Trions and biexcitons in a nanowire

Trions and biexcitons in a nanowire

R. Ya. Kezerashvili,  Z. S. Machavariani, B. Beradze, and T. Tchelidze Physics Department, New York City College of Technology, The City University of New York, Brooklyn, New York 11201, USA
Department of Exact and Natural Sciences, Tbilisi State University, 0179 Tbilisi, Georgia
July 13, 2019
Abstract

A theory of the trion and biexciton in a nanowire (NW) in the framework of the effective-mass model using the Born-Oppenheimer approximation is presented. We consider the formation of trions and biexcitons under the action of both the lateral confinement and the localization potential. The analytical expressions for the binding energy and eigenfunctions of the trion and biexciton are obtained and expressed by means of matrix elements of the effective one-dimensional cusp-type Coulomb potentials whose parameters are determined self-consistently by employing the same eigenfunctions of the confined electron and hole states. Our calculations for the ZnO/ZnMgO, CdSe/ZnS and CdSe/CdS core/shell cylindrical shaped NWs show that the trion and biexciton binding energy in NWs are size-dependent and for the same input parameters the biexciton binding energy in NWs is always larger than the binding energy of the trion. The trion and biexciton remain stable in CdSe/ZnS NW with the increase of the dielectric shell, while in ZnO/ZnMgO NW they become unstable when the surrounding dielectric shell exceeds 2.5 nm and 2 nm for each, respectively. The associative ionization of biexciton antibonding states into trion bonding states that leads to the formation of trions is studied. Based on the results for size dependence of biexciton binding energy and probability associative ionization an optimal radius for optoelectronic application NW is suggested.

I Introduction

The optical properties of quantum nanostructures have been increasingly investigated over the past decades. This is connected to the fact that the system characteristics that govern optical response, such as electronic level structure, oscillator strength, Coulomb interaction between charged carriers and electron-phonon interaction dramatically changes with size variation at nanometer scale Science2013 (); Nanophotonics (). Consequently, the issue of tuning of optical response by means of size and shape control has become important. Excitons and excitonic complexes in quantum nanostructures have been one of the hot topics since the early days of quantum nanostructures Efros (). The reason for this is that on the one hand excitons are main intrinsic emitters in short wavelength region and therefore, optimization of excitonic emission is very important for emitting device fabrication. On the other hand, the investigation of excitons and their complexes can provide deep insight into the peculiarities of inter particle interaction at low dimensions. Multi-particle states, like single particle ones, are strongly affected by space and dielectric confinement Giblin (). Therefore, the new possibilities of controlling their characteristics appear in quantum nanostructures.

In the late 1950s Lampert L () predicted the existence of charged and neutral exciton complexes formed when an electron in a conduction band or a hole in a valence band is bound to a neutral exciton or two single excitons are correlated. This idea gave rise to many publications in the 60s and the 70s in bulk materials (see, for example, the works Stebe1974 (); Stebe1977 (); Stebe1987 () and citations therein). The binding energies of the exciton complexes are very small in bulk at room temperature, but they are substantially enhanced in structures of reduced dimensionality. Theoretical calculations performed at the end of the 1980s Stebe1989 () predicted a considerable (up to tenfold) increase of the trion binding energy in quantum well heterostructures compared with bulk. Trions were first observed in quantum wells (QW) QW1 () in 1993 and shortly thereafter in GaAs-AlGaAs quantum wells QW2 (); QW3 (); QW4 ().

In the last two decades these complexes have been the subject of an extensive theoretical and experimental studies in QW Hawrylak1995 (); Varga1999 (); RPV2000 (); Stebe20001 (); Varga2001 (); Dacal2002 (); Lozovik2004 (); BonitzCD (), quantum dots (QD) OgawaTakagahara (); Klimov1994 (); Patton (); Szafran2005 (); Peng2010 (); Kaniber (); Jovanov (), quantum nanotubes  Pedersen (); Pedersen2 (); Kammerlander (); Pedersen3 (); DifMC (); Matsunaga (); Santos (); Bondarev (); Watanabe (); Pedersen4 (); ColombierPRL (); Yuma (); Bondarev2 (); Bondarev3 () and quantum wires QW1 Baars (); Zimmermann (); Tsu2001 (); 1DEXP (); ExQW (); PeetersPRB05 (); PeetersFewBody2006 (); PeetersPRB2008 (); Piermarocchi (); QW 2012 Phys Lett (). We cited these articles, but the recent literature on the subject is not limited by them. The reduced dimensionality considerably increases the binding energy of trions and biexcitons and, thus facilitates the formation of these exciton complexes in semiconductor quantum wells, quantum wires with different confinement geometry, quantum nanotubes and quantum dots.

Whilst in bulk materials excitonic characteristics are defined by dielectric constant and effective masses of electrons and holes, in quantum nanostructures, new controlling parameters, such as size, shape and material distribution profile become crucial. According to numerous investigations the main trend that has been revealed is that with size reduction binding energy of excitons and excitonic complexes, as well as their decay probability are strongly enhanced Chia (); Nanoscale2014 (); Pokutnii (), which seems very promising, and is crucial for applications in optoelectronics Wang2012 (), such as light-emitting diodes Cheng2014 (), photovoltaics Furchi2014 (), and phototransistors Lembke2013 (), to cite just a few.

