Security and energy efficiency are two critical design metrics in the future wireless communication networks. In this paper, a Reconfigurable Intelligent Surface (RIS) assisted multiple-input single-output (MISO) downlink network is investigated. In order to counteract the multiple randomly distributed eavesdroppers, an artificial noise (AN) jamming scheme is proposed. For achieving a desirable performance trade-off between security and energy efficiency, a new metric of secrecy energy efficiency (SEE) is proposed. In order to maximize the SEE, an optimization problem is then formulated, subject to the maximum transmit power limit and minimum required data rate. For tackling the challenging non-convex fractional order problem with multiple mutually coupled variables, an efficient alternating optimization algorithm based on Dinkelbach and semi-definite programming (SDP) relaxation is proposed. This corresponds to a joint design of the transmit pre-coding matrix, the covariance matrix of AN, and the phase shifts of RIS. Simulation results demonstrate that an inherent trade-off exists between the secrecy rate and SEE, and significant performance gains in terms of SEE are achieved by the proposed scheme as compared to existing schemes. Furthermore, the introduction of AN into the transmit beamforming plays a crucial role in enhancing the SEE, especially as the number of eavesdroppers increases.