In this work, AlN and nanocrystalline diamond thin films as well as multi-layer structures on their basis are characterized towards their mechanical properties. In particular, the Young's modulus E and the residual stress σ are obtained by wafer bow measurements of thin films as well as by bulge experiments and vibration measurements of freestanding membranes. Depending on the growth conditions, the AlN thin films, deposited by reactive magnetron sputtering, revealed values of σ ∼ +300 up to +400 MPa and E ∼ 370 GPa, while the diamond films, grown by microwave plasma CVD, showed values of σ ∼ −60 to +60 MPa and E ∼ 870 up to 1000 GPa. The values and the accuracy of the characterization techniques used are discussed and their limits are demonstrated.

AlN and nanocrystalline diamond (NCD) are both widely used materials for micro-electro-mechanical systems (MEMS). In particular, properties of sputtered AlN thin films for piezoelectric transducers in sensing and microwave applications have been intensely investigated 1–3. In this way, stress-compensating techniques have led to improved piezoelectric radio frequency-MEMS switches using sputtered AlN thin films 4, 5.

In addition, NCD films deposited by microwave plasma enhanced chemical vapor deposition (MW-PECVD) have been much vaunted for their extraordinary material properties similar to single crystal diamond 6, 7, showing many advantages like low cost, ease of fabrication, and the material compatibility with AlN. In detail, NCD films have shown high mechanical qualities such as Young's modulus of ∼1100 GPa, fracture strength of ∼5.3 GPa, and low mass density of ∼3.5 g/cm^{3}8, also a high thermal conductivity of up to 14 W/cm · K 9 aside from a high optical transparency from IR into deep UV 10, 11.

Recently, several AlN-NCD heterostructures, made of sputtered AlN on CVD grown NCD films, have been reported for the use in MEMS resonators, surface acoustic wave devices and micro-optics 12–14. In the development of such structures, the mechanical properties of the thin films play a key role. In most studies, however, a single measurement technique is selected to determine the residual stress and/or the Young's modulus. Rarely the values have been confirmed by a second technique.

In this work, we compare three characterization techniques, namely the radius-of-curvature measurement, the bulge test and laser vibrometry of both, sputtered AlN and MW-PECVD grown NCD samples. From this, values of residual stress and Young's modulus are derived and the instrumental accuracy of the measurement techniques is discussed.

2 Thin film properties

2.1 AlN mechanical properties

The residual stress of thin films in MEMS is an important matter of research, as it determines the properties of the fabricated devices. A detailed understanding of the residual stress and with it the strain becomes even more important, when different materials are combined in heterostructures such as in AlN/NCD bilayer membranes used in this work.

The 2H structure, most common for AlN, has a hexagonal unit cell and thus, two lattice constants, a = 3.112 Å and c = 4.982 Å 15. The elastic properties of 2H single crystalline films are described by five elements of the elastic stiffness matrix C_{ij} denoted C_{11}, C_{12}, C_{13}, C_{33}, C_{44}16.

In the limit of small deviations from the lattice equilibrium 17, one can relate the in plane isotropic stress σ_{xx} to the strain ε_{xx} through the Young's modulus, E = (C_{11} + C_{12}–[2C_{13}]^{2})/C_{33}, according to the following relation: σ_{xx} = Eε_{xx}/(1 − ν), where ν is the Poisson's ratio. As the strain arising in the c-plane is usually isotropic (ε_{xx} = ε_{yy}), one can denote the components of the elastic stress tensor in the following way: σ_{xx} = σ_{yy} = (C_{11} + C_{12})ε_{xx} + ε_{zz}C_{13}.

For the non-epitaxial (either textured or polycrystalline) films, the mentioned properties are very sensitive to the fabrication method. Table 1 summarizes the elastic parameters for AlN thin deposits, calculated as well as experimental ones. It is obvious, that the elasticity of epitaxially grown and sputtered AlN films, which show single crystalline and c-oriented textured properties, respectively, are quite different. It is affected by the crystallographic disorder in the lattice, which, in turn, depends on the crystallization conditions, e.g. deposition method and temperature, growth chemistry and substrate surface conditions.

Table 1. Elastic constants of nanocrystalline and epitaxial AlN.

