In this paper, we consider the design of joint resource blocks (RBs) and power allocation for dual-mode base stations operating over millimeter wave (mmW) band and microwave ($μ$W) band. The resource allocation design aims to minimize the system energy consumption while taking into account the channel state information, maximum delay, load, and different types of user applications (UAs). To facilitate the design, we first propose a group-based algorithm to assign UAs to multiple groups. Within each group, low-power UAs, which often appear in short distance and experience less obstacles, are inclined to be served over mmW band. The allocation problem over mmW band can be solved by a greedy algorithm. Over $μ$W band, we propose an estimation-optimal-descent algorithm. The rate of each UA at all RBs is estimated to initialize the allocation. Then, we keep altering RB's ownership until any altering makes power increases. Simulation results show that our proposed algorithm offers an excellent tradeoff between low energy consumption and fair transmission.