GaAlAs double quantum well under the external fields

Accepted Manuscript Binding energy and optical absorption of donor impurity states in “12-6” tuned GaAs/ GaAlAs double quantum well under the external fields E. Kasapoglu, S. Sakiroglu, H. Sari, I. Sökmen, C.A. Duque PII:

S0921-4526(18)30682-3

DOI:

https://doi.org/10.1016/j.physb.2018.11.006

Reference:

PHYSB 311141

To appear in:

Physica B: Physics of Condensed Matter

Received Date: 24 July 2018 Revised Date:

1 November 2018

Accepted Date: 3 November 2018

Please cite this article as: E. Kasapoglu, S. Sakiroglu, H. Sari, I. Sökmen, C.A. Duque, Binding energy and optical absorption of donor impurity states in “12-6” tuned GaAs/GaAlAs double quantum well under the external fields, Physica B: Physics of Condensed Matter (2018), doi: https://doi.org/10.1016/ j.physb.2018.11.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Binding energy and optical absorption of donor impurity states in ”12-6” tuned GaAs/GaAlAs double quantum well under the external fields E. Kasapoglu1 , S. Sakiroglu2 , H. Sari3 , I. S¨okmen2 , and C. A. Duque4∗ 1

Faculty of Science, Department of Physics, Cumhuriyet University, 58140 Sivas, Turkey 2 Dokuz Eylul University, Faculty of Science, Physics Department, 35390 Izmir, Turkey 3 Faculty of Education, Department of Mathematical and Natural Science Education, Cumhuriyet University, 58140 Sivas, Turkey and 4 Grupo de Materia Condensada-UdeA, Instituto de F´ısica, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia UdeA, Calle 70 No. 52-21, Medell´ın, Colombia (Dated: November 5, 2018)

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We have investigated the binding energies of the s-symmetric ground and first excited shallow donor impurity states and the total 1s → 2s absorption coefficient, including the first and third order corrections, in ”12-6” tuned GaAs/GaAlAs double quantum well as a function of the impurity position, size of the structure, and the electric and magnetic field intensities. The obtained results show that the electronic and optical properties of tuned GaAs/GaAlAs double quantum well can be adjustable by an appropriate choice of the sample geometry, material parameters and applied external fields which will lead to new potential applications in optoelectronics.

magnetic field.

I.

INTRODUCTION

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Keywords: Double quantum well; Absorption coefficient; Donor impurity; Electric and

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The quasi-dimensional behavior of confined carriers in low-dimensional semiconductor systems -such as quantum wells (QW), quantum well-wires, quantum dots, and superlattices- is one of the most important achievements that have been reported in the last 10 years from both theoretical and experimental points of view. Growth techniques have emerged with a high degree of precision and control over the characteristics of the systems to be formed. The calculation techniques, both analytical and numerical and those where quasi-analytical and numerical procedures are combined, have also developed enormously during the last four decades. Confining the carriers in quasi-two-dimensional systems allow sub-bands of energy associated with the discrete spectrum arising from confinement in one direction to emerge. This type of systems is classified as 2D type and among them we can mention the QWs and the superlattices. In the case of QWs, since the 80s, with the pioneering work of Bastard, intensive studies have been done that consider different forms of the structure, different compositions into the well and barrier materials, presence of impurities both donors and acceptors, excitonic complexes and excitonic systems linked to impurities, among others [1–5]. Likewise, in such a class of systems, multiple effects such as

∗ Corresponding

author (C. A. Duque): [email protected]

stationary electric and magnetic fields, hydrostatic pressure, temperature, etc. have been reported in the literature. The manipulation of the discrete spectrum allows obtaining a set of states and/or sub-bands of energy that can be adjusted at discretion to obtain a convenient response to certain studies or purposes [6–11]. Among the different types of quantum wells we can mention the particular cases of: i ) the well with square barriers, or simply abrupt [12, 13], ii ) the wells with parabolic confinement that can have finite spatial extension, with a confining potential given by the band-offset between well and barrier materials, and iii ) parabolic quantum wells that consider only one type of material, which extends infinitely in space. The latter simulates a one-dimensional harmonic oscillator for the minima of the energy sub-bands that appear from the 2D movement of the confined carriers [14, 15]. There are also other types of quantum wells, a little more exotic in their shapes and geometries, which have been proposed to explain the different experimental results. Among them we can mention: the half parabolic QW [16], graded QWs [17], V-shaped QWs [18], and inverse parabolic QWs [19]. Structures mentioned above play a major role in research and application of artificial semiconductor structures since the electronic properties of these lowdimensional systems can be used for a wide range of applications which permit the design of advanced devices. The existence of bound states of two interacting particles has an important effect on the optical properties of these structures. The spatial confinement of the

