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Abstract
Non-uniform self-heating and temperature hotspots are major concerns compromising the performance and reliability of submicron electronic and optoelectronic devices. At deep submicron scales where effects such as contact-related artifacts and diffraction limits accurate measurements of temperature hotspots, non-contact thermal characterization can be extremely valuable. In this work, we use a Bayesian optimization framework with generalized Gaussian Markov random field (GGMRF) prior model to obtain accurate full-field temperature distribution of self-heated metal interconnects from their thermoreflectance thermal images (TRI) with spatial resolution 2.5 times below Rayleigh limit for 530nm illumination. Finite element simulations along with TRI experimental data were used to characterize the point spread function of the optical imaging system. In addition, unlike iterative reconstruction algorithms that use ad hoc regularization parameters in their prior models to obtain the best quality image, we used numerical experiments and finite element modeling to estimate the regularization parameter for solving a real experimental inverse problem.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Understanding thermal transport in state-of-the-art devices with nanoscale features is indispensable for their optimization [1,2]. Examples include performance and reliability of nanoscale field-effect transistors [3,4], high-frequency high-power operation of high electron mobility transistors [5,6], thermal fluctuations and cross-talk in photonic integrated circuits [7,8], and localized heating in heat-assisted magnetic recording (HAMR) and in phase change memories [1,9]. In the last two decades, nanoscale thermal characterization techniques have advanced significantly. Scanning thermal microscopy uses an atomic force microscope (AFM) tip along with a resistive sensor or a thermocouple to measure temperature with spatial resolution below 50nm [10–14]. While impressive results are obtained in imaging heat distribution along single carbon nanotube or nanowire devices, scanning probe techniques are often slow and sensitive to sample roughness. Also contact related artifacts need to be carefully eliminated.
On the other hand, non-contact optical techniques such as Raman [6,15,16], Infrared [3] and thermoreflectance have excellent temperature resolution and fast response in sub nanosecond range, but their spatial resolution is limited by the diffraction. Thermoreflectance thermal imaging (TRI) detects a small change in surface reflectivity as a function of temperature and is proven useful for 2D mapping of thermal profile and qualitative evaluation of energy dissipation at submicron scales [4,17]. TRI is employed to create temperature map of electrically biased single crystalline VO$_2$ nanobeams, observing strongly localized alternating Peltier heating and cooling as well as Joule heating at the metal-insulator domain boundaries [17]. Furthermore, the technique was used to observe super-Joule self-heating at the transport bottlenecks in networks of silver nanowires and silver nanowire/single layer graphene hybrid films [18]. The high temporal resolution of TRI, 50 ns with a light emitting diode (LED) and 800ps with laser, makes it an appropriate technique for capturing fast transient self-heating effects in high speed switching devices. Despites higher spatio-temporal resolution compared to other non-contact techniques such as infrared (IR) and micro-Raman, TRI can still be limited by the optical diffraction for characterizing nanoscale devices. To overcome this issue, a high spatial resolution temperature measurement technique based on transmission electron microscope is developed [19]. Atomic scale resolution is achieved, but, this technique is slow and limited by the sample thickness and material composition.
In this paper, by combining finite element (FEM) and analytical modeling with a Bayesian image reconstruction approach, we demonstrate that the diffraction barrier can be overcome; and thus, using light in visible range, we are able to extract temperature profiles of devices that are 1.8x smaller diffraction limit in real experiment (2.5x below diffraction limit in numerical experiments). First, we systematically investigate the effect of optical diffraction on the TRI measurements for electron-beam fabricated self-heated metal interconnects. We show that thermoreflectance thermal images below diffraction limit can be obtained, and address the impact of optical blurring on the accuracy of the temperature measurement and how one can compensate for it. Further, image processing techniques are used to treat the ill-posed inverse problem of accurately reconstructing the temperature profiles of devices with sub-diffraction features. We combine image processing and thermoreflectance thermal imaging to accurately obtain thermal images of self-heated metal interconnects at scales below diffraction limit. We present Bayesian framework using Markov Random Field (MRF) priors [20], to formulate the problem, and utilize an iterative coordinate descent (ICD) algorithm to obtain a Maximum-a-posteriori (MAP) estimate of true thermal images from the diffraction-limited ones. Reconstruction is performed both on numerically-designed experiments and experimental thermal images. Independent measurements of local temperature using resistive sensors are used to verify the accuracy of the imaging results. Here is a summary of the main contributions:
- 1. We leveraged and extended the Bayesian image reconstruction framework in [21] to perform and demonstrate accurate image reconstruction of real experimental results for sub-diffraction nanoscale metal interconnects, that is verified by independent electrical measurements.
- 2. A key parameter in the Bayesian framework is its regularization parameter. This parameter is an ad-hoc parameter that is typically chosen empirically, to produce the best quality images (with high contrast and low noise) without regards for the underlying physical phenomenon. We use FEM and analytical modeling of the imaging system to model the impact of diffraction, and hence, simulate the temperature profiles of the sub-diffraction size devices. The proposed image reconstruction algorithm was then used to reconstruct the true temperature profiles of the simulated data with an empirically chosen regularization parameter that works best for these simulations. Once the regularization parameter is selected, the same parameter is used in the Bayesian image reconstruction algorithm to reconstruct the temperature profile of an experimentally measured sub-diffraction size metal interconnect. In other words, knowing materials and device structure, we leveraged FEM simulations to accurately extract temperature profiles of the sub-diffraction size metal interconnects without the need to use ad-hoc hyperparameters in the image reconstruction algorithm. It is worth noting that using physics-based simulations as prior model [22,23] and to extract prior model parameters has been an active area of research [24,25] and has recently been used for image reconstruction of a black-hole [24] .
