Residual stress analysis of all perovskite oxide cantilevers
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DOI: 10.1007/s10832-011-9663-6
Abstract
We have used a method to experimentally determine the curvature of thin film multilayers in all oxide cantilevers. This method is applicable for large deflections and enables the radius of curvature of the beam, at a certain distance from the anchor, to be determined accurately. The deflections of the suspended beams are measured at different distances from the anchor point using SEM images and the expression of the deflection curve is calculated for each cantilever. With this expression it is possible to calculate the value of the radius of curvature at the free end of the cantilever. Together with measured values for the Youngs Modulus, this enabled us to determine the residual stress in each cantilever. This analysis has been applied to SrRuO _{3}/BaTiO _{3}/SrRuO _{3}, BaTiO _{3}/MgO/SrTiO _{3} and BaTiO_{3}/SrTiO_{3} piezoelectric cantilevers and the results compared to two models in which the stresses are determined by lattice parameter mismatch or differences in thermal expansion coefficient. Our analysis shows that the bending of the beams is mainly due the thermal stress generated during the cooling down stage subsequent to the film deposition.
Keywords
Residual stress Thermal stress Cantilever Thin film stress1 Introduction
There is a wide range of sensors and actuators based on micromachined cantilevers. These devices convert the mechanical vibration of a suspended beam, which may be terminated with a proof mass, into an electrical signal, which may be used to measure acceleration, vibrational frequency or even as a source of electrical power. Usually, the freedom of movement of the cantilever is guaranteed by etching a pit into the substrate and the cantilever is designed so that the curvature is minimized. When a cantilever is fabricated using a multilayer thin film structure, mechanical stresses arise between the layers as a result of a mismatch of lattice constants and thermal expansion coefficients. These result in the beam being curved rather than straight, which is undesirable for some applications. In order to control the curvature of the beam, an understanding of the stresses in the beam is required, which can be obtained from beam theory.
We calculated the deflection curve of the cantilever by measuring the deflection of the beam at different distances from the anchor point. This enabled us to evaluate the radius of curvature at the free end of the structure.
The measurement of the deflection profile has been used elsewhere to estimate the curvature of suspended beams containing phase-change materials [4]. In that work the Stoney equation was used to calculate the elastic strain at the interface between a thick cantilever and a thin film grown on its top. Such treatment is only valid if the film thickness is negligible compared with the cantilever thickness [4, 5]. This restriction does not apply to our method and we demonstrate its effectiveness using multilayers to which the Stoney equation would not apply.
In this work the equations of beam theory [1], are used to estimate the residual stress on the top surface of the fabricated cantilevers; we have used this analysis to obtain values for residual stress in all-oxide piezoelectric cantilevers. These consisted of SrRuO _{3}/ BaTiO _{3}/SrRuO _{3} and BaTiO _{3}/MgO/SrTiO _{3} multilayers which are based on lead free materials and have applications as energy harvesting devices. The use of an all oxide structure allows the layers to be grown epitaxially, which enables us to exploit the anisotropy of piezoelectric oxides in device design. We found that the first structure can be used to produce energy harvesting devices working in the d _{31} mode because the polar axis is perpendicular to the surface of the film (out of plane), while the second structure can be used for the d _{33} energy harvesting mode [6, 7] as the polar axis is parallel to the surface of the film (in plane). The maintenance of epitaxy throughout the structure requires us to use an oxide sacrificial layer. Residual stress investigations have been also performed on BaTiO_{3}/SrTiO_{3} cantilevers, these structures were developed at the beginning of the project to investigate the growing conditions and the properties of such material, however they do not find practical applications as energy harvesting devices.
For the analysed structures, the residual stress produces an upward bending of the beam, after their release. We will show how the application of our stress analysis method enables us to identify the dominant sources of stress in our cantilevers and hence illustrate how it may be applied more generally.
The residual stress in a multilayer beam can be predicted given certain key parameters. We have measured the lattice parameter and Youngs modulus for our samples and used these together with published values for Poissons ratio and thermal expansion coefficients to calculate the residual stress in our samples. We compare these to the stresses determined from the curvature of the beam.
