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Optical corrections in measurements of photoionization cross sections from solids in the soft X-ray range
Journal of Electron Spectroscopy and Related Phenomena, 37 (1986) 389-393 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
Short communication
OPTICAL CORRECTIONS IN MEASUREMENTS OF PHOTOIONIZATION CROSS SECTIONS FROM SOLIDS IN THE SOFI’ X-RAY RANGE
U. DEL PENNINO*, and I. LINDAU???
S. NANNARONE **, L. BRAICOVICHt,
I. ABBATI?,
G. ROSSItt
*Dipartimento di Fisica dell’llniversiti, Modena (Italy) **Dipartimento di Fisica dell’Universit& Roma (Italy) tlstituto di Fisica de1 Politecnico, Milan0 (Italy) t fGroup PS2, L. U. R.E. Universiti de ParisSud, Orsay (France) tftStanfoni Electronics Laboratories, CA 94305 (U.S.A.) (First received 29 April 1985; in final form 20 September 1985)
ABSTRACT We present measurements of photoelectron flux from the 4d band of Pd in the photon energy range 70-200eV. A comparison between measurements at different light incidence angles demonstrates the effect of the optical corrections (reflectivity and refraction) needed in the measurement of photoionization cross sections. It is shown that simple model calculations account satisfactorily for this effect.
In photoelectron spectroscopy of solids the energy dependence of photoionization cross sections is assuming an increasing importance in both the assignment of the electron states [ 1, 21 and the understanding of solid state effects on the cross sections [3, 4). For most situations the hv dependence of the cross section, o(hv), is of significance, i.e., it is sufficient to determine o(hv) apart from a constant multiplicative factor. We present an experimental assessment of the effect of the optical constants of the sample on u measurements of this type in the soft X-ray region (typically 70-200 eV), one of the most interesting for solid state effects [ 3, 41. This is important because the correction due to optical properties, when needed, is very sensitive [5] and, being sample-dependent, is not cancelled in the comparison between u values measured from different substances in the same experiment. Optical properties of samples have two effects in competition with each other: the effective photon flux is reduced by reflectivity losses at the sample surface, while refraction concentrates photoemission beneath the surface with an increase in photoelectron escape probability. Obviously 036%2048/861$03.50
o 1986 Elsevier Science Publishers B.V.
390
these effects are much larger at grazing incidence and ,thus incidence near the normal to the sample is preferable in u measurements. On the other hand, it is also very important to be able to extract the maximum experimental information on cross sections from experiments which, for other reasons, cannot be carried out at near-normal incidence. To this end a theory of optical corrections has been developed by Henke [6] on the basis of a rather ideal model. A perfectly flat surface of a homogeneous sample is assumed, and structural effects on outgoing electrons (for example photoelectron diffraction) are neglected. For any photoelectron momentum, q, within the crystal the outgoing electron flux is given by Henke [6] as c,F
(1)
where F is a corrective function which accounts for optical effects and depends on the angle of incidence of the light, x, the complex refractive index (n + ih) of the sample, the escape depth and on the direction of q. The usefulness of this simplified model in the interpretation of experiments has never been tested in the present hv range, and a very limited test has been given in ref. 6 at 1486 eV where the effect of optical corrections is smaller. Moreover, the experiments do not meet all the conditions of the model. The samples which are most convenient to eliminate photoelectron diffraction are polycrystals or evaporated films, but these samples do not, in general, guarantee a flat surface. Another difficulty arises in partially integrated measurements: formula (1) cannot be used exactly since its integration would require a knowledge of the q dependence which would in turn