We present a theory of superconducting $p-n$ junctions. To this end, we consider a two band model of doped bulk semiconductors with attractive interactions between the charge carriers and derive the superconducting order parameter, the quasiparticle density of states, and the chemical potential as a function of the semiconductor gap ${\mathrm{\Delta }}_0$ and the doping level $\varepsilon$. We verify previous results for the quantum phase diagram for a system with constant density of states in the conduction and valence band, which show BCS-superconductor to Bose-Einstein-condensation (BEC) and BEC-to-insulator transitions as a function of doping level and the size of the band gap. Then, we extend this formalism to a density of states which is more realistic for 3D systems and derive the corresponding quantum phase diagram, where we find that a BEC phase can only exist for small band gaps ${\mathrm{\Delta }}_0<{\mathrm{\Delta }}_0^{*}$. For larger band gaps, we find rather a direct transition from an insulator to a BCS phase. Next, we apply this theory to study the properties of superconducting $p-n$ junctions. We derive the spatial variation of the superconducting order parameter along the $p-n$ junction. As the potential difference across the junction leads to energy band bending, we find a spatial crossover between a BCS and BEC condensate, as the density of charge carriers changes across the $p-n$ junction. For the two-dimensional system, we find two possible regimes, when the bulk is in a BCS phase, a BCS-BEC-BCS junction with a single BEC layer in the space charge region, and a BCS-BEC-I-BEC-BCS junction with two layers of BEC condensates separated by an insulating layer. In three dimensions we find that there can also be a conventional BCS-I-BCS junction for semiconductors with band gaps exceeding ${\mathrm{\Delta }}_0^{*}$. Thus, we find that there can be BEC layers in the well controlled setting of doped semiconductors, where the doping level can be varied to change and control the thickness of BEC and insulator layers, making Bose-Einstein condensates thereby possibly accessible to experimental transport and optical studies in solid-state materials.

中文翻译：

---

超导PN结中的空间BCS-BEC交叉

我们提出超导理论 $p-ñ$路口。为此，我们考虑了带有电荷载流子之间有吸引力的相互作用的掺杂体半导体的两个能带模型，并得出了超导阶参数，状态的准粒子密度和化学势随半导体间隙的变化情况${\mathrm{\Delta }}_0$ 和掺杂水平 $\varepsilon$。我们验证了在导带和价带中具有恒定态密度的系统的量子相图的先前结果，该系统表明BCS超导体到Bose-Einstein凝聚（BEC）和BEC到绝缘体的跃迁是掺杂的函数水平和带隙的大小。然后，我们将这种形式主义扩展到对于3D系统更现实的状态密度，并得出相应的量子相位图，在该图中我们发现BEC相位只能存在于较小的带隙中${\mathrm{\Delta }}_0<{\mathrm{\Delta }}_0^{*}$。对于较大的带隙，我们发现从绝缘体到BCS相直接过渡。接下来，我们运用这一理论来研究超导的性质$p-ñ$路口。我们推导了超导阶参数沿空间的空间变化$p-ñ$交界处。由于跨结的电势差导致能带弯曲，我们发现BCS和BEC冷凝物之间存在空间交叉，因为载流子的电荷密度在整个BCS和BEC上变化。$p-ñ$交界处。对于二维系统，我们发现两种可能的状态，即当主体处于BCS相时，在空间电荷区域中具有单个BEC层的BCS-BEC-BCS结和BCS-BEC-I-BEC-具有两层BEC冷凝物的BCS结由绝缘层隔开。在三个维度上，我们发现对于带隙超过10％的半导体，也可以使用常规的BCS-I-BCS结${\mathrm{\Delta }}_0^{*}$。因此，我们发现在掺杂半导体良好控制的环境中可能存在BEC层，可以改变掺杂水平以改变和控制BEC和绝缘体层的厚度，从而使玻色-爱因斯坦凝聚物易于实验传输和注入。固态材料的光学研究。