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A new class of spin-transfer torque magnetic random access memory (STT-MRAM) is discussed, in which writing is achieved using thermally initiated magnonic current pulses as an alternative to conventional electric current pulses. The magnonic pulses are used to destabilize the magnetic free layer from its initial direction, and are followed immediately by a bipolar electric current exerting conventional spin-transfer torque on the free layer. The combination of thermal and electric currents greatly reduces switching errors, and simultaneously reduces the electric switching current density by more than an order of magnitude as compared to conventional STT-MRAM. The energy efficiency of several possible electro-thermal circuit designs have been analyzed numerically. As compared to STT-MRAM with perpendicular magnetic anisotropy, magnonic STT-MRAM reduces the overall switching energy by almost 80%. Furthermore, the lower electric current density allows the use of thicker tunnel barriers, which should result in higher tunneling magneto-resistance and improved tunnel barrier reliability. The combination of lower power, improved reliability, higher integration density, and larger read margin make magnonic STT-MRAM a promising choice for future non-volatile storage.