Int. J. Nano Dimens., 10 (4): 375-390, Autumn 2019

ORIGINAL ARTICLE

Combined effects of pressure, temperature, and magnetic field on the ground state of donor impurities in a GaAs/AlGaAs quantum heterostructure

Samah Abuzaid, Ayham Shaer, Mohammad Elsaid*

Department of Physics, An-Najah National University, Nablus,West Bank, Jordan

Received 09 February 2019; revised 15 July 2019; accepted 20 July 2019; available online 30 July 2019 Abstract In the present work, the exact diagonalization method had been implemented to calculate the ground state energy of shallow donor impurity located at finite distance along the growth axis in GaAs/AlGaAs heterostructure in the presence of a magnetic field taken to be along z direction. The impurity binding energy of the ground state had been calculated as a function of confining frequency and magnetic field * * strength. We found that the ground state donor binding energy (BE) calculated at ωc =2 R and ω0 = 5.421R , decreases from BE=7.59822 R * to BE=2.85165 R * , as we change the impurity position from d=0.0 a* to d=0.5 a* , respectively .In addition, the combined effects of pressure and temperature on the binding energy, as a function of magnetic field strength and impurity position, had been shown using the effective-mass approximation. The numerical results show that the donor impurity binding energy enhances with increasing the pressure while it decreases as the temperature decreases.

Keywords: Binding Energy; Donor Impurity; Exact Diagonalization Method; Heterostructure; Magnetic Field.

How to cite this article Abuzaid S, Shaer A, Elsaid M. Combined effects of pressure, temperature, and magnetic field on the ground state of donor impurities in a GaAs/AlGaAs quantum heterostructure. Int. J. Nano Dimens., 2019; 10 (4): 375-390.

INTRODUCTION Low dimensional heterostructure have been of the impurity and the coulombic interaction studied intensively in the last decades, both between the system charge carrier and the donor theoretically and experimentally [1-3]. Quantum impurity change the heterostructure energy gap confined semiconductor systems such as quantum [6, 7]. The donor impurity binding energy had well (QW), quantum well wire (QWW), quantum been investigated from bulk to the quantum dot dots (QD) and quantum rings (QR) are important where it depends on the dimensionality of the elements in the present electronic devices. The system, shape, and the impurity position [1, 8, 9]. electrical, optical and transport properties of these Also, it is affected the heterostructure properties semiconductor heterostructure systems can be by the presence of the magnetic or electrical changed by external effects like: donor impurities fields, pressure, and temperature [9, 10]. For the near the heterostructure surface, electric field, quantum dot, the donor impurity binding energy magnetic field, pressure, and temperature [3,4]. increases continuously with decreasing the dot Moreover, the effect of the donor impurity on the radius, and the applied magnetic field strength, in properties of the heterostructure materials is very addition, it depends on the impurity position [11, interesting subject which has been investigated. 12]. Adding the donor impurity atoms change the The donor binding energy had been studied effective charge and mass of it, which alter the for the quantum well by using variation method, performance of the quantum devices and the where the ground state of donor binding energy transport properties [5]. The binding energy had been computed as a function of the impurity * Corresponding Author Email: [email protected] position and the QW width under different electric

This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. S. Abuzaid et al.

field strengths [13, 14]. The donor impurity (CQD). Alfonso et al. in [33] had investigated the energy for the ground and excited state of two- energy states of an electron confined in a two- dimensional heterostructure had been computed dimensional (2D) plane and bound to an off- as a function of magnetic field strength, for both plane shallow donor center in the presences of weak and strong magnetic field limit, using exact an external magnetic field by using variation and and perturbation methods [15]. Chuu et al. in Ref. numerical approaches. The effect of impurity on [16] obtained the donor impurity energy levels energy levels in double quantum rings had been analytically for both donor impurity located at considered by Khajeh Salehani et al. [34]. the quantum dots center, and for donor impurity In this work, we have investigated the combined located at the axis of the quantum-well wire. Zhu effects of pressure, temperature, magnetic field and Gu in Ref. [17] investigated the dependence strength and the impurity position on the ground of the energy transition on shallow donor impurity state binding energy of the donor impurity in states in a harmonic QD by using analytical heterostructure materials. We have computed method with the presence of the magnetic field. the ground state energy level of donor impurity The energy transition changes with the magnetic in a heterostructure by solving the donor impurity field strength, where the result shows the high Hamiltonian using exact diagonalization method. effects of magnetic fields on donor impurity The Hamiltonian theory of donor impurity energy states transition [17]. The dependence of located along the growth axis under the effect of the diamagnetic susceptibility and the binding a magnetic field, pressure and temperature had energy of the donor impurity on the pressure and been presented, followed by numerical results the temperature had been shown analytically. and discussion. The conclusion is given at the end Khordad and Fathizadeh found in their recent of the work. study the diamagnetic susceptibility increases by increasing the pressure and it decreases with THEORY increasing the temperature [18]. Peter in Ref. [19] This section describes the main parts of the 2. Theory reported the binding energy levels of shallow donor impurity formulation: i) the quantum hydrogenic impurities in a parabolic quantum dot heterostructure Hamiltonian, ii) the exact This section describes the main parts of the donor impurity formulation: i) the (QD)with pressure effect using variation approach. diagonalization method and iii) the pressure and Where, they are found that the ionization energy, temperaturequantum heterostructure effects Hamiltonian, on ii)the the exact computed diagonalization donor method and iii) the is purely pressure-dependent. Merchancano et al. impurity ground state binding energy by the pressure and temperature effects on the computed donor impurity ground state binding in Ref. [2] had also calculated the2. Theory binding energy of effective-mass approximation. The system is a the hydrogenic impurities in a spherical QD using quantumenergy by the heterostructure effective-mass approximation. confined inThe the system x-y isplane a quantum heterostructure This section describes the main parts of the donor impurity formulation: i) the the variation and perturbation approaches as a withconfined parabolic in the x-y confinement plane with parabolic potential confinement of potential confining of confining strength quantum heterostructure Hamiltonian, ii) the exact diagonalization method and1 iii) the function of pressure, QD size, and the impurity = *ω 22 strength ω0 , Vr( ) m0 r , and in the presence 0 position. The resultspressure of andthe temperature study effects show on thatthe computed the , donor( ) =impurity ground , and state 2 inbinding the presence of a donor impurity along the z- axis located𝜔𝜔𝜔𝜔 of a donor1 ∗ 2 impurity2 along the z- axis located binding energy increases with increasing the 0 energy by the effective-mass approximation. Theat system𝑉𝑉𝑉𝑉 distance𝑟𝑟𝑟𝑟 is2 a𝑚𝑚𝑚𝑚 quantum𝜔𝜔𝜔𝜔 d,𝑟𝑟𝑟𝑟 under heterostructure the influence of a uniform pressure [20]. The combined effects of pressure at distance d, under the influence of a uniform magnetic field strength, B= × , confined in the x-y plane with parabolic confinementmagnetic potential of fieldconfining strength, strength B= ∇×A , applied along the and temperature on the binding energy of donor applied along the z -direction. A= ( , , 0) is the vector potential. 𝛁𝛁𝛁𝛁 𝑨𝑨𝑨𝑨 B 𝜔𝜔𝜔𝜔0 impurity in a spherical, ( QD) = with the , and presence in the presence of of a donorz -direction. impurity along A= the z (-− axisyx, located ,0) is 𝐵𝐵𝐵𝐵the vector potential. 1 ∗ 2 2 2 2 −𝑦𝑦𝑦𝑦 𝑥𝑥𝑥𝑥 the electric field or without 2 had 0 been investigated at𝑉𝑉𝑉𝑉 distance𝑟𝑟𝑟𝑟 𝑚𝑚𝑚𝑚 d, 𝜔𝜔𝜔𝜔under𝑟𝑟𝑟𝑟 the influence of a uniform magneticTheThe field interaction interaction strength, B=between ×between the, electron the in the electron GaAs layer inand the the donor impurity, [5, 21]. GaAs layer and the donor impurity, located at The exact diagonalizationapplied along the zmethod -direction. A =had ( , been, 0) is thelocated vector potential.at distance d along𝛁𝛁𝛁𝛁 the𝑨𝑨𝑨𝑨 z-direction in AlGaAs barrier, is attractive coulomb 𝐵𝐵𝐵𝐵 distance d along the z-direction in AlGaAs barrier, used to solve the problem of two interacting2 −𝑦𝑦𝑦𝑦 𝑥𝑥𝑥𝑥 isinteraction attractive energy. The coulomb Hamiltonian interaction of the donor impurity energy. can be The written in an operator electrons in the QD includingThe interaction the pressure between the andelectron in the GaAs layer and the donor impurity, Hamiltonian of the donor impurity can be written temperature effects. The magnetization and form as: located at distance d along the z-direction in AlGaAsin an operatorbarrier, is attractive form as:coulomb magnetic susceptibility of confined electrons interaction energy. The Hamiltonian of the donor impurity can be written in an operator in parabolic quantum dot was considered in 1 1 1 2 form as: = + + + (1) both: experimental and theoretical studies 2 2 4 4 2 | | [22, 23]. Very recently, Elsaid et al. [24-32], has −1⁄2 𝜕𝜕𝜕𝜕 1⁄2 𝜕𝜕𝜕𝜕 2 2 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 𝜕𝜕𝜕𝜕 𝐻𝐻𝐻𝐻� − �𝜌𝜌𝜌𝜌 2 𝜌𝜌𝜌𝜌 2 � 2 �� ω ρ − 𝑖𝑖𝑖𝑖 − 1 1 1 𝜕𝜕𝜕𝜕𝜌𝜌𝜌𝜌 2 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 𝜙𝜙𝜙𝜙 (1) 𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 𝛒𝛒𝛒𝛒 − 𝐝𝐝𝐝𝐝 studied the electronic,= thermodynamic+ and+ + in Eq. (1) can be separated into (1) two parts as: 2 2 4 4 2 | | magnetic properties of −two1⁄2 𝜕𝜕𝜕𝜕 electrons1⁄2 𝜕𝜕𝜕𝜕 confined 2 2 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 𝜕𝜕𝜕𝜕 𝐻𝐻𝐻𝐻� − �𝜌𝜌𝜌𝜌 2 𝜌𝜌𝜌𝜌 2 � 2 �� 𝐻𝐻𝐻𝐻� ω ρ − 𝑖𝑖𝑖𝑖 − in a single quantum and coupled𝜕𝜕𝜕𝜕 𝜌𝜌𝜌𝜌 quantum𝜌𝜌𝜌𝜌 dots𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 𝛒𝛒𝛒𝛒 − 𝐝𝐝𝐝𝐝 = + ( ) (2) in Eq. (1) can be separated into two parts as: Hˆ in Eq. (1) can be separated into two parts as: ⊥ � where: Ĥ Ĥ 𝑉𝑉𝑉𝑉 𝜌𝜌𝜌𝜌 𝐻𝐻𝐻𝐻 = + ( ) (2)

376 where: Ĥ Ĥ⊥ 𝑉𝑉𝑉𝑉 𝜌𝜌𝜌𝜌 Int. J. Nano Dimens., 101 (4): 375-390,1 Autumn1 2019 = + + + (3) 2 2 4 4 2 1 1 1 −1⁄2 𝜕𝜕𝜕𝜕 1⁄2 𝜕𝜕𝜕𝜕 2 2 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 𝜕𝜕𝜕𝜕 = + + ⊥+ 2 (3) 2 2 2 2 4 Ĥ 4 − �𝜌𝜌𝜌𝜌 2 𝜌𝜌𝜌𝜌 � �� ω ρ − 𝑖𝑖𝑖𝑖 −1⁄2 𝜕𝜕𝜕𝜕 1⁄2 𝜕𝜕𝜕𝜕 and, 2 2 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜌𝜌𝜌𝜌 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 Ĥ⊥ − �𝜌𝜌𝜌𝜌 2 𝜌𝜌𝜌𝜌 2 � 2 �� ω ρ − 𝑖𝑖𝑖𝑖 and, 𝜕𝜕𝜕𝜕𝜌𝜌𝜌𝜌 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙

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2. Theory 2. Theory This section describes the main parts of the donor impurity formulation: i) the 2. Theory This section describes the main parts of the donor impurity formulation: i) the quantum heterostructure Hamiltonian, ii) the exact diagonalization method and iii) the This section describesquantum heterostructurethe main parts Hamiltonian,of the donor ii) impurity the exact formulation: diagonalization i) t hemethod and iii) the pressure and temperature effects on the computed donor impurity ground state binding quantum heterostructurepressure Hamiltonian, and temperature ii) the effects exact ondiagonalization the computed methoddonor impurity and iii) theground state binding energy by the effective-mass approximation. The system is a quantum heterostructure pressure and temperatureenergy byeffects the effectiveon the computed-mass approximation. donor impurity The ground system state is a binding quantum heterostructure confined in the x-y plane with parabolic confinement potential of confining strength confined in the x-y plane with parabolic confinement potential of confining strength energy by the effective-mass approximation. The system is a quantum heterostructure 0 ( ) = 𝜔𝜔𝜔𝜔 , , and in the presence of a donor0 impurity along the z- axis located confined in the x-y, plane( ) = with parabolic , and confinement in the presence potential of1 aof donor∗ confining2 2 impurity strength along the z- axis located𝜔𝜔𝜔𝜔 0 1 ∗ 2 2 𝑉𝑉𝑉𝑉 𝑟𝑟𝑟𝑟 2 𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔 𝑟𝑟𝑟𝑟 at distance d, under the influence 0of a uniform magnetic field strength, B= × , 2 0 𝜔𝜔𝜔𝜔 , ( ) = at𝑉𝑉𝑉𝑉 , distanceand𝑟𝑟𝑟𝑟 in 𝑚𝑚𝑚𝑚 thed, 𝜔𝜔𝜔𝜔 underpresence𝑟𝑟𝑟𝑟 the ofinfluence a donor ofimpurity a uniform along magnetic the z- axis field located strength, B= × , 1 ∗ 2 2 applied along the z -direction. A= ( , , 0) is the vector potential. 𝛁𝛁𝛁𝛁 𝑨𝑨𝑨𝑨 𝑉𝑉𝑉𝑉 𝑟𝑟𝑟𝑟 2 𝑚𝑚𝑚𝑚 𝜔𝜔𝜔𝜔0𝑟𝑟𝑟𝑟 at distance d, underapplied the influence along the ofz - direction.a uniform A =magnetic ( , ,field0) is thestrength, vector B=potential.× 𝐵𝐵𝐵𝐵 , 𝛁𝛁𝛁𝛁 𝑨𝑨𝑨𝑨 𝐵𝐵𝐵𝐵 2 −𝑦𝑦𝑦𝑦 𝑥𝑥𝑥𝑥 applied along the z -direction. A= ( , , 0) is the2 vector−𝑦𝑦𝑦𝑦 𝑥𝑥𝑥𝑥potential.The interaction between𝛁𝛁𝛁𝛁 𝑨𝑨𝑨𝑨the electron in the GaAs layer and the donor impurity, The interaction𝐵𝐵𝐵𝐵 between the electron in the GaAs layer and the donor impurity, 2 −𝑦𝑦𝑦𝑦 𝑥𝑥𝑥𝑥 located at distance d along the z-direction in AlGaAs barrier, is attractive coulomb The interactionlocated between at distance the electron d along in thethe zGaAs-direction layer inand AlGaAs the donor barrier impurity, is attractive, coulomb interaction energy. The Hamiltonian of the donor impurity can be written in an operator located at distanceinteraction d along energythe z-direction. The Hamiltonian in AlGaAs of thebarrier donor, is impurityattractive can coulomb be written in an operator form as: interaction energyform. The asHamiltonian: of the donor impurity can be written in an operator 2 2 form as: 1 (1) = 1 = 2 (4) = + + + | | + (1) 1 1 1 2 2 42 4 2 | | = + + ⁄+ ⁄ 𝑉𝑉𝑉𝑉 𝜌𝜌𝜌𝜌 − (1) − 𝑐𝑐𝑐𝑐 2 2 2 2 −1 2 𝜕𝜕𝜕𝜕 1 2 𝜕𝜕𝜕𝜕 𝛒𝛒𝛒𝛒 −2𝐝𝐝𝐝𝐝2 𝜔𝜔𝜔𝜔 𝜕𝜕𝜕𝜕 � 4 The4 2potential (22𝑐𝑐𝑐𝑐) represents2 | the c|oulomb interaction� between𝜌𝜌𝜌𝜌 𝑑𝑑𝑑𝑑 the electron in the GaAs −1⁄2 𝜕𝜕𝜕𝜕 1⁄12 1𝐻𝐻𝐻𝐻𝜕𝜕𝜕𝜕 − �𝜌𝜌𝜌𝜌 2𝜌𝜌𝜌𝜌 2 2 𝜔𝜔𝜔𝜔 �𝜕𝜕𝜕𝜕 �� ωS.ρ Abuzaid− 𝑖𝑖𝑖𝑖 et al. − = 𝐻𝐻𝐻𝐻� − �+𝜌𝜌𝜌𝜌 2 𝜌𝜌𝜌𝜌+ +2 � in2 Eq. (1�)� can 𝜕𝜕𝜕𝜕be𝜌𝜌𝜌𝜌ω separatedρ − 𝑖𝑖𝑖𝑖 into𝜌𝜌𝜌𝜌 (1 two𝜕𝜕𝜕𝜕) 𝜙𝜙𝜙𝜙− parts as: 𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 𝛒𝛒𝛒𝛒 − 𝐝𝐝𝐝𝐝 2 𝜕𝜕𝜕𝜕𝜌𝜌𝜌𝜌2 4 𝜌𝜌𝜌𝜌 4𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 2 layer and| the impurity𝑉𝑉𝑉𝑉| 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 ion, 𝛒𝛒𝛒𝛒located− 𝐝𝐝𝐝𝐝 at distance d along the z-direction in AlGaAs barrier −1⁄2 𝜕𝜕𝜕𝜕 in 1Eq.⁄2 (1) can be𝜕𝜕𝜕𝜕 separated into two2 parts2 as:𝜔𝜔𝜔𝜔 𝑐𝑐𝑐𝑐 𝜕𝜕𝜕𝜕 by the dielectric constant of GaAs medium. The 𝐻𝐻𝐻𝐻� − �𝜌𝜌𝜌𝜌 2 𝜌𝜌𝜌𝜌 2 � 2 �� 𝐻𝐻𝐻𝐻�ω ρ − 𝑖𝑖𝑖𝑖 − = + ( ) (2) 𝜕𝜕𝜕𝜕𝜌𝜌𝜌𝜌 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 𝜕𝜕𝜕𝜕screened𝜙𝜙𝜙𝜙 𝛒𝛒𝛒𝛒 by− the𝐝𝐝𝐝𝐝 dielectric constant of GaAs medium. TheHamiltonian Hamiltonian is aa harmonic harmonic oscillator type with in Eq. (1) can be𝐻𝐻𝐻𝐻� separated into two parts as: = + ( ) (2) (2) well-known solution.⊥ The effective confinement where: oscillator type with Ĥwell - known Ĥ ⊥ solution. 𝑉𝑉𝑉𝑉 𝜌𝜌𝜌𝜌 The effective confinement Ĥfrequency, is a 𝐻𝐻𝐻𝐻� where: where: = + ( )Ĥ Ĥ ⊥ 𝑉𝑉𝑉𝑉 𝜌𝜌𝜌𝜌 (2) frequency, is a combination of the magnetic combination of the magnetic field cyclotron frequency and parabolic confining 1 1 1 field cyclotron frequency and parabolic confining ⊥ ω 2 where: Ĥ Ĥ 𝑉𝑉𝑉𝑉 𝜌𝜌𝜌𝜌1 = 1 1 + + + (3) ωω22= + c frequency,2 given as: = 2 + 4. Generally4 , there isfrequency, no 2analytical givensolution2 as: available 0 24 . Generally, there is = + + ⁄ + ⁄ 2(3) 𝑐𝑐𝑐𝑐 ( ) = = (4) 2 2 −1 2 𝜕𝜕𝜕𝜕 1 2 𝜕𝜕𝜕𝜕 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 2 2 𝜔𝜔𝜔𝜔 𝜕𝜕𝜕𝜕 ⊥ 4 42 222 2 no analytical| solution| available+ for the complete −11⁄2 1⁄21 Ĥ 1 − �𝜌𝜌𝜌𝜌 𝜌𝜌𝜌𝜌2 2 𝑐𝑐𝑐𝑐� 0 4 �� ω ρ − 𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 for the complete donor𝜔𝜔𝜔𝜔𝜔𝜔𝜔𝜔 impurity𝜕𝜕𝜕𝜕𝜔𝜔𝜔𝜔 Hamiltonian. In this work, we will use the exact = ⊥ + 2 + +2 2 𝜕𝜕𝜕𝜕𝜌𝜌𝜌𝜌 (3)𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 donor𝜕𝜕𝜕𝜕 impurity𝜙𝜙𝜙𝜙 − Hamiltonian.2 In 2this work, we will Ĥ 2 − �𝜌𝜌𝜌𝜌 2𝜌𝜌𝜌𝜌 and�, �� ω ρ − 𝑖𝑖𝑖𝑖 (3) 𝑉𝑉𝑉𝑉 𝜌𝜌𝜌𝜌 𝛒𝛒𝛒𝛒 − 𝐝𝐝𝐝𝐝 − 𝜕𝜕𝜕𝜕𝜌𝜌𝜌𝜌 4 𝜌𝜌𝜌𝜌 4𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 2𝑐𝑐𝑐𝑐 𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 The potential ( ) represents the coulomb interaction� between𝜌𝜌𝜌𝜌 𝑑𝑑𝑑𝑑 the electron in the GaAs −1⁄a2nd𝜕𝜕𝜕𝜕, 1⁄2 𝜕𝜕𝜕𝜕 2 2 diag𝜔𝜔𝜔𝜔 onalization𝜕𝜕𝜕𝜕 technique to solve the donor impurity Hamiltonianuse the exact given by diagonalization Eq. (1) and technique to solve Ĥ⊥ − �𝜌𝜌𝜌𝜌 2 𝜌𝜌𝜌𝜌 2 � 2 �� ω ρ − 𝑖𝑖𝑖𝑖 𝜕𝜕𝜕𝜕𝜌𝜌𝜌𝜌 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 and, 𝜕𝜕𝜕𝜕𝜙𝜙𝜙𝜙 layer and the impurity𝑉𝑉𝑉𝑉 𝜌𝜌𝜌𝜌 the ion, donorlocated at impurity distance d along Hamiltonian the z-direction given in AlGaAs by Eq. barrier (1) and, study its electronic properties. and study its electronic properties. screened by the dielectric constant of GaAs medium. The Hamiltonian is a harmonic 2 For zero donor2 impurity case , Eq. (1) reducesFor to harmonic zero donoroscillator impuritytype case , Eq. (1) ٥ (4) = = ( ) reduces to harmonic oscillator type with⊥ a well- | with a| well-known eigenstates+ |n,m> oscillator and ⊥ typeeigenener (4) withgies well. -Theknown adopted solution. basis The (|n,m> effective = confinement Ĥfrequency, is a known eigenstates |n,m> and eigenenergies. The ٥ Ĥ − , ( , ) ) are harmonic2 2 oscillator type wave functions that have been used to 𝑉𝑉𝑉𝑉 𝜌𝜌𝜌𝜌 𝛒𝛒𝛒𝛒 − 𝐝𝐝𝐝𝐝 − combination of the magnetic field cyclotron frequency=ψ ρϕ,and parabolic confining adopted basis (|n,m> nm, ( ) ) are harmonic ٥ ( V (ρ The potential ( ) representsThe the coulomb potentialdiagonalize𝑛𝑛𝑛𝑛 interaction𝑚𝑚𝑚𝑚 the � represents betweentotal𝜌𝜌𝜌𝜌 Hamiltonian𝑑𝑑𝑑𝑑 the theelectron and coulomb obtain in the the GaAs ground state energy of the impurity 𝜓𝜓𝜓𝜓 𝜌𝜌𝜌𝜌 𝜑𝜑𝜑𝜑 oscillator type wave functions that have been used interaction between the electronfrequency, in giventhe GaAs as: = + . Generally, there is no analytical solution available system. 2 to diagonalize𝑐𝑐𝑐𝑐 the total Hamiltonian and obtain layer and the impurity𝑉𝑉𝑉𝑉 𝜌𝜌𝜌𝜌 ion,layer located and at distancethe impurity d along ion, the locatedz-direction at indistance AlGaAs dbarrier 2 2 𝜔𝜔𝜔𝜔 The basis wave functions are: 0 4 along the z-direction in AlGaAs barrierfor the complete screened donor𝜔𝜔𝜔𝜔the impurity ground𝜔𝜔𝜔𝜔 Hamiltonian. state energy In this of thework, impurity we will usesystem. the exact screened by the dielectric constant of GaAs medium. The Hamiltonian| , > = is( a, harmonic) = ( ) (5) diag,onalization technique, to solve the donor impurity Hamiltonian given by Eq. (1) and 1 𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝜑𝜑𝜑𝜑 Ĥ𝑛𝑛𝑛𝑛⊥𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 oscillator type with well-known solution. The effective confinement𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝜓𝜓𝜓𝜓 frequency𝜌𝜌𝜌𝜌| |𝜑𝜑𝜑𝜑 | ,| is√𝟐𝟐𝟐𝟐 𝝅𝝅𝝅𝝅a 𝑅𝑅𝑅𝑅 ! 𝜌𝜌𝜌𝜌| |𝑒𝑒𝑒𝑒 where, , ( )= study its electronic properties. ( ) (6) 1 2 2 ( |2 |)! −2𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 2 2 combination of the magnetic field cyclotron frequency𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 and parabolic confining and eigen energies 𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 𝑒𝑒𝑒𝑒 𝜌𝜌𝜌𝜌For 𝛼𝛼𝛼𝛼zero �donor𝑛𝑛𝑛𝑛+ 𝑚𝑚𝑚𝑚 impurity𝐿𝐿𝐿𝐿𝑛𝑛𝑛𝑛 𝜌𝜌𝜌𝜌case𝛼𝛼𝛼𝛼 , Eq. (1) reduces to harmonic oscillator type =with(2 a +well| -known| + 1) eigenstates |n,m> Ĥand⊥ eigenenergies. The adopted basis (|n,m> = frequency, given as: = + . Generally, there is no analytical, solution available + m (7) 2 ( , ) ) are harmonic oscillator𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 type wave functions that have been used to 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 , 2 2 | | 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 2 0 where L ( ) is the standard𝐸𝐸𝐸𝐸 associated𝑛𝑛𝑛𝑛 Laguerre𝑚𝑚𝑚𝑚 polynomialsℏ𝜔𝜔𝜔𝜔 [34]. And is an 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 4 diagonalize𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 the total Hamiltonian and obtain the ground state energy of the impurity for the complete donor impurity Hamiltonian.m In 2this2 work, we will𝜓𝜓𝜓𝜓 use𝜌𝜌𝜌𝜌 𝜑𝜑𝜑𝜑 the exact inverse lengthn dimension constant which is given by: 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 system. 𝛼𝛼𝛼𝛼 m diagonalization technique to solve the donor impurity Hamiltonian givenThe basisby Eq.= wave (1) functions and are: (8) ω α � | , > = , ( , ) = , ( ) (5) study its electronic properties. ℏ 1 𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝜑𝜑𝜑𝜑 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝜓𝜓𝜓𝜓 𝜌𝜌𝜌𝜌| |𝜑𝜑𝜑𝜑 | | √𝟐𝟐𝟐𝟐𝝅𝝅𝝅𝝅 𝑅𝑅𝑅𝑅 ! 𝜌𝜌𝜌𝜌| |𝑒𝑒𝑒𝑒 For zero donor impurity case , Eq. (1) reduces to harmonic where, oscillator type , ( )= ( ) (6) 1 2 2 ( |2 |)! −2𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 2٦𝛼𝛼𝛼𝛼 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 2 2 ⊥ =𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 with a well-known eigenstates |n,m> Ĥand eigenenergies. The adoptedand basiseigen energies(|n,m> 𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 𝑒𝑒𝑒𝑒 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 � 𝑛𝑛𝑛𝑛 + 𝑚𝑚𝑚𝑚 𝐿𝐿𝐿𝐿𝑛𝑛𝑛𝑛 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 ( , ) ) are harmonic oscillator type wave functions that have been used to , , = (2 + | | + 1) + m (7) Fig. 1.a 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 diagonalize𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 the total Hamiltonian and obtain the ground state energy of the| | impurity 𝜓𝜓𝜓𝜓 𝜌𝜌𝜌𝜌 𝜑𝜑𝜑𝜑 where L ( ) is the standard𝐸𝐸𝐸𝐸𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 associated𝑛𝑛𝑛𝑛 Laguerre𝑚𝑚𝑚𝑚 polynomialsℏ𝜔𝜔𝜔𝜔 2 [34]. And is an system. m 2 2 inverse lengthn 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼dimension constant which is given by: 𝛼𝛼𝛼𝛼 The basis wave functions are: m = (8) ω | , > = , ( , ) = , ( ) (5) α � ℏ 1 𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝜑𝜑𝜑𝜑 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝜓𝜓𝜓𝜓 𝜌𝜌𝜌𝜌| |𝜑𝜑𝜑𝜑 | | √𝟐𝟐𝟐𝟐𝝅𝝅𝝅𝝅 𝑅𝑅𝑅𝑅 ! 𝜌𝜌𝜌𝜌| |𝑒𝑒𝑒𝑒 where, , ( )= ( ) (6) ٦ !(| 2| ) 2 2 1 −2𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 2 2 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 and eigen energies 𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 𝑒𝑒𝑒𝑒 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 � 𝑛𝑛𝑛𝑛+ 𝑚𝑚𝑚𝑚 𝐿𝐿𝐿𝐿𝑛𝑛𝑛𝑛 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼

, = (2 + | | + 1) + m (7) 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 | | where L ( ) is the standard𝐸𝐸𝐸𝐸𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 associated𝑛𝑛𝑛𝑛 Laguerre𝑚𝑚𝑚𝑚 polynomialsℏ𝜔𝜔𝜔𝜔 2 [34]. And is an m 2 2 inverse lengthn 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼dimension constant which is given by: 𝛼𝛼𝛼𝛼 m = Fig. ( 1.b8)

Fig. 1. The ground state energy of the quantum heterostructure for fixed value of magnetic field ω * Figure (1) * strength (ω )= 2 R and parabolic confinment strength ω= 5.412 R against the number of α � c 0 ℏ * * basis for a) donor impurity at the origin (d = 0 a ) and b) donor impurity at (d = 0.5 a ). Fig. 2. The ground state energy of the quantum heterostructure for absence and presence of

the impurity against the magnetic field strength ωc the dashed line with impurity and solid line for no impurity cases for ω= 5.412 R * . 0 377 ٦ Int. J. Nano Dimens., 10 (4): 375-390, Autumn 2019 Fig. 3. The ground-state for fixed value of ω= 2 R * against the distance for two c ** ω00 , ( ω= 3.044 R dashed line and ω0 = 5.412 R ) the solid line).