2

12

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ACCEPTED MANUSCRIPT are widely used in molecular and material physics, they carriers leads to a strong increase of the binding energy have been extensively studied recently. The use of interand the oscillator strength as compared to the bulk semiatomic interaction potential functions is quite realistic conductors. As a consequence of quantum confinement and can be the source of QW-systems with much more effects, optical absorption lines in semiconductor strucadequate forms and characteristics than those that have tures shift towards the blue and provide the information been reported so far in the literature. For this reason, a about the confinement of carriers. The electronic and tuned GaAs/GaAlAs double quantum well (DQW) which optical properties of the low dimensional systems such is consisting of the asymmetric wells such as Tietz-Hua as QWs, quantum-well wires, and quantum dots can be and Morse QWs has been chosen so that it can be more adjustable by an appropriate choice of the sample gerealistic. There is no evidence in the literature of studometry, material parameters and applied external fields ies where the opto-electronic properties associated with which will lead to new potential applications in optoelecconfined carriers are analyzed in coupled well systems of tronics. So, electron states bounded to hydrogenic imthe Tietz-Hua and Morse type. The well potential will purities and the binding energies have been extensively be called ”12-6” like potential, which corresponds to a investigated for different semiconductor heterostructures even function that gives rise to two asymmetric quantum in order to understand their dependence on the external wells coupled by a central barrier whose height can be fields, material, and geometry parameters [20–27]. chosen according to the band-offset of the materials that 1 are usually used to form the regions of wells and barriers V(z) = 4 V [(z/ ) -(z/ ) ] in the different heterostructures. In Fig. 1 is depicted the 12-6 like DQW used in this work to simulate two coupled QWs which spatial extension is 2 σ and depth V0 . Note that in the case of very small value of σ, the 2 0 model system corresponds to two δ-like coupled QWs. 6

V

0

-1 -1.0

-0.5

0.0

z (

0.5

)

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0

V (V )

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0

1.0

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FIG. 1: (color online)z-dependent profile of a ”12-6” coupled double quantum well studied in this work. The extension of the structure is 2 σ and the depth V0 . Horizontal axis is in units of σ and vertical axis in units of V0 . Note that the z = 0 point corresponds to the symmetry center of the structure.

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The knowledge of the realistic profile of the confinement potential is necessary for a theoretical description of the electronic properties of QWs and what is more important to a fabrication of the well-designed devices, where transport properties are relevant physical characteristics. It is a strategic matter the appropriate choice of the QW profile when the main goal is to manipulate the electronic structure (wave functions and energy levels). This is known as wave function engineering. Wave function engineering that refers to a unique ability to adjust the spatial distribution of the electron wave function in quantum nanostructures through the control of their growth, geometry and external fields provides a degree of freedom for manipulating the physical properties of nanocrystals by confining the charge carriers in a specific domain of the structure. In contrast to real atoms, quantum nanostructures allow us to design new devices and flexible control over the confinement potential which gives rise to wave function engineering [28, 29]. For this reason, the QWs [30–33] formed by the use of interaction potential functions such as Tietz-Hua and Morse which

In GaAs bulk material the electron affinity is 4.07 eV, which corresponds to GaAs-vacuum interface [34]. Such kind of almost infinite confinement potential can be modeled via the barrier at the left and right extremes in Fig. 1. The central barrier in Fig. 1 corresponds to a Al0.3 Ga0.7 As material, whereas the minima at z = ±0.9 σ can be associated to GaAs. The variation of the confinement potential in the range −1 ≤ z ≤ +1 represents continuous changes of the Aluminum concentration from 0 to 0.3. We note that the spatial extension 2 σ can be achieved experimentally, e.g., applying a gate voltage between the quantum wells [35]. Thus, the effect of such gate voltages is described in the present potential profile by changing the spatial extension 2 σ. By considering all the points above, in the present work, and for such kind of ”12-6” double heterostructure, the binding energies of the ground and first excited donor impurities and also the total absorption coefficient for the transitions between the related impurity states have been investigated. The calculations are made in the effective mass approximation with parabolic conduction bands and using a variational procedure, with one-parameter hydrogeniclike trial wave functions, to deal with the electronimpurity Coulomb correlation for a donor center which has been placed along the growth direction of the heterostructure. The organization of the paper is as follows: Section II contains the presentation of the theoretical framework, in Section III, we discuss the obtained results and, in Section IV, we give our conclusions.

3 II.

ACCEPTED THEORETICAL MODEL

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The research problem in this work corresponds to an electron confined in a double QW of type 12-6, as shown in Fig. 1, under the presence of a donor impurity located along the growth direction. The system will be under the simultaneous influence of an electric field, directed along the growth direction, and a magnetic field perpendicular to the previous one, that is, parallel to the interfaces. Taking into account that the reference system is located at the symmetry point of the structure and considering the z-axis along the growth direction, the electric field is → − → − F =Fu bz and the magnetic field is, for example, B = Bu by . Under these considerations, and making use of the effective mass approximation, the Hamiltonian for the electron-hole system is given by

V (z) = 4 V0



eB c

2 

c ~ kx z+ eB

EP

,

− → − − Φ(→ r ) = exp(k⊥ · → ρ ) ϕ(z) ,

(2)

(3)

− → where k⊥ = (kx , ky ) and ϕ(z) satisfies the following 1D eigenvalues equation:

2 #

)

