A new bulge test technique for the determination of Young's
`modulus and Poisson's ratio of thin films
`J. J. Vlassak and W. D. Nix
`Department of Materials Science and Engineering, Stanford University, Stanford, California 94305
`
`(Received 26 May 1992; accepted 6 August 1992)
`
`A new analysis of the deflection of square and rectangular membranes of varying aspect
`ratio under the influence of a uniform pressure is presented. The influence of residual
`stresses on the deflection of membranes is examined. Expressions have been developed
`that allow one to measure residual stresses and Young's moduli. By testing both square
`and rectangular membranes of the same film, it is possible to determine Poisson's ratio
`of the film. Using standard micromachining techniques, free-standing films of LPCVD
`silicon nitride were fabricated and tested as a model system. The deflection of the silicon
`nitride films as a function of film aspect ratio is very well predicted by the new analysis.
`Young's modulus of the silicon nitride films is 222 ± 3 GPa and Poisson's ratio is
`0.28 ± 0.05. The residual stress varies between 120 and 150 MPa. Young's modulus and
`hardness of the films were also measured by means of nanoindentation, yielding values
`of 216 ± 10 GPa and 21.0 ± 0.9 GPa, respectively.
`
`I. INTRODUCTION
`The mechanical properties of thin films and the
`residual stresses in them have long been recognized to
`be important in the fabrication of electronic devices and
`microsensors.1 This has provided a motivation for the
`study of mechanical properties of thin films. Unfortu-
`nately, the techniques commonly used to measure these
`properties in bulk materials are not directly applicable to
`thin films. Thus, specialized mechanical testing methods
`have been sought.
`The bulge test was one of the first techniques intro-
`duced for the study of thin film mechanical properties.2
`In its original form, a circular film or membrane is
`clamped over an orifice and a uniform pressure is applied
`to one side of the film. The deflection of the film is
`then measured as a function of pressure allowing a
`determination of the stress-strain curve and the residual
`stress of the film. The stress state in the film is biaxial so
`that only properties in the plane of the film are measured.
`Traditionally the test has been plagued by a number
`of problems. The results are rather sensitive to small
`variations of the dimensions of the film and may be
`affected by twisting of the sample when it is mounted.
`Sample preparation is therefore crucial and special steps
`need to be taken to minimize these effects. The residual
`stresses in the film also have to be tensile. Finite element
`studies3'4 have shown that for films in compression, the
`circumferential stress near the edge of the film remains
`compressive even at high applied pressures, causing
`the film to buckle. Wrinkles in such films disappear
`only gradually as the pressure on the film is increased,
`leading to erroneous results. Finally, failure to take
`
`into account the initial height of the membrane in the
`analysis leads to apparent nonlinear elastic behavior of
`the film.5
`Developments in micromachining techniques and
`better analysis methods have made it possible to over-
`come many of the problems associated with the bulge
`test. In this paper, a new analysis of the deflection of rect-
`angular membranes is presented and it is demonstrated
`how Young's modulus, Poisson's ratio, and residual
`stress can be accurately measured by testing both square
`and rectangular films with large aspect ratios. We also
`describe a technique to fabricate free-standing films
`of silicon nitride on silicon substrates, using standard
`lithography and anisotropic etching techniques. The di-
`mensions of the films can be controlled precisely if the
`membranes are made rectangular in shape and oriented
`with the crystal axes of the Si substrate. Silicon nitride
`is used as a model system, but the technique can be
`extended to a large number of films with only minor
`modifications. The results obtained from the bulge test
`are compared to results from nanoindentation experi-
`ments performed on the same material.
`
`II. ANALYSIS OF THE DEFLECTION
`OF A MEMBRANE
`A. Square films
`Calculation of the deflection of a membrane under a
`uniform pressure is a difficult problem. For the large de-
`flections that are typical in bulge tests, the membrane be-
`haves nonlinearly. Let u, v, and w be the components of
`the displacement parallel to the x, y, and z directions (see
`
`3242
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`J. Mater. Res., Vol. 7, No. 12, Dec 1992
`
`© 1992 Materials Research Society
`
`IPR2015-01087 - Ex. 1041
`Micron Technology, Inc., et al., Petitioners
`1
`
`

`
`J. J. Vlassak and W. D. Nix: A new bulge test technique for the determination of Young's modulus and Poisson's ratio
`
`Fig. 1). The strains in the membrane are then given by 6:
`
`isotropic material is'
`
`_ bu
`
`1 (bw\J
`
`bv
`by
`bu
`7xy ~ by
`
`1 /bw
`~2
`bv
`~bx
`
`bw bw
`bx by
`
`(1)
`
`The nonlinear terms in these expressions arise from the
`fact that the deflection of the membrane in the z direction
`is large. Using the equilibrium equations, Hooke's law,
`and the appropriate boundary conditions, the deflection
`can be calculated. The problem can be reduced to the si-
`multaneous solution of two nonlinear partial differential
`equations.6 This, however, is a nontrivial task.
