edfas.org 5 ELECTRONIC DEVICE FAILURE ANALYSIS | VOLUME 27 NO. 4 thickness of relatively thin 3D structures such as 3D NAND flash and backside metallized devices. Another technique, thermo-reflectance (TR),[18-20] has started to gain attention in the FA community because of its potentially higher spatial resolution compared to LIT by means of shorter probing beam wavelength and higher modulation frequency of kHz to MHz range, which makes the technique a candidate for relatively thin 3D structures such as 3D NAND flash and BS-PDN devices. This article focuses on the latter two techniques, and technical details including working principles, performance, limitations, as well as future perspectives, are discussed. THEORY OF HEAT DIFFUSION Both TR and LI-OBIRCH techniques rely on heat diffusion, but they measure different physical properties of the sample. A key distinction lies in their heat sources: in TR, heat is generated internally within the sample, whereas in LI-OBIRCH, the laser spot serves as the external heat source. To better understand the behavior and effectiveness of these methods, it is useful to briefly review the principles of heat diffusion. Temperature, T, in isotropic and uniform medium is described by the well-known second order derivative equation: (Eq 1) where κ, ρ, and c are thermal conductivity, density, specific heat capacity of the medium, respectively. Consider a point heat source at the origin of coordinate in the medium. At the same time, the heat source is modulated at frequency, f, (or angular frequency of ω = 2πf ). A quasisteady state solution of the temperature field outside of the heat source, which satisfies Eq. 1, is obtained by omitting the time-dependent term ejwt: (Eq 2) where , and j is a unit of imaginary number. A is a constant complex number, which denotes the amplitude and phase of heat source temperature and is determined by heat source power and modulation frequency. Equation 2 can be broken down to amplitude and phase expression as follows: (Eq 3) Equation 3 means that thermal diffusion length and phase is proportional to and distance. Table 1 summarizes ld for some materials and frequencies. Thermal properties were cited from references 21 and 22. One can notice that heat diffusion in SiO2 is 10 times worse than Si. When layers of Si and SiO2 are stacked, for example, heat diffusion in stacked axis is dominated by SiO2. For single digit µm thickness, 10 to 100 kHz would be available, while 100 Hz to 10 kHz would be required to measure two digits of µm range. Although Eq 2 is valid just for a point heat source, it’s worth noting that Eq 2 is quite useful because convolving Eq 2 and heat source distribution gives a temperature field for general cases. For example, when the heat source is uniformly distributed in plane and its size is relatively smaller than or comparable to the z-dimension of interest, the expression of temperature field becomes (Eq 4) Heat source 0 otherwise which exactly matches a one-dimensional solution of Eq 1 with an infinite limit of heat source areas. Phase dependency against z is the same as Eq 3. THERMO-REFLECTANCE The authors’ TR technique, ThermoDynamic Imaging (TD Imaging), uses a focused low noise incoherent optical beam and laser scanning optics. Figure 1 shows the working principle and measurement system of TD Imaging, where a circuit with an anomaly or defect under a metal surface produces significant Joule heating when powered up, while the Joule heating in normal circuit is assumed to be negligible. A focused probing beam is used, and the reflected light power changes around the hotspot. TD Imaging uses fiber-coupled low noise incoherent light sources (high intensity lasers or HILs, Hamamatsu Photonics K.K.) as a probing beam. The probing beam is guided to the sample surface via scanning optics, with some portion reflected back. Pin represents the incident Table 1 Thermal diffusion length of Si and SiO2 at frequencies of 100 Hz to 100 kHz[21,22] Medium and thermal properties (ρ, c, κ) ld at 100 Hz ld at 1 kHz ld at 10 kHz ld at 100 kHz Si (2330 kg/m3, 703 J/kgK, 126 W/mK)[21] 540 µm 170 µm 54 µm 17 µm SiO2 (2200 kg/m3, 745 J/kgK, 1.38 W/mK)[22] 52 µm 16 µm 5.2 µm 1.6 µm
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