Modeling Methods. Michał Szermer Department of Microelectronics and Computer Science DMCS Lodz University of Technology TUL

Transcription

1 Modeling Methods Michał Szermer Department of Microelectronics and Computer Science DMCS Lodz University of Technology TUL

2 Introduction 2 Design path Modeling methods Materials used in microsystems Hypotheses used in modeling Finite Element Method (FEM)

3 3 Design Path MEMS specification NO Accept? Global simulation (microsystem + packaging) YES Behavioral model Circuit diagram Equivalent circuit Fabrication and Tests (reduced model) FEM simulation (ANSYS, COMSOL,...) (equations) Geometrical model 3D Behavioral simulations (SPICE, Verilog,...) Technological process Simulation (ATHENA,...) Masks design (GDSII, CIF) DRC (CADENCE,...)

4 Design Path cont d 4 Many coupled physical phenomena Strong nonlinearity Dimensional uncertainty Material properties Magnetic Fluidic Thermal Electromagnetic Electrostatic Electrical Structural

5 Analytical Modeling 5 The microsystem is considered to be a simplified system corresponding to the real system The microsystem is described by several simple equations (mechanical, electrical) Membrane F kx c x t

6 Numerical Modeling 6 The microsystem is modeled using a full 3D geometrical model (in some cases a 2D model can be used) The model is divided into many small parts (elements). Each part is described by the equation Equations form a matrix Membrane F M x C x K x

7 Comparison of Modeling Methods 7 Analytical model Numerical model Complexity Low High Geometry Basic shapes Unrestricted shapes Mathematical description A few equations Matrix equation Calculations complexity Hand-made Computer program Precision Moderate (model simplicity) High (model complexity, convergence) Calculation time Short Long Easiness of use Simple (combination of a few domains) Complex (coupled multidomains)

8 Modeling Methods cont d 8 Question Which method should be used? Answer One which meets the requirements (calculation time, precision, ).

9 Modeling Methods cont d 9 Simple microsystem analytical model Complex microsystem numerical model But what about time? The method which is as fast as possible. Time is money!

10 Project Optimization 10 One or a few FEM simulations

11 Statistical Simulations 11 Question How to design the optimal microsystem? Answer Find the solution of the function describing the relationship between the performance of the microsystem and its input parameters (dimensions, force, ).

12 Statistical Simulations cont d 12 Define the range of the input parameter Perform a lot of iterations in order to find the optimal value (analytical model is preferred) Estimate the production yield before the fabrication process (costs reduction)

13 Materials 13 Classification by the nature of the bonds between atoms: Metals metallic bond Ceramics ionic bond Polymers covalent bond Composites - modified materials made from two or more component materials with significantly different physical or chemical properties

14 Hypotheses on Materials 14 Continuity Homogeneity Uniform structure MEMS NEMS

15 Hypotheses on Materials cont d 15 Physical properties Isotropy the same properties in all crystallographic directions (glass) Anisotropy properties depends on the crystallographic direction (graphite) Orthotropy symmetric anisotropy the same properties in perpendicular crystallographic directions (silicon, germanium)

16 Hypotheses on Materials cont d 16 Linear elasticity Potential energy W of atoms is a function of their relative distance obeys the quadratic law if small deformations are considered W r w k r 2 0 r 0 F W r r 2 k r r 0

17 Hypotheses on Materials cont d 17 Bernoulli hypothesis Flat plane remains unchanged after bending. (valid for various clamped structures, beams, cantilevers, membranes, ) x x

18 Hypotheses on Materials cont d 18 Displacement hypothesis Relation between length and displacement of the structure Small deflections if h << l Large deflections if h ~ l Small deflections Additional stress from bending can be neglected (linear analysis) l h

19 Strain and Stress 19 Strain [dimensionless] relative elongation of the solid l l 0 Stress [Pa] average force per unit area F a v g S

20 Strain and Stress cont d 20 Tension longitudinal elongation by pulling force Compression longitudinal compression by pushing force Confining compression equal in all directions

21 Strain and Stress cont d 21 Normal stress Force per unit area acting normal Shear stress Force per unit area acting tangent

22 Strain and Stress cont d 22 Torsion the shear stress acting at the distance from the center of the shaft (twisting due to applied torque) Pure bending there is an axis within the material where the stress and strain are zero (neutral axis). Stress is no longer uniform over the cross section of the structure (it varies) Simple bending pure bending with the consideration of the specific conditions (homogeneity, continuity, elasticity, symmetry, )