require angular information. The purpose of the present work, the first analysis of the optical effects in the soft X-ray region (up to 200 eV), is to show that: (i) in the case of evaporated films the Henke theory works satisfactorily for measurements of cross sections partially integrated over the angles, (ii) in the integration of expression (1) it is possible to assume that F is independent of the direction of q and it is possible to calculate F at the average acceptance angle of the spectrometer (in the present case, a cylindrical mirror analyzer). We have chosen evaporated Pd films because 4d cross sections have been studied extensively [3, 41 and considerable corrections are expected since the optical constants [7] still vary strongly in this hv range. All experimental details are as described previously [4]. We measured the count rate C(hv, x) from the 4d band photoemission vs. hv at the angle x; o(hv) integrated over the acceptance angle ASJ of the spectrometer is defined, apart from a constant factor, by the relation C(hv, x)lc(hv)A(hv)F
(2)
where A comprises the incoming photon flux, the escape depth correction
391
and the detector efficiency (for details see ref. 4). The hv dependence of u comes essentially from the hv dependence of the total cross section, with a minor modulation in our experimental setup coming from the integral over AS2 of the angular factor containing the asymmetry parameter. At sufficiently high hv, the F becomes 1 at normal incidence; this condition is satisfied in our experiment as can be seen from the evaluation by Henke for Au, a more severe case than Pd because of its higher reflectivity. Thus the ratio of the count rates, R, near normal incidence C(hv, 90), and at an angle x, C(hv, x), is given by
R(hv, x) /
WV, 90) = 1 C(hv, xl F
(3)
In our experimental setup the absolute value of the detection efficiency depends on x, but the hv-dependence of the efficiency is not influenced by x so that R is a measurement of l/F apart from a multiplicative factor. The factor l/F, apart from the other corrections contained in A, transforms the measured count rates into cross sections. Thus, neglecting the optical corrections would underestimate the cross section at a low hv where reflectivity losses are relevant, and would overestimate the cross section at higher hv where refraction is relevant. This is shown clearly by
l/F R1.5 -
l-
0.5 -
01 50
100
hv
150
(eV)
200
Fig. 1. Comparison between experimental (R) and theoretical (l/F) values of the multiplication factor for the measured u that gives the values corrected for the effect of optical constants of the sample (Pd 4d). The shaded areas give the uncertainties of the experimental and theoretical values (due to the uncertainties in the optical constants). The inset gives the count rate vs. hp at near-normal incidence (solid line) and at grazing incidence (dashed line).
392
the comparison of the count rates at 15” grazing incidence and at normal incidence in Fig. 1 (inset). The comparison between experimental R and theoretical (l/F) is shown in Fig. 1 as a function of photon energy. The shaded areas represent the experimental uncertainties and the limits of confidence of the calculations connected with the accuracy of available optical constants. The R scale has been adjusted to give the best superposition with theoretical l/F; this adjustment is consistent with the fact that R is determined experimentally apart from a constant factor. The values of l/F,have been calculated using the theory of Henke (ref. 6, paragraph III-B) using the escape depth of ref. 8, the optical absorption of Pd from ref. 7 and generating n from k with a KramerKronig transformation. In order to show the sensitivity of the calculated l/F to the optical
7
5
‘g In 0
L Zr
-6
1.5-a _
l/F l-
01 50
II
81
\ \ \
-\ \
\
\
‘\_______________
I1 100
I1
100
hv
I
I 150
I
11
1
I
2oc
\
\ \ ‘, Pd \ \ \ 1
150
(eV)
20
Fig. 2. The sensitivity of the correction factor to the values of the optical constants. Lower panel, comparison between the theoretical correction factors in Zr and in Pd. Upper panel, optical absorption coefficients of Zr and Pd.
