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The current limits of the laser-acoustic test method to characterize low-k films
Microelectronic Engineering 82 (2005) 393–398 www.elsevier.com/locate/mee
The current limits of the laser-acoustic test method to characterize low-k films D. Schneider b
a,*
, S. Fru¨hauf b, S.E. Schulz b, T. Gessner
b,c
a Fraunhofer Institute for Material and Beam Technology IWS, 01277 Dresden, Germany Centre for Microtechnologies, Chemnitz University of Technology, 09107 Chemnitz, Germany c Fraunhofer Institute for Reliability and Microintegration IZM, 09126 Chemnitz, Germany
Available online 24 August 2005
Abstract The intention to make isolator films with dielectric constants <2.2 has initiated the development of porous siloxanebased films like silica xerogel and silsesquioxane-type materials. Although, the dielectric properties achieved are promising, introducing the technology still requires adapting their mechanical stability to the subsequent chemo-mechanical polishing (CMP). Porosity up to 50% causes the mechanical resistance to reduce drastically. According to several investigations, a value of more than 2 GPa seems to be required for the elastic modulus quantifying the stiffness of the film material. Efforts are currently undertaken to make high porous low-k films with an elastic modulus as high as possible. This requires the elastic modulus of thin soft films to be measured reliably. Surface acoustic waves have been shown to be very sensitive to thin surface films down to thickness of few nano-meters. This technique has been used to study the properties of low-k silica xerogel SiCOH-films. Until now it is necessary to prepare thicker films about 1000 nm for any elastic measurement in order to ensure, that elastic measurement is reliable. The goal of the investigation was to evaluate the extensibility of laser-acoustic based elasticity analysis to films with thickness below 500 nm. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Low-k films; Youngs modulus; Density; Surface acoustic waves; Laser-acoustics
1. Introduction The rapid introduction of new materials to reduce the dielectric permittivity in interconnect * Corresponding author. Tel.: +49 0 351 2583 451; fax: +49 0 351 2583 300. E-mail addresses: [email protected], schneider@ iws.fhg.de (D. Schneider).
systems creates challenges of integration and materials characterization [1–3]. One of the most critical points is the low mechanical stability of insulator films in copper damascene structures. Chemical–mechanical polishing of metal lines impacts high stress to the insulating film stacks. Since the low-k materials of choice with k < 2.2 consist mainly of Si–O-networks (altered with organic functional groups) with up to 50 vol% porosity,
0167-9317/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2005.07.073
394
D. Schneider et al. / Microelectronic Engineering 82 (2005) 393–398
their mechanical stability is significantly deteriorated compared to conventional silicon dioxide. Processing such layers well adapted to the integrated structure requires the knowledge of their mechanical behaviour. Additionally, an optimized material development demands reliable characterization methods to evaluate progress and backstrokes. Techniques based on surface acoustic waves, like Brillouin scattering [4], laser grating [5] or laser-acoustic spectroscopy [6] have more and more gained acceptance, due to their sensitivity to very thin films and their more straightforward physical interpretation. The laser-acoustic spectroscopy is fast and easy to use. Comprehensive results for low-k films have already been published [7–10]. The investigation presented deals with the limitations of applying this technique for low-k films.