- 3. We show that our accurate modeling of the imaging system and the point spread function (PSF), results in good agreement between the predicted coefficient of thermoreflectance (C$_{TR}$) and the real experimental data for sub-diffraction devices.
- 4. We study the impact of the noise and metal interconnect width on the performance of our image reconstruction algorithm.
2. Methods
In this section we describe thermoreflectance thermal imaging method for thermal measurement of nanoscale features. In addition, we briefly discuss the diffraction limit in optical measurements such as TRI. We next provide details on the samples designed and fabricated to perform sub-diffraction TRI.
2.1 Thermoreflectance thermal imaging (TRI)
Thermoreflectance thermal imaging [26] was used to obtain the temperature profile of electrically biased self-heated metal interconnects. Principle of TRI along with an optical image and a temperature profile of a self-heated metal interconnect is shown in Fig. 1(a). TRI measures the change in materials’ reflection coefficient as a function of temperature. The change in the reflectivity ($\Delta R$) of a material is related to its temperature change ($\Delta$T) through the thermoreflectance coefficient (C$_{TR}$), by Eq. (1).
Fig. 1. (a) Principle of thermoreflectance thermal imaging (TRI) along with an optical image and a thermal image of a 200 nm self-heated metal interconnect. (b) SEM of a 200nm wide metal interconnect.
In this technique, we illuminate the device under test (DUT) with a LED light in visible range through an objective lens that is appropriate for the dimension of the device that is being measured. In the case of quasi-steady state temperature measurement, the DUT is electrically biased with a low-frequency (7.5$\,$Hz) sinusoidal signal and heats up and cools down periodically. The reflected light ($R$) from the DUT is captured in a charge-coupled device (CCD) camera. From the change in reflectivity ($\Delta R$), knowing the calibrated value of thermoreflectance coefficient ($C_{TR}$), one can obtain the temperature change of the device $\Delta T$. More details on the technique are provided in the method section and in [27,28]. The magnitude of the thermoreflectance coefficient (C$_{TR}$) is usually on the order of 10$^{-2}$ - 10$^{-5}$ $\frac {1}{K}$ depending on the materials. This coefficient is influenced by sample composition, wavelength of probing light, and numerical aperture of the imaging system. For a typical device size, that is above the diffraction limit, we can simply use systematic calibration methods [29] to obtain the $C_{TR}$ for different materials in the device.
Typically, 530 nm illumination wavelength is used for gold samples but the wavelength can be tuned to maximize the signal [29]. Using a lock-in technique, this method is sensitive to Milli-Kelvin temperature variations ($\sim$ 0.1 K) with sub-nanosecond temporal resolution (800 ps) [17,26,27].
2.2 Diffraction limit
As any optical imaging techniques, TRI is limited by the Rayleigh criterion for the diffraction. Based on the Rayleigh criterion two point objects that are spaced closer than distance D (radius from the point source) are barely resolved [30]. Here D is given by Eq. (2), known as the diffraction limit.
where $N$ is the f-number of the objective and $\lambda$ is the wavelength of the light reflected from the surface. $N$ is approximately (1/2N.A.), where N.A. is the numerical aperture of the lens. Diffraction limit is determined by the emitted or reflected light from a point source since higher spatial frequencies are filtered by the optical system. Airy disk determines the minimum distance from a point source at which the imaged light intensity given by the Bessel function goes to zero [31]. This intensity profile can be approximately fit with a Gaussian profile with full-width-half-maximum 2.44 times the Airy disk diameter, which is convenient for further mathematical formulation [32]. The light intensity (E) as a function of distance from a point source is described as $E = E_{0}f(r)$ with,2.3 Device fabrication
To investigate the impact of diffraction on TRI, we designed and fabricated a set of nanoscale metal interconnects. A 5 microns thick In$_{0.53}$Ga$_{0.47}$As on 100 nm In$_{0.52}$Al$_{0.48}$As buffer layer grown on 500$\mu$m InP substrate by molecular beam epitaxy. After removing the native oxide on the InGaAs film with one-minute dip in dilute HF solution, 20 nm Al$_{2}$O$_{3}$ insulation layer was deposited using the atomic layer deposition technique at 200$^\circ$C followed by rapid thermal annealing at 450$^\circ$C for 30 s. These samples were then processed using electron beam lithography (EBL), metalization followed by lift-off to obtain 85 nm thick Au interconnects on top of 5 nm Ti used as sticking layer. Interconnects of different widths were fabricated. Aspect ratio of each device, i.e. ratio of length to width, was fixed to 20. Four large contact pads, each 80$\times$80 $\mu$m$^2$, were fabricated for each Au interconnect, so that the samples can be probed easily for the four-probe electrical measurement, and also the thermal measurement can be further confirmed using the interconnect resistance as thermometer [34]. A scanning electron micrograph (SEM) of a 200 nm wide, 8 $\mu$m long gold interconnect is shown in Fig. 1(a).
3. Proposed framework
In this work, we propose a framework that aims to use an image reconstruction algorithm to restore temperature maps of metal interconnects that are below diffraction limits when measured with light in the visible range. To accomplish this, we divide the proposed framework into two main parts, forward and inverse problems, which are described below.
In the case of forward problem, we aim to model the impact of the diffraction limit on thermoreflectance thermal images of nanoscale metal interconnects. Here are the steps need to be taken (these steps will be verified through real experiments in the next section):
- 1. Use FEM modeling to model temperature maps of nanoscale metal interconnects. Convert the temperature map to reflectance map, using calibrated $C_{TR}$ for the materials of the devices under test (Here Au and InGaAs).