١ Fig. 4. The ground-state binding energy against ωc , where * * * * ( ω0 = 3.044 R for dashed line, and ω0 = 5.412 R for solid line ) (a) d = 0 a , (b) d = 0.1 a , and (c) d=0.5a* . * Fig. 5. The binding energy for d=0 a at constant pressure (P=10 kbar) as a function of ωc for ** three temperatures (5K, 100K, and 200K) for (a ) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 6. The binding energy change for d=0 a at constant pressure (P=10 kbar) and ωc = 2 R * with respect to the temperature for( ω0 = 5.412 R * solid line and for ω0 = 3.044 R for the dashed line ). . * Fig. 7. The variation of ground-state binding energy for d=0a against the ωc at fixed temperature (20K) and for three different values of pressure (0, 10, and 20 kbar) a) ** ω00= 3.044 Rb ) ω= 5.412R . Fig. 8. The variation of ground-state binding energy for d = 0 a* against the pressure at fixed * ** temperature (20K) and ωc = 2 R for ( ω00= 3.044 R and ω= 5.412R ). Fig. 9. The ground-state binding energy for d=0.1a* at constant pressure (P=10 kbar) as a

function of ωc and for three temperatures (5K, 100K, and 200K): ** a) ω00= 3.044 R and b) ω= 5.412 R . Fig. 10. The ground-state binding energy for d = 0.1 a* at constant pressure (P=10 kba and * * ωc = 2 R , and temperature for (ω0 = 3.044 R dashed line * and ω0 = 5.412 R for the solid line). * Fig. 11. The variation of binding energy for d=0.1a against the ωc at fixed temperature (20K) ** for three pressure values (0, 10, and 20 kbar) a) for ω00= 5.412 Rb ) and for ω= 3.044 R . Fig. 12. The variation of ground-state binding energy for d=0.1a* as a function of the pressure * ** at fixed temperature (20K) and ωc = 2 R for ω00= 3.044 R and ω= 5.412 R . * Fig. 13. The binding energy for d=0.5a at constant pressure (P=10 kbar) as a function of ωc for ** three temperatures (5K, 100K, and 200K). a) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 14. The binding energy for d=0.5 a at constant pressure (P=10 kbar) and ωc = 2 R against * * temperature for( ω0 = 3.044 R for the dashed line and for ω0 = 5.412 R solid line). Using these harmonic oscillator bases |n, m > , the matrix elements of the full donor 2 2 ( ) = = impurity Hamiltonian (4;) , ( ) , ( ) , can be diagonalized numerically to | | + obtain the desired eigenenergies.𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 In each𝑛𝑛𝑛𝑛́ 𝑚𝑚𝑚𝑚 calculation step the number of basis |n, m > 2 2 �𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 �Ĥ�𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 � 𝑉𝑉𝑉𝑉 𝜌𝜌𝜌𝜌 − − 2 Using2 these harmonic oscillator bases |n, m > , the matrix elements of the full donor ( ) = 𝛒𝛒𝛒𝛒 −=𝐝𝐝𝐝𝐝 will be varied ( until4) a satisfied converging eigen energies are achieved. The stability The potential ( ) represents the |coulomb| interaction�+ between𝜌𝜌𝜌𝜌 𝑑𝑑𝑑𝑑impurity the electron Hamiltonian in the GaAs; ( ) ( ) , can be diagonalized numerically to 2 2 , , − converging procedure is displayed in Fig.1. ( ) = = 2 2 2 2 (4) 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛́ 𝑚𝑚𝑚𝑚 layer and the impurity𝑉𝑉𝑉𝑉 𝜌𝜌𝜌𝜌 𝑉𝑉𝑉𝑉 ion,𝜌𝜌𝜌𝜌 located 𝛒𝛒𝛒𝛒at− distance𝐝𝐝𝐝𝐝 − d along the z-directionobtain the in desiredAlGaAs eigenenergies. barrier�𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 �InĤ �each𝑅𝑅𝑅𝑅 calculation𝜌𝜌𝜌𝜌 � step the number of basis |n, m > ( ) | | ( ) = + =�𝜌𝜌𝜌𝜌 𝑑𝑑𝑑𝑑 The donor impurity (4) binding energy (BE) which is defined as the difference between the The potential represents the coulomb interaction| | between+ the electron in the GaAs − will be varied until a satisfied converging eigen energies are achieved. The stability screened𝑉𝑉𝑉𝑉 by𝜌𝜌𝜌𝜌 the dielectric constant− of− 2GaAs 2medium. The Hamiltonianenergy statesis a harmonicof the Hamiltonian (Eq. 1) without the presence of the impurity (E) and layer and the impurity𝑉𝑉𝑉𝑉 𝜌𝜌𝜌𝜌 ion,𝛒𝛒𝛒𝛒 −located𝐝𝐝𝐝𝐝 𝑉𝑉𝑉𝑉 at𝜌𝜌𝜌𝜌 distance𝛒𝛒𝛒𝛒 d− along𝐝𝐝𝐝𝐝 −the z-direction2 2 converging in AlGaAs procedure barrier is displayed in Fig.1. The potential ( ) representsThe potential the coulomb ( ) represents interaction the� cbetweenoulomb𝜌𝜌𝜌𝜌 𝑑𝑑𝑑𝑑 interaction the electron� between𝜌𝜌𝜌𝜌 in the𝑑𝑑𝑑𝑑 the GaAs electron in the GaAs Using these harmonic oscillator bases |n, m > , the matrix elements of the full donor with its⊥ presence (E0). The donor impurity binding energy (BE) which( ) is defined( ) as the difference between the oscillator type with well-known solution. The effective confinement Using these Ĥfrequency harmonicimpurity, oscillatoris a Hamiltonian base; s |n, , m > , ,the matrix, can be diagonalizedelements numericallyof the full to donor layer and the impurity𝑉𝑉𝑉𝑉screened𝜌𝜌𝜌𝜌 ion, by thelocated dielectric at𝑉𝑉𝑉𝑉 distance𝜌𝜌𝜌𝜌 constant d along of GaAs the zmedium.-direction The in AlGaAsHamiltonian barrier is a harmonic = 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 ́ 𝑚𝑚𝑚𝑚 (9) layer and the impurity ion, located at distance d along the z-directionenergy in AlGaAsstates of barrierthe Hamiltonianobtain the desired (Eq. eigenenergies. 1)�𝑅𝑅𝑅𝑅 without𝜌𝜌𝜌𝜌 �InĤ � each𝑅𝑅𝑅𝑅the calculationpresence𝜌𝜌𝜌𝜌 � step of thethe number impurity of basis (E) |n ,andm > impurity Hamiltonian; ( Using) these( )harmonic, can be diagonalizedoscillator bas numericallyes |n, m > , tothe matrix elements of the full donor combination of the magnetic field cyclotron frequency and parabolic⊥ confiningwill be varied, until a satisf, ied converging eigen energies are achieved. The stability oscillator type with well-known solution. The effective confinementwith Ĥ frequencyits presence, (Eis 0).a 0 screened by the dielectricscreened constant by theof GaAsdielectric medium. constant The of GaAsHamiltonian medium. Theis Hamiltonian a harmonic is a harmonicconverging 𝐵𝐵𝐵𝐵𝐸𝐸𝐸𝐸 procedureimpurity𝐸𝐸𝐸𝐸 −is displayed 𝐸𝐸𝐸𝐸Hamiltonian in Fig.1. ; ( ) ( ) , can be diagonalized numerically to obtain the desired eigenenergies.�𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝜌𝜌𝜌𝜌 �InĤ� each𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛́ 𝑚𝑚𝑚𝑚 calculation𝜌𝜌𝜌𝜌 � step the, number of, basis |n, m > The effectsThe ofdonor =the impurity pressure binding (P)energy and (BE )the which temperature is defined as the (T)difference on thebetween energy the of the = + ⊥ ⊥ 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 (́ 𝑚𝑚𝑚𝑚9) combinationfrequency,oscillator of given thetype magneticas:with well -knownfield cyclotron2solution.. Generally The frequency, effectivethere is noconfinementand analytical parabolic Ĥfrequency solution confining, availableis a obtain the desired eigenenergies. In each calculation step the number of basis |n, m > oscillator type with well-known solution. The effective confinement Ĥfrequencywill, is bea varied until a satisfenergy iedstates converging of the Hamiltonian eigen (Eq. energies1) without� 𝑅𝑅𝑅𝑅the are presence achieved.𝜌𝜌𝜌𝜌 � Ĥof� the𝑅𝑅𝑅𝑅 impurity The𝜌𝜌𝜌𝜌 stability� (E) and 2 2 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 0 ground state can be investigatedwill be0 varied using until aeffective satisfied convergingmass approximation eigen energies method are achieved. The stability combination of the𝜔𝜔𝜔𝜔 magnetic𝜔𝜔𝜔𝜔 4field cyclotron frequency andconverging parabolic procedureconfiningwith 𝐵𝐵𝐵𝐵is 𝐸𝐸𝐸𝐸 displayedits presence𝐸𝐸𝐸𝐸 − (E 𝐸𝐸𝐸𝐸in0). Fig.1. combination offrequency, thefor magneticthe given complete as:field donor=cyclotron impurity+ .frequency Generally Hamiltonian. , andthere parabolic isIn no this analytical work,confining wesolution willThe availableuseeffects the of exact the pressure (P) and the temperature (T) on the energy of the 2 The donor impurity binding energy converging ( BE )= which procedure is defined is displayed as the difference in F ig . (19.) between the 2 2 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 (EMA). The pressure and temperature dependence of the material electron effective frequency, given as: 0 = + . Generally, there is no analytical solution available 0 diagonalization technique to 4solve the2 donor impurity Hamiltonian given by Eq. (1) and 𝐵𝐵𝐵𝐵𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 − 𝐸𝐸𝐸𝐸 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 𝑐𝑐𝑐𝑐 energyground states state of thecan Hamiltonian be investigatedThe (Eq. donor 1)using without impurity effective the binding presence mass energy ofapproximation the ( BEimpurity) which (E)method is anddefined as the difference between the frequency, givenfor as:the complete= + donor. Generally impurity2 , there 2Hamiltonian. 𝜔𝜔𝜔𝜔is no analytical In this solution work, availablewe will use the exact The effects of the pressure (P) and the temperature (T) on the energy of the 2 0 mass, ( , ) and dielectric constant ( , ) are inserted in the impurity 2 2 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 4 energy states of the Hamiltonian (Eq. 1) without the presence of the impurity (E) and studyfor its the electronic0 complete properties. donor impurity Hamiltonian. In this work, withwe( EMA).will its presenceuse The the ∗pressure exact(E0). ground and temperaturestate can be investigateddependence using of effective the material mass approximation electron effective method diagonalization𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 technique4 to solve the donor impurity Hamiltonian given by Eq. (1) and 𝑟𝑟𝑟𝑟 for the complete donor impurity Hamiltonian. In this work, we will use the exactHamiltonian 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 as𝑇𝑇𝑇𝑇 shown belowwith its presence (E𝜖𝜖𝜖𝜖0). 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 (EMA).= The pressure and temperature dependence of the material electron (9) effective diagForonalization zero donor technique impurity to solve case the donor, Eq. impurity (1) reduces Hamiltonian to harmonic mass,given by Eq.(oscillator (1), ) and and type dielectric constant ( , ) are inserted in the impurity diagonalizationstudy technique its electronic to solve properties. the donor impurity Hamiltonian given by Eq. (1) and ∗ mass, ( 1 , ) and dielectric constant ( ,= ) 1are inserted in the impurity (9) 0 𝑟𝑟𝑟𝑟 study its electronic properties. ⊥ = ∗ with a well-known eigenstates |n,m> and eigenenergies. The adoptedHamiltonian𝑚𝑚𝑚𝑚 basis𝑃𝑃𝑃𝑃 as𝑇𝑇𝑇𝑇 shown(|n,m>( ) 𝐵𝐵𝐵𝐵= below𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 − 𝐸𝐸𝐸𝐸 ( )𝜖𝜖𝜖𝜖+ 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇( )𝑟𝑟𝑟𝑟 + ( , ) Ĥ Hamiltonian2𝑚𝑚𝑚𝑚 (𝑃𝑃𝑃𝑃 as𝑇𝑇𝑇𝑇 ,shown) below 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃2 𝑇𝑇𝑇𝑇 2 0 For zero donor impurity case , Eq. (1) reduces to harmonic oscillatorThe effects type of the pressure (P) and the temperature (T) ∗on the energy2 2 of the study its electronic properties.( , ) ) are harmonic oscillator type wave functions that have been used to 𝑒𝑒𝑒𝑒 𝐵𝐵𝐵𝐵𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 − 𝐸𝐸𝐸𝐸 , For zero donor impurity case , Eq. (1) reduces to harmonic oscillator type 1 ∗ 1 ⃗ 1 1 S. AbuzaidĤ 𝜌𝜌𝜌𝜌 et al. ( ) = �The𝑝𝑝𝑝𝑝⃗ 𝜌𝜌𝜌𝜌 effects( )𝐴𝐴𝐴𝐴 +of𝜌𝜌𝜌𝜌 the(�) pressure + 𝑚𝑚𝑚𝑚( (P),𝑃𝑃𝑃𝑃) 𝑇𝑇𝑇𝑇and𝜔𝜔𝜔𝜔 the𝜌𝜌𝜌𝜌 temperature (T) on the energy of the with a well-known eigenstates |n,m> and⊥ eigenenergies. The adopted basis (|n,m>( ) = = 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇( ) + ( 𝑐𝑐𝑐𝑐) + 2 ( , ) (10) 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚with a well-known eigenstates |n,m>Ĥ and⊥ eigenenergies. The adoptedground basis state (|n,m> can =be investigated2 (using, )2 effective2 mass2 approximation method For zero donor𝜓𝜓𝜓𝜓diagonalize impurity𝜌𝜌𝜌𝜌 𝜑𝜑𝜑𝜑 the case total Hamiltonian, Eq. (1) reduces andĤ obtainto harmonic the ground oscillator state typeenergy of the impurity2 ( , ) ( , ) + 𝑒𝑒𝑒𝑒 2 ∗ 2 2 Ĥ 𝜌𝜌𝜌𝜌 ∗ 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 �𝑝𝑝𝑝𝑝⃗ 𝜌𝜌𝜌𝜌 𝐴𝐴𝐴𝐴⃗ 𝜌𝜌𝜌𝜌 � ∗ 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇2𝜔𝜔𝜔𝜔2𝜌𝜌𝜌𝜌 , ( , ) ) (are )harmonic oscillator type wave functions that have been used to ∗ ground𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃state 𝑇𝑇𝑇𝑇 can 𝑐𝑐𝑐𝑐be investigated (10using) effective mass approximation method system., , ) are harmonic oscillator type wave functions that have beenĤ used𝜌𝜌𝜌𝜌 to −�𝑝𝑝𝑝𝑝⃗ 𝜌𝜌𝜌𝜌 𝐴𝐴𝐴𝐴⃗ 𝜌𝜌𝜌𝜌2 �2 2 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝜔𝜔𝜔𝜔 𝜌𝜌𝜌𝜌 with a well-known eigenstates |n,m> and⊥ eigenenergies. The adopted basis (|n,m>(EMA). = The pressure and temperature𝑟𝑟𝑟𝑟 dependence( , ) + of the material (10) electron effective Ĥ For quantum heterostructure𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 made2 of𝑐𝑐𝑐𝑐 GaAs𝑒𝑒𝑒𝑒 the dielectric constant ( , ) and the diagonalize𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚the total HamiltonianThe basis and obtainwave functionsthe ground are:state energy of the impurity For quantum𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃heterostructure−𝑇𝑇𝑇𝑇 �𝜌𝜌𝜌𝜌 2𝑑𝑑𝑑𝑑 2 made of GaAs the 𝜓𝜓𝜓𝜓 The𝜌𝜌𝜌𝜌 𝜑𝜑𝜑𝜑 𝜓𝜓𝜓𝜓diagonalizebasis𝜌𝜌𝜌𝜌 wave𝜑𝜑𝜑𝜑 the functions total Hamiltonian are: and obtain the ground state energy of the impurity ((EMA)., ) The𝑟𝑟𝑟𝑟+ pressure and temperature dependence of the material electron effective , ( , ) ) are harmonic oscillator type wave functions that have been used to ( , ) For quantum heterostructure𝑒𝑒𝑒𝑒 𝜖𝜖𝜖𝜖made𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇of( �GaAs𝜌𝜌𝜌𝜌 , th)𝑑𝑑𝑑𝑑e dielectric constant ( , ) and the mass, anddielectric dielectric− constant constant2 2 areand inserted the electronin the𝑟𝑟𝑟𝑟 impurity system. system. electron effective mass 𝑟𝑟𝑟𝑟( , ) are presented by [10] 𝜖𝜖𝜖𝜖𝑟𝑟𝑟𝑟 𝑃𝑃𝑃𝑃 𝜖𝜖𝜖𝜖𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 diagonalize𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 the total Hamiltonian and obtain| , the> =ground state( , energy) = of the impurity( For) quantum ∗ heterostructure electron effective 𝜖𝜖𝜖𝜖mademass,𝑃𝑃𝑃𝑃 mass 𝑇𝑇𝑇𝑇of �GaAs𝜌𝜌𝜌𝜌( (, ) are th,𝑑𝑑𝑑𝑑 epresented) dielectric and bydielectric [10 constant] constant ( , ) and( the , ) are inserted in the impurity 𝜓𝜓𝜓𝜓 𝜌𝜌𝜌𝜌 𝜑𝜑𝜑𝜑 , , (5) effective(5) ∗ mass ∗ 𝑟𝑟𝑟𝑟 are presented by [10] The basisThe wave basis functions wave functions are: are: Hamiltonian𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 as𝑇𝑇𝑇𝑇 shown below 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 1 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃∗ 𝑇𝑇𝑇𝑇 𝑟𝑟𝑟𝑟 system. 𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝜑𝜑𝜑𝜑 ( , ) 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑟𝑟𝑟𝑟 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 electron( effective) mass ( , )Hamiltonian are presented𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 as by𝑇𝑇𝑇𝑇 shown [10] below 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑛𝑛𝑛𝑛| 𝑚𝑚𝑚𝑚, > =𝜓𝜓𝜓𝜓 , (𝜌𝜌𝜌𝜌, 𝜑𝜑𝜑𝜑) = √𝟐𝟐𝟐𝟐𝝅𝝅𝝅𝝅 𝑅𝑅𝑅𝑅, !( )𝜌𝜌𝜌𝜌| |𝑒𝑒𝑒𝑒 , (5) 12.74 exp( 1.73 × 10 ) exp[9.4 × 10 ( 75.6)] for T < 200 K | , >where, = ( , )| =| | | ( ) 𝑟𝑟𝑟𝑟 ∗1 1 The basis wave functions where, are: , ( )= , , ( ) (5) ∈= (6)𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 (11) 1 2 2 1( |2 |)! 𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝜑𝜑𝜑𝜑 12.74( exp) =( 1𝑚𝑚𝑚𝑚13.73.18𝑃𝑃𝑃𝑃×exp𝑇𝑇𝑇𝑇10( 1.73( )×)exp10+−3[9) exp.4( ×[20)10.4 × +10−(5 ( 75(300,.6)])forT)] for T200< K200 K 1 𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝜑𝜑𝜑𝜑 𝑟𝑟𝑟𝑟 − 𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇 − −2𝜌𝜌𝜌𝜌 𝑛𝑛𝑛𝑛𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝟐𝟐𝟐𝟐𝝅𝝅𝝅𝝅2𝛼𝛼𝛼𝛼𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 ∈=( 2 𝑃𝑃𝑃𝑃)2𝑇𝑇𝑇𝑇 2 ( , ) −3 12−5 1 (11) 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚𝜓𝜓𝜓𝜓 𝜌𝜌𝜌𝜌| |𝜑𝜑𝜑𝜑 | | √ 𝑛𝑛𝑛𝑛 𝑅𝑅𝑅𝑅𝑚𝑚𝑚𝑚 ! 𝜌𝜌𝜌𝜌| |𝑒𝑒𝑒𝑒 , 13.18 exp( 1�.73 × 10−3 ) exp[20( ).4=× 10−5 ( ∗ 300( )])forT+2 2 ( 200) K+ ( , ) | , where,> = 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 ( 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚, , )( =)𝜓𝜓𝜓𝜓= 𝜌𝜌𝜌𝜌 𝜑𝜑𝜑𝜑 ( ) √𝟐𝟐𝟐𝟐𝝅𝝅𝝅𝝅 𝑅𝑅𝑅𝑅 𝑛𝑛𝑛𝑛 ! + 𝑚𝑚𝑚𝑚 𝜌𝜌𝜌𝜌| (11) | 𝑒𝑒𝑒𝑒 𝑛𝑛𝑛𝑛 ( (5) ) (6) (6) − 𝑒𝑒𝑒𝑒𝑃𝑃𝑃𝑃 2 𝑇𝑇𝑇𝑇 −1 ≥ and eigen energies, 𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 𝑒𝑒𝑒𝑒 1 | |,𝜌𝜌𝜌𝜌 | |𝛼𝛼𝛼𝛼 �( |2 |)! 𝐿𝐿𝐿𝐿 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 ∗ ( , ) = 1 + 7.51 2 + ( , ) −1 2 (12) 2 where, ( )= 2 2 ( 12) . 74 exp Ĥ 𝜌𝜌𝜌𝜌 ( 1.73− × 10 )−�exp𝑝𝑝𝑝𝑝3⃗𝑃𝑃𝑃𝑃𝜌𝜌𝜌𝜌[9.4 × 𝐴𝐴𝐴𝐴10⃗ 𝜌𝜌𝜌𝜌 (� , ) −755𝑇𝑇𝑇𝑇( 𝑚𝑚𝑚𝑚.6,−))]+for𝑃𝑃𝑃𝑃0.341𝑇𝑇𝑇𝑇 T <𝜔𝜔𝜔𝜔200𝜌𝜌𝜌𝜌 K , 1 1−2𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 2𝛼𝛼𝛼𝛼 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑟𝑟𝑟𝑟 (6) ∗ (10) ∗ 2 2 2 2 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚(𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝜑𝜑𝜑𝜑|2 |)! ∈= 2 𝑃𝑃𝑃𝑃2�𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑐𝑐𝑐𝑐 г г 0 𝑒𝑒𝑒𝑒 (11) −3 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2� �−5𝑔𝑔𝑔𝑔 ∗ 𝑔𝑔𝑔𝑔 �� 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 −2𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 𝑚𝑚𝑚𝑚𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛+ 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 2 132 .18 exp( 1.73− × 10 ) exp𝑃𝑃𝑃𝑃 [20.4Ĥ×𝜌𝜌𝜌𝜌102 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃(𝑇𝑇𝑇𝑇 𝐸𝐸𝐸𝐸300𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇−1)] �forT𝑝𝑝𝑝𝑝⃗ 𝜌𝜌𝜌𝜌 200𝐴𝐴𝐴𝐴 ≥⃗K𝜌𝜌𝜌𝜌 � 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝜔𝜔𝜔𝜔 𝜌𝜌𝜌𝜌 and eigen energies 𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 , 𝑒𝑒𝑒𝑒= (2 𝜌𝜌𝜌𝜌+ |𝛼𝛼𝛼𝛼 | +� 1) +𝐿𝐿𝐿𝐿 m 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 (7) ( , ) + 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝜓𝜓𝜓𝜓𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝜌𝜌𝜌𝜌 and𝜑𝜑𝜑𝜑 eigen√𝟐𝟐𝟐𝟐𝝅𝝅𝝅𝝅 energies𝑅𝑅𝑅𝑅 ! 𝜌𝜌𝜌𝜌| |𝑒𝑒𝑒𝑒 ( , ) = 1 + 7(.51, ) = 1.519 5.405+ × 10 + + −1 ( 13 ) ( 12 ) (10) | | | | 𝑛𝑛𝑛𝑛+ 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 − −3 𝑃𝑃𝑃𝑃 𝑒𝑒𝑒𝑒 ( , −)5𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚− ( 𝑃𝑃𝑃𝑃, 𝑇𝑇𝑇𝑇+) +2042 0.341 2 𝑐𝑐𝑐𝑐 where, and eigen, (energies)= 𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 𝑒𝑒𝑒𝑒 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 �( ) 𝐿𝐿𝐿𝐿 𝜌𝜌𝜌𝜌 (6)𝛼𝛼𝛼𝛼� г 2 2 −4 𝑇𝑇𝑇𝑇 2 1 2 2 ( |2 |)! ∗ − 𝑔𝑔𝑔𝑔 ( , ) + | | , = (2 + | | + 1) + m 2 − (7) 𝑃𝑃𝑃𝑃𝑟𝑟𝑟𝑟 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇2 � г − 𝑇𝑇𝑇𝑇 −1г � 𝑏𝑏𝑏𝑏𝑃𝑃𝑃𝑃𝑒𝑒𝑒𝑒≥ 𝑐𝑐𝑐𝑐𝑃𝑃𝑃𝑃 0 L ( −2𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 2𝛼𝛼𝛼𝛼 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 For quantum heterostructure ( 𝑚𝑚𝑚𝑚 )𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝜖𝜖𝜖𝜖made�𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇of �GaAs𝜌𝜌𝜌𝜌� th𝑑𝑑𝑑𝑑e dielectric𝑇𝑇𝑇𝑇 constant �� ( 𝑚𝑚𝑚𝑚, ) and the where ) is the𝑚𝑚𝑚𝑚 standard𝐸𝐸𝐸𝐸𝑚𝑚𝑚𝑚 associated𝑛𝑛𝑛𝑛 Laguerre𝑚𝑚𝑚𝑚 2 2 polynomialsℏ𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 [34]. And is an, = 1 + 7.51 𝐸𝐸𝐸𝐸𝑔𝑔𝑔𝑔 +𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝐸𝐸𝐸𝐸𝑔𝑔𝑔𝑔 −𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 −1 2 2 (12) 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 , = (2 + | |𝑛𝑛𝑛𝑛+ 1) + m (12) (7) (7) ( , ) ( , ) + 0.341𝑟𝑟𝑟𝑟 𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 m 𝑒𝑒𝑒𝑒| |2 2 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝑛𝑛𝑛𝑛+ 𝑚𝑚𝑚𝑚 𝐿𝐿𝐿𝐿 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 2 ∗ ( ,For) =quantum1.519 heterostructure5.405 × 10 made of+ GaAs+ th e dielectric constant (13) ( , ) and the and eigen energies where L ( ) is the standard� associated Laguerre polynomials𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 [ 34 ]. And is an г г 𝜖𝜖𝜖𝜖 2𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 �0 𝜌𝜌𝜌𝜌𝑟𝑟𝑟𝑟 𝑑𝑑𝑑𝑑 ٧ inverse lengthn dimension constant𝐸𝐸𝐸𝐸 which𝑛𝑛𝑛𝑛 is given𝑚𝑚𝑚𝑚 by:ℏ 𝜔𝜔𝜔𝜔 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 (� , ) � + 204�� 𝑚𝑚𝑚𝑚 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 | | m𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 electron effective𝛼𝛼𝛼𝛼 mass г are presented𝑔𝑔𝑔𝑔 by𝑔𝑔𝑔𝑔 [10] −4 2 2 2 𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 m 2 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 where L inverse( lengthn) =is the( 2dimension standard+𝐸𝐸𝐸𝐸 where| constant| associated+ 1L()𝑛𝑛𝑛𝑛 whichρα22 +Laguerre𝑚𝑚𝑚𝑚) m is isgiven the by:polynomials ℏ 𝜔𝜔𝜔𝜔 standard [ 34 associated].(7) And is an ( 𝐸𝐸𝐸𝐸,∗𝑔𝑔𝑔𝑔)𝑃𝑃𝑃𝑃=𝑇𝑇𝑇𝑇1.519� 5.405− × 10 + �+ 𝑏𝑏𝑏𝑏𝑃𝑃𝑃𝑃 (13) 𝑐𝑐𝑐𝑐 𝑃𝑃𝑃𝑃 (13) 𝑟𝑟𝑟𝑟 , 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 n m 𝛼𝛼𝛼𝛼 electron effective mass 2( , ) are presented by [10] 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 m 𝑐𝑐𝑐𝑐 + 204𝑇𝑇𝑇𝑇 2 2 =𝜔𝜔𝜔𝜔 α г(𝑚𝑚𝑚𝑚8) 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 −4 2 inverse lengthn dimension constantLaguerre which polynomials is given by: [34].m And is an inverse 𝑔𝑔𝑔𝑔 𝑇𝑇𝑇𝑇∗ | | 𝜌𝜌𝜌𝜌𝑛𝑛𝑛𝑛𝛼𝛼𝛼𝛼𝑚𝑚𝑚𝑚 =2 Where ( , ) is 𝛼𝛼𝛼𝛼 the free (8 electron)𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 mass,� −( , ) is the pressure and� 𝑏𝑏𝑏𝑏temperature𝑃𝑃𝑃𝑃 𝑐𝑐𝑐𝑐𝑃𝑃𝑃𝑃 where L ( ) is the standard𝐸𝐸𝐸𝐸 associated𝑛𝑛𝑛𝑛length Laguerre𝑚𝑚𝑚𝑚 dimension polynomialsℏ𝜔𝜔𝜔𝜔 constant [34ω]. which And is is given an by: Where is the free electron mass,𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃( 𝑇𝑇𝑇𝑇, ) is the pressure and temperature m α m � ω 12.74 exp( 1.73 × 10 )(exp,г [)9.4 × 10 ( 75.6)] for T < 200 K ٧ ( , ) ℏ Where 𝑟𝑟𝑟𝑟is the 0free electron mass, ( m, ) 𝑔𝑔𝑔𝑔is the pressure and temperatureг 2 2 inverse lengthn dimension constant which is given by: = α � dependent Where ∈= energy 𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 is band the free gap (electron for8) GaAsWhere mass, quantum0 Where 0𝐸𝐸𝐸𝐸is( heterostructure, the𝑃𝑃𝑃𝑃 is)𝑇𝑇𝑇𝑇 isthe free the free pressure electron electron at Г 𝑔𝑔𝑔𝑔and point, mass,mass, temperature b= 1.26× isis the pressure (11 and) temperature 𝜌𝜌𝜌𝜌 𝛼𝛼𝛼𝛼 ℏ 𝛼𝛼𝛼𝛼 ( 𝑚𝑚𝑚𝑚−г3 ) [ ( −5 ( 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇))] [ ( )] 13.18 expdependent1.73 energy× 10 band𝑟𝑟𝑟𝑟 12exp gap.74 20 forexp. 4 GaAs× 101 . quantum73 × 10 300 heterostructureexpforTг 9.4 200× at 10 Г K point, b=75 1.26 .6 ×for T < 200 K (11) ٧ ω 0 the pressure𝑔𝑔𝑔𝑔= гand0 temperature dependent𝑔𝑔𝑔𝑔 energy m dependent energy𝑚𝑚𝑚𝑚 band 0 gap for GaAs− quantum𝐸𝐸𝐸𝐸−3 ∈𝑃𝑃𝑃𝑃 heterostructure𝑇𝑇𝑇𝑇𝑔𝑔𝑔𝑔𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑚𝑚𝑚𝑚 ( at Г− 5 point,𝑇𝑇𝑇𝑇 − b=−3 1.26) ×𝐸𝐸𝐸𝐸 [ 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 −5 ( )] = α � dependent 10 eV GP energy a �(8𝑚𝑚𝑚𝑚and) band c = - gap3.77 for× 10 GaAsdependenteV quantumGPa energy𝐸𝐸𝐸𝐸.13 heterostructure.𝑃𝑃𝑃𝑃18 𝑇𝑇𝑇𝑇 bandexp gap1 .73 for at × GaAs Г10 point, quantum b=exp 1.26 20 heterostructure×.4 × 10 at Г300 point, forT b= 1.26200× K ℏ 10(8) −eV GPbanda and gap c 𝑃𝑃𝑃𝑃= -3.77 for × GaAs102 eV− quantum GPa 𝑇𝑇𝑇𝑇. −1 heterostructure𝑃𝑃𝑃𝑃 ≥ 𝑇𝑇𝑇𝑇 − −5 −−11−3 −2 ٦ −3 −1 −1 (12) −1 + 51.�7 + 1 = ( , ٦ ) ω 10 eV GPa and c = -3.77× 10−1 eV GPat a− Г1 . point, b= 1.26( −, 3×)10 eV(− 2,GPa) + 0and.341 c = -3.77 10 eV GPa and c = -3.77× 10∗ 10eV GPeVa GP.a and c = -3.77− × 10 eV GP𝑃𝑃𝑃𝑃a . 2 𝑇𝑇𝑇𝑇 −1 ≥ α � −−32 ( ) Usingℏ these Using harmonicthese harmonic− 1 oscillator oscillator The−1 baseseffective base s | nRydberg, m > − ,3 thein termmatrix×10− of2 eV elementspressure GPa . ofand theг temperature full donorг , is =used1 +as7 .the51 energy0 + −1 (12) −1 −1 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃−3𝑇𝑇𝑇𝑇 −1 � −2 −1 � 𝑔𝑔𝑔𝑔 𝑔𝑔𝑔𝑔 −3 −2�� 𝑚𝑚𝑚𝑚( , ) ( , ) + 0.341 of the full Rydberg donor in𝐸𝐸𝐸𝐸 term𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 of ∗𝐸𝐸𝐸𝐸pressure𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 and temperature is used as the energy Using these harmonic oscillator bases |n, m > , the matrix elements The effective٦ , the matrix elements of the full donor impurity The effective Rydberg in term of pressureг and г 0 impurity Hamiltonian The; effective, ( ) Rydberg, ( in) term, can ofbe pressurediagonalized( , ) =and1 . temperature519numerically5.405 is×to𝑚𝑚𝑚𝑚 10 used𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 as the� energy+ �+ (13) �� 𝑚𝑚𝑚𝑚 unit. The effective Rydberg in term of Thepressure effective and Rydbergtemperature in termis used+ of2042 pressureas the energy and𝐸𝐸𝐸𝐸𝑔𝑔𝑔𝑔 𝑃𝑃𝑃𝑃temperature 𝑇𝑇𝑇𝑇 𝐸𝐸𝐸𝐸𝑔𝑔𝑔𝑔 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇is used as the energy impurityHamiltonian; Hamiltonian Using ;t hese ,harmonic( ) oscillator, ( ) , can bas becanes |diagonalizedn , munit.be> , the numericallymatrixtemperatureг elements to isof used the full as donor the −energy 4 𝑇𝑇𝑇𝑇 unit. 2 obtain the desired eigenenergies.𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 In each𝑛𝑛𝑛𝑛́ 𝑚𝑚𝑚𝑚 calculation step𝑔𝑔𝑔𝑔 the number of basis |n, m > ( , ) = 1.519 5.405 × 10 + + (13) 𝜌𝜌𝜌𝜌 � 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 � − � 𝑏𝑏𝑏𝑏𝑃𝑃𝑃𝑃 𝑐𝑐𝑐𝑐𝑃𝑃𝑃𝑃 + 2042 ٦ Using these harmonic oscillator bases |n, m > , the matrix elements of unit.the full donor�𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 �Ĥ�𝑅𝑅𝑅𝑅 diagonalized numerically𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 to 𝑛𝑛𝑛𝑛́ 𝑚𝑚𝑚𝑚 obtain the desired unit. |n, m > г 𝑇𝑇𝑇𝑇 −4 2 obtain the desiredimpurity eigenenergies. �Hamiltonian𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 �InĤ� eachunit.𝑅𝑅𝑅𝑅; calculation,𝜌𝜌𝜌𝜌 (� ) step , ( the) , numbercan be diagonalized of basis numerically to 𝑔𝑔𝑔𝑔 𝑇𝑇𝑇𝑇 ( ) ( ) eigenenergies.will be In varied each untilcalculation a satisfied step converging the number eigen energies are achieved. The stability 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 � − (14) � 𝑏𝑏𝑏𝑏𝑃𝑃𝑃𝑃 𝑐𝑐𝑐𝑐𝑃𝑃𝑃𝑃 impurity Hamiltonian; , , will, becan varied be diagonalized until a satisf numericallyied converging to eigen energies are achieved. The( , stability) = (14) 𝑇𝑇𝑇𝑇 Using these harmonic oscillatorof basis base sobtain |convergingn, m >the , willdesiredthe procedure matrix be eigenenergies. varied elements is�𝑅𝑅𝑅𝑅 displayed𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 until𝜌𝜌𝜌𝜌 of � InĤthe � ineach𝑅𝑅𝑅𝑅 full 𝑛𝑛𝑛𝑛Fá 𝑚𝑚𝑚𝑚 ig satisfied calculationdonor.𝜌𝜌𝜌𝜌1. � step the number2 (of, basis)2 (|(n,,m, ))>= (14) Using𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 these harmonic Using𝑛𝑛𝑛𝑛́ 𝑚𝑚𝑚𝑚 toscillatorhese harmonic bases oscillator |n, m > ,bas thee smatrix |n, m elements> , |then, mmatrix of> the elementsfull donor of the full donor ∗ 𝑒𝑒𝑒𝑒 2 ( , ) 2 ( , ) ٧ (obtain the desired eigenenergies.�𝑅𝑅𝑅𝑅 𝜌𝜌𝜌𝜌 �InĤ� each𝑅𝑅𝑅𝑅 calculationconverging𝜌𝜌𝜌𝜌 � step procedure the number is displayed of basis in Fig.1. ( , )𝑦𝑦𝑦𝑦= ∗ ∗ (14 convergingwill eigenbe varied energies until a satisf areied converging achieved. eigen The energies𝑅𝑅𝑅𝑅( are,𝑃𝑃𝑃𝑃 )𝑇𝑇𝑇𝑇achieved.(= )2 ( The )stability𝐵𝐵𝐵𝐵 ( , ) 𝑒𝑒𝑒𝑒 = (14) (14) impurity Hamiltonian; , ( ) ,The( )donor, can impurity be diagonalized binding numericallyenergy( , (BE) ) to which is defined 2as the, difference𝜖𝜖𝜖𝜖* 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇,2 𝑎𝑎𝑎𝑎 between𝑦𝑦𝑦𝑦𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 the ∗2 ( , )2 ( , ) ٧ impurity Hamiltonianimpurity; Hamiltonian, ( ) ; , ( , )(, can) be diagonalized, ( ) , can beWhere numerically diagonalized to numerically is the effective to∗ Bohr radius2 which𝑒𝑒𝑒𝑒a( , PT ),given𝑅𝑅𝑅𝑅( ,as:𝑃𝑃𝑃𝑃) 𝑇𝑇𝑇𝑇 ∗ 𝐵𝐵𝐵𝐵 will be varied until a satisfied converging eigen energies are achieved. The stability Where𝑦𝑦𝑦𝑦 (∗ ,Where) is the effectiveB ∗( )Bohr is radius the effective which𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇given𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃as: 𝑇𝑇𝑇𝑇Bohr 𝑒𝑒𝑒𝑒 radius The stabilitydonor impurity convergingconverging binding procedure energy ( BEis isdisplayed) whichdisplayed∗ is indefined F inig .Fig.1. as 1. the𝑅𝑅𝑅𝑅 difference𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝑦𝑦𝑦𝑦 between the𝑒𝑒𝑒𝑒 ∗ 𝑦𝑦𝑦𝑦 ∗ obtain the desired eigenenergies.𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 In each𝑛𝑛𝑛𝑛́ energy𝑚𝑚𝑚𝑚 calculation stateś ofstep the the Hamiltonian number of basis(Eq. 1) |n without, m > the presence of the 𝐵𝐵𝐵𝐵impurity (E) and 𝑅𝑅𝑅𝑅 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 obtain the desiredobtain eigenenergies. �the𝑅𝑅𝑅𝑅�𝑅𝑅𝑅𝑅 desired𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌 � Ineigenenergies.Ĥ� �each𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑛𝑛𝑛𝑛́ �𝑚𝑚𝑚𝑚𝑅𝑅𝑅𝑅 calculation𝑛𝑛𝑛𝑛𝜌𝜌𝜌𝜌𝑚𝑚𝑚𝑚𝜌𝜌𝜌𝜌�𝜌𝜌𝜌𝜌� �InĤ �each𝑅𝑅𝑅𝑅 step𝑛𝑛𝑛𝑛 𝑚𝑚𝑚𝑚 calculation 𝜌𝜌𝜌𝜌theWhere� number step (of basis, the) 𝐵𝐵𝐵𝐵numberis |n the, m >effective of basis | nBohr, m > radius∗𝑅𝑅𝑅𝑅which𝑃𝑃𝑃𝑃 which𝑇𝑇𝑇𝑇𝜖𝜖𝜖𝜖 given𝑃𝑃𝑃𝑃 ( 𝑇𝑇𝑇𝑇given, 𝑎𝑎𝑎𝑎as:) 𝑃𝑃𝑃𝑃as:𝑇𝑇𝑇𝑇 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵 converging procedure is displayed in Fig.1. The donor impurity bindingWhere energy 𝑎𝑎𝑎𝑎( (BE),𝑃𝑃𝑃𝑃 )𝑇𝑇𝑇𝑇 is which the effective is Bohr𝐵𝐵𝐵𝐵 Where radius which𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇givenis 𝑎𝑎𝑎𝑎the 𝑃𝑃𝑃𝑃as:effective𝑇𝑇𝑇𝑇 Bohr radius which𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇given𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃as:𝑇𝑇𝑇𝑇 energy states ofThe the donor Hamiltonian impurity (Eq. binding 1)∗ without energy the (BE presence) which of is thedefined𝑎𝑎𝑎𝑎 impurity𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 as the (E) difference and ( between, ) the will bewill varied be varied until untilwill a satisfabe satisf variediedied untilconvergingconverging a satisfwith iedeigen eigen convergingits energies presence energies areeigen achieved.(Eare energies0). achieved. 𝐵𝐵𝐵𝐵 The are∗ stability achieved. The stability The stability ∗( ) The donor impurity binding energy (BE) whichdefined is defined as asthe the difference difference between between𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃 the𝑇𝑇𝑇𝑇 𝐵𝐵𝐵𝐵the energy states 𝐵𝐵𝐵𝐵 , = ( , ) (15) with its presence (E0). 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ( , ) 2 ( , ) = (15) converging procedureconverging is displayed procedure in Fisigenergy displayed.1. states in Fig of.1 .the Hamiltonian (Eq. 1) without the presence of∗ the( impurity, ) (E) and 2 converging procedure is displayedof the inHamiltonian Fig . 1 . (Eq. 1) without = the presence of 𝐵𝐵𝐵𝐵 ((𝜖𝜖𝜖𝜖9,𝑃𝑃𝑃𝑃) )𝑇𝑇𝑇𝑇 ħ ( , ) ( , ) energy states of the Hamiltonian (Eq. 1) without the presence of the impurity (E) and ( , )𝑎𝑎𝑎𝑎= 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ∗ 2 ∗ 𝜖𝜖𝜖𝜖 ( 𝑃𝑃𝑃𝑃 (,15𝑇𝑇𝑇𝑇 ))ħ = (15) The donor impurity binding energy (BE) which is defined as the difference between the ( , )(=, ) 2 𝐵𝐵𝐵𝐵 ∗ 2( 15 ) (15) 2 The donor impurity binding the energy impurity ( BE with) which (E) its is andpresence defined =with as (E theits0). differencepresence Finally, betweenthe (E effective ). the Rydberg can ∗ be (9 written) as: 𝑚𝑚𝑚𝑚( ,𝑃𝑃𝑃𝑃 )𝑇𝑇𝑇𝑇2 𝑒𝑒𝑒𝑒 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ( , ) The donor impurity binding energy (BE) which is defined as the difference0 between the ∗ 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ħ 𝑚𝑚𝑚𝑚∗ 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑒𝑒𝑒𝑒 0 Finally, the𝐵𝐵𝐵𝐵 effective Rydberg∗ 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃can2 𝑇𝑇𝑇𝑇 beħ written as: 𝐵𝐵𝐵𝐵 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ħ with its presence (E0). energy states ofenergy the Hamiltonian states of the (Eq. Hamiltonian 1) without (Eq.the presence 1) without of thethe𝐵𝐵𝐵𝐵 presenceimpurity𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 of(E)− the and𝐸𝐸𝐸𝐸 impurity (E) and 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇𝐵𝐵𝐵𝐵 ∗ 2 𝑎𝑎𝑎𝑎 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ∗ 2 energy states of the Hamiltonian (Eq. 1) without the presence Finally, 0 ofthe the effective= impurity Rydberg (E) and can be written Finally, as:𝑎𝑎𝑎𝑎 the𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 effective 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 (9𝑒𝑒𝑒𝑒) Rydberg can be written as: 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑒𝑒𝑒𝑒 = − 𝐵𝐵𝐵𝐵The𝐸𝐸𝐸𝐸 effects𝐸𝐸𝐸𝐸 − 𝐸𝐸𝐸𝐸ofFinally, the pressure the effective (P) andRydberg the temperature can beFinally, written (T) as:the on effective 𝑚𝑚𝑚𝑚the 𝑃𝑃𝑃𝑃energy𝑇𝑇𝑇𝑇 𝑒𝑒𝑒𝑒 ofRydberg the can be written as: with =its presencewith (E 0its). presence BE (E E0 ). E 0 ( 9 ) (9) ( , ) 0 The effects of the pressure (P) and the temperature (T) on the( energy, ) = of the ( , ) (16) with its presence (E ). 0 4 ∗ = = 𝐵𝐵𝐵𝐵 𝐸𝐸𝐸𝐸 ( 9 )𝐸𝐸𝐸𝐸 − 𝐸𝐸𝐸𝐸 (9) (2 , () , ) ( , ) = (16) 0 ground state can be investigated using effective ∗ mass approximation𝑒𝑒𝑒𝑒 𝑚𝑚𝑚𝑚( ,𝑃𝑃𝑃𝑃 )𝑇𝑇𝑇𝑇 method 4 ∗ ( , ) 𝐵𝐵𝐵𝐵𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 − 𝐸𝐸𝐸𝐸 = The effectsThe of effects the of the pressure pressure (P) (P) and and the the temperature( , )𝑦𝑦𝑦𝑦= (T) on the energy 2 ∗ of the 2 ( , ()16) ground state0 can be 0investigated using effective (9) mass approximation𝑅𝑅𝑅𝑅( ,𝑃𝑃𝑃𝑃 )𝑇𝑇𝑇𝑇4= method∗ 2 𝑒𝑒𝑒𝑒( 𝑚𝑚𝑚𝑚, ) 𝑃𝑃𝑃𝑃 = 𝑇𝑇𝑇𝑇 (16) (16) 𝐵𝐵𝐵𝐵𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 − 𝐸𝐸𝐸𝐸 𝐵𝐵𝐵𝐵𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 − 𝐸𝐸𝐸𝐸 2 ( , 4) ∗ 𝑦𝑦𝑦𝑦 2 4 ∗ (16) The effects of the pressure (P) and thetemperature temperature (T)(T) on the the energy energy of the of the ground ∗ 𝑒𝑒𝑒𝑒 𝑚𝑚𝑚𝑚2 𝑃𝑃𝑃𝑃�(𝜖𝜖𝜖𝜖𝑇𝑇𝑇𝑇,𝑃𝑃𝑃𝑃 )𝑇𝑇𝑇𝑇 𝑅𝑅𝑅𝑅� ħ𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ∗ 2 2( , ) (EMA). The pressure and temperature dependence𝑦𝑦𝑦𝑦 of∗ the material𝑒𝑒𝑒𝑒 𝑚𝑚𝑚𝑚 2electron𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 effective 𝑒𝑒𝑒𝑒 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 The effects of Thethe pressureeffects of (P) 0the and pressure the temperature (P) and the (T) temperature on the energy (T) onof the energy𝑅𝑅𝑅𝑅 of the𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑦𝑦𝑦𝑦 2 2 𝑦𝑦𝑦𝑦�𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 � ħ 2 ground state can be investigated using effective𝑅𝑅𝑅𝑅 mass𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 approximation 2 method𝑅𝑅𝑅𝑅 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2 𝐵𝐵𝐵𝐵𝐸𝐸𝐸𝐸(EMA).state𝐸𝐸𝐸𝐸 − The can𝐸𝐸𝐸𝐸 pressure be investigated and temperature using dependence effective of the material mass electron�𝜖𝜖𝜖𝜖 effective𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 � ħ ground state can be investigated using effective mass approximation Tablemethod (1) shows the variation of the material� 𝜖𝜖𝜖𝜖physical𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 � ħparameters, like effective�𝜖𝜖𝜖𝜖 mass𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 � ħ groundThe effectsstate cangroundof thebe stateinvestigatedpressure can be(P) using investigatedmass,and effectivethe temperature ( using mass, )effective andapproximation (T) dielectric onmass the approximation method energyconstant ofT able the method ( (1) , shows ) Table are the inserted (1) variation shows in of the thethe impurity material variation physical of parameters the material, like effective mass approximation(EMA). Themethod pressure (EMA). and temperature The pressure dependence and of the material electron effective mass, ( , ) and dielectric∗ Table constant(1) shows the( variation, ) are ofinserted the materialphysical in the physical parameters,impurity parameters like, like effective mass mass and (EMA). The pressure and temperature dependencetemperature of the dependencematerial electron ofT ableandtheeffective dielectricmaterial(1) shows constant, electronthe variation in 𝑟𝑟𝑟𝑟terms of oftheTable the material pressure(1) shows physical and the temperature. variationparameters of , thelike material effective physical mass parameters , like effective mass ground(EMA). state The can pressure(EMA). be and investigatedThe temperature pressure∗ and usingdependenceHamiltonian temperature effective𝑚𝑚𝑚𝑚 of𝑃𝑃𝑃𝑃 dependenceasthe𝑇𝑇𝑇𝑇 shownmaterialmass below ofapproximationelectron the material effective electron method 𝜖𝜖𝜖𝜖effective 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑟𝑟𝑟𝑟 and dielectricdielectric constant, constant, in terms of in the terms pressure of andthe temperature. pressure and Hamiltonianeffective𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 as𝑇𝑇𝑇𝑇mass, shown mass, below( , ) andand dielectricdielectric𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃constant constant𝑇𝑇𝑇𝑇 ( , ) are inserted in the impurity ( , ) ( , ) ( , ) and dielectric( , ) constant, in terms of the pressureand dielectric and temperature. constant, in terms of the pressure and temperature. mass, ( , ) and mass,dielectric constantmass, and dielectric ( , andconstant ) arearedielectric insertedinserted constant∗ inare thetheinsertedand impurityimpurity dielectric in are the inserted Hamiltonian impurityconstant, in the in termsimpurity oftemperature. the pressure and temperature. (EMA). The∗ pressure and∗ temperature dependence of the material1 electron3. Results effective and𝑟𝑟𝑟𝑟 discussion1 s ∗ Hamiltonian𝑟𝑟𝑟𝑟 𝑚𝑚𝑚𝑚 1𝑃𝑃𝑃𝑃 (as𝑇𝑇𝑇𝑇 )shown=𝑟𝑟𝑟𝑟 below ( ) +1 𝜖𝜖𝜖𝜖 ( 𝑃𝑃𝑃𝑃) 𝑇𝑇𝑇𝑇 3+. Results ( ,and ) discussion s Hamiltonian𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 as𝑇𝑇𝑇𝑇 Hamiltonianshown𝑚𝑚𝑚𝑚 belowas𝑟𝑟𝑟𝑟𝑃𝑃𝑃𝑃 asshown 𝑇𝑇𝑇𝑇 shown belowbelow 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ( ) 2 Hamiltonian𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 as𝑇𝑇𝑇𝑇 shown below 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ( ) = (2 ) +3. ,Results( ) and + discussion( , sRESULTS) 2 AND DISCUSSIONS mass, ( , ) and dielectric constant ( ( , )) are inserted 3in. Resultsthe2 impurity and𝑒𝑒𝑒𝑒 discussion s ∗ 3. Results2 2 and discussions 2 , ∗1 2 1 ∗ 1 1 Ĥ 1𝜌𝜌𝜌𝜌 For heterostructure1𝑒𝑒𝑒𝑒 �𝑝𝑝𝑝𝑝⃗ 𝜌𝜌𝜌𝜌 system𝐴𝐴𝐴𝐴⃗∗ 𝜌𝜌𝜌𝜌 made� Forfrom2 𝑚𝑚𝑚𝑚 2 heterostructure GaAs𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 the𝜔𝜔𝜔𝜔 material𝜌𝜌𝜌𝜌 system parameters made are: fromdielectric GaAs 1 ( ) = ( ) = ( )1+ ( 𝑟𝑟𝑟𝑟) ∗( )+(+ ) =(( ), 𝑚𝑚𝑚𝑚) +𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇⃗ ( ,( ) ) + For𝑐𝑐𝑐𝑐 ( heterostructure) + ( system , ()10 )made from GaAs the material parameters are: dielectric Hamiltonian𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 as𝑇𝑇𝑇𝑇 shown2 below( , ) Ĥ2 𝜌𝜌𝜌𝜌 ( , )𝜖𝜖𝜖𝜖 2𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇2 �𝑝𝑝𝑝𝑝⃗2𝜌𝜌𝜌𝜌2 ( 2, 𝐴𝐴𝐴𝐴) 𝜌𝜌𝜌𝜌 � 2 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇2 𝜔𝜔𝜔𝜔2𝜌𝜌𝜌𝜌 ( ) = ( ) + ( ) + (𝑚𝑚𝑚𝑚, )𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇For ∗heterostructure2𝑐𝑐𝑐𝑐 2 ∗ ( system , ) 2 *made 2 + from (the10 GaAs) material the material parameters parameters are: are: dielectric dielectric constant 𝑒𝑒𝑒𝑒 Fig. 15𝑒𝑒𝑒𝑒. The Forbindingconstant 2heterostructure energy for= 12d=0.5.4 system,a effective against made the Rydber ωForfromc at heterostructurefixed GaAs∗ temperature= 5the.926 material2 system 2(20 K) and forparameters made threethe effective from are: GaAs massdielectric the of materialan parameters are: dielectric 2 ( , ) ∗ 2 ∗2 ⃗ (⃗ , ) + 𝑒𝑒𝑒𝑒 𝑒𝑒𝑒𝑒 * Ĥ 𝜌𝜌𝜌𝜌 Ĥ 𝜌𝜌𝜌𝜌 �𝑝𝑝𝑝𝑝⃗ 𝜌𝜌𝜌𝜌 𝐴𝐴𝐴𝐴∗ 𝜌𝜌𝜌𝜌�𝑝𝑝𝑝𝑝⃗� 𝜌𝜌𝜌𝜌Ĥ 𝜌𝜌𝜌𝜌2𝑚𝑚𝑚𝑚𝐴𝐴𝐴𝐴2 𝜌𝜌𝜌𝜌𝑃𝑃𝑃𝑃 �𝑇𝑇𝑇𝑇∗ 𝜔𝜔𝜔𝜔 𝜌𝜌𝜌𝜌𝑚𝑚𝑚𝑚 �𝑃𝑃𝑃𝑃𝑝𝑝𝑝𝑝⃗𝑇𝑇𝑇𝑇𝜌𝜌𝜌𝜌 𝜔𝜔𝜔𝜔 𝜌𝜌𝜌𝜌 2constant𝐴𝐴𝐴𝐴⃗ 𝜌𝜌𝜌𝜌 2� = 12𝑚𝑚𝑚𝑚.4, **𝑃𝑃𝑃𝑃,effective effective𝑇𝑇𝑇𝑇 𝜔𝜔𝜔𝜔 𝜌𝜌𝜌𝜌Rydber Rydber = R5.926= 5.926 meVand the and effective mass of an 𝑚𝑚𝑚𝑚𝑒𝑒𝑒𝑒 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃𝑐𝑐𝑐𝑐 𝑇𝑇𝑇𝑇 pressure 𝑐𝑐𝑐𝑐 values 𝑒𝑒𝑒𝑒 ( 10 )− (0, 10, and (10 20) kbar): a) for ( ω00= 3.044 Rb∗ ) and ω= 5.412 R ). ∗ 1 2 2 1 𝑚𝑚𝑚𝑚 =𝑃𝑃𝑃𝑃 12 𝑇𝑇𝑇𝑇 𝑟𝑟𝑟𝑟 . 4 (10)𝑐𝑐𝑐𝑐 = 5 . 926 (10) Ĥ 𝜌𝜌𝜌𝜌 �𝑝𝑝𝑝𝑝⃗ 𝜌𝜌𝜌𝜌 𝐴𝐴𝐴𝐴⃗ 𝜌𝜌𝜌𝜌 (� , ) For𝑚𝑚𝑚𝑚 +quantum(𝑃𝑃𝑃𝑃 , 𝑇𝑇𝑇𝑇−) 𝜔𝜔𝜔𝜔 heterostructureconstant𝜌𝜌𝜌𝜌+ 2 𝜖𝜖𝜖𝜖made,𝑃𝑃𝑃𝑃2 effective 𝑇𝑇𝑇𝑇of �2GaAs𝜌𝜌𝜌𝜌 Rydber th𝑑𝑑𝑑𝑑e dielectric constant * = 12 .and4 ( the, )effective and∗ the mass of= an5.926 ( ) = ( 𝑒𝑒𝑒𝑒) + (Fig.) 𝑒𝑒𝑒𝑒16. The+𝑟𝑟𝑟𝑟 variationconstantelectron( ,of ground ) (𝜖𝜖𝜖𝜖= =12,-state0.)4.067, bindingeffective + atenergy ambientRydber forconstant d = zero0.5𝑅𝑅𝑅𝑅 =a temperature 5against.926 the,𝑚𝑚𝑚𝑚 effective 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒pressure andand thepressure. at Rydber fixedeffective The mass impurity of an is and the effective mass of an 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 For𝑐𝑐𝑐𝑐( quantum) heterostructure Table 1 (102 )made of GaAs the dielectric constant𝜖𝜖𝜖𝜖 ∗ = (0.067, ) and the 𝑅𝑅𝑅𝑅 𝑚𝑚𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 22 −, −2 2 𝜖𝜖𝜖𝜖2 𝑃𝑃𝑃𝑃2 2𝑇𝑇𝑇𝑇 �𝜌𝜌𝜌𝜌 𝑑𝑑𝑑𝑑∗ 𝑒𝑒𝑒𝑒 electron **∗ at ambient zero temperature∗ and pressure. The impurity is ( , ) + 𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟 temperature∗ (20K)2 2and ω2c = for ω00= 3.044 R and ω= 5.412 R𝑟𝑟𝑟𝑟 . For quantum heterostructureFor 𝑒𝑒𝑒𝑒quantum heterostructure𝜖𝜖𝜖𝜖made𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇of �electronGaAs𝜌𝜌𝜌𝜌 𝜖𝜖𝜖𝜖𝑒𝑒𝑒𝑒made th𝑑𝑑𝑑𝑑𝑃𝑃𝑃𝑃e effective dielectric𝑇𝑇𝑇𝑇of �GaAs𝜌𝜌𝜌𝜌electron massconstantth𝑑𝑑𝑑𝑑e dielectric 𝜖𝜖𝜖𝜖 − =((0, ,.constant067)) areand presentedthe at ( 2ambient, 𝑒𝑒𝑒𝑒) and by2 [thezero10𝑅𝑅𝑅𝑅 ∗] temperature𝜖𝜖𝜖𝜖 𝑚𝑚𝑚𝑚=𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 0and.067𝜖𝜖𝜖𝜖 pressure.𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 The impurity𝑅𝑅𝑅𝑅 is 𝑚𝑚𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 Ĥ 𝜌𝜌𝜌𝜌 ∗ �𝑝𝑝𝑝𝑝⃗ 𝜌𝜌𝜌𝜌 𝐴𝐴𝐴𝐴⃗ 𝜌𝜌𝜌𝜌 �Table 1. The𝑚𝑚𝑚𝑚electronlocated dependence𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 along𝜖𝜖𝜖𝜖𝑟𝑟𝑟𝑟𝑚𝑚𝑚𝑚𝜔𝜔𝜔𝜔= of𝜌𝜌𝜌𝜌 0 zthe. 067axis physical at𝑚𝑚𝑚𝑚 the at parameters distanceambientelectron 𝑟𝑟𝑟𝑟ond,zero 𝑅𝑅𝑅𝑅thewith temperaturepressure the𝑒𝑒𝑒𝑒 presence and𝑚𝑚𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 temperature. and of at apressure. uniformambient Theexternalzero impurity temperature magnetic is and pressure. The impurity is − 2 2 For quantum( heterostructure, ) ∗ 𝜖𝜖𝜖𝜖made𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇of �GaAs𝜌𝜌𝜌𝜌 th𝑑𝑑𝑑𝑑e dielectric𝑚𝑚𝑚𝑚 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 constant𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚 ( , ) and the 𝑚𝑚𝑚𝑚electron𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 effective mass𝑐𝑐𝑐𝑐 are presented ∗ 𝑟𝑟𝑟𝑟 (10 )by [10] 𝑟𝑟𝑟𝑟 located along z axis at∗ the distance d, with the presence of a uniform external magnetic electron effective𝑟𝑟𝑟𝑟 electron mass effective( , ) mass are presented 2( , )by are [Table10 presented] 2. The( , byground) [10] 𝜖𝜖𝜖𝜖state𝑃𝑃𝑃𝑃 ∗(m=0)𝑇𝑇𝑇𝑇 energy𝑒𝑒𝑒𝑒 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 (in𝑇𝑇𝑇𝑇 R*) computed by the exact diagonalization method For quantum heterostructure 𝜖𝜖𝜖𝜖made𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇of �GaAs𝜌𝜌𝜌𝜌 th𝑑𝑑𝑑𝑑e (dielectric, ) constant+ ∗ located along 𝑚𝑚𝑚𝑚and𝑚𝑚𝑚𝑚 the𝑃𝑃𝑃𝑃z axis𝑇𝑇𝑇𝑇 at𝑚𝑚𝑚𝑚 the distance𝑒𝑒𝑒𝑒 d, with the presence of a uniform𝑒𝑒𝑒𝑒 external magnetic 1 locatedfield (B)along𝑚𝑚𝑚𝑚 along z axis the at 𝑚𝑚𝑚𝑚z the direction. distance locatedIn d, thewith fialong 𝑚𝑚𝑚𝑚rstthe steppresence z axis we𝑟𝑟𝑟𝑟 at have𝑚𝑚𝑚𝑚of the a distanceuniformcalculated externald, withthe ground the magnetic presence state of a uniform external magnetic ∗ electron𝑒𝑒𝑒𝑒∗ ( , effective𝑚𝑚𝑚𝑚against) 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 mass expansion ( method, ) are for presented various range by of[10 magnetic] field strength𝜖𝜖𝜖𝜖 (1γ𝑃𝑃𝑃𝑃=𝑇𝑇𝑇𝑇 6.75B (Tesla)). 𝑚𝑚𝑚𝑚−𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃2𝑇𝑇𝑇𝑇 2 𝑟𝑟𝑟𝑟N field (B) along the z direction. In the first step we have calculated the ground state electron effective mass ( , ) are presented by𝑟𝑟𝑟𝑟 [10] 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ∗ For quantum( , ) heterostructure( , ) ( ,made) of GaAs the dielectricfield( (B) constant along the ) (z direction., [) and theIn the( fi*rst step )we] have calculated the ground state 𝜖𝜖𝜖𝜖 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 �𝑟𝑟𝑟𝑟𝜌𝜌𝜌𝜌12.74𝑑𝑑𝑑𝑑 Tableexp 3. The1.73field donoreigenenergy× (B)10 impurity along ground(whereexp the 9z state .m=0)4direction.× energy10 for the (E(InfieldR donorthe), and 75 (B)fi rstdonor.impurity6 along stepfor impurity weTthe of< haveGaAs/AlGaAsz binding200 direction. calculated K energy In heterostructure thethe figroundrst step state weas ahave calculated the ground state ∗ = * 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 eigenenergy (where m=0) for the donor impurity (11 of) GaAs/AlGaAs heterostructure as a 12.74 exp( 112.73.74× exp1012.(741)exp.exp73 ×[(∈910.41×.73𝑃𝑃𝑃𝑃10)𝑇𝑇𝑇𝑇×exp(10[9(BE(.475×R).))610exp )against] for([9 T. the4<75×− 200magnetic3.106) ]Kfor( Tfield< strength20075.6 K)] − for5ωc andT < various200 impurityK position (d) for =𝑚𝑚𝑚𝑚𝑟𝑟𝑟𝑟 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 =𝑟𝑟𝑟𝑟 =𝑟𝑟𝑟𝑟 (13, .18) exp( 1.73 × 10 𝑟𝑟𝑟𝑟) exp [ 20 (11.4) × 10 ( (11) 300)] forT (11200) K electron∈ effective𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ∈mass𝑃𝑃𝑃𝑃 ∈ 𝑇𝑇𝑇𝑇 (𝑃𝑃𝑃𝑃−, 𝑇𝑇𝑇𝑇3 ) are presented−3 −by5 [10−eigenenergy3] −5 (where𝜖𝜖𝜖𝜖 −m=0)𝑃𝑃𝑃𝑃5 𝑇𝑇𝑇𝑇 for the donor eigenenergy*impurity of GaAs/AlGaAs(where m=0) for heterostructure the donor impurity as a of GaAs/AlGaAs heterostructure as a 13.18 exp( 113.73.18× exp1013.(181)exp.exp73 ×[(20101.4.73× 10)×exp10([20.4−300×) exp10eigenenergy)] forT[20( .4300200×𝑃𝑃𝑃𝑃 10(where) K] forT( m=0)200 ω3000 =for K 3.044 ) ]theforTR 𝑇𝑇𝑇𝑇donor. − 200impurity K of GaAs/AlGaAs heterostructure as a ( , ) 12.74 exp( 1.73 × 10 −3 ) exp[9.4 × 10 −(5 75.6)] for T < 200 K − ∗ −3−𝑃𝑃𝑃𝑃 −𝑟𝑟𝑟𝑟 �−3 𝑃𝑃𝑃𝑃 −5𝑇𝑇𝑇𝑇 − 𝑃𝑃𝑃𝑃 −5𝑇𝑇𝑇𝑇 − 𝑇𝑇𝑇𝑇 − * * 12.74 exp( 1.73 ×�10 ) exp�[9.4 �× 10 ( ∈=75𝑃𝑃𝑃𝑃.6)𝑇𝑇𝑇𝑇] forTable T− 43<.− The200 donor K impurity𝑃𝑃𝑃𝑃 energy−5 (E(R2), and donor impurity𝑇𝑇𝑇𝑇 −1 binding energy≥ (BE(R )) against ( 11) 𝑟𝑟𝑟𝑟 − 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇−𝑃𝑃𝑃𝑃 213𝑃𝑃𝑃𝑃.18 exp𝑇𝑇𝑇𝑇(−12 1.73( ,×)10𝑇𝑇𝑇𝑇=−≥1−13+)7exp.51 [20≥.4 × 10+−5 ( 300)] forT 200 = K * (12) ٨ . the𝑃𝑃𝑃𝑃 magnetic+ field 2 strength ( 11 ) ω 𝑇𝑇𝑇𝑇 c and(− 112 ) various impurity (12≥) position (d) −for1 ω0 5.412 R ++7.51 1 =−( , )7.51 + 1 = ( , ) = ∈ 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ( ) −3 378 −5 ( ) ( () , ) (−1, ) + 0.341 −1 ( , ) ( , ) + 0.341Int. J. Nano Dimens., 10 (4): 375-390, Autumn 2019 ٨ (exp( 1.73 ×, 10 ) exp[20.4 × 10 ( 300, ( ,)])forT=, 1−+ 07200.∗341.51 K −3 𝑃𝑃𝑃𝑃+ −5𝑇𝑇𝑇𝑇−1− (12 13.18 ∗ ∗ ( , ) ( , ) + 0.341 0 ٨ г � г г 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 г𝑇𝑇𝑇𝑇 0� � г0 г �� 𝑚𝑚𝑚𝑚 ٨ ≤ exp−3 𝑃𝑃𝑃𝑃( 1𝑚𝑚𝑚𝑚.73𝑃𝑃𝑃𝑃×𝑇𝑇𝑇𝑇 10� 𝑚𝑚𝑚𝑚−)5exp𝑇𝑇𝑇𝑇𝑃𝑃𝑃𝑃 �𝑇𝑇𝑇𝑇−[𝑔𝑔𝑔𝑔9 .�4∗ × 10𝑔𝑔𝑔𝑔 � 𝑔𝑔𝑔𝑔( − 75�.𝑔𝑔𝑔𝑔6� )]𝑚𝑚𝑚𝑚for 𝑃𝑃𝑃𝑃T <�200� 𝑚𝑚𝑚𝑚 𝑔𝑔𝑔𝑔K2 𝑔𝑔𝑔𝑔 𝑇𝑇𝑇𝑇 −1 12.74 − ٨ (𝑟𝑟𝑟𝑟 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃𝐸𝐸𝐸𝐸𝑇𝑇𝑇𝑇 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ( 𝐸𝐸𝐸𝐸, )𝑃𝑃𝑃𝑃г=𝑇𝑇𝑇𝑇 1 + 7.51г 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 + 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 0 (12 � ∈= 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2( , ) = 1.519( 1,5𝑚𝑚𝑚𝑚.405) =𝑃𝑃𝑃𝑃×1𝑇𝑇𝑇𝑇10.519 � 5.405 ×+10�(𝑔𝑔𝑔𝑔+ ) + 𝑔𝑔𝑔𝑔 + (13 ) ( � 11 � ) ( 13𝑚𝑚𝑚𝑚) −1 − 𝑃𝑃𝑃𝑃( −3 ) 𝑇𝑇𝑇𝑇 −[ −5 (≥2 𝐸𝐸𝐸𝐸),]𝑃𝑃𝑃𝑃 2𝑇𝑇𝑇𝑇= 1𝐸𝐸𝐸𝐸.519𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇(5.405, ) × 10( , ) + 0.341+ + (13) ( 13, ).18= exp1 + 7.511.73 × 10 + exp 20.4 × 10 + 204 ∗ 300 + forT 204 (12) 200 K + 2042 г г ( −,4−1) =𝑇𝑇𝑇𝑇 1.519г−4 5.405𝑇𝑇𝑇𝑇2 × 10 2 + −4+ (132 ) − (𝑔𝑔𝑔𝑔 , ) 𝑃𝑃𝑃𝑃 ( ,𝑔𝑔𝑔𝑔 ) + 0.341 𝑇𝑇𝑇𝑇 − г 2 г 𝑇𝑇𝑇𝑇 0 ∗ 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇−3 � 𝐸𝐸𝐸𝐸 −𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 � −5 𝑚𝑚𝑚𝑚 �𝑃𝑃𝑃𝑃 𝑏𝑏𝑏𝑏𝑇𝑇𝑇𝑇𝑔𝑔𝑔𝑔𝑃𝑃𝑃𝑃 𝑐𝑐𝑐𝑐�𝑃𝑃𝑃𝑃 � 𝑏𝑏𝑏𝑏𝑃𝑃𝑃𝑃� 𝑔𝑔𝑔𝑔𝑐𝑐𝑐𝑐𝑃𝑃𝑃𝑃+ 204 𝑔𝑔𝑔𝑔 �� 𝑚𝑚𝑚𝑚 � г 𝑇𝑇𝑇𝑇 0 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 � −4𝐸𝐸𝐸𝐸− 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 2 � 𝑏𝑏𝑏𝑏𝑃𝑃𝑃𝑃 𝑐𝑐𝑐𝑐𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 � − � г 𝑃𝑃𝑃𝑃г 2 𝑔𝑔𝑔𝑔 �� 𝑚𝑚𝑚𝑚𝑇𝑇𝑇𝑇 −1 ≥ 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 ( 𝐸𝐸𝐸𝐸,𝑔𝑔𝑔𝑔 )𝑃𝑃𝑃𝑃=𝑇𝑇𝑇𝑇 1 +𝐸𝐸𝐸𝐸𝑔𝑔𝑔𝑔7.51𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝐸𝐸𝐸𝐸 +𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 � (−, ) = 1.519 5 . 405 × � 10 (12𝑏𝑏𝑏𝑏𝑃𝑃𝑃𝑃) 𝑐𝑐𝑐𝑐𝑃𝑃𝑃𝑃 + + (13) −1 𝑇𝑇𝑇𝑇 + 2042 ( , ) = 1.519 5.405 × 10 ( , +) +( , ) + 0 г . 341 (13) −4 2 ∗ + 2042 𝑔𝑔𝑔𝑔 𝑇𝑇𝑇𝑇 𝑏𝑏𝑏𝑏𝑃𝑃𝑃𝑃 𝑐𝑐𝑐𝑐𝑃𝑃𝑃𝑃 � ٧ ٧ − г −4 г г 2 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 � 0 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 � �𝑇𝑇𝑇𝑇𝑔𝑔𝑔𝑔 𝑔𝑔𝑔𝑔 �� 𝑚𝑚𝑚𝑚 𝑇𝑇𝑇𝑇 ٧ 𝐸𝐸𝐸𝐸𝑔𝑔𝑔𝑔 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 � − 𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃 �𝑇𝑇𝑇𝑇 𝑏𝑏𝑏𝑏𝑃𝑃𝑃𝑃𝐸𝐸𝐸𝐸 𝑃𝑃𝑃𝑃𝑐𝑐𝑐𝑐𝑃𝑃𝑃𝑃𝑇𝑇𝑇𝑇 ٧ (1.519𝑇𝑇𝑇𝑇 5.405 × 10 + + (13 = ( , ) + 2042 г −4 𝑇𝑇𝑇𝑇 2 ٧ 𝐸𝐸𝐸𝐸𝑔𝑔𝑔𝑔 𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 � − � 𝑏𝑏𝑏𝑏𝑃𝑃𝑃𝑃 𝑐𝑐𝑐𝑐𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 ٧