393
properties, in Fig. 2 we compare the theoretical values of l/F for Pd and for Zr, whose optical absorption [7] varies much less than that of Pd (Fig. 2, upper panel). The above results deserve the following comments. (i) The general trend with the dominance of reflectivity losses at lower hv and of refraction corrections at higher hv is very well reproduced by the theory: the general agreement is good in spite of the numerous approximations inherent in the use of the theory. (ii) Quantitatively the agreement is less satisfactory in the region below 100 eV where the onset of the reflectivity effect is smoother in the experimental results and is displaced by -10 eV. The smoothing could be due in part to the polycrystalline nature of the sample which produces some kind of average over the angles. (iii) As a consequence the theory is more reliable in correcting for refraction effects than for reflectivity. This allows the theory to be used safely in many cases in the soft X-ray region; an example is Zr which has a more constant reflectivity (see Fig. 2) so that the optical correction is essentially a refraction effect. In conclusion, we have shown that the theory by Henke is very effective for extracting experimental information on cross sections in an energy interval which can be ‘dangerous’ because of the hv dependence of optical constants. This can be of great help because it expands the possibility of assessing cross section effects, and contributes to a deeper discussion of the results of photoemission experiments carried out with synchrotron radiation. ACKNOWLEDGEMENTS
The authors are indebted to M.H. Hecht for illuminating discussions during this research project. This work has been supported by the Office of Naval Research through the Contract N 0014-82-K-0524. The experiments were performed at SSRL which is supported by DOE and NSF. REFERENCES 1 2 3 4
5 6 7 8
K. Codling, J. Electron Spectrosc. Relat. Phenom., 17 (1979) 279. G. Rossi, I. Abbati, L. Braicovich, I. Lindau and W.E. Spicer, Solid State Commun., 39 (1981) 195. I. Abbati, L. Braicovich, G. Rossi, I. Lindau, U. de1 Pennino and S. Nannarone, Phys. Rev. Lett., 50 (1983) 1799. G. Rossi, I. Lindau, L. Braicovich and I. Abbati, Phys. Rev. B,28 (1983) 3031. M.H. Hecht, PhD Thesis, Stanford University, 1982; M.H. Hecht and I. Lindau, J. Electron Spectrosc. Relat. Phenom., 35 (1985) 211. B.L. Henke, Phys. Rev. A, 6 (1972) 94. J.H. Weaver and C.G. Olson, Phys. Rev. B, 14 (1976) 3251. M.P. Seah and W.E. Dench, Surf. Interface Anal., 1 (1979) 1.
Short communication
OPTICAL CORRECTIONS IN MEASUREMENTS OF PHOTOIONIZATION CROSS SECTIONS FROM SOLIDS IN THE SOFI’ X-RAY RANGE
U. DEL PENNINO*, and I. LINDAU???
S. NANNARONE **, L. BRAICOVICHt,
I. ABBATI?,
G. ROSSItt
*Dipartimento di Fisica dell’llniversiti, Modena (Italy) **Dipartimento di Fisica dell’Universit& Roma (Italy) tlstituto di Fisica de1 Politecnico, Milan0 (Italy) t fGroup PS2, L. U. R.E. Universiti de ParisSud, Orsay (France) tftStanfoni Electronics Laboratories, CA 94305 (U.S.A.) (First received 29 April 1985; in final form 20 September 1985)
ABSTRACT We present measurements of photoelectron flux from the 4d band of Pd in the photon energy range 70-200eV. A comparison between measurements at different light incidence angles demonstrates the effect of the optical corrections (reflectivity and refraction) needed in the measurement of photoionization cross sections. It is shown that simple model calculations account satisfactorily for this effect.
In photoelectron spectroscopy of solids the energy dependence of photoionization cross sections is assuming an increasing importance in both the assignment of the electron states [ 1, 21 and the understanding of solid state effects on the cross sections [3, 4). For most situations the hv dependence of the cross section, o(hv), is of significance, i.e., it is sufficient to determine o(hv) apart from a constant multiplicative factor. We present an experimental assessment of the effect of the optical constants of the sample on u measurements of this type in the soft X-ray region (typically 70-200 eV), one of the most interesting for solid state effects [ 3, 41. This is important because the correction due to optical properties, when needed, is very sensitive [5] and, being sample-dependent, is not cancelled in the comparison between u values measured from different substances in the same experiment. Optical properties of samples have two effects in competition with each other: the effective photon flux is reduced by reflectivity losses at the sample surface, while refraction concentrates photoemission beneath the surface with an increase in photoelectron escape probability. Obviously 036%2048/861$03.50
o 1986 Elsevier Science Publishers B.V.
