- Email: [email protected]
Fundamentals of total reflection X-ray fluorescence
S ectrochimicaAC@ Vol. 468, No. 10, pp.133~1340, 1991 &ted in Great B&in.
0%4-8547/91 s3.00 + .lm 0 1991 Pergamon Press pk.
Fundamentals of total reflection X-ray fluorescence PETER KREGSAMER Atominstitut der osterreichischen
Universititen,
SchtittelstraOe 115, A-1020 Wien, Austria
(Received 14 December 1990; accepted 13 March 1991) Abstract-In the last years, total reflection X-ray fluorescence (TXRF) has shown to be an analytical technique for trace-element analysis. Theoretical considerations for the angular behaviour of scattered and fluorescence intensities are reviewed for the reflector material quartz and Mo-K_ excitation. The main influencing parameters are the critical angle, the transmission coefficient and the angle of refraction. Also the interference effect between incoming and reflected beam (standing wave) for total reflection is described.
1. INTRODUCTION A COLLIMATED X-ray beam impinging upon a surface at a grazing angle of incidence is totally reflected [l]. The development of the phenomenon from an interesting effect for physicists to an analytical tool was done by various groups of authors [2-51, while the interference effect on top of reflectors described hereafter was first observed and explained in diffraction measurements ([6] and references therein). There are six kinds of application of X-ray total reflection:
(9 (ii) (iii) (iv)
(v) (vi)
“Conventional” TXRF, where small samples are deposited on suitable reflectors (substrates) and subsequently analysed. Surface analysis, where the reflector itself is inspected for contaminations. The high-energy cutioff effect, which allows it to alter the primary spectral distribution. Capillaries, which can be used as wave guides for X-rays, produce microbeams with a few pm diameter, with improved intensities compared to ordinary collimators. X-ray optical elements (lenses, focussing devices). The standing wave effect, which enables the measurement of small distances of atoms above surfaces.
The present work is devoted to the description of those total reflection effects that relate to the sample directly, while the interested reader is referred to Refs [7-lo] for capillaries, to Refs [ll-141 for the cut-off effect and to Refs [15,16] for X-ray optics.
2. THEORETICAL The theoretical fundamentals of TXRF can be deduced in a way analogous to that in light optics, One has to solve the problem of an electromagnetic plane wave that hits the plane boundary between vacuum and a medium described by its dielectric constant e, which takes into account both scattering and absorption and. can be calculated by quantum mechanics (e.g. Ref. [17]). The electromagnetic waves of the incoming, reflected and refracted beam can be defined in the usual way, and Fresnel’s formulae will give the desired ratios of the amplitudes. For exact calculations, the polarization state of the primary radiation and the propagation of the refracted beam as a so-called inhomogeneous wave have to be considered ([l&19] and references therein). The inhomogeneity relates to the fact that the planes of equal phases are perpendicular to the respective wave vector, whereas the planes of equal amplitudes are parallel to the surface of the medium. For the primary as well as the reflected waves they are identical; for the refracted beam they 1333
P. KREOSAMER
1334 1T
---------__________________, 0
10’
h 0.
da
13
1.
2.i Incident
3.0 3.b Angle (mrad)
Fig. 1. The transmission coefficient T and reflection coefficient R have their inflection points near (pE= 1.8 mad for MO-K, totally reflected on quartz.
are distinct. There exists a direction, inside the medium (depending on the state of polarization), that describes the energy flow (Poynting vector) and is not perpendicular in the general case to the two above mentioned planes. With such a strict derivation one can obtain a refractive index n, which is the ratio of the vacuum velocity of light and the phase velocity of the wave inside the medium. This refractive index is a real quantity, in general less than 1 (though it may be different for energies close to an absorption edge of the reflector), and depends in a complex way on the dielectric constant (n* # e) and also depends on the incident angle [18,20]. Fo~unately, the effects of neglecting the polarization state and simpli~~tion of the refraction index for, e.g. Mo-I& primary radiation and a quartz reflector are insignificant for the calculation of transmission and reflection coefficients as well as refraction angles. (In principle one should show this for every new combination of exciting radiation and reflector material [21].) 