٧

* Fig. 15. The binding energy for d=0.5a against the ωc at fixed temperature (20 K) for three ** pressure values (0, 10, and 20 kbar): a) for ( ω00= 3.044 Rb ) and ω= 5.412 R ). Fig. 16. The variation of ground-state bindingS. Abuzaid energy et for al. d = 0.5 a* against the pressure at fixed ** temperature (20K) and ω2c = for ω00= 3.044 R and ω= 5.412 R . Table 1. The dependence of the physical parameters on the pressure and temperature. Table 2. The ground state (m=0) energy (in R*) computed by the exact diagonalization method 1 against Table expansion 2 method for various range of magnetic field strength (1γ = 6.75B (Tesla)). N Table 3. The donor impurity ground state energy (E(R*), and donor impurity binding energy * (BE(R )) against the magnetic field strength ωc and various impurity position (d) for * ω0 = 3.044 R . Table 4. The donor impurity energy (E(R*), and donor impurity binding energy (BE(R*)) against * the magnetic field strength ωc and various impurity position (d) for ω0 = 5.412 R .

Fig. 1. The ground state energy of the quantum heterostructure for fixed value of magnetic field * * strength (ωc )= 2 R and parabolic confinment strength ω0 = 5.412 R against the number of * * basis for a) donor impurity at the origin (d = 0 a ) and b) donor impurity at (d = 0.5 a ). Fig. 2. The ground state energy of the quantum heterostructure for absence and presence of Figure (2) the impurity against the magnetic field strength ωc the dashed line with impurity and solid line * for no impurity cases for ω0 = 5.412 R . * Fig. 3. The ground-state for fixed value of ωc = 2 R against the distance for two ** ω00 , ( ω= 3.044 R dashed line and ω0 = 5.412 R ) the solid line).

Fig. 4. The ground-state binding energy against ωc , where * * * * ( ω0 = 3.044 R for dashed line, and ω0 = 5.412 R for solid line ) (a) d = 0 a , (b) d = 0.1 a , and (c) d=0.5a* . * Fig. 5. The binding energy for d=0 a at constant pressure (P=10 kbar) as a function of ωc for ** three temperatures (5K, 100K, and 200K) for (a ) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 6. The binding energy change for d=0 a at constant pressure (P=10 kbar) and ωc = 2 R * with respect to the temperature for( ω0 = 5.412 R * solid line and for ω0 = 3.044 R for the dashed line ). . * Fig. 7. The variation of ground-state binding energy for d=0a against the ωc at fixed temperature (20K) and for three different values of pressure (0, 10, and 20 kbar) a) ** ω00= 3.044 Rb ) ω= 5.412R . Fig. 8. The variation of ground-state binding energy for d = 0 a* against the pressure at fixed * ** temperature (20K) and ωc = 2 R for ( ω00= 3.044 R and ω= 5.412R ). Fig. 9. The ground-state binding energy for d=0.1a* at constant pressure (P=10 kbar) as a Fig. 1. The ground state energy of the quantum heterostructure for fixed value of magnetic field function of ωc and for three temperatures (5K, 100K, and 200K): * * strength (ωc )= 2 R and parabolic confinment strength **ω0 = 5.412 R against the number of a) ω00= 3.044 R and b) ω= 5.412 R . * * Fig.basis 10 . forThe a) ground donor- impuritystate binding at the energy origin for (d =d 0= 0.1a ) anda* at b) constant donor impurity pressur eat (P=10 (d = 0.5 kba a and). Fig. 2. The ground state energy* of the quantum heterostructure* for absence and presence of ωc = 2 R , and temperature for (ω0 = 3.044 R dashed line the impurity against the magnetic field strength ωc the dashed line with impurity and solid line and ω= 5.412 R* for the solid line). 0 * for no impurity cases for ω0 = 5.412 R . Fig. 11. The variation of binding energy for d=0.1 a* against the ω at fixed temperature (20K) Fig. 3. The ground-state for fixed value of ω= 2 R * againstc the distance for two Figure c(3) ** for three pressure values (0, 10, **and 20 kbar) a) for ω00= 5.412 Rb ) and for ω= 3.044 R . ω00 , ( ω= 3.044 R dashed line and ω0 = 5.412 R ) the solid line). Fig. 12. The variation of ground- state binding energy for d=0.1a* as a function of the pressure Fig. 4. The ground-state binding* energy against ω**c , where at fixed temperature (20K) and ωc = 2 R for ω00= 3.044 R and ω= 5.412 R . ٢ * * * * ( ω0 = 3.044 R for dashed line, and *ω0 = 5.412 R for solid line ) (a) d = 0 a , (b) d = 0.1 a , Fig. 13. The binding energy for d=0.5a at constant pressure (P=10 kbar) as a function of ωc for * and (c) d=0.5a .= **= Int. J. Nano Dimens., 10 (4): 375-390,three temperatures Autumn 2019 (5K, 100K, and 200K). a) ω00 3.044 R and b) ω 5.412 R . 379 Fig. 5. The binding energy for d=0 a* at constant pressure (P=10 kbar) as a function of ω for Fig. 14. The binding energy for d=0.5 a* at constant pressure (P=10 kbar) and ω= 2 R * againstc **c three temperatures (5K, 100K, and* 200K) for (a ) ω00= 3.044 R and b) ω*= 5.412 R . temperature for( ω0 = 3.044 R for the dashed line and for ω0 = 5.412 R solid line). * * Fig. 6. The binding energy change for d=0 a at constant pressure (P=10 kbar) and ωc = 2 R * with respect to the temperature for( ω0 = 5.412 R * solid line and for ω0 = 3.044 R for the dashed line ). . * Fig. 7. The variation of ground-state binding energy for d=0a against the ωc at fixed temperature (20K) and for three different values of pressure (0, 10, and 20 kbar) a) ** ω00= 3.044 Rb ) ω= 5.412R . Fig. 8. The variation of ground-state binding energy for d = 0 a* against the pressure at fixed * ** temperature (20K) and ωc = 2 R for ( ω00= 3.044 R and ω= 5.412R ). Fig. 9. The ground-state binding energy for d=0.1a* at constant pressure (P=10 kbar) as a

function of ωc and for three temperatures (5K, 100K, and 200K): ** a) ω00= 3.044 R and b) ω= 5.412 R . Fig. 10. The ground-state binding energy for d = 0.1 a* at constant pressure (P=10 kba and * * ωc = 2 R , and temperature for (ω0 = 3.044 R dashed line * and ω0 = 5.412 R for the solid line). * Fig. 11. The variation of binding energy for d=0.1a against the ωc at fixed temperature (20K) ** for three pressure values (0, 10, and 20 kbar) a) for ω00= 5.412 Rb ) and for ω= 3.044 R . Fig. 12. The variation of ground-state binding energy for d=0.1a* as a function of the pressure * ** at fixed temperature (20K) and ωc = 2 R for ω00= 3.044 R and ω= 5.412 R . * Fig. 13. The binding energy for d=0.5a at constant pressure (P=10 kbar) as a function of ωc for ٣ ** three temperatures (5K, 100K, and 200K). a) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 14. The binding energy for d=0.5 a at constant pressure (P=10 kbar) and ωc = 2 R against * * temperature for( ω0 = 3.044 R for the dashed line and for ω0 = 5.412 R solid line). S. Abuzaid et al.

Fig 4.a

Fig 4.b

Fig. 1. The ground state energy of the quantum heterostructure for fixed value of magnetic field * * strength (ωc )= 2 R and parabolic confinment strength ω0 = 5.412 R against the number of basis for a) donor impurity at the origin (d = 0 a* ) and b) donor impurity at (d = 0.5 a* ). Fig. 2. The ground state energy of the quantum heterostructure for absence and presence of

the impurity against the magnetic field strength ωc the dashed line with impurity and solid line * for no impurity cases for ω0 = 5.412 R . * Fig. 3. The ground-state for fixed value of ωc = 2 R against the distance for two ** ω00 , ( ω= 3.044 R dashed line Figand 4 ω.c0 = 5.412 R ) the solid line).

Fig. 4. The ground-state binding energy against ωc , where ( ω= 3.044 R * for dashed line, and ω= 5.412 R * for solid line ) (a) d = 0 a* , (b) d = 0.1 a* , 0 0 Figure 4 and (c) d=0.5a* . * Fig. 5. The binding energy for d=0 a at constant pressure (P=10 kbar) as a function of ωc for ** three temperatures (5K, 100K, and 200K) for (a ) ω00= 3.044 R and b) ω= 5.412 R . * * * Fig. 6. The binding energy change for d=0 a at constant pressure (P=10 kbar) and ωc = 2 R the effective mass of an electron mm= 0.067 e field (B) along the z direction. In the first step we with respect to the temperature for( ω= 5.412 R * have 0 calculated the ground state eigenenergy at ambient zero temperature and pressure. The * solid line and for ω0 = 3.044 R for the dashed line ). . impurity is located along z axis at the distance d, (where m=0)* for the donor impurity of GaAs/ Fig. 7. The variation of ground-state binding energy for d=0a against the ωc at fixed with the presence of a uniformtemperature external (20K) and formagnetic three different valuesAlGaAs of pressure heterostructure (0, 10, and 20 kbar) a) as a function of the ٤ ** ω00= 3.044 Rb ) ω= 5.412R . Fig. 8. The variation of ground-state binding energy for d = 0 a* against the pressure at fixed * ** temperature (20K) and ωc = 2 R for ( ω00= 3.044 R and ω= 5.412R ). 380 Fig. 9. The ground-state binding energy for d=0.1a* at constantInt. pressure J. Nano (P=10 Dimens., kbar) as a 10 (4): 375-390, Autumn 2019

function of ωc and for three temperatures (5K, 100K, and 200K): ** a) ω00= 3.044 R and b) ω= 5.412 R . Fig. 10. The ground-state binding energy for d = 0.1 a* at constant pressure (P=10 kba and * * ωc = 2 R , and temperature for (ω0 = 3.044 R dashed line * and ω0 = 5.412 R for the solid line). * Fig. 11. The variation of binding energy for d=0.1a against the ωc at fixed temperature (20K) ** for three pressure values (0, 10, and 20 kbar) a) for ω00= 5.412 Rb ) and for ω= 3.044 R . Fig. 12. The variation of ground-state binding energy for d=0.1a* as a function of the pressure * ** at fixed temperature (20K) and ωc = 2 R for ω00= 3.044 R and ω= 5.412 R . * Fig. 13. The binding energy for d=0.5a at constant pressure (P=10 kbar) as a function of ωc for ** three temperatures (5K, 100K, and 200K). a) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 14. The binding energy for d=0.5 a at constant pressure (P=10 kbar) and ωc = 2 R against * * temperature for( ω0 = 3.044 R for the dashed line and for ω0 = 5.412 R solid line). * Fig. 15. The binding energy for d=0.5a against the ωc at fixed temperature (20 K) for three ** pressure values (0, 10, and 20 kbar): a) for ( ω00= 3.044 Rb ) and ω= 5.412 R ). Fig. 16. The variation of ground-state binding energy for d = 0.5 a* against the pressure at fixed ** temperature (20K) and ω2c = for ω00= 3.044 R and ω= 5.412 R .

Table 1. The dependence of the physicalS. Abuzaid parameters et al. on the pressure and temperature. Table 2. The ground state (m=0) energy (in R*) computed by the exact diagonalization method 1 against expansion method for various range of magnetic field strength (1γ = 6.75B (Tesla)). N Table 3. The donor impurity ground state energy (E(R*), and donor impurity binding energy * (BE(R )) against the magnetic field strength ωc and various impurity position (d) for Table 3 * ω0 = 3.044 R . Table 4. The donor impurity energy (E(R*), and donor impurity binding energy (BE(R*)) against * the magnetic field strength ωc and various impurity position (d) for ω0 = 5.412 R .

* Fig. 15. The binding energy for d=0.5a against the ωc at fixed temperature (20 K) for three ** pressure values (0, 10, and 20 kbar): a) for ( ω00= 3.044 Rb ) and ω= 5.412 R ). Fig. 16. The variation of ground-state binding energy for d = 0.5 a* against the pressure at fixed ** temperature (20K) and ω2c = for ω00= 3.044 R and ω= 5.412 R . Table 1. The dependence of the physical parameters on the pressure and temperature. Table 2. The ground state (m=0) energy (in R*) computed by the exact diagonalization method 1 against expansion method for various range of magnetic field strength (1γ = 6.75B (Tesla)). N Table 3. The donor impurity ground state energy (E(R*), and donor impurity binding energy * functionfunction of the of magnetic the magnetic field fieldstrength strength with withimpurity impurity located located at the at origin the origin (d=0) (d=0) for for (BE( R )) against the magnetic field strength ωc and various impurity position (d) for * ω= 3.044 R . 𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐 functiontwo specifictwo0 of specific the values magnetic values of the fieldof confinemen the strength confinemen t 𝜔𝜔𝜔𝜔frequency witht 𝜔𝜔𝜔𝜔frequency impurity strength strength located = at 5the.412= origin5 .412, and(d=0) , and for= = TableTable 44 . The donor impurity energy (E(R*), and donor impurity binding energy (BE(R*)) against 𝑐𝑐𝑐𝑐 * ∗ ∗ the magnetic field strength ω c and various impurity position (d) for ω0 = 5.412 R . 0 0 0 0 two3.044 specific3 .R044. The Rvalues .accuracy The ofaccuracy the of confinemenour of obtained our obtainedt 𝜔𝜔𝜔𝜔 frequencyresults results are strengthtested are tested against ω against= theω5. 412corresponding the 𝑅𝑅𝑅𝑅corresponding, and𝑅𝑅𝑅𝑅 ω ones=ω ones ∗ ∗ ∗ 0 0 3.044 R . The accuracy of our obtained results are tested againstω the corresponding𝑅𝑅𝑅𝑅 ω ones producedproduced by by expansion expansion method method [35] [.T35able] .T able2 shows 2 shows the comparisonthe comparison between between the the ∗ 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝑵𝑵𝑵𝑵 producedground state by (m=0) expansion 𝑵𝑵𝑵𝑵computed method energy [35 for] .T theable present 2 shows exact the diagonalization comparison between method theand ground state𝟏𝟏𝟏𝟏 (m=0) computed energy for the present exact diagonalization method and 𝑵𝑵𝑵𝑵 ground state (m=0) computed energy for the present exact diagonalization method and the availablethe available corresponding corresponding energy energy produced produced by expansionby expansion method. method. The comparisonThe comparison 𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏 𝑵𝑵𝑵𝑵 theshows available a good corresponding agreement between energy bothproduced methods. by Toexpansion test𝑵𝑵𝑵𝑵 the method.convergence The comparisonissue of our shows a good agreement between both methods.𝟏𝟏𝟏𝟏 To test the convergence issue of our 𝑵𝑵𝑵𝑵 showsexactexact diagonalizationa good diagonalization agreement technique, between technique, we both have we methods. haveplotted plotted To in testF igin .the1a F ig ,convergence1.1ab, the,1b ,computed the issuecomputed groundof our ground

exactstate stateenergiesdiagonalization energies (E) of (E) thetechnique, of donor the donor impuritywe haveimpurity Hamiltonianplotted Hamiltonian in F igagainst.1a against,1 bthe, the number thecomputed number of basis groundof basis(n) (n)

statefrom energiesfrom1 to 38 1 tofor (E) 38 frequency forof thefrequency donor = impurity 5.412= 5 . 412Hamiltonian , impurity , impurity distanceagainst distance thed= 0number d= 0and of 0.5 and basis 0.5 , and(n) , and ∗ ∗ ∗ ∗ ∗ ∗ o o fromat ma 1atgnetic to ma 38gnetic fieldfor frequency fieldstrength strength ω == 2ω 5 =.412. 2The 𝑅𝑅𝑅𝑅 .figures The, impurity𝑅𝑅𝑅𝑅 figures clearly distance clearly show dshow the= 0 numerical𝑎𝑎𝑎𝑎 the and numerical𝑎𝑎𝑎𝑎 0.5 stability𝑎𝑎𝑎𝑎 ,stability and𝑎𝑎𝑎𝑎 in in * * function of the magnetic field strength with impurityω located at the origin (d=0)distance for d= 0 a and 0.5∗ a , and at magnetic field∗ ∗ magnetic field strength c with impurity located o ∗ ∗ at magnetic fieldstrength strength ωωc= ω2 c𝑅𝑅𝑅𝑅 . The The𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 figures figures clearly clearly show theshow numerical𝑎𝑎𝑎𝑎 the stability𝑎𝑎𝑎𝑎 in function of the magnetic field strength with impurityat locatedthe origin at the (d=0) origin𝑐𝑐𝑐𝑐 for (d=0) two for specific valuesour ofcomputedour the computed scheme. scheme. The groundThe ground state stateapproaches approaches a limiting a limiting value value as the as number the number of of two specific values of the confinement 𝜔𝜔𝜔𝜔frequency strength = 5.412 , and = ∗ confinement frequency strength numerical stabilityc in our computed scheme. The 𝑐𝑐𝑐𝑐 ourbasis computed increases.∗ scheme. Theω ground𝑅𝑅𝑅𝑅 state approaches a limiting value as the number of two specific values of the confinement 𝜔𝜔𝜔𝜔frequency strengthand = 53.044.412 R* , Theand accuracy= of our0 obtainedbasis increases.ground0 state approaches a limiting value as the 3.044 R . The accuracy of our obtained results are tested againstω the corresponding𝑅𝑅𝑅𝑅 ω ones ∗ number of basis increases. ∗ results 0are tested against the0 correspondingbasis ones increases. 3.044 R . The accuracy of our obtained results are tested againstω the corresponding𝑅𝑅𝑅𝑅 ω ones Fig. 2 displays the energy of the donor impurity 1 Fig.2Fig display.2 displays the senergy the energy of the of donor the donor impurity impurity energy energy as a functionas a function of the of magnetic the magnetic ∗ produced by expansionproduced method by [ 35] .T expansionable 2 shows method the comparison [35] .Table between the 𝟏𝟏𝟏𝟏 energy as a function of the magnetic field strength N Fig.2 displays the energy of the donor impurity energy as a function of the magnetic produced by expansion method [35] .Table 𝑵𝑵𝑵𝑵2 shows2 showsthe comparison the comparison between the between the field ground fieldstrength strength , for for , confinement confinementfor confinement frequency frequencyfrequency = 5. 412= 5R.412 . TheR . . solidThe solidplot ofplot the of the 𝟏𝟏𝟏𝟏 ground state (m=0) computed energy for the present exact diagonalization method and ∗ ∗ 𝑵𝑵𝑵𝑵 state (m=0) computed energy for the present Thec solidc plot of the system indicates0 0 the absence ground state (m=0) computed energy for the present exact diagonalization method and fieldsystem strengthsystem indicates indicates ω the, forω absence theconfinement absence of the of impurity, frequencythe impurity, and ω the and=ω dashed5 the.412 dashedR one . The indicates one solid indicates theplot presence theof thepresence exact diagonalization method and the available of the impurity, and the dashed one indicates∗ the the available corresponding energy produced by expansion method. The comparisonc 0 𝟏𝟏𝟏𝟏 1 system indicatesωpresence the absence of the of the impurity. impurity, It isand ωclear the dashedfrom Fig. one 2 indicatesthat the presence the available corresponding energy produced by expansioncorresponding method. The energy comparison produced by expansionof theof impurity. the impurity. It is clearIt is clearfrom fromFig. 2Fig. that 2 thethat effect the effect of the of impurity the impurity is to isdecreas to decrease the e the 𝑵𝑵𝑵𝑵 N the effect of the impurity is to decrease the energy shows a good agreement𝟏𝟏𝟏𝟏 between both methods. To test the convergence issue of our method. The comparison shows a good agreement 𝑵𝑵𝑵𝑵 ofenergy theenergy impurity. of the ofof system.Itthe theis clearsystem. system. The from Thepresence Fig. The presence 2 presence thatof thedonor of effect donor ofimpurity donorof impuritythe impuritylowers impurity lowers the is to enerthe decreas gyener ofegy thethe of the shows a good agreement between both methods. To test the convergence issue of our exact diagonalizationbetween technique, both we havemethods. plotted To in test Fig .the1a , 1convergenceb, the computed groundlowers the energy of the heterostructure energy issue of our exact diagonalization technique,energyheterostructureheterostructure of the system.energy energy due The to due presenceit negativeto it negative of coulomb donor coulomb attraction.impurity attraction. lowers the energy of the exact diagonalization technique, we have plotted in Fig.1a ,1b, the computed ground due to it negative coulomb attraction. state energies (E) ofwe the havedonor plottedimpurity Hamiltonian in Fig. 1aagainst ,1b, the the number computed of basis (n) heterostructure energyThe due energy to it negative of the coulomb heterostructure attraction. shows a state energies (E) of thefunction donor ofimpurity the magnetic Hamiltonian field strength groundagainst the state with number energiesimpurity of basis located (E) (n) of at the the donor origin impurity(d=0) for significant dependence on the impurity position. = 5.412 The energyThe energy of the of heterostructurethe heterostructure shows shows a significant a significant dependence dependence on the on impuritythe impurity from 1 to 38 for frequencyHamiltonian against ,the impurity number distance of basis d= 0 (n) fromand 0.5 Increasing , and the impurity distance (d), changing the 𝑐𝑐𝑐𝑐 ∗ ∗ ∗ from 1 to 38 for frequencytwo specific= 5. 412values of, impurity the confinemen distance1 to t 38 𝜔𝜔𝜔𝜔dofrequency= for0 frequencyand strength 0.5 , and= 5.412 , andThe impurityposition. energyposition.= Increasing of Increasingthe heterostructurethe impuritythe impurity distanceshows distance a(d) significant, changing(d), changing dependence the systemthe system onfrom the from 2Dimpurity to2D 3D to 3D at magnetic field strength ω = 2 . The𝑅𝑅𝑅𝑅 figures clearly show the numerical𝑎𝑎𝑎𝑎 stability𝑎𝑎𝑎𝑎 system in from 2D to 3D (bulk), as displayed clearly in ∗ ∗ ∗ ∗ o ∗ 0 0 at magnetic field strength3.044 ω =R 2. The . Theaccuracy𝑅𝑅𝑅𝑅 figures of clearlyour obtained showc theresults numerical𝑎𝑎𝑎𝑎 are tested stability𝑎𝑎𝑎𝑎 againstω in the corresponding𝑅𝑅𝑅𝑅 position.(bulk),ω ones(bulk), a sIncreasing displayed as displayed theclearly impurity clearly in F igindistance.3 .F ig For.3. (d) fixedFor, changing fixedvalues values ofthe impurity systemof impurity fromposition position2D (d),to 3D the(d), the our computed scheme. Theω ground𝑅𝑅𝑅𝑅 state approaches a limiting value as the number of ∗ ∗ c Int. J. Nano Dimens., 10 (4): 375-390, Autumn 2019 381 our computed scheme. Theω ground𝑅𝑅𝑅𝑅 state approaches a limiting value as the number of (bulk), as displayed clearly in Fig.3. For fixed values of impurity position (d), the produced bybasis increases.expansion method [35] .Table 2 shows the comparison between the 𝟏𝟏𝟏𝟏 basis increases. 𝑵𝑵𝑵𝑵 ground state (m=0) computed energy for the present exact diagonalization method and ٩ ٩ Fig.2 displays the energy of the donor impurity energy as a function of the magnetic Fig.2 displays the energy of the donor impurity energy as a function of the magnetic the available corresponding energy produced by expansion method.= 5.412 TheR comparison ٩ field strength , for confinement frequency𝟏𝟏𝟏𝟏 . The solid plot of the ∗ field strength , for confinement frequency c= 5.412R . The solid𝑵𝑵𝑵𝑵 plot of 0the shows a goodsystem agreement indicates ωbetween the absence both ofmethods. the impurity, To test and theω the convergence dashed one issue indicates of our the presence ∗ c 0 system indicatesω the absence of the impurity, andω the dashed one indicates the presence exact diagonalizationof the impurity. technique, It is clear we havefrom plottedFig. 2 thatin F theig. 1aeffect ,1b , ofthe the computed impurity groundis to decreas e the of the impurity. It is clear from Fig. 2 that the effect of the impurity is to decrease the state energiesenergy (E) ofof thethe donorsystem. impurity The presence Hamiltonian of donoragainst impurity the number lowers of basisthe ener(n) gy of the energy of the system. The presence of donor impurity lowers the energy of the from 1 to 38heterostructure for frequency energy = due5.412 to it negative , impurity coulomb distance attraction. d= 0 and 0.5 , and ∗ ∗ ∗ o heterostructure energy dueat ma tognetic it negative field strengthcoulomb attraction.ω = 2 . The𝑅𝑅𝑅𝑅 figures clearly show the numerical𝑎𝑎𝑎𝑎 stability𝑎𝑎𝑎𝑎 in The energy of the heterostructure∗ shows a significant dependence on the impurity ωc 𝑅𝑅𝑅𝑅 The energy of the heterostructureour computed shows scheme. a significantThe ground dependence state approaches on the a limitingimpurity value as the number of position. Increasing the impurity distance (d), changing the system from 2D to 3D position. Increasing thebasis impurity increases. distance (d), changing the system from 2D to 3D (bulk), as displayed clearly in Fig.3. For fixed values of impurity position (d), the

(bulk), as displayed clearlyFig.2 displayin Fig.3s . the For energy fixed ofvalues the donor of impurity impurity position energy (d),as athe function of the magnetic field strength , for confinement frequency = 5.412R . The solid plot of the ∗ c 0 ٩ system indicatesω the absence of the impurity, andω the dashed one indicates the presence

٩ of the impurity. It is clear from Fig. 2 that the effect of the impurity is to decrease the energy of the system. The presence of donor impurity lowers the energy of the heterostructure energy due to it negative coulomb attraction.