2 Fabrication and characterization of all oxide cantilevers
We used a KrF excimer laser with a wavelength of 248 nm to grow our films by pulsed laser deposition. A 4 Hz pulse repetition rate is used, the distance between the material target and the substrate is set to 5.7 cm and all the devices were fabricated on (001) oriented SrTiO _{3} substrates.
PLD deposition parameters for the deposited films
Material |
Fluence [J/cm^{2}] |
Deposition temperature [°C] |
Deposition pressure [Torr] |
---|---|---|---|
YBa _{2} Cu _{3} O _{7} |
3.4 |
740 |
0.15 O _{2} flow |
SrRuO _{3} |
3.2 |
780 |
0.3 O _{2} flow |
BaTiO _{3} |
3.5 |
740 |
0.15 O _{2} flow |
MgO |
3.5 |
740 |
0.20 O _{2} flow |
SrTiO _{3} |
3.2 |
740 |
0.21 O _{2} flow |
In these structures, BaTiO _{3} is the piezoelectric layer, SrRuO _{3} acts as an electrode for the OP stacks whilst for the IP stacks, the MgO film is introduced to grow the BaTiO _{3} with the polar axis parallel to the surface of the film, and the SrTiO _{3} works as buffer layer to improve the interface between the YBa _{2} Cu _{3} O _{7} and the MgO film in the IP structure [8].
The IP structure needs the deposition of a gold top electrode. However the gold deposition is performed at room temperature, so it does not influence the thermal stress analysis which will be developed in the next sections. The cantilever geometry is then defined by contact photolitograpy and argon ion beam milling. To suspend the beam the sample is undercut in 0.1% HNO _{3}, rinsed in distilled water and dried using critical point drying in order to avoid stiction problems.
Indentations were performed from the top surface down to the desired depth through the multilayer. The indenter was held at load for 60 s, then retracted from the sample at a rate of 0.5 nm/s. Each indentation was separated by 50 μm, to avoid any influence of the previous indentation.
Thermal expansion coefficient, Poisson’s ratio and Young Modulus of the deposited films. The Young Modulus has been measured by nanoindentation technique
Material |
Thermal expansion coefficient [10^{ − 6}1/K] |
Young modulus [GPa] |
Poisson ratio |
---|---|---|---|
YBa _{2} Cu _{3} O _{7} |
13.4 |
129 |
0.3 |
SrRuO _{3} |
8 |
190 |
0.3 |
BaTiO _{3} (on MgO or SrRuO _{3}) |
11.3 |
175 |
0.35 |
BaTiO _{3} (on SrTiO _{3}) |
11.3 |
79 |
0.35 |
MgO |
8 |
233 |
0.18 |
SrTiO _{3} |
9.4 |
130 |
0.25 |
Lattice constants of the different materials used in the cantilever stack. The lattice parameters have been masured by X-ray diffraction technique
Material |
a Measured (a theoretical) |
b Measured (b theoretical) |
c Measured (c theoretical) |
---|---|---|---|
BaTiO _{3} (on SrRuO _{3}) |
\(3.951\dot{A}\) \((3.992\dot{A})\) |
3.951 \((3.992\dot{A})\) |
4.069 \((4.036\dot{A})\) |
BaTiO _{3} (on MgO) |
\(4.002\dot{A}\) |
\(4.002\dot{A}\) |
\(3.997\dot{A}\) |
BaTiO _{3} (on SrTiO _{3}) |
\(3.996\dot{A}\) \((3.992\dot{A})\) |
\(3.996\dot{A}\) \((3.992\dot{A})\) |
\(4.015\dot{A}\) \((4.036\dot{A}) \) |
SrRuO _{3} (on YBa _{2} Cu _{3} O _{7}) |
\(5.519\dot{A}\) (\(5.567\dot{A})\) |
\(5.519\dot{A}\) \((5.530\dot{A})\) |
\(7.825\dot{A}\) \((7.845\dot{A})\) |
MgO (on SrTiO _{3}) |
\(4.220 \dot{A}\) \((4.211\dot{A})\) |
\(4.220\dot{A}\) \((4.211\dot{A})\) |
\(4.220\dot{A}\) \((4.211\dot{A})\) |
To validate the SEM measurements, the deflection of the cantilevers was also measured with a MicroXAM2 interferometer (Omniscan, UK). The interferometer was operated in phase mode employing light of wavelength 510 nm, and at a magnification of 100X. Interferograms were found to be unreliable, due to differences in the reflectivities of the various materials. Instead, the interferometer was operated manually, employing a motorised x, y, z stage with ±0.1 μm resolution. The interferometric fringes were focused on the sample surface and the position of peak light intensity was considered to be the x, y position of interest, whose position could be determined with approximately ± 3 μm accuracy due to the manual positioning involved. The corresponding z-position was recorded for each surface location assessed.