- 2. Apply the impact of imaging system, by convolving the reflectance map with diffraction function described in section 2.2 and add noise to the final image. The output is a noisy-blurry image that mimics real experimental thermoreflectance thermal images.
Next, we need develop an inverse problem framework to reconstruct the true temperature maps of the metal interconnects from their noisy-blurry images. This means, we have to extract $f$ from the measured $g$, using the modeled $\textbf {H}$, calibrated $C$ and estimated $\sigma _W$. This is an ill-posed problem for which may not have a unique solution [35–38]. Therefore, one needs to add apriori information to regularize this inverse problem. Fortunately, there is a vast literature on solving this type of problems in signal and image processing [39,40]. As noted, we formulated the reconstruction problem in the Bayesian framework as the maximum $a$ $posteriori$ (MAP) estimate [20,41–43]:
Here, $P(.)$ denotes the probability. The first term is the log likelihood and the second term is prior model assumed for the image distribution. We assumed $P(g|f)$ has the following distribution: We define $norm(x) \equiv ||x||^2 = \sum _{i\in S} x_i^2$, and $S$ correspond to set of all pixels in the image.For the choice of prior, we used a non-Gaussian prior model, namely generalized Gaussian Markov random fields (GGMRF) to enforce spatial correlation of the reflectance map [44–46]. The GGMRF prior model $P(f)$ can be written as:
In Eq. (7), $\aleph = \{\{i,j\}\, |\, i \in \partial j, \, j \in S\}$ is the set of all neighboring pixel pairs, and $\partial j$ denotes the neighborhood of $j$. $b_{i-j}$ is the coefficient linking pixels $i$ and $j$, $Z$ is a normalizing constant and $p$ is a number in $[1,2]$ that controls the smoothness of the reconstruction. We used $p = 1$ to maintain the sharp edges in the temperature profile [34] near the edges of self-heated metal interconnect. The $\sigma _x$ is a scale regularization parameter and is chosen empirically in numerical simulations to maximize the signal-to-noise ratio of the output temperature in comparison with the ground truth.For real measurements, such as in TRI, the variance of the noise ($\sigma _w^2$) is typically unknown. Thus, we will need to estimate $\sigma _w^2$ and $f$ jointly. The corresponding optimization problem will become:
4. Results and discussion
In this section, first, we designed an experiment to exemplify the impact of the diffraction limit on the thermoreflectance thermal imaging of nanoscale metal interconnects. Then we design a set of numerical experiments, and replicate the impact of diffraction limit on numerical thermal maps of nanoscale metal interconnects. We use the proposed image reconstruction algorithm to reconstruct thermal images of sub-diffraction metal interconnects in numerical measurements. Further, the image reconstruction of numerical measurements guide us to extract regularization parameters for image reconstruction in real TRI measurements. Finally, we demonstrate image reconstruction of sub-diffraction TRI images using the same parameters obtained from numerical measurements.
4.1 Forward problem: effect of diffraction on TRI thermal images
We ran independent thermal measurements using electrical resistivity of metal interconnects along with their thermoreflectance thermal imaging, and observed that the apparent temperature change of sub-diffraction size devices measured by TRI was lower than its actual temperature change [32]. This is because the diffraction function (PSF) blurs the reflectance map of devices, and in turn results in a change in the coefficient of thermoreflectance (C$_{TR}$). We designed experiments to compare TRI thermal images above and below Rayleigh criterion for a given illumination wavelength ($\lambda$) and numerical aperture ($N.A.$) of the imaging system.
Figure 2 shows experimental results for a 1$\mu$m wide metal interconnect with two different objective lenses, 100x and 10x. The LED illumination wavelength is 530 nm. The numerical apertures for 100x and 10x objective lenses are 0.75 and 0.2, which correspond to Rayleigh radii of 353 nm and 1330 nm, respectively (Eq. (2)). This means that the 1 $\mu$m interconnect is well above diffraction at 100x, and is below diffraction at 10x. It is worth noting that at 100x each pixel in the image is about $\sim$ 160 nm, while it is $\sim$ 1.6 $\mu$m at 10x. Thus, a 1 $\mu$m interconnect at 100x contain 7 pixels, while a fraction of a pixel is filled at 10x as shown in the optical CCD images in Figs. 2(a) and 2(b).
Fig. 2. Effect of diffraction on a 1$\mu$m metal interconnect at different objective lens magnifications. CCD images under (a) 100x, and (b) 10x, objective lenses. The optical resolution limit (Eq. (2)) at 10x and 100x are 0.35$\mu$m and 1.33$\mu$m, respectively. Corresponding TRI thermal images for the 1$\mu$m self-heated interconnect under (c) 100x, and (d) 10x. (e) Taking the effect of diffraction function into account by blurring the 100x temperature profile to obtain 10x results. (f) A comparison between the vertical cross sections along $A-A'$ TRI images at 100x, 10x and the blurred 100x.
Figures 2(c) and (d) show the experimental temperature maps of the 1 $\mu$m device under the two magnifications. Calibration showed that the C$_{TR}$ for gold for 100x objective for 1 $\mu$m interconnect is $\sim$ -2.2$\times$10$^{-4}$ K$^{-1}$, and that of the substrate area outside the gold lines is found to be $\sim$ +2$\times$10$^{-4}$ K$^{-1}$. Calibration is done independently using both electrical and TRI measurements [34]. Cross sections of the temperature profiles along the width direction ($A-A'$ Cross Section) are compared in Fig. 2(f).