The energy of the heterostructure shows a significant dependence on the impurity position. Increasing the impurity distance (d), changing the system from 2D to 3D

(bulk), as displayed clearly in Fig.3. For fixed values of impurity position (d), the

٩

S. Abuzaid et al.

Fig. 5.a

Fig. 1. The ground state energy of the quantum heterostructure for fixed value of magnetic field * * strength (ωc )= 2 R and parabolic confinment strength ω0 = 5.412 R against the number of basis for a) donor impurity at the origin (d = 0 a* ) and b) donor impurity at (d = 0.5 a* ). Fig. 2. The ground state energy of the quantum heterostructure for absence and presence of

the impurity against the magnetic field strength ωc the dashed line with impurity and solid line * for no impurity cases for ω0 = 5.412 R . * Fig. 3. The ground-state for fixed value of ωc = 2 R against the distance for two ** ω00 , ( ω= 3.044 R dashed line and ω0 = 5.412 R ) the solid line).

Fig. 4. The ground-state binding energy against ωc , where = * = * * * ( ω0 3.044 R for dashed line, and ω0 5.412 R for solid line ) (a) d = 0 a , (b) d = 0.1 a , and (c)Fig. d=0.5 5.b a* .

Fig. 5. The binding energy for d=0 a* at constant pressure (P=10 kbar) as a function of ω for Figure 5 c ** three temperatures (5K, 100K, and 200K) for ( a ) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 6. The binding energy change for d=0 a at constant pressure (P=10 kbar) and ωc = 2 R * with respect to the temperature for( ω0 = 5.412 R * solid line and for ω0 = 3.044 R for the dashed line ). . * Fig. 7. The variation of ground-state binding energy for d=0a against the ωc at fixed temperature (20K) and for three different values of pressure (0, 10, and 20 kbar) a) = **= value of magnetic field Fig. ω1.00 The 3.044 groundRb state ) ω energy 5.412R of the. quantum heterostructure for fixed٥ Fig. 8. The variation of ground-state binding energy* for d = 0 a* against the pressure at fixed * Fig. 1. The ground state energyFig. 1. ofThe the ground quantum state heterostructure energy strengthof the quantum for(ω cfixed )= 2 heterostructurevalueR and of parabolic confinment strength magnetic for field fixed value of magneticω0 = 5.412 field R against the number of * * *** * * * * strengthtemperature(ω )= 2 R and (20K) parabolic confinment strength and ωc = 2 R for ( ω00= 3.044 ωR= 5.412 and ωR against= 5.412R the number ). of strength (ωc )= 2 R and parabolic confinment strength c ωbasis0 = 5.412 for a)R donoragainst impurity the number0 at the of origin (d = 0 a ) and b) donor impurity at (d = 0.5 a ). * * * basis for a) donor impurityFig.basis at 9 . the Thefor origin a)ground donor (d- =state impurity 0 a *binding) andFig. at b)the2 .energy donorThe origin ground impurityfor (d d=0.1= state0 a ata) energyand(d at = constant b)0.5 donorof a *the). pressure impurityquantum (P=10 at heterostructure (d =kbar) 0.5 aas) .a for absence and presence of Fig. 2. The ground state energyFig. 2. ofThe the ground quantumfunction state heterostructureof energy ωthec and impurityof for the three quantum for against absencetemperatures heterostructurethe and magnetic presence (5K, 100K,field for of strength absenceand 200K) ωandc: the presencedashed of line with impurity and solid line ** * the impurity against the magnetic field strength ωc the dashed line with impurity and solid line the impurity against the magnetic field strength ωc a) the dashedω00= 3.044 lineR with and b) ω impurityfor and= 5.412 Rno solid impurity line . cases for ω0 = 5.412 R . * * * * for no= impurity cases for ω0 = 5.412 R . forFig. no 10 impurity. The ground cases-state for binding ω0 5.412 energyFig.R 3. . The for dground = 0.1 -astate at constant for fixed pressurvalue ofe (P=10ωc = 2 kbaR against and the distance for two * * * * = ** Fig. 3. The ground-state forFig. fixed 3. The value ω groundc = of 2 Rω-state,= and 2 R for temperatureagainst fixed value the distance for (of ωωc 0 = for 2 3.044 R twoagainstR dashed line the distance for two c ω00 , ( ω= 3.044 R dashed line and ω0 = 5.412 R ) the solid line). ****= * ω , ω= 3.044 R dashed line ω00 , ( ωand= ω 3.044 R=and 5.412 ω dashed line 0 R 5.412 R the solidand for the solid line). ωline).0 = 5.412 R ) the solid line). 00( 0 ) Fig. 4. The ground-state binding energy against ωc , where * * * * * Fig. 11. The variationFig. 4of. Thebinding ground energy-state for binding d=0.1a energy against against the ω cω atc , fixedwhere temperature (20K) Fig. 4. The ground-state binding energy ( ω0 against= 3.044 ωRc , for dashed line, where and ω0 = 5.412 R for solid line ) (a) d = 0 a , (b) d = 0.1 a , ** * * * ** * * * ( ωfor= three 3.044 pressureR for dashed line, values (0, 10, and and ω20 =kbar) 5.412 a) Rfor for solid line ω00= 5.412 Rb) (a) ) and for ω d = 0 a , = (b)* 3.044 R d = 0.1 .a , ( ω0 = 3.044 R for dashed line,0 and ω0 = 5.412 R for solid line0 ) (a) d = 0 a , (b) d = 0.1 aand, (c) d=0.5a . * Fig. 12. The variation of ground* -state binding energy for* d=0.1a * as a function of the pressure and (c) d=0.5a Fig.. 5. Theand binding (c) d=0.5 energya . for d=0 a at constant pressure (P=10 kbar) as a function of ωc for * * ** * = = = ** Fig. 5. Theat fixed binding temperature energy for (20K) d=0 and at ω constantc 2 R pressurefor ω00 3.044 (P=10R kbar) and ω as a function 5.412 R of. ωc for Fig. 5. The binding energy for d=0 a at constant pressurethree (P=10 temperatures kbar) as a function (5K, 100K, of andωc for 200K) for (a ) ω00= 3.044 R and b) ω= 5.412 R . * Fig. 13. The binding energy for d=0.5a at** constant pressure (P=10** kbar) as a function of ω for three temperatures (5K, 100K,three andtemperatures 200K) for (5K,a 100K,ω= 3.044 and 200K)R and b) ω for (a ) ω00== 5.412 R 3.044 R . and b) ω* = 5.412 Rc . * ( ) 00Fig. 6. The binding energy change for d=0 a at constant pressure (P=10 kbar) and ωc = 2 R * *** three temperatures* (5K, 100K, and 200K). a) ω= 3.044 R and b) ω* = 5.412 R . * Fig. 6. The binding energy change for d=0 a at constant00 pressure= (P=10 kbar) and ωc = 2 R Fig. 6. The binding energy change for d=0 a at constant pressure (P=10 kbar)with and respect ωc to 2 Rthe temperature for( ω0 = 5.412 R * Figure 6 * * Fig. 14. The binding energy for d=0.5 a at constant* pressure (P=10 kbar) and ω*= 2 R against with respect to the= temperature for( ω0 = 5.412 R c with respect to the temperature for( ω0 5.412 R solid line and for ω0 = 3.044 R for the dashed line ). . * * * temperature for( ω=* 3.044 R for the dashed line and for ω= 5.412 R solid line). * = solid line and for 0 ω0 = 3.044 R for the dashed line ).0 . solid line and for ω0 3.044 R for the dashFig. 7. ed line ).The variation. of ground-state binding energy for d=0a against the ωc at fixed Fig. 7. The variation of ground-state* binding energy for d=0a* against the ω at fixed Fig. 7. The variation of ground-state binding energy for temperatured=0a against (20K) the andωc at for fixed three different valuesc of pressure (0, 10, and 20 kbar) a) temperature (20K) and for three different values* of pressure= (0, 10,* **and 20 kbar)= a) Fig. 3. Fortemperature fixed values (20K) and forof impurity three different position values of pressure (d), (0,the 10, andR 20 kbar) to ω 5.412Ra)00 3.044 Rb. This) ω 5.412R energy. behavior agrees with ** ** * ω= 3.044 Rb ) ω= 5.412R ω00= 3.044 . Rb ) ω= 5.412R . energy of the donor Hamiltonian00 enhancesFig. 8 .as The the variation ofour ground expectation.-state binding energy for d = 0 a against the pressure at fixed * * * ** Fig. 8. The variation of groundFig. 8.- state The vabindingriation energy of ground for -dstate = 0 a bindingtemperature against energy the pressure(20K) for d and = 0at aω fixedc against= 2 R forthe (pressureω00= 3.044 at Rfixed and ω = 5.412R ). ) * ** energy of the donor Hamiltonian enhances as theconfinement confinement strengthstrength ( temperature increasesincreases* (20K) from and ω 3.044**= 2 R for ( ω= 3.044 Furthermore,R and ω= 5.412R we* ).have investigated the impurity temperature (20K) and ωc = 2 R for ( ω00= 3.044 Fig.R 9c . and ωThe ground= - 5.412Rstate00 binding ). energy for d=0.1a at constant pressure (P=10 kbar) as a Fig. 9. The ground-state binding* energy forfunction d=0.1a of* atω constant and for threepressure temperatures (P=10 kbar) (5K, as a 100K, and 200K): * * Fig. 9. The ground-state binding0 energy for d=0.1a at constant pressure (P=10 kbar)c as a ** from 3.044 R to 5.412R . This energy behavior agrees with our expectation.ω function of ωc and for three temperatures (5K, 100K, and 200K): function of ωc and for three temperatures (5K, 100K, and 200K): a) ω00= 3.044 R and b) ω= 5.412 R . ** ** * = a) ω=00= 3.044 R and b) ω= 5.412 R . 382 a) ω00 3.044 R and b) ωFig. 5.412 R 10. The .ground -state binding energyInt. J. for Nano d = 0.1 Dimens., a at constant 10 (4): pressur 375-390,e (P=10 Autumn kba and 2019 * ** * Fig. 10. The ground-state Fig.binding 10. The energy ground for -dstate = 0.1 binding a at constant energy for pressur ωd c== 0.1e 2 R(P=10 a , andat kbaconstant temperature and pressur for (e (P=10ω0 = kba 3.044 andR dashed line * * * * * Furthermore, we have investigated the impurity ground -state eigenenergy= (E) and ω cthe= 2 R , and= temperature for (ω0 = 3.044 R dashed line ωc 2 R , and temperature for (ω0 3.044 R dashed line and ω0 = 5.412 R for the solid line). * * * = and ω0 = 5.412 R for the solid line). and ω0 5.412 R for the solid line).Fig. 11. The variation of binding energy for d=0.1a against the ωc at fixed temperature (20K) * * ** binding energy (BE) against the magnetic field strength , for specificFig. 11 values. The variation of of binding energy for d=0.1a against the ωc at fixed temperature (20K) Fig. 11. The variation of binding energy for d=0.1a againstfor three the ω pressurec at fixed values temperature (0, 10, and (20K) 20 kbar) a) for ω00= 5.412 Rb ) and for ω= 3.044 R . ** ** * for three pressure values for(0, three10, and pressure 20 kbar) values a) for (0, ω 10,= 5.412 and 20Rb kbar) ) and for ω a) for ω00== 3.044 R 5.412 Rb . ) and for ω= 3.044 R . c 0 Fig.00 12. The variation of ground-state binding energy for d=0.1a as a function of the pressure * * * ** and versus values of (d). Fig.4a ,4b,4c,4d show theFig. dependence 12.ω The variations of ground Fig.the 12 -ground.state The variationbinding stateω energy of ground for -d=0.1stateat abinding fixed as a temperaturefunction energy forof thed=0.1 (20K) pressurea and as aω cfunction= 2 R forof the ω00 pressure= 3.044 R and ω= 5.412 R . * ** * *** at fixed temperature= (20K)= and ωc = 2 R for= ω00= 3.044 R and ω= 5.412 R . at fixed temperature (20K) and ωc 2 R for Fig.ω00 13 3.044 . TheR binding and ω energy 5.412 R for d=0.5. a at constant pressure (P=10 kbar) as a function of ωc for * * ** Fig. 13. The binding energy for d=0.5a at constant pressure (P=10 kbar) as a function of ωc for binding energy (BE) on the magnetic field strength Fig. ,13 for. The different binding energy values for d=0.5 of aimpurity at constant pressurethree (P=10 temperatures kbar) as a function (5K, 100K, of ω cand for 200K) . a) ω00= 3.044 R and b) ω= 5.412 R . ** *** * three temperatures (5K,= 100K, and 200K). a) ω= 00= 3.044 R and b) ω= 5.412 R . three temperatures (5K, 100K, and 200K). a) ωFig.00 3.044 14. TheR binding and b) ω energy 5.412 R for d=0.5 . a at constant pressure (P=10 kbar) and ωc = 2 R against * * c * * * * Fig. 14. The binding energyFig. for 14 d=0.5. The abinding at constant energy pressure for d=0.5 (P=10 a at kbar)constant and pressure ω== 2 R (P=10 against kbar) and ωc = 2 R against = distances : (a) d=0 , (b) d=0.1 , and (c) d=0.5 ω and two confinement frequencies temperature for( ω0 c 3.044 R for the dashed line and for ω0 5.412 R solid line). * * * * = temperature for( ω0 = 3.044 R for the =dashed line and for ω0 = 5.412 R solid line). ∗ ∗ ∗ temperature for( ω0 3.044 R for the dashed line and for ω0 5.412 R solid line). ( =3.044 R* and 𝑎𝑎𝑎𝑎 =5.412R*)𝑎𝑎𝑎𝑎. The binding energy𝑎𝑎𝑎𝑎 (BE), against the magnetic field

0 0 strength𝜔𝜔𝜔𝜔 , is greatly reduced𝜔𝜔𝜔𝜔 as the distances (d) increases as shown by Figs. 2, 3 and 4. ٦ In addition, the figures demonstrate that the binding energy increases as magnetic field

and the parabolic confinement strengths increase.

ωc 𝜔𝜔𝜔𝜔0 In Table 3, we have listed the donor impurity energy and binding energy as a function of magnetic field strength for = 3.044 and for various values of d. ∗ c 0 For = 3.044 and d=0 a*, theω bindingω energy increases𝑅𝑅𝑅𝑅 as the magnetic field ∗ 0 strengthω increases.𝑅𝑅𝑅𝑅 This behavior persists for all d-values. However, the binding energy decreases as the impurity (d) increases. We can see a significant decrease in the binding energy as d increases from d=0.1 a* to d=0.5 a*. For example, at = 3.044 and ∗ 0 = 2 the binding energy reduces significantly from 4.62499 ωR* to 2.55177𝑅𝑅𝑅𝑅 R*. ∗ c Thisω result𝑅𝑅𝑅𝑅 is due to the great reduction in the coulomb impurity energy (Eq. 4), as we mentioned earlier. The same qualitative behavior can also be observed in Table 4 with different quantitative behavior for = 5.412 . ∗ ω0 𝑅𝑅𝑅𝑅 To see the effects of pressure and temperature parameters on the binding energy of the donor impurity, we have inserted the pressure and temperature dependent material

١٠

S. Abuzaid et al.

function of the magnetic field strength with impurity located at the origin (d=0) for

𝜔𝜔𝜔𝜔𝑐𝑐𝑐𝑐 function of the magnetic field strength with impurity located at the origin (d=0) fortwo specific values of the confinement frequency strength = 5.412 , and = ∗ energyenergy of ofthe the donor donor Hamiltonian Hamiltonian enhances enhances as asthe the confinement0 confinement strength strength (0 ( ) )increases increases 𝑐𝑐𝑐𝑐 3.044 R . The accuracy of our obtained results are tested againstω the corresponding𝑅𝑅𝑅𝑅 ω ones two specific values of the confinement 𝜔𝜔𝜔𝜔frequency strengthenergy of= the5energy.412 donor of , Hamiltoniantheand donor = Hamiltonian enhances as enhances the confinement as the confinement strength ( strength ) increases ( ) increases ∗ 0 0 ∗ fromfrom 3.044 3.044 R* R to* to5.412R 5.412R*. This*. This energy energy behavior behavior agrees agrees with with our our expectation. expectation.ω ω function of the magnetic field strength with0 impurity located *at the0 origin* (d=0) for 0 0 3.044 R . The accuracy of our obtained results are testedfrom againstω 3.044 the fromR corresponding*Fig.7.a to 3.044 5.412R𝑅𝑅𝑅𝑅 R* .to ω Thisones 5.412R energyproduced . This behavior by energy agreesexpansion behavior with agreesmethod our expectation. with [35 our] ω.T expectation.able 2 showsω the comparison between the 𝑐𝑐𝑐𝑐 𝟏𝟏𝟏𝟏 ∗ two specific values of the confinement 𝜔𝜔𝜔𝜔frequency strength = 5.412 , and = Furthermore,Furthermore,𝑵𝑵𝑵𝑵 we we have have investigated investigated the the impurity impurity ground ground -state -state eigenenergy eigenenergy (E) (E) and and the the produced by expansion method [35] .Table 2 shows the comparison between theground state (m=0) computed energy for the present exact diagonalization method and Furthermore,Furthermore, we have investigated we have investigated∗ the impurity the ground impurity -state ground eigenenergy -state eigenenergy (E) and the (E) and the 𝟏𝟏𝟏𝟏3.044 R . ω0 𝑅𝑅𝑅𝑅 ω0 The accuracyFig. of 1 .our The groundobtained state resultsenergy of arethe quantumtested heterostructureagainst the corresponding for fixed value of magnetic onesbindingbinding field energy energy (B (EB)E against) against the the magnetic magnetic field field strength strength , for , for specific specific values values of of 𝑵𝑵𝑵𝑵 strength (ω )= 2 R * and parabolic confinment strength ω= 5.412 R * against the number of ground state (m=0) computed∗ energy for the presentc exact diagonalizationbinding energymethod0 ( BandE)the against available the magneticcorresponding field strengthenergy produced , for specificby expansion values of method. The comparison binding energy (BE*) against the magnetic field* strength , for specific values of produced by expansionbasis method for a) donor [35 impurity] .Table at the 2 originshows (d = 0the a ) andcomparison b) donor impurity between at (d = 0.5 the a ). 𝟏𝟏𝟏𝟏 ωcωc ω0ω0 Fig. 2. The ground state energy of the quantum heterostructure for absence and presenceandand versus of versus values values of of(d) (d). Figc. Fig.4a.4 ,4b,4c,4da ,4b,4c,4d show show the the dependence0 dependences ofs ofth eth grounde ground state state 𝟏𝟏𝟏𝟏 c ω 𝑵𝑵𝑵𝑵 0 ω the available corresponding energy producedthe impurity by against expansion the magneticand versus method.field strength andvalues Theversus ω ofthe comparison (d)dashedvalues. Fig line .of 4withshows a (d),4b,4c,4d impurity. Figa good .and4a showsolid,4b,4c,4d agreement line the dependenceshow ωbetween the dependence sboth of th methods.e grounds of Toth statee ω testground the convergencestate issue of our energy of the donor Hamiltonian enhancesc as the confinement strength ( ) increases 𝑵𝑵𝑵𝑵 𝟏𝟏𝟏𝟏 * ground state (m=0) computed energy for thefor no present impurity casesexact for diagonalization ω0 = 5.412 R . method bindingandbinding energy energy (B (EB)E on) on the the magnetic magnetic field field strength strength , for, for different different values values of ofimpurity impurity function of the magnetic field strength with impurity* 𝑵𝑵𝑵𝑵 located* atbinding the origin energy= (d=0)* (B forE) exacton the diagonalization magnetic field 0technique,strength, we, for have different plotted values in F igof. 1aimpurity ,1b, the computed ground Fig. 3. The groundbinding-state for fixedenergy value (ofB ωEc) on 2 R theagainst magnetic the distance field for two strength ω for different values of impurity shows a good agreement between fromboth 3.044methods. R toTo 5.412R test the. Thisconvergence energy behavior issue of agrees our with our expectation. c c ω , ω= 3.044 R** dashed line and ω= 5.412 R the solid line). ω ω the available corresponding𝑐𝑐𝑐𝑐 energy00 produced( by expansion0 method.) The comparisondistancedistance s s: (a): (a) d=0 d=0c, (b), (b) d=0.1 d=0.1, and, and (c) (c) d=0.5 d=0.5 and and two two confinement confinement frequencies frequencies two specific values of the confinement 𝜔𝜔𝜔𝜔frequency strength = 5distance.412 s , and: (a) d=0= state, (b) energies d=0.1 (E), and ofω (c)thec d=0.5 donorω impurity and two Hamiltonianconfinement againstfrequencies the number of basis (n) exact diagonalization technique, we have plotted Fig.in 4F. igThedistance.1a ground ,1b-s,state𝟏𝟏𝟏𝟏 the : (a)binding computed d=0 energy, against (b)ground d=0.1 ωc , where, and (c) d=0.5 and two confinement∗ ∗ frequencies∗ ∗ ∗ ∗ * ∗* ∗ * ∗ * * * ∗ * * Furthermore, ( ω0 = 3.044 R we for dashed line, have investigated and𝑵𝑵𝑵𝑵 ω0 = 5.412 the Rimpurity for solid line∗ ground) (a) d =∗ -0state a , (b) eigenenergy( d =( 0.1= a3.044=, 3.044∗ (E)R R andand and 𝑎𝑎𝑎𝑎the 𝑎𝑎𝑎𝑎= 5.412R=5.412R)𝑎𝑎𝑎𝑎. The)𝑎𝑎𝑎𝑎. The binding binding energy energy𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 (BE (BE), against), against the the magnetic magnetic field field 0 * 0 * shows a good agreement between both methods.ω To test* 𝑅𝑅𝑅𝑅the *convergenceω 𝑎𝑎𝑎𝑎* issue of our𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 3state.044 energies R . The accuracy(E) of the of donor our obtained impurity results Hamiltonian are tested against( against=3.044 the the and(number correspondingR (c) =andd=0.53.044 ofa𝑎𝑎𝑎𝑎. basis=R5.412R andones (n) from )𝑎𝑎𝑎𝑎.= The5.412R 1 to binding 38 )for. The frequencyenergy binding𝑎𝑎𝑎𝑎 (BE energy), against= 5 .(BE412 the), againstmagnetic , impurity the field magnetic distance fieldd= 0 and 0.5 , and Fig. 5. The binding energy for d=0 a* at constant pressure (P=10 kbar) as a function of ω0 for0 0 0 ∗ ∗ ∗ ∗ binding energy (BE) against the magnetic field strength , forstrength𝜔𝜔𝜔𝜔 specificstrength𝜔𝜔𝜔𝜔c , is ,values isgreatly greatly of reduced𝜔𝜔𝜔𝜔 reduced𝜔𝜔𝜔𝜔 as asthe the distances distances (d) (d) increases increases as asshown shown by by Fig Fsig. s2,. 2,3 and3 and 4. 4. 0 0 0 **0 o exact diagonalization technique,three temperatures we have (5K, 100K,plotted and 200K)in𝜔𝜔𝜔𝜔 F forig . 1a(a ) ω,1b=,3.044 the Rcomputed and b) ω𝜔𝜔𝜔𝜔 ground= 5.412 R . ω = 2 𝑅𝑅𝑅𝑅 𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 producedfrom 1 to 38by for frequencyexpansion method= 5. 412[35] .T ,able impurity 2 shows distancestrength𝜔𝜔𝜔𝜔 the comparisond,= is 0strength greatly and ,betweenreduced 0.5𝜔𝜔𝜔𝜔is00 greatly , asandthe reducedtheat madistancesgnetic as the field(d) distances increases strength (d) as increases shown by. asThe Fshownig figuress. 2, by3 clearly andFig s4. . 2, show 3 and the 4 .numerical stability in * c * 0 𝟏𝟏𝟏𝟏 Fig. 6. The binding energy change for d=0 a at constant pressure (P=10 kbar)ω and ωc = 2 R ω ∗ and versus∗ values of (d). Fig.4a ∗,4b,4c,4d show∗ the dependenceIns Inaddition, of addition, the ground the the figuresc figuresstate demonstrate demonstrate that that the the binding binding energy energy increases increases as asmagnetic magnetic field field state energieso (E) of the donor impurity Hamiltonian against the number *of basis (n) 𝑵𝑵𝑵𝑵 with respect to theIn temperature addition, for( theω 0figures= 5.412 ourR demonstrate computed scheme. that the Thebindingω ground energy𝑅𝑅𝑅𝑅 state increases approaches as magnetica limiting field value as the number of at magnetic field strength ω = 2 . The𝑅𝑅𝑅𝑅 figures clearlyIn show addition, the numerical𝑎𝑎𝑎𝑎 theFig.7.b figures stability𝑎𝑎𝑎𝑎 demonstrate in that the binding energy increases as magnetic field ground state (m=0) computed energy for the present solid line and for exact diagonalizationω= 3.044 R *method for the dash anded line ). . binding∗ energy (BE) on the magnetic0 field strength , for different and values and the the parabolicof parabolic impurity confinement confinement strength strengths s increase. increase. from 1 to 38 for frequency = 5.412 , impurity distance d= 0 * and 0.5 , and c Fig. 7. The variation of ground-state bindingFigure energy 7 for d=0a against the ωc at fixed our computed scheme. Theω ground𝑅𝑅𝑅𝑅 state approaches a limiting and valuethe parabolic asand the the numberconfinement parabolic ofbasis confinement strength increases.s strength increase.s increase. the available corresponding energy producedtemperature by expansion (20K) and∗ for threemethod. different The values comparison of pressure∗ (0, c10, and 20∗ kbar) a)c c 0 0 o ω ω ω 𝜔𝜔𝜔𝜔 𝜔𝜔𝜔𝜔 distances : (a) d=0 , (b) d=0.1 c **, and (c) d=0.5 and two confinement frequencies0 at magnetic field strength ω = 2 𝟏𝟏𝟏𝟏 . The𝑅𝑅𝑅𝑅 figuresc ω00= 3.044 clearlyRb show ) ω= 5.412Rthe numerical𝑎𝑎𝑎𝑎. stability𝑎𝑎𝑎𝑎 in0 ω∗ ω ∗ ∗ 𝜔𝜔𝜔𝜔 In InTable Table𝜔𝜔𝜔𝜔 3, 3we, we have have listed listed the the donor donor impurity impurity energy energy and and binding binding energy energy as asa a basis increases. 𝑵𝑵𝑵𝑵 * Fig. 8. The variation* ∗ of ground-state binding * energyIn forTable d = 0 a3 ,against weFig havethe.2 displaypressure listed ats fixedthethe donorenergy impurity of the donor energy impurity and binding energy energy as a functionas a of the magnetic shows a good agreement between both methods. To test𝑎𝑎𝑎𝑎 the convergenceIn Table𝑎𝑎𝑎𝑎* 3, issuewe have of** ourlisted𝑎𝑎𝑎𝑎 the donor impurity energy and binding energy as a ( =3.044c temperature R and (20K)= and5.412R ω= 2 R). forThe ( ω binding= 3.044 R energy and ω (BE= 5.412R), against ). the magnetic field our computed scheme. Theω ground𝑅𝑅𝑅𝑅 state approachesc a limiting00 value as the number of * functionfunction of ofmagnetic magnetic field field strength strength for for = =3.0443.044 and and for for various various values values of ofd. d. 0 Fig. 9. The ground-state0 binding energy for d=0.1a at constant pressure (P=10 kbar) as a = 5.412R exact diagonalizationground technique, -state eigenenergy we have plotted (E) in and Fig function the.1a ,1 bindingb , ofthefunction magnetic computed magneticof field magnetic ground strength field fieldfield strength strengthstrength for = ,3 forforfor.044 confinement = and3.044 for variousfrequency andand for values various of d. values of. d.The ∗solid∗ plot of the Fig.2 displays the energy of the donorstrength𝜔𝜔𝜔𝜔 impurity, is greatly functionenergy reduced𝜔𝜔𝜔𝜔 of asωc anda function asfor threethe distancestemperatures of the magnetic(d) (5K, increases 100K, and 200K) as :shown by Figs. 2, 3 and 4. * * c c 0 0 basis increases. ** ∗ ∗ ω ω ω ω ∗ 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 energy (BE) against the magnetic a) fieldω= strength3.044 R and b) ωfor =various 5.412 R . values of d.For For c c = =3.04430.044 and andand d=0 d=0d=0 a ,a the, the 0binding binding energy energy increases increases as asthe the magnetic magnetic field field ٧ c * 0 *00 For =*3* .044 systemand ωd=0 indicates a ,ω theωω thebinding absenceω𝑅𝑅𝑅𝑅 energy of the increases 𝑅𝑅𝑅𝑅impurity, as and ωthe the magnetic dashed onefield indicates the presence fieldstate energiesstrength (E) , offor the confinement donor impurity frequencyFig. Hamiltonian 10. The ground=-state 5against.For412 binding R the .energy =The 3number.044 forsolid d = 0.1 of plot anda basis at constantofd=0 the(n) a pressur, thee (P=10binding kba and energy increases∗ ∗ as the magnetic field energy of the donor Hamiltonian enhances as the, for confinement specific valuesstrengthIn addition, of( )theand increases figures versus demonstrate values of that athe, thebinding binding energy energy increases increases as0 0magnetic as the field magnetic = * ∗ = * ∗ ω ω 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 ωc 2 R , and temperature∗ for (0 ω0 3.044 R dashed line strengthstrength increases. increases. This This behavior behavior persists persists for for all all d- values.d-values. However, However, the the binding binding energy energy (d).Figc .Fig.2 display 4a ,4b,4c,4ds the energy show of thethe dependencesdonor0 impurity0 of energy the as afield function strength of the magneticincreases. This behavior persists * * 0 and ωω= 5.412 Rstrengthω* for the solid line).𝑅𝑅𝑅𝑅 increases.𝑅𝑅𝑅𝑅 Thisof the behavior impurity. persists It is forclear all from d-values. Fig. 2However, that the effectthe binding of the energyimpurity is to decrease the from 3.044 R to 5.412R . Thissystemfrom energy 1 toindicates behavior38 forω frequencythe agrees absence with of our= the 5 expectation. . 412impurity, and theω, impurity and parabolic ω the dasheddistance strengthconfinement one d0 =increases. indicates0 strength and Thisthe0.5s presence behavior , iandncrease. persists for all d-values. However, the binding energy ground state binding energy (BE) on the magnetic * for all d-values. However, the binding energy Fig. 11∗ . The variation of binding energy for∗ d=0.1a against∗ the ωc at fixed temperaturedecreasesdecreases (20K) as asthe the impurity impurity (d) (d) increases. increases. We We can can see see a significanta significant decrease decrease in inthe the binding binding field strengtho , for confinement frequency = 5.412R . The solid plot of the field strength for c different values of impurity decreases decreases as =the0 impurity** energy as the (d) of impurityincreases.= the system. (d)We increases.can The see presence a significant We canof donor decrease impurity in the bindinglowers the energy of the at magnetic field strength ω = 2 . Thefor𝑅𝑅𝑅𝑅 threefigures pressure clearly valuesdecreases show (0, 10, theand as20numerical𝑎𝑎𝑎𝑎 kbar)the impuritya) for ωstability00𝑎𝑎𝑎𝑎 5.412 (d) Rb increases.in ) and for ω We 3.044 R can see . a significant decrease in the binding of the impurity. It is clear from Fig.ω *2 that the effect of the impurity is to decreas𝜔𝜔𝜔𝜔 e the * * ∗ * * * * * Furthermore, we have investigated the impuritydistances ground : -(a)state d=0c eigenenergy ∗ a ,Fig. (b) 12In d=0.1. The Table (E) variationa and 3, ,and of wethe ground (c) have -d=0.5state listed0 bindinga the energy donorsee for d=0.1 aimpurity significanta as a function energy ofdecrease theandenergy pressureenergy binding in as the asd energyincreases d bindingincreases as afroenergy from md=0.1 d=0.1as a ato tod=0.5 d=0.5 a .a For. For example, example, at at = =3.0443.044 and and system indicatesc ω the absence of the impurity, andω the dashed= * one= indicates** the=* presence ** * ** at fixed temperature (20K) and ωenergyc * 2 R foras ωd00 increases 3.044 R and ωheterostructure from 5.412 R d=0.1. a energyto d=0.5 due ato. itFor negative example, =coulomb3 .at044 attraction.= 3.044 and ourenergy computed of the and scheme.system. two The Theconfinementω ground presence𝑅𝑅𝑅𝑅 state of frequenciesapproaches donor impurity a limiting(ωenergy0 = lowers3.044 asvalue d Rtheincreases as enerthed numbergyincreases fro ofm d=0.1the of from a to d=0.1 d=0.5 a ato. d=0.5For example, a . For atexample, and ∗ ∗ Fig. 13. The binding energy for d=0.5a* at constant pressure (P=10 kbar) as a function of ω for 0 0 binding energy (BE) against the magnetic field strength , *forfunction specific of values magnetic of field strength for = 3.044 and for various=c =2 2 thevalues the binding bindingof d. energy energy reduces reduces∗ significantly significantly∗ from from 4.62499 4.62499 ωR ω*R to* to2.55177 2.55177𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 R *R. *. and ω = 5.412R ). The binding energy (BE), against at **and the binding energy 0 of the 0impurity. It is clear fromthree temperatures Fig. 2 that (5K, the 100K, effect and 200K) of .the a) ωimpurity= 3.044 R is and b) ω to decreas= 5.412 Re the . 0* ω * * 𝑅𝑅𝑅𝑅 * basisheterostructure increases. energy due to it negative coulomb attraction. = 2 the =binding2 00 the energy binding reduces energy∗ significantly reduces significantly∗ from ∗ 4.62499* from ωR 4.62499 to 2.55177 R𝑅𝑅𝑅𝑅 to R 2.55177. R . the magneticc field strength, is greatly0 reduced* as c reduces0 significantlyThe energy= from*ofc cthe 4.62499heterostructure R to 2.55177 shows a significant dependence on the impurity Fig. 14. The binding energy for d=0.5 a *at constant pressure∗ (P=10 kbar) and ωc 2 R against and versus values of (d). Fig.4a ,4b,4c,4d show the dependenceω Fors of th=e ground3.044 stateω and d=0 a∗ , theω binding* ω energy increases𝑅𝑅𝑅𝑅 This ωasThisω theresult result 𝑅𝑅𝑅𝑅magnetic 𝑅𝑅𝑅𝑅is isdue due tofield tothe the great great reduction reduction in inthe the coulomb coulomb imp impurityurity energy energy (Eq. (Eq. 4), 4), as aswe we theenergy distances of the (d)system. increases Thetemperature presenceas shown for( ωof =by 3.044 Rcdonor Figs.* for the impurity2, 3c dashed R lowers line and for . This the resultω ener= 5.412 isgy dueR * solid line).of tothe the great reduction in the ∗ 0Thisω result𝑅𝑅𝑅𝑅 Thisisω due result to𝑅𝑅𝑅𝑅 the is greatdue to reduction 0the great in reduction the coulomb in the imp coulomburity energy impurity (Eq. energy 4), as (Eq.we 4), as we 0 position. Increasing the impurity distance (d), changing the system from 2D to 3D binding energy (BE) on the magneticFigThe. 2energy display field ofsstrength andt hethe energy 4.heterostructure In, addition,forof the different donorstrength shows the ωvaluesimpurity increases.a figures significantof impurityenergy𝑅𝑅𝑅𝑅 This as dependencedemonstrate behavior a function persists thatonof thecoulomb for magneticimpurity all d- values.impurity However, energymentioned the(Eq. binding 4), earlier. as weenergy The mentioned same qualitative behavior can also be observed in Table 4 with heterostructure energy due to it negative coulomb attraction. mentioned earlier. The same qualitative behavior can also be observed in Table 4 with the binding energy increases as magneticmentioned mentionedearlier.field Theearlier. earlier. same The qualitativeThe same same qualitativequalitativebehavior can behavior behavioralso be can observed can also also be in observed beTable 4 inwith Table 4 with fieldposition. strength Increasing , for theω confinementc impurity distance frequency (d) , changing= 5.412 theR system. The solidfrom plot2D ofto the3D(bulk), as displayed clearly in Fig.3. For fixed values of impurity position (d), the distances : (a) d=0 , (b) d=0.1 , and (c) d=0.5 and theand two parabolic confinementdecreases confinement as frequencies the impurity strengths (d) increases. ω0 Weobserved can see a in significant Table 4 different decrease withdifferent different quantitativein quantitative the binding behaviorquantitative behavior for for = =5.4125.412 . . ∗ ∗ ∗ ∗ different quantitative behavior = 5 .for412 = 5.412 . increase.Thec energy of the heterostructure0 showsdifferent a significant quantitative dependencebehavior behavior foronfor the impurity . ∗ ∗ * * ω ω * * 0 0 ( =3.044 R and 𝑎𝑎𝑎𝑎 =5.412Rsystem(bulk),)𝑎𝑎𝑎𝑎. The indicatesas bindingdisplayed the energy absenceclearly𝑎𝑎𝑎𝑎 (BE inof), theFagainstigenergy .impurity,3. theFor as magnetic fixed dand increases thevalues fielddashed fro of m impurityone d=0.1 indicates apositionto thed=0.5 To presence(d), seea .the For the example, effects atof pressure∗ = 3.044 and ∗ temperature and ω ω 𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅 In Table 3, we have listed the donor impurity 0 ω0 𝑅𝑅𝑅𝑅 position. Increasing the impurity distance (d), changing the system fromω 2D to To3DTo see 𝑅𝑅𝑅𝑅 see the the effect effects ∗ofs ofpressure pressure and and temperature temperature parameters parameters on on the the binding binding energy energy of ofthe the 0 0 energy and binding energy as a functionTo ofsee theparameters effects of pressure on the and binding temperature0 energy parameters of the donor on the binding energy of the strength𝜔𝜔𝜔𝜔 , is greatly reduced𝜔𝜔𝜔𝜔 asof the the distances impurity. (d) It increasesis clear from as shown Fig. 2 by =that 2F ig thes. the 2,effect 3binding and of To 4the. energysee impurity the reduceseffect is tos of decreassignificantly pressuree the and from temperature 4.62499 ωparametersR* to 2.55177 on𝑅𝑅𝑅𝑅 the R* .binding energy of the material material ٩ bulk), as displayed clearly in∗ Fig.3. For fixed values of impurity position (d), donorthedonor impurity, impurity, we we have have inserted inserted the the pressure pressure and and temperature temperature dependent dependent) c donor impurity, we have inserted the pressure and temperature dependent material In addition, the figures demonstrateenergy thatof the bindingsystem. energyThe presence increasesThisω result ofas 𝑅𝑅𝑅𝑅donormagnetic is due impurity tofield thedonor greatlowers impurity, reduction the enerwe in gy havethe of coulomb insertedthe imptheurity pressure energy and (Eq. temperature 4), as we dependent material 383 ٩ Int. J. Nano Dimens., 10 (4): 375-390, Autumn 2019 and the parabolic confinementheterostructure strengths energy increase. due to it negative coulomb attraction. ١٠ ١٠ mentioned earlier. The same qualitative behavior can also be observed in Table 4 with ١٠ ١٠ c 0 ω 𝜔𝜔𝜔𝜔 different quantitative behavior for = 5.412 . The energy of the heterostructure shows a significant dependence on the impurity ٩ ∗ In Table 3, we have listed the donor impurity energy and binding energy as a ω0 𝑅𝑅𝑅𝑅 function of magnetic field strengthposition. Increasingfor = 3the.044 impurity and forTodistance varioussee the (d) effectvalues, changings of d.pressure the system and temperature from 2D toparameters 3D on the binding energy of the ∗ c 0 For = 3.044 and d=0 (bulk),a*, theω a sbinding displayedω energy clearly increases𝑅𝑅𝑅𝑅 in Fig donor.as3. theFor impurity, magnetic fixed values we field have of impurityinserted positionthe pressure (d), theand temperature dependent material ∗ 0 strengthω increases.𝑅𝑅𝑅𝑅 This behavior persists for all d-values. However, the binding energy

