The increasing global push for carbon reduction highlights the importance of integrating renewable energy into the supply chain of cellular networks. However, due to the stochastic nature of renewable energy generation and the uneven load distribution across base stations, the utilization rate of renewable energy remains low. To address these challenges, this paper investigates the trade-off between carbon emissions and downlink throughput in cellular networks, offering insights into optimizing both network performance and sustainability. The renewable energy state of base station batteries and the number of occupied channels are modeled as a quasi-birth-death process. We construct models for the probability of channel blocking, average successful transmission probability for users, downlink throughput, carbon emissions, and carbon efficiency based on stochastic geometry. Based on these analyses, an energy-based cell association scheme is proposed to optimize the carbon efficiency of cellular networks. The results show that, compared to the closest cell association scheme, the energy-based cell association scheme is capable of reducing the carbon emissions of the network by 13.0% and improving the carbon efficiency by 11.3%.