MS6021 Scientic Computation Project 1 Project 1. Strains in a silicon chip. This project is adapted from a problem submitted to MACSI for a study group. An electronics company designs and manufactures Integrated Circuits (ICs) miniature circuit boards consisting of millions of electronic devices such as transistors, capacitors, resistors and diodes. These components are built up and connected together in layers on the surface of a silicon substrate, known as a die, via etching, deposition and photolithography. The die are then packaged in plastic to make chips: silicon chips perform the underlying functions of most modern electronic devices. A particular type of silicon chip produced by the company is loaded with an arbitrary force of the order of 100 N during testing. A key component in the chip is an oscillator circuit whose frequency must reach a 1% accuracy specication over a temperature range of 40 to +105 C and 100% of the chips are tested. However, the chips in question were not reaching this level of precision and the company attributed this to the force exerted on the chip during testing. The piezoresistive effect could account for this lack of accuracy if strains of the order of 104 are produced during testing. A rst estimate of the magnitude of the strains can be obtained by modelling the chip as a beam made of silicon. (These estimates can then be used as a sanity check for a more sophisticated nite element elasticity calculation.) The vertical displacements u of a beam are given by 4u EI 4 = f (1) x where E is Young's modulus, I is second moment of area, f is force per unit 3 2 length applied to the beam. The boundary conditions are u = 0 and EI u = x2 x3 F , where F is the force applied to the edges and the applies the the left and right edges. Figure 1: Beam geometry. The depth of the chip is 2A William Lee william.lee@ul.ie 1 MS6021 Scientic Computation Project 1 The upper load placed on the chip is modelled as two point forces while the loads exerted by the pins of the chip are modelled as (localised) Hookean springs as shown in gure 1. Thus the equations become EI 4u = F [(x B) + (x + B)] k2 u(0) (x) x4 (2) Boundary conditions 3u = k1 u(A) x3 3u EI 3 = k1 u(A) x 2u =0 x2 EI x = A (3) x=A (4) x = A (5) Given the displacement the tensile strain xx can be calculated from the equation h 2u (6) xx = 2 x2 where h is the height of the chip. Estimates of the parameters are given in table 1. Parameter Value A 3 mm B 2 mm h 1 mm F 50 N k1 40 MN m1 k2 4 MN m1 Table 1: Estimated (or guessed) values of parameters. Show that calculating the displacements can be reduced to the problem of solving a set of linear equations. Solve the equations and produce graphs of the displacements and the strains. Email me a one page report explaining your method and including the graphs, giving references to anything you look up (don't cite Wikipedia!). Also send me matlab m-les that solve the equations and produce the graphs. Deadline: 8am Monday 12th November. William Lee william.lee@ul.ie 2 \f\f