In modern applications, the shrinking available space on terminals coupled with the increasing number of frequency bands has prompted the greatest demand for antenna miniaturization and multi-band features. This work introduces an innovative method for designing a dual-band planar antenna with an exceptionally compact size, addressing the pressing requirement. Two half-mode patches, both operating in two modes (TM0.5,0 of 3.5 GHz frequency band and TM0.5,2 of 4.9 GHz frequency band) are placed back-to-back. The dual-mode operation in the 3.5 GHz band is achieved using the two TM0.5,0 modes and the dual-mode operation in the 4.9 GHz band is obtained from the TM0.5,2 and strip modes. Then, based on the elementary antenna design, a 4×4 MIMO system is designed and measured. The isolation between the elements is found to be greater than 15.5 dB. The proposed antenna achieves a bandwidth coverage of 3.36-3.70 GHz and 4.79-5.01 GHz, corresponding to N78 and N79 frequency bands of the fifth generation (5G) wireless communication systems, respectively. The proposed antenna, based on a single-layer design without any air spacing, features a compact size and high integration density, making it an interesting solution for 5G terminals.