`A number of researchers have derived approximate
`solutions using an energy minimization method. 6"9 In
`this approach, one assumes a displacement field for
`the membrane that contains a number of unknown
`parameters and satisfies
`the boundary conditions.
`According to the principle of virtual displacements,
`the unknown parameters are then determined by the
`condition that the total potential energy of the system
`is minimum with respect
`to
`the parameters. The
`displacement field used most often is the first term
`in the Fourier expansion of the actual deflection. The
`same method with a different displacement field is
`used in the present study in order to derive a more
`accurate expression for
`the load-deflection behavior
`of a square membrane. The displacement field for a
`square film with side 2a can be approximated by
`
`u = AJ-5(a2 - x2)(a2 - y2)
`2)(a2
`
`v = A^(a2
`
`- x2)(a2 -
`2)(a2 -
`
`y2)
`y2)
`
`w = w^{a2
`
`- x2){a2 - y2)[l +
`
`+ y2)]
`(2)
`
`where A, w0, and R are the unknown parameters. The
`potential energy of the membrane in the case of an
`
`Wo
`
`2a
`FIG. 1. Schematic diagram of a membrane with a uniform pressure
`applied to one side.
`
`V =
`
`Et
`2(1 - v2)
`
`e2x + e2 + 2vexey
`
`+ ^(1 - v2)yxy)dxdy - ff qwdxdy
`
`(3)
`
`where t, E, and v are the thickness, Young's modulus,
`and Poisson's ratio of the film, respectively, and q is the
`pressure applied to the membrane. The first term in this
`expression represents the strain energy of the membrane
`due to the stretching in the plane of the membrane. The
`contribution of bending to the strain energy has been
`neglected. This is valid because the deflection is much
`larger than the thickness of the membrane. The second
`term represents the potential energy of the pressure
`applied to the membrane. Minimization of Eq. (3) with
`respect to the undetermined parameters leads to a set of
`three simultaneous nonlinear equations in A, w 0, and R,
`that can be readily solved. The deflection of the center
`of the membrane is then given by
`
`= f{v)
`
`1/3
`
`qaA{\ - v)
`Et
`
`(4)
`
`is a complicated function of Poisson's ratio
`where f{v)
`which can be approximated by f{v) ~ 0.800 + 0.062*/.
`The form of Eq. (4) is the same as that found by other
`researchers, except for the function f(v). A few remarks
`about Eq. (4) are in order. First, for a given pressure
`and displacement, Young's modulus is proportional to
`the fourth power of a. Therefore, if one wants to
`measure Young's modulus by means of the bulge test,
`the dimensions of the film have to be measured very
`accurately. This is often impossible if the film is taken
`off the substrate and glued onto a sample holder. Second,
`the fact that f(p)
`is a function of Poisson's ratio arises
`from the fact that the stress state in the membrane is
`not entirely equal-biaxial. The assumption of an equal-
`biaxial stress state has often been made in the derivation
`of the deflection of circular membranes.2-10'11 The strain
`state actually varies from equal-biaxial in the center of
`the membrane to plane strain at the edges, where the
`film is clamped. The biaxial modulus E/{\ — v) alone
`is insufficient to characterize the load-deflection behavior
`of the membrane and the membrane is more compliant
`than one would expect based on the assumption of
`an equal-biaxial stress state. In Fig. 2 the variation
`of f{v) with Poisson's ratio is compared to a finite
`element calculation of the same quantity7 and an energy
`minimization calculation using the first term of the
`Fourier expansion of the deflection.7*8 Agreement with
`the finite element calculation is excellent. The Fourier
