ZHENG Wei-min, HUANG Hai-bei, LI Su-mei, CONG Wei-yan, WANG Ai-fang, LI Bin, SONG Ying-xin

(1. School of Space Science and Physics, Shandong University(Weihai), Weihai 264209, China;2. School of Chemistry, The University of Melbourne, Victoria 3010, Australia;3. School of Information Engineering, Shandong University(Weihai), Weihai 264209, China;4. Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China;5. Jinan Semiconductor Research Institute, Jinan 250014, China)

Abstract: A GaAs/AlAs quantum well, with a 15 nm-thick GaAs well surrounded by a 5 nm-thick AlAs barrier, is δ-doped with Be acceptors of various doping levels at the well center. The ionized acceptor diffused profiles within the quantum well are solved by the diffusion equation. The additional potential, due to both the ionized acceptor diffused profile and hole distribution in valence-band subbands, is incorporated into the quantum well potential. The self-consistent solution and converged hole energy eigenvalue for the Schr?dinger’s and Poisson’s equations are looked for by an iterative method. It is found through calculations that the energy of the heavy-hole ground state hh has changed by about 1 meV, while it is red-shifted towards the valence-band top with increasing Be acceptor doping concentrations. The calculated results are in a good agreement with experimental results.

Key words: doping levels; δ-doped; GaAs/AlAs quantum wells; acceptor diffused profiles

1 Introduction

Semiconductor quantum wells have attracted a large interest due to their size tunable electronic structures and optical properties, which make them be ideal building blocks for a wide variety of opto-electronic devices[1-3]. In particular, the GaAs/Ga1-x-AlxAs quantum well with a mature material growth and optoelectronic device fabricated techniques is one of the most interesting quantum-well systems, thus it has been widely investigated[4-6]. For the GaAs/Ga1-xAlxAs quantum wells which are doped with acceptors in the GaAs well or Ga1-xAlxAs barrier layers, the majority of researches focus on an influence of the quantum well width on acceptor energy state structures. Gammonetal.[7]reported the resonant Raman spectra of the shallow acceptors Be in GaAs-Ga1-xAlxAs quantum-well structures where Be acceptors were doped at the center and the edge of the wells with well widths in the range 7-16.5 nm, respectively. Reederetal.[8]investigated the odd-parity transitions between energy levels of Be acceptors by far-infrared absorption spectra at 4.2 K, which are doped over the center 1/3 of each well of GaAs/Al0.3Ga0.7As multiple quantum wells. In addition, the effect of quantum-well sizes on transitions between energy states of the acceptors confined in quantum wells was also studied widely by photoluminesence (PL) and PL excitation spectra[9-12]. Moreover, by the time-resolved pump-probe measurement for the far-infrared spectroscopy, Zhengetal.[13]studied the effect of quantum-well confinement on the excite state lifetime of shallow Be acceptors-doped at the well center of the GaAs/AlAs multiple quantum wells. Theoretically, Masselinketal.[14-15]calculated the acceptor binding energies in GaAs-AlxGa1-xAs quantum wells with and without the application of electric, magnetic, and uniaxial stress fields by the variation method. The literatures mentioned above only report that the quantum-well confinement has an effect on energy states of the acceptors doped in each well layer. This is only one aspects of the problem. However, another aspect, that the doped acceptors have an influence on subbands of the GaAs/Ga1-xAlxAs quantum wells, is seldom taken into account. The diffusion could significantly affect the operational lifetime of opto-electronic devices based on the GaAs/Ga1-xAlxAs quantum wells. In this paper, we will report that the diffusion of acceptors doped at the well center effects on the subbands of a GaAs/AlAs quantum well. In general, the acceptors doped within the well layers are ionized completely at elevated temperatures, then there exists a electric file around each of ionized acceptors. Therefore, both diffusion profiles of ionized acceptors in the well and distributions of holes in the valence-band subbands, supplied by the ionized Be acceptors, give rise to an additional potential, by which the original subbands of the GaAs/AlAs quantum well without acceptor doping are altered. The ionized acceptor diffused profile is calculated by the diffusion equation. The additional potential is incorporated into the Schr?dinger’s equation. Then the Schr?dinger’s and Poisson’s equations are simultaneously solved by a self-consistent method. Finally, the calculated results of the quantum-well subbands are compared with experimental results.