3 Analysis of the cantilevers
Device type, length, width and co-ordinates (x,y) of three example cantilevers (one for each layer sequence); x is the position of the measurement along the cantilever and y is the deflection. The parabolic equations describing the deflection of the cantilevers through the points (A; C; E) are reported for both the SEM and the interferometer data, the radius of curvatures of each cantilever calculated from Eq. 8 with the parabola obtained from SEM data are also given
Device |
Cantilever type |
Length [μm] |
Width [μm] |
SEM coordinates (A ; B ; C ; D ; E) \(\left(x[\mu \rm{m}],y[\mu \rm{m}]\right)\) |
Interferometer coordinates (A ; B ; C ; D ; E) \(\left(x[\mu \rm{m}],y[\mu \rm{m}]\right)\) |
SEM Parabola (Interferometer parabola) y = ax ^{2} + bx + c |
Radius of curvature [μm] |
---|---|---|---|---|---|---|---|
OP |
U-shape |
120 μm |
20 μm |
(0,1.74); (20,3.35);(50,8.44); (90,19.5); (118,30.5); |
(0,1.5); (20,3.5);(50,8.2); (90,18); (118,31); |
\(y_{a}=1620x^{2}+0.053x+1.74\cdot 10^{-6}\) (y = 1706x ^{2} + 0.0487x + 1.5·10^{ − 6}) |
403±7.15 |
IP |
U-shape |
120 μm |
20 μm |
(0,1.02); (20,2.3);(50,5.72); (90,13.5); (122,21.6); |
(0,2); (20,3.6);(50,5.8); (90,12.3); (118,18.6); |
\(y_{b}=1040x^{2}+0.042x+1.02\cdot 10^{-6}\) (y = 834x ^{2} + 0.0343x + 2·10^{ − 6}) |
551±37 |
Bi |
Beam |
44 μm |
10 μm |
(2.33,1.57); (10,2.23);(20,3.2); (30,4.82); (39,6.68); |
(0,1.1); (/,/);(20,3.4); (30,4.5);(44,7.9) |
\(y_{c}=2500x^{2}+0.0339x+1.51\cdot 10^{-6}\) (y = 1650x ^{2} + 0.082x + 1.1·10^{ − 6}) |
217±57 |
Errors on the points B and D consequence of the parabolic approximation used for the de-ection of the cantilever. Also reported are the percentage errors e _{ a } and e _{ b } of the parabola coefficients obtained for the interferometers measurements respect to the SEM data. The reported data are relative to the three example devices
Device |
Cantilever type |
Length [μm] |
Width [μm] |
Error on B for SEM par. [μm] |
Error on D for SEM par. [μm] |
e _{ a } SEM–INT |
e _{ b } SEM–INT |
---|---|---|---|---|---|---|---|
OP |
U-shape |
120μm |
20μm |
+ 0.1 |
+ 0.22 |
5.3% |
8.11% |
IP |
U-shape |
120μm |
20μm |
+ 1.22 |
− 0.28 |
20% |
18.3% |
Bi |
Beam |
44μm |
10μm |
− 0.13 |
− 0.04 |
51.5% |
142% |
These represent the experimental values of the longitudinal stress at the cantilever top surface.
