This paper presents a proposal for a new memory polynomial modeling technique with non-uniform delay taps to capture the thermal memory effects in power amplifiers. In the proposed modeling structure, each order of the memory polynomial is assigned a different memory delay. The delay profile is an exponentially shaped function, instead of equal unit delays for all memory polynomial branches. Three different metrics, the memory effect modeling ratio (MEMR) and the normalized mean square error (NMSE) and spectrum error (SE) are used to benchmark the proposed exponentially shaped delay profile memorial polynomial model performance against previously published models, namely the memoryless, the unit delay and sparse delay memory based polynomial models. The model coefficients of four models are extracted for three different excitation signals, which were selected as a 64-QAM signal around 2.14 GHz with 20 kHz, 30 kHz and 40 kHz bandwidths, ensuring that the thermal memory effects dominate the electrical memory effects. It is shown that the proposed model outperforms all the previously published models for all three excitation signals used in the experiment.