Electronic Structure of a Hydrogenic Acceptor Impurity in Semiconductor Nano-structures

The electronic structure and binding energy of a hydrogenic acceptor impurity in 2, 1, and 0-dimensional semiconductor nano-structures (i.e. quantum well (QW), quantum well wire (QWW), and quantum dot (QD)) are studied in the framework of effective-mass envelopefunction theory. The results show that (1) the energy levels monotonically decrease as the quantum confinement sizes increase; (2) the impurity energy levels decrease more slowly for QWWs and QDs as their sizes increase than for QWs; (3) the changes of the acceptor binding energies are very complex as the quantum confinement size increases; (4) the binding energies monotonically decrease as the acceptor moves away from the nano-structures’ center; (5) as the symmetry decreases, the degeneracy is lifted, and the first binding energy level in the QD splits into two branches. Our calculated results are useful for the application of semiconductor nano-structures in electronic and photoelectric devices. Introduction Impurity states play a very important role in the semiconductor revolution. Hydrogenic impurities, including donors and acceptors, have been widely studied in theoretical and experimental approaches [1]. Recently, Mahieu et al. investigated the energy and symmetry of Zn and Be dopant-induced acceptor states in GaAs using cross-sectional scanning tunneling microscopy and spectroscopy at low temperatures [2]. The ground and first excited states were found to have a non-spherical symmetry. In particular, the first excited acceptor state has Td symmetry. Bernevig and Zhang proposed a spin manipulation technique based entirely on electric fields applied to acceptor states in p-type semiconductors with spin-orbit coupling. While interesting on its own, the technique could also be used to implement fault-resilient holonomic quantum computing [3]. Loth et al. studied tunneling transport through the depletion layer under a GaAs surface with a low temperature scanning tunneling microscope. Their findings suggest that the complex band structure causes the observed anisotropies connected with the zinc blende symmetry [4]. Kundrotas et al. investigated the optical transitions in Be-doped GaAs/AlAs multiple quantum wells with various widths and doping levels [5]. The fractional dimensionality model was extended to describe free-electron acceptor (free hole-donor) transitions in a quantum well (QW). The measured photoluminescence spectra from the samples were interpreted within the framework of this model, and acceptor-impurity induced effects in the photoluminescence line shapes from multiple quantum wells of different widths were demonstrated. Buonocore et al. presented results on the ground-state binding energies for donor and acceptor impurities in a deformed quantum well wire (QWW) [6]. The impurity effective-mass Schrödinger equation was reduced to a onedimensional equation with an effective potential containing S.-S. Li J.-B. Xia CCAST (World Lab.), P. O. Box 8730, Beijing 100080, P.R. China S.-S. Li (&) J.-B. Xia State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, P. O. Box 912, Beijing 100083, P.R. China e-mail: [email protected] 123 Nanoscale Res Lett (2007) 2:554–560 DOI 10.1007/s11671-007-9098-9 both the Coulomb interaction and the effects of the wire surface irregularities through the boundary conditions. Studying the ground-state wave functions for different positions of the impurity along the wire axis, they found that there are wire deformation geometries for which the impurity wave function is localized either on the wire deformation or on the impurity, or even on both. For simplicity, they only considered hard wall boundary conditions. Lee et al. calculated the magnetic-field dependence of low-lying spectra of a single-electron magnetic quantum ring and dot, formed by inhomogeneous magnetic fields using the numerical diagonalization scheme [7]. The effects of on-center acceptor and donor impurities were also considered. In the presence of an acceptor impurity, transitions in the orbital angular momentum were found for both the magnetic quantum ring and the magnetic quantum dot when the magnetic field was varied. Galiev and Polupanov calculated the energy levels and oscillator strengths from the ground state to the odd excited states of an acceptor located at the center of a spherical quantum dot (QD) in the effective mass approximation [8]. They also used an infinite potential barrier model. Using variational envelope functions, Janiszewski and Suffczynski computed the energy levels and oscillator strengths for transitions between the lowest states of an acceptor located at the center of a spherical QD with a finite potential barrier in the effective mass approximation [9]. Climente et al. calculated the spectrum of a Mn ion in a p-type InAs quantum disk in a magnetic field as a function of the number of holes described by the Luttinger-Kohn Hamiltonian [10]. For simplicity, they placed the acceptor at the center of the disk. In this paper, we will study the electronic structures and binding energy of a hydrogenic acceptor impurity in semiconductor nano-structures in the framework of effective-mass envelope-function theory. In our calculations, the finite potential barrier and the mixing effects of heavyand light-holes are all taken into account. Theoretical Model Throughout this paper, the units of length and energy are given in terms of the Bohr radius a 1⁄4 h 0=m0e and the effective Rydberg constant R 1⁄4 h=2m0a ; where m0 and e0 are the mass of a free electron and the permittivity of free space. For a hydrogenic acceptor impurity located at r0 1⁄4 ðx0; y0; z0Þ in a semiconductor nano-structure, the electron envelope function equation in the framework of the effective-mass approximation is H0 2a jr r0j þ VðrÞ wnðrÞ 1⁄4 E nwnðrÞ; ð1Þ where H 0 1⁄4 Pþ R Q 0 R P 0 Q Q 0 P R 0 Q R Pþ 2 664 3 775 ð2Þ with P 1⁄4ðc1 c2Þðp2x þ p2yÞ þ ðc1 2c2Þp2z ; Q 1⁄4 i2 ffiffiffi 3 p c3ðpx ipyÞpz; R 1⁄4 ffiffiffi 3 p c2ðp2x p2yÞ 2ic3pxpy h i : ð3Þ In the above equations, c1,c 2, and c 3 are the Luttinger parameters and jr r0j 1⁄4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x x0Þ þ ðy y0Þ þðz z0Þ q : The subscript n = 0, 1, 2,... correspond to the ground-, first excited-, second excited-,... states, respectively. The quantum confinement potential VðrÞ can be written in different forms for various nano-structures. In Eq. 1, a is 0 when there are no acceptors and 1 when there are acceptors in the nano-structure. The binding energy of the n-order hydrogenic donor impurity state is explicitly calculated by the following equation: Eb 1⁄4 E 0 E n: ð4Þ We express the wave function of the impurity state as [11] Wh rh ð Þ 1⁄4 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi LxLyLz p X nxnynz anxnynz bnxnynz cnxnynz dnxnynz 2 66664 3 77775 ei1⁄2ðkxþnxKxÞxþðkyþnyKyÞy þðkzþnzKzÞz ; ð5Þ where Lx, Ly, and Lz are the side lengths of the unit cell in the x, y, and z directions, respectively. Kx = 2 p /Lx,Ky = 2 p /Ly,Kz = 2 p /Lz, nx [{ – mx,..., mx }, ny [{ – my,..., my }, and nz [{ – mz,..., mz}. The plane wave number is Nxyz = Nx Ny Nz = (2 mx + 1)(2 my + 1)(2 mz + 1), where mx, my, and mz are positive integers. We take Lx = Ly = Lz = L = Wmax + 25 nm,Kx = Ky = Kz = K = 2 p/L, and Nx = Ny = Nz = 7 in the following calculation, where Wmax is the maximum side length of the nano-structures. If we take larger Nx, Ny, and Nz, the calculation precision will be increased somewhat. The matrix elements for solving the energy latent root of the impurity states can be found from Eqs. 1 and 5. The Nanoscale Res Lett (2007) 2:554–560 555