Experimental stress, theoretical thermal stress and theoretical stress due to combined effect of thermal and misfit stress evaluated at the top surface of each cantilever. For each cantilever the device type corresponding to different layer sequence is indicated, also the cantilever shapes and the cantilever lengths and widths are reported
Device |
Cantilever type |
Length [μm] |
Width [μm] |
Experimental stress \(\left[10^{9}~N/m^{2}\right]\) |
Theoretical thermal stress \(\left[10^{9}~N/m^{2}\right]\) |
Theoretical thermal−misfit stress \(\left[10^{9}~N/m^{2}\right]\) |
---|---|---|---|---|---|---|
OP |
U-shape |
120 μm |
20 μm |
− 0.104±0.002 |
− 0.051 |
− 0.0013 |
OP |
U-shape |
128 μm |
20 μm |
− 0.093±0.009 |
− 0.051 |
− 0.0013 |
OP |
U-shape |
234 μm |
20 μm |
− 0.051±0.026 |
− 0.051 |
− 0.0013 |
IP |
U-shape |
120 μm |
20 μm |
− 0.153±0.011 |
− 0.145 |
− 7.21 |
IP |
U-shape |
122 μm |
20 μm |
− 0.131±0.009 |
− 0.145 |
− 7.21 |
IP |
U-shape |
180 μm |
20 μm |
− 0.112±0.028 |
− 0.145 |
− 7.21 |
Bi |
Beam |
32 μm |
10 μm |
− 0.173±0.136 |
− 0.071 |
1.2 |
Bi |
Beam |
44 μm |
10 μm |
− 0.200±0.069 |
− 0.071 |
1.2 |
Bi |
Beam |
55 μm |
10 μm |
− 0.190±0.112 |
− 0.071 |
1.2 |
There are three sources of uncertainty in this analysis. The first source is due to the measurement technique and it is linked to the resolution of the measurement instruments. The second is the curve chosen to approximate the deflection of the beam. The third source of error is due to the fabrication process, this kind of error arises from the undercut and resist resolution.
The uncertainties on the values of the residual stresses \(\tilde{\sigma_{a}}\), \(\tilde{\sigma_{b}}\) and \(\tilde{\sigma_{c}}\) have been calculated developing an error analysis on the radius of curvature.
Considering the canonical form of the parabola and Eq. 8, the radius of curvature depends only on the parameters a and b, so the uncertainty in the radius of curvature is dependent on the uncertainty in a and b but not in c.
The radius of curvature of each cantilever was calculated at the free end of the structure, so an error on the length of the cantilever also affects its final value. The principle errors on the cantilever lengths is due to the undercut at the anchors points. Undercuts between 9 μm and 20 μm have been measured.
The error in the experimental values of the residual stress can be calculated from the error on the radius of curvature, they are reported in Table 6.
4 Theoretical values for the residual stress
By using the measured lattice parameters we have found that for our layer combinations, the critical thickness is less than 10 nm. The film thicknesses in the two stacks are in the order of hundred of nm, this means that only the thermal stress will be considered in the following stress analysis, as the misfit is released by the generated dislocations. Similar considerations have been also reported in [21].
The deposition of the cantilever films is performed at temperatures of 740°C and 780°C, then the film stack is cooled down to room temperature. The thermal stress will be the result of the differences in the film thermal expansion coefficients.
According to Eq. 1, the longitudinal stress in each layer varies linearly with the film thickness.
The cantilevers under investigation have total thicknesses of 500 nm and 670 nm, widths between 10 μm and 20 μm and lengths between 32 μm and 230 μm. So at a first approximation it is possible to consider each layer in the stack as a two dimensional system, this means that the stress will occur only at the interfaces.
When the contributions of the lattice mismatch are included in the calcultation of the residual stress there is a clear disagreement between theoretical and experimental values, this is why the residual stress is mainly attributed to the differences in the thermal expansion coefficients between the materials of the different layers.