١٠ decreases as the impurity (d) increases. We can see a significant decrease in the binding

٩ * * energy as d increases from d=0.1 a to d=0.5 a . For example, at = 3.044 and ∗ 0 = 2 the binding energy reduces significantly from 4.62499 ωR* to 2.55177𝑅𝑅𝑅𝑅 R*. ∗ c Thisω result𝑅𝑅𝑅𝑅 is due to the great reduction in the coulomb impurity energy (Eq. 4), as we mentioned earlier. The same qualitative behavior can also be observed in Table 4 with different quantitative behavior for = 5.412 . ∗ ω0 𝑅𝑅𝑅𝑅 To see the effects of pressure and temperature parameters on the binding energy of the donor impurity, we have inserted the pressure and temperature dependent material

١٠ energy of the donor Hamiltonian enhances as the confinement strength ( ) increases parameters of GaAs , ( (parameters, ) of (GaAs, ) ), ( , in (the, present ) calculations( , ) ) ,. inIn theFig present. 5 calculations. In Fig. 5 * * ω0 from 3.044 Renergy to 5.412R of the . donorThis energy Hamiltonian behavior enhances agrees withas the our confinement expectation. strength ( )energy increases of the donor Hamiltonian∗ enhances as the∗ confinement strength ( ) increases to Fig.16 , we have shown𝑚𝑚𝑚𝑚 explicitly𝑃𝑃𝑃𝑃to 𝑇𝑇𝑇𝑇Fig.𝑎𝑎𝑎𝑎16𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 the, we∈ behavior have𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 shown of𝑚𝑚𝑚𝑚 the explicitly𝑃𝑃𝑃𝑃 donor𝑇𝑇𝑇𝑇 𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛binding the∈ behavior𝑃𝑃𝑃𝑃 energy𝑇𝑇𝑇𝑇 of (B theE) donoras a binding energy (BE) as a energy0 of the donor Hamiltonian enhances as the confinement strength ( ) increases0 from 3.044 R* to 5.412R*. This energy behavior agrees with our expectation.ω from 3.044 R* to 5.412R*. This energy behavior agrees with our expectation.ω Furthermore, we have investigated the impurity ground -state eigenenergy (E) and the function of the magnetic field strength 0 , taking in consideration the impurity from function3.044 R * ofto 5.412Rthe magnetic*. This energyfield strength behavior agrees , taking with in our consideration expectation.ω the impurity c ωc binding energyFurthermore, (BE) against we havethe magnetic investigated field the strength impurity ,ground for specific -state valueseigenenergy of position (E)Furthermore, and (d), the temperature we have (T), positioninvestigated pressure (d), (P) temperaturethe andω impurity confinement (T), ground pressure frequency -state (P) eigenenergyand confinement. Fig. 5 a,(E) 5b and frequency the . Fig. 5a, 5b c Furthermore,0 we have investigated the impurity ground -state eigenenergy (E) and the ω ω 0 0 and versus bindingvalues of energy (d). Fig (B.E4)a against,4b,4c,4d the show magnetic the dependence field strengths of th ,e forground specific stateshow values binding the of donor energy binding (BE )energ againstshowies the againstthe donor magnetic the binding magnetic field energ strengthfieldies strengthagainst , fortheforω specificmagneticthree different values field strength of forω three different