+ V (z) + |e| F z

Because we are interested in the donor impurity states associated to the minimum − → of the first noncorrelated electron subband, we take k⊥ = (kx , ky ) = (0, 0). Under such restriction, the Eq. (4) becomes:   ~2 d2 e2 B 2 z 2 − + + V (z) + |e| F z ϕ(z) = Ez ϕ(z) . 2 m∗ dz 2 2 m∗ c2 (5) The study under consideration involves the two states of donor impurity with lower energies and that preserve azimuthal symmetry. This last aspect, given that the presence of the potential barriers along the z-direction breaks the spherical symmetry of the 1s-like and 2s-like states. In order to obtain the wave functions and the corresponding energies of the 1s-like and 2s-like states, the well-known variational method is used, in which case hydrogenoid-type test functions will be used. Considering in the Hamiltonian of Eq. (1) the electrostatic interaction term between impurity and electron, and taking − → into account the restriction of k⊥ = 0, the total wave

AC C

σ

SC

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d2 −~ + dz 2 2

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"

−

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i2 − → 1 h→ e→ e2 − − p + A ( r ) + V (z) + |e| F z − . (1) 2 m∗ c εr → − In the previous equation, A = B z u bx is the vector po− tential, → p is the 3D linear momentum operator, m∗ is the electron effective mass, e is the elementary electron charge, ε is the static dielectric constant, and r = p ρ2 + (z − zi )2 is the distance between the electron and impurity. The 3D spatial points (x, y, z) and (0, 0, zi ) are the coordinates of electron and impurity, respectively, p with ρ = x2 + y 2 being the in-plane electron-impurity distance. V (z) is the type ”12-6” confinement potential,

1 2 m∗

σ

 z 6 

where, V0 and σ define the well depth and width, respectively, (see Fig. 1). Note that Eq. (1) was written considering constant values of the electron effective mass and static dielectric constant in the whole range of the heterostructure. Clearly, such approximation avoids the presence of the image charge effects associated to variations of the dielectric constant. Considering z-dependent effective masses the electronic spectrum exhibits a red shift due to the increasing character of the electron effective mass as we evolve from GaAs to Al0.3 Ga0.7 As material. The choice for the present potential profile is motivated by the experimental fact that the spectrum of single dots in GaAs is well described by a parabolic confinement potential [36, 37]. The noncorrelated electron wave function corresponding to the Hamiltonian in Eq. (1), i.e. without the last term at the right hand side, can be written as [38]

H=

(

  z 12

"

~2 ky2 ϕ(z) = Ez − 2 m∗

# ϕ(z) .

(4)

function Ψ will be expressed as the next product Ψ(ρ, φ, z, λ) = ϕ(z) χ(ρ, φ, z, λ) ,

(6)

where χ(ρ, φ, z, λ) is the hydrogenic wave function. The hydrogenic type wave functions for the two lowest axialsymmetric states are given by [39] χ1s (ρ, φ, z, λ) = N1 exp(−r/λ1s ) , χ2s (ρ, φ, z, λ) = N2 (2 − r/λ2s ) exp(−r/λ2s ) ,

(7) (8)

respectively. Here N1 and N2 are the normalization constants, λ1s and λ2s are the variational parameters. The method used for calculating the wave functions and energy levels has been previously used in many studies and accuracy degree is quite sufficient, see for instance [1–4, 17, 18, 30–33]. In the case of te noncorrelated energy levels, the number of wave functions used in the expansion [38] has been extended to guarantee individual variations less than 0.1 meV, which means variations

4

2 I β (1) (Ep ) |M12 |2 ε0 nr c (E12 − Ep )2 + Ω212

  |T12 |2 (E12 − Ep )2 − Ω212 + 2 E12 (E12 − Ep ) 1− , 2 + Ω2 |2 M12 |2 E12 12

(11)

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EP

β (3) (Ep , I) = −

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ACCEPTED MANUSCRIPT potential well with the L-width (L is large enough in less than 0.01 meV in the transition energies. With that comparing with the spatial extension 2 σ of the tuned degree of precision, we have observed that the variations GaAs/GaAlAs DQW of our model). We provided that in the dipole moments, where the wave functions are inthe eigenvalues are independent of L-width, and the wave volved, are lower than 0.01 %. functions are localized in the well regions. The binding energy of the two lowest impurity states is given by the well-known definition After the energies and wave functions for the 1s-like and 2s-like states have been obtained, we proceed to 1s,2s Eb = Ez1 − minλ1s,2s hΨ1s,2s |H|Ψ1s,2s i , (9) study, among others, the process of optical absorption between them, which in the present study will include where, Ez1 is the lowest value of the spectrum associated first and third order corrections. Following the standard to Eq. (5) and second term gives the impurity energies density matrix formalism combined with the perturbaE1s and E2s , which are found by minimizing the expection expansion method, the first-order, third-order, and tation value of the Hamiltonian in Eq. (1) with respect total absorption coefficients, for the 1s → 2s transition, to the variational parameters. are respectively [40–44] To solve the one-dimensional differential Eq. (5), r which means to find ϕ(z) and Ez1 , we follow the work |M12 |2 σv Ω12 Ep µ0 of Xia and Fan [38] and use a diagonalization procedure β (1) (Ep ) = (10) ~ εr (E12 − Ep )2 + Ω212 where the base for expansion of wave the function comes from the problem of one electron confined in an infinite