In Fig. 2(b), the optical image of the 1 $\mu m$ metal interconnect is obscured, while the corresponding thermal image in Fig. 2(d) shows the temperature map clearly. This raises the question that why the TRI is able to detect temperature signal for sub-diffracted device sizes? TRI detects the thermal signal because of the sign difference between thermoreflectance coefficient of gold and the semiconductor substrate region at this illumination wavelength. This behavior is akin to the operation of phase shift lithography. Therefore, two objects that are closer than the diffraction limit, a phase-shift mask inverts the phase of the wave passes through one of them resulting in a destructive interfere in electric field, and in turn discerning the intensity of the two objects [48]. The zero crossing (dip in the cross-section) near the edge of metal line in the thermal cross sections of both 100x and 10x images shown in Fig. 2(f) is due to the change in the sign of C$_{TR}$ in the metal and semiconductor regions. In the 10x image, there are not only edge effects but also the apparent temperature change on top of the device is significantly reduced.
Fig. 3. Independent calibration of C$_{TR}$ for Au metal interconnects of different widths compared with the theoretical calculations [34] at $\lambda = 530~nm$. (-C$_{TR}$ is plotted as Au has a negative $C_{TR}$ at this $\lambda$). Theoretical curve is obtained by blurring of C$_{TR}$ of metal and neighboring semiconductor using the approximated point spread function (optical diffraction function) of the imaging system defined by the numerical aperture of the imaging system.
To reproduce the 10x thermal image, we can convolve the 100x thermal map with the PSF of imaging system (diffraction function approximated by Gaussian in Eq. (3)) before normalizing with the C$_{TR}$. The temperature profile from this procedure is shown in Fig. 2(e). It is evident that the reduction in apparent maximum temperature can be fully explained by the diffraction filtering. This also confirms that our modeling of the imaging system based on the diffraction function is accurate and capture the physics of the problem. Temperature profiles’ vertical cross sections are plotted in Fig. 2(f) showing marked agreements.
4.2 Impact of diffraction on coefficient of thermoreflectance
As it is already shown, a direct consequence of diffraction is the reduction in the apparent temperature measured by the TRI. To account for this change, one has to re-calibrate the $C_{TR}$ coefficient and obtain an effective $C_{TR}$ for sub-diffraction size devices. Under typical condition and device sizes (i.e. above diffraction limit), calibration is done by putting the sample on a temperature controlled stage and by slowly modulating its temperature in sub-Hertz regime [29]. Accurate calibration for submicron size devices, however, is difficult due to problems with thermal expansion and movements of the sample under test when the stage temperature changes. Therefore, for the sub-diffraction metal interconnect in this work, we used current-voltage characteristics at different ambient temperature to extract the average temperature on top of the interconnect at different applied currents [34]. The extracted C$_{TR}$ for each device is plotted against width in Fig. 3 (red dots). It is evident that as the width of interconnect reaches sub-diffraction scale, C$_{TR}$ reduces dramatically.
We devised numerical experiments to simulate the impact of diffraction. To that end, we obtained the temperature profiles of metal interconnects of different width, converted them to reflectance map, blurred them with diffraction function and re-scale them to blurred temperature map. From the ratio of the blurred temperature map of the metal line to the original ground truth temperature map, theoretically extracted $C_{TR}$. The calculated $C_{TR}$ is plotted as the solid line. The good agreement between the calculated $C_{TR}$ and the experimental results further confirms the accuracy of our modeling of the imaging system and its PSF (optical diffraction function).
4.3 Inverse problem: temperature map reconstruction
4.3.1 Numerical experiments
In practice, it is far more reaching to obtain the true temperature profile from the measured apparent temperature map when the image is blurred due to diffraction. Thus far, we have shown that diffraction does play a significant role in the experimental temperature profile obtained from TRI. In the aforementioned case studies for self-heated interconnects, we confirmed this observation by extracting the average temperature and C$_{TR}$ independently using electrical resistivity measurement. However, there is a major challenge to have an independent temperature sensor near each nanoscale hot spot. Sometimes it is not possible to measure independently the local temperature for e.g. HAMR devices or a hot spot near the source of a nanoscale transistor. Also, forward modeling is not always possible due to unknown material properties and electron flow pattern. One needs an optical image reconstruction algorithm to restore the true temperature profile from the measured apparent temperature map when the image is blurred due to diffraction. We use the Bayesian framework described in section 3 to obtain a maximum-a-posteriori (MAP) estimate to the cost function that formulates the inverse problem, and in turn to reconstruct the true temperature profile of the nanoscale metal interconnects.
Figure 4 shows the results for a set of numerical experiments designed to verify the proposed reconstruction algorithm. We first used FEM to create the temperature profile of a 200 nm self-heated metal interconnects. This is shown in Fig. 4(a). The dimension and material properties were set based on measured device dimension and properties [34]. The image was then filtered using the diffraction function of a 100x objective lens with N.A. = 0.75, to which random Gaussian noise with variance $\sigma _w^2 = 20$ (i.e. the signal-to-noise-ratio (SNR) value of 2.3 dB). This blurry noisy image (numerically produced sub-diffraction temperature map) is shown in Fig. 4(b). We then used the proposed MAP reconstruction technique to reconstruct the true thermal image from the sub-diffraction (blurry-noisy) thermal image. The reconstructed image is shown in Fig. 4(c). A comparison between the cross sections along horizontal and vertical axis (Figs. 4(d) and (e)) demonstrate excellent agreement between the reconstructed and the ground truth thermal images.
Fig. 4. Numerical experiment to verify the sub-diffraction reconstruction algorithm and to study the impact of signal to noise on image reconstruction. Results are for 200 nm self-heated metal interconnect. Each row corresponds to a given signal-to-noise-ratio (SNR). Ground truth, noisy-blurry, and reconstructed images along with horizontal and vertical temperature cross sections are shown in different columns. (a-e) Excellent reconstruction for sub-diffraction thermal image with 2.3dB SNR. Reconstruction is reasonably good even after decreasing SNR to -0.7 dB (f-j), and -4.7 dB (k-o). Reconstruction fails as SNR reaches -7.7 dB (p-t).