c binding energy (B0E) against the magnetic field strength , for specificc values of 0 binding energyand (versusBE) on values the magnetic of (d). fieldFig. 4strengtha ,4b,4c,4d , forshow different the dependence ωvalues ofs impurityof thetemperatures ground and versus stateω (5K, values 100K, of and(d)temperatures. 200K)Fig.4a and,4b,4c,4d (5K, fixed 100K, valuesshow and theof 200K) pressure,dependenceω and impurityfixeds of valuesth positione ground of pressure, stateω impurity position c 0 c and versus values of (d). Fig.4a ,4b,4c,4d show the dependenceω s of the ground stateω distances : binding(a) d=0 energy, (b) (d=0.1BE) on ,the and magnetic (c) d=0.5 fieldω and strength two confinement , for different frequencies valuesd=0 of binding impurity and confinementenergy (BE) frequencyond=0 the magnetic and confinement= 3 field.044 strengthR . frequencyFor fixed, for temperature, different= 3.044 valuesR the. Forfigure of fixed impurity temperature, the figure ∗ ∗ ∗ ∗ ∗ ∗ energy of the donor∗ Hamiltonian0 enhances as the confinement strength ( ) increases * * c binding energy (BE) on the magnetic𝑎𝑎𝑎𝑎 field0 strength , for differentc ω values of impurity ( =3.044 distanceR and s𝑎𝑎𝑎𝑎 : =(a)5.412R d=0 )𝑎𝑎𝑎𝑎., The(b) d=0.1binding ,energy and𝑎𝑎𝑎𝑎 (c) (BE d=0.5), againstω and thetwo magnetic confinement fieldclearly frequenciesdistance𝑎𝑎𝑎𝑎 showss the: (a) enhancement d=0 clearly, (b) d=0.1of shows theω binding, theand enhancement (c) energy d=0.5 asω theandof the magnetictwo binding confinement field energy strength frequenciesas the magnetic field strength * * 0 ∗ ∗ ∗ ∗ from 3.044∗ R to c5.412R∗. This energy behavior agrees with our expectation.ω 0 0 energy of thedistance donor Hamiltonians : (a) d=0 enhances, (b) d=0.1 as the, andconfinement (c) d=0.5 strengthω and two ( confinement) increases frequencies strength𝜔𝜔𝜔𝜔 , is greatly reduced𝜔𝜔𝜔𝜔 * as the𝑎𝑎𝑎𝑎 distances *(d)𝑎𝑎𝑎𝑎 increases as shown𝑎𝑎𝑎𝑎 by Figs. 2, 3 and 4. increases.= This *enhancement𝑎𝑎𝑎𝑎 = increases. in the*𝑎𝑎𝑎𝑎 donor This enhancementbinding energy𝑎𝑎𝑎𝑎 in canthe donorbe attributed binding to energy the can be attributed to the ( =3.044 R and =5.412R ). The binding energy (BE), against the magnetic( field3.044 R∗ and 5.412R∗ ). The binding∗ energy (BE), against the magnetic field * * * *c 0 0 0 from 3.044 R( to =5.412R3.044c 0 R. This and energy 𝑎𝑎𝑎𝑎 =5.412R behavior0ω )𝑎𝑎𝑎𝑎. Theagrees binding with our energy 𝑎𝑎𝑎𝑎expectation. (BE),ω against the magnetic field In addition, strengththe𝜔𝜔𝜔𝜔 figures, is greatlydemonstrate reduced𝜔𝜔𝜔𝜔 that as the the binding distances energy (d) increasesincreases asas shown magnetic by Ffieldigparabolicωs. 2,strength𝜔𝜔𝜔𝜔 3 and magnetic , 4is. greatly confinement reduced𝜔𝜔𝜔𝜔 parabolic asFurthermore,term the magnetic distances inconfinementwe E(d) q.have (increases3). investigated For term fixed as shown values the inimpurityby of E Fq.magneticig (s3).. 2,ground For 3 and fixed -4state. values eigenenergy of magnetic (E) and the energy of the donor Hamiltonian enhances as the confinement1 2 strength2 ( ) increases 𝜔𝜔𝜔𝜔0 𝜔𝜔𝜔𝜔0 1 2 2 energy of the donor Hamiltonian enhances as the confinement strength ( ) increasesstrength , is greatly reduced as the distancesbinding4 (d)energy increases (BE) asagainst shown the by4 ωmagnetic Figρ s. 2, 3field and strength4. , for specific values of and the Inparabolic addition, confinement the figures strength demonstrates thatincrease. the binding energyFurthermore, increases we as magnetichaveIn investigated addition, field the the figures impurityfield demonstrate groundstrength ω -thatstateρ , the theeigenenergy bindingbinding energyenergy (E) and increasesdecreases the as when magnetic0 the temperature field increases, as field strengthfrom 3.044 ,R *the to 5.412Rbinding* .energy This energy decreases behavior when agrees the temperature with our expectation. increasesω, as energy of the donorenergy Hamiltonian of0 the donor enhances Hamiltonian as the confinementenhances as strengththe confinement ( ) increases strength ( ) increases c * * 0 ω In addition, the figures demonstrate that the bindingc energy increases as magnetic field c 0 fromω 3.044 R to 5.412R . Thisenergy energy of behavior the donor𝜔𝜔𝜔𝜔 agrees Hamiltonian with our expectation.enhances as the confinement strengthc ( ) increasesclearlyand shown versus ωin Fvaluesig.5a. Similarof (d). Figbehavior.4a ,4b,4c,4d of the donor show binding the dependence energyω is sdisplayed of the ground in stateω and the parabolic confinement strengths increase.binding energy (BclearlyE) against shownand thethe ωin parabolicmagnetic Fig.5 a. Similar confinementfield strength behavior strength 0 of , thefors donorspecific increase. binding 0values energy of is displayed in In Table 3, we have listed the donor impurityfrom energy3.044 R and* to from5.412Rbinding 3.044* .energy This R* energyto as 5.412R a behavior*. This agreesenergy with behavior our expectation. agrees withω our expectation.ω Furthermore, we have investigated the impurity ground -state eigenenergy (E) and the c * * 0 and thec parabolic confinement0 Fig .strength5bbinding for different,s energyc increase. (B=E0)5 on. 412 the magnetic. 0field strength , for different values of impurity Furthermore,ω we have investigatedfrom 3.044the impurity R to 5.412R ground. 𝜔𝜔𝜔𝜔This-state energy eigenenergyand behavior versus (E) agrees valuesand the Fwith igof . 5ω (d)bour for. Figexpectation. different,.4a ,4b,4c,4dω = 5 show.412 the. dependenceω 𝜔𝜔𝜔𝜔s of the ground stateω function of magneticIn fieldTable strength 3, we have for listed the= 3 donor.044 impurity and for energyvarious and values binding cof d. energy asIn aTable 3, we have listed the0 donor impurity energy∗ and binding energy as a energy of the donor Hamiltonian enhances as the confinement strength ( ) increasesω binding energy (BE) against∗ the magnetic𝜔𝜔𝜔𝜔 0 field strength , for specific valuesc of Furthermore,∗ we Furthermore,have investigated we have the impurityinvestigated ground the 0impurity-state eigenenergy grounddistance -states (E) : (a) andeigenenergyω d=0 the , (b) 𝑅𝑅𝑅𝑅(E) d=0.1 and the, and (c) d=0.5 ω and two confinement frequencies c 0 binding energy (BE)In on Table the magnetic 3, we haveω field listed strength the𝑅𝑅𝑅𝑅 donor, for impurity different energy values and of impurity binding energy as a binding =energy3.044 (B E) against the* magneticω fieldω strength𝑅𝑅𝑅𝑅 , for specific values of In Fig.6 we have shown the dependence of the donor binding energy on the For function* andof magnetic * d=0 Furthermore,a , fieldthe bindingstrength we energy have for investigated increases= 3.044 asthe the impurity and magnetic0 for groundvarious field - valuesstatefunctionIn eigenenergy ofFig d.of.6 magneticwe have (E) shownandfield the strength the dependence for of =the∗3 .044donor binding andc ∗ for energy various on values the ∗ of d. 0 from 3.044 R to 5.412R . This energy behavior agrees with our expectation.ω and versus values of (d). Fig.4a ,4b,4c,4d* show the dependenceω* s of the ground stateω ∗ binding energyc (BbindingE∗) against energy the (magneticBE0) against field the strength magnetic( field,) for strengthspecific( (c=3.044 values) , Rfor of andspecific 𝑎𝑎𝑎𝑎 = values5.412R∗ of)𝑎𝑎𝑎𝑎 . The binding energy𝑎𝑎𝑎𝑎 (BE), against the magnetic field 0 ω distances : (a)functionparameters d=0ω of, (b)ofmagnetic GaAs d=0.1 , (field , and strength ,(c) d=0.5 ωfor and , two )c= , 3inconfinement.044 the0 present and frequenciesforcalculations various . values In Fig .of 5 d. and versusω values𝑅𝑅𝑅𝑅 of (d). Fig.4a ,4b,4c,4d *show cthe dependence0 s of the ground statetemperature for fixed valuestemperature of pressure* for (P=10fixed valuesKbar), of p=ressure2 ,(P=10 = 0Kbar), and = 2 , = 0 and strength increases.For This= 3 .behavior044 binding and persists d=0 energy fora , all the(B ωdE -)bindingvalues. againstω However, energythe magnetic increases 𝑅𝑅𝑅𝑅the binding field energyas strength theenergy ofmagnetic the For donor, for field specific=Hamiltonian 3.044 values∗ c andenhances of d=0 aas, thetheω confinementbindingω energy0 strength increases𝑅𝑅𝑅𝑅 ( ) as increases the magnetic field ∗ binding energy∗ (BωE) on the magnetic0∗ ωc field strengthω ∗0 , for differentω 0 values of impurity ∗ ∗ ∗ and versus valuesand of versus(d). Fig values.4*a ,4b,4c,4d of (d). Figshow.4a *the,4b,4c,4d𝑚𝑚𝑚𝑚 dependence𝑃𝑃𝑃𝑃∗ 𝑇𝑇𝑇𝑇 show*𝑎𝑎𝑎𝑎𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛s strength the𝜔𝜔𝜔𝜔ofc∈ thdependence𝑃𝑃𝑃𝑃e𝑇𝑇𝑇𝑇 ,ground 0is greatly sstate of reduced𝜔𝜔𝜔𝜔 the ground as the state distances∗ (d)∗ increasesc as shown by Figs. 2, 3 and 4. Furthermore, we have investigated the impurity ground -state eigenenergy= (E) andto Fig. the16 𝑎𝑎𝑎𝑎= c, =3we.044 have shown𝑎𝑎𝑎𝑎 explicitly theω0 𝑎𝑎𝑎𝑎 behaviorω of the donor𝑅𝑅𝑅𝑅 bindingc energy (BE) as a binding energy (BE0 ) on the magnetic field strength , for different( values3.044 of impurity RFor and 5.412R0 )and. The d=0( binding a=various, 3the. 044energy bindingconfinements (BEand ),energy against= ( 5 .increases412 the= 3R magnetic.044)ω as the and 𝑅𝑅𝑅𝑅field0 magnetic 𝑑𝑑𝑑𝑑= 5.412 𝑎𝑎𝑎𝑎field R )ω. The binding𝑅𝑅𝑅𝑅 𝑑𝑑𝑑𝑑 energy𝑎𝑎𝑎𝑎 again decreases asstrength the impurityω increases. (d) increases.𝑅𝑅𝑅𝑅 andThis versus behavior We values can persists see of a(d) significantfor. Figall .d4-avalues. ,4b,4c,4ddecrease However, inshow the bindingthe bindingdependenceS.various Abuzaid ωstrength* energyconfinementsω ets al.increases.of *the ground𝑅𝑅𝑅𝑅 This behaviorstateω persists for all d-values.. cThe However, bindingω energythe binding again energy from 3.044 R to 5.412R∗ . This energy behavior agrees with our expectation.ω ∗ ∗ binding energy (BbindingE)0 on the energy magnetic (BE)0 fieldon0 distancethe strength magnetics : (a) ,field for d=0 differentstrength, (b) d=0.1values∗, for of,different and impurity (c) d=0.5values ∗ of and impurity two confinement frequencies c functionω of the 𝑅𝑅𝑅𝑅magnetic field0 strengthIn addition, , taking the0 figures in0 consideration demonstrate thethat 0impurity the binding energy increases as magnetic field distancebinding senergy : (a) d=0(BE ) against, (b) d=0.1 the magnetic,* and (c) field d=0.5 *strengthω and two , forconfinement specificstrength𝜔𝜔𝜔𝜔 values , frequenciesis greatlystrength of reduced𝜔𝜔𝜔𝜔 increases. as the This distances behaviorω shows(d)∗ persists increases a 𝑅𝑅𝑅𝑅clear for asall∗decreasing ω shownd-values.ω by behavior However,Figs.∗ 2, 𝑅𝑅𝑅𝑅 3as andthe the binding 4ωtemperature. energy of the system increases. The energy as ddecreases increases asfro them impurityd=0.1binding a (d)to energy d=0.5increases. ( BaE. ) ForWeon the example,can magnetic see a atsignificant field= strength3. 044decrease and ,shows infor thedecreases different bindinga clear as valuesdecreasing the impurity of impurity behavior (d) increases. as the temperatureWe can see ofa significant the system decrease increases. in theThe binding ∗ ∗ ∗ ( =3.044ω Rc * and 𝑎𝑎𝑎𝑎 =5.412Rωc *)c𝑎𝑎𝑎𝑎. The binding energy𝑎𝑎𝑎𝑎 (BE), against the magnetic field impurity, we distance have inserteds c : (a) d=0 distance the , pressure(b)s Furthermore, :d=0.1 (a)position ∗d=0 and,0 and (d), , we (b)(c) temperature have d=0.1ofd=0.5 the investigated , system andand (T), two(c) pressure increases. d=0.5theconfinement impurity (P) andand andω The the two groundfrequenciesconfinement parabolicdependence confinement -state confinement frequencyeigenenergy frequencies of the strength (E). Fig and. 5sa, the 5b increase. * and =versus values* of𝑎𝑎𝑎𝑎 (d)=. Fig.4a* 𝑎𝑎𝑎𝑎,4b,4c,4d show the𝑎𝑎𝑎𝑎 dependenceω s Inof 0 addition,the ground thedecreases statefigurescω as demonstrate the impurity that (d) the increases.dependence binding We energy ofcan the seeincreases material a significant as parameter magnetic decreases* likefield in the the effective binding mass m (P,T) and dielectric ( 3.044 R and 5.412R ). The binding *energy (BE),* againstω the*∗ magnetic𝑅𝑅𝑅𝑅ω field∗dependence* ∗ of the∗∗ material parameter∗ s like* the effective* mass* m (P,T) and dielectric = 2 theenergy binding as d energyincreasesdistance reduces temperaturefroms d=0.1significantly: (a) d=0a to dependent d=0.5, from(b) d=0.1 a4.62499. For material , example,and R (c)to 2.55177d=0.5 parametersat = andR3.. 044energytwo confinement0 material andas d increases parameters frequencies fro0 m d=0.1 like the a toeffective d=0.5 a . Formass mexample,(P,T) at = 3.044 and = * 𝑎𝑎𝑎𝑎 = * *𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 strength𝜔𝜔𝜔𝜔 , *is𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 greatly reduced𝜔𝜔𝜔𝜔 𝑎𝑎𝑎𝑎asc the distances (d) increases as shown0 by Fig0 s. 2, 3 and 4. 0 ∗ 0 * ( ∗ 3.044 R ∗and( =3.0445.412Rbinding Rshow and∗) . energyThe the bindingdonor= 5.412R(B∗E )binding energyagainst). The energ (BEthe binding ),iesmagnetic against against energy*ω thefield the (BEmagnetic magneticstrength), *against field field the , forstrength magnetic specific for ωfield threevalues different of 𝜔𝜔𝜔𝜔 ) ∗ * binding𝜔𝜔𝜔𝜔c energy (BE)𝜔𝜔𝜔𝜔 on the magneticof GaAsfield strength , ( m(PT ,, )for and different∈ ( PT ,) )values ,and in the theof impurityparabolic energy present as confinement denergy increasesand( of dielectric) the strengthfro donorm d=0.1 sconstant constant Hamiltonian a increase.to d=0.5( , enhances )a on.on For thethe asexample, temperature temperaturethe confinement at and = 3strengthpressure.044 ( andexplain * increases in Table 1, where m strength, is greatly reduced as the distances (d)* increases as shown* by Figs. 2, 3 and0* constant4. *, on the temperature andIn Table pressure 3, we explain have listedin Table the donor1, 0*where impurity m energy* and binding energy as a Thisωparameters result𝑅𝑅𝑅𝑅 is of due GaAs= to2 the , (the great (binding ,(reduction ) = energy3.044 in ( R thereduces, and coulomb) ) 𝑎𝑎𝑎𝑎 , insignificantly= the5.412R imp presenturity)𝑎𝑎𝑎𝑎. Theenergy from calculations binding 4.62499 (Eq. 4),energy. ωInR 𝑎𝑎𝑎𝑎as Fig towe (BE . 2.55177 5 ), against𝑅𝑅𝑅𝑅= 2 R . thethe magneticbinding energyfield reduces significantly* from 4.62499 ωR to 2.55177𝑅𝑅𝑅𝑅 R . calculations. In 0Fig. 5 to Fig. 160 ,0 we have shown0 and pressure explain in Table 1, wherec m decreases ∗0 0 strength𝜔𝜔𝜔𝜔 c , is greatlystrength𝜔𝜔𝜔𝜔 reduced𝜔𝜔𝜔𝜔c , is asgreatly thetemperatures distances reduced𝜔𝜔𝜔𝜔 In addition, (5K,(d)as theincreases 100K, distances the* andfigures as shown200K)(d) 0 demonstrateincreases* byand F𝑟𝑟𝑟𝑟 igfixed sas. 2, that shownvalues 3 andtheω by ofbinding4. Fpressure,igs. 2,energy 3 impurityand0 increases4. position asω magnetic field ∗ ∗ ω ω and versus valuesfrom 𝑟𝑟𝑟𝑟of 3.044 (d)∗ . RFig to.4 a5.412R ,4b,4c,4ddecreases𝜔𝜔𝜔𝜔 . This 𝜖𝜖𝜖𝜖 showand energy𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 the increases behaviordependence with agrees sincreasing of with thωe * ourground T expectation. which state𝑅𝑅𝑅𝑅 diminish ω* donor impurity binding Indistance addition,s : (a)thec d=0figures, demonstrate(b) d=0.10explicitly , thatand the(c) the bindingd=0.5 behavior0 energy and twoof increases the confinement donor as bindingmagnetic frequencies energy field=decreases2 c the𝜖𝜖𝜖𝜖 andbinding𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 increases increases energy withreduces with increasing increasing significantly T which T whichfrom diminish 4.62499 diminish donor R toimpurity 2.55177 binding =R 3. .044 mentionedto Fig.16 ,earlier. Thisweω have result The𝑅𝑅𝑅𝑅 shown issame𝑚𝑚𝑚𝑚 due explicitly𝑃𝑃𝑃𝑃 qualitativetostrength𝑇𝑇𝑇𝑇𝜔𝜔𝜔𝜔 the𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛great the, is ∈behavior behaviorgreatlyreduction𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 reducedcan𝜔𝜔𝜔𝜔 of in thealso the donor ascoulombbe the observed binding distances imp inenergyurity (d)Table energyincreases ( B4E) with as(Eq. aasThis ω 4), shown asresult we𝑅𝑅𝑅𝑅 by is F dueigs. to2, 3the and great 4function. reduction of magneticin the coulomb field strength impurity energy for (Eq. 4), as we and for various values of d. ∗ ∗ ∗ In Table 3, we ∗ have listed the donor impurity energy and binding energy as a (BE) as a functionIn addition, of the the magneticfiguresIn addition, demonstrate bindingfield thed=0 cfiguresstrength energy that and demonstratethe (confinementB andEbindingdonor) on the 𝑟𝑟𝑟𝑟the impurityparabolic energythat magneticfrequency the increases binding confinementbinding field = strengthenergyas energy3 .magnetic044 strength 𝑟𝑟𝑟𝑟 increases R (BE). , Forfor fields . fixeddifferent as imagneticncrease. temperature, values field of theimpurity figure ∗ ( =3.044 R* and 𝑎𝑎𝑎𝑎 =5.412Renergy*)𝑎𝑎𝑎𝑎. The of bindingthe donor energy Hamiltonian𝑎𝑎𝑎𝑎 (BE), against enhances the magneticas the confinementω field 𝑅𝑅𝑅𝑅 strength𝜖𝜖𝜖𝜖 ( ) increasesenergy (BE) . 𝜖𝜖𝜖𝜖 c 0 and the parabolic confinement strengths increase. This energyresult∗ is(B dueE) . to the great reduction in the= coulomb3.∗044 impurity energy* ω(Eq. 4),ω as we 𝑅𝑅𝑅𝑅 different quantitativementioned behavior earlier. InforThe addition, same,= taking5 qualitative. 412the figuresin .consideration behavior demonstrate can alsothethat impuritybethe observed binding position energyin Tablementioned increasesFurthermore, 4c withIn earlier. asFig.7a magnetic we The,7b, have same thefield investigated= Forbehavior 3 qualitative.044 theof behavior theimpurity donor 0and can ground d=0bindingalso a be-state, observedthe eigenenergybinding in Tableenergy (E) 4 andwithincreases the as the magnetic field function of the magnetic field strength , taking in considerationfunction theof magneticimpurity𝑎𝑎𝑎𝑎 fieldω strength for ω0 andc for various𝜔𝜔𝜔𝜔 values of d. c0 0 * 0 ∗ and* the parabolic and confinement thedistance parabolicclearly strengths : showsconfinement(a) d=0s the ienhancement,ncrease. (b) strength d=0.10 s ,of and theincrease. (c) binding d=0.5 energyω and twoas∗ the confinement magnetic fieldfrequencies strength strengthω𝜔𝜔𝜔𝜔 , is greatly reduced𝜔𝜔𝜔𝜔 as fromthe (d),distances 3.0440 temperature R (d) 𝜔𝜔𝜔𝜔to 5.412Rincreases (T),. Thispressure as shown energy (P)by behavior andFigs .confinement 2, agrees 3mentioned and with4. our earlier. expectation.energyIn The Table ωsame had 3 , qualitative beenwe have shown listed behavior0 ∗ asthe adonorcan function also impurity be observed ofenergy the in and Table binding 4 with energy as a ω c𝑅𝑅𝑅𝑅 binding∗ *energyc (BEI)∗n 0against Figstrength.7a ,the7bω, themagneticincreases. behavior∗ 𝑅𝑅𝑅𝑅field This of the strengthbehavior donor binding persists , for energyspecific for all had dvalues-values. been ofshown However, as a functionthe binding of energy positionIn (d),Tabledifferent temperature 3, we quantitative have (T), listed pressure behavior frequencyand the thedonor (P) parabolicfor andω impurity confinement =. Fig. confinement5c.412 energy5a, 5b . frequency andshow strength bindingFor the s donor energy =. Fig3 increase..044 .binding 5asa, I n5ba F and different ig .7 ad=0 ,7bmagnetic0 ,quantitative athe, thebehaviorω binding fieldbehavior ofω the energy strength donor for increasesbinding𝑅𝑅𝑅𝑅 = for5. 412energy differentas the .had magnetic been values shown field as a function of To see the effects of pressure and temperature parametersω on the bindingωc energy of increases.the * This𝜔𝜔𝜔𝜔 enhancement* 𝜔𝜔𝜔𝜔 in0 the donor binding energy can be attributed to the ∗ ( =3.044 R and 𝑎𝑎𝑎𝑎 =5.412R )𝑎𝑎𝑎𝑎. The binding energy𝑎𝑎𝑎𝑎 (BE),∗ against the magnetic field In addition, the figures demonstratec energies that the againstbinding energy the magnetic increases as fieldmagnetic0 strengthdifferent field∗ quantitative for functionof parameters:of behavior magnetic for field =strength5.412 . for+ = 3.044 ,and c for various values of d.0 0 In Table 3 , we have0In0 Table listed 3 ,the we donor have impuritylisted the energydonorthe impurityand magnetic decreasesbinding energy0 energy fieldas theand strengthimpurity asbinding a (d)energy , increases. ωfor as adifferent We can values see a significantof parametersω decrease : in the= binding show the donor binding energFurthermore,ωies against the weω magnetic =have3.044 investigated field𝑅𝑅𝑅𝑅 strength thestrength impurityforωω three𝜔𝜔𝜔𝜔 increases. ground differentc 𝑅𝑅𝑅𝑅the This- state magnetic andbehavior eigenenergy versus field persists values (E)strength for andof all(d) the d. - Figvalues. , .4forωa ,4b,4c,4d However,different 𝑅𝑅𝑅𝑅show thevalues binding the ofdependence energyparameters s of : the ground= state donorfunction impurity, of magnetic we havefield strengthinserted three theIn for differentTablepressure 3, weand temperatures havetemperature listedand for the (5K,variousdependent donor 100K, valuesimpurity 0materialandω of 200K) energyd. andpressure binding0 energyP=0, 10, as anda 20 kbar,∗ impurity position∗ To see the effects of pressure and temperature parameters on thestrength 𝜔𝜔𝜔𝜔bindingparabolic, isenergy greatlyTo magnetic see of reducedthe𝜔𝜔𝜔𝜔 effectconfinement ass ofthe pressure distances term0 and (d)* temperature increases in Ecq. (3). as parameters0 Forshown fixed by values Fonig thes. 2,of binding 3magnetic and 4energy. of the 5.412∗ and = 3.044 R , pressure P=0,= 10,3*.044 and 20 kωbar,c impurity𝑅𝑅𝑅𝑅ω positionω c 𝑅𝑅𝑅𝑅 0 0 and the parabolic confinement strengths increase.function of magneticfunction field of strength magnetic field forFor strength = 3a.044 for and and for= d=0 3various1.044 2a ,2 thevalues and binding for of various d. energy ωvalues increases of d.* as the magnetic* field ω * andc fixed0 values of pressure, impurity position bindingd=0 energy, and (BE ) temperature on ωtheenergy magnetic as d T=20 fieldincreases K.strength The fro m donor d=0.1, for differenta to d=0.5 values ωa . Forof impurity example, at = 3.044 and Fortemperatures = 3.044 (5K, 100K,and d=0 and bindinga 200K), theω bindingenergy and* ωfixed (BenergyE )values against increases𝑅𝑅𝑅𝑅 of the pressure,∗ magnetic as decreasesthe impurity magnetic field as strength Totheposition∗ field seeimpurity the ,effect for(d) specificincreases.s of pressure values We ∗and can of temperaturesee a significant parameters decrease on in the the binding binding energy of the function of magnetic field strength for0 = 3.044 and for various values∗ of d. 4 ω ρ ∗ c donor impurity, we haved=0 ainserted and 0confinement the pressure 𝑅𝑅𝑅𝑅 andfrequency temperatureω In addition,dependentfield strengthc donorthe material figures binding0impurity,0 , thedemonstrate binding energycwe have energy 0that shows inserted the decreases binding a significantthe pressureenergywhen the increases and increasetemperature temperature as magnetic as increases dependent field, as material ∗ * ١١ *ω ∗ 𝜔𝜔𝜔𝜔 d=0= ,3 and.044 temperature T=20* K. Theω cdonorstrengthω ωbinding *increases. ωenergy𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 This showsω behavior 0a significant𝑅𝑅𝑅𝑅 persists increase for all d-values.c However, the binding energy 0 . from 4.62499 ωR to 2.55177𝑅𝑅𝑅𝑅 R ١١ For For and =d=03. 044a , the and∗binding d=0 energya , the* increasesbinding energyas* the= increasesmagnetic2 the asbindingfield the magnetic energyω reducesfield significantly 0 d=0ω In and Table confinement 3𝑅𝑅𝑅𝑅, we have frequency listedand . versustheFor donorfixed =values3 .044impurity temperature, of R (d). Forenergy. Fig fixed∗.4 aandthe ,4b,4c,4dtemperature, bindingenergyfigurec show as energy0clearly dthe donorincreasesthe figureas dependenceshows aimpurity, ω frodistancemthe d=0.1 s weof s magnetic thhave :a e(a) toground d=0 insertedd=0.5 state field, aω (b)the . Ford=0.1 pressureincreases; example,, and and at(c)as temperature d=0.5we= increase3.044 and dependent two and confinement material frequencies 𝑅𝑅𝑅𝑅 c∗١٠ strength increases. This behavior persists = 3for.044 all d-values. However,* ω the∗ bindingω energy For and d=00 𝑎𝑎𝑎𝑎 a , the binding energy andclearly theincreases shownparabolic ωinas Fconfinementtheig.5 a.magnetic Similar strength ∗behaviorfield s of theincrease.∗∗ donorω , binding energy∗ is displayed in ∗ the enhancement∗ as ofthe magnetic the binding field0 energy increases as; as decreases we theincreas magneticase the magneticimpurity field field(d)c increases. strengthstrength Wec , the can the see electron- a significant∗ decrease in the binding 0 ∗strengthω increases.strength𝑅𝑅𝑅𝑅 Thisω behavior increases. persists𝑅𝑅𝑅𝑅 This behaviorfor all d -persistsvalues.* However,for all d-ωvalues. the binding* 𝑅𝑅𝑅𝑅However, energy the0 binding energy function𝑎𝑎𝑎𝑎 of magnetic field strengthbinding ω energy for (B=E3) .on044 the magnetic and for variousfield= strength2 values of , d.for different( =3.044 values R ofand impurity 𝑎𝑎𝑎𝑎 =This5.412R result)𝑎𝑎𝑎𝑎. The is due binding toω the* energy great𝑎𝑎𝑎𝑎 (BEreduction𝑅𝑅𝑅𝑅 ), against* in the the coulomb magnetic imp fieldurity energy (Eq. 4), as we ١٠ . reduces separation significantly distance from0 decrease 4.62499𝑐𝑐𝑐𝑐 Rwhich to 2.55177in turn R ١٠ decreasesclearly shows as the theimpurity enhancement (d) increases. theof 0the magneticWe binding can see energy a significantfield as strengththe decreasemagnetic increases. in field thec cthe bindingstrength binding This energyatom strengthω increases.𝑅𝑅𝑅𝑅 This behaviorelectron-atom persists separation for ωallω ddistance-Fvalues.ig.5b for However,decrease different, whichthe binding= in5. 412turn energy i.ncrease 𝜔𝜔𝜔𝜔s the electron𝜔𝜔𝜔𝜔 ∗ ∗ 0 0 * * = 3.044 enhancementc 0 decreasesin the donor as the binding impuritydecreasesc energy (d) as increases.the can impurityc be Weenergy (d) canincreases increases. seeas da significantincreases theWe can electron fro decreaseseem ad=0.1 significant confinement. in atheto binding d=0.5decrease a The .in For the bindingexample, binding at and ١٠s .can 2, 3also and be 4 . observed in Table 4 with ω* ω * 𝑅𝑅𝑅𝑅 ωIn Tablestrength𝜔𝜔𝜔𝜔 3, we , haveis greatly listed reduced 𝜔𝜔𝜔𝜔the∗ donormentioned as theimpurity distances earlier. energy The(d) increasesand same binding qualitative as shownenergy behavior byas Faig * Forenergy increases. as= d3 .increases044 This enhancementand fro d=0m d=0.1 distancea , theina thetosbinding d=0.5 :donor (a) d=0 aenergy binding. For, (b)increasesexample, energy d=0.1 canatas, This andω thebe = (c)attributedresultmagnetic3 d=0.5.𝑅𝑅𝑅𝑅044 is due tofield andand tothe thetwo great confinement reduction0 frequencies in the coulomb impurity energy (Eq. 4), as we ∗ decreasesattributed as the toimpurity the confinement.parabolic (d) increases. magnetic The We binding can confinementsee energy a significant again shows decreaseenergy ωa great againin the enhancement bindingshows𝑅𝑅𝑅𝑅 a great as the enhancement pressure as the 0 ∗ ∗ ∗ ∗∗ * In Fig.6= we2* have the* shownbinding the energy* dependence reduces of significantlythe donor binding from 4.62499energy on ωR *the to 2.55177𝑅𝑅𝑅𝑅 R*. c 0 * energy as d increases* energy0 fro asmfunction dd=0.1 increases aof tomagnetic frod=0.5m pressured=0.1 a field. For a strength toexample, increases, d=0.5 a at for. For while =example,= 3 keeping.0443.044 at and the and = magneticfor3.044 various and values= 5. 412of d. strengthω ω increases.𝑅𝑅𝑅𝑅 This behavior( persists=3.044 for R alland d -𝑎𝑎𝑎𝑎values.=5.412R However,)𝑎𝑎𝑎𝑎. The ωthe binding* binding energy energy𝑅𝑅𝑅𝑅𝑎𝑎𝑎𝑎 (BE* ),In against addition, the magneticthe figures field demonstratedifferent quantitative that the binding behavior energy for increases as magnetic. field parabolic= 2 magneticthe binding confinement energy reduces termterm significantly in Eq.Eq. from (3).(3). For For4.62499 fixedfixed*mentioned Rvaluesvalues to 2.55177 ofof earlier.* magneticmagnetic R The. same qualitative∗ behavior can also be observed∗ in Table 4 with energy as d increases froincreases,m d=0.1 while a to keepingd=0.5 athe. Formagnetic example, fieldc strengthat = unchanged.3.044 and ∗ ∗ ∗ 1 field strength unchanged.c 0* 0 *0 ∗ 0 2 2 0 = 2 temperatureThisω for result fixed𝑅𝑅𝑅𝑅 is valuesdue *to oftheω p greatressureωω reduction (P=10 𝑅𝑅𝑅𝑅Kbar),in𝑅𝑅𝑅𝑅 theω *coulomb = 2 imp𝑅𝑅𝑅𝑅 , urity 0*= 0energy and (Eq. 4), as we c 𝜔𝜔𝜔𝜔 field strength 𝜔𝜔𝜔𝜔 , the bindingthe binding energy= 2energy For decreasesthe reducesbinding= 3.044 significantlyenergy and and reducesthe d=0 parabolicfrom asignificantly , 4.62499the ∗confinement binding R from to energy 2.55177 strength4.62499 increases sR R. to i ncrease. as2.55177 the magnetic ωR . field𝑅𝑅𝑅𝑅 Thisdecreasesω result𝑅𝑅𝑅𝑅 as is the due impurity to the great (d) increases. strengthreduction, Weis 4in ωgreatly thecanρ coulomb see reduced a significant imp as uritythe decreasedistances energydifferent (Eq.in (d) thequantitative increases4), binding as we as behavior shown Inbyfor 0 F Fig.igs=. 2, 8,5 . 3412 weand take4.. the temperature T=20K,∗ ∗ field strength , the binding energywhen= 2 decreasesthe binding temperature when energy the∗ reducestemperature increases, significantly increases∗ as clearly from, as 4.62499∗ ωR* to 2.55177𝑅𝑅𝑅𝑅 R*To. see the effects of pressurec and temperature parameters on the binding energy of the ωc In F𝑅𝑅𝑅𝑅ig.8, we takec the temperaturevarious0 T=20 confinementsmentionedK,c = 2 (earlier. = 3 The.044= same0 ∗ and parameters,parameters,qualitative = 5 .412whilebehavior while R 0)ω. Thechangingcan bindingalso𝑅𝑅𝑅𝑅 be 𝑑𝑑𝑑𝑑 energyobserved𝑎𝑎𝑎𝑎 again in Table 4 with * ∗ * This result is dueThis ωto the result great𝑅𝑅𝑅𝑅strength is reductiondueω toincreases. the in greatω the𝑅𝑅𝑅𝑅 This coulombreduction behavior0 imp in persiststheurity coulomb energy for all (Eq.imp d-values.urity 4), as energy weHowever,𝜔𝜔𝜔𝜔 (Eq. 4),the asbinding we energy mentionedenergy as dearlier. increasesc The fro samem d=0.1 qualitativeIn caddition,shown a to d=0.5behavior inthe Fig. figuresa . can 5a.For demonstrate also Similarexample, be observed behavior at that the=in 3 ofTablebinding.044 the 4 donorwithenergy and increasesthe pressure. asω magnetic∗ We field𝑅𝑅𝑅𝑅 observe∗∗ a great enhancement∗ clearly shown ωin Fig.5a. SimilarThisω behaviorresult𝑅𝑅𝑅𝑅 is dueof the to thedonor great binding reduction energy in theis displayed coulomb impin urity energyc In Table (Eq.0 34),, we as havewedonor listed impurity,0 the donor we impurityhave inserted energy the and pressure binding andenergy temperature as a dependent material binding energy is displayedchanging the in pressure.Fig.To see 5b thefor We effectdifferent, observeshows∗ s of a pressure greatclear enhancementωdecreasing and 𝑅𝑅𝑅𝑅temperatureω𝑎𝑎𝑎𝑎 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛behavior in 𝑑𝑑𝑑𝑑the donor parameters 𝑅𝑅𝑅𝑅𝑎𝑎𝑎𝑎as thebinding ωtemperature on energy the binding as of the energy system of increases. the The mentioned earlier.mentioned The0 same earlier. qualitative The samebehaviordifferentin qualitative the quantitativecan donoralso behavior be behaviorobserved binding can alsoforin energy Table be =observed 54 as .412with the in . pressureTable 4 with = 2 and the parabolic confinement strengthω * s decreases increase.𝑅𝑅𝑅𝑅 *as the impurity (d) increases. We can see a significant decrease in the binding different quantitativethe binding behavior energy for reduces = 5significantly.412 . from 4.62499 R to 2.55177 R . =∗ 3.044 Fig.5b for different, = 5.412mentioned . earlier. The samethe pressure qualitative increasesdonor behavior impurity, for canconfinement alsowe havebefunction observedstrength: insertedincreases of in magneticthe Table =for 3pressure .044confinement 4 field with andstrength temperature =strength: 05.412 for R dependent. * material and for various values of d. ∗ ∗ dependence of the material parameters like theω effective 𝑅𝑅𝑅𝑅mass m (P,T) and dielectric c c In Fig. 6 wedifferent have shownquantitative the behavior dependence0 for of= 5.412 . = 5.412* * ∗ ω ∗0 different𝜔𝜔𝜔𝜔 quantitativeenergy as d behavior increasesand for from d=0.1 a to∗ . d=0.5 This a behavior. For example, ∗ is expected; at = 3.044 and ١٠ Thisω result𝑅𝑅𝑅𝑅 is due to the0 great reductionω in the 𝑅𝑅𝑅𝑅coulomb impurity energy (Eq. 4), as we To see the effect0 s of pressure and*0 temperaturec 0 parameters on the binding energy of the ω differentthe𝑅𝑅𝑅𝑅 donor quantitative binding behavior energy for on the= 5 temperature.412 . for For ∗ = ω3.044 and𝑅𝑅𝑅𝑅∗ d=0 ωa , theω bindingω* energy increases𝑅𝑅𝑅𝑅 as ∗the *magnetic field To see the effects of pressure and temperatureIn Table 3parameters, we haveThis on listedbehavior the thebinding isdonor expected; energy impurity constant sinceof the energy0 as we( andincreas,since) binding one asthethe0 we energypressure,temperature increase as the a effectiveand the pressure pressure, mass explainm the effectivein Table 1, where m In Fig.6 we have shown the dependence of the donor binding energy onω the 𝑅𝑅𝑅𝑅 0* * fixed values of pressure (P=10 Kbar), ∗ = 2 the, binding energyω * reduces∗ 𝑅𝑅𝑅𝑅 significantly from 4.62499 ωR to 2.55177𝑅𝑅𝑅𝑅 R . mentioned earlier. The same qualitative behavior Tocan see also the be effect observed0sTo of see pressurein theTable effect and4 withs temperature of pressuredonormass parametersandimpurity,0 temperaturem increases weon thehave parameters binding whileinserted energythe on the thedielectric pressureof binding the andconstantenergy temperature of the dependent material the binding energy ١٠ ,function of *magnetic fieldincreases strength whileω forthe dielectric𝑅𝑅𝑅𝑅= 3.044 constant ∗ 𝜖𝜖𝜖𝜖and𝑟𝑟𝑟𝑟strength 𝑃𝑃𝑃𝑃 for 𝑇𝑇𝑇𝑇 ωdecreases. various increases. valuesThese𝑅𝑅𝑅𝑅 This ofchanges behaviord. in thepersists material for all d-values. However donor impurity, we have inserted da the= 0 pressure and various and temperature confinements dependent cmaterialdecreases and increases decreases. with increasing These changes T which indiminish the materialdonor impurity binding temperature for fixed valuesTo of see p ressurethe effect (P=10s of pressure Kbar), and =temperature2 , This ω=parameters0 result 𝑅𝑅𝑅𝑅and is on due the to binding the great energy reduction of the in the coulomb impurity energy (Eq. 4), as we different quantitative behavior for = 5.412 . The binding energy again shows∗ 𝑟𝑟𝑟𝑟 donor impurity,* donorwec ∗have impurity,0 inserted ∗we the have pressure 𝜖𝜖𝜖𝜖insertedparameters𝑟𝑟𝑟𝑟 and the temperature pressure lead to and dependent a temperature reduction material dependent in the material kinetic For = 3.044 and parametersd=0 a ,c the leadω binding to a ωreduc energytion in increases the𝑅𝑅𝑅𝑅 kineticdecreases 𝜖𝜖𝜖𝜖as energy the as magnetic ofthe the impurity electron field (d) while increases. the attractive We can see a significant decrease in the binding ∗ ω 𝑅𝑅𝑅𝑅 𝑑𝑑𝑑𝑑 energy𝑎𝑎𝑎𝑎 (BE) . various confinements ( =donor3.044a 0 clear impurity, and decreasing we= have5.412 behavior inserted R ). The the asbinding thepressure temperatureenergy and again temperature energy dependent of thematerial electron while the attractive ١٠ ω 𝑅𝑅𝑅𝑅 ∗ mentioned earlier. The same qualitative behavior can also be observed in Table 4 with 0 ∗ coulomb∗ energy of the electron enhances. * * However,energy asthe d binding increases energy from d=0.1 a to d=0.5 a . For example, at = 3.044 and ١٠ To see the effects of pressure0 strengthand temperatureω increases. parameters0𝑅𝑅𝑅𝑅 This behavior on the persists binding for energy all d- values.of the shows a clear decreasingω behavior 𝑅𝑅𝑅𝑅as the ωtemperature of the system increases.In F igThe.7a ,7b, the behavior of the donor binding energy had been shown as a function of ∗ ١٠ ١٠ . different quantitative behavior for = 5.412 0* * = 2 the binding energy reduces∗ significantly from 4.62499 ωR to 2.55177𝑅𝑅𝑅𝑅 R . at fixed value , ١٠ donor impurity, we have inserteddecreases the aspressure the impurity and Intemperature (d) Fig increases..9a and 9dependent*b, We we can have see materialshow a significantn the effect decrease of changing in the the binding temperature0 dependence of the material parameters like the effective mass m (P,T) and dielectricthe magnetic field ∗strength ω , for different𝑅𝑅𝑅𝑅 values of parameters : = c * To* see the* effectThisωs of result pressure𝑅𝑅𝑅𝑅 is due and to temperature the great reduction parameters in theon coulombthe binding imp energyurity energy of the (Eq. 4), as we energy as d increases froofm the d=0.1 impurity a to position d=0.5 a(d=. For0.1 example,a )*, on the at binding = energy.3.044 Again,c and the binding energy 0 constant ( , ) on the temperature and pressure explainFig. 1. The groundin Table state energy1, where of the quantum m heterostructure for fixed valueω of magnetic field ω ∗ = * = * we haveω00* 5.412 insertedR against the the *numberpressure of and temperature dependent material ١٠ ,strength (ωc ) 2 R and donorparabolic confinment strength impurity 𝑟𝑟𝑟𝑟 = 2 the binding energyshows great reduces enhancement significantly as we fromincrease 4.62499mentioned the magnetic ωR earlier.to field2.55177 Thestrength𝑅𝑅𝑅𝑅 Rsame. qualitative for fixed values behavior can also be observed in Table 4 with ١١ .( *decreases𝜖𝜖𝜖𝜖 and𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 increases with increasing T which diminishbasis fordonor a) donor impurity impurity at binding the origin (d = 0 a* ) and b) donor impurity at (d = 0.5 a ∗ Fig. 2. The ground state energy of the quantum heterostructure for absence and presencec of c of the parameters: Temperature (T= 5,different 100, and quantitative 200K), Pressure behaviorω (P for=10 kbar)= 5 .and412 . 𝑟𝑟𝑟𝑟 Thisω result𝑅𝑅𝑅𝑅 is due to the greatthe impurityreduction against in the the magnetic coulomb field strength impurity ωc the energydashed line(Eq. with 4), impurity as we and solid line energy (BE) . 𝜖𝜖𝜖𝜖 * for no impurity cases for ω= 5.412 R . ∗ ١٠ 0 0 confinement frequencies = 3.044 and * = 5.412 R . We display inω figure 10 𝑅𝑅𝑅𝑅 mentioned earlier. The same qualitativeFig. 3. The groundbehavior-state forcan fixed also value be of observed ωc = 2 R against in Table the distance 4 with for two To∗ see the effects of∗ pressure and temperature parameters on the binding energy of the ω , ω= 3.044 R0 ** dashed line and ω= 5.412 0 R the solid line). In Fig.7a ,7b, the behavior of the donor binding energythe had donor been binding shown00 ( energyas a functionω against ofthe 𝑅𝑅𝑅𝑅temperature0 ω (T),) while the rest of the physical different quantitative behavior for = Fig.5.412 4. The ground. -state binding energy against ω , where donor impurity,c we have inserted the pressure and temperature dependent material ( ω= 3.044 R * for dashed line,∗ and ω= 5.412 R * for solid line ) (a) d = 0 a* , (b) d = 0.1 a* , the magnetic field strength , for different parametersvalues0 of of0 parametersthe system are: kept0= fixed. The binding energy shows an important ω 𝑅𝑅𝑅𝑅 and (c) d=0.5a* . To seec the effects of pressureFig. and5. The temperature binding energy forparameters d=0 a* at0 constant on the pressure binding (P=10 energykbar) as a functionof the of ω for ω dependence on the temperature,ω and the impurity binding energy EB c decreases with three temperatures (5K, 100K, and 200K) for (a ) ω= 3.044 R ** and b) ω= 5.412 R . ١٠ 00 donor impurity, we have insertedFig. 6. The bindingthe pressure energy change and for temperatured=0 a* at constant dependent pressure (P=10 material kbar) and ω= 2 R * c ١١ .increasing the temperature with respect to the temperature for( ω= 5.412 R * 0 * solid line and for ω0 = 3.044 R for the dashed line ). . * Fig. 7. The variation of ground-state binding energy for d=0a against the ωc at fixed (kbar) a ١٠ temperature (20K) and for three different values of pressure (0, 10, and 20 ** ω00= 3.044 Rb ) ω= 5.412R . * Fig. 8. The variation of ground-state bindingFigure energy for8 d = 0 a against the pressure at fixed ١٢ .( temperature (20K) and ω= 2 R * for ( ω= 3.044 R ** and ω= 5.412R c 00 Fig. 9. The ground-state binding energy for d=0.1a* at constant pressure (P=10 kbar) as a

function of ωc and for three temperatures (5K, 100K, and 200K): ** a) ω00= 3.044 R and b) ω= 5.412 R . 384 * Int. J. Nano Dimens., 10 (4): 375-390, Autumn 2019 Fig. 10. The ground-state binding energy for d = 0.1 a at constant pressure (P=10 kba and * * ωc = 2 R , and temperature for (ω0 = 3.044 R dashed line * and ω0 = 5.412 R for the solid line). * Fig. 11. The variation of binding energy for d=0.1a against the ωc at fixed temperature (20K) ** for three pressure values (0, 10, and 20 kbar) a) for ω00= 5.412 Rb ) and for ω= 3.044 R . Fig. 12. The variation of ground-state binding energy for d=0.1a* as a function of the pressure * ** at fixed temperature (20K) and ωc = 2 R for ω00= 3.044 R and ω= 5.412 R . * Fig. 13. The binding energy for d=0.5a at constant pressure (P=10 kbar) as a function of ωc for ** three temperatures (5K, 100K, and 200K). a) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 14. The binding energy for d=0.5 a at constant pressure (P=10 kbar) and ωc = 2 R against * * temperature for( ω0 = 3.044 R for the dashed line and for ω0 = 5.412 R solid line).