However, the situation is not always so straightforward; e.g. in Ref. Sarkar () it was reported that with decreasing size biexciton binding energy was dropped to zero in InAs/AlAs quantum dots. Much more challenging is the fact, that, with size reduction, some nonradiative processes also become more intensive. In nanocrystal quantum dots/rods, nonradiative carrier losses are dominated by surface trapping and multiparticle Auger relaxation Mikhailovsky (). Nonradiative processes connected to surface traps increase with the size reduction due to the increase in surface-to-volume ratio. However, these processes can be suppressed by using core-shell structures for passivating surface traps. Auger recombination is a nonradiative process in which the electron–hole recombination energy is transferred to a third particle Kurzmann (). Auger recombination has a relatively low efficiency in bulk semiconductors. However, Auger decay is greatly enhanced in quantum-confined systems, which is mainly connected to the increase of particle wave functions overlap and breakdown of translational symmetry Grim (). Auger recombination is effective at high excitation regime when number of excited carrier pairs exceeds one per quantum dot Nozik () and when multi-exciton effects are intensive as well. This is why trions and biexcitons still are subjects of extensive investigations KezFBS2017 (); Falko (); Falko2 (); KezerashviliTrion2018 (); Kez2018 ().

Recent studies of one-dimensional (1D) nanostructures as nanotubes Santos (); Yuma (); Bondarev2 (); Bondarev3 () and nanowires PeetersPRB2008 (); Efros 2010 (); QW 2012 Phys Lett (); Xiong 2013 ()  show that the trion and biexciton binding energy depend on the electron to hole mass ratio and the geometric characteristics of a nanostructure. Although the exciton complexes like trions in solid state physics are very similar to the few-body bound systems in atomic and nuclear physics there is a major difference related to band effects, which make the effective masses of the electrons and holes smaller than the bare electron mass, and screening effects, resulting from the host lattice, which make the Coulomb force much weaker than in atomic systems.

The majority of recent research is conducted with core/shell NWs in which the emitting core is overcoated with a thin layer of a semiconductor material that has much higher band gap which enhances carrier localization and suppresses Auger recombination. In group II–IV materials ZnO/ZnMgO, CdSe/ZnS and CdSe/CdS are prime examples of such core/shell NWs which are the subject of the present study.

In this paper, we present a theoretical approach to study a trion and a biexciton in a NW in the framework of the effective-mass model using Born-Oppenheimer approximation. We consider the formation of trions and biexciton under the action of both the lateral confinement and the localization potential.  Our approach allows us to obtain analytical expressions for the binding energy and eigenfunctions of the trion and biexciton. The corresponding energies are expressed by means of matrix elements of effective one-dimensional cusp-type Coulomb potentials whose parameters are determined self-consistently by employing the same eigenfunctions of the confined electron and hole states. We calculated the exciton, trion and biexciton binding energies in ZnO/ZnMgO, CdSe/ZnS and CdSe/CdS core/shell NWs of cylindrical shape and their dependence on NW radius. Because of high exciton binding energy ZnO/ZnMgO, CdSe/ZnS and CdSe/CdS are a very good candidates for achieving efficient excitonic laser action at room temperature and Auger recombination is expected to be reduced for this class of materials Nitrid Semiconductor () as well as for elongated core-shell structures Klimov3 (). Having chosen the system where Auger recombination is expected to be low, we aimed to find size dependence of trion and biexciton binding energies in order to define size/composition optimal for effective lasing. We also investigated the process of autoionization during which the biexciton transforms to a trion. We propose that as numbers of photo generated electrons and holes are the same, this process should be the main source of generation of trions, which in its turn are suspected to be the main reason of photoluminescence intermediacy and efficiency drop Alicia2009 ().

The paper is organized in the following way. In Sec. II, we provide the theoretical model for a trion and biexciton in a core-shell NW and obtain single-particle wavefunctions for confined electrons and holes, which allows us to find the effective electronhole, holehole and electronelectron interactions in 1D. Using these interactions in the framework of the effective-mass model we solve the one-dimensional Schrödinger equations within the fixed center approximation for an exciton, trion and biexciton and obtain the analytical expressions for the binding energies and wavefunctions for these exciton complexes in Subsections II.3, II.4, and II.5, respectively. In Sec. III the results of calculations and discussion are presented. Conclusions follow in Sec. IV.

Ii Theoretical formalism

ii.1 Setting the model

We consider a formation of trions and biexcitons in a core-shell nanowire. The system represents a cylindrical core of radius , surrounded by a cylindrical  shell of thickness The trion and biexciton are a three- and four-body system and the corresponding Schrödinger equations cannot be solved analytically, while a solution of the Faddeev equations for a few-body system widely used in nuclear and atomic physics is a challenging task, which involves complex numerical computations. To overcome this difficulty, for the core-shell NW we consider the theoretical model that is based on two assumptions:

•  Coulomb interaction is assumed to be decisive only along the NW axis and in radial direction motion of carriers is governed by the strong lateral confinement perpendicular to the NW.

•  Heavy holes on the average, move appreciably more slowly than the electrons, which allows one to use the Born-Oppenheimer approximation and solve Schrödinger equation for fixed interhole distances.