2.1. APrgularde~ende~~i~ for TXRF One of the main features of TXRF is the strong dependence on the angle of incidence &-,of both the scattered as well as the fluorescence signals that originate from the substrate itself. One will be interested in minimizing the scattering contribution of the reflector and thus measure small (non-reflecting) samples on top of it. On the other hand, for the inspection of wafer surfaces, the fluorescence signal of contamination atoms should be m~imum. The following factors have to be considered: (1) For TXRF we generally can assume that the area on the reflector surface effectively seen by the detector is smaller than the area hit by the primary radiation, therefore a geometry factor proportional to sin J10has to be used. For when this assumption does not hold more detailed considerations can found in Ref. [22]. For total reflection, the angles of incidence and refraction are so small, that the sine can be replaced by its argument. (2) Only those primary photons that are not totally reflected on the (ideal flat) surface are able to penetrate and induce interactions. This leads to a transmission coefficient T (Fig. l), which, owing to energy conservation, sums up with the reflection coefficient R to unity (T + R = 1). The so-called critical angle cpCwill be close to the infIection point of these coefficients. Also the surface quality of the reflector must not be disregarded: insuffi~ent flatness will lead to only partial reflection of the primary beam, whereas roughness can be treated mathematically to some extent (see Refs [23,24] and references therein), but can quench total reflection completely. (3) If we consider the phenomenon of elastic or inelastic scatttering, the differential
Fundamentals
of TURF
Scattsring
1335
Angle (degraas)
Fip, 2. Elastic (full line) and inelastic (dashed line) differential scattering cross sections for a primary radiatkm of Mo-K_ scattered on atomic (amorphous) Si. The scattering angle a, which is the sum of incident and detection angle, will be in the range of 90” for TXRF. crofs sections ~~~~~~~~, which depend on the scattering angle 8, have to be calculated. Using atumie form factors and incoherent scattering functiuns [Zs] (s~pli~ca~on of unbound, irregular silicon atoms, especially for scattering into the forward direction) one obtains the respective angular dependencies in Fig. 2, which show a smooth behaviour near a value of 90”. Although there does not exist a fixed angle of detection in TXRF (there is a spread in angles because of the relatively large sample and detector diameters compared to their distance) an averge scattering angle af 90” is a good approximation, resulting in constant scattering probabilities. (4) The absorption of the refracted beam inside the substrate to the location where an interaction takes place and of the resulting radiation from there again to the surface (mass attenuation coefficient h) into the direction of the detector may not be neglected. The first absorption path (mass atteuuation coefficient tar) can be described by the penetration depth zP This quantity can be underst~ as the depth from where information comes out of the substrate. Because of its direct pro~~ionali~ to the angle of refraction #’ (Fig. 3), it shows the same sudden decrease near the critical angle of total-reflection cpC(e,g. Ref, [26]): scattering
+&I) = JI’ ($0) * bp
w
where p is the density. zP will roach approximately 3 nm for the theoretical incidence
0.
1
0.6
1.
lb
h
b
kT&nt A&(rnmd3j
Fig, 3. Refraction ar@e $8 as function of the incidence an&e lffc for MO-K, and a quartz reflector. The dashed line shows JI’ = Jlo (no refraction).
P. &tRGSAMER
1336 3.0 -
3 2
2.5 -
B c" s S 2 ii a 8 m
2.0 .' .' ** .' ..' .' .' .' 1.0 I' ..' .' .' .' 0.s .' I* *' .' .0.0 . 1. 0.0 Oh 1.h 1.5 -
incident
I
Lila
I
(mad;
Fig. 4. The background intensity for TXRF measurements, if Mo-K, primary radiation undergoes total reflection on a quart2 reflector. The dashed line represents the case of no total reflection.
angle of 0”, independent of the primary energy and only slightly dependent on the reflector material. For practical purposes, the value at the angle (pFis usually given. (5) If the concentration c of the (trace) elements investigated varies as a function of the distance to the surface z, a depth profile K(z) has to be defined. (6) The path length of the incident radiation along a depth dz is dz/JI’. If all factors mentioned are collected, a general intensity formula for both scattering and fluorescence (only the scattering cross sections have to be replaced by the absorption coefficient r divided by 4~) can be given:
44~0~4
= exp{-z~(l&,(Jlo)+
p2. p)} =
exp [- z * p &j+c2)].
(3)
2.2. Spectral background Only small samples are used for “conventional” TXRF so that, they should not significantly contribute to scattering if their residues and size are chosen properly. Otherwise .another kind of substrate and (or) excitation geometry will perhaps be advantageous. If we are interested in the angular dependence of the scattering processes originating from the substrate, the integration over the depth can be performed under the assumption of an “infinite” thickness with constant concentrations. As the attenuation coefficients for the primary and scattered radiation are equal or only slightly different (inelastic scattering), only two factors (called energy transfer in Refs [27,28]) govern the angular behaviour: scatter Z(Jro),,, a Jlo *T(Go). $ . %(90’, reflector). For angles larger than the critical angle, the geometry factor is responsible for the increase in the spectral background, whereas the sudden reduction (Fig. 4) for smaller angles originates from the almost perfect total reflection; the reflection coefficient is nearly 1 in this range. The dashed line, which describes the case of the same arrangement but without total reflection (e.g. a scratched reflector) reveals an improvement by a factor of T(JIo) for Jr0 < CPC).