Investigations of sputtered AlN films have shown, that the gas pressure, the rf-power and the Ar/N_{2} flow ratio have the strongest influence on the residual stress 21–25. By varying one of the growth parameters the resulting residual stress of the AlN film can be tuned from compression to tension. It has been found, that the parameters' influence on the ionic bombardment of the film and with it the energy of the atoms is responsible for a strong change of the residual stress rather than the substrate material 24.

2.2 NCD properties

The residual stress in NCD films generally consists of two parts, thermal and intrinsic. Thermal stress originates from differences in the thermal expansion coefficients of the film and the substrate while different lattice constants of the film and the substrate as well as defects within the film are responsible for intrinsic stress. Additionally, thickness dependent stress gradients occur in the case of changing crystallographic structure during the film growth. This is the case for CVD grown NCD layers, where the grains grow with the layer thickness. Larger grains lead to a smaller non-diamond carbon fraction that is responsible for stress relaxation between the grains, described by the grain boundary relaxation model 26.

Studies of thin diamond films, grown by chemical vapor deposition, have shown that temperature and methane fraction during growth have the strongest influence on the resulting residual stress 26 and Young's modulus due to the incorporation of impurities and the presence of non-diamond phases at the grain boundaries 27. For NCD films, grown by CVD, all kinds of residual stress states have been reported: from highly tensile to strong compressive stressed films, depending on the growth method and the choice of parameters 26, 28–32. Also the Young's modulus can be tuned over a wide range, from 400 to 1100 GPa, by changing the methane gas flow and the microwave power density (Table 2).

Table 2. Comparison of Young's modulus values of CVD grown NCD films found in the literature.

The AlN films were grown on 3” Si (001) substrates with resistivities of 10–20 mΩ cm in a commercial Leybold Z550 RF magnetron sputtering system. More details on the AlN growth can be found in a previous work 34. For typical deposition conditions, the substrate temperature rises during growth from room temperature to almost 150 °C. The film thickness was determined using a rotating-analyzer variable-angle spectroscopic ellipsometer (VASE™ of J. A. Woolam Co.) in the spectral range from 190 to 1700 nm (6.52–0.73 eV).

Prior to NCD growth, 3” Si (001) wafers were seeded with a monodisperse diamond colloid known to yield NCD nucleation densities up to 10^{12} cm^{−2}35. Then the NCD layers were prepared by microwave plasma CVD at temperatures above 750 °C using a H_{2}/CH_{4} gas mixture following a standard deposition procedure 36. For the evaluation of the NCD mechanical properties, three films with a thickness of ∼220 nm were grown using methane concentrations of 0.5, 1.5, and 3%, respectively. For the exact determination of σ and E the thickness values of the used structures have been measured by means of an omt mm-202 normal incidence reflectometer equipped with an omt mm-102 video microscope.

AlN/NCD heterostructures have been made by smoothing the as-grown NCD layer using a chemo-mechanical polishing process followed by the sputtering of the AlN layer on top.

The Si etching for membrane fabrication is performed with an inductively coupled plasma multiplex system from Surface Technology Systems. The cyclic deep reactive ion etching process (c-DRIE) consists of an alternating sequence of etching (SF_{6} and O_{2} plasma) and passivation (c-C_{4}F_{8} plasma) 37.

The resonant frequencies of thin membranes have been measured by a laser Doppler vibrometry (LDV) method 38. In these studies, a Polytec MSA-500 vibrometer has been used, equipped with a velocity (0–1.5 MHz) and an amplitude (30 kHz–20 MHz) detector. The out-of-plane displacement of the membranes surfaces has been achieved by external mechanical excitation of the membranes.

For this purpose, the samples have been mounted onto a piezo stack actuator. The external electrical signal for driving the piezo stack has been generated using an Agilent 33250A function generator. The experimental LDV set-up is shown schematically in Fig. 1. The mounted sample was placed in a vacuum chamber equipped with a turbo pump and a needle valve system, that allows a precise control of the operational pressure in the range of 10^{−5}–1 bar. For the detailed spectral frequency studies, a network analyzer has been used with the excitation frequencies ranging from 10 kHz up to 500 MHz.