To solve the problem of a positive trion (two holes and one electron) and biexciton (two holes and two electrons) laterally confined in a quantum NW we adopt the Born-Oppenheimer approximation, very well known in physics of molecules Atkins Friedman (). The Born-Oppenheimer approximation BornOppen () accounts for a difference in masses of light and heavy particles and assumes that the light particles can respond almost instantly to heavy particles’ displacement Heitler and London (). The best example for a such system is a hydrogen molecular ion H and a hydrogen molecule H, which as a positively charged trion and biexciton consist from two heavy and one light and two heavy and two light particles, respectively Atkins Friedman (); Davidov (). Therefore, instead of solving the three-body Schrödinger equation for all particle simultaneously one can treat heavy particles as motionless and solve the Schrödinger equation for a definite position of heavy particles, taking the interparticle separation as a parameter . After that calculations are carried out for different .

The application of the Born-Oppenheimer approximation naturally separates the calculation into the following steps: due to the strong lateral confinement perpendicular to the NW one first calculates the two-dimensional (2D) energies and wave functions of the electron and hole, while neglecting the Coulomb interaction between them. Therefore, the fast transverse motions of charge carriers remain independent of each other. Next, using these wave functions of transverse electron and hole motion, one can average the three-dimensional (3D) Coulomb potential to a 1D Coulomb interaction between the charge carriers along the NW. Finally, after an appropriate modeling of these potentials by functions which depend on the distance between the charge carriers, one should find energies and wavefuctions for a trion or a biexciton for each fixed position of the holes by solving the corresponding reduced 1D Schrödinger equations. 

Let us assume that the conduction and highest valence bands are decoupled, which is a reasonable approximation for the below considered type of a NW because of the large direct band gap of the ZnO/ZnMgO, CdSe/ZnS and CdSe/CdS materials. The full Hamiltonian for a three- or four-particle excitonic complexes in a confinement within the single-band effective-mass approximation can be written as

(1)

where and are the position and effective mass of the -th particle, correspondingly, is a number of particle that is equal to 3 and 4 for the trion and biexciton, respectively, and are confinement potentials for each of particles. For example, one can consider a lateral confinement potential shown in Fig.  1a for a core-shell NW. In the Hamiltonian (1) describes the Coulomb interactions between particles with the electric charge in a medium with a dielectric constant and is

(2)

in case of the positively charged trion and

(3)

in case of the biexciton.

Figure 1: (Color online) (a) A lateral confinement potential. (b) Schematic of the two 1D excitons sharing the same electron in a nanorwire to form a positive trion. Schematically the electron in the field of two one-dimensional cusp-type Coulomb potentials of the holes is shown in Fig. 1b.

Below we consider the formation of trions and biexcitons in a core-shell NW. The system represents a cylindrical core of radius , surrounded by a shell of thickness , as schematically shown in Fig. 1b. To find the binding energies, calculate the energy spectra of trions and biexcitons and find eigenfunctions needed for calculation their optical properties, we must solve the Schrödinger equation with Hamiltonian (1). We assume that the lateral confinement is strong, so that only the lowest subband for the electron and hole is occupied. This assumption allows a reduction of the Schrödinger equation to an effective one-dimensional form. Assuming strong lateral confinement, we are allowed to separate the motion from the lateral motion in the plane. In other words, we assume that the Coulomb interaction does not affect the motion of the particles, so that we can separate the electron and hole motion confined in the lateral direction from the electron-hole relative motion. Therefore, the envelope function for a trion or biexciton can be approximated as

(4)

where is the envelope function describing the electron-hole relative motion in the trion or biexciton in the axis along the NW and is the radial single-particle wave functions for an electron (hole) for the lateral motion in the plane, which due to the axial symmetry of the system depend on cylindrical coordinates for each of particles. Here and below the Cartesian coordinates, wavefunctions and masses of the electron (hole) are denoted with the suffix and , respectively.

By averaging the Schrödinger equation with Hamiltonian (1) by using function (4) after the separation of variables we obtain

(5)

which is the Schrödinger equation of single-particle states for electrons and holes confined in a NW and the following Hamiltonian

(6)

in the case of trion, while in the case of biexciton the corresponding Hamiltonian reads

(7)

In Hamiltonians (6) and (7) , and are effective electronhole, holehole and electronelectron interactions that are defined in Appendix A. Conceptually a positively charged 1D trion can be considered as two ground-state 1D excitons sharing the same electron to form a positive trion state and 1D biexciton as two 1D excitons that are sharing the same two interacting electrons to form a biexciton bound state. A schematic of this concept for the trion is depicted in Fig. 1b.

Thus, finding the eigenfunctions and eigenenergies for the trion and biexciton for the Hamiltonian (1) is reduced to the solution of the Schrödinger equation for single-particle states of the electrons and the hole confined in a NW (5) and a solution of the Schrödinger equation with the Hamiltonian (6) for the trion and Hamiltonian (7) for the biexciton with the effective potentials (28) - (30). The effective potentials (28) - (30) between charged particles are free from the singularity of the bare Coulomb potential at the origin as a result of averaging with the lateral subband wave functions. However, these effective potentials can be given only numerically. Following Ogawa and Takagahara OgawaTakagahara () the effective potentials (28) - (30) for 1D semiconductors usually are modeled by effective one-dimensional cusp-type Coulomb potentials approximated by the first order rational function , where is interparticle distance in -direction and and are fitting parameters. These parameters are defined by wave functions that are the solutions of (5), which in its turn, depend on a NW geometry, particles’ effective masses, and band offsets . Consequently, they vary with the NW radius and are different for the electronhole, holehole and electronelectron interactions (see Appendix A).

