The two-terminal III-V/Si double junction solar cells were recognized to achieve comparable performance with lower cost of fabrication than the III-V/III-V tandems. They have already yielded an efficiency of 35.9% of power conversion. However, this performance still remains below the theoretically limit which is stated to be beyond 43%. Considering monolithic InGaP/Si based tandem designed with double back surface field layers, this work dealt with optimization of efficiency as a function of the structure layers thicknesses and dopings. Numerical simulations were conducted for this purpose by using Silvaco/ATLAS TCAD software and were sorted according to a multi-step optimization procedure. The obtained optimum tandem InGaP/Si solar cell reached an unprecedented power conversion efficiency of 38.16% under 1.5G spectrum. Furthermore, a realistic analysis of performance of this structure was performed under different temperatures and illumination levels. The effect of InGaP band gap variation on efficiency was also investigated.