Fundamentals of TXRF
1337
To compare TXRF with conventional XRF measurements one would put the sample on a suitable thin foil (45” geometry). Under the assumption that the excitation and detection would be equally efficient and the rest of the setup can be kept constant (especially the very short distance sample-detector), the background intensity will only be represented by: scatter m da I XRF a k;. -&9O”,foil)
(5)
where m/F is the mass per area of this foil. The dependence on the angle of incidence of absorption and geometry factor cancels. We assume for conventional XRF: m/F = 1 mg/cm2 and for TXRF an adjusted incident angle of 1 mrad, resulting in a transmission coefficient of T(JI, = 1 mrad) = 6.6 x 10m3. The absorption coefficient for Mo-K, in quartz is CL,= 3.8 cm2/g. The scattering cross sections are practically equal for both materials: 0.01 cm2/(g-sr) at 90”. If we insert those values in Eqns (4) and (5), the resulting ratio of the backgroud intensities would favour TXRF at this working point by a factor of approximately 500. 2.3. Contaminations in surfaces The previously defined function (2) shows an interesting behaviour, if pi/~ < 1 is assumed and pi/+’ < l.~~;follows [21] a needle-like maximum near the critial angle of the substrate, which, in the optimum case, can reach four times the intensity values at larger angles. This is partly fulfilled if, e.g. Mo-K, primary radiation produces lowenergetic fluorescence radiation. The effect can be understood, if one has in mind the very shallow path of the refracted beam until it can excite fluorescence radiation, which hence only has to cover a very short distance in the direction of the detector. For angles J10well below cpC,the decrease in the transmission coefficient T will compensate this and the intensity goes to zero. A suitable depth profile can support this tendency of peaking for the more realistic case that the absorption coefficients for the exciting radiation and the produced tluorescence radiation are of the same order of magnitude. For a thin zone of contamination atoms i (thickness ti being in the range of nm, so that in formula (3) A(&,z) 2: 1) with constant concentration (K(r) * ci) the fluorescence intensity (2) can be simplified:
Figure 5 shows such a behaviour, which in practice can be realized for wafer contaminations. (The divergence of the primary radiation will smooth the needle [28] and should be considered.) In principle, depth profiles of, e.g. contaminations of wafers can be extracted from measurements of such angular dependencies, at least, by fitting parameters of assumed distributions. For uncertainties in the measured intensites and lack of mechanical precision, the theoretical deconvolution (Laplace transformation) seems to be problematic owing to the inherent rapid changes in the angular behaviour. Non-reflecting residues on top of suitable surfaces give a doubled fluorescence intensity, when measurements are made at angles of incidence smaller than the respective critical angle of the substrate, owing to the twofold excitation [27], (Fig. 6). On the other hand, atoms within the range of the penetration depth can show a pronounced peak near (oC(see, e.g. Ref. [29]).
1338
P. KaxoSAMxR
Incident Angie (mrad)
Fig. 5. For primary radiation of MO-&, with 0” divergency, the inspection of a silicon surface wig yield an increased fluorescence signal of contamination atoms near (oe, compared to the case where no total reflection will occur (dashed line). A rectangular (constant) profile with a depth of 1 nm has been assumed.
Fig. 6. Twofold excitation of the sample for TXRF by the primary beam (1) and the reflected beam (2), for a case where the reflector is long enough so that beam (2) can occur.
2.4. Standing wave
Only very recently the interference effect on top of the reflector between the incoming and reflected beam has been recognized. The undisturbed coherent superposition of the plane electromagnetic waves results in a variation of the intensity pattern, depending on the distance above the surface, called a standing wave. Figure 7 displays the fundamental facts; the length D gives the distance between two maxima of the standing wave (A = wavelenth of the radiation): A
WJlo)= m,
(7)
D
Fig. 7. The interference wne (standing wave) between the incident (lo) and reflected (Z,J plane waves with wavelength A shows nodes and antinodes with a period of D.
and is typically in the range between 10 and 100 nm. If allowance is made for a phase factor Q, between incoming and reflected beam [30], the intensity of the standing wave as a function of the height z above the reflector surface and Jlo can be written as
Fundamentals of TXRF
@o,t)
= 10 * 1+ Wd
I
+ 24m
- 2+-J} * cos @blJ010)
1339
(8)
if absorption of any possible sample is excluded. This intensity can vary between 0 and 4 times the primary intensity IO. A proportional variation of the fluorescence radiation of atoms within this interference zone has already been used to determine the position of a zinc film above a gold surface to 20 mn with a precision of 0.2 nm 1301.For more complex problems such as thin (multiple) layers and multi-layer mirrors the reader is referred to Refs [23,31,32]. For conventional TXRF where granular residues on top of reflectors should be inspected, the problem might arise that samples with different heights (small compared with D) above the surface of the substrate (otherwise equal conditions) can give different fluorescence intensities. For homogeneous residues where many of those maxima and minima will occur within the sample height, this effect will level out to a large extent. If an internal standard can be homogeneously mixed with the sample, the ratio of the respective fluorescence intensities will lead to an acceptable precision in any case. Therefore the use of an internal standard normally applied for easier quantification of unsown samples seems to be inherently necessary for TXRF measurements. 3.