2. Laser-acoustic technique The laser-acoustic method uses surface acoustic waves. Elastic vibrations propagate along the material surface with amplitude decaying within the material exponentially. The wave amplitude is highest at the surface, making the wave very sensitive to surface films even much thinner than the penetration depth of the wave. Unlike the nanoindentation test, the laser-acoustic method is a dynamic technique. The indentation performs a slow, quasi-static deformation with high local deformation in the zone beneath indenter tip. The surface acoustic wave makes a very small completely linear-elastic displacement, only in the range of few Angstroms. The deformation alternates very fast with frequencies in the MHz range. Therefore, the result is not influenced by creeping effects and represents the completely linear-elastic stiffness of the test material. The wave parameter to be measured is the propagation velocity of the wave, also termed phase velocity. Because the acoustic wave propagation is a dynamic process, apart from the elastic constants of the material the mass density influences the wave velocity. For the simple case of a homogeneous and isotropic material described by
Youngs modulus E, Poisson ratio m and density q, the following relation represents a good approximation for the dependence of phase velocity c of the surface acoustic wave on the material properties: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:87 þ 1:12m E c¼ . ð1Þ 1þm 2qð1 þ mÞ The penetration depth of the surface acoustic wave is defined by that depth from the surface where the amplitude has decayed to 1/e of the amplitude at the surface, approximately corresponding to the wavelength k. Increasing the frequency reduces the penetration depth of the wave. Higher frequency waves are more influenced by the film and propagate with a velocity closer to that of the film material. Lower frequency waves penetrate deeper into the substrate material and propagate with the velocity closer to that of the substrate. This phenomenon results in the phase velocity of the surface acoustic waves to depend on frequency, termed dispersion. By the laser-acoustic technique the dispersion curve is measured, i.e., the phase velocity depending on frequency, and the material parameters are deduced by fitting a theoretical curve. Fig. 1 shows the schematic representation of the test equipment. A focused nitrogen pulse laser is used to generate wide-band surface acoustic wave impulses in the test material. The short heating rapidly expands the material in the focus of the laser beam, resulting in a high frequency vibration that propagates along the surface as surface acoustic
Fig. 1. Schematic equipment.
representation
of
the
laser-acoustic
D. Schneider et al. / Microelectronic Engineering 82 (2005) 393–398
wave. This wave is detected by a piezoelectric transducer with a bandwidth of 200 MHz [11]. A digitizing oscilloscope records the laser-acoustic signal. Both test sample and detector are fixed on a translation stage that moves perpendicular to the laser beam with an accuracy of ±1.5 lm. In this way, the distance x between the laser focus line and the transducer is precisely varied which is required for an accurate dispersion measurement. The laser-acoustic signals uj(t) (j = 1 or 2 for laser position 1 and 2) are captured for at least two different distances x1 and x2 between the laser focus line and the transducer. Fourier-transforming the signals enables the amplitude Uj(f) and phase spectra Uj(f) to be calculated. The phase spectra Uj(f) are used to calculate the phase velocity c(f) of the surface acoustic waves depending on frequency f Cðf Þ ¼
ðx2 x1 Þf 2p . U2 ðf Þ U1 ðf Þ
ð2Þ
Youngs modulus E, density q and/or the thickness of the film can be deduced from the dispersion curve by means of the theory of the surface acoustic wave propagation in coated material. The elastic constant C11 of the silicon substrate is yielded additionally. The theory is not as straightforward as relation (1) is, but can exactly be derived from physical principles of the elasto-dynamic theory [12]. Because the theory can only be derived in implicit form, a fit procedure has to be applied to determine the material parameters, making use of the Levenberg-Marquardt algorithm [13] to minimize the least-square-error. How many parameters can be fitted depends on the shape of the dispersion curve and the bandwidth of the measurement. The details of the procedure to measure the dispersion curve and the theoretical analysis is described elsewhere [14–16].
395
studied have been thick enough, in the range of one micrometer, to make sure that both E and q can be gained. In order to find the lower limit above both film parameters can still be fitted, a sample series with varying film thickness, d = 72, 99, 155, 422, 821 nm, has been prepared. The dispersion curves measured are shown Fig. 2 revealing that they have a distinct curvature for the thickness of 821 and 422 nm. Both Youngs modulus E and density q of the films can be fitted. The thinner the film, the more the curve gets a straight line, not enabling one to fit more than one film parameter. Fitting the curves has yielded the film parameters presented in Table 1. The samples were measured at least five times to get mean value and standard deviation. The standard deviation increases with reducing film thickness especially for the Youngs modulus. The density scatters considerably less than the elastic modulus does. Below d = 155 nm the fit procedure does not converge for two free film parameters any more. The film density obtained from the mean values of the thicker films had to be put into the fit procedure to yield the film modulus. Whereas for the films with thickness from d = 99 to 821 nm film moduli from 1.66 to 1.96 GPa were calculated, for the 72 nm film a distinctly lower value E = 0.26 GPa was yielded.
