Using a variational approach, we have investigated the effects of the magnetic field, the impurity position, and the nitrogen and indium concentrations on impurity binding energy in a Ga_{1−x}In_{
x
}N_{
y
}As_{1−y}/GaAs quantum well. Our calculations have revealed the dependence of impurity binding on the applied magnetic field, the impurity position, and the nitrogen and indium concentrations.

Keywords

ImpuritiesQuantum wellDilute nitride

Review

Background

Over the past decade, the GaInNAs-based quantum-well structures have emerged as a subject of considerable theoretical and experimental research interest due to their very unique physical properties and due to a wide range of possible device applications. GaInNAs exhibits interesting new properties and differs considerably from the conventional III to V alloys. Significant changes occur in the electronic band structure compare with GaInAs with incorporation of small amounts of nitrogen into GaInAs. These include a large redshift of the bandgap, a highly nonlinear pressure dependence of the bandgap, an increase in the electron effective mass, and the N-induced formation of new bands [1–10]. This new material has received considerable attention due to the growing interest in its basic physical properties. Shan et al. showed that interaction between the conduction band and narrow resonant band formed by nitrogen states in GaInNAs alloys leads to a splitting of conduction band into sub-bands and a reduction of the fundamental bandgap [11]. Fan et al. have investigated the electronic structures of strained Ga_{1−x}In_{
x
}N_{
y
}As_{1−y}/GaAs quantum wells [12]. Hetterrich et al. investigated the electronic states in strained Ga_{0.62}In_{0.38}N_{0.015}As_{0.985}/GaAs multiple quantum-well structures [13]. Pan et al. have investigated the optical transitions in Ga_{1−x}In_{
x
}N_{
y
}As_{1−y}/GaAs single and multiple quantum wells using photovoltaic measurements at room temperature [14]. Several studies have been done on detailed optical characterization of Ga_{1−x}In_{
x
}N_{
y
}As_{1−y}. These papers include the temperature dependence of photoluminescence, absorption spectrum, and low-temperature photoluminescence [15–19].

There are many studies associated with the hydrogenic binding of an electron to a donor impurity which is confined within low-dimensional heterostructures [20–25]. The understanding of the electronic and optical properties of impurities in such systems is important because the optical and transport properties of devices made from these materials are strongly affected by the presence of shallow impurities. Also, it is well known that a magnetic field considerably affects the optical and electronic properties of semiconductors. Thus, the effects of magnetic field on the impurity binding energy are a very important problem [26–28]. However, up to now, to the best of our knowledge, no theoretical studies have been focused on impurity binding energies in single GaInNAs/GaAs quantum well (QW) under the magnetic field.

In this paper, using a variational technique within the effective mass approximation, we have investigated the effects of the magnetic field, the impurity position, and the nitrogen (N) and indium (In) concentrations on impurity binding energy in a Ga_{1 − x}In_{
x
}N_{
y
}As_{1 − y}/GaAs QW.

Theoretical overview

The growth axis is defined to be along the z-axis and takes the magnetic field to be applied to the z-axis, i.e., B = (0, 0, B). We choose a vector potential A in written form $\text{A=}\frac{1}{2}\left(\frac{-{B}_{y}}{2},\frac{{B}_{x}}{2},0\right)$ to describe the applied magnetic field. Within the framework of the effective mass approximation, the Hamiltonian of a hydrogenic donor impurity in a Ga_{1 − x}In_{
x
}N_{
y
}As_{1 − y}/GaAs quantum well in the presence of magnetic fields can be written as follows:

where m^{*} is the effective mass, e is the elementary charge, p_{e} is the momentum of the electron, ε_{
o
} is the dielectric constant of the system, r_{
e
} and r_{
i
} are the electron and impurity atom positions, and V(z) is the confinement potential of the electron in the z-direction. Using cylindrical coordinates (x = ρ cos ϕ, y = ρ cos ϕ z = z) , we obtain the Hamiltonian as follows:

where the term $\rho =\sqrt{{\left({x}_{e}-{x}_{i}\right)}^{2}+{\left({y}_{e}-{y}_{i}\right)}^{2}}$ is the distance between the electron and impurity in the (x and y) plane. The location of the hydrogenic donor in the structure is given as (0, 0, z_{
i
}). The electron confining potential, V(z), is taken as follows:

with L as the well width, and V_{0} is the conduction band offset which is taken to be 80% of the total discontinuity between the bandgap of GaAs and Ga_{1 − x}In_{
x
}N_{
y
}As_{1 − y} grown on GaAs [13].

