Background: Dual-energy CT (DECT) and material decomposition play vital roles in quantitative medical imaging. However, the decomposition process may suffer from significant noise amplification, leading to severely degraded image signal-to-noise ratios (SNRs). While existing iterative algorithms perform noise suppression using different image priors, these heuristic image priors cannot accurately represent the features of the target image manifold. Although deep learning-based decomposition methods have been reported, these methods are in the supervised-learning framework requiring paired data for training, which is not readily available in clinical settings. Purpose: This work aims to develop an unsupervised-learning framework with data-measurement consistency for image-domain material decomposition in DECT.