β(Ep , I) = β (1) (Ep ) + β (3) (Ep , I) ,

(12)

2 where T12 = M22 − M11 , Ω12 = ~ Γ√ 12 , εr = nr ε0 is the real part of the permittivity, nr = ε, σv is the carrier density, µ0 is the vacuum permeability, E12 = E2s − E1s , −1 Mij = hΨi |e z|Ψf i (i, j = 1, 2), Γ12 = T12 (where T12 is

relaxation time of the interlevel transition), c is the light speed in free space, and I is the optical intensity of the zpolarized incident photon with ω-angular frequency, and energy Ep = ~ ω.

5

ACCEPTED MANUSCRIPT = 15 nm

= 7.5 nm

300 (a)

(b)

B = 0

150

B = 0 F = 50 kV/cm

F = 0

-300 300 (c)

(d)

B = 10 T

150

F = 0

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-150

B = 10 T

F = 50 kV/cm

SC

V (z) (meV)

0

0

-300 -20

-10

0

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-150

10

20

-20

-10

0

10

20

z (nm)

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FIG. 2: (color online)z-position dependence of the confinement potential profile and the squared wave function, corresponding to the ground state energy level of the electron confined in the tuned GaAs/GaAlAs DQW, for several setups of σ, electric, and magnetic field: (a) F = 0, B = 0, (b) F = 50 kV/cm, B = 0, (c) F = 0, B = 10 T, and (d) F = 50 kV/cm, B = 10 T.

RESULTS AND DISCUSSION

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In this section we proceed to present our numerical results for the binding energy of the 1s-like and 2slike states for a donor impurity confined in a tuned GaAs/GaAlAs DQW that is subject to the combined effects of electric and magnetic fields. The results of the optical absorption associated with 1s → 2s transition are also presented. The parameters used in numerical calculations are ε = 12.58, m∗ = 0.0665 m0 (where m0 is the free electron mass), V0 = 228 meV, T12 = 0.2 ps, σv = 3 × 1022 m−3 , and I = 5 × 108 W m−2 [8]. In the Figs. 2(a-d), the changes of the confinement potential profile and the squared wave function corresponding to ground state energy level of the electron confined in the tuned GaAs/GaAlAs DQW as a function of the position are plotted with respect to different σ, electric, and magnetic field values (where, σ is the well-width and it will be called as a structure parameter). Blue (red) line is for σ = 15 nm (σ = 7.5 nm). As seen in these figures, each of wells is symmetric and the tuned GaAs/GaAlAs DQW becomes narrower as the σ decreases. Clearly, the elec-

tronic localization in the wells increases with the increase of the σ. Note that for σ = 15 nm, the two wells are essentially decoupled, which implies that the ground state of the system is quasi degenerate, with energy lower than the height of the central barrier. The state of lowest energy corresponds to a symmetric function while the first excited state (not shown here) is an antisymmetric function with the same configuration of the density of probability as the ground state. In the case of σ = 7.5 nm, interaction between the two wells is presented (note how the probability density takes finite values in the central barrier region) and the energy of the ground state is close to the value taken by the central barrier. When the electric field is turned on the electron is pushed towards the left barrier, increasing the spatial location of the wave function in one of the wells, and with a redshift of the energy values. This effect of decreasing energy, despite a greater location of the electron, is simply the result of having chosen the origin of coordinates in the middle of the central barrier. It is well known that when applying a magnetic field a greater confinement of the carriers appears and therefore there must be a shift

6

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b

(meV)

