In this work, we investigate the secrecy energy efficiency (SEE) optimization problem for a multiple-input single-output (MISO) cognitive radio (CR) network based on a practical nonlinear energy-harvesting (EH) model. In particular, the energy receiver (ER) is assumed to be a potential eavesdropper due to the open architecture of a CR network with simultaneous wireless information and power transfer (SWIPT), such that the confidential message is prone to be intercepted in wireless communications. The aim of this work is to provide a secure transmit beamforming design while satisfying the minimum secrecy rate target, the minimum EH requirement, and the maximum interference leakage power to primary user (PU). In addition, we consider that all the channel state information (CSI) is perfectly known at the secondary transmitter (ST). We formulate this beamforming design as a SEE maximization problem; however, the original optimization problem is not convex due to the nonlinear fractional objective function. To solve it, a novel iterative algorithm is proposed to obtain the globally optimal solution of the primal problem by using the nonlinear fractional programming and sequential programming. Finally, numerical simulation results are presented to validate the performance of the proposed scheme.