ii.2 Single-particle states for confined electrons and holes

Let us solve the Schrödinger equation (5) for single-particle states of the electron and the hole confined in a core-shell NW. We assume that confinement potentials for the electron and hole depicted in Fig. 1a have a stepped well shape, are different and have an axial symmetry

(8)

where is the conduction (valence) band offset between core and shell materials and, obviously, it is different for electrons and holes due to the different bands that leads to their different masses. Proceeding from the cylindrical symmetry the solution of Eq. (5) should have the form

(9)

By substituting (9) into (5) we obtain

(10)

The solutions of this equation are Bessel functions: within the core we have the Bessel function of first kind , which is finite at , while within the shell the solution is a linear combination of the Bessel functions of second type and

(11)

where

(12)

The and coefficients in (11) are defined from the condition of continuity and smoothness of the wavefunction at the boundary, and from the condition of its orthonormality. Consequently, the energy of radial motion of electrons (holes) are defined by means of equation Having found , we can define the in-plane energy and the wave functions of a confined non-interacting electron and hole, respectively. The solutions of a single-particle radial Schrödinger equation (5) are used to average three-dimensional potentials in (28) - (30) over in-plane motion. As a result, one-dimensional effective potentials are obtained for the electronhole, holehole and electronelectron interactions that then are parameterized in the form  as shown in Appendix A.

ii.3 Exciton in a nanowire

In the ideal limit the 1D electron-hole system with a perfect confinement, can be treated as a ”one-dimensional hydrogen-atom” problem in the framework of the effective-mass approximation. As a first step let us consider the interacting electron and hole in a 1D NW and model an exciton by using an effective one-dimensional cusp-type Coulomb potential. Following OgawaTakagahara () one can write the equation that describes relative motion of the electron and hole bound with the cusp-type Coulomb interaction in one-dimensional radially confined NW in the form

(13)

Here is the reduced effective mass of electron-hole pair, are the fitting parameters for the effective electronhole one-dimensional cusp-type Coulomb potentials obtained through the parametrization of Eqs. (28), is the relative electron-hole motion coordinate, is the binding energy of the electron and hole that form the exciton and is the corresponding wavefunction. Eq. (13) has the same form as the equation for one-dimensional hydrogen atom studied by Loudon Loudon1 (). One can introduce the following notations

(14)

and reduce (13) to the Whittaker’s equation

(15)

The solution of (15) is the Whittaker function as shown in Refs. Loudon1 (); Loudon2 (). The value of which defines  and , is determined by the boundary condition stating that for even states the derivative of wave function at must turn to zero

(16)

The energy of exciton in a quantum NW is a sum of energies of radial motion of electron and hole , and the energy of the exciton

ii.4 Positive trion in a nanowire

Following Atkins Friedman () for positively charged trions bound by the effective one-dimensional cusp-type Coulomb potentials the Hamiltonian (6) in the Born-Oppenheimer approximation can be written as

(17)

where is electron-hole reduced mass,  is distance between two holes, which are assumed to be motionless at , and and are the fitting parameters for the effective electronhole and holehole one-dimensional cusp-type Coulomb potentials obtained through the parametrization of Eqs. (28) and (29), correspondingly. Schematically the electron in the field of two 1D cusp-type Coulomb potentials is shown in Fig. 1b.

For solution of the Schrödinger equation for the trion with Hamiltonian (17) we use the method of linear combination of atomic orbitals (LCAO) Atkins Friedman (). In the framework of  the LCAO method the eigenfunction of trion is presented as a linear combination of single exciton wave functions centered at

(18)

where and are the solutions of the Schrödinger equation

(19)

Close to the hole located at the wavefunction will resemble a single electron orbital, while the wavefunction will represent the electron orbital near the hole located at . Thus, the linear combination (18) represents both cases. The states and are degenerate due to symmetry. Following the well-known perturbation theory for the degenerate states Landau (), one can obtain the normalized wavefunctions for the first two states

(20)

where is an overlap integral. The wavefunction corresponds to the localization of the electron density between holes (bonding orbital) and the accumulation of electron density in the interhole region is simulated due to the constructive interference that takes place between the two electron waves centred on neighboring holes, while the wavefunction  has a node between the holes (antibonding orbital). Trion energies corresponding to these states are:

(21)
(22)

where the expressions for and are given in Appendix B. The last term in (21) and (22) is the energy of interaction between two holes

(23)

Thus, to find the dependence of binding energy of the trion on interhole separation  one needs to evaluate the relevant matrix elements and and calculate the interaction energy between two holes (23).

ii.5 Biexciton in a nanowire

Now let us consider a 1D biexciton for which the model of the hydrogen molecule H is used. Following Davidov () for the biexciton, where two electrons and two holes are bound via the effective one-dimensional cusp-type Coulomb potentials, the Hamiltonian (7) in the Born-Oppenheimer approximation can be written as

(24)

where and and are the fitting parameters for the effective one-dimensional cusp-type Coulomb potentials in the Hamiltonian (7) for the electronelectron interaction obtained from (30), while the other fitting parameters are already defined in Subsec. II.4. The wave function of the biexciton can be constructed by means of single exciton wave functions again. If the electron–electron and hole-hole interactions are ignored one can use as an approximation for the biexciton wavefunction. However, the symmetry of the wavefunction has to be taken into account. Following Davidov () we construct symmetric and antisymmetric wave functions coinciding to antiparallel (singlet state) and parallel orientation (triplet state) of spins of electrons as:

(25)

The energies of bonding symmetric and antibonding antisymmetric states in the first order perturbation theory are found as mean values of (24). The corresponding energies for these states are:

(26)
(27)

where and are given in Appendix C.

