石墨烯是21世纪以来热门的研究材料之一，实验上一经成功制备，立即引发了研究者的关注。能带理论是固体物理中研究材料电磁学性质的一个重要理论，我们从单层石墨烯出发，用紧束缚近似下的原子轨道线性组合方法，推导得到单层石墨烯的能带结构，在不考虑次近邻项的情况下，石墨烯是一种带隙为零的半导体，具有电子—空穴的对称性，当考虑次近邻项后，对称性被破坏。为了研究石墨烯的磁关联，我们在哈伯德模型下建立实空间的哈密顿量，并使用行列式的量子蒙特卡洛方法计算在电子掺杂的情况下，具有锯齿形边界的二维石墨烯纳米带的自旋磁化率。当电子浓度趋于半满时，锯齿形边界以铁磁性为主导，并且发现库伦相互作用能对边界磁化率有显著的促进作用，从而可以改变库伦相互作用能来调控边界铁磁性。与边界磁化率不同的是，体系整体的磁化率在电子浓度小于0.8时，呈现出了一定的铁磁涨落。考虑到双层石墨烯的层间耦合，在一定的电子浓度的条件下，整体可能会呈现出更强的磁化率，所以我们从能带结构出发，类比单层石墨烯的推导方法，得到了双层石墨烯的能带结构。双层石墨烯的能带从单层的两条分裂成四条，在狄拉克点处，导带和价带有所交叠，使得双层石墨烯呈现出一定的半金属性，不具有电子—空穴的对称性。接下来我们需要建立哈伯德模型下实空间的哈密顿量，用行列式的量子蒙特卡洛方法对双层石墨烯磁化率进行计算，这是我们还未完成的工作。本文的推导过程、结果和讨论，可供石墨烯的初学者进行参考，而磁性方面的研究结果可作为材料应用方面，对寻找室温下的铁磁性半导体提供一定的参考。

Graphene is becoming more and more popular as successfully produced in experiment since the 21st century. And energy band theory is fundamental to study electronic and magnetic properties of material. With linear combination of atomic orbitals in tight binding approximation, we derived energy band of graphene. Without considering the next-nearest-neighbor hopping energy, graphene is a zero-gap semiconductor and electronic - hole symmetry. But considering next nearest neighbor hopping energy, symmetry is destroyed. In order to study graphene magnetic correlation, we build up Hamiltonian of Hubbard Model in real space. Determinant Quantum Monte Carlo(DQMC) method is used to calculate spin susceptibility in doped graphene with zigzag edge. It is found that ferromagnetic fluctuations at the zigzag edge dominates around half filling, and it is strengthened markedly by the on-site Coulomb interaction U. And bulk graphene is weakly magnetic but ferromagnetic fluctuations may also dominate as the electron filling is lower than 0.8. Due to the coupling of two layers, a stronger magnetic susceptibility may be induced under certain electron filling in bilayer graphene. Therefore we derived the energy band of bilayer according to the way of monolayer graphene. As a result, Bilayer graphene’s band split to four. At the Dirac point, the conduction band overlaps with valence band so that bilayer graphene exhibits a clear semimetallic behavior and show electron - hole asymmetry at the same time. Next we need to establish Hamilton of bilayer in real space and use DQMC to calculate spin susceptibility. The beginners of graphene can take the derivation process, results and discussion in this paper as a reference. And the results of magnetic correlation may help to seek room-temperature ferromagnetic semiconductor.