3. Results and discussion The previous investigations with the laseracoustic technique have shown that Youngs modulus E and density q can be deduced from the dispersion curves measured for low-k films. The film thickness has to be measured ellipsometrically and introduced into the fit procedure. The films
Fig. 2. Dispersion curves measured for low-k films with different film thickness d.
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D. Schneider et al. / Microelectronic Engineering 82 (2005) 393–398
Table 1 Youngs modulus E and density q of low-k films with reducing films thickness Film thickness (nm)
q – Film (g/cm3)
E – Film E (GPa) Mean value
Fitting Youngs modulus E and density q of the film 821 1.8 422 1.69 155 1.96
Standard deviation
Mean value
Standard deviation
0.01 0.03 0.6
0.866 0.889 0.87
0.002 0.002 0.013
0.875 0.875
– –
Fitting Youngs modulus E of the film with density q as input parameter 99 1.66 0.19 72 0.26 0.04
The experimental results in Table 1 confirm that there is a lower limit for the film thickness below that only one film parameter of both elastic modulus E and density q can be determined. To illustrate the effect of the geometrical relation between the penetration depth of the wave (about the wavelength k) and the thickness of the film d, it is useful to present the dispersion curve in a normalized form as shown in Fig. 3. The frequency axis can be replaced by the ratio of film thickness to wavelength k, dividing each frequency value f by the related phase velocity c and multiplying the film thickness d, f Æ d/c. In the normalized diagram, all dispersion curves measured for different film thicknesses would fit a general curve on the condition the films had the same properties apart from differing in the film thickness. Fig. 3 shows
Fig. 3. Normalized diagram obtained from the dispersion curves in Fig. 2 showing the phase velocity c versus the ratio of film thickness d to wavelength k determining the penetration depth of the wave.
the test samples nearly meet this requirement. Near the origin d/k = 0 the general dispersion curve is nearly linear and can be approached by a simple relation c ¼ c0 þ f ðE=q; C 011 =q0 Þ ðd f Þ with c0 denoting the wave velocity of the substrate and f ðE=q; C 011 =q0 Þ denoting a function of the ratio of elastic constants and density of the film and substrate. Measuring in this range cannot provide both E and q of the film. A two-film-parameter fit can successfully be done only if the measurement can be extended into the range where summands of higher order contribute to the dispersion that reveal in a distinct curvature of the dispersion curve. This can be achieved by either increasing the thickness of the film or applying higher frequencies of the wave. In the film thickness range below the lower limit where only one film parameter can be deduced, the method is more suitable for determining the film density instead of determining the Youngs modulus of the film. Table 2 shows also the effect of varying the density as input parameter on the value yielded for the film modulus. Increasing the density by 3% from q = 0.865 g/cm3 to q = 0.89 g/cm3 causes the result for the modulus to jump by more than 80% from E = 1.2 to 2.25 GPa. Table 3 shows the opposite procedure putting in the Youngs modulus to calculate the density. With a rough assumption for the elastic modulus between E = 1.3 and 2 GPa, the density yielded changes only from q = 0.868 to 0.89 g/cm3. It is to note that this behaviour only applies for such very compliant films as the low-k films are. For hard coatings as diamond-like carbon or TiN, the uncertainty of the density value has a less
D. Schneider et al. / Microelectronic Engineering 82 (2005) 393–398
397
Table 2 Effect of varying the film density on the Youngs modulus fitted for the 99 nm film Film thickness (nm)
99 99 99
q – Film (g/cm3)
E – Film E (GPa) Mean value
Standard deviation
Mean value
Standard deviation
1.2 1.66 2.25
0.17 0.19 0.25
0.865 0.875 0.89
– – –
Table 3 Effect of varying the Youngs modulus on the film density fitted for the 99 nm film Film thickness (nm)
99 99 99
q – Film (g/cm3)
E – Film E (GPa) Mean value
Standard deviation
Mean value
Standard deviation
1.3 1.7 2.0
– – –
0.868 0.873 0.882
0.004 0.004 0.005
effect on the results for the film modulus, because their stiffness usually distinctly higher than that of the substrate dominates the dispersion of the acoustic surface wave [16,17]. Fig. 3 suggests the way to lower the limit of the thickness for the two-parameters fit: Improving accuracy of the measurement of the phase velocity. Higher frequency range (bandwidth).