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MANUSCRIPT and so the maximums of the binding energies of 1s-like to the blue of the energy. This effect isACCEPTED not clear in figures and 2s-like impurity states shifts towards the left well. (c) and (d) where B = 10 T for two reasons: i) the dimenAs known, the electric field effect is very pronounced in sions of the two considered tuned GaAs/GaAlAs DQWs wider wells or weak confinement regime, while the elecare in both cases of the order of magnitude of an effective tron is completely localized in the left well for σ = 15 nm, Bohr radius and ii) the infinite potential barriers, located there is still the possibility of finding the electron in the in ±σ, overlap the effect of the applied magnetic field. right well for σ = 7.5 nm. Thus, the maximum of bind18 ing energies of the impurity states taken into account are blue = 15 nm; red = 7.5 nm (a) F = 0 found in different impurity positions. In the Fig. 3(b), solid B = 0; dashed B = 10 T 15 it is easy to see that the electric field effect is quite pronounced when both electric and magnetic fields are ap12 plied together. Magnetic field effect is seen in the well 9 edges as in the Fig. 3(a). Panels (a) and (b) clearly show small changes in the binding energy as an effect of the 1s-like 6 applied magnetic field, which is consistent with the small variations that the magnetic interaction induces on the 3 2s-like uncorrelated ground state that has been shown in the Fig. 2. 0 After analyzing the effects that the σ-parameter, the 18 impurity position, the externally applied electric field, (b) F = 50 kV/cm and the applied magnetic field generate on the bind15 ing energy of a donor impurity confined in a tuned 12 GaAs/GaAlAs DQW, we proceed to make a study of the optical absorption coefficient between the two lower 9 states of s-symmetry. Based on the definition of bind1s-like ing energy and taking into account that for both states 6 the binding energy refers to the same uncorrelated state, the transition energy is given by E12 = E2s − E1s = 3 2s-like Eb1s − Eb2s . In Figs. 4, 5, and 6 the total optical absorp0 tion coefficient, see Eq. (12), is presented as a function -20 -10 0 10 20 of the energy of the incident photon considering different configurations of the system: i ) the Fig. 4 is for F = 0 z (nm) with several setus of the applied magnetic field, the σparameter, and the impurity position, ii ) in Fig. 5 the FIG. 3: (color online)Binding energies of the 1s- and 2s-like results follow the same presentation of Fig. 4, but with shallow donor impurity states versus the impurity position-zi B = 0 and F = 50 kV/cm, and iii ) in Fig. 6, fixed valfor different σ and magnetic field values. In (a) the results are ues of non-zero electric and magnetic field are combined for zero applied electric field whereas in (b) F = 50 kV/cm. with several impurity positions. Solid (dashed) lines are for B = 0 (B = 10 T), whereas red (blue) lines are for σ = 7.5 nm (σ = 15 nm). In Fig. 5(a) it is observed that as the value of zi increases, a shift to the blue of the resonant peak (RP) of the AC appears first and then a shift to red. This behavThe binding energies of the 1s-like and 2s-like shalior is in consonance with the vertical separation between low donor impurity states versus the impurity position the blue curves in Fig. 3(a), where the high variations of for several setups of the σ-parameter and applied exterthe binding energy of the 1s-like state are responsible for nal electric and magnetic fields are given in the Fig. 3. the shifts of the RPs of the spectra. In the case of Fig. By analysing the electronic localization in the Figs. 2(a5(b), it can be seen that at any time when the value of d) with respect to the σ-parameter and applied external zi increases, a redshift of the RP appears. This is assofield values, it is seen easily that the binding energy beciated with the behavior shown by E12 between the red comes maximum (minimum) for impurities located where curves of Fig. 3(a). Note that Eb1s and Eb2s are decreasthe probability of finding an electron shows a maximum ing functions of |zi | with a more pronounced variations on (minimum) since the impurity position dependence of the the 1s-like state that goes from 8.5 meV to 3 meV when binding energy follows the same behaviour of the spatial zi goes from zero to 17.5 nm. In the case of the 2s-like distribution of the wave function of the electron confined state there are only 1 meV variations in the whole range in the tuned GaAs/GaAlAs DQW. The binding energies of variations of the impurity position. of both levels are virtually unaffected by the magnetic From Eqs. (10) and (11), for the case of structures that field except for the well edges since the geometric confinepreserve the symmetry of the potential, that is, when ment is more dominant, whereas the electric field effect is there is no applied electric field applied or when the imquite pronounced. The electric field pushes the electron purities are located at the point of symmetry of the poto the opposite direction to the electric field direction

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i

7

ACCEPTED MANUSCRIPT tential, the magnitude of the RP of first and third order Under such conditions above, the two s-like involved are proportional to states in the transition states have even symmetry and consequently T12 = 0. For finite values of the applied β (1) (Ep ) ∼ (Eb1s − Eb2s ) |M12 |2 (13) electric field and/or for off-center impurities, the third order correction of the RP should be written as and

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EP AC C

20

-1

30

A = -54146

20

1

A = +14491 2

10

A = -605 3

0 6

9

15 nm

12

15

z (nm) i

5 nm

-1

m )

m )

10 nm

i

3

z =

30

10

(15)

= 15 nm

(10

(a)

R

40

M AN U 1.2

= 7.5 nm z = i

5 nm

1.2

A = +2002

-1

(b)

1.0

m )

(10

3

0

3

The Eqs. (13-15) are the direct mechanism to find the numerical data of the dipole matrix elements associated with the transitions. The reason is that the interlevel energy transition is provided by Fig. 3 (which corresponds to the energy position of the RP in Fig. 5) and the magnitude of RP is provided by Fig. 5, so with very little numerical effort can be obtained the values of M12 . The insets in Figs 4(a) and 4(b) are for the magnitude of the RP (β = β R when Ep = E12 ) as a function of the ±zi -impurity position. The numbers in the insets are the coefficients of a quadratic adjust for the curve: β R = A1 +A2 zi +A3 zi2 . Clearly the degree of complexity of the curves show the relevance of this study because it is evident the impossibility of making predictions about the behavior of the RP. Two points can be concluded from the Fig. 4: i ) E12 has a dominant character on the behavior of the magnitude of the RP, ii ) despite the negative character of the third order correction of the AC, the intensity used for the incident radiation preserves the behavior of the correction of first order and is not large enough to induce a splitting of its central peak.