5 Discussion
There are two different techniques used to evaluate the residual stress in thin films. In the first method the curvature of a flat substrate is measured after the film deposition and the residual stress is evaluated using the Stoney formula [4, 7, 14, 29].
The assumptions in this method are: the properties of the film-substrate system are such that the film materials contribute negligibly to the overall elastic stiffness; the change in film stress due to substrate deformation is small; the thickness of the film is small compared to the thickness of the substrate and the curvature of the substrate midplane is spatially uniform [5]. Corrections to the original Stoney formula can be made in the case of films having a thickness which is not negligible with respect to the thickness of the substrate.
Thus the Stoney formula may not accurately describe the bending of our cantilever, for example the radius of curvature of the fabricated cantilevers is not uniform along the length of the beam and furthermore in multilayer cantilevers all the films contribute to the overall elastic stiffness.
When this method is applied to a multilayer structure grown on a certain substrate it is assumed that the individual layers in the film are added sequentially and that the mismatch strain in each layer depends only on the substrate but not on the order in which the layers are formed [5]. To evaluate the stress in a multilayer the substrate curvature has to be measured before and after each layer deposition [5, 7].
All these assumptions can lead to systematic errors in the evaluation of the local residual stresses.
The second technique involves the measurement of the bending of the multilayer structure when all the constraints are removed [14]. This is the method usually used to evaluate the residual stress in released cantilevers. Under the assumption of small deflections, the deflection of the free end is measured [2, 3], and is used to calculate an approximated value for the radius of curvature (Eq. 2). This is only valid for small deflections of the cantilever free end, and so to calculate a more precise value for the radius of curvature, the method which we proposed can be followed. Measurement of the deflections at different distances from the anchor point is a straighfroward way to experimentally evaluate the residual stress in multilayers cantilevers.
In the case of Bi devices the magnitude of the absolute error on the parabolic approximations (error on points B and D, Table 5) is comparable with the absolute error of the IP and OP structures. The Bi devices experiences deflections which are between three and six times smaller than the deflections experienced by the IP and OP cantilevers, this produces a larger relative error on the parabolic approximation used for the Bi structures. This explains why a bigger error on the residual stress of these cantilevers is present.
For the Bi cantilevers, as already reported in Section 4, the measured value of the Young modulus is smaller than the value measured for BaTiO _{3} films grown on SrRuO _{3} or on MgO. This smaller value of 79 GPa is in agreement with that reported in [28]. When the BaTiO _{3} is grown on MgO or on SrRuO _{3} its Young’s modulus is larger. Further investigation is necessary to understand the origin of such disagreement. It is known that MgO and SrRuO _{3} are stiffer than SrTiO _{3} and so an influence of the underlying layer on the Yong’s modulus of BaTiO _{3} can not be excluded. Furthermore it has to be highlighted that in the Bi structure all the thin films are deposited in situ while for the IP and OP stacks this is not possible because our deposition system allows the deposition of only three layers in situ. In the IP and OP devices before the deposition of the BaTiO _{3} the thin film stacks are heated up in oxygen at the deposition temperature and annealed for about 30 mininutes before the deposition. The annealing step can improves the quality of the top surface of the film stack which acts as a seed for the BaTiO _{3} deposition. Alternatively, since it is known that heating and cooling cycles can reduce stresses in ceramics like BaTiO3 [13], stresses in the Bi structures may be higher than those in the IP and OP because the IP and OP structures were grown with extra heating/cooling steps. Thirdly, the Bi structures are smaller than the IP and OP devices so edges effects might contribute to the total longitudinal stress [5].
Nevertheless for two of the Bi structures the theoretical stress lies with the error range of the experimental value. Therefore we consider that the overall agreement with theory is good and that the mismatch between the thermal expansion coefficients is the main cause of the residual stress in these devices. If instead the lattice mismatches are used to calculate the longitudinal stresses at the top surfaces of the cantilevers, there is adifference of one or two orders of magnitude between theoretical and experimental values. This rules out the lattice mismatch as the cause of residual stress.