We used $p=1$ and the regularization parameter $\sigma _x = 21$. It should be emphasized that the regularization parameter $\sigma _x$ is chosen empirically, so that it will produce the most accurate reconstruction of nanoheater temperature profile, that is a reconstruction with minimum average temperature error on the nanoheater line in comparison with the ground truth temperature map. There is a vast literature on choosing the regularization parameter [41,49–57]. Most of these approaches are designed for natural images and to produce the best quality image in term of signal to noise (sharp and smooth) without regards for an underlying physical property that can be extracted. Such ad-hoc values, however, may not be suitable when the goal is not only the best quality image output, but also quantitative information, such as actual temperature values, that need to be extracted. To demonstrate this important point, Fig. 5 compares the temperature profiles obtained with optimum $\sigma _x$ value proposed in [41] and with the reconstruction obtained in this work. While using the optimum $\sigma _x$ in our algorithm results in a much smoother image with lower overall noise, it is severely underestimating the true temperature of the nanoscale metal interconnects (average temperature rise is 14$^\circ$C versus 29$^\circ$C in the ground truth image). In the next section we discuss how we leveraged the simulations, and the same empirically extracted regularization parameters used in the simulations, to reconstruct the temperature profile from a real sub-diffraction TRI experiment.
Fig. 5. Impact of regularization parameter on the reconstruction algorithm. a) Ground truth Temperature map. b) noisy sub-diffraction temperature map. c) Reconstructed thermal map using equation proposed in [41] for optimum regularization parameter. Note that the average temperature rise is underestimated by more than a factor of 2. d) Reconstructed thermal map using the empirically chosen $\sigma _x$. It is worth noting, we also used equations proposed in other references such as in [57] to obtain the optimum $\sigma _x$, but the results (not shown here) are not very different from panel c.
4.3.2 Physical experiments: actual thermoreflectance thermal imaging
The same image reconstruction algorithm was applied to a set of experimental (TR) thermal images. Results for a 200 nm wide self-heated metal interconnect (8 $\mu$m long) under an applied current of 3.65 mA is shown in Fig. 6. Figure 6(a) shows the TRI thermal image. As already noted, the Rayleigh criterion is 353 nm for 100x lens using 530 nm LED light. Thus, the 200 nm metal interconnect is below diffraction. Using the same reconstruction algorithm, i.e. with the regularization parameter as the simulated data in Figs. 4(a–e), we performed thermal image reconstruction for the real data. The reconstructed thermal image is shown in Fig. 6(b) (top panel). The numerically simulated temperature profile is also shown in Fig. 6(b) (bottom panel). A cross section comparison along horizontal axis in Fig. 6(c) shows excellent agreement on the metal interconnect. It is noted that the true maximum temperature is almost four times higher than the apparent value. This was also measured independently using electrical characterization, confirming the simulated results [34]. The simulated temperature profile was obtained using the hydrodynamic model proposed in reference [34], and addresses the quasi-ballistic effects near the nanoscale size heat source. It should be noted that the accuracy of the hydrodynamic model is further verified in [58].
Fig. 6. Sub-diffraction image reconstruction using measured TRI thermal images. (a) Thermal image of the 200 nm self-heated metal interconnects. The image is calibrated with nominal C$_{TR}$ of gold (-2.2$\times$ 10$^{-4}$ K$^{-1}$). (b) Reconstructed image (Top) using the MAP estimation framework described in the text compared with the simulated temperature profile (bottom) [34]. (c) Comparison of cross sections along A-A’. The average temperature of the metal interconnect is verified independently using electrical resistivity measurement [34].
4.3.3 Impact of noise
While TRI is able to measure thermal images of interconnects as small as 70 nm wide, the proposed reconstruction technique has limitations in resolving the true temperature values from the measured apparent thermal signals. To address these limitations, we performed several FEM simulations along with analytical modeling that simulates the impact of imaging systems and diffraction (section 2.2). In these simulations, we systematically investigated the impact of signal-to-noise-ratio (SNR) on reconstruction performance. Further, we challenged the proposed method by simulating interconnects of different widths (minimum feature size) until the method is not further able to restore true temperature profiles of the lines. SNR is defined as $10~log(\frac {\Delta T}{v_{noise}})$, where $\Delta T$ and $v_{noise}$ are the average temperature change over central region on the self-heated interconnect and variance of the noise, respectively.
First, in order to examine the effect of noise on the proposed reconstruction technique, we decreased the SNR systematically until the reconstruction technique could not be further used to resolve the thermal signal of the metal interconnect. In all these simulations, the width of metal interconnect was kept at 200 nm. The results for SNR = -0.7, -4.7 and -7.7 dB are plotted in Figs. 4(f)–4(j), 4(k)–4(o), and 4(p)–4(t), respectively. Note that in Fig. 4(g), almost no signal is visible in the sub-diffracted image, while the reconstructed image has a marked agreement with the ground truth temperature distribution and peak temperature. The analysis further demonstrates that only after decreasing the signal by larger than a factor of 10, it was difficult for the proposed algorithm to reconstruct the true thermal image.
Second, to investigate the effect of signal-to-noise at different widths, we performed several numerical experiments in which we varied the width of the metal interconnect from 100 nm to 215 nm. For each case study the SNR was decreased until the true temperature profile from the noisy-blurry input could not be reconstructed. Figure 7 summarizes the results. It can be seen that above 150 nm even with a thermal map that was buried in noise 0dB SNR), the true temperature profile was recovered within error less than 20$\%$. Even above 135 nm, with a moderate noise level (20dB SNR), the signal was recovered with less than 20$\%$ error.