2 Theoretical Models

2.1 Diffusion Equation

The one-dimensional diffusion equation for the acceptor distribution represented byx(z,t) is[16]:

(1)

wheretis the time. The diffusion coefficientDwill, in general, have a temporalt, spatialzand concentrationxdependence. In order to calculate the numerical solution of the Eq.(1), the finite difference approximations to first and second derivatives are deduced as follows:

(2)

and

(3)

Then Eq.(1) is expanded in terms of the Eq.(2) and (3), resulting in the following:

(4)

in a common case, the functionx(z,t) is known whent=0, which is referred to as the initial profile of doped acceptors. In order to proceed with the calculation, the entityDmust be prescribed,i.e., its functional dependence on the variablesx,zandtmust be known. Given this, it is apparent from Eq.(4) that the concentrationxat any pointzcan be calculated a short time intervalδtinto the future, if the concentrationxis known at a small spatial stepδzon either side ofz.

2.2 Poisson’s Equation

Assuming that the acceptors doped at the center of the GaAs/AlAs quantum well are completely ionized, then the number of holes in the valence-band subbands supplied by the ionized Be acceptors would be equal to that of negatively charged ionized acceptors in the quantum well, due to the charge conservation law. Both holes and ionized acceptors will produce the electric field around themselves, which causes a significant additional potentialVρ. Since the energy states of holes in the valence-band subbands of the GaAs/AlAs quantum well are dependent on the quantum-well-barrier potentialVplus the additional potential, so it is necessary to look for this additional potentialVρ. The additional potential termVρ(z) can be written as by Poisson’s equation[17]:

(5)

whereεis the permittivity of materials. The solution of Eq.(5) can be expressed by the electric fieldE:

(6)

since the GaAs/AlAs quantum well is the one-dimensional confinement for holes in thez-axis direction, thus the holes are free in thex-yplane. The hole volume densityρ(z) can be considered as an infinite plane with areal densityσ(z) and thicknessδ(z). The quantum well can be viewed as consisting of a large number of such infinite planes, and the total field strength at any point in the well layer is the sum of individual contributions of each infinite slabs,i.e.

(7)

where sign(z-z′) is a function sign. The net areal charge densityσ(z) comes from two-part contributions. The first arises from the ionized acceptors, which can be known from the doping density in each semiconductor layer, as defined at growth time. The second is due to the distribution of holes in the valence-band subbands of the GaAs/AlAs quantum well, which can be calculated from the probability distributions of holes. Therefore, the net areal charge densityσ(z) has the form as follows:

whereρ(z) is the volume density of diffused ionized acceptors at positionz.Niis the number of holes in the valence-band subband with the energy eigenenergyEi, whileψiis the wave function ofEi.

2.3 Schr?dinger’s Equation

Under effective-mass and envelop-function approximations, the Schr?dinger equation, with the additional potential due to the net areal charge densityσ(z), can be expressed for a hole confined in the GaAs/AlAs quantum well as:

(9)

whereV(z) denotes the barrier potential of the GaAs/AlAs quantum well, andVρ(z) is the additional potential arising from the net areal charge densityσ(z). Eq.(9) can be solved by the numerical shooting method[18]. Becauseσ(z) andVρ(z) are related to wave functions, so it is apparent that a iterative loop of calculations is formed in order to find the solution to satisfy both Schr?dinger’s and Poisson’s equations. However, the first iterative loop usually produces the main change of hole energy eignvaluesEifrom the zero-doping system to the doped system. Subsequent iterations yield only minor refinements toEi.