CONCLUSIONS
The description of the theoretical fundamentals of TXRF leads to a better understanding of the effect and enables a complete comparison of experimental results with theoretical calculations for samples on top of reflectors (“conventional” TXRF), impurities and contaminations in and on wafer surfaces, thin (multiple) layers on substrates, multi-layer mirrors, and the standing wave effect. For all those examples the angle of incidence of a well collimated primary beam plays an important role for the different kinds of observed intensities, and its controlled variation can give depth information in the nm-range. Ack~w~dgemen~~ would like to thank the “Fonds zur Forderung der ~~n~h~tichen the financial support of the work (parts of project P711.5).
Forschung” for
REFERENCES [l] [2] [3] [4] [S] [6] [7] [8] [9] [lo]
A. H. Compton, Phil. Mug. 45, 1121 (1923). Y. Yoneda and T. Horiuchi, Rev. Sci. Instrum. 42, 1069 (1971). H. Aiginger and P. Wobrauschek, Nucl. Instrum. Met/z. 114, 157 (1974). P. Wobrauschek and H. Aiginger, Anal, Chem. 47, 852 (1975). J. Kuoth and H. Schwenke, Frese&s’ 2. Anal. Chem. 291, 200 (1978). hf. Bedzyk, D. Bilderback, G. Bommarito, M. Caffrey and J. Schikikraut, Science 241, 1788 (1988). A. Rindby, NucI. instrum. Meth. A 249, 536 (1986). E. Stem, Z. Kalman, A. Lewis and K. Lie~~nn, Appl. Optics 27, 513.5 (1988). A. Carpenter, X-Ray Spectrom. 18, 253 (1989). P. Engstr&n, S. Larsson, A. Rindby, P. Voghs, G. Nilsson and B. Stocklassa, X-Ray Microbeam Spectroscopy, technical report, University of Goteborg, Sweden (1989). [ll] G. Damaschun, G. Kley and J. Miiller, Actu Phys. Austriaca 28, 233 (1968). 1121 E. Schnabel, R. Hosemann and B. Rode, J. Appi. Phys. 43, 3237 (1972). [13] H. Schwenke and J. Knoth, Fresenius’ 2. Anaf. Chem. 301, 7 (1980). [14] B. Lairson and D. Bilderback, Nucl. Instrum. Meth. 195, 79 (1982). [U] S. Wilkins, A. Stevenson, K. Nugent, H. Chapman and S. Steenstrup, Rev. Sci. Instrum. 60, 1025 (1989). [16] M. Kumakhov and F. Komarov, Whys. Reports 191, 289 (1990). [17] B. Henke, P. Lee, T. Tanaka, R. Shimabukuro and B. Fujikawa, Atom. Data Nucl. Data Tables 27, 1 (1982). [18] M. Levevere and M. Montel, Optica Actu 20, 97 (1973). flQ] P. Lee, Am. 1. Phys. 53, 885 (1985). [20] P. Kregsamer, Thesis, Technische Universitiit Wien, Austria (1990). [21] B. Henke, Phys. Rev. A 6, 94 (1972).
1340
P.
KltEQSAMER
[22] D. K. G. de Boer, X-Ray Spectrom. 18, 119 (1989). [23] A. Krol, C. Sher and Y. Kao, Phys. Rev. B 38, 8579 (1988). [24] F. Stanglmeier, Ph.D. Thesis, Forschungszentrum Jtilich, F.R.G. (1990). [25] J. Hubbel, W. Veigele, E. Briggs, R. Brown, D. Cromer and R. Howerton, J. Whys. Chem. Refi Data 4, 471 (1975). [26] L. Parratt, Phys. Rev. 95, 359 (1954). [27] H. Schwenke, W. Bemeike, J. Knoth and U. Weisbrod, Adv. X-Ray Anal. 32, 105 (1989). [28] H. Schwenke and J. Knoth, Handbook on X-Ray Spectrometry, Ed. R. van Grieken, to be published. [29] A. Iida, K. Sakurai, A. Yoshinaga and Y. Goshi, Nucl. Instrum. Meth. A 246, 736 (1986). [30] M. Bedzyk, G. Bommarito and J. Schildkraut, Phys. Rev. Letters 62, 1376 (1989). [31] D. K. G. de Boer and W. W. van de Hoogenhof, X-Ray Spectrom., accepted for publication. [32] D. K. G. de Boer and W. W. van de Hoogenhof, Spectrochim. Acta 46B, (1991).