`expansion, however, overestimates the compliance of the
`membrane significantly. The shape of the deflected mem-
`
`J. Mater. Res., Vol. 7, No. 12, Dec 1992
`
`3243
`
`2
`
`

`
`J. J. Vlassak and W. D. Nix: A new bulge test technique for the determination of Young's modulus and Poisson's ratio
`
`Normalized deflection f(v) in eq. (4)
`as a function of Poisson's ratio
`
`0.75
`
`0.70
`
`0.65
`
`0.60
`
`r
`
`-
`
`o
`
`Finite element calculation [7]
`First Fourier term
`This study
`
`0.2
`0.3
`Poisson's ratio
`
`for square films with Poisson's
`FIG. 2. Variation of the function f{v)
`ratio. The open diamonds correspond to results obtained from finite el-
`ement calculations.7
`
`Shape of membrane in the xz-plane
`
`Experimental data [7]
`"This analysis, v=0.3
`- This analysis, v=0.2
`First Fourier term
`
`0.8
`
`0.4
`
`0.6
`x-coordinate
`(a)
`
`brane is depicted in Fig. 3. Both the deflection in the
`xz -plane and along the membrane diagonal are plotted
`and show very good agreement with experimental data. 7
`Even though the film is clamped along the edges, the
`slope of the deflection at the edges is not zero because
`for very thin films and large deflections the bending
`stiffness of the film can be neglected. According to
`finite element calculations7 the shape of a membrane
`for a given deflection is independent of Poisson's ratio.
`Although in this analysis the parameter R in Eq. (2) is
`a weak function of Poisson's ratio, the shape calculated
`varies only very slightly with Poisson's ratio.
`
`B. Rectangular films
`The load-deflection behavior of a rectangular film
`with sides 2a and 2b can be derived using the same
`energy minimization technique as for square films. The
`displacement field is very similar to that of square films
`in Eq. (2) but contains five unknowns instead of three.
`The resulting load-deflection relation is then
`
`b\(qa\l - v)
`
`1/3
`
`(5)
`
`is a function of Poisson's ratio and the
`where g{v,b/a)
`aspect ratio of the membrane. Figure 4 shows the change
`of g{v, b/a) with membrane aspect ratio for three differ-
`ent values of Poisson's ratio. Apparently, a membrane
`shows a rapidly increasing deflection as its aspect ratio
`increases above unity, but once the aspect ratio exceeds
`5, the deflection is independent of the aspect ratio. There
`are two limiting cases for which this calculation can be
`checked. First, for an aspect ratio of 1.0, the solution
`has to be the same as the one derived previously in
`this paper. Second, for an infinitely long membrane the
`
`I o
`
`lor q 5
`
`3
`•§
`
`ized
`
`rmal o
`
`1
`
`0.8
`
`0.6
`
`0.4
`
`0.2
`
`1.40
`
`1.30
`
`1.20
`
`1.10
`
`1.00
`
`0.90
`
`0.80
`
`0.70
`
`:
`
`0.60
`
`g(v,b/a) in eq. (6) as a
`function of aspect ratio
`
`Plane strain solution
`
`v=0.25
`v=0.30
`v=0.35
`Fourier term v=0.25 \S]
`
`2b
`
`2a
`
`'
`
`i
`
`•> Experiment [7]
`— " T h is analysis, v=0.3 -
`— *— This analysis, v=0.2
`— -First Fourier term
`
`- -
`
`Shape of membrane
`along diagonal
`
`X
`
`0
`
`•
`
`0
`
`0.2
`
`0.4
`
`0.6
`
`0.8
`
`1
`
`1.2
`
`.
`
`.
`
`,
`
`Coordinate along film diagonal
`
`0
`
`2
`
`4
`6
`8
`Membrane aspect ratio b/a
`
`10
`
`12
`
`(b)
`FIG. 3. The deflection of a thin film in the xz-plane (a) and along the
`diagonal (b). The shape of the membrane is virtually independent of
`Poisson's ratio of the film.
`
`FIG. 4. Variation of the function g{v,b/a) with membrane aspect
`ratio for three different values of Poisson's ratio. The solid lines at
`the right-hand side are the plane strain solutions (see appendix) for
`the same Poisson's ratios. For a given aspect ratio, g(v, b/a) increases
`with increasing Poisson's ratio.