  |T12 |2 Ω212 1+ , |2 M12 |2 (Eb1s − Eb2s )2 + Ω212

1

0.8

A = -239 2

(10

−

Eb2s ) |M12 |4

0.8

R

(Ep , I) ∼ − I

(Eb1s

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β

(3)

(14)

SC

β (3) (Ep , I) ∼ − I (Eb1s − Eb2s ) |M12 |4 ,

0.6

A = +8 3

0.4

0.0 6

9

12

15

z (nm)

0.4

i

10 nm

0.2 15 nm

0.0 0

10

20

30

40

Photon energy (meV)

FIG. 4: (color online)The effect of the magnetic field on the total AC for the transitions between the first two impurity states for different impurity positions versus the incident photon energy: (a) σ = 15 nm and (b) σ = 7.5 nm. Results are for zero applied electric field with zero magnetic field (solid lines) and B = 10 T (dashed lines). The insets in (a) and (b) are for the magnitude of the AC-resonant peak (β = β R when Ep = E12 ) as a function of the ±zi -impurity position. The numbers in the insets are the coefficients of a quadratic adjust for the curve: β R = A1 + A2 zi + A3 zi2 .

8

ACCEPTED MANUSCRIPT energies of the

= 15 nm

-15 nm

-12.5 nm

-10 nm

-5 nm

50

A = -26150 1

60

A = -12857 2

50

A = -550

(10

3

-1

m )

40

3

R

30

40 30

20

20 -12

-9

-6

z (nm) i

-1

m )

-15

10

(10

3

0

12

(b)

= 7.5 nm

-15 nm

-12.5 nm

-10 nm

-5 nm

10 A = +19935

12 -1

m )

8

1

A = +2120

9

2

(10

3

A = +70 3

6

R

6

3

4

0 -15

-12

-9

-6

z (nm)

2

i

0 10

20

30

Photon energy (meV)

40

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0

1s- and 2s-like states of donor impurities and also the absorption coefficient for the transitions between the related impurity states in ”12-6” tuned GaAs/GaAlAs double quantum well. The main findings can be summarized as follows: i ) the dominant character of the resonant peak of the AC follows the behaviour of the transition energies, ii ) the structures where the dimensions are in the order of magnitude of the effective Bohr radius (a0 ) show AC with RP one order of magnitude smaller than those cases where structural dimensions are at least 1.5 a0 , iii ) tuned GaAs/GaAlAs DQW with dimensions in the order of 1.0 a0 are almost insensible to the magnetic field effects because the dominant character of the ”12-6” potential, and iv ) the RP has almost a quadratic character with the impurity position, even in cases where the electric and magnetic fields have been considered.

RI PT

(a)

SC

60

TE D

FIG. 5: (color online)The effect of the electric field on the total AC for the transitions between the first two impurity states for different impurity positions versus the incident photon energy: (a) σ = 15 nm and (b) σ = 7.5 nm. Results are for zero magnetic field with F = 50 kV/cm. Dashed lines are the corresponding results as solid lines represent, but for positive values of +zi -impurity position. The insets in (a) and (b) are for the magnitude of the AC-resonant peak (β = β R when Ep = E12 ) as a function of the zi -impurity position (only negative values of the impurity position have been represented in the curve). The numbers in the insets are the coefficients of a quadratic adjust for the curve: β R = A1 + A2 zi + A3 zi2 .

AC C

EP

Finally, when both the magnetic and electric fields are applied together, Fig. 6, the electric field effect is dominant and only a slight decrease in peak amplitudes is observed. These results are in perfect agreement with those shown in Fig. 3 and are clear evidence that in the range of the magnetic fields studied it is not possible for them to induce variations in the location of the carriers. In this case, the shape of the ”12-6” potential is dominant over the parabolic character that adds the presence of the magnetic field.

IV.

CONCLUSIONS

Using the effective mass and parabolic conduction band approximations combined with the standard density matrix formalism, the perturbation expansion method, and a variational technique, we have investigated the effects of the magnetic and electric fields, impurity position, and σ-tune parameter on the binding

Our results may be useful in the exploration of new ways to manipulate the opto-electronic properties of quantum-well devices. The relatively simplicity of the model probably hinder its direct comparison with any particular experimental dataset but, on the other hand, lends generality to our observations. Further, the methodology can be easily understood by a broad audience. The model, despite its simplicity, is useful to explain the optical properties associated with the active region in quantum cascade laser devices and in solid state lasers with intentional doping, which allows modulating the effective gap and radiative recombinations. Additionally, the infinite barriers at the ends of the heterostructure give a good account of the real situation where the devices can simply be immersed either in air or in a vacuum.”

