Energy efficiency (EE) has been regarded as an essential design metric in 5G mobile communications. However, the massive antenna numbers, larger bandwidths, and higher density of base stations have been severely deteriorating the power consumption of 5G networks. In this article, the fundamental EE-spectrum efficiency (SE) joint design issues are thoughtfully considered. The relationships between EE and SE in single-input single-output (SISO), multiple-input multiple-output, and hybrid beamforming structures are first reviewed, where the increase of SE (EE) will unfortunately bring a reduction of EE (SE). To resolve this contradiction, a non-orthogonal multiple access (NOMA) scheme with energy-efficient user scheduling and resource allocation is investigated, which promises a simultaneous increase of EE and SE. Furthermore, a precoded waveform overlapping multiplexing scheme is presented, with significant improvements in EE and SE over the traditional Nyquist-criterion-based SISO systems. The waveform overlapping multiple access scheme is further proposed, which promises large performance improvements over the NOMA schemes discussed in 5G New Radio. The application of the waveform overlapping scheme in multiple-antenna systems is also investigated. Finally, the challenges of the waveform overlapping scheme and several other research topics are discussed. For sustainable development of the wireless communications industry, greener design in physical layer technologies should be second to none in the future 6G networks and beyond.