We believe that the variation in stress values can arise from variations in deposition parameters, associated with the alignment of the laser ablation plume with the centre of the substrate. For the sample containing the OP structures, two of the devices were much closer to each other than the third device. If the thickness of each layer deposited on the SrTiO _{3} substrate were not uniform all over the sample, this would give errors in the position of the neutral axis and according to Eq. 1 an error on the value of the residual longitudinal stress.
Another potential source of variation is the value of the Young’s modulus. Measurements show that the values of the Young modulus are not constant through the film thicknesses with variations up to 10% around the central value across the thickness of each film. Another source of error might be due to the not perfect undercut in fact it is possible that material from the top part of the sacrificial layer might remain attached to the bottom surface of the cantilever. Finally other sources of stress like microscopic voids, incorporation of impurities and recrystallization [5, 15] could be present in the deposited films.
To validate the method, the measurements of the deflection of the cantilevers have been also performed with the interferometry technique. The measured valued together with the corresponding parabola are reported in Table 4. Table 5 reports the percentage errors on the parabola coefficients a and b, e _{ a } and e _{ b } respectively, obtained from the interferometer measurements respect to the parobola coeffients relative to the SEM data. For the IP and OP devices errors on the coefficients between 5% and 20% seem to validate the method applied on the SEM measurements. For the Bi devices errors on the coefficients over 50% are reported, this is attributed to the smaller resolution associated with the interferometer measurements. The Bi devices have lenghts between 32 μm and 55 μm and experience a maxmimun deflection of the free end equal to 10 μm. For these small deflections, the resolution of the method as consequence of the manual measurement involved does not allow the application of the method on these data.
The SEM measurements present a better resolution than the interferometers data, this is why the radius of curvature and the values of the residual stresses at the top surface of the cantilevers have been calculated from the SEM measurements.
6 Conclusions
We have shown a method to experimentally determine the curvature of thin film multilayer suspended cantilever structures. This method is applicable for beams with large deflections and which do not present a constant radius of curvature. It enables the radius of curvature at a certain distance from the anchor to be determined accurately. The deflection of the suspended beams is measured at different distances from the anchor point using SEM and interferometer images in this way the expression of the deflection curve is calculated for each cantilever. With this expression it is possible to calculate the value of the radius of curvature at the cantilever free end. Together with measured values for the Youngs Modulus, this enabled us to determine the residual stress in a cantilever. This analysis has been applied to SrRuO _{3}/BaTiO _{3}/SrRuO _{3} and BaTiO _{3}/MgO/SrTiO _{3} piezoelectric cantilevers. These thin film sequences produce BaTiO _{3} layers with polar axes oriented out of plane(OP) or in plane(IP) respectively. The OP structures are suited to energy harvesting applications where the d _{31} mode is used whilst the IP structures are suited to the d _{33} mode. Investigations have been also performed on BaTiO _{3}/SrTiO _{3} bilayer cantilevers. The results were compared to two models in which the stresses are determined by lattice parameter mismatch or differences in thermal expansion coefficient. The experimentally determined residual stresses of the IP and OP devices were found to agree with the calculated thermal stresses, suggesting that the latter is the source of the curvature, rather than the lattice mismatch. For the Bi structures the experimental stress is three times bigger than the theoretically calculated thermal stress, however in this case a large uncertainty is associated with the experimental values. For energy harvesting applications, the output power of a cantilever increases when its swinging amplitude increases. So in some cases, the bending up, can be used to increase the swinging amplitude of the released beam. In this way it is possible to have swinging amplitudes in the orders of 20 μm without the need to etch the substrate. Using the methods described in this paper, the upward curvature of such cantilevers can be better understood and even tuned by appropriate selection of oxide layers to enhance their performances.
Acknowledgements
The Interferometer and Nanoindenter used in this research were obtained, through Birmingham Science City: Innovative Uses for Advanced Materials in the Modern World (West Midlands Centre for Advanced Materials Project 2), with support from Advantage West Midlands (AWM) and part funded by the European Regional Development Fund (ERDF).
This research has been funded by the UK EPSRC under EP/E026494/1 and by The University of Birmingham.
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