The bandgap energy and electron effective mass of Ga_{1 − x}In_{
x
}N_{
y
}As_{1 − y}/GaAs is calculated using the band anti-crossing model (BAC). The electron effective mass of Ga_{1 − x}In_{
x
}N_{
y
}As_{1 − y}/GaAs as predicted by the BAC model is given by the following [29, 30]:

where y is the N composition in Ga_{1 − x}In_{
x
}N_{
y
}As_{1 − y}; E_{C0} is the energy in the absence of N; E_{
C
}, E_{
N
}, and V_{
NC
} are the bandgap energies of InGaAs at Г point, the energy of the isolated N level in the InGaAs host material, and the coefficient describing the coupling strength between E_{
N
} and the InGaAs conduction band, respectively. The band structure parameters used in this study are listed in Table 1[31–36].

Table 1

Parameters of the binary compounds used for the calculation

Material

GaAs^{a}

InAs^{a}

GaN^{b}

InN^{b}

Electron effective mass m^{*}(m_{0})

0.067

0.026

0.15

0.14

Dielectric constant

12.53

14.55

10.69^{c}

7.46^{d}

Energy gap E_{g} (eV)

1.420

0.417

3.299

1.94

^{a}Vurgaftman et al. [33], ^{b}Vurgaftman and Meyer [34], ^{c}Chow et al. [35], and ^{d}Gavrilenko and Wu [36].

Using the variational method, it is possible to associate a trial wave function, which is an approximated eigenfunction of the Hamiltonian described in Equation 2. The trial wave function is given by the following:

where λ is the variational parameter, ψ(z) is the wave function of the donor electron which is exactly obtained from the Schrödinger equation in the z-direction without the impurity, and ϕ(ρ,λ) is the wave function in the (x and y) plane, and it is given by the following:

The ground state impurity energy is evaluated by minimizing the expectation value of the Hamiltonian in Equation 2 with respect to λ. The ground state donor binding energy is calculated using the following equation:

where E_{
z
} is the confinement ground state energy of the electron.

Results and discussion

In this paper, we have theoretically investigated the effects of the magnetic field, the impurity position, and the nitrogen and indium concentrations on impurity binding energy in a Ga_{1 − x}In_{
x
}N_{
y
}As_{1 − y}/GaAs QW.

The variation of the impurity binding energy as a function of the well width for different values of the magnetic field and for two different nitrogen concentrations (y = 0, y = 0.005, and y = 0.01) is given in Figure 1. As seen in this figure, when the magnetic field increases, the binding energy also increases. Magnetic field strength gives an additional parabolic magnetic confinement term. Under this additional magnetic confinement, the probability of the electron and impurity atom being on the same plane increases, and therefore, the binding energy increases. In this figure, one sees that when the well width increases, the impurity binding energies increase until they reach a maximum value, and they bind to decrease. This behavior is related to the change of the electron and impurity atom confinement in the quantum well. It should be noted that the donor binding energy increases with nitrogen concentration. As the nitrogen concentration increases, both the electron effective mass and the band discontinuity increase, while dielectric constant decreases. In this case, the coulombic interaction between the electron and a donor impurity increases, and therefore, the impurity binding energy increases.

In Figure 2, we have presented the variation of the binding energy for a donor impurity placed on the center of a Ga_{1−x}In_{
x
}N_{
y
}As_{1−y}/GaAs quantum well as a function of the well width for different values of the magnetic field and for two different indium concentrations (× = 0.15 and × = 0.30). It is seen that the impurity binding energy decreases with the well width for the considered values of the indium concentration. This behavior can be explained as follows: the band discontinuity and the dielectric constant increase with the increasing indium concentration; on the other side, the electron effective mass decreases. Thus, by increasing the indium concentration, particles get to be more energetic and they can penetrate into the potential barriers easily where the wave functions of the particles reflect their three-dimensional character, and the probability of finding the electron and hole in the same plane weakens. This behavior weakens the Coulomb interaction between the electron and impurity atom, and the binding energy begins to decrease.

The calculated impurity binding energy for a hydrogenic donor in Ga_{1−x}In_{
x
}N_{
y
}As_{1−y}/GaAs quantum well as a function of the magnetic field (B) is given in Figure 3 for different impurity positions (z_{i}) and well width (L =100 Å). As expected, when the magnetic field increases, the binding energy also increases. It can be clearly seen that the impurity binding energy decreases as the position of the impurity approaches the potential barriers. This is because the electron impurity distance increases when the position of the impurity approaches the potential barriers. This leads to the weakening of the electrostatic interaction, therefore to the decreasing in value of impurity binding energy.

Conclusions

As a summary, we have investigated the effects of the magnetic field, the impurity position, and the nitrogen and indium concentrations on the impurity binding energy in a Ga_{1−x}In_{
x
}N_{
y
}As_{1−y}/GaAs quantum well in this study. The calculations were performed within the effective mass approximation. We have found the impurity binding energy on the magnetic field, the impurity position, and the nitrogen and indium concentrations. This case gives a new degree of freedom in device applications, such as near-infrared electro-absorption modulators and quantum well infrared detectors, and all optical switches. We hope that our results will stimulate further investigations of the related physics as well as device applications of group III nitrides.