Fig. 7. Impact of SNR on the minimum feature size for which temperature profile can be reconstructed. The accuracy of the reconstruction is assumed to be from 1$\%$ to 40$\%$.
It should be noted that the error is defined as $100 \times \frac {|1 - \Delta T_{ground_truth}|}{\Delta T_{reconstructed}}$, where $\Delta T$ is the average temperature on top of the interconnect over the central region where SNR was defined. As the size of the self-heated interconnect decreases below 100 nm the thermal signal could not be recovered even with very large SNR. At 100 nm, the thermal signal was recovered, with smaller SNR (less than 20 dB) and a qualitatively good thermal image, but the reconstructed signal was still much weaker than the true temperature.
5. Conclusions
In summary, in this work we used numerical simulation combined with a Bayesian framework for image reconstruction to extract temperature profiles of self-heated metal interconnects with feature sizes less than the diffraction limit. The numerical simulations were used to verify the point spread function (diffraction function) of the optical system as well as to extract the regularization parameter for solving the inverse problem. The extracted regularization parameter was leveraged in the same algorithm for reconstruction of temperature profiles of sub-diffraction size metal interconnects. The measured temperature rises are independently verified using electrical resistor sensors. Excellent reconstruction has been demonstrated even under low SNR values ($\sim$-5 dB). Since in practice to characterize hot spots in small scale electronic devices, calibration and independent sensor measurement may not be applicable, the image reconstruction technique can be a powerful tool for accurate non-contact temperature measurement. This provides an alternative to characterize nanoscale features compared to very successful scanning probe techniques that are often slower, more intrusive and quite sensitive to surface roughness [12,14,59,60].
Acknowledgments
This manuscript has been authored by UT-Battelle, LLC, under Contract No. DEAC05-00OR22725 with the U.S. Department of Energy. The United States Government and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).
Disclosures
The authors declare no conflicts of interest.
References
1. L. Shi, C. Dames, J. R. Lukes, P. Reddy, J. Duda, D. G. Cahill, J. Lee, A. Marconnet, K. E. Goodson, J.-H. Bahk, A. Shakouri, R. S. Prasher, J. Felts, W. P. King, B. Han, and J. C. Bischof, “Evaluating Broader Impacts of Nanoscale Thermal Transport Research,” Nanoscale Microscale Thermophys. Eng. 19(2), 127–165 (2015). [CrossRef]
2. D. G. Cahill, P. V. Braun, G. Chen, D. R. Clarke, S. Fan, K. E. Goodson, P. Keblinski, W. P. King, G. D. Mahan, A. Majumdar, H. J. Maris, S. R. Phillpot, E. Pop, and L. Shi, “Nanoscale thermal transport. II. 2003-2012,” Appl. Phys. Rev. 1(1), 011305 (2014). [CrossRef]
3. M. H. Bae, S. Islam, V. E. Dorgan, and E. Pop, “Scaling of high-field transport and localized heating in graphene transistors,” ACS Nano 5(10), 7936–7944 (2011). [CrossRef]
4. S. Shin, S. Member, M. A. Wahab, M. Masuduzzaman, K. Maize, J. Gu, M. Si, S. Member, A. Shakouri, P. D. Ye, M. A. Alam, and A. G.-a.-a. Gaa, “Direct Observation of Self-Heating in III–V Gate-All-Around Nanowire MOSFETs,” IEEE Trans. Electron Devices 62(11), 3516–3523 (2015). [CrossRef]
5. C. H. Lin, T. A. Merz, D. R. Doutt, M. J. Hetzer, J. Joh, J. A. Del Alamo, U. K. Mishra, and L. J. Brillson, “Nanoscale mapping of temperature and defect evolution inside operating AlGaN/GaN high electron mobility transistors,” Appl. Phys. Lett. 95(3), 033510 (2009). [CrossRef]
6. Z. Yan, G. Liu, J. M. Khan, and A. A. Balandin, “Graphene quilts for thermal management of high-power GaN transistors,” Nat. Commun. 3(1), 827 (2012). [CrossRef]
7. D. Liang, X. Huang, G. Kurczveil, M. Fiorentino, and R. G. Beausoleil, “Integrated finely tunable microring laser on silicon,” Nat. Photonics 10(11), 719–722 (2016). [CrossRef]
8. K. Padmaraju and K. Bergman, “Resolving the thermal challenges for silicon microring resonator devices,” Nanophotonics 3(4-5), 269–281 (2014). [CrossRef]
9. S. Bhargava and E. Yablonovitch, “Lowering HAMR Near-Field Transducer Temperature via Inverse Electromagnetic Design,” IEEE Trans. Magn. 51(4), 1–7 (2015). [CrossRef]
10. A. Majumdar, J. Lai, M. Chandrachood, O. Nakabeppu, Y. Wu, and Z. Shi, “Thermal imaging by atomic force microscopy using thermocouple cantilever probes,” Rev. Sci. Instrum. 66(6), 3584–3592 (1995). [CrossRef]