0%4-8547/91 s3.00 + .lm 0 1991 Pergamon Press pk.
Fundamentals of total reflection X-ray fluorescence PETER KREGSAMER Atominstitut der osterreichischen
Universititen,
SchtittelstraOe 115, A-1020 Wien, Austria
(Received 14 December 1990; accepted 13 March 1991) Abstract-In the last years, total reflection X-ray fluorescence (TXRF) has shown to be an analytical technique for trace-element analysis. Theoretical considerations for the angular behaviour of scattered and fluorescence intensities are reviewed for the reflector material quartz and Mo-K_ excitation. The main influencing parameters are the critical angle, the transmission coefficient and the angle of refraction. Also the interference effect between incoming and reflected beam (standing wave) for total reflection is described.
1. INTRODUCTION A COLLIMATED X-ray beam impinging upon a surface at a grazing angle of incidence is totally reflected [l]. The development of the phenomenon from an interesting effect for physicists to an analytical tool was done by various groups of authors [2-51, while the interference effect on top of reflectors described hereafter was first observed and explained in diffraction measurements ([6] and references therein). There are six kinds of application of X-ray total reflection:
(9 (ii) (iii) (iv)
(v) (vi)
“Conventional” TXRF, where small samples are deposited on suitable reflectors (substrates) and subsequently analysed. Surface analysis, where the reflector itself is inspected for contaminations. The high-energy cutioff effect, which allows it to alter the primary spectral distribution. Capillaries, which can be used as wave guides for X-rays, produce microbeams with a few pm diameter, with improved intensities compared to ordinary collimators. X-ray optical elements (lenses, focussing devices). The standing wave effect, which enables the measurement of small distances of atoms above surfaces.
The present work is devoted to the description of those total reflection effects that relate to the sample directly, while the interested reader is referred to Refs [7-lo] for capillaries, to Refs [ll-141 for the cut-off effect and to Refs [15,16] for X-ray optics.
2. THEORETICAL The theoretical fundamentals of TXRF can be deduced in a way analogous to that in light optics, One has to solve the problem of an electromagnetic plane wave that hits the plane boundary between vacuum and a medium described by its dielectric constant e, which takes into account both scattering and absorption and. can be calculated by quantum mechanics (e.g. Ref. [17]). The electromagnetic waves of the incoming, reflected and refracted beam can be defined in the usual way, and Fresnel’s formulae will give the desired ratios of the amplitudes. For exact calculations, the polarization state of the primary radiation and the propagation of the refracted beam as a so-called inhomogeneous wave have to be considered ([l&19] and references therein). The inhomogeneity relates to the fact that the planes of equal phases are perpendicular to the respective wave vector, whereas the planes of equal amplitudes are parallel to the surface of the medium. For the primary as well as the reflected waves they are identical; for the refracted beam they 1333
P. KREOSAMER
1334 1T
---------__________________, 0
10’
h 0.
da
13
1.
2.i Incident
3.0 3.b Angle (mrad)
Fig. 1. The transmission coefficient T and reflection coefficient R have their inflection points near (pE= 1.8 mad for MO-K, totally reflected on quartz.
are distinct. There exists a direction, inside the medium (depending on the state of polarization), that describes the energy flow (Poynting vector) and is not perpendicular in the general case to the two above mentioned planes. With such a strict derivation one can obtain a refractive index n, which is the ratio of the vacuum velocity of light and the phase velocity of the wave inside the medium. This refractive index is a real quantity, in general less than 1 (though it may be different for energies close to an absorption edge of the reflector), and depends in a complex way on the dielectric constant (n* # e) and also depends on the incident angle [18,20]. Fo~unately, the effects of neglecting the polarization state and simpli~~tion of the refraction index for, e.g. Mo-I& primary radiation and a quartz reflector are insignificant for the calculation of transmission and reflection coefficients as well as refraction angles. (In principle one should show this for every new combination of exciting radiation and reflector material [21].) 