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60

(a)

ACCEPTED MANUSCRIPT FIG. 6: (color online)The effects of the electric and magnetic

= 15 nm

50

-15 nm

40

-10 nm

-12.5 nm

-5 nm

30

+5 nm

20

+10 nm

10

+15 nm

+12.5 nm

0 = 7.5 nm

10

-15 nm -12.5 nm

8

-10 nm -5 nm

6

+5 nm

4

+10 nm

2

+15 nm

0

+12.5 nm

0

10

20

30

Photon energy (meV)

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EP

TE D

[1] G. Bastard, Phys. Rev. B 24, 4714 (1981). [2] G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. B 28, 3241 (1983). [3] G. Bastard, Superlattice Microst. 1, 265 (1985). [4] R. L. Greene and K. K. Bajaj, Solid State Commun. 53, 1103 (1985). [5] H. Mathieu, P. Lefebvre, and P. Christol, J. Appl. Phys. 72, 300 (1992). [6] H. Odhiambo Oyoko, N. Porras-Montenegro, S. Y. L´ opez, and C. A. Duque, Phys. Status Solidi C 4, 298 (2007). [7] C. M. Duque, M. E. Mora-Ramos, and C. A. Duque, J. Nanopart. Res. 13, 6103 (2011). [8] U. Yesilgul, F. Ungan, S. Sakiroglu, M. E. Mora-Ramos, C. A. Duque, E. Kasapoglu, H. Sari, and I. S¨ okmen, J. Lumin. 145, 379 (2014). [9] X. Yao and A. Belyanin, J. Phys.: Condens. Matter 25, 054203 (2013). [10] I. V. Bondarev and M. R. Vladimirova, Phys. Rev. B 97, 165419 (2018). [11] H. M. Baghramyan, M. G. Barseghyan, A. A. Kirakosyan, J. H. Ojeda, J. Bragard, and D. Laroze, Sci. Rep. 8, 6145 (2018). [12] T. B. Boykin and G. Klimeck, Eur. J. Phys. 26, 865 (2005). [13] J. Juang, W.-W. Lin, and S.-F. Shieh, SIAM J. Matrix Anal. Appl. 23, 524 (2001). [14] W.-P. Yuen, Phys. Rev. B 48, 17316 (1993). [15] R. C. Miller, A. C. Gossard, D. A. Kleinman, and O. Munteanu, Phys. Rev. B 29, 3740 (1984). [16] Y.-B. Yu, S.-N. Zhu, and K.-X. Guo, Phys. Lett. A 335,

AC C

This research was partially supported by Colombian Agencies: CODI-Universidad de Antioquia (Estrategia de Sostenibilidad de la Universidad de Antioquia) and Facultad de Ciencias Exactas y Naturales-Universidad de Antioquia (CAD-exclusive dedication project 20172018). This work used resources of the Centro Nacional de Processamento de Alto Desempenho em S˜ ao Paulo (CENAPAD-SP). The authors are grateful to The Scientific Research Project Fund of Cumhuriyet University (CUBAP) under the project number F-557.

SC

(b)

M AN U

(10

12

ACKNOWLEDGMENTS

RI PT

V.

3

-1

m )

fields on the total AC for the transitions between the first two impurity states for different impurity positions versus the incident photon energy: (a) σ = 15 nm and (b) σ = 7.5 nm. Results are for F = 50 kV/cm with B = 10 T.

175 (2005). [17] E. Kasapoglu, H. Sari, and I. S¨ okmen, Surf. Rev. Lett. 13, 397 (2006). [18] E. Kasapoglu and I. S¨ okmen, Physica E 27, 198 (2005). [19] W. Q. Chen, S. M. Wang, T. G. Andersson, and J. Thordson, J. Appl. Phys. 74, 6247 (1993). [20] S. Baskoutas and A. F. Terzis, Eur. Phys. J. B 69, 237 (2009). [21] S. Baskoutas and A. F. Terzis, Physica E 40, 1367 (2008). [22] S. Baskoutas, A. F. Terzis, and E. Voutsinas, J. Comp. Theor. Nanoscience 1, 317 (2004). [23] D. N. Quang, L. Tuan, and N. T. Tien, Phys. Rev. B 77, 125326 (2008). [24] B. Chen, K.-X. Guo, R.-Z. Wang, Y.-B. Zheng, and B. Li, Eur. Phys. J. B 66, 227 (2008). [25] K. Efstathiou, O. V. Lukina, and D. A. Sadovskii, Phys. Rev. Lett. 101, 253003 (2008). [26] A. Abramov, World Journal of Condensed Matter Physics 2, 188 (2012). [27] E. C. Niculescu, L. Burileanu, and M. Cristea, U.P.B. Sci. Bull., Series A 69, 81 (2007). [28] L. R. Ram-Mohan and J. R. Mever, J. Nonlinear Opt. Phys. 4, 191 (1995). [29] J. M¨ uller, J. M. Lupton, P. G. Lagoudakis, F. Schindler, R. Koeppe, A. L. Rogach, and J. Feldmann, Nano Lett. 5, 2044 (2005). [30] E. Kasapoglu, S. Sakiroglu, F. Ungan, U. Yesilgul, C. A. Duque, and I. S¨ okmen, Physica B 526, 127 (2017). [31] E. B. Al, E. Kasapoglu, S. Sakiroglu, C. A. Duque, and I. S¨ okmen, Journal of Molecular Structure 1157, 288 (2018).

