Hertzian Indentation

Section 4.4.2 further separates the polishing mode of abrasion into two submodes, that of Hertzian indentation based wear and fluid-based wear. The difference between these two polishing modes is the nature of the slurry fluid layer between the pad and the wafer. This area of CMP is still poorly understood, yet has important implications as to the removal mechanisms of CMP. 

Whether CMP occurs as Hertzian indentation or fluid-based wear is not clear and has been the subject of some debate. The difference between the two wear modes is in the slurry fluid layer between the pad and wafer. As discussed in Section 4.2, if the fluid layer is not continuous, then pad-wafer contact occurs. Note, however, that the pad does not contact the wafer surface directly, but rather the pad presses abrasive particles against the surface. In such instances, the pad will drag the abrasives across the surface, resulting in Hertzian indentation. 

If the fluid layer is continuous, then the pad does not contact the wafer surface, and Hertzian indentation will not occur. Instead, the collisions between abrasive particles and the pad accelerate the abrasive particles. The particles then impinge on the wafer surface, resulting in fluid-based wear. The velocity and angle of approach of the abrasive particles will determine the kinetic... 

There are several models discussed in the literature of CMP planarization of oxides. In this section we shall review two of these models that assume different modes of polishing, i.e., Hertzian indentation, by Wamock and fluid based wear, by Runnels. In addition to predicting planarity, these models gives good insight into the planarization mechanisms for CMP. However, these models are not necessarily the most accurate models. Other models include those by Burke et al., Yu et al, and Renteln and Coniff.< >... 

This model implicitly assumes that, at least to some extent, the pad makes contact with the wafer surface (i.e., the pad directly presses the abrasive against the surface) and exerts pressure directly to the surface. The abrasive then moves across the surface as a Hertzian indenter. As discussed in Chapter 4 however, it is also possible that a continuous fluid layer exists between the wafer and the pad. The pad compresses the fluid layer, which in turn exerts hydrodynamic pressure on the surface. The existence of a hydrodynamic fluid layer is an important distinction because the wear mechanisms are different for fluid-based wear as opposed to Hertzian indenter-based wear (see Chapter 4). 

Quasi-static Young s modulus measured by Hertzian indentation (b) Data taken from ref [5] (c) Measured by Dynamic Mechanical Thermal Analysis (D.M.T.A) at 1 Hz (T is taken as the temperature of the maximum in tan 5) (d) (7y and Oy are the yield stress under uniaxial and plane strain compression, respectively, for an equivalent strain rate of 5x10" s" (see ref... 

Baig MS, Dowling AH, Fleming GJ. Hertzian indentation testing of glass-ionomer restoratives A reliable and clinically relevant testing approach. J Dent. 41(11) (2013) 968-973. 

Scherer G W (1996) Bending of gel beams Effect of deflection rate and Hertzian indentation. Journal of Non-Crystalline Solids 201 1-25... 

Figure 1337 Schematic comparison of the brittle-ductile temperature transition in four different tests ( ) Hertzian indentation (lower transition), (2) plastic-elastic indentation (upper transition), (3) Double cantilever beam (lower transition) and (4) notched bar (upper transition). (Reproduced from Puttick, K.E. (1980) The correlation of fracture transitions. ). Phys. D, 13, 2249. Copyright (1980) Institute of Physics.)...

Fig. 8. (a) Schematic of a Vickers indentation-induced Hertzian cone crack, (b) View from the bottom of an aluminosihcate glass block of a Vickers... 

An alternative explanation has been proposed by Quesnel [62]. Assuming that the adhesion-induced deformation could be treated as a Hertzian indentor, with the load due to the force arising from the surface energy, Quesnel calculated the indentation in a self-consistent manner. That is to say, Quesnel recognized that the attractive force would vary as the particle or substrate deformed, owing to the increased circumference of the contact patch. He also recognized that, due to... 

LGMs of the AT/alumina and AT/ZTA displayed some very interesting properties which include excellent machinability, low thermal expansion coefficient, improved thermal shock resistance, low hardness (about 5 GPa), low Young s modulus (E) (250 GPa) and excellent flaw tolerance [Pratapa, 1997 Pratapa Low, 1998 Skala, 2000 Manurung, 2001], These materials appeared to display a large degree of near-surface quasi-plasticity under the Hertzian or the Vickers indenter which effectively inhibits the formation and propagation of cracks. The ductile behaviour of these materials was... 

Fig. 17 Contact mechanics analysis of Herztian cracks within brittle materials.a Schematic description of a Hertzian cone crack induced under normal indentation by a rigid sphere, b Reduced plot of JC-field as function of cone crack length and for increasing loads pf < p// < pm during sphere-on-flat normal indentation of brittle materials. Arrowed segments denote stage of stable ring crack extension from Cf to cc (initiation), then unstable to ci at P = P,n (cone-crack pop-in) (From [67]). Branches (1) and (3) correspond to unstable crack propagation (dK/dc > 0), branches (2) and (4) to stable crack propagation (dK/dc < 0)...

Stress distribution in indentation is largely affected by the indenter tip geometry, which is a vital factor in determining the boundary conditions for the field. The major types of indenter tips shown schematically in Figure 3 may be separated into two groups, viz. point-force (pyramidal and conical) and spherical indenters. Correspondingly, Boussinesq and Hertzian stress fields will describe point-force and spherical indentation in the case of purely elastic loading (Fig. 4). To account for possible elastic compliance of the indenter, a reduced elastic modulus Er is... 

FIGURE 16.15 Plasticity under the indenter (the shaded area) causes the deviation from Hertzian behavior. 

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