The frequencies applied for the external mechanical excitation in the LDV experiments were principally limited by the inertia and capacitance of the piezo-stack assembly to values below 500 kHz. In addition, under typical experimental conditions in air, the maximal velocity of the membrane's center for the fundamental resonant frequency f_{10} ∼ 124 kHz was estimated to be V_{max} = ωA ∼ 1.2 × 10^{−2} m/s that surely do not exceed the critical value leading to adiabatic processes caused by the resistance of the pressurized gas 39. This is important for the validity of the used formalism of the vibrometry measurements.

4 Methodological approach

4.1 Wafer curvature measurements

Magnitude and sign of the residual stress in as-grown layers can be determined from wafer curvature measurements. This technique has been frequently used for the characterization of both AlN 22–25, 40 and NCD thin films 26, 28, 31. In this case, scanning laser reflectometry (SLR) is used to measure the substrate curvature before (1/R_{0}) and after (1/R) film deposition.

The stress in the layer on the substrate with thicknesses d and d_{S}, respectively, is then calculated from the differences in curvatures using modified Stoney's equation 41:

((1))

where E/(1 − ν) is the biaxial modulus of the substrate.

The equation is derived under the assumptions that the film thickness d is much smaller than the thickness of the substrate d_{S} and that the latter behaves elastically for induced deformations with a constant Poisson's ratio ν. This method gives the mean stress value over the scan length.

4.2 Static deflection of circular membranes

A method that allows a more local determination of the residual stress and also of the Young's modulus is given by bulge experiments of thin membranes. It has been used for AlN 42 as well as NCD membranes 6, 33. In these studies, N_{2} gas was used to induce an overpressure at the backside of the membrane of radius R. White light interferometry (WLI) was used to measure the maximal deflection, z ≪ R, in the center of the membrane for every discrete value of the differential pressure Δp applied to the membrane. Thus, values of E and σ have been derived by fitting the experimental data using the formula for circular thin plates 43–45:

((2))

with the coefficients f(ν) = 1 and c_{2} = 4 for circular geometry and under the assumption of a spherical deflection (c_{1} = 8/3), given by Ref. 46. The restrictions related to the application of this formalism to thin membranes are considered in the discussion section.

4.3 Vibrometry measurements

The vibration of elastic bodies having one degree of freedom was thoroughly analyzed in the literature (see for instance Ref. 47). If an ideal membrane of radius R, uniformly distended in all directions due to a tension T, is vibrating in a viscous medium with a small damping factor 2n, the deflection of membrane surface z(r, θ, t) can be written in form of the wave equation 48:

((3))

with the wave speed v = (T/hρ)^{1/2} depending on the membrane tensile tension, T (N/m), and mass per unit area, hρ (kg/m^{2}). Both of these are assumed to be uniform. F_{0} is the amplitude of the periodic external force of angular frequency ω; n = cg/(2W) is the damping coefficient with the weight W and the constant c that is equal to the magnitude of the damping force when the velocity is equal to unity where the phase difference α between the disturbing force and the forced vibration is α ∼ arctan [2nω/(p^{2} − ω^{2})]. Equation (3) can be solved using separation of variables with the boundary conditions such that z(r, θ, t) = z(r, θ+2π, t) and z(R, θ, t) = 0 along the outer circumference and assuming that the membrane is initially at rest z(r, θ, 0) = 0 49.

In this case, the eigenfrequencies of the membrane are

((4))

where the matrix β_{ji} represents the jth zero of the ith Bessel function of the first kind, J_{i}48. It implies that due to the viscous gas damping, the eigenfrequencies slightly increase, while for the in vacuo case, the damping n can be neglected implying

((5))

By selective excitation of the (1,0) mode of a tensile stressed membrane, only the deflection of the membrane center (Eq. (3)) has to be taken into account, which reduces the vibration analysis to the 1D case of steady forced vibrations z(t) = Asin(ωt−α) under viscous damping.

5 Experimental results

5.1 Wafer curvature

The growth of high quality AlN thin films has been optimized regarding their piezoelectric and mechanical properties 52. The dependence of the residual stress of the films on the deposition parameters and the substrate material has been evaluated by wafer curvature measurements using standard SLR technique and can be found elsewhere 34.