`
`3244
`
`J. Mater. Res., Vol. 7, No. 12, Dec 1992
`
`3
`
`

`
`J. J. Vlassak and W. D. Nix: A new bulge test technique for the determination of Young's modulus and Poisson's ratio
`
`center of the membrane can be written as12
`
`-Wo
`
`16a2
`
`X
`
`n=l,3,5
`
`(rot
`
`1 -
`
`nirb
`
`-l
`
`(7)
`
`whereas Eq. (5) or (6) can be used for q2, depending
`on the aspect ratio of the membrane. The load-deflection
`relationship for a stressed membrane is then given by
`Et
`
`aot
`
`C2"
`
`(8)
`
`q = q\
`
`strain state must be one of plane strain since any plane
`perpendicular to the axis of the film is a mirror plane. In
`this case an exact solution for the membrane deflection
`can be readily derived (see appendix). The deflection in
`the center of the membrane is given by
`
`-
`= (6qa4(l
`\
`SEt
`
`v2)\113
`)
`
`(6)
`
`so that for large aspect ratios g{v,b/a) must approach
`[6(1 + v)/8]m. The virtual energy solution indeed ap-
`proaches a limit value which is within 3.7% of the correct
`solution. The maximum in the plot at an aspect ratio
`of about two is most likely an artifact arising from the
`approximations used in the energy method. For small
`aspect ratios, one should use Eq. (5) for the membrane
`deflection, whereas Eq. (6) is better for large aspect
`ratios. In the plane strain case the load-deflection be-
`havior of the membrane is fully determined by the ratio
`E/{\ — v2). As a result, the deflection does not depend
`as strongly on Poisson's ratio as for square membranes.
`Testing of long rectangular membranes therefore allows
`a more accurate determination of Young's modulus when
`Poisson's ratio is not exactly known. A similar observa-
`tion has been made by Tabata et al.8
`Comparing the results for square and rectangular
`membranes, an interesting observation can be made. If
`both a square and a much longer rectangular film are
`tested, the coefficients of q in Eqs. (4) and (6) can be
`determined. Elimination of Young's modulus from the
`two coefficients makes it possible to calculate Poisson's
`+ v). This
`ratio of the film from the ratio f(vf/(l
`method will give at least a very good estimate of a
`quantity that is otherwise very difficult to measure.
`Since Poisson's ratio is rather sensitive to propagation
`of experimental errors in the calculations, a sufficient
`number of films should be tested.
`
`C. The influence of residual stress
`on the deflection of a membrane
`Until now, only membranes without residual stress
`were considered. The presence of such a stress, ao, can
`alter the deflection behavior of a membrane considerably.
`The energy minimization technique used in the two pre-
`vious sections fails to give a straightforward formula for
`the deflection in this case, since the nonlinear equations
`derived from the minimization of the total potential
`energy of the system have to be solved numerically for
`each value of the pressure q and stress cr0. However, if
`one assumes that the pressure can be resolved into two
`components qx and q2 such that q\ is balanced by the
`residual stress in the membrane and q2 by the stretching
`of the membrane, a solution can be readily derived. An
`expression for q\ as a function of the deflection of the
`
`3 or 8/6(1 + v), de-
`where c2 is given by g{v,b/a)
`pending on the aspect ratio. Figure 5 shows c\ as a
`function of membrane aspect ratio. The constant is
`independent of material properties and has a value of
`3.393 for square membranes, decreases rapidly as the
`aspect ratio increases, and reaches a value of 2 for
`infinitely long membranes. The conditions in which
`Eq. (8) holds are that c\ does not change for large
`deflections and that c2 is not a function of o^. Again,
`Eq. (2) can be checked for two limiting cases. In the case
`of plane strain, the problem can be solved analytically
`and Eq. (8) gives the correct solution (see appendix).
`Finite element calculations have been done to study
`the influence of residual stress on the deflection of
`circular and square membranes.3'4'7 According to Lin,7
`c\ is constant and equal to 3.41. This is in very close
`agreement with Eq. (8). The same study also shows
`that at least for circular films, c2 is independent of the
`residual stress in the film. More recent calculations for
`circular films,3'4 however, have shown that c2 is a weak
`function of residual stress. One would expect Eq. (8) to
`
`Residual stress coefficient as a
`function of membrane aspect ratio
`
`32
`
`Plane strain solution
`
`Aspect ratio
`
`FIG. 5. The residual stress coefficient c\ as a function of film as-
`pect ratio.