11. K. L. Grosse, M.-H. Bae, F. Lian, E. Pop, and W. P. King, “Nanoscale Joule heating, Peltier cooling and current crowding at graphene-metal contacts,” Nat. Nanotechnol. 6(5), 287–290 (2011). [CrossRef]
12. F. Menges, H. Riel, A. Stemmer, and B. Gotsmann, “Nanoscale thermometry by scanning thermal microscopy,” Rev. Sci. Instrum. 87(7), 074902 (2016). [CrossRef]
13. K. Kim, J. Chung, G. Hwang, O. Kwon, and J. S. Lee, “Quantitative measurement with scanning thermal microscope by preventing the distortion due to the heat transfer through the air,” ACS Nano 5(11), 8700–8709 (2011). [CrossRef]
14. F. Menges, P. Mensch, H. Schmid, H. Riel, A. Stemmer, and B. Gotsmann, “Temperature mapping of operating nanoscale devices by scanning probe thermometry,” Nat. Commun. 7(1), 10874 (2016). [CrossRef]
15. M. S. Liu, L. a. Bursill, S. Prawer, K. W. Nugent, Y. Z. Tong, and G. Y. Zhang, “Temperature dependence of Raman scattering in single crystal GaN films,” Appl. Phys. Lett. 74(21), 3125–3127 (1999). [CrossRef]
16. D. Majumdar, S. Biswas, T. Ghoshal, J. D. Holmes, and A. Singha, “Probing Thermal Flux in Twinned Ge Nanowires through Raman Spectroscopy,” ACS Appl. Mater. Interfaces 7(44), 24679–24685 (2015). [CrossRef]
17. T. Favaloro, J. Suh, B. Vermeersch, K. Liu, Y. Gu, L.-Q. Chen, K. X. Wang, J. Wu, and A. Shakouri, “Direct observation of nanoscale peltier and joule effects at metal–insulator domain walls in vanadium dioxide nanobeams,” Nano Lett. 14(5), 2394–2400 (2014). [CrossRef]
18. K. Maize, S. R. Das, S. Sadeque, A. M. S. Mohammed, A. Shakouri, D. B. Janes, and M. A. Alam, “Super-Joule heating in graphene and silver nanowire network,” Appl. Phys. Lett. 106(14), 143104 (2015). [CrossRef]
19. M. Mecklenburg, W. A. Hubbard, E. R. White, R. Dhall, S. B. Cronin, S. Aloni, and B. C. Regan, “Nanoscale temperature mapping in operating microelectronic devices,” Science 347(6222), 629–632 (2015). [CrossRef]
20. C. A. Bouman, Model Based Image Processing (available online at https://engineering.purdue.edu/~bouman/grad-labs/, 2013).
21. A. Ziabari, Y. Xuan, J.-H. Bahk, M. Parsa, P. Ye, and A. Shakouri, “Sub-diffraction thermoreflectance thermal imaging using image reconstruction,” in 2017 16th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems (ITherm), (IEEE, 2017), pp. 122–127.
22. A. Ziabari, J. M. Rickman, L. F. Drummy, J. P. Simmons, and C. A. Bouman, “Physics-based regularizer for joint soft segmentation and reconstruction of electron microscopy images of polycrystalline microstructures,” IEEE Transactions on Computational Imaging (2019).
23. M. U. Sadiq, J. P. Simmons, and C. A. Bouman, “Model Based Image Reconstruction with Physics Based Priors,” in IEEE Internation Conference on Image Processing (ICIP), (2016), pp. 3176–3179.
24. The Event Horizon Telescope Collaboration, “First m87 event horizon telescope results. iv. imaging the central supermassive black hole,” The Astrophys. J. Lett. 875(1), L4 (2019). [CrossRef]
25. M. Kellman, E. Bostan, N. Repina, and L. Waller, “Physics-based learned design: Optimized coded-illumination for quantitative phase imaging,” IEEE Transactions on Computational Imaging (2019).
26. M. Farzaneh, K. Maize, D. Lüerßen, J. A. Summers, P. M. Mayer, P. E. Raad, K. P. Pipe, A. Shakouri, R. J. Ram, and J. A. Hudgings, “CCD-based thermoreflectance microscopy: principles and applications,” (2009).
27. B. Vermeersch, J.-H. Bahk, J. Christofferson, and A. Shakouri, “Thermoreflectance imaging of sub 100ns pulsed cooling in high-speed thermoelectric microcoolers,” J. Appl. Phys. 113(10), 104502 (2013). [CrossRef]
28. K. Maize, J. Christofferson, and A. Shakouri, “Transient Thermal Imaging Using Thermoreflectance,” in 2008 IEEE Twenty-fourth Annual Semiconductor Thermal Measurement and Management Symposium, Semi-Therm., (2008), pp. 55–58.
29. A. Shakouri, A. Ziabari, D. Kendig, J.-H. Bahk, Y. Xuan, D. Y. Peide, K. Yazawa, and A. Shakouri, “Stable thermoreflectance thermal imaging microscopy with piezoelectric position control,” in 2016 32nd Thermal Measurement, Modeling & Management Symposium (SEMI-THERM), (IEEE, 2016), pp. 128–132.
30. W. Steen, Principles of Optics, vol. 32 (Cambridge University Press, Cambridge, 2000), 7th ed.
31. G. P. Karman, M. W. Beijersbergen, A. van Duijl, D. Bouwmeester, and J. P. Woerdman, “Airy pattern reorganization and subwavelength structure in a focus,” (1998).
32. A. Ziabari, J.-H. H. Bahk, Y. Xuan, P. D. Ye, D. Kendig, K. Yazawa, P. G. Burke, H. Lu, A. C. Gossard, and A. Shakouri, “Sub-diffraction Limit Thermal Imaging for HEMT Devices,” 31th Annu. IEEE Therm. Meas. Model. & Manag. Symp. (SEMI-THERM) pp. 1 –6 (2015).