The distance has been limited here by the design of the translation stage. Extending the bandwidth to higher frequencies is another way to make the two-parameters fit possible for films thinner than d = 155 nm. The bandwidth of the laser-acoustic device used is limited by the piezoelectric detector. Efforts to improve the detector design are worth to do for the low-k films.
4. Conclusions The phase velocity in the dispersion curve is derived from measuring the time the wave needs to pass the distance x2 x1. The measuring uncertainty depends on the accuracy of the devices used in the equipment, the oscilloscope with time error of |Dt| 6 0.5 ns and translation stage with distance error of |Dx| 6 1.5 lm. Taking into account that the phase velocity is calculated from relation (2) representing the procedure of a difference measurement, the uncertainty of the phase velocity can be estimated from |Dc/c| 6 2|Dx/(x2 x1)| + 2|Dt/ (t2 t1)|. This relation suggests that without using better, however, also more expensive measuring devices the uncertainty can be reduced by simply increasing the measuring distance x2 x1, accompanied with an increase of the time t2 t1 the wave needs to pass this distance. The curves in Fig. 2 were measured with a measuring distance of x2 x1 = 20 mm, resulting in |Dc/c| 6 3.104.
The laser-acoustic test device used, specified by an accuracy of |Dc/c| 6 3.104 and a bandwidth of 200 MHz, can independently determine both elastic modulus and density of low-k films thicker than 150 nm. The uncertainty of the results reduces with increasing film thickness. For the Youngs modulus, values in the range from E = 0.25 to 2 GPa have been measured and for the density values from q = 0.866 to 0.89 g/cm3. The very low scattering of the values yielded for the film density is to emphasize (|Dq/q| 6 1%). Therefore, below the lower limit for the twoparameters fit the test method is more suitable for measuring the film density than for measuring the film modulus. This applies only for these very compliant low-k films. For hard coatings it is opposite.
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D. Schneider et al. / Microelectronic Engineering 82 (2005) 393–398
The limit of the film thickness for the twoparameters fit can be lowered by increasing the bandwidth and improving the measuring accuracy.
References [1] International Roadmap for Semiconductors (ITRS) Edition 2003, Sematech. [2] ITRS 2003, Update 2004. [3] K. Maex, M.R. Baklanov, D. Shamiryan, F. Iacopi, S.H. Brongersma, Z.S. Yanovitskaya, J. Appl. Phys. 93 (2003) 8793. [4] C.M. Flannery, T. Wittkowski, K. Jung, B. Hillebrands, M.R. Baklanov, Appl. Phys. Lett. 80 (2002) 4594. [5] A. Maznev, A. Mazurenko, L. Zhuoyun, M. Gostein, Rev. Sci. Instrum. 74 (2003) 667. [6] P. Hess, Proc. SPIE 4703 (2002) 1.
