In situ bulge testing in an atomic force microscope: Microdeformation experiments of thin film membranes

Abstract

A new bulge testing setup for the measurement of the mechanical properties of thin films is presented. This self-built device can be incorporated in an atomic force microscope (AFM), which allows the recording of topographic images of the observed sample membranes under load conditions. Bulge test experiments on different silicon nitride films are presented and compared to nanoindentation experiments. The measured elastic moduli from nanoindentation and bulge testing are in good agreement. Apart from that, the ability to extract stress–strain data from AFM scans is shown, and the results are compared to standard bulge testing experiments. Imaging of the sample microstructure under load conditions is demonstrated on a thin Cu film.

Appendices

Appendix A: Stress and Strain in Long Rectangular Membranes19

Imagine a cylindrical pipe bearing an internal pressure p which is cut in two halves along its axis of symmetry [Fig. A1(a)]. The wall thickness, the pipe length, and the radius of curvature are represented by t, l, and R, respectively. By balancing the forces, one finds

$$\eqalign{2Rlp = 2tl{\sigma }}\,\,\,\,\,, \;\;\;\;\;\;\;\, \Rightarrow {\sigma }\;{\text{ = }}\;\frac{{Rp}}{t} $$

((A1))

FIG. A1

Equilibrium of the wall forces and the force resulting from the internal pressure; effect of angular misalignment α on the displacement detection. (b) Derivation of the film strain ϵ_x from geometrical parameters.

where σ is the stress in circumferential direction. Simple geometric relations show that, for h ≪ a:

$$\eqalign{{R^2} = {\left( {R - h} \right)^2} + {a^2} \,\,\,\,\,\,\, = {R^2} - 2Rh + {h^2} + {a^2} \,\,\,\,\,\,\, \Rightarrow R = \frac{{{h^2} + {a^2}}}{{2h}} \approx \frac{{{a^2}}}{{2h}} \cr} $$

((A2))

Inserting Eq. (A2) in Eq. (A1) yields

$${{\sigma }_x} = \frac{{p{a^2}}}{{2th}}$$

((A3))

The strain is obtained purely from geometric considerations as well [Fig. A1(b)]:

$$ = \frac{{R{\theta } - a}}{a} = \frac{{{\theta } - \left( {a - R} \right)}}{{a/R}}$$

((A4))

The Taylor expansion of θ is

$${\theta } = \arcsin \left( {a/R} \right) = \left( {a/R} \right) + \frac{{{{\left( {a/R} \right)}^3}}}{6} + \cdots $$

((A5))

Therefore

$$ = \frac{{\left( {{a \mathord{\left/ {\vphantom {a R}} \right. } R}} \right) + \frac{{{{\left( {{a \mathord{\left/ {\vphantom {a R}} \right. } R}} \right)}^3}}}{6} - \left( {{a \mathord{\left/ {\vphantom {a R}} \right. } R}} \right)}}{{\left( {{a \mathord{\left/ {\vphantom {a R}} \right. } R}} \right)}} = \frac{{{a^2}}}{{6{R^2}}}$$

((A6))

Substituting R by Eq. (A2) yields

$$ = \frac{{2{h^2}}}{{3{a^2}}}$$

((A7))

Appendix B: Displacement Detector Alignment

During system operation the laser detector can be misaligned in two ways: lateral misalignment in the x direction and angular misalignment by tilting the stage relative to the laser axis.

1. Lateral misalignment

According to the bulge model, the membrane shape in the x–z plane is described as a segment of a circle. To have the origin of the coordinate system where it previously has been defined, the circular equation reads

$${x^2} + {\left( {z + R + h} \right)^2} = {R^2}$$

((A8))

where R is the radius of the circle. Using Pythagoras’ theorem, one finds

$$R = \frac{{{h^2} + {a^2}}}{{2h}}$$

((A9))

insertion in Eq. (A1) yields

$$z\left( {x,a,h} \right) = \frac{1}{{2h}}\left( {{h^2} - {a^2} + \sqrt {{h^4} + 2{h^2}{a^2} + {a^4} - 4{x^2}{h^2}} } \right)$$

((A10))

for all z > 0. Negative values of z are not relevant in this consideration.

To check the lateral alignment accuracy of the setup, the light microscope of the AFM was focused on a prominent surface feature, and its coordinates relative to the origin of the AFM stage were recorded. Afterward, the stage was first moved to an arbitrary position and then repositioned to the defined coordinates. The effective distance between this point and the position of the prominent feature was determined to Δx = 70.7 μm and Δy = 59.0 μm, Δx and Δy being the average misalignment in the x and y directions, respectively. If the alignment process is made strictly unidirectional, thus eliminating the backslash of threads, these values decrease to 39.7 μm and 27.0 μm, respectively. Assuming a membrane half width of a = 1 mm and a membrane displacement of h = 70 μm, which are reasonable values, Eq. (A3) yields a relative error in displacement detection of 0.157% and 0.0084% for Δx = 39.7 μm and Δy = 29.0 μm, respectively.

2. Angular misalignment

To study the effect of sample tilting, the circle segment model is extended by a straight line representing the laser beam which passes the origin and is tilted by α. We assume perfect lateral alignment of the laser beam [Fig. A1(a)]. Hence, the trajectory of the laser beam is described by the simple equation

$$l\left( x \right) = \tan \left( {90^\circ - {\alpha }} \right)x = \cot \left( {\alpha} \right)x$$

((A11))

The x coordinate of the point of intersection is

$${x_{\operatorname{in}}} = \frac{1}{{2h\left( {\tan \left( {\alpha } \right) + 1} \right)}}\left[ {{h^2} - {a^2} + A\,\,\tan \left( {\alpha } \right)} \right]$$

((A12))

$$A = \sqrt {{h^4} + 2{h^2}{a^2} + {a^4} + 4{h^2}{a^2}{{\tan }^2}\left( {\alpha } \right)} $$

((A13))

The corresponding y coordinate is

$${y_{\operatorname{int} }} = {x_{\operatorname{int} }}\cot \left( {\alpha} \right)$$

((A14))

The measured displacement is the distance between the origin and the point of intersection

$${h_{{\text{meas}}}} = \sqrt {x_{\operatorname{int} }^2 + y_{\operatorname{int} }^2} $$

((A15))

Therefore, the error in the measurement equals

$$\eqalign{\Delta h = \frac{{{h_{{\text{meas}}}} - h}}{h} \;\;\;\;\, = \frac{1}{{2h\left( {{{\tan }^2}\left( {\alpha } \right) + 1} \right)}}\sqrt {{{\left( {{h^2} - {a^2} + A} \right)}^2}{{\tan }^2}\left( {\alpha } \right)\left( {{{\cot }^2}\left( {\alpha } \right) + 1} \right)}\;\;\;\;\;\;\;\, - 1 } $$

((A16))

To stay below a detection error of 1%, a misalignment from the normal direction of 8° is tolerable, assuming the same values for a and h as mentioned above. This criterion is easily met in this experimental setup.