This work presents a very low-power digitally controlled oscillator (DCO) with a fast switching functionality applied for wake-up receivers. Based on a complementary -Gm LC cross-coupled topology, the DCO employs a 2-bit capacitor bank for course tuning as well as a varactor along with a 5-bit digital-to-analog converter (DAC) for fine tuning. A circuit prototype has been fabricated in a 45 nm RFSOI CMOS technology, where the DCO core occupies a silicon area of 0.05 mm2. According to measurement results, the DCO's overall tuning frequency ranges from 5.4 GHz to 8.4 GHz. The DCO reaches a settling time of 20 ns, enabling its operation under an aggressively duty-cycled mode with a duty cycle below 0.1 %. The output power of the DCO driven by an output buffer is above -9 dBm in the whole frequency span. The DCO core including its bias circuitry, DAC and limiting amplifier dissipates only 1.3 mW from a 1V supply.