[7] C.M. Flannery, D.C. Hurley, AIP Conf. Proc. 657B (2003) 1463. [8] C.M. Flannery, M.R. Baklanov, J. Mater. Res. 15 (2000) 2036. [9] C.M. Flannery, Ultrasonics 40 (2002) 237. [10] K. Schulze, S.E. Schulz, S. Fruehauf, H. Koerner, U. Seidel, D. Schneider, T. Gessner, Microelectr. Eng. 76 (2004) 38. [11] H. Coufal, R. Grygier, P. Hess, A. Neubrand, J. Acoust. Soc. Am. 92 (1992) 2980. [12] W. Farnell, E.L. Adler, in: W.P. Mason, R.N. Thurston (Eds.), Physical Acoustics, vol. IX, Academic Press, New York, 1972, p. 3. [13] D. Marquardt, J. Appl. Math. 11 (1963) 431. [14] T. Schwarz, D. Schneider, Experimentelle Technik der Physik 36 (1988) 7. [15] D. Schneider, T. Schwarz, B. Schultrich, Thin Solid Films 219 (1992) 92–102. [16] D. Schneider, T. Schwarz, H.J. Scheibe, M. Panzner, Thin Solid Films 295 (1997) 107. [17] D. Schneider, P. Siemroth, T. Schulke, J. Berthold, B. Schultrich, H.H. Schneider, R. Ohr, B. Petereit, H. Hillgers, Surf. Coat. Technol. 153 (2002) 252.
The current limits of the laser-acoustic test method to characterize low-k films D. Schneider b
a,*
, S. Fru¨hauf b, S.E. Schulz b, T. Gessner
b,c
a Fraunhofer Institute for Material and Beam Technology IWS, 01277 Dresden, Germany Centre for Microtechnologies, Chemnitz University of Technology, 09107 Chemnitz, Germany c Fraunhofer Institute for Reliability and Microintegration IZM, 09126 Chemnitz, Germany
Available online 24 August 2005
Abstract The intention to make isolator films with dielectric constants <2.2 has initiated the development of porous siloxanebased films like silica xerogel and silsesquioxane-type materials. Although, the dielectric properties achieved are promising, introducing the technology still requires adapting their mechanical stability to the subsequent chemo-mechanical polishing (CMP). Porosity up to 50% causes the mechanical resistance to reduce drastically. According to several investigations, a value of more than 2 GPa seems to be required for the elastic modulus quantifying the stiffness of the film material. Efforts are currently undertaken to make high porous low-k films with an elastic modulus as high as possible. This requires the elastic modulus of thin soft films to be measured reliably. Surface acoustic waves have been shown to be very sensitive to thin surface films down to thickness of few nano-meters. This technique has been used to study the properties of low-k silica xerogel SiCOH-films. Until now it is necessary to prepare thicker films about 1000 nm for any elastic measurement in order to ensure, that elastic measurement is reliable. The goal of the investigation was to evaluate the extensibility of laser-acoustic based elasticity analysis to films with thickness below 500 nm. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Low-k films; Youngs modulus; Density; Surface acoustic waves; Laser-acoustics
1. Introduction The rapid introduction of new materials to reduce the dielectric permittivity in interconnect * Corresponding author. Tel.: +49 0 351 2583 451; fax: +49 0 351 2583 300. E-mail addresses: [email protected], schneider@ iws.fhg.de (D. Schneider).
systems creates challenges of integration and materials characterization [1–3]. One of the most critical points is the low mechanical stability of insulator films in copper damascene structures. Chemical–mechanical polishing of metal lines impacts high stress to the insulating film stacks. Since the low-k materials of choice with k < 2.2 consist mainly of Si–O-networks (altered with organic functional groups) with up to 50 vol% porosity,
0167-9317/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2005.07.073
394
D. Schneider et al. / Microelectronic Engineering 82 (2005) 393–398
their mechanical stability is significantly deteriorated compared to conventional silicon dioxide. Processing such layers well adapted to the integrated structure requires the knowledge of their mechanical behaviour. Additionally, an optimized material development demands reliable characterization methods to evaluate progress and backstrokes. Techniques based on surface acoustic waves, like Brillouin scattering [4], laser grating [5] or laser-acoustic spectroscopy [6] have more and more gained acceptance, due to their sensitivity to very thin films and their more straightforward physical interpretation. The laser-acoustic spectroscopy is fast and easy to use. Comprehensive results for low-k films have already been published [7–10]. The investigation presented deals with the limitations of applying this technique for low-k films.