2.1. APrgularde~ende~~i~ for TXRF One of the main features of TXRF is the strong dependence on the angle of incidence &-,of both the scattered as well as the fluorescence signals that originate from the substrate itself. One will be interested in minimizing the scattering contribution of the reflector and thus measure small (non-reflecting) samples on top of it. On the other hand, for the inspection of wafer surfaces, the fluorescence signal of contamination atoms should be m~imum. The following factors have to be considered: (1) For TXRF we generally can assume that the area on the reflector surface effectively seen by the detector is smaller than the area hit by the primary radiation, therefore a geometry factor proportional to sin J10has to be used. For when this assumption does not hold more detailed considerations can found in Ref. [22]. For total reflection, the angles of incidence and refraction are so small, that the sine can be replaced by its argument. (2) Only those primary photons that are not totally reflected on the (ideal flat) surface are able to penetrate and induce interactions. This leads to a transmission coefficient T (Fig. l), which, owing to energy conservation, sums up with the reflection coefficient R to unity (T + R = 1). The so-called critical angle cpCwill be close to the infIection point of these coefficients. Also the surface quality of the reflector must not be disregarded: insuffi~ent flatness will lead to only partial reflection of the primary beam, whereas roughness can be treated mathematically to some extent (see Refs [23,24] and references therein), but can quench total reflection completely. (3) If we consider the phenomenon of elastic or inelastic scatttering, the differential
Fundamentals
of TURF
Scattsring
1335
Angle (degraas)
Fip, 2. Elastic (full line) and inelastic (dashed line) differential scattering cross sections for a primary radiatkm of Mo-K_ scattered on atomic (amorphous) Si. The scattering angle a, which is the sum of incident and detection angle, will be in the range of 90” for TXRF. crofs sections ~~~~~~~~, which depend on the scattering angle 8, have to be calculated. Using atumie form factors and incoherent scattering functiuns [Zs] (s~pli~ca~on of unbound, irregular silicon atoms, especially for scattering into the forward direction) one obtains the respective angular dependencies in Fig. 2, which show a smooth behaviour near a value of 90”. Although there does not exist a fixed angle of detection in TXRF (there is a spread in angles because of the relatively large sample and detector diameters compared to their distance) an averge scattering angle af 90” is a good approximation, resulting in constant scattering probabilities. (4) The absorption of the refracted beam inside the substrate to the location where an interaction takes place and of the resulting radiation from there again to the surface (mass attenuation coefficient h) into the direction of the detector may not be neglected. The first absorption path (mass atteuuation coefficient tar) can be described by the penetration depth zP This quantity can be underst~ as the depth from where information comes out of the substrate. Because of its direct pro~~ionali~ to the angle of refraction #’ (Fig. 3), it shows the same sudden decrease near the critical angle of total-reflection cpC(e,g. Ref, [26]): scattering
+&I) = JI’ ($0) * bp
w
where p is the density. zP will roach approximately 3 nm for the theoretical incidence
0.
1
0.6
1.
lb
h
b
kT&nt A&(rnmd3j
Fig, 3. Refraction ar@e $8 as function of the incidence an&e lffc for MO-K, and a quartz reflector. The dashed line shows JI’ = Jlo (no refraction).
P. &tRGSAMER
1336 3.0 -
3 2
2.5 -
B c" s S 2 ii a 8 m
2.0 .' .' ** .' ..' .' .' .' 1.0 I' ..' .' .' .' 0.s .' I* *' .' .0.0 . 1. 0.0 Oh 1.h 1.5 -
incident
I
Lila
I
(mad;
Fig. 4. The background intensity for TXRF measurements, if Mo-K, primary radiation undergoes total reflection on a quart2 reflector. The dashed line represents the case of no total reflection.
angle of 0”, independent of the primary energy and only slightly dependent on the reflector material. For practical purposes, the value at the angle (pFis usually given. (5) If the concentration c of the (trace) elements investigated varies as a function of the distance to the surface z, a depth profile K(z) has to be defined. (6) The path length of the incident radiation along a depth dz is dz/JI’. If all factors mentioned are collected, a general intensity formula for both scattering and fluorescence (only the scattering cross sections have to be replaced by the absorption coefficient r divided by 4~) can be given:
44~0~4
= exp{-z~(l&,(Jlo)+
p2. p)} =
exp [- z * p &j+c2)].
(3)
2.2. Spectral background Only small samples are used for “conventional” TXRF so that, they should not significantly contribute to scattering if their residues and size are chosen properly. Otherwise .another kind of substrate and (or) excitation geometry will perhaps be advantageous. If we are interested in the angular dependence of the scattering processes originating from the substrate, the integration over the depth can be performed under the assumption of an “infinite” thickness with constant concentrations. As the attenuation coefficients for the primary and scattered radiation are equal or only slightly different (inelastic scattering), only two factors (called energy transfer in Refs [27,28]) govern the angular behaviour: scatter Z(Jro),,, a Jlo *T(Go). $ . %(90’, reflector). For angles larger than the critical angle, the geometry factor is responsible for the increase in the spectral background, whereas the sudden reduction (Fig. 4) for smaller angles originates from the almost perfect total reflection; the reflection coefficient is nearly 1 in this range. The dashed line, which describes the case of the same arrangement but without total reflection (e.g. a scratched reflector) reveals an improvement by a factor of T(JIo) for Jr0 < CPC).