390
these effects are much larger at grazing incidence and ,thus incidence near the normal to the sample is preferable in u measurements. On the other hand, it is also very important to be able to extract the maximum experimental information on cross sections from experiments which, for other reasons, cannot be carried out at near-normal incidence. To this end a theory of optical corrections has been developed by Henke [6] on the basis of a rather ideal model. A perfectly flat surface of a homogeneous sample is assumed, and structural effects on outgoing electrons (for example photoelectron diffraction) are neglected. For any photoelectron momentum, q, within the crystal the outgoing electron flux is given by Henke [6] as c,F
(1)
where F is a corrective function which accounts for optical effects and depends on the angle of incidence of the light, x, the complex refractive index (n + ih) of the sample, the escape depth and on the direction of q. The usefulness of this simplified model in the interpretation of experiments has never been tested in the present hv range, and a very limited test has been given in ref. 6 at 1486 eV where the effect of optical corrections is smaller. Moreover, the experiments do not meet all the conditions of the model. The samples which are most convenient to eliminate photoelectron diffraction are polycrystals or evaporated films, but these samples do not, in general, guarantee a flat surface. Another difficulty arises in partially integrated measurements: formula (1) cannot be used exactly since its integration would require a knowledge of the q dependence which would in turn require angular information. The purpose of the present work, the first analysis of the optical effects in the soft X-ray region (up to 200 eV), is to show that: (i) in the case of evaporated films the Henke theory works satisfactorily for measurements of cross sections partially integrated over the angles, (ii) in the integration of expression (1) it is possible to assume that F is independent of the direction of q and it is possible to calculate F at the average acceptance angle of the spectrometer (in the present case, a cylindrical mirror analyzer). We have chosen evaporated Pd films because 4d cross sections have been studied extensively [3, 41 and considerable corrections are expected since the optical constants [7] still vary strongly in this hv range. All experimental details are as described previously [4]. We measured the count rate C(hv, x) from the 4d band photoemission vs. hv at the angle x; o(hv) integrated over the acceptance angle ASJ of the spectrometer is defined, apart from a constant factor, by the relation C(hv, x)lc(hv)A(hv)F
(2)
where A comprises the incoming photon flux, the escape depth correction
391
and the detector efficiency (for details see ref. 4). The hv dependence of u comes essentially from the hv dependence of the total cross section, with a minor modulation in our experimental setup coming from the integral over AS2 of the angular factor containing the asymmetry parameter. At sufficiently high hv, the F becomes 1 at normal incidence; this condition is satisfied in our experiment as can be seen from the evaluation by Henke for Au, a more severe case than Pd because of its higher reflectivity. Thus the ratio of the count rates, R, near normal incidence C(hv, 90), and at an angle x, C(hv, x), is given by
R(hv, x) /
WV, 90) = 1 C(hv, xl F
(3)
In our experimental setup the absolute value of the detection efficiency depends on x, but the hv-dependence of the efficiency is not influenced by x so that R is a measurement of l/F apart from a multiplicative factor. The factor l/F, apart from the other corrections contained in A, transforms the measured count rates into cross sections. Thus, neglecting the optical corrections would underestimate the cross section at a low hv where reflectivity losses are relevant, and would overestimate the cross section at higher hv where refraction is relevant. This is shown clearly by
l/F R1.5 -
l-
0.5 -
01 50
100
hv
150
(eV)
200
Fig. 1. Comparison between experimental (R) and theoretical (l/F) values of the multiplication factor for the measured u that gives the values corrected for the effect of optical constants of the sample (Pd 4d). The shaded areas give the uncertainties of the experimental and theoretical values (due to the uncertainties in the optical constants). The inset gives the count rate vs. hp at near-normal incidence (solid line) and at grazing incidence (dashed line).
