Redalyc.Electronic Structure of Pn Delta-Doped Quantum Wells in Gaas

Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México

Rodriguez-Vargas, I.; Gaggero-Sager, L.M. Electronic structure of pn delta-doped quantum wells in GaAs Revista Mexicana de Física, vol. 53, núm. 7, diciembre, 2007, pp. 109-111 Sociedad Mexicana de Física A.C. Distrito Federal, México

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How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative REVISTA MEXICANA DE FISICA´ S 53 (7) 109–111 DICIEMBRE 2007

Electronic structure of pn delta-doped quantum wells in GaAs

I. Rodriguez-Vargas Unidad Academica´ de F´ısica, Universidad Autonoma´ de Zacatecas, Calzada Solidaridad Esquina con Paseo la Bufa S/N, 98060 Zacatecas, ZAC., Mexico,´ e-mail: [email protected] L.M. Gaggero-Sager Facultad de Ciencias, Universidad Autonoma´ del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, 62209 Cuernavaca, MOR., Mexico.´ Recibido el 30 de noviembre de 2006; aceptado el 8 de octubre de 2007

We present the electron and hole subband structure in a coupled p- and n-type delta-doped quantum well in GaAs. The study is performed in the framework of the envelope function approximation via analytical expressions for the Hartree potential previously obtained through the lines of the Thomas-Fermi approximation. The calculations are focused on the electronic structure and specially on the determination of the one-dimensional p- and n-type Bohr radius as a function of the two dimensional acceptor and donor concentrations, respectively, in order to obtain the minimum distance between the delta-doped layers that avoids or generates a reduction in the potential barriers, which is essential in the design of devices based on the coupling of p- and n-type delta-doped structures.

Keywords: Electronic structure; pn δ-doped quantum wells; Thomas-Fermi approximation.

Presentamos la estructura de subbandas de electrones y huecos en pozos delta dopados acoplados tipo n y p en GaAs. El estudio se realiza dentro de la aproximacion´ de funcion´ envolvente via expresiones anal´ıticas para el potencial Hartree previamente obtenidas a traves´ de los lineamientos de la aproximacion´ Thomas-Fermi. Los calculos´ se enfocan en la estructura electronica´ y especialmente en la determinacion´ del radio de Bohr unidimensional como funcion´ de la concentracion´ bidimensional de aceptores y donores, respectivamente, con tal de obtener la distancia m´ınima entre las capas delta dopadas que evite o genere una reduccion´ en las barreras de potencial, lo cual es esencial en el diseno˜ de dispositivos basados en el acoplamiento de estructuras delta dopados tipo p y n.

Descriptores: Estrcutura electronica;´ pozos delta dopados pn; aproximacion´ Thomas-Fermi.

PACS: 73.21.Fg; 73.61.Ey

1. Introduction In the present paper we study the electron and hole level structure in pnDD quantum wells in GaAs within the effec- The everyday improvement in doping and growth techniques tive mass envelope function approximation. The Thomas- allows the realization of semiconductor structures with delta- Fermi approximation has been applied to obtain analytical function doping profiles. In particular p- and n-type delta- expressions for the bending of the conduction and valence doped structures present electron and hole confinement due band. The electronic structure for electrons and holes is to the bending of the conduction and valence band caused by analyzed as a function of the distance between the accep- the ionized impurities. Systems in which p- and n-type delta- tor and donor layers. We paid special attention to the one- doped (pnDD) layers are used as conductive channel and dimensional Bohr radius (1DBR) of the electron and hole punchthrough stopper (ALD-FET), respectively, are of spe- ground levels and its variation with the acceptor and donor cial interest due to the high transconductance observed [1,2]. concentration. With the data presented here, it is possible to pnDD quantum wells are also important because they find the minimum distance between the donor and acceptor form the basic elements in delta-doped nipi superlattices layers to avoid a reduction in the potential barrier whether for (also called sawtooth superlattices), which have the follow- electrons or holes, which is essential in parallel and vertical ing features: tunable band gaps at energies below that of the transport devices based on these structures. bulk, long carrier recombination life times, strong nonlin- earity and improved optical properties over homogeneously 2. Theoretical framework doped nipi superlattices [3]. All these properties make the sawtooth superlattices attractive for technological applica- The scheme of calculation starts by modeling the conduction tions in light emitting diodes and lasers [4], infrared detec- and valence band profile, within the local density Thomas- tors and modulators, [5] and optical computing [6]. The nipi Fermi approximation. The outcome of this approach is an an- delta-doping structures are also ideal systems to investigate alytical expression for the one-dimensional potential energy interesting phenomena like the disorder effects caused by the function describing the band bending in a single delta-doped random distribution of impurities [7, 8], the band edge relax- quantum well (SDDQW) [13, 14]. ation effects [9,10], high field electron transport [11] and the For a single n-type delta-doped quantum well centered in metal-insulator transition [12]. z = a, the confining potential can be written as [13], 110 I. RODRIGUEZ-VARGAS AND L.M. GAGGERO-SAGER

2 ∗ ∗ β VHn(z) − µn = − 4 , (1) (β |z − a| + z0n) with µ ¶ 2 β3 1/5 β = and z0n = au . 15π πn2D ∗ ∗ ∗ ∗ VHn = VHn/Ryn and µ = µ/Ryn are given in units of the effective Bohr radius and effective Rydberg,