`
`J. Mater. Res., Vol. 7, No. 12, Dec 1992
`
`3245
`
`4
`
`

`
`J. J. Vlassak and W. D. Nix: A new bulge test technique for the determination of Young's modulus and Poisson's ratio
`
`be quite accurate as long as the residual stress is not too
`high, i.e., smaller than 0.5 GPa.
`According to Eq. (8) a plot of load versus deflection
`at the center of a membrane is a cubic parabola, the
`slope of which at zero deflection is determined by
`the residual stress in the film. The nonlinear term, on the
`other hand, yields information about Young's modulus.
`By fitting Eq. (8) to experimental data from the bulge
`test, both Young's modulus and residual stress can be
`calculated.
`
`III. EXPERIMENTAL
`A. Measuring apparatus
`A schematic of the bulge tester used in this study
`is shown in Fig. 6. The sample to be tested is glued
`onto a sample holder and pressure is applied to one side
`of the film, by pumping water into the cavity under
`the film. The deflection of the film is measured by
`means of a laser interferometer with a He-Ne laser
`light source. The displacement resolution is half the
`wavelength of the light, i.e., 0.3164 yum. The pressure is
`measured with a pressure transducer with a resolution of
`70 Pa. A maximum pressure of 100 kPa can be applied.
`The experiment is controlled by computer via a data
`acquisition system.
`
`B. Sample preparation
`The nitride films examined in this study were de-
`posited by means of LPCVD. The deposition tempera-
`ture was 785 °C and the gas pressure was 300 mTorr.
`The ratio of dichlorosilane to ammonia was 5.2:1. The
`deposition rate was 30 A/min and the final thickness
`
`of the films was approximately 2900 A. The substrates
`were (100) oriented n-type Si wafers, between 200 and
`250 fim thick. The films were deposited on both sides of
`the wafers. Both square and rectangular windows with
`various aspect ratios were etched in the silicon nitride
`on one side of the wafers using standard lithographic
`techniques and plasma etching. In order to make free-
`standing films, the exposed silicon was etched using
`an anisotropic etchant containing potassium hydroxide
`and methanol at a temperature of 65 °C. A typical film
`is shown in Fig. 7. The thickness of each membrane
`was measured by means of ellipsometry. Finally, a
`thin aluminum coating was deposited onto the silicon
`nitride to enhance the reflectivity of the films. Tests were
`performed on each of five square membranes and on
`eight rectangular films with aspect ratios varying from
`1.2 to 4.9.
`In order to have an independent check of the results
`of the bulge test, Young's modulus of the silicon nitride
`was also measured by means of continuous indentation
`testing. The indentations were performed on the film
`using a Nanoindenter, a high-resolution depth-sensing
`hardness tester, the description of which can be found
`elsewhere in the literature.13'14 Both applied load and
`displacement were continuously recorded during the
`experiments. A total of 36 indentations were made to
`plastic depths ranging from 20 to 60 nm. The depths
`of the indentations were small enough that only film
`properties were measured. The velocity of the indenter
`upon loading was between 3 and 6 nm/s. When the
`desired indentation depth was reached, the load was held
`constant for 15 s and then decreased at a rate equal to the
`last loading rate. Hardness and Young's modulus were
`calculated from the load-displacement curves using the
`analysis given by Doerner and Nix.13
`
`interfero-
`meter
`
`_ 'limp
`
`:sample clamp
`'///////.+fiSm
`
`computer+
`data acquisition
`
`FIG. 6. A schematic of the bulge testing apparatus.
`
`FIG. 7. Scanning electron micrograph of a square SiN x window in a
`silicon wafer.
`
`3246
`
`J. Mater. Res., Vol. 7, No. 12, Dec 1992
`
`5
`
`

`
`J. J. Vlassak and W. D. Nix: A new bulge test technique for the determination of Young's modulus and Poisson's ratio
`
`IV. RESULTS AND DISCUSSION
`A. Bulge test results
`The results of the square films were used to calculate
`the elastic modulus of the silicon nitride. In Fig. 8 a
`typical load-deflection plot of a square membrane is
`depicted. The plot consists of a number of loading
`cycles. Since loading and unloading segments trace each
`other, no plastic deformation is taking place and curve
`fitting can be used to determine Young's modulus and
`residual stress. The curve fit is very sensitive to the
`initial height of the film, which if not taken into account,
`can lead to very large errors.3"5 However, with the
`laser interferometer one can simultaneously obtain an
`interference pattern of film and substrate so that it is
`possible to start the tests with a perfectly flat film.
