InAlN/GaN异质结二维电子气波函数的变分法研究

Variational study of the 2DEG wave function in InAlN/GaN heterostructures

使用变分法推导了InAlN/GaN异质结二维电子气波函数和基态能级的解析表达式,并讨论了InAlN/GaN异质结结构参数对二维电子气电学特性的影响.在假设二维电子气来源于表面态的前提下,使用了一个包含两个变分参数的尝试波函数推导电子总能量期望值,并通过寻找能量期望极小值确定变分参数.计算结果显示,二维电子气面密度随InAlN厚度的增大而增大,且理论结果与实验结果一致.二维电子气面密度增大抬高了基态能级与费米能级,并保持二者之差增大以容纳更多电子.InAlN/GaN界面处的极化强度失配随着In组分增大而减弱,二维电子气面密度随之减小,并导致基态能级与费米能级减小.所建立的模型能够解释InAlN/GaN异质结二维电子气的部分电学行为,并为电子输运与光学跃迁的研究提供了解析表达式.

The variational method has been widely used to study the electronic structures of heterostructure materials in spite of this method being less accurate than the numerical method, because analytical formulas for some electrical parameters can be derived using this method. However, effects of surface states on the two-dimensional electron gas（2 DEG） have not been taken into account in the variational studies of GaN-based heterostructures. In the present study, analytical formulas for the electron wave function and ground state energy level of the 2 DEG in InAlN/GaN heterostructures are derived using the variational method, and the influences of structural parameters of InAlN/GaN heterostructures on the electrical properties are discussed. In the theoretical model, evenly distributed surface states below the conduction band are assumed to be the origin of the 2 DEG, and the polarization charges at the InAIN surface and the InAlN/GaN interface due to spontaneous and piezoelectric polarization effects in InAlN/GaN heterostructures are taken into account.A trial envelope wave function with two variational parameters is used to derive the expectation value of the total energy per electron. The variational parameters are determined by minimizing the expectation value. The model predicts a linear conduction band profile in InAIN barrier layer and an approximately triangular-shaped potential well on the GaN side of the InAlN/GaN interface. Electrons released from the surface states are confined in the potential well,forming the 2 DEG. The 2 DEG sheet density for the lattice-matched InAlN/GaN heterostructure with a 15 nm InAIN layer is 1.96×10^（13） cm^（-2）, and the average distance from the InAlN/GaN interface of electrons is 2.23 nm. The 2 DEG sheet density increases rapidly with InAIN thickness increasing when the InAIN layer exceeds the critical thickness, and starts to be saturated above 15 nm. The dependence of the calculated 2 DEG sheet density on the InAIN thickness quantitatively agrees with recently reported experimental data. The increasing 2 DEG sheet density results in increasing the ground state energy level and Fermi energy, and the energy spacing between the two also increases for containing more electrons. The polarization discontinuity at the InAlN/GaN interface decreases with increasing In mole fraction,causing the 2 DEG sheet density to decrease, and thus the ground state energy level and the Fermi energy to decrease.This model is conducive to understanding the electrical behaviors of InAIN/GaN heterostructures and providing readily applicable formulas for studying the electron transport and optical transitions.