The binding energy of biexciton is defined as maximal value of , and when this difference is positive biexciton is a stable system.

Iii
Results of calculations and discussion

Our calculations are based on the assumption that the Coulomb interaction strength in the radial direction is much weaker than the lateral confinement effect and the Coulomb interaction is decisive only along the z-axis and is presented by the 1D cusp-type Coulomb potentials for the electronhole, holehole and electronelectron interactions. The Born-Oppenheimer approximation successfully used in the physics of molecules assumes that the ratio of mass of the atomic nuclei and electrons is large enough. In the case of positively charged trions and biexcitons the ratio of hole-electron mass is much less than the proton-electron mass ratio, which makes questionable not only the validity of the Born-Oppenheimer approximation, but also the existence of bound states. In Aven Prener () it is stated that exciton bound to neutral/ionized donor for which models of H and H are also used, the binding energy sharply increases when the hole-electron mass ratio varies from 1 to 3. For II-VI semiconductors, where the mass ratio varies between 35, for calculation of trion and biexciton states the Born-Oppenheimer approximation is often used AP (). Even in the case when satisfactory values are obtained for the binding energy of the two- and three-dimensional biexcitons Lozovik ().

We calculate the trion and biexciton binding energies in ZnO/ZnMgO, CdSe/ZnS and CdSe/CdS core/shell quantum structures of a cylindrical shape. The calculations are performed for the set of parameters listed in Table 1. Here the effect of the differences in the effective masses in core and shell material is not taken into account. For ZnO/ZnMgO and CdSe/CdS structures this difference is not significant. In CdSe/ZnS structure the barrier height is very high, penetration of carriers into the barrier is not substantial. Because of high exciton binding energy ZnO/ZnMgO is a very good candidate for achieving efficient excitonic laser action at room temperature. ZnO/ZnMgO is a wide bandgap (3.4 eV) material, has sufficient hole-electron mass ratio. The most important part of the ZnO/ZnMgO band structure, which is the bottom of the conduction band and the top of the valence bands in the vicinity of the point, can be well described within the effective-mass approximation Zn Oxide ().

CdSe/ZnS and CdSe/CdS are also promising direct-bandgap II–VI semiconductors active in the visible range, with potential applications in electronics and optoelectronics Shalev2017 (). Nanostructures on the base of CdSe are one of the extensively investigated low dimensional semiconductor structures with the aim of application field-effect transistors, photodetectors, and light-emitting diodes Zhou2007 (). The bandgap of CdSe is lower than that of ZnO 1.74 eV Trindade1997 (). Exciton binding energy in bulk CdSe also is significantly lower than that in bulk ZnO – 0.13 eV Meulenberg2009 (). However, investigations show possibility of effective control/enhancement of stability of excitonic complexes in nanostrudtures Meulenberg2009 (); Hawrylak2010 ().

eV eV
ZnO/ZnMgO 0.24 Kuzmina () 0.86 Kuzmina () 0.37 Janotti () 0.31 Janotti () 8.13
CdSe/ZnS 0.13 Aven Prener () 0.45 Aven Prener () 1.2 Reiss () 0.7 Reiss () 10.2
CdSe/CdS 0.13 0.45 0.30 Reiss () 0.44 Reiss () 10.2

[0pt]

Table 1: Input parameters for ZnO/ZnMgO and CdSe/ZnS NWs. and are the mass of the electron and hole, respectively, and are the lateral confinement potentials for a conduction and valence band offset between core and shell materials, respectively, and is dielectric constant. is the mass of free electron.
Figure 2: The dependence of trion energy in ZnO/ZnMgO and CdSe/ZnS NWs on interhole distance for different radii of a NW. The thickness of the surrounding dielectric shell is nm.
Figure 3: The dependence of biexciton energy in ZnO/ZnMgO and CdSe/ZnS NWs on interhole distance for different radii of a NW. The thickness of the surrounding dielectric shell is nm.

As a first step we calculate the effective interactions (28) - (30) by averaging over the electron and hole wavefunctions, which reduce the 3D Coulomb potential to a one-dimensional potential that depends only on the coordinate of two particle relative motion. The corresponding computational modeling allows one to find the numerical values of the fitting parameters and for the 1D cusp-type Coulomb potentials. Once these constants are known one can use them as the input parameters and evaluate  and  given in (31) and (32), and using Eqs. (21) and (21) find the trion energy for given interhole separation . In the case of biexciton using the same matrix elements  and  given in (31) and (32), and the matrix elements and given in (33) and (34), one can find and defined in (35) and determine the biexciton energy by mean of Eqs. (26) and (27).

We study the dependence of the trion and biexciton binding energy on the interhole distance, the NW radius and the thickness of the surrounding dielectric shell using Eqs. (21) and (22), and (26) and (27), respectively. When the trion or biexciton energies in ZnO/ZnMgO, CdSe/ZnS and CdSe/CdS as a function of interhole distance exhibit minimums at the particular interhole distances and for the particular NW radius it can be a signature of possible existence of the bound state of the trion or biexciton. If the minimum lies below the energy of a separated exciton and a hole its indicates the formation of the trion, while if the minimum energy is below the energy of two separated excitons, it is an indication of the biexciton formation.