33. M. Shahram and P. Milanfar, “Imaging Below the Diffraction Limit : A Statistical Anlysis,” IEEE Trans. on Image Process. 13(5), 677–689 (2004). [CrossRef]
34. A. Ziabari, P. Torres, B. Vermeersch, Y. Xuan, X. Cartoixà, A. Torelló, J.-H. Bahk, Y. R. Koh, M. Parsa, D. Y. Peide, X. F. Alvarez, and A. Shakouri, “Full-field thermal imaging of quasiballistic crosstalk reduction in nanoscale devices,” Nat. Commun. 9(1), 255 (2018). [CrossRef]
35. X. Wang, S. Farsiu, P. Milanfar, and A. Shakouri, “Power Trace: An Efficient Method for Extracting the Power Dissipation Profile in an IC Chip From Its Temperature Map,” IEEE Trans. Comp. Packag. Technol. 32(2), 309–316 (2009). [CrossRef]
36. H. W. Engl and C. W. Groetsch, Inverse and ill-posed problems, vol. 4 (Elsevier, 2014).
37. V. B. Glasko, “Inverse problems in mathematical physics,” Moscow Izdatel Moskovskogo Universiteta Pt (1984).
38. O. M. Alifanov, E. A. Artiukhin, and S. V. Rumiantsev, Extreme methods for solving ill-posed problems with applications to inverse heat transfer problems (Begell house, New York, 1995).
39. C. Byrne, “Iterative algorithms in inverse problems,” UMass Library (2006).
40. O. Scherzer, Handbook of mathematical methods in imaging (Springer Science & Business Media, 2010).
41. S. S. Saquib, C. A. Bouman, and K. Sauer, “ML parameter estimation for Markov random fields with applications to Bayesian tomography,” IEEE Trans. on Image Process. 7(7), 1029–1044 (1998). [CrossRef]
42. S. Sreehari, S. V. Venkatakrishnan, B. Wohlberg, L. F. Drummy, J. P. Simmons, and C. A. Bouman, “Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolation,” IEEE Trans. Comput. Imaging 2(4), 1 (2016). [CrossRef]
43. J.-B. Thibault, K. D. Sauer, C. A. Bouman, and J. Hsieh, “A three-dimensional statistical approach to improved image quality for multislice helical ct,” Med. Phys. 34(11), 4526–4544 (2007). [CrossRef]
44. C. A. Bouman and K. D. Sauer, “A generalized gaussian image model for edge-preserving map estimation,” IEEE Trans. on Image Process. 2(3), 296–310 (1993). [CrossRef]
45. C. A. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Transactions on image processing 5(3), 480–492 (1996). [CrossRef]
46. C. J. Pellizzari, R. Trahan, H. Zhou, S. Williams, S. E. Williams, B. Nemati, M. Shao, and C. A. Bouman, “Optically coherent image formation and denoising using a plug and play inversion framework,” Appl. Opt. 56(16), 4735–4744 (2017). [CrossRef]
47. Z. Yu, J.-B. Thibault, C. A. Bouman, K. D. Sauer, and J. Hsieh, “Fast model-based x-ray ct reconstruction using spatially nonhomogeneous icd optimization,” IEEE Trans. on Image Process. 20(1), 161–175 (2011). [CrossRef]
48. L. R. Harriott, “Limits of lithography,” Proc. IEEE 89(3), 366–374 (2001). [CrossRef]
49. D. Girard and L. Desbat, “The minimum reconstruction-error choice of regularization parameters: some effective methods and their application to deconvolution problems,” SIAM J. Sci. and Stat. Comput. 8(6), 934–950 (1987). [CrossRef]
50. S. Ramani, T. Blu, and M. Unser, “Monte-carlo sure: A black-box optimization of regularization parameters for general denoising algorithms,” IEEE Trans. on Image Process. 17(9), 1540–1554 (2008). [CrossRef]
51. C. M. Stein, “Estimation of the mean of a multivariate normal distribution,” The annals of Statistics pp. 1135–1151 (1981).
52. J. Rice, “Choice of smoothing parameter in deconvolution problems,” Contemp. Math. 59, 137–152 (1986). [CrossRef]
53. V. Solo, “A sure-fired way to choose smoothing parameters in ill-conditioned inverse problems,” in Proceedings of 3rd IEEE International Conference on Image Processing, vol. 3 (IEEE, 1996), pp. 89–92.
54. J.-C. Pesquet, A. Benazza-Benyahia, and C. Chaux, “A sure approach for digital signal/image deconvolution problems,” IEEE Trans. Signal Process. 57(12), 4616–4632 (2009). [CrossRef]
55. F. Lucka, K. Proksch, C. Brune, N. Bissantz, M. Burger, H. Dette, and F. Wübbeling, “Risk estimators for choosing regularization parameters in ill-posed problems-properties and limitations,” Inverse Probl. & Imaging 12(5), 1121–1155 (2018). [CrossRef]
56. F. Luisier, T. Blu, and M. Unser, “A new sure approach to image denoising: Interscale orthonormal wavelet thresholding,” IEEE Trans. on Image Process. 16(3), 593–606 (2007). [CrossRef]
57. J. L. Mead, “Statistical tests for total variation regularization parameter selection,” (2015).
58. P. Torres, A. Ziabari, A. Torelló, J. Bafaluy, J. Camacho, X. Cartoixà, A. Shakouri, and F. Alvarez, “Emergence of hydrodynamic heat transport in semiconductors at the nanoscale,” Phys. Rev. Mater. 2(7), 076001 (2018). [CrossRef]
59. F. Menges, H. Riel, A. Stemmer, and B. Gotsmann, “Quantitative thermometry of nanoscale hot spots,” Nano Lett. 12(2), 596–601 (2012). [CrossRef]
60. C. D. S. Brites, P. P. Lima, N. J. O. Silva, A. Millán, V. S. Amaral, F. Palacio, and L. D. Carlos, “Thermometry at the nanoscale,” Nanoscale 4(16), 4799–4829 (2012). [CrossRef]
