The problem of an exciton trapped in a Gaussian quantum dot (QD) of GaAs is studied in both two and three dimensions in the presence of an external magnetic field using the Ritz variational method, the 1/N expansion method and the shifted 1/N expansion method. The ground state energy and the binding energy of the exciton are obtained as a function of the quantum dot size, confinement strength and the magnetic field and compared with those available in the literature. While the variational method gives the upper bound to the ground state energy, the 1/N expansion method gives the lower bound. The results obtained from the shifted 1/N expansion method are shown to match very well with those obtained from the exact diagonalization technique. The variation of the exciton size and the oscillator strength of the exciton are also studied as a function of the size of the quantum dot. The excited states of the exciton are computed using the shifted 1/N expansion method and it is suggested that a given number of stable excitonic bound states can be realized in a quantum dot by tuning the quantum dot parameters. This can open up the possibility of having quantum dot lasers using excitonic states.

中文翻译：

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磁场对GaAs量子点中激子能级的影响：激子激光器的应用。

使用里兹变分方法，1 / N展开方法和位移的1 / N，在存在外部磁场的情况下，在二维和三维中研究了GaAs的高斯量子点（QD）中捕获的激子问题。扩展方法。获得了激子的基态能和结合能，它是量子点大小，约束强度和磁场的函数，并与文献中的那些作了比较。变分法给出了基态能量的上限，而1 / N扩展法给出了下限的能量。从移位1 / N扩展方法获得的结果显示出与从精确对角化技术获得的结果非常匹配。还研究了激子尺寸的变化和激子的振子强度与量子点尺寸的关系。利用移位的1 / N展开方法计算激子的激发态，并建议通过调节量子点参数可以在量子点中实现给定数量的稳定激子束缚态。这可以开辟使用激子态的量子点激光器的可能性。