The results of calculations for the trion energies in ZnO/ZnMgO and CdSe/ZnS NWs as a function of interhole distance  are presented in Fig. 2. The calculation is performed for different radii of a NW. The energy curves vary with the interhole distance. The upper branches of curves correspond to the antibonding orbital for the antisymmetric state, while the lower branches of curves correspond to the bonding orbital for the symmetric state. The steep rise dependence in energy at  for both ZnO/ZnMgO and CdSe/ZnS NWs is largely due to the increase in the hole-hole potential energy as the two holes are brought close together. When a curve exhibits minimum and its energy is below the energy of a separated exciton and a hole one can confirm the existence of a bound trion. For all values of a NW radius the typical behavior of a symmetric bonding state for ZnO/ZnMgO NW is observed with minimum at some particular hole-hole distance, while one can see the minimum of the energy for the antisymmetric state only for the NW with the smallest radius nm. The minimums are not sharp, nevertheless bound states of trions exist. In the case of CdSe/ZnS the curves for the trion energies as a function of interhole distance exhibit minimum for the symmetric bonding states as well as for the antibonding antisymmetric states. The minimums are pronounced for antisymmetric states for the radius of NW nm and nm, however, there is no minimum in the antibonding state for the other considered values of NW radii.

Let us now present results of calculations for the biexciton in ZnO/ZnMgO and CdSe/ZnS NWs. Using Eqs. (26) and (27) the dependence of biexciton energies on interhole distance for different NW radii are calculated and presented in Fig. 3. The lower branches of curves correspond to the bonding symmetric state, while the upper branches of curves correspond to the antibonding antisymmetric state. In the antisymmetric state when the holes approach one another the energy increases monotonically for all considered radii of the NW. It follows from Fig. 3 for the symmetric state when the holes approach one another the energies are lowered down to a particular distance after which the energies rise steeply for when the distance is further decreased. The minimums of energy for the symmetric state are sharply pronounced and lie below the energy of two separated excitons for all considered NW radii. The minimums are much deeper for ZnO/ZnMgO NW and occur when nm, while in the case of CdSe/ZnS the minimums appear at nm. The confinement potentials for the electrons and holes in ZnO/ZnMgO and CdSe/ZnS NWs are significantly different. The comparison of Fig. 2a and 2b and 3a and 3b illustrates the effect of the confinement on the energy dependence on interhole distance for trions and biexciton, respectively. The dependences of trion and biexciton energies on interhole distance for CdSe/CdS are different compared to CdSe/CdS due to the smaller confinement potentials and have similarity to that for ZnO/ZnMgO NW.

Figure 4: Dependence of the binding energies of excitonic complexes on the radius of a nanowire in ZnO/ZnMgO () (a), CdSe/ZnS () and CdSe/CdS ( ) (b) NWs, respectively, The thickness of the surrounding dielectric shell is nm.

The results of our calculations demonstrate an appreciable dependence of the exciton, trion and biexciton binding energy on the radius of NW. In Fig. 4 are presented dependence of the binding energies of the exciton, trion and biexciton in ZnO/ZnMgO, CdSe/ZnS and CdSe/CdS on the radius of NW. We perform calculations for the lateral confinement potentials and for a conduction and valence band offset between core and shell materials presented in Table 1. The comparison of the binding energies for the exciton, trion and biexciton shows that for the same hole to electron mass ratio the binding energy of biexciton is larger than for the trion and smaller than for the exciton in all NWs. Moreover, for the same input parameters the exciton, trion and biexciton have the maximum binding energy in ZnO/ZnMgO and CdSe/ZnS NW for the same radius nm. In CdSe/CdS for the set of the lateral confinement potentials meV and =0.44 meV the maximum of the binding energies for these excitonic complexes are pronounced at nm for the exciton and biexciton, while the trion has the maximum binding energy for about larger radius. The increase of the confinement potentials in CdSe/ZnS qualitatively and quantitatively changes the binding energy of the excitonic complexes: i) the binding energies of excitonic complexes are increased; ii) they are bound for a smaller NW radius, nm; iii) while the biexciton binding energy is monotonically decreases, the maximum of binding energy for the exciton and trion occurs at the radii of NW nm and nm, respectively. Our calculations show that the biexciton binding energy exceeds that for the trion rather significantly at a small NWs radius ( nm in ZnO/ZnMgO and CdSe/CdS, and nm in CdSe/ZnS) and with the increase of the radius the difference between binding energies decreases. Moreover, in both CdSe/ZnS and CdSe/CdS NWs the biexciton-to-trion binding energy ratio is greater than unity, decreasing with the NW radius increase. The same tendency one can observe for in ZnO/ZnMgO NW when the radius nm.

Figure 5: Comparison of the dependence of the binding energies of the exciton (a), trion (b) and biexciton (c) in ZnO/ZnMgO () and CdSe/ZnS () NWs on radius of a NW. The thickness of the surrounding dielectric shell is nm.
Figure 6: Dependence of the exciton, trion and biexciton binding energy in ZnO/ZnMgO (a) and CdSe/ZnS (b) on the thickness of the surrounding dielectric shell.

