edfas.org 23 ELECTRONIC DEVICE FAILURE ANALYSIS | VOLUME 26 NO. 2 is generated by the electric current given by: Qt = ς · ∇V · ∇V (Eq 3) Cooling or heating occurs at the junction of two dissimilar thermoelectric materials when an electric current flow through the junction. Peltier heat is proportional to the current, and changes sign if the current direction is reversed.[11] A potential gradient generates both a charge flow and a heat flow, giving the local equation:[10] (Eq 4) where is the surface heat flux, π the Peltier coefficient, and the charge flux. This phenomenon can be understood qualitatively by noticing that the particles that conduct the current are also those that transport the energy. Thus, if an electric current appears then the particles move along with their energy. The global equation:[10] QPeltier = (πa - πb) · I (Eq 5) where Q is the heat flux, I the electric current, πa and πb respectively the Peltier coefficients of materials a and b. The flow of the current I leads to a heating of the junction (case πa > πb). Reversing the direction of the current or having πa < πb, would have obtained a cooling. [9] Concerning the Seebeck effect, it is a voltage produced in a thermoelectric material by a temperature difference. The induced voltage is proportional to the temperature difference. The proportionality coefficient is known as the Seebeck coefficient (S).[11] More specifically, using the global equation 6: ∆V = S · ∆T = QPeltier/I (Eq 6) where ∆V is the generated electric field and ∆T the temperature difference. Equation 7 describes the resistivity heating effect in the conductor, where P represents the power (or heat) dissipation, and the parameters I, R, j, ϱ, and V denote the current, resistance, current density, resistivity, and volume, respectively. Equation 8 shows that increasing the stress current can increase heat dissipation of solder joint temperature tremendously and hence of the whole system: P = R · I2 = j2 · ϱ · V (Eq 7) Qj = P/A (Eq 8) THERMAL FIELD Heat removal from electronic packages is usually done through a combination of two distinct modes. The first mode is convection, which is defined as the heat transfer between a solid and a moving fluid and governed by: Qconv = hA (Ts - Tf) (Eq 9) where h is the heat transfer coefficient, A the crosssectional area for heat flow, Ts the surface temperature, Tf the fluid temperature. The heat equation comes from the law of conservation of thermal energy and Fourier’s law. dQ/dt = - λA(q(x0, t) - q(x0 + dx, t)) (Eq 10) ρcp dT/dt= λ(d2T/dx2) + Qj (Eq 11) dT/dt = k(d2T/dx2) + Qj (Eq 12) where k is the diffusivity (speed of temperature change) = λ/(ρcp) [m2/s], q is the density of material, c p is the specific heat capacity, T is the temperature, λ is the thermal conductivity, t denotes the time and Qj is the heat source shown in Equation 8. An approximate thermal resistor network for a BGA package is shown in Fig. 1. Thermal resistance definition differs depending on the mode of heat transfer. For conductive resistors, the thermal resistance is defined as L/Aλ while for convective resistors the resistance is defined as 1/Ah, where L is the length of the heat flow path, A is the cross-section area of the heat flow path, λ is the material thermal conductivity and h is the convection heat transfer coefficient. The thermal conductivity is the velocity at which the thermal flux flows. The reason for the slower heat propagation is the impurities and defects in the material and the viscoplastic character of the Sn-Ag-Cu material. The effusivity is defined as the capacity at the level of the external contact when the heat arrives on it, thus the capacity of the molecules of surface to accept the electrons and to exchange with them to allow them to transit toward the interior. The effusivity coefficient is: b = √(ρλcp ) (Eq 13) This coefficient specifies how many joules have penetrated on 1 m2 of surface of the material, 1 s after it is put in contact with another surface of 1m2 warmer by 1°C. A material absorbs more thermal power if heat can be easily conducted to it and that means λ is high and it absorbs more easily this power if its temperature rises little under the effect of heat. This article will only cover the effusivity and diffusivity toward the solder joints as depicted in Fig. 1. ELECTRO-THERMAL COUPLING A silicon chip cools down as heat is taken away from the environment. On the other hand, the copper layers and the solder joints are heating up. There is heat emission Qj. The Peltier coefficient is defined as the product of the temperature and the Seebeck coefficient. These quantities
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