Influence of substrate thickness on thermal impedance of microelectronic structures

Abstract

The thermal impedance Z th (jx) has been calculated numerically, using the boundary element method, for a silicon substrate with a uniform heat source on top. The key feature is that the dynamic thermal behaviour is calculated directly in the frequency domain. The calculations were performed for a wide range of values for the thickness of the substrate. By representing the thermal impedance in a Nyquist plot (i.e. Im[Z th (jx)] vs. Re[Z th (jx)] with x as parameter), mainly two circular arcs are observed. For the lower frequency arc, the impedance values as well as the frequency scale are found to be largely influenced by the substrate thickness. The arc corresponding to high frequencies on the other hand remains unchanged under thickness variations.Further analysis revealed an almost perfectly linear relationship between the thermal resistance R th = Z th (jx = 0) and the substrate thickness, even when the heat source is not centred on the substrate. Both the slope and intersection value obtained from the curve fitting can be explained by a simple geometrical model including the fixed-angle heat spreading approximation, used since many years in the literature.

Frequency Domain Dynamic Thermal Analysis in Gaas HBT for Power Amplifier Applications

“…The ac thermal impedance can be defined as the temperature rise ∆T in a volume for a unit quantity of heat supplied ∆Q, Figure 5 shows the ac thermal impedance Z th which is complex frequency dependent parameter at the junction radial distance r = a. Frequency-dependent nonlinear thermal impedance is clearly observed. This correlates well with thermal impedance analyzed by [7,22]. In reality, Z th varies with power through temperature-dependent thermal conductance.…”

“…However, the frequency response can be measured experimentally [7]. By substituting (3) into (1) and (2), it follows that T r (r) must satisfy,…”

“…However, they have limitation in predicting dynamic T j . In general, four methods: (i) Numerical Analysis [5][6][7], (ii) 3D Simulation [8][9][10], (iii) Measurement [11], and (iv) Behavior Modeling [12][13][14], have been used to predict dynamic T j for various applications. Among these, frequency-dependent dynamic T j analysis is conducted using different envelope frequencies of up to 100 kHz in [11], 1 MHz in [10], and up to 10 MHz in [5,9].…”

Thermographic detection of abraded pipeline walls in the industrial installations

“…5).  On the basis of the Nyquist characteristics, the angular frequency ωB should be approximately equal to 1/τ [14,15].…”

“…(12) Subsisting power P from the (7) into (4) may be calculate the thermal constant value (13), taking wall thickness d, thermal conductivity λ, surface of the heat transfer S into consideration. While the heat capacity is the result of the volume V, density ϱ and the specific heat cp multiplication (14). (13) (14) where the volume is equal (15).…”