Declarations

Acknowledgments

This work supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK) for a research grant COST 109T650 and was partially supported by the Scientific Research Project Fund of Cumhuriyet University under the project number F-360. MEMR acknowledges support from Mexican CONACYT through Research Grant CB-2008-101777 and 2011-2012 Sabbatical Grant No. 180636. He is also grateful to the Universidad de Antioquia for hospitality during his sabbatical stay. This research was partially supported by Colombian Agencies: CODI-Universidad de Antioquia (Estrategia de Sostenibilidad 2013-2014 de la Universidad de Antioquia) and Facultad de Ciencias Exactas y Naturales-Universidad de Antioquia (CAD-exclusive dedication project (2012-2013). The authors thank CONACYT (Mexico) and COLCIENCIAS (Colombia) for support under the 2012-2013 Bilateral Agreement "Estudio de propiedades opticas, electronicas y de transporte en sistemas de baja dimension basados en carbono y semiconductores III-V: efectos de campos externos,temperatura y presion hidrostatica". The work was developed with the help of CENAPAD-SP, Brazil.

Authors’ Affiliations

(1)

Physics Department, Cumhuriyet University

(2)

Physics Department, Dokuz Eylül University

(3)

Instituto de Fisica, Universidad de Antioquia

(4)

Faculty of Sciences, Morelos State University

References

Ribeiro F, Latge A: Impurities in a quantum dot: a comparative study. Phys Rev B 1994, 50: 4913. 10.1103/PhysRevB.50.4913View Article

Porras N, Perez S, Latge A: Binding energies and density of impurity states in spherical GaAs‐(Ga, Al)As quantum dots. J Appl Phys 1993, 74: 7624. 10.1063/1.354943View Article

Movilla J, Planelles J: Off-centering of hydrogenic impurities in quantum dots. Phys Rev B 2005, 71: 075319.View Article

Miller A, Chemla D, Schmitt S: Electroabsorption of highly confined systems: theory of the quantum–confined Franz–Keldysh effect in semiconductor quantum wires and dots. Appl Phys Lett 1988, 52: 2154. 10.1063/1.99562View Article

Nomura S, Kobayashi T: Clearly resolved exciton peaks in CdSxSe1−x microcrystallites by modulation spectroscopy. Solid State Commun 1990, 73: 425. 10.1016/0038-1098(90)90044-CView Article

Porras-Montenegro N, Perez-Merchancano S: Hydrogenic impurities in GaAs-(Ga, Al)As quantum dots. Phys Rev B 1992, 46: 9780. 10.1103/PhysRevB.46.9780View Article

Zhu J: Exact solutions for hydrogenic donor states in a spherically rectangular quantum well. Phys Rev B 1989, 39: 8780. 10.1103/PhysRevB.39.8780View Article

Zhu J, Xiong J, Gu B: Confined electron and hydrogenic donor states in a spherical quantum dot of GaAs-Ga_{1−x}Al_{
x
}As. Phys Rev B 1990, 41: 6001. 10.1103/PhysRevB.41.6001View Article

Pozina G, Ivanov I, Monemar B, Thordson J, Andersson T: Properties of molecular-beam epitaxy-grown GaNAs from optical spectroscopy. J Appl Phys 1998, 84: 3830. 10.1063/1.368562View Article

Bi W, Tu C: Bowing parameter of the band-gap energy of GaN_{
x
}As_{1−x}. Appl Phys Lett 1997, 70: 1608. 10.1063/1.118630View Article

Shan W, Walukiewicz W, Ager J, Haller E, Geisz J, Friedman D, Olson J, Kurtz S: Band anticrossing in GaInNAs alloys. Phys Rev Lett 1999, 82: 1221. 10.1103/PhysRevLett.82.1221View Article

Fan W, Yoon S: Electronic band structures of GaInNAs/GaAs compressive strained quantum wells. J Appl Phys 2001, 90: 843. 10.1063/1.1378336View Article

Hetterich M, Dawson M, Egorov A, Bernklau D, Riechert H: Electronic states and band alignment in GalnNAs/GaAs quantum-well structures with low nitrogen content. Applied Physics Lett 2000, 76: 1030. 10.1063/1.125928View Article

Pan Z, Li L, Lin Y, Sun B, Jiang D, Ge W: Conduction band offset and electron effective mass in GaInNAs/GaAs quantum-well structures with low nitrogen concentration. Appl Phys Lett 2001, 78: 2217. 10.1063/1.1362335View Article

Tournie E, Pinault M, Vezian S, Massies J, Tottereau O: Long wavelength GaInNAs/GaAs quantum-well heterostructures grown by solid-source molecular-beam epitaxy Appl. Phys Lett 2000, 77: 2189.

