Millimeter wave communications rely on narrow-beam transmissions to cope with the strong signal attenuation at these frequencies, thus demanding precise beam-alignment between transmitter and receiver. The resulting signaling overhead may become excessive, especially in mobile environments. This paper addresses the energy efficient design of the beam-alignment protocol, with the goal of minimizing power consumption under a transmission rate constraint. The optimality of fractional search is proved, which allocates a given fraction of the interval of uncertainty on the mobile user's angular coordinate during beam-alignment. The fractional value is shown to be a function of the communication-sensing energy ratio: when large, a wider beam is selected and the fractional value approaches 1/2 (bisection); when small, a narrower beam is used to reduce the energy cost of sensing; finally, when smaller than 1/2, sensing is suboptimal. It is proved that the energy consumption under fractional search is smaller than that under bisection by at least a quantity proportional to the product of the minimum energy per radian used during beam-alignment, and the initial uncertainty on the mobile user's angular coordinate. Numerical results demonstrate a 2dB reduction in the average power consumption compared to bisection, for a wide range of rates.