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 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 [13]. 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 [46]. 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 [810].
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 method [1113] 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 [1113]. 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 of Γ_{8 }heavy and light hole bands, and Γ_{7 }spinorbitsplithole band; 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 . For a Blochtype eigenfunction of the SL of energy E and wavevector , 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 }, , 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 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, 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].
Figure 1. Schematic representation of a SL band structure used in this study. Minibands for heavy hole levels, ε_{hh,1 }, minigaps, subbands, and Fermi level, E_{F}, are shown. The zero of energy was considered at the top of the Coulomb potential at the barrier. Horizontal dashed lines indicate the bottom of the first miniband and the top of the second miniband, respectively.
In this way, we have the following expression for :
The parameters used in these calculations are the same as those used in our previous studies [1113]. 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 2. Conductivity behavior for vertical transport in ptype GaAs/Al_{0.3}Ga_{0.7}As SLs with barrier and well widths equal to 2 nm, as a function of (a) the acceptor concentration N_{2D }and (b) the Fermi energy E_{F}.
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].
Figure 3. Conductivity behavior for vertical transport in ptype GaN/Al_{0.3}Ga_{0.7}N SLs with barrier and well widths equal to 2 nm, as a function of (a) the acceptor concentration N_{2D }and (b) the Fermi energy E_{F}.
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 . 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.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
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.
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.
References

Nakamura S: InGaNbased violet laser diodes.
Semicond Sci Technol 1999, 14:R27. Publisher Full Text

Sharma TK, Towe E: On ternary nitride substrates for visible semiconductor lightemitters.
Appl Phys Lett 2010, 96:191105. Publisher Full Text

Khmissi H, Sfaxi L, Bouzaïene L, Saidi F, Maaref H, BruChevallier C: Effect of carriers transfer behavior on the optical properties of InAs quantum dots embedded in AlGaAs/GaAs heterojunction.
J Appl Phys 2010, 107:074307. Publisher Full Text

Leite JR, Rodrigues SCP, Scolfaro LMR, Enderlein R, Beliaev D, Quivy AA: Electrical conductivity of δ doping superlattices parallel to the growth direction.
Mater Sci Eng B 1995, 35:220. Publisher Full Text

Sinyavskii EP, Khamidullin RA: Special features of electrical conductivity in a parabolic quantum well in a magnetic field.
Semiconductors 2002, 36:924. Publisher Full Text

Pusep YuA, Silva MTO, Galzerani JC, Rodrigues SCP, Scolfaro LMR, Lima AP, Quivy AA, Leite JR, Moshegov NT, Basmaji P: Raman measurement of vertical conductivity and localization effects in strongly coupled semiconductor periodical structures.
J Appl Phys 2000, 87:1825. Publisher Full Text

Kauser MZ, Osinsky A, Dabiran A, Chow PP: Enhanced vertical transport in ptype AlGaN/GaN superlattices.
Appl Phys Lett 2004, 85:5275. Publisher Full Text

Brandt O, Yang H, Kostial H, Ploog KH: High ptype conductivity in cubic GaN/GaAs(113)A by using Be as the acceptor and O as the codopant.
Appl Phys Lett 1996, 69:2707. Publisher Full Text

Kim JK, Waldron EL, Li YL, Gessmann Th, Schubert EF, Jang HW, Lee JL: Ptype conductivity in bulk Al_{x}Ga_{1x}N and Al_{x}Ga_{1x}N/Al_{y}Ga_{1y}N superlattices with average Al mole fraction > 20%.
Appl Phys Lett 2004, 84:3310. Publisher Full Text

Miller N, Ager N III, Smith HM III, Mayer MA, Yu KM, Haller EE, Walukiewicz W, Schaff WJ, Gallinat C, Koblmüller G, Speck JS: Hole transport and photoluminescence in Mgdoped InN.
J Appl Phys 2010, 107:113712. Publisher Full Text

Rodrigues SCP, d'Eurydice MN, Sipahi GM, Scolfaro LMR, da Silva LMR Jr: White light emission from pdoped quaternary (AlInGa)Nbased superlattices: Theoretical calculations for the cubic phase.
J Appl Phys 2007, 101:113706. Publisher Full Text

Rodrigues SCP, dos Santos OFP, Scolfaro LMR, Sipahi GM, da Silva EF Jr: Luminescence studies on nitride quaternary alloys double quantum wells.
Appl Surf Sci 2008, 254:7790. Publisher Full Text

Rodrigues SCP, Sipahi GM, Scolfaro LMR, Leite JR: Hole charge localization and band structures of pdoped GaN/InGaN and GaAs/InGaAs semiconductor heterostructures.
J Phys Condens Matter 2002, 14:5813. Publisher Full Text

Brimont C, Gallart M, Crégut O, Hönerlage B, Gilliot P, Lagarde D, Balocchi A, Amand T, Marie X, Founta S, Mariette H: Optical and spin coherence of excitons in zincblende GaN.
J Appl Phys 2009, 106:053514. Publisher Full Text

Novikov SV, Zainal N, Akimov AV, Staddon AV, Kent AJ, Foxon CT: Molecular beam epitaxy as a method for the growth of freestanding zincblende (cubic) GaN layers and substrates.
J Vac Sci Technol B 2010, 28:C3B1. Publisher Full Text

Rodrigues SCP, Sipahi GM: Calculations of electronic and optical properties in pdoped AlGaN/GaN superlattices and quantum wells.
J Cryst Growth 2002, 246:347. Publisher Full Text

Hu CY, Wang YJ, Xu K, Hu XD, Yu LS, Yang ZJ, Shen B, Zhang GY: Vertical conductivity of pAl_{x}Ga_{1x}N/GaN superlattices measured with modified transmission line model.
J Cryst Growth 2007, 298:815. Publisher Full Text

Li J, Yang W, Li S, Chen H, Liu D, Kang J: Enhancement of ptype conductivity by modifying the internal electric field in Mg and Siδcodoped Al_{x}Ga_{1x}N/Al_{y}Ga_{1y}N superlattices.
Appl Phys Lett 2009, 95:151113. Publisher Full Text

Park SH, Ahn D: Interband relaxation ime in wurtzite GaN/InAlN quantumwell.
Jpn J Appl Phys 1999, 38:L815. Publisher Full Text