Recent efforts have focused on training neural networks to replace density functional theory (DFT) calculations. However, prior neural network training methods required an extensive number of DFT simulations to obtain the ground truth (Hamiltonians). Conversely, when working with limited training data, deep learning models often exhibit increased errors in predicting Hamiltonians and band structures for testing data. This phenomenon carries the potential risk of yielding inaccurate physical interpretations, including the emergence of unphysical branches within band structures. To address this challenge, we introduce a novel deep learning-based method for calculating DFT Hamiltonians, specifically designed to generate accurate results with limited training data. Our framework not only employs supervised learning with the calculated Hamiltonian but also generates pseudo Hamiltonians (targets for unlabeled data) and trains the neural networks on unlabeled data. We compare our results with those obtained using the state-of-the-art method, which trains neural networks using atomic structures as inputs and DFT Hamiltonians as targets. We demonstrate the superior performance of our framework compared to the previous approach on various datasets, such as MoS2, Bi2Te3, HfO2, and InGaAs.