We consider the interaction between acceptor pairs in doped semiconductors in the limit of large interacceptor separation relevant for low doping densities. Modeling individual acceptors via the spherical model of Baldereschi and Lipari, we calculate matrix elements of the quadrupole tensor between the four degenerate ground states and show that the acceptor has a nonzero quadrupole moment. As a result, the dominant contribution to the large-separation acceptor-acceptor interaction comes from direct (charge-density) terms rather than exchange terms. The quadrupole is the leading nonzero moment, so the electric quadrupole-quadrupole interaction dominates for large separation. We calculate the matrix elements of the quadrupole-quadrupole interaction Hamiltonian in a product-state basis and diagonalize, obtaining a closed-form expression for the energies and degeneracies of the sixteen-state energy spectrum. All dependence on material parameters enters via an overall prefactor, resulting in surprisingly simple and universal results. This simplicity is due, in part, to a mathematical happenstance, the nontrivial vanishing of a particular Wigner 6-$j$ symbol, $\left\{\begin{array}{ccc}2& 2& 2\\ \frac{3}{2}& \frac{3}{2}& \frac{3}{2}\end{array}\right\}=0$. Results are relevant to the control of two-qubit interactions in quantum computing implementations based on acceptor spins, as well as calculations of the thermodynamic properties of insulating $p$-type semiconductors.

中文翻译：

---

受体对in掺杂半导体之间的四极相互作用

我们认为，在大的受主间间距限制下，掺杂半导体中的受主对之间的相互作用与低掺杂密度有关。通过Baldereschi和Lipari的球形模型对单个受体建模，我们计算了四个简并基态之间的四极子张量的矩阵元素，并表明该受体具有非零的四极矩。结果，对大间距受体-受体相互作用的主要贡献来自直接（电荷密度）项而不是交换项。四极杆是领先的非零矩，因此四极杆-四极杆的电相互作用对于较大的分离起主导作用。我们以产品状态为基础计算四极-四极相互作用哈密顿量的矩阵元素，并将其对角化，获得十六态能谱的能量和简并性的闭式表达式。对材料参数的所有依赖都通过一个整体因素来输入，从而产生令人惊讶的简单通用结果。这种简单性部分是由于数学上的偶然性，即特定Wigner 6-$Ĵ$ 符号， $\left\{\begin{array}{ccc}2& 2& 2\\ \frac{3}{2}& \frac{3}{2}& \frac{3}{2}\end{array}\right\}=0$。结果与基于受体自旋的量子计算实现中的两个量子位相互作用的控制以及绝缘体的热力学性质的计算有关$p$型半导体。