Study of the vertical transport in pdoped superlattices based on group IIIV semiconductors
 Osmar FP dos Santos^{1},
 Sara CP Rodrigues^{1}Email author,
 Guilherme M Sipahi^{2},
 Luísa MR Scolfaro^{3} and
 Eronides F da SilvaJr^{4}
DOI: 10.1186/1556276X6175
© dos Santos et al; licensee Springer. 2011
Received: 5 July 2010
Accepted: 25 February 2011
Published: 25 February 2011
Abstract
The electrical conductivity σ has been calculated for pdoped GaAs/Al_{0.3}Ga_{0.7}As and cubic GaN/Al_{0.3}Ga_{0.7}N thin superlattices (SLs). The calculations are done within a selfconsistent approach to the $\overrightarrow{k}\cdot \overrightarrow{p}$ theory by means of a full sixband LuttingerKohn Hamiltonian, together with the Poisson equation in a plane wave representation, including exchange correlation effects within the local density approximation. It was also assumed that transport in the SL occurs through extended minibands states for each carrier, and the conductivity is calculated at zero temperature and in lowfield ohmic limits by the quasichemical Boltzmann kinetic equation. It was shown that the particular minibands structure of the pdoped SLs leads to a plateaulike behavior in the conductivity as a function of the donor concentration and/or the Fermi level energy. In addition, it is shown that the Coulomb and exchangecorrelation effects play an important role in these systems, since they determine the bending potential.
Introduction
The transport phenomena in semiconductors in the direction perpendicular to the layers, also known as vertical transport, have been investigated in recent years from both experimental and theoretical points of view because of their increased application in the development of electrooptical devices, lasers, and photodetectors [1–3]. The theoretical decsription of the electron transport phenomena in several quantized systems, such as quantum wells, quantum wires, and superlattices (SLs), has been given in earlier studies, and it is mainly based on the solution of the Boltzmann equation [4–6]. The use of SLs is important since increasing the dispersion relation of the minibands for carriers is possible [7]. Therefore, this means that different origins of the periodic electron/hole potential, which take place in the compositional SLs and in the SLs formed by selective doping, can cause different consequences, influencing the formation of the miniband structures, altering the electrical conductivity, and affecting the electron scattering [6]. However, most of those studies treat only ntype systems, and very little has been reported in the literature regarding ptype materials, including experimental results [8–10].
In this study, the behavior of the electrical conductivity in ptype GaAs/Al_{0.3}Ga_{0.7}As and cubic GaN/Al_{0.3}Ga_{0.7}N SLs with thin barrier and well layers is studied. A selfconsistent $\overrightarrow{k}\cdot \overrightarrow{p}$ method [11–13] is applied, in the framework of the effectivemass theory, which solves the full 6 × 6 LuttingerKohn (LK) Hamiltonian, in conjunction with the Poisson equation in a plane wave representation, including exchangecorrelation effects within the local density approximation (LDA). The calculations were carried out at zero temperature and lowfield limits, and the collision integral was taken within the framework of the relaxation time (τ) approximation.
The IIIN semiconductors present both phases: the stable wurtzite (w) phase, and the cubic (c) phase. Although most of the progress achieved so far is based on the wurtzite materials, the metastable cphase layers are promising alternatives for similar applications [14, 15]. Controlled ptype doping of the IIIN material layers is of crucial importance for optimizing electronic properties as well as for transportbased device performance. Nevertheless, this has proved to be difficult by virtue of the deep nature of the acceptors in the nitrides (around 0.10.2 eV above the top of the valence band in the bulk materials), in contrast with the case of GaAsderived heterostructures, in which acceptor levels are only few meV apart from the band edge [9, 11]. One way to enhance the acceptor doping efficiency, for example, is the use of SLs which create a twodimensional hole gas (2DHG) in the well regions of the heterostructures. Contrary to the case of wurtzite material systems, in pdoped cubic structures, a 2DHG may arise, even in the absence of piezoelectric (PZ) fields [16]. The emergence of the 2DHG, is the main reason for the realization of our calculations in cubic phase; the PZ fields can decrease drastically the dispersion relation and consequently the conductivity [17, 18].
The results obtained in this study constitute the first attempt to calculate electron conductivity in ptype SLs in the direction perpendicular to the layers and will be able to clarify several aspects related to transport properties.
Theoretical model
The calculations were carried out by solving the 6 × 6 LK multiband effective mass equation (EME), which is represented with respect to a basis set of plane waves [11–13]. One assumes an infinite SL of squared wells along <001> direction. The multiband EME is represented with respect to plane waves with wavevectors K = (2π/d)l (l integer, and d the SL period) equal to reciprocal SL vectors. Rows and columns of the 6 × 6 LK Hamiltonian refer to the Blochtype eigenfunctions $jmj\overrightarrow{k}\u3009$ of Γ_{8} heavy and light hole bands, and Γ_{7} spinorbitsplithole band; $\overrightarrow{k}$ denotes a vector of the first SL Brillouin zone.
where T is the effective kinetic energy operator including strain, V _{HET} is the valence and conduction band discontinuity potential, which is diagonal with respect to jm _{ j } , $j\text{'}{m}_{j}^{\text{'}}$, V _{A} is the ionized acceptor charge distribution potential, V _{H} is the Hartree potential due to the hole charge distribution, and V _{XC} is the exchangecorrelation potential considered within LDA. The Coulomb potential, given by contributions of V _{A} and V _{H}, is obtained by means of a selfconsistent procedure, where the Poisson equation stands, in reciprocal space, as presented in detail in refs. [11, 12].
The parameters used in these calculations are the same as those used in our previous studies [11–13]. In the above calculations, 40% for the valenceband offset and relaxation time τ = 3 ps has been adopted [19].
Results and discussion
Comparing both the systems (Figures 2 and 3), one can observe higher conductivity values for the nitride; several factors can contribute to this behavior, such as the many body effects as well as the values of effective masses, involved in the calculations of the densities ${n}_{q,\nu}^{\text{eff}}({E}_{\text{F}})$. Experimental results for pdoped cubic GaN films, which use the concept of reactive codoping, have obtained vertical conductivities as high as 50/Ωcm [8]. Those results corroborate with those of this study, since in the case of SLs, higher values for the conductivity are expected. Another interesting point concerning the arsenides relates to the higher values found for their conductivity in the case of systems, e.g., ntype delta doping GaAs system. The reason is the same as that given earlier.
Conclusions
In conclusion, this investigation shows that the conductivity behavior for heavy holes as a function of N _{2D} or of the Fermi level depicts a plateaulike behavior due to fully occupied levels. A remarkable point refers to the relative importance of the Coulomb and exchangecorrelation effects in the total potential profile and, consequently, in the determination of the conductivity. These results presented here are expected to be treated as a guide for vertical transport measurements in actual SLs. Experiments carried out with good quality samples, combined with the theoretical predictions made in this study, will provide the way to elucidate the several physical aspects involved in the fundamental problem of the conductivity in SLs minibands.
Abbreviations
 2DHG:

twodimensional hole gas
 EME:

effective mass equation
 LDA:

local density approximation
 PZ:

piezoelectric
 SLs:

superlattices.
Declarations
Acknowledgements
The authors would like to acknowledge the Brazilian Agency CNPq, CTAção Tranversal/CNPq grant #577219/20081, Universal/CNPq grant #472.312/20090, CNPq grant #303880/20082, CAPES, FACEPE (grant no. 10771.05/08/APQ), and FAPESP, Brazilian funding agencies, for partially supporting this project.
Authors’ Affiliations
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