2. Laser-acoustic technique The laser-acoustic method uses surface acoustic waves. Elastic vibrations propagate along the material surface with amplitude decaying within the material exponentially. The wave amplitude is highest at the surface, making the wave very sensitive to surface films even much thinner than the penetration depth of the wave. Unlike the nanoindentation test, the laser-acoustic method is a dynamic technique. The indentation performs a slow, quasi-static deformation with high local deformation in the zone beneath indenter tip. The surface acoustic wave makes a very small completely linear-elastic displacement, only in the range of few Angstroms. The deformation alternates very fast with frequencies in the MHz range. Therefore, the result is not influenced by creeping effects and represents the completely linear-elastic stiffness of the test material. The wave parameter to be measured is the propagation velocity of the wave, also termed phase velocity. Because the acoustic wave propagation is a dynamic process, apart from the elastic constants of the material the mass density influences the wave velocity. For the simple case of a homogeneous and isotropic material described by
Youngs modulus E, Poisson ratio m and density q, the following relation represents a good approximation for the dependence of phase velocity c of the surface acoustic wave on the material properties: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:87 þ 1:12m E c¼ . ð1Þ 1þm 2qð1 þ mÞ The penetration depth of the surface acoustic wave is defined by that depth from the surface where the amplitude has decayed to 1/e of the amplitude at the surface, approximately corresponding to the wavelength k. Increasing the frequency reduces the penetration depth of the wave. Higher frequency waves are more influenced by the film and propagate with a velocity closer to that of the film material. Lower frequency waves penetrate deeper into the substrate material and propagate with the velocity closer to that of the substrate. This phenomenon results in the phase velocity of the surface acoustic waves to depend on frequency, termed dispersion. By the laser-acoustic technique the dispersion curve is measured, i.e., the phase velocity depending on frequency, and the material parameters are deduced by fitting a theoretical curve. Fig. 1 shows the schematic representation of the test equipment. A focused nitrogen pulse laser is used to generate wide-band surface acoustic wave impulses in the test material. The short heating rapidly expands the material in the focus of the laser beam, resulting in a high frequency vibration that propagates along the surface as surface acoustic
Fig. 1. Schematic equipment.
representation
of
the
laser-acoustic
D. Schneider et al. / Microelectronic Engineering 82 (2005) 393–398
wave. This wave is detected by a piezoelectric transducer with a bandwidth of 200 MHz [11]. A digitizing oscilloscope records the laser-acoustic signal. Both test sample and detector are fixed on a translation stage that moves perpendicular to the laser beam with an accuracy of ±1.5 lm. In this way, the distance x between the laser focus line and the transducer is precisely varied which is required for an accurate dispersion measurement. The laser-acoustic signals uj(t) (j = 1 or 2 for laser position 1 and 2) are captured for at least two different distances x1 and x2 between the laser focus line and the transducer. Fourier-transforming the signals enables the amplitude Uj(f) and phase spectra Uj(f) to be calculated. The phase spectra Uj(f) are used to calculate the phase velocity c(f) of the surface acoustic waves depending on frequency f Cðf Þ ¼
ðx2 x1 Þf 2p . U2 ðf Þ U1 ðf Þ
ð2Þ
Youngs modulus E, density q and/or the thickness of the film can be deduced from the dispersion curve by means of the theory of the surface acoustic wave propagation in coated material. The elastic constant C11 of the silicon substrate is yielded additionally. The theory is not as straightforward as relation (1) is, but can exactly be derived from physical principles of the elasto-dynamic theory [12]. Because the theory can only be derived in implicit form, a fit procedure has to be applied to determine the material parameters, making use of the Levenberg-Marquardt algorithm [13] to minimize the least-square-error. How many parameters can be fitted depends on the shape of the dispersion curve and the bandwidth of the measurement. The details of the procedure to measure the dispersion curve and the theoretical analysis is described elsewhere [14–16].