The stress state of thin NCD films can be tuned from tensile to compressive values by changing the methane content and layer thickness. As shown in Fig. 2, for a given thickness the NCD films are tensile stressed for low methane flow rates and compressively stressed at higher ones. This characteristic has been explained by a higher sp^{2} content with a remarkable larger specific volume that produces a compressive stress field in the diamond film 26, 33.

5.2 Bulge test

Microfabricated AlN and NCD membranes have been tested in a static deflection mode towards their mechanical properties by means of bulge experiments employing WLI to measure the surface deflection.

Figure 2 shows the residual stress and Young's modulus of NCD membranes, made from the same wafers that have been measured by SLR before. To ensure the comparability of both measurement techniques all membranes are taken from the central region of the wafers. The stress values show the same tendency as those obtained by the SLR technique. In detail, the sample grown at 1.5% methane shows the best agreement of the stress values obtained from the wafer and membrane measurements. In the case of the sample grown at the lowest methane concentration, the bulge test revealed a higher tension of the film than the wafer curvature measurement. It can be explained by the remarkably higher thickness inhomogeneity of this sample, which leads to a higher deviation of the mean stress value obtained from wafer curvature and the more local stress value from the membrane bulging.

In addition, the bulge experiments revealed a linear decrease of the Young's modulus of the NCD films with increasing methane concentration, shown in Fig. 2. This descent of the film strength is due to a higher sp^{2} content in the diamond films grown at higher methane concentrations, which has been proven by Raman spectrometry 6, 33.

Figure 3a shows the results of some NCD membranes of variable radii processed from the wafer grown with 1.5% methane. From variations in the surface deflection as a function of the differential pressure, the Young's modulus and the tension in the films were calculated by fitting the pressure load curves using Eq. (2) and E and σ as fit parameters. From Eq. (2), it follows that for low-pressure values the first power term dominates the slope of the curves, leading to the approximation at the low pressure asymptotic (curve (i) in Fig. 3b).

In the case of higher values of the differential pressure, the 3rd power term representing the Young's modulus becomes more dominant. This leads to the nonlinear slope of the curves, shown in Fig. 3, resulting in the approximation of the curves to the asymptotic of the 3rd power term (curve (ii) in Fig. 3b). Thus, E can be determined by fitting the obtained pressure deflection curves as long as the membranes can stand differential pressures of several hundred mbar.

Bulge test results of NCD membranes with different radii, grown at 3% methane, along with an AlN/NCD heterostructure are shown in Fig. 4a. In the range of lower differential pressure values, the curves show a different bending than those in Fig. 3. This is due to the compression of the films, which leads to a wrinkling of the membranes. In this case, the formalism of the thin plate theory is not applicable and consequently a correct determination of the residual stress is not possible. Figure 4a shows that the influence of the wrinkling on the membrane surface increases with decreasing radius. For higher pressures the membrane surfaces smoothes out by the bulging of the membrane, which allows the straightforward determination of the Young's modulus.

In Fig. 4a, the comparison of the pressure load curves of the AlN/NCD heterostructure and the NCD membrane with the same radius but different layer thickness revealed nearly the same curvature shifted to lower deflections. In contrast, the comparison of the pressure load curves of the AlN membranes shown in Fig. 4b with those of the NCD membranes (Fig. 4a) having the same radii of 0.8 and 1.0 mm shows a clearly different behavior of the weaker AlN membranes. It means that the mechanical behavior of the AlN/NCD stack is mainly determined by the NCD film.

The approximated values for the 200 nm thick NCD membranes of E ∼ 852 GPa as well as for the 400 nm thick AlN/NCD membrane of E ∼ 492 GPa have been derived. The derived values for the 220 nm thick AlN membranes are σ ∼ +390 ± 13 MPa and E ∼ 374 ± 11 GPa 42, which is in good agreement with the literature (cf. Table 1).

5.3 Laser Doppler vibrometry

Resonant mode shapes and frequencies of contact-free circular AlN membranes have been analyzed by LDV in order to evaluate the mechanical properties with a higher precision. The quality factor Q = ω_{10}/Δω_{10} > 40000 was obtained for the best resonators in vacuo indicating negligible deviations of the material and geometrical properties over the membrane.