23 Hooke s Law 23 Stress and strain values are related by the known relation, called the Hooke s law The coefficient of this relation is called the Young s modulus and it is linear

24 Hooke s Law cont d 24 Hooke s law E E Young s modulus also known as elastic modulus [Pa] F S E l l F E l l S F ks

25 Stress vs. Strain Dependency 25 1 Elastic region 2 Plastic region 3 Strain hardening region 4 Necking region σ y Yield strength (yield point) σ m Ultimate strength (maximal stress) σ f Fracture

26 Stress vs. Strain Dependency cont d 26 Elastic zone Plastic and strain hardening zone Necking zone A C D

27 Stress vs. Strain Dependency cont d 27 elastic zone plasticity new elastic zone 0 y 0 y y σ y yield point ε p plastic strain

28 Stress vs. Strain Dependency cont d 28 rigid perfectly plastic elastic perfectly plastic elastic with strain hardening

29 Normal Strain 29 Small displacements

30 Shear Strain 30 Small rotations

31 Generalized Hooke s Law 31 Each element of a cubic shape crystal lattice under mechanical loading is being exposed to normal and shear stresses The stress in a small volume has to be expressed as a tensor with nine independent elements Generalized Hooke s law for anisotropic coefficients has to be rewritten in more general form z <001> σ zz σ zy σ xz σ xx σ zx σ xy σ yz σ yx σ yy y <010> C x <100> C Stiffness matrix

32 Generalized Hooke s Law cont d 32 C Stiffness matrix S Compliance matrix 81 coefficients in general case

33 Generalized Hooke s Law Simplifications 33 Equilibrium state when no torsional movement appears some stress components have the same value Stress and strain vectors may be reduced to only six components symmetry ij ji

34 Generalized Hooke s Law Simplifications cont d 34 Tension Hydrostatic pressure p F p p

35 Generalized Hooke s Law Simplifications cont d 35 Diamond-like structures have the same mechanical properties in the orthogonal directions (orthotropic materials) If the silicon wafer is cut along the (100) plane what is usually done it has diamond-like structures z <001> (100) (001) (010) y <010> x <100>

36 Generalized Hooke s Law Simplifications cont d The stress-strain relation for crystalline silicon G G G E 1 E ν E ν E ν E 1 E ν E ν E ν E 1 Stiffness and compliance matrices reduces to 12 non-zero elements E Young modulus n Poisson s ratio G Shear modulus

37 Typical Properties of Silicon 37 Mechanical parameters of silicon in a function of crystallographic direction Direction Young modulus [GPa] Shear modulus [GPa] Poisson coefficient Anisotropy coefficient <100> <110> Poisson s ratio (n or nu) is the signed ratio of transverse strain to axial strain (named after Siméon Denis Poisson ( ) a French mathematician, engineer, and physicist)

38 Finite Element Method 38 Numerical technique for finding approximate solutions of partial differential equations Based on meshing of surface or space into finite element Can be used in 1-D, 2-D and 3-D spaces Usage: Material resistance, simulation of deformations, stresses, displacements Heat and liquid flow Domains: dynamics, kinematics, static, electrostatic, magnetic, electromagnetic,

39 Finite Element Method cont d 39 Meshing: Space division into finite number of elements In mathematic language: discretization replace of infinite dimensional linear problem into finite dimensional problem Continuous Geometrical Model Ideal Discrete Model Discrete Model for Calculations

40 Calculation Time Finite Element Method cont d 40 Number of Elements Discretization Function Meshing

41 Finite Element Method cont d 41 FEM modelling is based on partial differential equations. The function is formulated for all nodes of each element. F M x C x K x [M] inertia matrix [C] damping matrix [K] stiffness matrix [F] load vector [x] displacement vector [ẋ] velocity vector [ẍ] acceleration vector Each element has only a few neighbor. Thus, resultant matrices are sparse.