Now let us present, discuss and compare the results for the binding energies of the exciton, trion and biexciton. In Fig. 5a the single exciton binding energies in ZnO/ZnMgO and CdSe/ZnS calculated by means of Eqs. (16) and (14) are presented. The binding energy varies between 56.375.3 meV when the NW radius varies between 1.03.5 nm, and has maximum at about 1.5 nm NW radius and at nm the exciton binding energy drops to that of bulk materials. The same behavior of exciton binding energy is observed for ZnO/ZnMgO quantum wells Sun2002 (), with maximum at about 2 nm QW width. The dependence of trion binding energy on the NW radius is depicted in Fig. 5b. The maximum of the binding energy 12.8 meV is obtained for the NW radius nm. With the further decrease of wire radius the binding energy sharply drops. With increasing of the wire radius from 2.0 nm to 3.5 nm the binding energy decreases again.

The dependences of the binding energy of biexciton in ZnO/ZnMgO and CdSe/ZnS on the NW radius is plotted in Fig. 5c. The biexciton has the maximum binding energy 21 meV for the wire radius nm. Interestingly, in Ref Chia () biexcitons were investigated in the ZnO/ZnMgO/ZnMgO multiple quantum wells. Experimentally determined binding energies for biexcitons vary between 17.5 to 30.9 meV depending on the width of quantum well and the maximum value was obtained for 2 nm width and binding energy drops to that of bulk materials.

We also study the influence of the thickness of dielectric shell on the binding energy of excitonic complexes. In Fig. 6 are presented the dependence of the binding energy of the exciton, trion and biexciton binding energy on the thickness of the surrounding dielectric shell in ZnO/ZnMgO and CdSe/ZnS. We have calculated the binding energy as a function of barrier width for the exciton, trion and biexciton in ZnO/ZnMgO and CdSe/ZnS NWs. One can conclude that in the case of CdSe/ZnS NW all excitonic complexes remain stable with the increase of dielectric shell thickness, while in ZnO/ZnMgO NW biexcitons become unstable when the surrounding dielectric shell exceeds 2 nm. The trion remains stable for the thickness of the dielectric shell nm. The stability of excitonic complexes in CdSe/ZnS NW can be explained by the high lateral confinement potentials for the electron and hole. The relatively low potential barriers for the lateral confinement of electrons and holes in ZnO/ZnMgO NW allow the penetration of the corresponding electronic wave functions in the surrounding dielectric shell area which leads to the strong decrease of the binding energies of the exciton and biexciton when nm. Our calculations demonstrate that the size of the core of the NW has stronger influence on the binding energy of trions and biexcitons compared to the thickness of interfacial alloying. Let us mention that observation for CdSe/CdS Nanoscale2014 () suggested that the size of the surrounding shell has equal or less influence on Auger suppression compared to the radius of the core of NW.

Figure 7: The dependence of the trion and biexciton energies on the interhole separation for NWs of 1.03.5 nm radius. The notations and indicate bounding states energy for the trion and biexciton, respectively, while and represents trions and biexciton antibonding states. indicates the interhole distance of the crossing of the biexciton antibonding state energy curve with the trion bonding state energy curve. Results for ZnO/ZnMgO NW.
Figure 8: The dependence of the trion and biexciton energies on the interhole separation for NWs of 1.03.5 nm radius. The notations and indicate bounding states energy for the trion and biexciton, respectively, while and represents trions and biexciton antibonding states. indicates the interhole distance of the crossing of the biexciton antibonding state energy curve with the trion bonding state energy curve. Results for CdSe/ZnS NW.

Our calculations show a radius dependence of characteristics of excitons, trions and biexcitons and enhancement of their binding with size reduction as expected in nanostructures. With the radius reduction quantitative change of energy level alignment is also observed as one can see in Fig. 7. In Fig. 7 biexciton and trion energies for the same radius of wire are depicted in the same figure to investigate associative ionization (AI), the process in which two excitons interact to produce a free electron and a bound trion. Indeed, a biexciton in the antibonding state can dissociate into a trion bonding state and an electron. When two excitons approach one another along Born-Oppenheimer potential energy curve the ionization occurs only after this reactant pair enters a region of the plane in which the bound initial electronic state becomes embedded in the continuum associated with the final state, trion-electron. One can see that for the radii nm, nm, nm and nm of wire the energy of the antibonding biexciton state sharply increases and the energy curve crosses the trion bonding state energy curve at the interhole distance . At distances the biexciton energy in the antibonding state becomes larger compared to the trion energy. Therefore, it is possible to transition from the biexciton antibonding state to the trion bonding state with release of an electron - e.a., the associative ionization. During photoexcitation the antibonding states of biexciton are created at different interhole distances . Such states will survive if holes go away from each other on the distance larger than before AI, and then stabilize (dissociate). Quantitatively the probability of surviving can be estimated by means of . The larger , the smaller the probability would be. Fig. 7 shows that the probability should drop with the NW size reduction. In Fig. 8 are presented the crossings of energy curves for the biexciton in the antibonding states and trion in the bonding states energies for the same radii for CdSe/ZnS NW. These results also demonstrate a possible transition from the biexciton antibonding state to the trion bonding state with release of an electron at some particular interhole distances .

Taking into account the behavior of the biexciton binding energy with the NW size variation, we can propose that there exists an optimal radius of elongated ZnO/ZnMgO quantum wire, for which biexciton binding energy is still larger than the bulk value, whilst associative ionization into trion state (which in its turn has strong tendency to the Auger decay) is weakened. This radius ranges between nm. At the same time for the elongated CdSe/ZnS quantum wire this range is nm due to the stronger lateral confinement.