2 ∗ ²r~ ∗ e a0n = ∗ 2 and Ryn = ∗ . m e 2²ra0n In the case of a single p-type delta-doped quantum well centered in z = b, the conﬁning potential can be written FIGURE 1. Potential proﬁle of pnDD quantum wells in GaAs as [14], for various distances between the donor and acceptor layers. The 12 −2 α2 donor and acceptor densities are: n2D = 1 × 10 cm and ∗ ∗ 13 −2 VHp(z) − µp = 4 , (2) p2D = 1 × 10 cm . (α |z − b| + z0p) with

3/2 µ 3 ¶1/5 2ma α α = and z0p = au . 15π πp2D ∗ ∗ ∗ ∗ VHp = VHp/Ryp and µp = µ/Ryp are given in units of the effective Bohr radius and effective Rydberg,

2 ∗ ²r~ ∗ e a0p = ∗ 2 and Ryp = ∗ . mhhe 2²ra0p The next step is the construction of the pnDD potential well, so the potential can be written as

2 2 ∗ β α VH (z) = − 4 + 4 (3) (β |z| + z0n) (α |z − l| + z0p) where a and b have been taken as 0 and l. FIGURE 2. Electron and hole levels versus the interlayer distance The latter equation summarized the model for the con- of pnDD quantum wells in GaAs. The impurity density in the donor duction and valence band bending proﬁle. Instead of carrying and acceptor layers is the same as that in Fig. 1. out numerically troublesome self-consistent calculations, we simply solve the Schrodinger-like¨ effective mass equations at the zone center k = 0, thus obtaining the corresponding subband electron and hole levels.

3. Results and Discussion

The input parameters for the delta-doped quantum wells are: ∗ ∗ ∗ me=0.067m0, mhh=0.52m0, mlh=0.087m0, ²r=12.5 and 12 14 −2 10 ≤ p2D ≤ 10 cm . In Fig. 1 the potential profiles of the pnDD quantum wells are depicted for various interwell distances for fixed impurity 12 −2 density in the delta-doped layers, n2D = 1 × 10 cm and 13 −2 p2D = 1 × 10 cm . We can see that the width of the confining region increases as the potential barrier is moved away from the n-type delta-doped layer. This is reflected in the number of electron and hole states and their effective con- FIGURE 3. Electron 1DBR versus the donor density (n-type) in a finement, see Fig. 2. delta-doped quantum well.

Rev. Mex. F´ıs. S 53 (7) (2007) 109–111 ELECTRONIC STRUCTURE OF PN DELTA-DOPED QUANTUM WELLS IN GAAS 111

of heavy holes, p-type delta-doped wells. The range of acceptor densities considered is larger compared to electrons due to the experimental interest. The main difference between Figs. 3 and 4 comes from the heavy hole effective mass that is approximately six times greater than that of the electron, which affects primarily the hole conﬁnement and reﬂects on a reduction of the 1DBR. This reduction is nearly of 50 % as compared to electrons for the same impurity range. Figures 3 and 4 also permit us to discern the distance between the donor and acceptor layers to avoid or generate a shrink in the potential barrier whether for electrons or holes, which is essential in devices based on pnDD quantum wells, ALD-FET and sawtooth superlattices.

FIGURE 4. Heavy hole 1DBR versus the acceptor density (p-type) 4. Conclusions in a delta-doped quantum well. In summary, we have presented the electronic structure in ∗1D In Fig. 3 we present the 1DBR (a0 ) for the electron pnDD quantum wells within the envelope function approxi- ground level as a function of the donor density. The impu- mation. The Thomas-Fermi approach has been used to model rity concentrations managed are in the experimental range of the bending of the conduction and valence band for electrons ˚ interest. The 1DBR changes from 47 to 22 A for the lower and holes, respectively. The electronic structure is analyzed and the higher donor densities considered, which constitute a as a function of the distance between the donor and accep- reduction of 50 %. In general, the 1DBR depends on the ef- tor layers. Special attention is paid to the 1DBR, where we fective mass, the relative dielectric constant, the confinement find the optimum interwell distances to avoid a reduction in of the carriers and the impurity density. In the case of delta- the potential barriers of the pnDD quantum wells, which is doped wells, 1DBR is inversely proportional to the density essential in electronic devices based on these structures. as it is seen in Figs. 3 and 4. At first sight, we can expect that the dependency of 1DBR on the impurity density to be proportional to some power law of the latter, and inversely Acknowledgments proportional to the confinement as in the 3D case. However, in delta-doped wells, the confinement of the carriers is pro- This work was supported by the Consejo Nacional de Ciencia 2/3 y Tecnolog´ıa through grant 51504-CB. I.R.-V. and J.M.-M. portional to the impurity density (E ≈ n2D ); as a conse- quence, the 1DBR inversely depends on the impurity density. are specially indebted to the Secretar´ıa General de la UAZ This behavior resembles what happened in atoms in which and Consejo Zacatecano de Ciencia y Tecnolog´ıa, and S.J.V. the atomic number takes the role of the impurity density. In to the Universidad Autonoma´ del Estado de Morelos for sup- Fig. 4 we depict the same as in Fig. 3 but for the ground level port

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