42 Finite Element Method cont d 42 Boundary Conditions loads clamping conditions Degree of Freedom (DOF) displacement (movement, rotation) pressure temperature electrical voltage magnetic potential 3 DOF define the location of material point in space 6 DOF define the position of the rigid body in space

43 Finite Element Method cont d 43 Task division pre-processor task preparation solver performing calculations post-processor visualization of the results Software Ansys Comsol Coventor Ware Abaqus

44 Materials 44 Silicon Germanium Gallium Arsenide Indium Antimonide Boron Phosphorus Arsenic Diamond Polymers Piezorezistive Piezoelectric

45 Silicon 45 Good electrical properties semiconductor Good mechanical properties high Young s modulus low density wide range of linear elasticity high yield stress low thermal expansion melting point at 1685 [K] Oxidation number: 4 Direction Young modulus [GPa] Shear modulus [GPa] Poisson coefficient Anisotropy coefficient <100> <110>

46 Boron, Phosphorus and Other Dopants 46 Materials used in silicon doping in order to change its electrical properties N-type semiconductor by adding pentavalent impurities like phosphorus, arsenic, N-type semiconductor by adding trivalent impurities like boron, Boron Oxidation number: 3 Phosphorus Oxidation number: 5

47 Diamond 47 Atoms are arranged in the cubic lattice (allotrope of carbon) Properties very high hardness (stiffness) high thermal conductivity optical transparency range chemical stability erosion resistance

48 Polymers 48 They consist of a large molecule, or macromolecule, composed of many repeated subunits Broad range of properties Synthetic and natural Polymer applications Medicine Electronics Apparatus and machine parts Cosmetology

49 Polymers cont d 49 Molecular imprinting on polymer surface After prof. B. Wandelt, DMP, TUL

50 Smart Materials 50 These materials are frequently combined with biological components in order to create material that changes its properties when the environment changes (i.e. stress, temperature, moisture, ph, electrical, magnetic field, ) Piezoelectric Magnetostrictive Magnetic shape memory (react to changing the magnetic field) Shape memory alloys (react to changing the temperature or stress) ph sensitive polymers Temperature-responsive polymers Dielectric elastomers

51 Smart Materials cont d 51 Medical applications Diagnostic tests in vivo monitoring The Macro Fiber Composite (MFC) a lowprofile actuator and sensor offering high performance, flexibility and reliability in a costcompetitive device After

52 Smart Materials cont d 52 ph paper strips one of the simplest sensing devices Several smooth color changes over a ph value range from 0 to 14 in order to indicate the acidity or alkalinity of measured solutions Used for detection and diagnostics for home diagnostic kits Another type of sensor used for measurement the concentration of hydrogen ions is ISFET The sensor which measures other ions than hydrogen ions is a modified ISFET with the membrane sensitive to proper type of ions called CHEMFET

53 Piezoresistivity 53 It is a change in the electrical resistivity of a semiconductor or metal when mechanical strain is applied Applications piezoresistive pressure sensors Applied force induced strain resistivity change σ l σ t Δ R R π π l Longitudinal piezoresistivity coefficient π t Transversal piezoresistivity coefficient l σ l π t σ t

54 Piezoelectricity 54 It is the electric charge that accumulates in certain solid materials in response to applied mechanical stress It is a reversible process in materials exhibiting the direct piezoelectric effect Used in sensors that measure changes in: Pressure Acceleration Temperature Strain Force Applied force induced stress electric potential generation

55 Summary 55 Microsystems (MEMS) consists of: Sensors Actuators Electronic circuits Analytical and numerical methods are used in microsystems modeling The most important materials used in microsystems manufacturing are: Silicon Boron (typical p-type dopant) Phosphorus, Arsenic (typical n-type dopants)

56 Sources 56 ANSYS 14.0 Documentation COMSOL Multiphysics 4.3 Documentation Maluf N., Williams K.: An Introduction to Microelectromechanical Systems Engineering (2nd edition), Artech House, Inc., 2004 Leondes C.T.: MEMS/NEMS Handbook, Techniques and Applications, vol. 5, Medical Applications and MOEMS, Springer, 2006 Sze S.M.: Semiconductor Devices, Physics and Technology, Wiley, 2002 Landau E., Lifschitz F., Theory of elasticity (2nd edition), Pergamon Press, 1970 Wortman J., Evans R., Young s modulus, Shear modulus and Poisson ratio in Silicon and Germanium, Journal of Applied Physics, vol. 36, no. 1, Jan 1965, pp Timoshenko S. P., and Woinowsky-Krieger S., Theory of Plates and Shells (2nd edition), McGraw-Hill, New York, 1959