`is
`The elastic modulus of
`the silicon nitride
`222 ± 3 GPa, assuming a Poisson's ratio of 0.28. The
`contribution of the aluminum coating on the silicon
`nitride amounts to approximately 3% and has been
`taken out. It should be noted that the measurement
`was very reproducible and the scatter
`in the data
`extremely small. Young's modulus of LPCVD silicon
`nitride has been measured previously using a variety
`of different
`techniques,
`including bulge
`testing,8'11
`nanoindentation,15'16 and beam deflection techniques.16'17
`The values vary over a wide range from 150 to
`373 GPa depending on the deposition temperature and
`the stoichiometry, but for a low stress nitride deposited
`under conditions similar to this nitride, a value of
`235 GPa has been reported.16
`The results of the rectangular films make it possible
`to examine how accurately expression (8) describes the
`deflection of a membrane. In Fig. 9, measured values of
`g{v,b/a)
`are plotted versus aspect ratio. For compari-
`son, the plane strain solution and the solution given by
`
`Pressure-deflection curve
`- for a SiNx membrane
`
`SiNx + Al coating
`a=2.11 mm
`tsii*.=290 nm
`tAi—31 nm
`
`0
`
`20
`
`40
`
`60
`
`80
`
`100
`
`120
`
`Height (am)
`
`FIG. 8. A typical pressure versus height plot for a square, 290 nm
`SiNj membrane.
`
`Experimental values of g(v,b/a)
`
`plane strain, v=0.28
`
`• Experiment, v=0.28
`Theory, v=0.25
`Theory, v=0.30
`
`1.00
`
`0.95
`
`0.90
`
`0.85
`
`0.80
`
`0.75
`
`0.70
`
`Film aspect ratio b/a
`
`FIG. 9. Experimental values of g{v, b/a) as a function of film aspect
`ratio, assuming a Poisson's ratio of 0.28.
`
`Eq. (8) are also plotted. Agreement between experimen-
`tal results and calculated values is excellent. For aspect
`ratios greater than two, g{v,b/a)
`does not vary with
`aspect ratio and is closer to the plane strain solution. This
`suggests that films with aspect ratios greater than two can
`be used in combination with the results of the square
`films to calculate Poisson's ratio of the film, yielding
`a value of 0.28 ± 0.05. This corresponds well with the
`Poisson's ratios of polycrystalline (0.27) and amorphous
`silicon nitride (0.30) reported in Ref. 15 and justifies the
`use of this value for the calculation of Young's modulus.
`The average residual stress calculated from the tests
`of the square membranes is 124 ± 14 MPa. The tests
`of the rectangular films yield 147 ± 25 MPa. The fact
`that the experimental scatter is greater in the latter case
`can be attributed to some slight twisting of the samples
`that may have occurred during sample mounting. Long
`rectangular films are more prone to this than square films
`and the residual stress, which is determined by the initial
`slope of the load-deflection curve, is more affected than
`Young's modulus. The residual stress in LPCVD silicon
`nitride depends primarily on the deposition temperature
`and the ratio of dichlorosilane to ammonia, and decreases
`when either of these quantities increases.17 Based on this
`observation one would expect a residual stress in the
`range of 100 to 200 MPa.
`
`B. Nanoindenter results
`Figure 10 shows a load-displacement plot for a
`typical indentation in the silicon nitride film. Using
`the analysis first given by Doerner and Nix, 13 Young's
`modulus of the nitride can be determined from the
`unloading slope of this plot. However, since silicon
`nitride shows a substantial amount of elastic recovery
`upon unloading, a more refined analysis developed by
`Oliver and Pharr was used to determine the contact
`
`J. Mater. Res., Vol. 7, No. 12, Dec 1992
`
`3247
`
`6
`
`

`
`J. J. Vlassak and W. D. Nix: A new bulge test technique for the determination of Young's modulus and Poisson's ratio
`
`is independent of the indentation depth. The substrate
`therefore does not significantly affect the measurement.
