi velocity of sodium and r is the readius of the spherical particle, then r is determined and plotted in fig. 7b. The results of hw, are given in fig. 8a. It can be seen that fiw, decreases from the bulk value when dw decreases to less than 1 8. If they are attributed to the number of free electrons per stom, it also decreases with decrease of thickness as shown in fig. 8b.
6. Discusions
The sensitivity for detecting adsorbed sodium in this experiment exceeds that of the usual torsion balance by more than two orders. Further, enhancement of one order can be made with relative ease, by thinning the substrate plate and taking the path length in the substrate longer. The validity of the obtained high sensitivity
304
T. Ymmg~chi,
H. Takahash~ / Sensitive i~ves~~g~~ion of initialgrowth
can be confirmed by fig. 4. If the sensitivity is low, i.e., the slope of the broken line is small, then the observed growth curve with the highest incident rate of 7.2 A/min will intersect the broken line. Since the broken line is considered to be a growth curve with sticking coefficient of unity, the intersection should not occur. It must be noted that the weight sensitivity of lo-” g/cm’ can apply only to alkali metals. The sensitivity for silver will be by one order lower because of its high density. It can be seen in fig. 4 that the sticking coefficient of sodium on a fused quartz substrate is still very small at the low temperature of 77 K in this small-thickness range. It was also found in another experiment that sodium films are not stable and re-evaporate completely in a few hours even at 77 K. As the re-evaporation was accelerated by a slight increase of the substrate temperature, a number of depositions could be made in a short time without changing the substrate. Although the decay of the optical absorption can occur by the film oxidation, the rate of the decay did not change by increasing the residual gas pressure by more than one order. It is understood from fig. 6b that island particles are optically equivalent to the rotational ellipsoid with axial ratio of 1.4-l S. Since n is less than unity as shown in fig. 6c, the particle shape may be rather cap-shape like. Decrease of 77 with decrease of & may be attributed to the decrease of contact angle of the cap-shaped island. The discontinuity of hw, in fig. 7a between long-path and short-path may be due to a small difference in the condition of deposition. It seems curious that the change in particle size with d, is small, as is seen in fig. 7b. Other causes might exist for the reason of the very large value of AU,. So far the statistical distribution in fil [6,7] or in r [8] has been taken into account for the explanation of the broad optical absorption peak metal island films much thicker than the present films. However, the fitting between observed and theoretical curves in fig. 5 seems too good to take the statistics distribution of particle shapes into consideration, because the theory used is based on a single particle shape. Possibly, the theory of the classical size effect, that is based on the limit of the mean free path of electrons by the particle surface, can not be applied to such extremely small particles. No matter what is the reason, a small change in fiiw, with d,., is a convenient thing for the purpose of using it as a detector of deposit, because the linear relation between &(p&) and d, in fig. 4 is due to this fact. The decrease of the number of free electrons per atom with an decrease of dw as shown in fig 8b seems reasonable, for the fraction of surface atoms increases with decrease of particle size. It is interesting that we can detect the change in the electronic properties of a small metal particle. In the case of silver, we must take into account the contribution not only from the free electrons but also from the bound electrons. We found that the contribution from the bound electrons must also differ from the bulk value for the explanation of the results of MATRS.
T. Yamaguchi, H. Takahashi/Sensitive investigationof initialgrowth
30.5
Short path --__ _ __
0 0
0
0
1
0
0
o
2
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4
5
d,(A)
Fig. 9. X~(peak)~X~i(peak) versus C& relations for spectra observed (crosses), calculated with the fictitious plane-panel f&n model (solid circle) and calculated by integration method (open circles).
In the previous paper [3] we obtained the ATR equation using a fictitious planeparallel ftim model as well as integrating waves scattered by island particles. The factor of XT in the second equation of (1) becomes X$ in the former calculation instead of X,X, in the latter. In order to check the validity of both calculations, ratios between peak values of xl” and xi; are compared in fig. 9. The observed values (crosses) are between values calculated using a fictitious plane-parallel model (solid circle) and those calculated by integrating scattered waves (open circles). For films thinner than 1 A, the calculation by the integration method looks better than the other.
7. Summary It was shown that MATRS is a sensitive and useful tech~que for the investigation of the initial growth process of metal ftims. The sensitivity for detecting adsorbed sodium was found to exceed that of the usual torsion balance by more than two orders. It was also shown that MATRS brings information about shape and electronic properties of small metal islands. Other merits of MATRS are, e.g., in mechanical stability, easy temperature control and continuous in situ study with quick responce.
References [ 1] S. Fujiwara and H. Terajima, Phil. Mag. 27 (1973) 853. [2] R.T. Holm and E.D. Palik, Appl. Opt. 17 (1978) 394.
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T. Yamaguchi, H. Takahashi j Sensitive investigation of initial growth
[3] T. Yamaguchi, H. Takahashi and A. Sudoh, J. Opt. Sot. Am., to be published.
[4] [S] [6] [7] [8]
T. Yamaguchi, S. Yoshida and A. Kinbara, Thin Solid Films 21 (1974) 173. T. Yamaguchi, S. Yoshida and A. Kinbara, Thin Solid Films 18 (1973) 63. E. David, Z. Physik 114 (1939) 389. H. Schopper, Z. Physik 130 (1951) 565. S. Yoshida, T. Yamaguchi and A. Kinbara, J. Opt. Sot. Am. 61 (1971) 62,463.