Large Language Models (LLMs) have demonstrated remarkable potential in code generation. The integration of Chain of Thought (CoT) reasoning can further boost their performance. However, current CoT methods often require manual writing or LLMs with over 100 billion parameters to generate, impeding their applicability in resource-constrained scenarios. In this study, we investigate lightweight Language Models (lLMs), which are defined to have fewer than 10 billion parameters. Empirically, we find that most lLMs cannot generate high-quality CoTs when prompted by the few-shot method, but can take advantage of high-quality CoTs generated elsewhere to improve their performance in code generation. Based on these findings, we design a novel approach COTTON which can leverage lLMs to automatically generate CoTs for code generation. We synthesize new datasets and conduct extensive experiments on various benchmarks. The results show that the CoTs generated by COTTON outperform the baselines in terms of automated and human evaluation metrics. In particular, the CoTs generated by COTTON boost various lLMs to achieve higher performance gains than those generated by LLMs such as ChatGLM (130B), and are competitive with those generated by gpt-3.5-turbo (175B). Our study also showcases the potential of lLMs in software engineering applications.