A series of the formation energies Ef of boron dopants in Si quantum dots with different shapes, including tetrahedron (TH), tetrahedron-centered (TC), and octahedron (OT), are investigated by the first-principle calculation. The site of B dopants can be simply divided into vertex (“′v”), edge (“e”), and facet (“f”) on the surfaces of the Si quantum dots. It is found that the Ef value is strongly relying on quantum dot shape. In particular, when the size is given, one finds that TH shape has the lowest Ef values, while OT shape possesses the largest ones, and thus there is a sequence of Ef(TH) < Ef(TC) < Ef(OT). This can be well explained by the shape factor λ since the same sequence of λ(TH) < λ(TC) < λ(OT) exists. Interestingly, for each shape of TH, TC, and OT, λ is not a constant, behaving as the increasing trend when the size increases, while it behaves in decreasing trend for Ef. This means that the shape and the size have the opposite roles in Ef. Thus, considering the shape influence, one can say that the Si quantum dot with a smaller λ value is in favor of B dopants in them.

中文翻译：

---

量子点形状对硅中硼掺杂剂形成能的影响

通过第一性原理计算研究了一系列不同形状的Si量子点中硼掺杂剂的形成能Ef，包括四面体(TH)、四面体中心(TC)和八面体(OT)。B掺杂剂的位置可以简单地分为Si量子点表面上的顶点（“'v”），边缘（“e”）和小平面（“f”）。发现 Ef 值强烈依赖于量子点形状。特别是，当给定大小时，发现 TH 形状的 Ef 值最低，而 OT 形状的 Ef 值最大，因此存在 Ef(TH) < Ef(TC) < Ef(OT) 的序列。这可以通过形状因子 λ 很好地解释，因为存在相同的 λ(TH) < λ(TC) < λ(OT) 序列。有趣的是，对于 TH、TC 和 OT 的每种形状，λ 不是常数，当大小增加时表现为增加趋势，而 Ef 表现为减少趋势。这意味着形状和大小在 Ef 中具有相反的作用。因此，考虑到形状的影响，可以说具有较小 λ 值的 Si 量子点有利于其中的 B 掺杂剂。