392
the comparison of the count rates at 15” grazing incidence and at normal incidence in Fig. 1 (inset). The comparison between experimental R and theoretical (l/F) is shown in Fig. 1 as a function of photon energy. The shaded areas represent the experimental uncertainties and the limits of confidence of the calculations connected with the accuracy of available optical constants. The R scale has been adjusted to give the best superposition with theoretical l/F; this adjustment is consistent with the fact that R is determined experimentally apart from a constant factor. The values of l/F,have been calculated using the theory of Henke (ref. 6, paragraph III-B) using the escape depth of ref. 8, the optical absorption of Pd from ref. 7 and generating n from k with a KramerKronig transformation. In order to show the sensitivity of the calculated l/F to the optical
7
5
‘g In 0
L Zr
-6
1.5-a _
l/F l-
01 50
II
81
\ \ \
-\ \
\
\
‘\_______________
I1 100
I1
100
hv
I
I 150
I
11
1
I
2oc
\
\ \ ‘, Pd \ \ \ 1
150
(eV)
20
Fig. 2. The sensitivity of the correction factor to the values of the optical constants. Lower panel, comparison between the theoretical correction factors in Zr and in Pd. Upper panel, optical absorption coefficients of Zr and Pd.
393
properties, in Fig. 2 we compare the theoretical values of l/F for Pd and for Zr, whose optical absorption [7] varies much less than that of Pd (Fig. 2, upper panel). The above results deserve the following comments. (i) The general trend with the dominance of reflectivity losses at lower hv and of refraction corrections at higher hv is very well reproduced by the theory: the general agreement is good in spite of the numerous approximations inherent in the use of the theory. (ii) Quantitatively the agreement is less satisfactory in the region below 100 eV where the onset of the reflectivity effect is smoother in the experimental results and is displaced by -10 eV. The smoothing could be due in part to the polycrystalline nature of the sample which produces some kind of average over the angles. (iii) As a consequence the theory is more reliable in correcting for refraction effects than for reflectivity. This allows the theory to be used safely in many cases in the soft X-ray region; an example is Zr which has a more constant reflectivity (see Fig. 2) so that the optical correction is essentially a refraction effect. In conclusion, we have shown that the theory by Henke is very effective for extracting experimental information on cross sections in an energy interval which can be ‘dangerous’ because of the hv dependence of optical constants. This can be of great help because it expands the possibility of assessing cross section effects, and contributes to a deeper discussion of the results of photoemission experiments carried out with synchrotron radiation. ACKNOWLEDGEMENTS
The authors are indebted to M.H. Hecht for illuminating discussions during this research project. This work has been supported by the Office of Naval Research through the Contract N 0014-82-K-0524. The experiments were performed at SSRL which is supported by DOE and NSF. REFERENCES 1 2 3 4
5 6 7 8
K. Codling, J. Electron Spectrosc. Relat. Phenom., 17 (1979) 279. G. Rossi, I. Abbati, L. Braicovich, I. Lindau and W.E. Spicer, Solid State Commun., 39 (1981) 195. I. Abbati, L. Braicovich, G. Rossi, I. Lindau, U. de1 Pennino and S. Nannarone, Phys. Rev. Lett., 50 (1983) 1799. G. Rossi, I. Lindau, L. Braicovich and I. Abbati, Phys. Rev. B,28 (1983) 3031. M.H. Hecht, PhD Thesis, Stanford University, 1982; M.H. Hecht and I. Lindau, J. Electron Spectrosc. Relat. Phenom., 35 (1985) 211. B.L. Henke, Phys. Rev. A, 6 (1972) 94. J.H. Weaver and C.G. Olson, Phys. Rev. B, 14 (1976) 3251. M.P. Seah and W.E. Dench, Surf. Interface Anal., 1 (1979) 1.