3 Results and Discussion

Fig.1 illustrates Be acceptor diffused profiles after 0, 15, 30, 60, 120, and 240 s of diffusion for the GaAs/AlAs single quantum well with a 15 nm well, sandwiched by a 5 nm AlAs barrier, where Be acceptors are-doped to 21018cm-3within a 0.2 nm-thick region at the well center. It can be seen clearly from Fig.1 that the diffused profile of Be acceptors broadens, and the height of peaks gradually falls as the diffusion time increases. This case is attributed to that the total amount of the Be acceptors doped within the quantum well remains the same in diffusion processes. Although the diffusion coefficientDis dependent on the variablesx,zandt, the analysis of the experimental data of exciton energies in CdTe/Cd1-xMnxTe quantum wells demonstrates that the diffusion of Mn from the barrier into well can be explained successfully in terms of a constant diffusion coefficientD0[19-20]. Hence, a constant diffused coefficient ofD0=0.01 nm2·s-1is employed here in the numerical calculation of Eq.(4). Meanwhile, It can be also observed that the diffusion profiles of Be acceptors is symmetric with respect to the axis through the quantum-well center, perpendicular to thez-axis. The acceptor concentrationx(z,t) at each interfaces between GaAs and AlAs layers approaches to zero, which results from the closed-system boundary condition.

Fig.1 Acceptor diffused profiles after 0, 15, 30, 60, 120, 240 s of the diffusion with a constant diffused coefficient ofD0=0.01 nm2·s-1, for the 15 nm GaAs/5 nm AlAs single quantum well with Be acceptors-doped to 2×1018cm-3within a 0.2 nm-thick region at the well center, whent=0.

At high temperatures, the Be acceptors ε-doped at the well center happen to diffuse and are simultaneously ionized, which introduced holes into the quantum well. Fig.2 shows the net areal charge densityσ(z) defined in Eq.(8), along thez-axis after 0, 15, 30, 60, 120, 240 s of diffusions for the GaAs/AlAs quantum-well structure as described in Fig.1. It can be seen obviously that theσ(z) curve broadens, and its peak depth is lowered with increasing diffusion time. Additionally, theσ(z) curve is also of symmetry, which comes from two-part contributions. The first is ascribed to the diffused profile of negatively charged ionized Be acceptors, and the second originates from the distribution of holes in valence-band subbands. According to the charge conservation law, it is known that the number of ionized Be acceptors would be equal to that of the holes. Therefore, the summation of the areal charge densityσ(z) over each thin slabs within the quantum well should be zero. It is also noticed in Fig.2 thatσ(z) at two edges of the GaAs well layer tends to zero. This is because in calculation, each holes is assumed to occupy the lowest heavy-hole state, and its wave function at the interface between the GaAs well and AlAs barrier approaches toward zero.

Fig.2 The net areal charge densityσ(z) within the GaAs well layer, due to the ionized-acceptor diffused profile and distribution of holes in valence-band subbands, after 0, 15, 30, 60, 120, 240 s of diffusion with a constant diffused coefficient ofD0=0.01 nm2·s-1for the 15 nm GaAs/5 nm AlAs single quantum well with Be acceptors-doped at the well center.

Fig.3 plots the electric field strengthEalong thez-axis, produced by the areal charge distributionσ(z) shown in Fig.2. It can be seen from Fig.3 that there are a few distinct features. First, all of curves are asymmetric with respect to the line through the well centre and perpendicular to thez-axis. It results from the axial symmetry of the net areal charge distributionσ(z), while these curves get broad and their peak depths decrease as diffusion time increases. Second, the electric field strengthEtends to zero at each edges of the GaAs well layer, which implies that the quantum well system remains charge neutrality. Finally, the electric field is zero at the center of the quantum well, which reflects the axial symmetry of the charge distribution.

Fig.3 The electric field strengthE, produced by the net areal charge distributionσ(z) shown in Fig.2, after 0, 15, 30, 60, 120, 240 s of diffusions with a constant diffused coefficient ofD0=0.01 nm2·s-1for the 15 nm GaAs/5 nm AlAs single quantum well with Be acceptors-doped at the well center.

Fig.4 displays the additional potentialVρ(z) due to the net areal charge densityσ(z), after the diffusion time of 0, 15, 30, 60, 120, 240 s. As the diffused profile of the doped acceptors broadens, the addition potentialVρ(z) depth gradually reduces. This is because that the diffused profile of the ionized acceptors begins to resemble the probability densityψ*(z)ψ(z) of the hole wave function, thus leading to an almost complete cancellation of the positive and negative potentials. It is also noticed that the addition potentialVρ(z) is also symmetric with respect to the line through the well centre and perpendicular to thez-axis, which reflects the symmetry of the acceptor diffused profile and probability distribution of the hole wave function.