Iv Conclusions

In summary, we presented the theoretical description of the trion and biexciton in a NW in the framework of the effective-mass model using Born-Oppenheimer approximation and considered both the lateral confinement and the localization potential. The analytical expressions for the binding energy and wavefunctions are obtained and expressed by means of matrix elements of the effective one-dimensional cusp-type Coulomb potentials which parameters are determined self-consistently by employing the same eigenfunctions of the confined electron and hole states. We investigated biexcitons and trions in ZnO/ZnMgO, CdSe/ZnS and CdSe/CdS quantum NWs of a cylindrical shape and study the dependence of their binding energies on the radius of the NW. It is found that for the same input parameters the biexciton binding energy in NWs is always larger than binding energy of the trion. For the same input parameters the exciton, trion and biexciton have the maximum binding energy for the same radius of ZnO/ZnMgO NW, while the trion has the maximum binding energy for about larger radius of a NW. We found an appreciable dependence of the trion binding energy on the radius of the quantum wire. It was revealed that a radius reduction up to 1.5 nm enhances binding energy of the exciton, trions and biexciton in ZnO/ZnMgO NW, while for the biexciton in CdSe/CdS quantum NW the maximum binding energy is obtained for the thinner NW with 1 nm radius. For very thin NWs binding energies of excitonic complexes decrease. The excitonic complexes remain stable in CdSe/ZnS NW with the increase of the dielectric shell, while in ZnO/ZnMgO NW the trion and biexciton become unstable when the surrounding dielectric shell exceeds 2.5 nm and 2 nm, respectively. We suggest  the mechanism of formation of the trion via associative ionization of a biexciton. As for probability of the associative ionization of biexciton into vulnerable to Auger decay trion states, it continually decreases with increasing the radius of NW. This leads us to the conclusion that nm radius of NW should be optimal for optoelectronic application at high excitation intensity.

Appendix A Effective interactions

The effective electronhole, holehole and electronelectron interactions that are defined as

(28)
(29)
(30)

where is the Coulomb potential. As a  result of averaging of over the electron and hole wave functions in the lateral directions, the effective potentials (28) - (30) are free from the singularity of the bare Coulomb potential at the origin. The modeling of the potentials (28) - (30) with the first order rational function , where is interparticle distance in -direction and and are fitting parameters, provides a slight modification of the long-range Coulomb potential by a cusp-type Coulomb potential. The values of the fitting parameters and depend on a dielectric constant of a NW material, a NW core radius and a shell thickness . In our particular case related to the ZnO/ZnMgO, CdSe/ZnS and CdSe/CdS quantum NWs the dependence of the fitting parameters and on the NW radius are presented in Fig. 9. From Fig. 9a one can conclude that the fitting parameters and that indicate the strength of the cusp-type Coulomb electron-electron, and hole-hole interactions converge with the increase of NW radius. The fitting parameters and presented in Fig. 9b display the linear dependence on NW radius. The slops of these dependences are almost the same for the same NW, but there are the significant differences for the slopes for ZnO/ZnMgO, CdSe/ZnS and CdSe/CdS NW. The variation of and parameters with a nanowire radius in ZnO/ZnMgO, CdSe/ZnS and CdSe/CdS NW can be explained the following way: these parameters are obtained by means of averaging of 3D Coulomb potential with in-plane (radial) wave functions, which describe the lateral confinement of electrons and holes. After averaging procedure Coulomb potentials depend only on the distance between carriers. should be a measure of their average lateral separation. Now it is obvious that must increase i) when barrier is decreased and/or ii) when carrier effective masses are reduced.

Figure 9: (Color online) Dependence of the fitting parameters and (a) and and (b) on the core radius for ZnO/ZnMgO, CdSe/ZnS and CdSe/CdS NWs. The shell thickness nm. Dotted curves - for electron-electron interaction; solid curves - for electron-hole interaction; dashed curves - for hole-hole interaction.

Appendix B Matrix elements for  and  

The value for  and  are given by the following matrix elements

(31)
(32)

The matrix elements (31) can be treated as the total energy of the cusp-type Coulomb interaction between the hole located at with the electron density or the hole located at with the electron density Numerically, these two matrix elements are equal to one another. The matrix elements (32) correspond to the energy of the cusp-type Coulomb interaction of the overlap charge density localized around the hole located at with the hole. By symmetry, the energy of interaction of the overlap charge density with the hole located at has the same value. The matrix elements  and  both depend on the interhole separation and are calculated for each fixed value of interhole distance.

Appendix C Matrix elements for  and  

For the biexciton, the values for  and  are the same as for the trion and are given by (31) and (32), while and are the matrix elements as defined below

(33)
(34)

The matrix element is the repulsion energy of the charge density  of electron 1 localized around the hole located at with the charge density  of electron 2 localized around the hole located at , when the repulsion occurs via the cusp-type Coulomb interaction. The matrix element presents the repulsion energy between the electrons due to the cusp-type Coulomb interaction, which is connected with the correlation in the motion of the electrons arising from the antisymmetrization of the wavefunctions in accordance with the Pauli principle. The matrix elements and are functions of the distance between the holes.

Once the value of  and and and are known one can find the value of   and  as

(35)

and obtain the biexciton energy as a function of the interhole distance using Eqs. (26) and (27).

Acknowledgements.
The authors are thankful to T. Kereselidze for the useful discussion. This work is supported by joint research grant from Shota Rustavely Georgian National Science Foundation (# 04/04) and Science and Technology center in Ukraine (# 6207). R.Ya.K. gratefully acknowledges support from the PSC – City University of New York Award # 61188-00 49.

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