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To address computing's carbon footprint challenge, designers of computing systems are beginning to consider carbon footprint as a first-class figure of merit, alongside conventional metrics such as power, performance, and area. To account for total carbon <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\text{tC})$</tex> footprint of a computing system, carbon footprint models must consider both embodied carbon <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\mathrm{C}_{\text{embodied}})$</tex> due to emissions during manufacturing, and operational carbon <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\mathbf{C}_{\text{operational}})$</tex> from day-to-day use. Models for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\mathbf{C}_{\text{operational}})$</tex> are relatively mature due to the direct relationship between <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\mathbf{C}_{\text{operational}})$</tex> and energy consumed while computing. In contrast, models for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{C}_{\text{embodied}}$</tex> primarily focus on today's silicon-based technologies, not capturing the wide range of beyond-Si technologies that are actively being developed for future computing systems, including emerging nanomaterials, emerging memory devices, and various three-dimensional (3D) integration techniques. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{C}_{\text {embodied }}$</tex> models for emerging technologies are essential for accurately predicting which technology directions to pursue without exacerbating computing's carbon footprint. In this paper, we (1) develop <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{C}_{\text {embodied }}$</tex> models for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{3D}$</tex>-integrated computing systems that leverage emerging nanotechnologies. We analyze an example fabrication process that is highly promising for energy-efficient computing: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$3\mathbf{D}$</tex> integration of carbon nanotube field-effect transistors (CNFETs) and indium gallium zinc oxide (IGZO) FETs fabricated directly on top of Si CMOS at a 7 nm technology node. We show that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{C}_{\text{embodied}}$</tex> of this process is, on average (considering various energy grids), <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1.31\times$</tex> higher per wafer vs. a baseline 7 nm node Si CMOS process. (2) As a case study, we quantify tradeoffs in power, performance, area, and tC footprint for an embedded system comprising an ARM Cortex-M0 processor and embedded DRAM, implemented in each of the above processes. For a representative lifetime of the system (running applications from the Embench suite for 2 hours per day over 24 months, with a clock frequency of 500 MHz), we show that the 3D IGZO/CNFET/Si implementation is 1.02 × more carbon-efficient per good die (considering yield) vs. the baseline Si implementation, quantified by the product of tC and application execution time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(tCDP$</tex>, an effective metric of carbon efficiency). (3) Finally, we show techniques to quantify carbon efficiency benefits of future computing systems, even when there is uncertainty in carbon footprint models. Specifically, we show how to robustly compare <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\text{tCDP}$</tex> for multiple computing systems, given underlying uncertainty in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{C}_{\text{embodied}}$</tex>, computing system lifetime, carbon intensity (in equivalent grams of CO<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> emissions per unit energy consumption), and yield.