Polimeni A, Capizzi M, Geddo M, Fischer M, Reinhardt M, Forchel A: Effect of temperature on the optical properties of (InGa)(AsN)/GaAs single quantum wells. Appl Phys Lett 2000, 77: 2870. 10.1063/1.1320849View Article

Perlin P, Winiewski P, Skierbiszewski C, Suski T, Kaminska E, Subramanya S, Weber E, Mars D, Walukiewicz W: Solar cells with 1.0 eV band gap, lattice matched to GaAs. Appl Phys Lett 2000, 76: 1279. 10.1063/1.126008View Article

Pinault M, Tournie E: On the origin of carrier localization in Ga InNAs/GaAs quantum wells. Appl Phys Lett 2001, 78: 1562. 10.1063/1.1354153View Article

Shirakata S, Kondow M, Kitatani T: Photoluminescence and photoreflectance of GaInNAs single quantum wells. Appl Phys Lett 2001, 79: 54. 10.1063/1.1374221View Article

Ungan F, Kasapoglu E, Sari H, Somken I: Dependence of impurity binding energy on nitrogen and indium concentrations for shallow donors in a GaInNAs/GaAs quantum well under intense laser field. The European Physical Journal B 2011, 82: 313. 10.1140/epjb/e2011-20081-6View Article

Sali A, Fliyou M, Loumrhari H: Photoionization of shallow donor impurities in finite-barrier quantum-well wires. Physica B 1997, 233: 196. 10.1016/S0921-4526(97)00305-0View Article

Duque C, Kasapoglu E, Sakiroglu S, Sari H, Sökmen I: Intense laser effects on donor impurity in a cylindrical single and vertically coupled quantum dots under combined effects of hydrostatic pressure and applied electric field. Appl Surf Sci 2010, 256: 7406. 10.1016/j.apsusc.2010.05.081View Article

Ruihaoi W, Wenfang X: Optical absorption of a hydrogenic impurity in a disc-shaped quantum dot. Curr Appl Phys 2010, 10: 757. 10.1016/j.cap.2009.09.010View Article

Kasapoglu E, Yesilgul U, Sari H, Sökmen I: The effect of hydrostatic pressure on the photoionization cross-section and binding energy of impurities in quantum-well wire under the electric field. Physica B 2005, 368: 76. 10.1016/j.physb.2005.06.039View Article

Yesilgul U, Kasapoglu E, Sari H, Sökmen I: Photoionization cross-section and binding energy of shallow donor impurities in GaInNAs/GaAs quantum wires. Solid State Communications 2011, 151: 1175. 10.1016/j.ssc.2011.04.029View Article

Sari H, Sökmen I, Yesilgul U: Photoionization of donor impurities in quantum wires in a magnetic field. J Phys D: Appl Phys 2004, 37: 674. 10.1088/0022-3727/37/5/005View Article

Terzis A, Baskoutas S: Binding energy of donor states in a GaAs quantum dot: effect of electric and magnetic field. J Phys Conf Ser 2005, 10: 77.View Article

Kasapoglu E, Sari H, Sökmen I: Binding energy of impurity states in an inverse parabolic quantum well under magnetic field. Physica B 2007, 390: 216. 10.1016/j.physb.2006.08.016View Article

Skierbiszewski C: Experimental studies of the conduction-band structure of GaInNAs alloys. Semicond Sci Technol 2002, 17: 803. 10.1088/0268-1242/17/8/309View Article

Ng ST, Fan W, Dang Y, Yoon S: Comparison of electronic band structure and optical transparency conditions of In_{
x
}Ga_{1−x}As_{1−y}N_{
y
}/GaAs quantum wells calculated by 10-band, 8-band, and 6-band k·p models. Phys Rev B 2005, 72: 115341.View Article

Chuang SL: Physics of Optoelectronic Devices. New York: Wiley; 1995.

Vurgaftman I, Meyer J, Ram-Mohan L: Band parameters for III–V compound semiconductors and their alloys. J Appl Phys 2001, 89: 5815. 10.1063/1.1368156View Article

Vurgaftman I, Meyer J: Band parameters for nitrogen-containing semiconductors. J Appl Phys 2003, 94: 3675. 10.1063/1.1600519View Article

Chow W, Wright A, Nelson J: Theoretical study of room temperature optical gain in GaN strained quantum wells. Appl Phys Lett 1996, 68: 296. 10.1063/1.116064View Article

Gavrilenko V, Wu R: Linear and nonlinear optical properties of group-III nitrides. Phys Rev B 2000, 61: 2632. 10.1103/PhysRevB.61.2632View Article

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