Attention-based Transformers have demonstrated strong adaptability across a wide range of tasks and have become the backbone of modern Large Language Models (LLMs). However, their underlying mechanisms remain open for further exploration. The energy-based perspective has long provided a valuable principle for understanding neural computation. In this paper, we revisit the principle of energy as a lens to understand attention-based Transformer models. We present a unified energy-based framework which is composed of three key components: the local energy $E_i$, the global energy $F$, and the employed optimization algorithms. We show that different attention forms including unnormalized linear attention, gated linear attention and standard softmax attention can be induced by choosing their corresponding recipes within this framework. Building on this framework, we propose energy-based modifications of attention structures. Inspired by classical gradient descent (GD) algorithms, we extend the original attention formulation based on standard GD to the momentum-based GD, Nesterov Accelerated Gradient (NAG), and Newton's method, each inducing a corresponding new attention structure. Our experiments provide preliminary support for the potential of the energy-based framework for designing attention mechanisms.