Fundamentals of TXRF
1337
To compare TXRF with conventional XRF measurements one would put the sample on a suitable thin foil (45” geometry). Under the assumption that the excitation and detection would be equally efficient and the rest of the setup can be kept constant (especially the very short distance sample-detector), the background intensity will only be represented by: scatter m da I XRF a k;. -&9O”,foil)
(5)
where m/F is the mass per area of this foil. The dependence on the angle of incidence of absorption and geometry factor cancels. We assume for conventional XRF: m/F = 1 mg/cm2 and for TXRF an adjusted incident angle of 1 mrad, resulting in a transmission coefficient of T(JI, = 1 mrad) = 6.6 x 10m3. The absorption coefficient for Mo-K, in quartz is CL,= 3.8 cm2/g. The scattering cross sections are practically equal for both materials: 0.01 cm2/(g-sr) at 90”. If we insert those values in Eqns (4) and (5), the resulting ratio of the backgroud intensities would favour TXRF at this working point by a factor of approximately 500. 2.3. Contaminations in surfaces The previously defined function (2) shows an interesting behaviour, if pi/~ < 1 is assumed and pi/+’ < l.~~;follows [21] a needle-like maximum near the critial angle of the substrate, which, in the optimum case, can reach four times the intensity values at larger angles. This is partly fulfilled if, e.g. Mo-K, primary radiation produces lowenergetic fluorescence radiation. The effect can be understood, if one has in mind the very shallow path of the refracted beam until it can excite fluorescence radiation, which hence only has to cover a very short distance in the direction of the detector. For angles J10well below cpC,the decrease in the transmission coefficient T will compensate this and the intensity goes to zero. A suitable depth profile can support this tendency of peaking for the more realistic case that the absorption coefficients for the exciting radiation and the produced tluorescence radiation are of the same order of magnitude. For a thin zone of contamination atoms i (thickness ti being in the range of nm, so that in formula (3) A(&,z) 2: 1) with constant concentration (K(r) * ci) the fluorescence intensity (2) can be simplified:
Figure 5 shows such a behaviour, which in practice can be realized for wafer contaminations. (The divergence of the primary radiation will smooth the needle [28] and should be considered.) In principle, depth profiles of, e.g. contaminations of wafers can be extracted from measurements of such angular dependencies, at least, by fitting parameters of assumed distributions. For uncertainties in the measured intensites and lack of mechanical precision, the theoretical deconvolution (Laplace transformation) seems to be problematic owing to the inherent rapid changes in the angular behaviour. Non-reflecting residues on top of suitable surfaces give a doubled fluorescence intensity, when measurements are made at angles of incidence smaller than the respective critical angle of the substrate, owing to the twofold excitation [27], (Fig. 6). On the other hand, atoms within the range of the penetration depth can show a pronounced peak near (oC(see, e.g. Ref. [29]).
1338
P. KaxoSAMxR
Incident Angie (mrad)
Fig. 5. For primary radiation of MO-&, with 0” divergency, the inspection of a silicon surface wig yield an increased fluorescence signal of contamination atoms near (oe, compared to the case where no total reflection will occur (dashed line). A rectangular (constant) profile with a depth of 1 nm has been assumed.
Fig. 6. Twofold excitation of the sample for TXRF by the primary beam (1) and the reflected beam (2), for a case where the reflector is long enough so that beam (2) can occur.
2.4. Standing wave
Only very recently the interference effect on top of the reflector between the incoming and reflected beam has been recognized. The undisturbed coherent superposition of the plane electromagnetic waves results in a variation of the intensity pattern, depending on the distance above the surface, called a standing wave. Figure 7 displays the fundamental facts; the length D gives the distance between two maxima of the standing wave (A = wavelenth of the radiation): A
WJlo)= m,
(7)
D
Fig. 7. The interference wne (standing wave) between the incident (lo) and reflected (Z,J plane waves with wavelength A shows nodes and antinodes with a period of D.
and is typically in the range between 10 and 100 nm. If allowance is made for a phase factor Q, between incoming and reflected beam [30], the intensity of the standing wave as a function of the height z above the reflector surface and Jlo can be written as
Fundamentals of TXRF
@o,t)
= 10 * 1+ Wd
I
+ 24m
- 2+-J} * cos @blJ010)
1339
(8)
if absorption of any possible sample is excluded. This intensity can vary between 0 and 4 times the primary intensity IO. A proportional variation of the fluorescence radiation of atoms within this interference zone has already been used to determine the position of a zinc film above a gold surface to 20 mn with a precision of 0.2 nm 1301.For more complex problems such as thin (multiple) layers and multi-layer mirrors the reader is referred to Refs [23,31,32]. For conventional TXRF where granular residues on top of reflectors should be inspected, the problem might arise that samples with different heights (small compared with D) above the surface of the substrate (otherwise equal conditions) can give different fluorescence intensities. For homogeneous residues where many of those maxima and minima will occur within the sample height, this effect will level out to a large extent. If an internal standard can be homogeneously mixed with the sample, the ratio of the respective fluorescence intensities will lead to an acceptable precision in any case. Therefore the use of an internal standard normally applied for easier quantification of unsown samples seems to be inherently necessary for TXRF measurements. 3.