`An average hardness of 21.0 ± 0.9 GPa was found. This
`is very close to the value of 23 GPa reported in Ref. 15.
`
`V. CONCLUSIONS
`Using an energy minimization technique we have
`derived new and more accurate expressions to determine
`the deflection of square and rectangular membranes
`under the influence of a uniform pressure. Membranes
`both with and without residual stress were considered.
`These formulas can be used to analyze bulge test results
`and to calculate Young's modulus and residual stress
`of thin films. By testing both square and rectangular
`films with a sufficiently large aspect ratio it is possible
`to determine Poisson's ratio of the film.
`Sample preparation in the bulge test is extremely
`important. If samples are prepared properly, the bulge
`test yields very reproducible results. Using standard
`lithography and anisotropic etching techniques, free-
`standing films of LPCVD silicon nitride were fabricated
`and tested as a model system. The deflection of the films
`as a function of film aspect ratio is very well predicted by
`the new analysis. Young's modulus of the silicon nitride
`films is 222 ± 3 GPa and Poisson's ratio is 0.28 ± 0.05.
`The residual stress varies between 120 and 150 MPa.
`Agreement with the literature is very good. Young's
`modulus and hardness were also measured by means of
`nanoindentation, yielding values of 216 ± 10 GPa and
`21.0 ± 0.9 GPa, respectively.
`
`ACKNOWLEDGMENTS
`The authors would like to thank M. K. Small for
`building the bulge test apparatus. This study was funded
`by the Department of Energy, Grant DE-FG03-89-
`ER45387.
`
`REFERENCES
`
`1. D.W. Burns and H. Guckel, J. Vac. Sci. Technol. A 8 (4),
`3606-3613 (1990).
`2. J. W. Beams, in Mechanical Properties of Thin Films of Gold and
`Silver (John Wiley and Sons, Inc.), p. 183.
`3. M.K. Small and W.D. Nix, J. Mater. Res. 7, 1553-1563 (1992).
`4. M. K. Small, J. J. Vlassak, and W. D. Nix, in Thin Films: Stresses
`and Mechanical Properties HI, edited by William D. Nix, John
`C. Bravman, Edward Arzt, and L. Ben Freund (Mater. Res. Soc.
`Symp. Proc. 239, Pittsburgh, PA, 1992).
`5. H. Itozaki, Mechanical Properties of Composition Modulated
`Copper-Palladium, Ph.D. Dissertation, Northwestern University
`(1982).
`6. S. Timoshenko and S. Woinowski-Krieger, in Theory of Plates
`and Shells (McGraw-Hill, New York, 1987).
`7. P. Lin, The In-Situ Measurement of Mechanical Properties of
`Multilayer-Coatings, Ph.D. Dissertation, Massachusetts Institute
`of Technology (1990).
`8. O. Tabata, K. Kawahata, S. Sugiyama, and I. Igarashi, Sensors
`and Actuators 20, 135-141 (1989).
`
`Typical mdentation plot for SiN
`
`Depth (nm)
`
`in a
`load-depth plot for a nanoindentation
`typical
`FIG. 10. A
`300 nm silicon nitride film. The indenter velocity upon
`loading
`was between 3 and 6 nm/s; the hold time at maximum load was
`15 s.
`
`area between indenter and sample.18 In this analysis
`it is assumed that upon unloading the indenter shape
`can be modeled as a paraboloid. In Fig. 11, the contact
`compliance, i.e., the reciprocal of the unloading slope
`at maximum load, is plotted versus the reciprocal of
`the square root of the projected contact area between
`indenter and sample. This is a straight line, the slope of
`which is inversely proportional to the elastic modulus
`of the film. The modulus is then 216 ± 10 GPa which
`is in excellent agreement with the value measured with
`the bulge test. For comparison, a Young's modulus of ap-
`proximately 250 GPa was measured with a Nanoindenter
`for a stoichiometric LPCVD nitride in Ref. 16. The
`same analysis also allows one to determine the hardness
`of the film. The hardness of the silicon nitride film
`
`Contact compliance for SiN x
`t=300nm
`v=0.28 and e=0.75
`E=216±10 GPa
`
`z
`
`•A
`
`C oo coo
`
`0.0020
`
`0.0025
`
`0.0030
`
`0.0035
`
`0.0040
`
`0.0045
`
`0.005C
`
`Projected contact area-1/2
`
`(1/nm)
`
`FIG. 11. Contact compliance as a function of the reciprocal of the
`square root of the projected contact area between indenter and sample.
