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.
Expanding the EME with respect to plane waves 〈zK〉 means representing this equation with respect to Bloch functions \u3008\overrightarrow{r}{m}_{j}\overrightarrow{k}+K{\widehat{e}}_{z}\u3009. For a Blochtype eigenfunction \u3008zE\overrightarrow{k}\u3009 of the SL of energy E and wavevector \overrightarrow{k}, the EME takes the form:
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].
According to the quasiclassical transport theory based on Boltzmann's equation with the collision integral taken within the relaxation time approximation, the conductivity for vertical transport in SL minibands at zero temperature and lowfield limit can be written as
where the relaxation time τ _{ qv } is ascribed to the band E _{ q,v } , and hh, lh, and so, respectively, denote heavy hole, light hole and splitoff hole. Introducing σ _{ q } (E _{F}) as the conductivity contribution of band E _{ q,v } , one can write
where
The prime indicates the derivative of ε _{ q,v } (k _{ z } ) with respect to k _{ z } . Once the SL miniband structure is accessed, σ _{ q } can be calculated, provided that the values of τ _{ q,v } are known. The relaxation time for all the minibands is assumed to be the same. In order to describe qualitatively the origin of the peculiar behavior as a function of E _{F} , Equation (5) is analyzed with the aid of the SL band structure scheme as shown in Figure 1. It is important to see that minibands are presented just for heavy hole levels, since only they are occupied. Let us assume that E _{F} moves down through the minibands and minigaps as shown in the figure. One considers the zero in the top of the Coulomb barrier. The density {n}_{q,\nu}^{\text{eff}}({E}_{\text{F}}) is zero if E _{F} lies up at the maximum (Max) of a particular miniband ε _{ q,v } . Its value rises continuously as E _{F} spans the interval between the bottom and the top of this miniband. For E _{F} smaller than the minimum (Min) of this miniband, {n}_{q,\nu}^{\text{eff}}({E}_{\text{F}}) remains constant. A straightforward analysis of Equation (5) shows that σ _{ q } increases as E _{F} crosses a miniband and stays constant as E _{F} crosses a minigap. Therefore, a plateaulike behavior is expected for σ _{ q } as a function of E _{F}. For a particular SL of period d, one moves the Fermi level position down through a minigap by increasing the acceptordonor concentration N _{A}, so the same behavior is expected for σ _{ q } as a function of N _{A}. This fact was reported previously for ntype delta doping SLs [4].
In this way, we have the following expression for {n}_{q,\nu}^{\text{eff}}({E}_{\text{F}}):
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
Figure 2a shows the conductivity for heavy holes (σ as a function of the twodimensional acceptor concentration, N _{2D}, for unstrained GaAs/Al_{0.3}Ga_{0.7}As SLs with barrier width, d _{1} = 2 nm, and well width, d _{2} = 2 nm). The conductivity increases until N _{2D} = 3 × 10^{12} cm^{2} because of the upward displacement of the Fermi level, which moves until the first miniband is fully occupied. Afterward, one observes a small range of concentrations with a plateaulike behavior for the conductivity; this is a region where there is no contribution from the first miniband or where the second band is partially occupied, but its contribution to the conductivity is very small. In the groupIII arsenides, the minigap is shorter due to the lower values of the effective masses. After N _{A} = 4 × 10^{12} cm^{2,} the conductivity increases again because of occupation of the second miniband, and this being very significant in this case. Figure 2b indicates the Fermi level behavior as a function of N _{2D}, where the zero of energy is adopted at the top of the Coulomb barrier, as mentioned before. It is observed that the Fermi energy decreases as N _{2D} increases. This happens because of the exchangecorrelation effects, which play an important role in these structures. These effects are responsible for changes in the bending of the potential profiles. The bending is repulsive particularly for this case of GaAs/AlGaAs, and so the Coulomb potential stands out in relation to the exchangecorrelation potential.
Figure 3a depicts the conductivity behavior of heavy holes as a function of N _{2D} for unstrained GaN/Al_{0.3}Ga_{0.7}N SLs with barrier width, d _{1} = 2 nm, and well width d _{2} = 2 nm. In this case, the conductivity increases until N _{2D} = 2 × 10^{12} cm^{2} and afterward it remains constant, until N _{2D} = 6 × 10^{12} cm^{2}. A simple joint analysis of Figure 3a,b can provide the correct understanding of this behavior. At the beginning, the first miniband is only partially occupied; once the band filling increases, i.e., as the Fermi level goes up to the first miniband value, the conductivity increases. When the occupation is complete (N _{2D} = 2 × 10^{12} cm^{2}), one reaches a plateau in the conductivity. After the second miniband begins to get filled up, σ is found to increase again. However, it is important to note that, for the nitrides, the Fermi level shows a remarkable increase as N _{2D} increases, a behavior completely different as compared to that of the arsenides. This can be explained in the following way: for thinner layers of nitrides, the exchangecorrelation potential effects are stronger than the Coulomb effects, and so the potential profile is attractive, and it is expected that the Fermi level goes toward the top of the valence band, as well as the miniband energies. This has been discussed in our previous study describing a detailed investigation about the exchangecorrelation effects in group IIInitrides with short period layers [13].
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.
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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.
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OFPS carried out the calculations. GMS, LMRS and EFSJ discussed the results and purposed new calculations and improvements. SCPR conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
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dos Santos, O.F., Rodrigues, S.C., Sipahi, G.M. et al. Study of the vertical transport in pdoped superlattices based on group IIIV semiconductors. Nanoscale Res Lett 6, 175 (2011). https://doi.org/10.1186/1556276X6175
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DOI: https://doi.org/10.1186/1556276X6175
Keywords
 Fermi Level
 Local Density Approximation
 Heavy Hole
 Vertical Transport
 Conduction Band Discontinuity