Fig.4 The additional potentialVρ(z) due to the net areal charge distributionσ(z), after 0, 15, 30, 60, 120, 240 s of diffusions with a constant diffused coefficient ofD0=0.01 nm2·s-1for the 15 nm GaAs/5 nm AlAs single quantum well with Be acceptorsδ-doped at the well center.

The energy eigenvalue of heavy-hole subbands is calculated from Eq.(9) by incorporating the additional potentialVρ(z) into the quantum-well potentialVVB(z) for the GaAs/AlAs quantum well without doping. TheVVB(z)=0.33ΔEg(x) is taken in the calculation, where ΔEg(x) is the band gap atk=0 between bulk GaAs and Ga1-xAlxAs[21], which is equal to 1247xmeV, wherexis the mole fraction of the bulk Ga1-xAlxAs. The effective massm*is used as the heavy-hole mass 0.62m0in bulk GaAs, wherem0is the electron mass. The relative dielectric permittivityεris set to be 17.20. Givenm*andεr, the calculated binding energy of Be acceptors is 28.00 meV confined in a GaAs/AlAs quantum well with a wider well size, which is in a good agreement with the experimental result of bulk GaAs[22]. Fig.5 shows the calculated energy of the heavy-hole ground state hh for a GaAs/AlAs quantum well as a function of diffusion time. It can be noticed that the upper point at 2.71 meV whent=0 corresponds to the energy of the heavy-hole ground state hh at zero doping, which is in a good agreement with that reported by Masselineetal.[14], while the lower at 1.73 meV is calculated through a iterative method for the same quantum-well structure, but with Be acceptors-doping at the well center. Apparently, there is a change of 0.98 meV in energy, when taking accounting of the contribution of the net charge distributionσ(z), which reduces as the diffusion time increases. Sunetal.[23]found experimentally that the PL peak of a heavy-hole free exciton for GaAs/Ga0.73Al0.27As quantum wells with Si donors doped uniformly to 1×1018cm-3in the well layers, is shifted by 1 meV towards the lower energy direction compared to that for the same quantum-well structure with a zero-doping. The inset (a) shows the valence-band schematic diagram for the GaAs/AlAs single quantum well with heavy- and light-hole states sketched in the GaAs quantum well. In addition, The inset (b) of Fig.5 shows the energy of the heavy-hole ground state hh as a function of the acceptor doped concentration whent=60 s. It can be seen that the energy of the heavy-hole ground state hh falls monotonously as the acceptorδ-doped concentration increases, which implied the heavy-hole subband is red-shifted towards the valence-band top of the quantum well. This red-shift phenomena are also observed experimentally by PL spectra for the GaAs/Al0.3Ga0.7As quantum wells with Be acceptor doped within the well region[24-26].

Fig.5 The energy of the heavy-hole ground state hh as a function of the diffusion time for the 15 nm GaAs/5 nm AlAs single quantum well with Be acceptors δ-doped at the well center. The inset (a) shows the valence-band schematic diagram with the heavy- and light-hole states in the GaAs quantum well. The inset (b) plots the energy of the heavy-hole ground state hhversusBe acceptor-doped concentration whent=60 s.

4 Conclusion

The GaAs/AlAs quantum well with a 15 nm-thick GaAs well, surrounded by a 5 nm-thick AlAs barrier, is δ-doped by Be acceptors at the quantum-well center with various doping levels. The acceptor diffused profile at any time is solved numerically by the diffusion equation described by a constant diffused coefficient. The additional potential, due to the ionized acceptor diffusion and the distribution of holes in valence-band subbands, is calculated by the Poisson’s equation, which is incorporated into the potential of the quantum well, then formed a new potential. The Schr?dinger equation with such a new potential is solved by a iterative method, until a converged energy eigenvalue of the hole subbands is found. The calculation indicates that the energy of the heavy-hole ground state hh is red-shifted towards the valence-band top with increasing acceptor-doping concentration, which is in a good agreement with experimental results.