`The slope of the plot is inversely proportional to the elastic modulus of
`the film.
`
`3248
`
`J. Mater. Res., Vol. 7, No. 12, Dec 1992
`
`7
`
`

`
`J. J. Vlassak and W. D. Nix: A new bulge test technique for the determination of Young's modulus and Poisson's ratio
`
`9. M. G. Allen, M. Mehregani, R. T. Howe, and S. D. Senturia, Appl.
`Phys. Lett. 51 (4), 241-243 (1987).
`10. R.J. Jaccodine and W.A. Schlegel, J. Appl. Phys. 37 (6),
`2429-2434 (1966).
`11. E.I. Bromley, J.N. Randall, D.C. Flanders, and R.W. Mountain,
`J. Vac. Sci. Technol. B 1 (4), 1364-1367 (1983).
`12. S. P. Timoshenko and J. N. Goodier, in Theory of Elasticity
`(McGraw-Hill, New York, 1970).
`13. M.F. Doerner and W.D. Nix, J. Mater. Res. 1, 601-609 (1986).
`14. M. F. Doerner, Mechanical Properties of Metallic Thin Films
`on Substrates Using Sub-micron Indentation Methods and Thin
`Film Stress Measurement Techniques, Ph.D. Dissertation, Stanford
`University (1987).
`15. J. A. Taylor, J. Vac. Sci. Technol. A 9 (4), 2464-2468 (1991).
`16. S. Hong, Free-standing Multi-layer Thin Film Microstructures
`for Electronic Systems, Ph.D. Dissertation, Stanford University
`(1991).
`17. M. Sekimoto, H. Yoshihara, and T. Ohkubo, J. Vac. Sci. Technol.
`21 (4), 1017-1021 (1982).
`18. W.C. Oliver and G.M. Pharr, J. Mater. Res. 7, 1564-1583
`(1992).
`
`APPENDIX: THE PLANE STRAIN DEFLECTION
`OF A THIN MEMBRANE
`The plane strain deflection of a thin membrane in
`the presence of a residual stress can be formulated as
`
`follows: H Jw\2
`
`= CTrrt
`
`d2w
`Ix2
`
`= 0
`
`dx)
`dax
`dx
`
`(Al)
`
`(A2)
`
`(A3)
`
`where q is the differential pressure applied to the mem-
`brane, (To is the residual stress in the membrane, and t is
`the membrane thickness, u and w are the displacements
`in the plane of the membrane and perpendicular to the
`membrane, respectively (see Fig. 1). Equations (Al) and
`(A2) are the equilibrium equations; Eqs. (A3) arise from
`the condition of plane strain and express that u and
`w are functions of x solely. Equations (A4) are the
`boundary conditions. For thin membranes, the bending
`stiffness can be neglected. In the case where the deflec-
`tion is much smaller than the width of the membrane,
`the first derivative in Eq. (Al) can be neglected and
`Eq. (Al) can be readily integrated. Taking into account
`Eqs. (A2-A4), one finds:
`
`w =
`
`q
`It a.
`
`-{a1 - x2)
`
`(A6)
`
`Using this expression for w, Eq. (A5) can be integrated
`to find u. Taking into account that w(0) = 0, this leads
`to:
`
`U
`
`=
`
`- v
`
`{(Txx
`
`-
`
`1 q2x3
`6 {taxxf
`
`(A7)
`
`The stress axx can be determined by setting u{±a) to
`zero:
`
`,2^2
`Eq2a
`6f2(l - v2
`
`(A8)
`
`Let w0 be the deflection along the center of the mem-
`brane. From Eq. (A6), one finds that
`
`w = fix)
`u = g(x)
`
`u(±a) = 0
`M(0) = 0
`w(±a) = 0
`
`Wo =
`
`2tax
`
`(A4)
`
`and Eq. (A8) becomes:
`
`(A9)
`
`(A10)
`
`1 - v2
`
`du
`
`1 / d w \ 2
`
`1 - v2
`-<TX
`
`(A5)
`
`2ta0
`
`w0 +
`
`q =
`
`%Et
`6a4(l - ^2)
`
`J. Mater. Res., Vol. 7, No. 12, Dec 1992
`
`3249
`
`8