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This paper highlights and experimentally verifies a new source of random threshold-voltage (V_th) fluctuation in emerging metal-gate transistors and proposes a statistical framework to investigate its device and circuit-level implications. The new source of variability, christened work-function (WF) variation (WFV), is caused by the dependence of metal WF on the orientation of its grains. The experimentally measured data reported in this paper confirm the existence of such variations in both planar and nonplanar high-k metal-gate transistors. As a result of WFV, the WFs of metal gates are statistical distributions instead of deterministic values. In this paper, the key parameters of such WF distributions are analytically modeled by identifying the physical dimensions of the devices and properties of materials used in the fabrication. It is shown that WFV can be modeled by a multinomial distribution where the key parameters of its probability distribution function can be calculated in terms of the aforementioned parameters. The analysis reveals that WFV will contribute a key source of V_th variability in emerging generations of metal-gate devices. Using the proposed framework, one can investigate the implications of WFV for process, device, and circuit design, which are discussed in Part II.