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ACCEPTED MANUSCRIPT [32] V. Mart´ınez-Rend´ on, C. Casta˜ no-Uribe, A. Giraldo[38] J.-B. Xia and W.-J. Fan, Phys. Rev. B 40, 8508 (1989). Mart´ınez, J. P .Gonz´ alez-Pereira, R. L. Restrepo, A. L. [39] D. J. Griffiths, in Introduction to Quantum Mechanics, Morales, and C. A. Duque, Revista EIA 12, E85 (2016). R. Kernan and F. Dahl, Editors, Prentice Hall Inc., New [33] S. Sakiroglu, E. Kasapoglu, R. L. Restrepo, C. A. Duque, Jersey (1995). and I. S¨ okmen, Phys. Status Solidi B 254, 1600457 [40] H. Dakhlaoui, J. Appl. Phys. 117, 135705 (2015). (2017). [41] H. M. Baghramyan, M. G. Barseghyan, A. A. Ki[34] S. M. Sze, Physics of semiconductor devices, 2nd edition, rakosyan, R. L. Restrepo, M. E. Mora-Ramos, and C. New York, Wiley-Interscience, p. 878 (1981). A. Duque, J. Lumin. 145, 676 (2014). ¨ [35] F. R. Waugh, M. J. Berry, D. J. Mar, R. M. Westervelt, [42] B. C ¸ akir, Y. Yakar, and A. Ozmen, Physica B 458, 138 K. L. Campman, and A. C. Gossard, Phys. Rev. Lett. (2015). 75, 705 (1995). [43] M. R. K. Vahdani and G. Rezaei, Phys. Lett. A 374, 637 [36] S. Tarucha, D. G. Austing, T. Honda, R. J. van der (2010). Hage, and L. P. Kouwenhoven, Phys. Rev. Lett. 77, 3613 [44] ´I. Karabulut and S. Baskoutas, J. Appl. Phys. 103, (1996). 073512 (2008). [37] G Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59, 2070 (1999).

TE D

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ACCEPTED MANUSCRIPT

1

0

AC C

0

V (V )

EP

V(z) = 4 V

0

[(z/

12

)

-(z/

6

)

]

2

V

0

-1 -1.0

-0.5

0.0

z (

0.5

)

1.0

TE D

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ACCEPTED MANUSCRIPT

= 15 nm

300 (a)

B = 0

150

-150

-300 300 (c)

AC C

0

V (z) (meV)

(b)

B = 0 F = 50 kV/cm

EP

F = 0

= 7.5 nm

(d)

B = 10 T

150

B = 10 T

F = 0

F = 50 kV/cm

0

-150

-300 -20

-10

0

10

20

-20

z (nm)

-10

0

10

20

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(a) F = 0

blue

15

= 15 nm; red

M AN U

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1s-like

6

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18

EP

0

(b) F = 50 kV/cm

15 12 9

AC C

(meV)

3

b

= 7.5 nm

solid B = 0; dashed B = 10 T

12

E

SC

18

1s-like

6 3 2s-like

0 -20

-10

0 z (nm) i

10

20

RI PT

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= 15 nm z =

30

10 nm

(10

3

-1

m )

i

R

20

M AN U

(a)

SC

40

30

A = -54146

20

1

A = +14491 2

10

A = -605 3

0

6

12

15

TE D

z (nm) i

5 nm

0

i

0.8 0.6

A = +2002

-1

m )

5 nm

AC C

z =

1.2

3

1.0

= 7.5 nm

1

0.8

A = -239 2

(10

(b)

R

1.2

EP

3

(10

9

15 nm

-1

m )

10

A = +8 3

0.4

0.0 6

9

12

15

z (nm)

0.4

i

10 nm

0.2 15 nm

0.0 0

10

20

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Photon energy (meV)

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(a)

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-15 nm

= 15 nm

-12.5 nm

50

-5 nm

M AN U

-10 nm

A = -26150 1

60

(10

3

-1

m )

40

R

30

A = -12857 2

50

A = -550 3

40 30

20

20

0

6

-1

z (nm) i

-12.5 nm

-10 nm

-5 nm

A = +19935

12 m )

8

-6

-15 nm

1

A = +2120

9

2

A = +70

3

AC C

10

= 7.5 nm

(10

(b)

-9

3

6

R

12

EP

3

(10

-12

TE D

10

-1

m )

-15

3

4

0 -15

-12

-9

-6

z (nm)

2

i

0 0

10

20

30

Photon energy (meV)

40

60

(a)

= 15 nm

50

M AN U

-15 nm

-12.5 nm

40

-10 nm -5 nm

30

+5 nm +10 nm +12.5 nm

TE D

10 0 12

(b)

= 7.5 nm

-15 nm

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10

+15 nm

EP

3

-1

m )

20

(10

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8 6

-12.5 nm -10 nm -5 nm +5 nm

4

+10 nm

2

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0

+12.5 nm

0

10

20

30

Photon energy (meV)

40