Question 7.19: LOW-κ POROUS DIELECTRICS FOR MICROELECTRONICS It was mention...
LOW_{- \kappa} POROUS DIELECTRICS FOR MICROELECTRONICS It was mentioned in Chapter 2 that today’s high transistor density ICs have multilayers of metal interconnect lines that are separated by an interlayer dielectric (ILD). The speed of the chip (as limited by the RC time constant) depends on the overall interconnect capacitance, which depends on the relative permittivity \varepsilon _{rILD} of the ILD. The traditional ILD material has been SiO_{2} with \varepsilon _{r} = 3.9. There is much research interest in finding suitable low_{- \kappa} materials for such ILD applications, especially in ultralarge scale integration (ULSI). Estimate the required porosity in SiO_{2} if its effective relative permittivity is to be 2.5? What would be the porosity needed if we start with a dielectric that has \varepsilon_{r} = 3.0?
The Maxwell–Garnett equation is particularly useful for such porous media calculations. Substituting \varepsilon_{r2} = 3.9, \varepsilon _{r1} = 1 (air pores), and setting \varepsilon_{reff} = 2.5 in Equation 7.97 we have
\frac{\varepsilon_{r eff }-\varepsilon_{r 2}}{\varepsilon_{r eff }+2 \varepsilon_{r 2}}=v_{1} \frac{\varepsilon_{r 1}-\varepsilon_{r 2}}{\varepsilon_{r 1}+2 \varepsilon_{r 2}} [7.97]
\frac{2.5-3.9}{2.5+2(3.9)}=v_{1} \frac{1-3.9}{1+2(3.9)}
and solving gives
v_{1}=0.412, \quad \text { or } \quad 41 \% \text { porosity }
Such porosity is achievable but it may have side effects such as poorer mechanical properties and lower breakdown voltage. (We should take the calculated porosity as an estimate since the volume fraction is higher than typical limits for Equation 7.97.) Note that the Lichtenecker formula gives 32.6 percent porosity. As apparent from this example, there is a distinct advantage in starting with a dielectric that has a low initial \varepsilon _{r}, and then using porosity to lower \varepsilon _{r} further. For example, if we start with \varepsilon _{r2}= 3.0, and repeat the calculation above for \varepsilon _{reff} = 2.5, then we would find v_{1} = 0.21 or 21 percent porosity. Many polymeric materials have \varepsilon_{r} values around 2.5 and have been candidate materials for low _{- \kappa} ILD applications in microelectronics. (The above ideas are explored further in Questions 7.35 and 7.36.)