Читать книгу Electronic Packaging Science and Technology - King-Ning Tu - Страница 14

An electronic device in operation is an open system because electrical charges flow in and out of the device. While the number of charges in transport is conserved, entropy production is not. The waste heat in entropy production is Joule heating on the basis of irreversible processes. [4, 5] For electrical conduction, Onsager’s Eq. (1.1) below shows that entropy production is the product of the conjugated flux of j (current density = coulomb/cm²‐sec) and the conjugated driving force of E (electric field E = jρ, where ρ is resistivity). Derivation of the Onsager equation will be given in Chapter 9.

(1.1)

where T is the temperature, V is the volume of sample, dS/dt is the entropy production rate, and j²ρ is the Joule heating per unit volume per unit time. Typically, the power from Joule heating is written as P = I²R = j²ρV, where I is the applied current and R is the resistance of the sample. Thus, j²ρ is power density or Joule heating per unit volume per unit time of the sample, in units of Watt/cm³, and I²R is Joule heating per unit time for the entire sample, in units of Watt. Clearly, this is the reason why we need low‐power devices or low entropy production devices.

While the cost of production of 3D IC can be reduced when it is in mass production, the problem of reliability due to over‐heating has to be solved fundamentally by a smart system design or by design‐for‐reliability (DfR) and by a critical selection in materials integration. To put it simply, we need to design low‐power devices, and also we need to understand heat production (Joule heating) in irreversible processes and heat dissipation in the device structure. [6] Hence, the science and engineering of electronic packaging come into focus.

Entropy production is the most relevant understanding of failure induced by electromigration, thermomigration, and stress‐migration in irreversible processes. [7] Statistical analysis of failure requires the knowing of mean‐time‐to‐failure (MTTF). An example is Black’s equation of MTTF for elctromigration. In Chapter 13, we shall present a unified model of MTTF for electromigration, thermomigration, and stress‐migration on the basis of entropy production.

Figure 1.7 shows an example of electromigration‐induced damage in Cu interconnects. The high current density in the interconnect has induced a flow of atoms along the electron flow direction, going from the cathode to the anode, leading to vacancy accumulation and void formation in the cathode region. Resistance of the interconnect increases gradually until an opening in the circuit occurs, where the resistance increases dramatically.

Figure 1.7 An example of electromigration electromigration‐induced failure of void formation in Cu interconnects. (a) Electrons drifted from the bottom Cu line to the top Cu line. (b) Electrons drifted from the top Cu line to the bottom Cu line.