Physical-layer key generation (PKG) establishes cryptographic keys from highly correlated measurements of wireless channels, which relies on reciprocal channel characteristics between uplink and downlink, is a promising wireless security technique for Internet of Things (IoT). However, it is challenging to extract common features in frequency division duplexing (FDD) systems as uplink and downlink transmissions operate at different frequency bands whose channel frequency responses are not reciprocal any more. Existing PKG methods for FDD systems have many limitations, i.e., high overhead and security problems. This paper proposes a novel PKG scheme that uses the feature mapping function between different frequency bands obtained by deep learning to make two users generate highly similar channel features in FDD systems. In particular, this is the first time to apply deep learning for PKG in FDD systems. We first prove the existence of the band feature mapping function for a given environment and a feedforward network with a single hidden layer can approximate the mapping function. Then a Key Generation neural Network (KGNet) is proposed for reciprocal channel feature construction, and a key generation scheme based on the KGNet is also proposed. Numerical results verify the excellent performance of the KGNet-based key generation scheme in terms of randomness, key generation ratio, and key error rate. Besides, the overhead analysis shows that the method proposed in this paper can be used for resource-contrained IoT devices in FDD systems.