CONCLUSIONS
The description of the theoretical fundamentals of TXRF leads to a better understanding of the effect and enables a complete comparison of experimental results with theoretical calculations for samples on top of reflectors (“conventional” TXRF), impurities and contaminations in and on wafer surfaces, thin (multiple) layers on substrates, multi-layer mirrors, and the standing wave effect. For all those examples the angle of incidence of a well collimated primary beam plays an important role for the different kinds of observed intensities, and its controlled variation can give depth information in the nm-range. Ack~w~dgemen~~ would like to thank the “Fonds zur Forderung der ~~n~h~tichen the financial support of the work (parts of project P711.5).
Forschung” for
REFERENCES [l] [2] [3] [4] [S] [6] [7] [8] [9] [lo]
A. H. Compton, Phil. Mug. 45, 1121 (1923). Y. Yoneda and T. Horiuchi, Rev. Sci. Instrum. 42, 1069 (1971). H. Aiginger and P. Wobrauschek, Nucl. Instrum. Met/z. 114, 157 (1974). P. Wobrauschek and H. Aiginger, Anal, Chem. 47, 852 (1975). J. Kuoth and H. Schwenke, Frese&s’ 2. Anal. Chem. 291, 200 (1978). hf. Bedzyk, D. Bilderback, G. Bommarito, M. Caffrey and J. Schikikraut, Science 241, 1788 (1988). A. Rindby, NucI. instrum. Meth. A 249, 536 (1986). E. Stem, Z. Kalman, A. Lewis and K. Lie~~nn, Appl. Optics 27, 513.5 (1988). A. Carpenter, X-Ray Spectrom. 18, 253 (1989). P. Engstr&n, S. Larsson, A. Rindby, P. Voghs, G. Nilsson and B. Stocklassa, X-Ray Microbeam Spectroscopy, technical report, University of Goteborg, Sweden (1989). [ll] G. Damaschun, G. Kley and J. Miiller, Actu Phys. Austriaca 28, 233 (1968). 1121 E. Schnabel, R. Hosemann and B. Rode, J. Appi. Phys. 43, 3237 (1972). [13] H. Schwenke and J. Knoth, Fresenius’ 2. Anaf. Chem. 301, 7 (1980). [14] B. Lairson and D. Bilderback, Nucl. Instrum. Meth. 195, 79 (1982). [U] S. Wilkins, A. Stevenson, K. Nugent, H. Chapman and S. Steenstrup, Rev. Sci. Instrum. 60, 1025 (1989). [16] M. Kumakhov and F. Komarov, Whys. Reports 191, 289 (1990). [17] B. Henke, P. Lee, T. Tanaka, R. Shimabukuro and B. Fujikawa, Atom. Data Nucl. Data Tables 27, 1 (1982). [18] M. Levevere and M. Montel, Optica Actu 20, 97 (1973). flQ] P. Lee, Am. 1. Phys. 53, 885 (1985). [20] P. Kregsamer, Thesis, Technische Universitiit Wien, Austria (1990). [21] B. Henke, Phys. Rev. A 6, 94 (1972).
1340
P.
KltEQSAMER
[22] D. K. G. de Boer, X-Ray Spectrom. 18, 119 (1989). [23] A. Krol, C. Sher and Y. Kao, Phys. Rev. B 38, 8579 (1988). [24] F. Stanglmeier, Ph.D. Thesis, Forschungszentrum Jtilich, F.R.G. (1990). [25] J. Hubbel, W. Veigele, E. Briggs, R. Brown, D. Cromer and R. Howerton, J. Whys. Chem. Refi Data 4, 471 (1975). [26] L. Parratt, Phys. Rev. 95, 359 (1954). [27] H. Schwenke, W. Bemeike, J. Knoth and U. Weisbrod, Adv. X-Ray Anal. 32, 105 (1989). [28] H. Schwenke and J. Knoth, Handbook on X-Ray Spectrometry, Ed. R. van Grieken, to be published. [29] A. Iida, K. Sakurai, A. Yoshinaga and Y. Goshi, Nucl. Instrum. Meth. A 246, 736 (1986). [30] M. Bedzyk, G. Bommarito and J. Schildkraut, Phys. Rev. Letters 62, 1376 (1989). [31] D. K. G. de Boer and W. W. van de Hoogenhof, X-Ray Spectrom., accepted for publication. [32] D. K. G. de Boer and W. W. van de Hoogenhof, Spectrochim. Acta 46B, (1991).