395
studied have been thick enough, in the range of one micrometer, to make sure that both E and q can be gained. In order to find the lower limit above both film parameters can still be fitted, a sample series with varying film thickness, d = 72, 99, 155, 422, 821 nm, has been prepared. The dispersion curves measured are shown Fig. 2 revealing that they have a distinct curvature for the thickness of 821 and 422 nm. Both Youngs modulus E and density q of the films can be fitted. The thinner the film, the more the curve gets a straight line, not enabling one to fit more than one film parameter. Fitting the curves has yielded the film parameters presented in Table 1. The samples were measured at least five times to get mean value and standard deviation. The standard deviation increases with reducing film thickness especially for the Youngs modulus. The density scatters considerably less than the elastic modulus does. Below d = 155 nm the fit procedure does not converge for two free film parameters any more. The film density obtained from the mean values of the thicker films had to be put into the fit procedure to yield the film modulus. Whereas for the films with thickness from d = 99 to 821 nm film moduli from 1.66 to 1.96 GPa were calculated, for the 72 nm film a distinctly lower value E = 0.26 GPa was yielded.
3. Results and discussion The previous investigations with the laseracoustic technique have shown that Youngs modulus E and density q can be deduced from the dispersion curves measured for low-k films. The film thickness has to be measured ellipsometrically and introduced into the fit procedure. The films
Fig. 2. Dispersion curves measured for low-k films with different film thickness d.
396
D. Schneider et al. / Microelectronic Engineering 82 (2005) 393–398
Table 1 Youngs modulus E and density q of low-k films with reducing films thickness Film thickness (nm)
q – Film (g/cm3)
E – Film E (GPa) Mean value
Fitting Youngs modulus E and density q of the film 821 1.8 422 1.69 155 1.96
Standard deviation
Mean value
Standard deviation
0.01 0.03 0.6
0.866 0.889 0.87
0.002 0.002 0.013
0.875 0.875
– –
Fitting Youngs modulus E of the film with density q as input parameter 99 1.66 0.19 72 0.26 0.04
The experimental results in Table 1 confirm that there is a lower limit for the film thickness below that only one film parameter of both elastic modulus E and density q can be determined. To illustrate the effect of the geometrical relation between the penetration depth of the wave (about the wavelength k) and the thickness of the film d, it is useful to present the dispersion curve in a normalized form as shown in Fig. 3. The frequency axis can be replaced by the ratio of film thickness to wavelength k, dividing each frequency value f by the related phase velocity c and multiplying the film thickness d, f Æ d/c. In the normalized diagram, all dispersion curves measured for different film thicknesses would fit a general curve on the condition the films had the same properties apart from differing in the film thickness. Fig. 3 shows
Fig. 3. Normalized diagram obtained from the dispersion curves in Fig. 2 showing the phase velocity c versus the ratio of film thickness d to wavelength k determining the penetration depth of the wave.