Figure 5 shows the in vacuo frequency-domain analysis of selected vibration modes for circular AlN membranes with different diameters. For illustration purposes, the shapes of the mechanically excited vibration modes recorded by LDV are shown in the right part of the figure. Here, i is the circumferential mode number and j is the radial mode number. As it follows from Eq. (5), the (j, i) mode has i diametrical nodal lines and j circular nodal lines, including the outermost boundary.

The experimental resonant frequencies (represented by the black rectangles) have been parametrically fitted by Eq. (5) using the tension, T, as parameter (the dashed lines). The averaged value of T ∼ +65.4 ± 1.8 N/m (σ = T/d_{AlN} ∼ 295 MPa) has been derived for the measured sample series. Thus, no significant impact of membrane size on the residual stress value was observed indicating a high reliability of the deposition and DRIE technologies.

The data set corresponding to the membrane diameter of ∼1.61 mm (marked by the open circles) is an exclusion demonstrating the impact of technological imperfection: an apparent frequency shift from the theoretical value of Δf ∼ 11.3 kHz is caused by silicon residue on the backside of the membrane.

By taking the correlation between frequency and mass for the circular membrane of mass m_{mem} = ρπR^{2}d_{AlN} = 5.7592 µg loaded with an extra mass m_{Si}in vacuo, one can calculate the mass of the silicon attached to the membrane of m_{Si} = 48.1 ng for this particular case. Measurements of the optical transmittance of the membrane and spectroscopic ellipsometry next to the membrane support the assumption of silicon residue on their backside.

A vibrating thin membrane is very sensitive to the properties of the surrounding media. The impact of the viscous gas damping on the vibration modes on thin circular AlN membranes is shown in Fig. 6. The frequency spectrum of the oscillating membrane changes drastically, when the ambient pressure is increased from the in vacuo state to low pressure until ambient pressure is achieved. This change is caused by the viscous gas damping on the vibration amplitude, resulting in two different effects, which are visible in the example of the (1,0) mode of a resonating membrane in Fig. 6.

The first effect is the frequency shift of the resonant mode. The damping leads to a worse signal-to-noise ratio and hence a higher impact of disturbance frequencies on the resonant mode. Indeed, the force of the gas acting on the membrane as it deflects is not negligible because the mass per unit area of the very thin membrane is roughly comparable to that of air. In particular, hρ_{AlN} ∼ 6.51 × 10^{−4} kg/m^{2} of a 220 nm thick AlN circular membranes is approximately equivalent to a 0.5 mm layer of dry air at ambient conditions (ρ_{air}(300 K) = 1.2041 kg/m^{3}).

The second effect of the damping at higher pressures on the frequency spectrum is the appearance of broadband disturbances such as those between 86 and 92 kHz in Fig. 6. When the pressure is increased further, these broadband disturbances can even extinguish several resonant modes, especially higher modes with a lower signal-to-noise ratio than the (1,0) mode shown in Fig. 6.

The complex damping effects related to internal friction in the membrane and the dynamic viscosity of the gas have been considered in a previous work 42. In particular, it has been demonstrated that LDV measurements are relevant in the pressure range below 0.1 mbar.

6 Discussion

All three interferometry methods used for the determination of the mechanical material properties (SLR, WLI, and LDV) deliver a relative instrumental accuracy δ = detection limit/deflection range = z_{lim}/Δz essential for reliable deflection measurements (Table 3). Therefore, the significant difference in the derivation of E and σ values is subject of geometrical restrictions along with approximations admitted by the physical models used for the treatment of the experimental results.

Table 3. Comparison of the instrumental accuracy and material parameters obtained by the three discussed methods.

The SLR measurements of as-grown wafers principally give a mean value of stress over the large scan length. Taking into account the gradients in the layer properties, the average value of stress obtained by SLR is suitable only for tendencies of residual stress dependencies on the growth conditions, as shown in Fig. 2. Compared to WLI and LDV the main advantage of the SLR technique is the possibility to determine values for the compressive stressed samples.

