Both spectrum efficiency and energy efficiency are two critical issues in the design of wireless communication networks. Energy harvesting cognitive radio networks have been recently proposed to attempt to solve both the issues simultaneously. In this paper, we consider a cognitive radio network in which the secondary transmitter equipped with an energy harvesting device is allowed to opportunistically access the primary channel. Here, we assume that energy arrival process and primary channel state are random process and two-state time-homogeneous discrete Markov process, respectively. By using the average throughput for the secondary network as a design criterion and jointly optimizing energy harvesting and spectrum sensing, we achieve both the energy efficiency and the spectrum efficiency. In addition, the corresponding optimal detection threshold and the maximum achievable throughput are obtained, which are substantiated by our comprehensive computer simulations.