the test samples nearly meet this requirement. Near the origin d/k = 0 the general dispersion curve is nearly linear and can be approached by a simple relation c ¼ c0 þ f ðE=q; C 011 =q0 Þ ðd f Þ with c0 denoting the wave velocity of the substrate and f ðE=q; C 011 =q0 Þ denoting a function of the ratio of elastic constants and density of the film and substrate. Measuring in this range cannot provide both E and q of the film. A two-film-parameter fit can successfully be done only if the measurement can be extended into the range where summands of higher order contribute to the dispersion that reveal in a distinct curvature of the dispersion curve. This can be achieved by either increasing the thickness of the film or applying higher frequencies of the wave. In the film thickness range below the lower limit where only one film parameter can be deduced, the method is more suitable for determining the film density instead of determining the Youngs modulus of the film. Table 2 shows also the effect of varying the density as input parameter on the value yielded for the film modulus. Increasing the density by 3% from q = 0.865 g/cm3 to q = 0.89 g/cm3 causes the result for the modulus to jump by more than 80% from E = 1.2 to 2.25 GPa. Table 3 shows the opposite procedure putting in the Youngs modulus to calculate the density. With a rough assumption for the elastic modulus between E = 1.3 and 2 GPa, the density yielded changes only from q = 0.868 to 0.89 g/cm3. It is to note that this behaviour only applies for such very compliant films as the low-k films are. For hard coatings as diamond-like carbon or TiN, the uncertainty of the density value has a less
D. Schneider et al. / Microelectronic Engineering 82 (2005) 393–398
397
Table 2 Effect of varying the film density on the Youngs modulus fitted for the 99 nm film Film thickness (nm)
99 99 99
q – Film (g/cm3)
E – Film E (GPa) Mean value
Standard deviation
Mean value
Standard deviation
1.2 1.66 2.25
0.17 0.19 0.25
0.865 0.875 0.89
– – –
Table 3 Effect of varying the Youngs modulus on the film density fitted for the 99 nm film Film thickness (nm)
99 99 99
q – Film (g/cm3)
E – Film E (GPa) Mean value
Standard deviation
Mean value
Standard deviation
1.3 1.7 2.0
– – –
0.868 0.873 0.882
0.004 0.004 0.005
effect on the results for the film modulus, because their stiffness usually distinctly higher than that of the substrate dominates the dispersion of the acoustic surface wave [16,17]. Fig. 3 suggests the way to lower the limit of the thickness for the two-parameters fit: Improving accuracy of the measurement of the phase velocity. Higher frequency range (bandwidth).
The distance has been limited here by the design of the translation stage. Extending the bandwidth to higher frequencies is another way to make the two-parameters fit possible for films thinner than d = 155 nm. The bandwidth of the laser-acoustic device used is limited by the piezoelectric detector. Efforts to improve the detector design are worth to do for the low-k films.
4. Conclusions The phase velocity in the dispersion curve is derived from measuring the time the wave needs to pass the distance x2 x1. The measuring uncertainty depends on the accuracy of the devices used in the equipment, the oscilloscope with time error of |Dt| 6 0.5 ns and translation stage with distance error of |Dx| 6 1.5 lm. Taking into account that the phase velocity is calculated from relation (2) representing the procedure of a difference measurement, the uncertainty of the phase velocity can be estimated from |Dc/c| 6 2|Dx/(x2 x1)| + 2|Dt/ (t2 t1)|. This relation suggests that without using better, however, also more expensive measuring devices the uncertainty can be reduced by simply increasing the measuring distance x2 x1, accompanied with an increase of the time t2 t1 the wave needs to pass this distance. The curves in Fig. 2 were measured with a measuring distance of x2 x1 = 20 mm, resulting in |Dc/c| 6 3.104.
The laser-acoustic test device used, specified by an accuracy of |Dc/c| 6 3.104 and a bandwidth of 200 MHz, can independently determine both elastic modulus and density of low-k films thicker than 150 nm. The uncertainty of the results reduces with increasing film thickness. For the Youngs modulus, values in the range from E = 0.25 to 2 GPa have been measured and for the density values from q = 0.866 to 0.89 g/cm3. The very low scattering of the values yielded for the film density is to emphasize (|Dq/q| 6 1%). Therefore, below the lower limit for the twoparameters fit the test method is more suitable for measuring the film density than for measuring the film modulus. This applies only for these very compliant low-k films. For hard coatings it is opposite.
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The limit of the film thickness for the twoparameters fit can be lowered by increasing the bandwidth and improving the measuring accuracy.
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