Due to the multiple measurements of membranes with varying diameter the bulge experiments reveal a much higher reproducibility of the stress determination than the wafer curvature measurements and are therefore more reliable. Additionally, in the case of inhomogeneous growth, as it has been observed for the NCD film grown at very low methane concentration, the more local determination of the mechanical layer properties via bulge test gets more important.

The formalism applied to the bulge experiments (Eq. (2)) is a subject of restriction given by the thin plate theory 50. Its use for the thin membranes of thickness d having the ratio R/d ∼ 10^{4} is not straightforward due to the negligible flexural rigidity, D, of the membrane. As a result, thin membranes should not resist any bending loads and can sustain only tensile strain. Moreover, the restoring force present in the membrane is the in plane tension only, while in the case of a thin plate, due to the finite D ≠ 0, stress develops along the thickness of the plate.

In practice, it means that for very thin membranes, the first part of Eq. (2) corresponding to z^{3}/R^{4} term is small. It does not allow exact determination of the Young's modulus for materials with low E in the typical range of pressures and deflections used for bulge experiments. In contrast, AlN and NCD membranes can sustain a significant pressure load. Indeed, the deviation of the curves from the linear behavior is already visible for pressures below 20 mbar (Fig. 3) while the cubic behavior is yet apparent at pressures above 0.2 bar, which allows the determination of E with high accuracy.

There is, however, the inaccuracy of the values of the coefficients c_{1}, c_{2}, and f(ν), used for the calculation of E and σ (see Eq. (2)). They are adjusted as a function of the membrane shape and depend on the associated analytical model.

An additional source of inaccuracy for static measurements is the technology dependent deviation of the membrane radius R. This drastically affects the validity of E ∝ 1/R^{4} and σ ∝ 1/R^{2} values, while in the case of dynamic measurements the resonant frequency is proportional to 1/R. Therefore, the vibration measurements lead to the most accurate determination of the residual stress on the assumption of tensile stressed membranes.

Due to the high instrumental accuracy of the in vacuo LDV method and the straightforward character of the measurements, the fit of eigenfrequencies versus membrane diameter curves (Fig. 5) should deliver the most exact results on the tension in free-standing films. In addition, the formalism represented by Eq. (3) is not a subject of any significant restrictions, which can be considered for thin membranes. The value of E, however, cannot be derived from the thin membrane theory in principle. The dominating factors limiting the accuracy of the tension determination are the inhomogeneities of the film thickness along with deviations of the membrane geometry from an ideal circular form (typically below 0.5%) defined by the relation ω_{ji} ∝ 1/R.

7 Summary

The comparison of wafer curvature measurements, bulge experiments and vibrometry method has shown that the dynamic measurements provide the highest accuracy in the stress determination of tensile stressed membranes. The Young's modulus cannot be derived by the vibration theory, which in turn, is possible for the bulge experiments if the thin membranes can sustain significant pressure loads. The accuracy of both methods is mainly limited by inhomogeneities in the film density and thickness as well as deviations of the membrane geometry from an ideal circular form, especially in bulge experiments.

Additionally, these experiments reveal extreme flexibility and robustness of thin AlN and NCD membranes showing no degradation after multiple bulge cycles with differential pressures up to 1 bar. It has been shown that the mechanical properties of the AlN actuators can be advanced by the combination with NCD layers. In doing so, the effective value of the Young's modulus rises from 370 GPa for the single AlN membrane to 492 GPa for the AlN/NCD heterostructure, while the stress is compensated (from ∼370 MPa to low compression).

Acknowledgements

This work was funded by BMBF and Fraunhofer research grants as well as by the German Research Foundation DFG within the Priority Program Active Micro-optics.

Biographical Information

Fabian Knöbber received his Diploma degree in Microsystems Engineering from the University of Freiburg in 2008. He is currently working towards his PhD degree at the Fraunhofer Institute for Applied Solid State Physics in Freiburg, Germany. His research interest lies in the development of piezoelectric tunable micro-optics using aluminum nitride and nanocrystalline diamond thin films. Therein he focuses primarily on the material characterization and development in order to tune the electrical, optical and mechanical properties with regard to adaptive micro-optical components.