٨

) energy of the donor Hamiltonian enhances as the confinement strength ( increases energy of the donor Hamiltonian enhances as the confinement strength ( ) increases energy of the donor Hamiltonian enhances as the confinement strength ( ) increases * * 0 0 from 3.044 R to 5.412R . This energy behavior agrees with our expectation.ω from 3.044 R* to 5.412R*. This energy behavior agrees with our expectation.ω energy of the donor Hamiltonian enhances as the confinement strength ( ) increases 0 from 3.044 R* to 5.412R*. This energy behavior agrees with our expectation.ω * * ω0 Furthermore, we have investigated thefrom impurity 3.044 ground R to 5.412R -state eigenenergy. This energy (E) behavior and the agrees with our expectation. Furthermore, we have investigated the impurity ground -state eigenenergy (E) and the Furthermore, we have investigated the impurity ground -state eigenenergyparameters (E) and of the GaAs , ( ( , ) ( , ) ) , in the present calculations. In Fig. 5 binding energy (BE) against the magnetic field strength , for specific values of Furthermore, we have investigated the impurity ground -state eigenenergy (E)binding and the energy (BE) against the magnetic∗ field strength , for specific values of binding energy (BE) against the magnetic field strength , for specificto Fig.values16 , ofwe have shown𝑚𝑚𝑚𝑚 explicitly𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑎𝑎𝑎𝑎𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 the∈ behavior𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 of the donor binding energy (BE) as a c 0 c 0 and versus values of (d). Fig.4a ,4b,4c,4d show the dependenceω s of the ground stateω and versus values of (d). Fig.4a ,4b,4c,4d show the dependenceω s of the ground stateω binding energy (BE) against the magnetic field strength , for specific valuesc of 0 and versus values of (d). Fig.4a ,4b,4c,4d show the dependenceω s of thfunctione ground of statetheω magnetic field strength , taking in consideration the impurity binding energy (BE) on the magnetic field strength , for different values of impurity c 0 and versus values of (d). Fig.4a ,4b,4c,4d show the dependenceω s of the groundbinding stateω energy (BE) on the magnetic field strength c, for different values of impurity binding energy (BE) on the magnetic field strength , for different valuesposition of impurity (d), temperature (T), pressure (P) andω confinement frequency . Fig. 5a, 5b ωc c distances : (a) d=0 , (b) d=0.1 , and (c) d=0.5 and two confinement frequencies distances : (a) d=0 , (b) d=0.1 , and (c) d=0.5 ω and two confinement frequencies0 binding energy (BE) on the magnetic field strength , for different valuesc of impurity show the donor binding energies against the magnetic field strength forω three different ∗ ∗ ∗ distances : (a) d=0 , (b) d=0.1 , and (c) d=0.5 ω and two confinement frequencies∗ ∗ ∗ * 𝑎𝑎𝑎𝑎 *𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎 * * ( =3.044 R and =energy5.412R of). theThe donor binding Hamiltonianenergy energy of (BE the enhances ),donor against Hamiltonian as the the magnetic confinement∗ enhances field strength as∗ cthe confinement( ) increases∗ strength ( ( = )3.044 increases R and 𝑎𝑎𝑎𝑎 =5.412R )𝑎𝑎𝑎𝑎. The binding energy𝑎𝑎𝑎𝑎 (BE), against the magnetic field distances : (a) d=0 , (b) d=0.1 , and (c) d=0.5 ω and two confinement frequencies temperatures (5K, 100K, and 200K) and fixed values of pressure, impurity position ( =3.044 R* and 𝑎𝑎𝑎𝑎 =5.412R*)𝑎𝑎𝑎𝑎. The binding energy𝑎𝑎𝑎𝑎 (BE), against the magnetic field 𝜔𝜔𝜔𝜔0 𝜔𝜔𝜔𝜔0 * * *∗ * ∗ ∗ 0 0 0 0 strength, is greatly reducedfrom as 3.044the distances R to 5.412R (d)from increases. This3.044* energy Ras shownto 5.412Rbehavior by F*. ig Thisagreess. 2, energy 3 with and ourbehavior4. expectation. agreesω with our expectation.strength𝜔𝜔𝜔𝜔ω , is greatly reduced𝜔𝜔𝜔𝜔 as the distances (d) increases as shown by Figs. 2, 3 and 4. ( =3.044 R and 0𝑎𝑎𝑎𝑎 =5.412R )𝑎𝑎𝑎𝑎. The0 binding energy𝑎𝑎𝑎𝑎 (BE), against the magnetic field d=0 and confinement frequency = 3.044 R . For fixed temperature, the figure strength𝜔𝜔𝜔𝜔 , is greatly reduced𝜔𝜔𝜔𝜔 as the distances (d) increases as shown by Figs. 2, 3 and 4. ∗ ∗ In addition, the figures demonstrate that the0 binding energy 0increases as magnetic field 0 strength𝜔𝜔𝜔𝜔 , is greatly reduced𝜔𝜔𝜔𝜔 as the distances (d) increases as shown by Figs. 2,In 3 addition,and 4. clearly the𝑎𝑎𝑎𝑎 figures shows demonstrate the enhancement that the of bindingtheω binding energy energy increases as the as magnetic magnetic field field strength Furthermore, we haveFurthermore, investigatedIn addition, we the have impurity the investigated figures ground demonstrate -thestate impurity eigenenergy that ground the binding(E) -state and energyeigenenergythe increases (E) asand magnetic the field and the parabolic confinement strengths increase. In addition, the figures demonstrate that the binding energy increases as magnetic and field the parabolic increases. confinement This enhancement strength ins the idonorncrease. binding energy can be attributed to the binding energy (BE) bindingagainst theenergy magnetic (BE) againstfield strength the magnetic , for field specific strength values of, for specific values of c 0 and the parabolic confinement strengths increase. ω 𝜔𝜔𝜔𝜔 ωc c 𝜔𝜔𝜔𝜔0 and the parabolic confinement strengths c increase. c 0 ω 0 In Table 3, we haveand listedversus the values donor of impurityand (d) . versusFig .energy4ωa valuesc,4b,4c,4d and of binding (d)show. Fig theenergy.4a dependence,4b,4c,4dω as a shows of ththee ground𝜔𝜔𝜔𝜔dependence0 ω stateω s of the groundIn Tableparabolic stateω 3, wemagnetic have listedconfinement the donor term impurity energyin Eq. (3).and For binding fixed energyvalues ofas magnetica In Table 3, we have listed the donor impurity energy and binding energy as a 1 2 2 c 0 4 function of magnetic fieldbinding strength energy ωfor (B E ) bindingon= the3.044 magnetic energy and ( Bfield forE) on variousstrength the magnetic values , for of𝜔𝜔𝜔𝜔 field differentd. strength values , offor impurity different values of impurityfield strength , the binding energy ωdecreasesρ when the temperature increases, as In Table 3, we have listed the donor impurity energy and binding energyfunction as a of magnetic field strength for = 3.044 and for various values of d. function∗ of magnetic field strength for = 3.044 and for various values of d. ∗ c 0 c c c For = 3.044 and d=0 a*, theω bindingω energy increases𝑅𝑅𝑅𝑅 as the magneticω field ω clearly shown ωin Fig.5*a. Similarc behavior0 of the donor binding energy is displayed in distances : (a) d=0 distance, (b) d=0.1s : (a), d=0and (c), (b)d=0.5 d=0.1 andS. ,Abuzaid twoand= (c)confinement3 et.044 d=0.5 al. and frequencies two confinement ∗ For frequencies= 3.044 and d=0 a , theω bindingω energy increases𝑅𝑅𝑅𝑅 as the magnetic field function of magnetic field strength for * c and0 for various values of d. ∗ ∗ For ∗ = 3∗.044 and∗ ∗d=0 a , theω binding∗ω energy increases𝑅𝑅𝑅𝑅 as the magnetic∗ field ω0 𝑅𝑅𝑅𝑅 * * * * ∗ 0 Fig.5b for different, = 5.412 . strength increases. This behavior( =3.044 persists R and for 𝑎𝑎𝑎𝑎all( =d-5.412R=values.3.044 )However,𝑎𝑎𝑎𝑎R. The and binding 𝑎𝑎𝑎𝑎 the=*5.412R binding energy∗ 𝑎𝑎𝑎𝑎c )𝑎𝑎𝑎𝑎 .(BEenergy The0), bindingagainst energythe𝑎𝑎𝑎𝑎 magnetic (BE), fieldagainst thestrength magneticω increases. field 𝑅𝑅𝑅𝑅 This behavior persists for all d-values. However, the binding energy For = 3.044 and0 d=0 a , theω bindingω energy increases𝑅𝑅𝑅𝑅 as the magnetic field strengthω increases.𝑅𝑅𝑅𝑅 This behavior persists for all d-values. However, the binding energy ∗ 0coulomb energy 0of0 the electron∗ enhances.0 the impurity binding energy E decreases with ω0 𝑅𝑅𝑅𝑅 decreases as the impuritystrength (d)𝜔𝜔𝜔𝜔 increases., is greatly We canreduced𝜔𝜔𝜔𝜔0strength𝜔𝜔𝜔𝜔 see a assignificant, isthe greatly distances decrease reduced𝜔𝜔𝜔𝜔 (d) increases inas thethe bindingdistances as shown (d) by increases Figs. 2, as3 andshown 4. by Figs. 2, 3B and 4. strengthω increases.𝑅𝑅𝑅𝑅 This behavior persists for all d-values. However, the bindingdecreases energy as the impurityIn Fig.6 (d)we increases.have shown We the can dependence see a significant of the decrease donor binding in the bindingenergy on the In Fig.9a and 9b, wedecreases have shown as the impurity the effect (d) increases.increasing We can the see temperature. a significant decrease in the binding * * energy as d increases froInm addition,d=0.1of changing a theto d=0.5figures theIn a temperature,demonstrateaddition,. For example, the thatfigures atat the fixed bindingdemonstrate= 3 . value044 energy that andof increases the bindingIn asFig. magnetic energy 11a and increases field 11b, theasenergy magnetic pressure as d fieldincreases effect onfro them d=0.1 a* to d=0.5 a*. For example, at = 3.044 and decreases as the impurity (d) increases. We can see a significant decrease in the binding temperature* for fixed values of pressure (P=10 Kbar), = 2 , = 0 and the impurity position (d= 0.1 a*), on the binding∗ donor* binding* energy for d=0.1 =a3 .044 against the energy as d increases0 from d=0.1 a to d=0.5 a . For example, at and ∗ ∗ ∗ ω * 𝑅𝑅𝑅𝑅 * c0 = 2 the binding energy energy.and reduces the parabolic Again, significantly theconfinement and binding fromthe parabolic strength 4.62499 energy sconfinement R shows to i ncrease.*2.55177 great strength R .* smagnetic increase. field strength = had2 been the binding illustrated.∗ energy reduces significantly from 4.62499 ωR* to 2.55177𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅𝑑𝑑𝑑𝑑 R*. 𝑎𝑎𝑎𝑎 energy as d increases from d=0.1 a to d=0.5 a . For example, at = 3.044 and0 various confinements ( = 3.044 and = 5.412 R ). The binding energy again ∗ = 2 ω * 𝑅𝑅𝑅𝑅 * c enhancement as we increase the the magnetic binding energy field reducesFig. 11a significantly obviously fromshows 4.62499 that theR ∗ tobinding 2.55177 energy R . ∗ ∗ ω 𝑅𝑅𝑅𝑅 c c 0 0 c ∗ 0 0 This result is due to the greatω reduction in the ωcoulomb impurity energy∗ 𝜔𝜔𝜔𝜔 (Eq. 4), as we 𝜔𝜔𝜔𝜔 0* Thisω result*𝑅𝑅𝑅𝑅 is due to the great reductionω in the𝑅𝑅𝑅𝑅 coulombω impurity energy (Eq. 4), as we strength = for2 fixed the binding valuesc energy of the reduces parameters: significantly increases from 4.62499 as the ω Rmagnetic to 2.55177 𝑅𝑅𝑅𝑅field R . strengthshows a clearenhances decreasing behavior as the temperature of the system increases. The In Table 3, we have Inlisted ThisωTable theresult 3 donor𝑅𝑅𝑅𝑅, we is duehave impurity to listed the energygreat the donor reduction and impuritybinding in the energyenergy coulomb asand a imp bindingurity energyenergy (Eq.as a 4), as we Temperature (T= 5,∗ 100, and 200K), Pressure (P=10 while the pressure (0, 10, and 20 kbar), T, d, and mentioned earlier. The same qualitative behaviorc can also be observed in Table 4 with mentioneddependence earlier. The of same the materialqualitative parameter behaviors likecan thealso effective be observed mass inm *Table(P,T) 4and with dielectric Thisω result𝑅𝑅𝑅𝑅 is due to the great reduction= 3.044 in the coulomb = imp3.044urity energy (Eq. 4), as we functionkbar) of and magnetic confinement functionfield strength offrequenciesmentioned magnetic for earlier. field strength The same andqualitative for for various behavior values andcan are of alsofor d. fixed. various be observed Fig. values in 11bof Table d. shows 4 with similar ∗ ∗ * different quantitative behaviorand for = 5.412 . We display inc Fig. 100 the donor c behavior0 of the (BE) but differentfor different quantitativeconstant confinement (behavior, ) on for the temperature= 5.412 and. pressure explain in Table 1, where m mentioned earlier. *Theω same qualitativeω * behavior𝑅𝑅𝑅𝑅ω canω also be observed𝑅𝑅𝑅𝑅 in Table 4 with For = 3.044 andFor∗ d=0 =a3, .044the binding and d=0energy a , increasesthe binding as =energythe5 .magnetic412 increases field as the magnetic field binding0 energy against thedifferent temperature quantitative (T), behavior while for . Effective-mass approximation ∗ ∗ ∗ 𝑟𝑟𝑟𝑟 0 ω 𝑅𝑅𝑅𝑅 ∗ decreases* 𝜖𝜖𝜖𝜖 and𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 increasesω with increasing𝑅𝑅𝑅𝑅 T which diminish donor impurity binding the0 rest ofdifferent the physical quantitative0 parameters behavior offor the =system5.412 . investigates the pressure effects on m and , To see the effects of pressurestrength ωand increases. temperature𝑅𝑅𝑅𝑅 Thisstrength parameters behaviorω increases. persistson the𝑅𝑅𝑅𝑅 Thisbinding for allbehavior denergy-values. persists of However, the forω all0 the d- values.binding𝑅𝑅𝑅𝑅 However, energy theTo binding see the energy effects of pressure and temperature parameters on the binding energy of the are kept fixed. The binding energy shows an ∗ as shown in Table 1 which explain the reason 𝑟𝑟𝑟𝑟of To see the effects of0 pressure and temperature parameters on the bindingenergy energy (B Eof) . the 𝜖𝜖𝜖𝜖 donor impurity, we havedecreases insertedimportant as the the pressure dependenceimpuritydecreases and(d) increases. temperature onas the the impurity temperature,We dependentcan (d) ωsee increases. a significant material and 𝑅𝑅𝑅𝑅We decreaseenhancingcan see a in significant theBE. binding decrease in the binding To see the effects of pressure and temperature parameters on the binding energydonor of theimpurity, we have inserted the pressure and temperature dependent material donor impurity, we have inserted the pressure and temperature dependent material * * * * = 3.044 = 3.044 In Fig.7a ,7b, the behavior of the donor binding energy had been shown as a function of energy as d increasesdonor impurity,energy from d=0.1as wed increases ahaveto d=0.5inserted fro ma .d=0.1 theFor pressureexample, a to d=0.5 and at atemperature. For example, dependentand at material and ∗ ∗ 0 0 ١٠ = :١٠ the binding energy= 2 reduces the binding significantly energy reducesfrom 4.62499 significantly ωR* to from2.55177 𝑅𝑅𝑅𝑅4.62499 R*. ωR* to 2.55177𝑅𝑅𝑅𝑅the Rmagnetic*. field strength , for different values of parameters 2 = ١٠ ∗ ∗ c 0 c c ω ω ١٠4), as we .Thisω result𝑅𝑅𝑅𝑅 is due to theThisω great result𝑅𝑅𝑅𝑅 reduction is due to in the the great coulomb reduction impurity in the energy coulomb (Eq. imp4), urityas we energy (Eq ١١ mentioned earlier. Thementioned same qualitative earlier. Thebehavior same canqualitative also be behaviorobserved canin Table also be4 withobserved in Table 4 with different quantitative differentbehavior quantitativefor = 5. 412behavior . for = 5.412 . ∗ ∗ ω0 𝑅𝑅𝑅𝑅 ω0 𝑅𝑅𝑅𝑅 To see the effects of Topressure see the and effect temperatures of pressure parameters and temperature on the binding parameters energy on of the the binding energy of the

donor impurity, we donorhave insertedimpurity, the we pressure have inserted and temperature the pressure dependent and temperature material dependent material

١٠ ١٠

Fig. 9.a Fig. 1. The ground state energy of the quantum heterostructure for fixed value of magnetic field * * strength (ωc )= 2 R and parabolic confinment strength ω0 = 5.412 R against the number of basis for a) donor impurity at the origin (d = 0 a* ) and b) donor impurity at (d = 0.5 a* ). Fig. 2. The ground state energy of the quantum heterostructure for absence and presence of

the impurity against the magnetic field strength ωc the dashed line with impurity and solid line * for no impurity cases for ω0 = 5.412 R . * Fig. 3. The ground-state for fixed value of ωc = 2 R against the distance for two ** ω00 , ( ω= 3.044 R dashed line and ω0 = 5.412 R ) the solid line).

Fig. 4. The ground-state binding energy against ωc , where * * * * ( ω0 = 3.044 R for dashed line, and ω0 = 5.412 R for solid line ) (a) d = 0 a , (b) d = 0.1 a , and (c) d=0.5a* . * Fig. 5. The binding energy for d=0 a at constant pressure (P=10 kbar) as a function of ωc for ** three temperatures (5K, 100K, and 200K) for (a ) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 6. The binding energy change for d=0 a at constant pressure (P=10 kbar) and ωc = 2 R * with respect to the temperature for( ω0 = 5.412 R * solid line and for ω0 = 3.044 R for the dashed line ). . * Fig. 7. The variation of ground-state binding energy for d=0a against the ωc at fixed temperature (20K) and for three different values of pressure (0, 10, and 20 kbar) a) ** ω00= 3.044 Rb ) ω= 5.412R . Fig. 8. The variation of ground-state binding energy for d = 0 a* against the pressure at fixed * Fig. 9.b ** temperature (20K) and ωc = 2 R for ( ω00= 3.044 R and ω= 5.412R ). * Fig. 9. The ground-state binding energy forFigure d=0.1 a 9 at constant pressure (P=10 kbar) as a function of ωc and for three temperatures (5K, 100K, and 200K): ** a) ω00= 3.044 R and b) ω= 5.412 R . Fig. 10. The ground-state binding energy for d = 0.1 a* at constant pressure (P=10 kba and * * ωc = 2 R , and temperature for (ω0 = 3.044 R dashed line * and ω0 = 5.412 R for the solid line). Int. J. Nano Dimens., 10 (4): 375-390, Autumn 2019 * 385 Fig. 11. The variation of binding energy for d=0.1a against the ωc at fixed temperature (20K) ** for three pressure values (0, 10, and 20 kbar) a) for ω00= 5.412 Rb ) and for ω= 3.044 R . ٩ Fig. 12. The variation of ground-state binding energy for d=0.1a* as a function of the pressure * ** at fixed temperature (20K) and ωc = 2 R for ω00= 3.044 R and ω= 5.412 R . * Fig. 13. The binding energy for d=0.5a at constant pressure (P=10 kbar) as a function of ωc for ** three temperatures (5K, 100K, and 200K). a) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 14. The binding energy for d=0.5 a at constant pressure (P=10 kbar) and ωc = 2 R against * * temperature for( ω0 = 3.044 R for the dashed line and for ω0 = 5.412 R solid line). Fig. 1. The ground state energy of the quantum heterostructure for fixed value of magnetic field * * strength (ωc )= 2 R and parabolic confinment strength ω0 = 5.412 R against the number of basis for a) donor impurity at the originS. Abuzaid (d = 0 a* et) and al. b) donor impurity at (d = 0.5 a* ). Fig. 2. The ground state energy of the quantum heterostructure for absence and presence of

the impurity against the magnetic field strength ωc the dashed line with impurity and solid line * for no impurity cases for ω0 = 5.412 R . * Fig. 3. The ground-state for fixed value of ωc = 2 R against the distance for two ** ω00 , ( ω= 3.044 R dashed line and ω0 = 5.412 R ) the solid line).

Fig. 4. The ground-state binding energy against ωc , where * * * * ( ω0 = 3.044 R for dashed line, and ω0 = 5.412 R for solid line ) (a) d = 0 a , (b) d = 0.1 a , and (c) d=0.5a* . * Fig. 5. The binding energy for d=0 a at constant pressure (P=10 kbar) as a function of ωc for ** three temperatures (5K, 100K, and 200K) for (a ) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 6. The binding energy change for d=0 a at constant pressure (P=10 kbar) and ωc = 2 R * with respect to the temperature for( ω0 = 5.412 R * solid line and for ω0 = 3.044 R for the dashed line ). . * Fig. 7. The variation of ground-state binding energy for d=0a against the ωc at fixed temperature (20K) and for three different values of pressure (0, 10, and 20 kbar) a) ** ω00= 3.044 Rb ) ω= 5.412R . Fig. 8. The variation of ground-state binding energy for d = 0 a* against the pressure at fixed * ** temperature (20K) and ωc = 2 R for ( ω00= 3.044 R and ω= 5.412R ). Fig. 9. The ground-state binding energy for d=0.1a* at constant pressure (P=10 kbar) as a

function of ωc and for three temperatures (5K, 100K, and 200K): a) ω= 3.044 R ** and b) ω= 5.412 R . 00 Fig. 10. The ground-state binding energy for d = 0.1 a* at constant pressure (P=10 kba and * Figure 10 * ωc = 2 R , and temperature for (ω0 = 3.044 R dashed line * and ω0 = 5.412 R for the solid line). * Fig. 11. The variation of binding energy for d=0.1a against the ωc at fixed temperature (20K) ** for three pressure values (0, 10, and 20 kbar) a) for ω00= 5.412 Rb ) and for ω= 3.044 R . Fig. 12. The variation of ground-state binding energy for d=0.1a* as a function of the pressure * ** at fixed temperature (20K) and ωc = 2 R for ω00= 3.044 R and ω= 5.412 R . * Fig. 13. The binding energy for d=0.5a at constant pressure (P=10 kbar) as a function of ωc for ** three temperatures (5K, 100K, and 200K). a) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 14. The binding energy for d=0.5 a at constant pressure (P=10 kbar) and ωc = 2 R against * * temperature for( ω0 = 3.044 R for the dashed line and for ω0 = 5.412 R solid line).

Fig. 1. The ground state energy of the quantum heterostructure for fixed value of magnetic field * * strength (ωc )= 2 R and parabolic confinment strength ω0 = 5.412 R against the number of basis for a) donor impurity at the origin (d = 0 a* ) and b) donor impurity at (d = 0.5 a* ). Fig. 2. The ground state energy of the quantum heterostructure for absence and presence of

the impurity against the magnetic field strength ωc the dashed line with impurity and solid line * for no impurity cases for ω0 = 5.412 R . * Fig. 3. The ground-state for fixed value of ωc = 2 R against the distance for two ** ω00 , ( ω= 3.044 R dashed line Fig.and 11.a ω0 = 5.412 R ) the solid line).

Fig. 4. The ground-state binding energy against ωc , where ( ω= 3.044 R * for dashed line, and ω= 5.412 R * for solid line ) (a) d = 0 a* , (b) d = 0.1 a* , ١٠ 0 0 * and (c) d=0.5a . * Fig. 5. The binding energy for d=0 a at constant pressure (P=10 kbar) as a function of ωc for ** three temperatures (5K, 100K, and 200K) for (a ) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 6. The binding energy change for d=0 a at constant pressure (P=10 kbar) and ωc = 2 R * with respect to the temperature for( ω0 = 5.412 R * solid line and for ω0 = 3.044 R for the dashed line ). . * Fig. 7. The variation of ground-state binding energy for d=0a against the ωc at fixed temperature (20K) and for three different values of pressure (0, 10, and 20 kbar) a) ** ω00= 3.044 Rb ) ω= 5.412R . Fig. 8. The variation of ground-state binding energy for d = 0 a* against the pressure at fixed * ** temperature (20K) and ωc = 2 R for ( ω00= 3.044 R and ω= 5.412R ). Fig. 9. The ground-state binding energy for d=0.1a* at constant pressure (P=10 kbar) as a

function of ωc and for three temperatures (5K, 100K, and 200K): ** a) ω00= 3.044 R and b) ω= 5.412 R . Fig. 10. The ground-state binding energy for d = 0.1 a* at constant pressure (P=10 kba and * * ωc = 2 R , and temperature for (ω0 = 3.044 R dashed line Fig.* 11.b and ω= 5.412 R for the solid line). 0 Fig. 11. The variation of binding energy for d=0.1a* against the ω at fixed temperature (20K) Figure 11 c ** for three pressure values (0, 10, and 20 kbar) a) for ω00= 5.412 Rb ) and for ω= 3.044 R . Fig. 12. The variation of ground-state binding energy for d=0.1a* as a function of the pressure * ** at fixed temperature (20K) and ωc = 2 R for ω00= 3.044 R and ω= 5.412 R . * Fig. 13. The binding energy for d=0.5a at constant pressure (P=10 kbar) as a function of ωc for 386 **Int. J. Nano Dimens., 10 (4): 375-390, Autumn 2019 three temperatures (5K, 100K, and 200K). a) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 14. The binding energy for d=0.5 a at constant pressure (P=10 kbar) and ωc = 2 R against * * ١١ .(temperature for( ω0 = 3.044 R for the dashed line and for ω0 = 5.412 R solid line

Fig. 1. The ground state energy of the quantum heterostructure for fixed value of magnetic field * * strength (ωc )= 2 R and parabolic confinment strength ω0 = 5.412 R against the number of basis for a) donor impurity at the origin (d = 0 a* ) and b) donor impurity at (d = 0.5 a* ). Fig. 2. The ground state energy of the quantum heterostructure for absence and presence of

the impurity against the magnetic field strength ωc the dashed line with impurity and solid line * for no impurity cases for ω0 = 5.412 R . * Fig. 3. The ground-state for fixed value of ωc = 2 R against the distance for two ** ω00 , ( ω= 3.044 R dashed line and ω0 = 5.412 R ) the solid line). S. Abuzaid et al. Fig. 4. The ground-state binding energy against ωc , where * * * * ( ω0 = 3.044 R for dashed line, and ω0 = 5.412 R for solid line ) (a) d = 0 a , (b) d = 0.1 a , and (c) d=0.5a* . * Fig. 5. The binding energy for d=0 a at constant pressure (P=10 kbar) as a function of ωc for ** three temperatures (5K, 100K, and 200K) for (a ) ω00= 3.044 R and b) ω= 5.412 R . * * Fig. 6. The binding energy change for d=0 a at constant pressure (P=10 kbar) and ωc = 2 R * with respect to the temperature for( ω0 = 5.412 R * solid line and for ω0 = 3.044 R for the dashed line ). . * Fig. 7. The variation of ground-state binding energy for d=0a against the ωc at fixed temperature (20K) and for three different values of pressure (0, 10, and 20 kbar) a) ** ω00= 3.044 Rb ) ω= 5.412R . Fig. 8. The variation of ground-state binding energy for d = 0 a* against the pressure at fixed * ** temperature (20K) and ωc = 2 R for ( ω00= 3.044 R and ω= 5.412R ). Fig. 9. The ground-state binding energy for d=0.1a* at constant pressure (P=10 kbar) as a

function of ωc and for three temperatures (5K, 100K, and 200K): ** a) ω00= 3.044 R and b) ω= 5.412 R . Fig. 10. The ground-state binding energy for d = 0.1 a* at constant pressure (P=10 kba and * * ωc = 2 R , and temperature for (ω0 = 3.044 R dashed line * and ω0 = 5.412 R for the solid line). * Fig. 11. The variation of binding energy for d=0.1a against the ωc at fixed temperature (20K) ** for three pressure values (0, 10, and 20 kbar) a) for ω00= 5.412 Rb ) and for ω= 3.044 R . * Fig. 12. The variation of ground-state bindingFigure energy 12 for d=0.1a as a function of the pressure at fixed temperature (20K) and ω= 2 R * for ω= 3.044 R ** and ω= 5.412 R . energy of the donor Hamiltonian enhances as thec confinement00 strength ( ) increases * Fig. 13. The binding energy for d=0.5a at constant pressure (P=10 kbar) as a function of ωc for three temperatures (5K, 100K, and 200K). a) ω= 3.044 R ** and b) ω= 5.412 R0 . from 3.044 R* to 5.412R*. This energy behavior agrees with00 our expectation.ω parameters of GaAs , ( ( , ) ( , ) ) , in the present calculations. In Fig. 5 Fig. 14. The binding energy for d=0.5 a* at constant pressure (P=10 kbar) and ω= 2 R * against c ∗ temperature for( ω= 3.044 R* for the dashed line and for ω= 5.412 R * solid line). 0 0 to Fig.16 , we have shown𝑚𝑚𝑚𝑚 explicitly𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 𝑎𝑎𝑎𝑎𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 the∈ behavior𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 of the donor binding energy (BE) as a Furthermore, we have investigated the impurity ground -state eigenenergy (E) and the function of the magnetic field strength , taking in consideration the impurity Fig. 1. The ground state energy of the quantum heterostructure for fixed value of magnetic field * * binding energy (BEstrength) against(ω ) =the 2 R andmagnetic parabolic confinment strength field strength ω =, 5.412 for Rspecificagainst the values number ofof c 0 c * * ω basis for a) donor impurity at the origin (d = 0 a ) and cb) donor impurity at (d position= 0.5 a ). (d),0 temperature (T), pressure (P) and confinement frequency . Fig. 5a, 5b and versus values Fig.of 2(d). The. groundFig.4 astate ,4b,4c,4d energy of the show quantum the heterostructure dependenceω fors absence of th ande ground presence ofstate ω the impurity against the magnetic field strength ω the dashed line with impurity and solid line 0 c show the donor binding energies against the magnetic field strength forω three different for no impurity cases for ω= 5.412 R * . binding energy (BE) on the magnetic field strength , for0 different values of impurity Fig. 3. The ground-state for fixed value of ω= 2 R * against the distance for two c temperatures (5K, 100K, and 200K) and fixed values of pressure, impurity position ω , ω= 3.044 R** dashed line andc ω= 5.412 R the solid line). distances : (a) d=0 , (b) d=0.100( , and (c) d=0.5 ω and0 two confinement) frequencies Fig. 4. The ground-state binding energy against ω , where ∗ ∗ ∗ c d=0 and confinement frequency = 3.044 R . For fixed temperature, the figure ( ω= 3.044 R * for dashed line, and ω= 5.412 R * for solid line ) (a) d = 0 a* , (b) d = 0.1 a* , = * 0𝑎𝑎𝑎𝑎 = *𝑎𝑎𝑎𝑎 0 𝑎𝑎𝑎𝑎 ( 3.044 R and 5.412R ). The binding energy (BE* ), against the magnetic∗ field ∗ and (c) d=0.5a . 0 energy of the donor Hamiltonian enhances as the confinement strength ( * ) increases clearly𝑎𝑎𝑎𝑎 shows the enhancement of theω binding energy as the magnetic field strength 0 Fig. 50. The binding energy for d=0 a at constant pressure (P=10 kbar) as a function of ωc for strength𝜔𝜔𝜔𝜔 , is greatly reduced𝜔𝜔𝜔𝜔 as the distances (d) increases as shown** by Figs. 2, 3 and 4 . three temperatures (5K, 100K, and 200K) for (a ) ω00= 3.044 R and b) ω= 5.412 R . * * 0 Fig. 13.a ω * * from 3.044 R to 5.412R . This energy behavior agreesFig. 6. withThe binding our energyexpectation. change for d=0 a at constant pressure (P=10 kbar) and ωcincreases.= 2 R This enhancement in the donor binding energy can be attributed to the In addition, the figures demonstratewith thatrespect the to thebinding temperature energy for( ωincreases= 5.412 R * as magnetic field 0 c = * ω solid line and for ω0 3.044 R for the dashed line ). . parabolic magnetic confinement term in Eq. (3). For fixed values of magnetic Furthermore, we have investigated andthe theimpurity parabolic ground Fig.confinement 7 .- Thestate variation eigenenergy strength of grounds-state (E) binding andincrease. energythe for d=0a* against the ω at fixed 1 2 2 ١٢ c temperature (20K) and for three different values of pressure (0, 10, and 20 kbar) a) c 0 4 ω ρ ** field strength , the binding energy decreases when the temperature increases, as binding energy (BE) against theω magnetic field strength , for specific ω00 =values 3.044 𝜔𝜔𝜔𝜔 Rb of ) ω = 5.412R . In Table 3, Fig.we 8. haveThe va riationlisted of theground donor-state binding impurity energy energy for d = 0 aand* against binding the pressure energy at fixed as a c temperaturec (20K) and ω= 2 R * for ( ω0= 3.044 R ** and ω= 5.412R ). ω and versus values of (d)In. FigFig.114aa ,4b,4c,4dand 11b, theshow pressure the dependence effectω on thes donorof th ebinding cground energy stateω00 for d=0.1 against clearly shown in Fig.5a. Similar behavior of the donor binding energy is displayed in Fig. 9. The ground-state binding energy for d=0.1a* at constant pressure (P=10 kbar) as a function of magnetic field strength for = 3.044 and for∗ various values of d. function of ω and for three temperatures (5K, 100K, and 200K): binding energy (BE) on the magnetic fieldfield strengthstrength had, for been different illustrated valuesc . Fig.11 of impuritya obviously ∗ shows𝑎𝑎𝑎𝑎 that the Fig.5b for different, = 5.412 . a) c ω= 3.044 0 R ** and b) ω= 5.412 R . = 3.044 * ω 00ω 𝑅𝑅𝑅𝑅 For and d=0 a , the binding energy increases* as the magnetic field ∗ Fig.c c 10. The ground-state binding energy for d = 0.1 a at constant pressure (P=10 kba and 0 binding energy increases as ω∗the magnetic field *strength enhances while the *pressure (0, ω 𝑅𝑅𝑅𝑅 distances : (a) d=0 , (b) d=0.1 , and (c) d=0.5 ω and two ωconfinement= 2 R , and temperature frequencies for (ω = 3.044 R dashed line 0 c 0 In Fig.6 we have shown the dependence of the donor binding energy on the ∗ strength∗ ω increases.𝑅𝑅𝑅𝑅 This∗ behavior persists for all d-values.* However, the binding energy and ω0 = 5.412 R for the solid line). * 𝑎𝑎𝑎𝑎 10, and 20*𝑎𝑎𝑎𝑎 kbar), T, d, and 𝑎𝑎𝑎𝑎 = 3.044 are fixed. Fig.11b shows* similar behavior of ( =3.044 R and =5.412R ). The binding energyFig. 11. The(BE variation), against of binding the energymagnetic for d=0.1 fielda against the ωc at fixed temperature (20K) decreases as the impurity (d) increases.∗ We can see a significant decrease** intemperature the binding for fixed values of pressure (P=10 Kbar), = 2 , = 0 and for0 three pressure values (0, 10, and 20 kbar) a) for ω00= 5.412 Rb ) and for ω= 3.044 R . 0 0 the (BE) but for different confinementω 𝑅𝑅𝑅𝑅 = 5.412 . Effective-mass approximation* ∗ ∗ strength𝜔𝜔𝜔𝜔 , is greatly reduced𝜔𝜔𝜔𝜔 as the distances (d) increasesFig. 12. The as variation shown of ground by F-igstates. binding2, 3 and energy 4. for d=0.1a as a function of the pressure c * ∗ *Fig.13.b* ** ( = 3.044 and = 5.412 R )ω 𝑅𝑅𝑅𝑅 𝑑𝑑𝑑𝑑 𝑎𝑎𝑎𝑎 energy as d increases froat fixedm d=0.1 temperature a to (20K) d=0.5 and ω ac =.2 RFor for example, ω00= 3.044 R at and ω == 5.412 R3various.044. confinementsand . The binding energy again * 0 investigates the pressure effects on m andω , as shown𝑅𝑅𝑅𝑅 * in Table 1 which explain the Fig. 13. The binding energy for d=0.5a at constant pressure (P=10 kbar) as a function of ωc∗ for ∗ ∗ In addition, the figures demonstrate that the binding energy increases as magneticFigure field 13 0 0 0 *** * ω 𝑅𝑅𝑅𝑅 ω = 2 three temperatures (5K, 100K, and 200K). a) ω00= 3.044 R and b) ωω = 5.412 Rshows .𝑅𝑅𝑅𝑅 a clear decreasing behavior as the temperature of the system increases. The the binding energy reduces𝜖𝜖𝜖𝜖𝑟𝑟𝑟𝑟 significantly from 4.62499 R to 2.55177 R . reason of enhancing BE. Fig. 14. The binding energy for d=0.5 a* at constant pressure (P=10 kbar) and ω= 2 R * against and the parabolic confinement strength∗s increase. c c temperature for( ω= 3.044 R* for the dashed line and for ω= 5.412 R * solid line). * Thisω result𝑅𝑅𝑅𝑅 is due to the great reduction0 in the coulomb impurity energy0 (Eq.dependence 4), as we of the material parameters like the effective mass m (P,T) and dielectric c In Fig. 12, the 0 ground state binding energy had been plotted . The binding energy, (BE) shows ω In Fig.12, the ground 𝜔𝜔𝜔𝜔state binding energy as a function of the pressure for fixed values , In Table 3, we haveas listed a mentioned function the donor earlier. ofimpurity theThe sameenergy pressure qualitative and forbinding behavior fixed energy values can as alsoa great be observed enhancement in Tableconstant as 4 the with pressure( , ) on increases the temperature for and pressure explain in Table 1, where m* ofof T=20K, d = 0.1 , = 2 , andand atat differentdifferent confinements:fixed values of =confinement frequency because of 𝑟𝑟𝑟𝑟 ١٣and𝑃𝑃𝑃𝑃 𝑇𝑇𝑇𝑇 increases with increasing T which diminish donor impurity binding function of magnetic field strengthdifferent for quantitative =∗ 3.044 behavior ∗and for for various= 5.412 values . of d. * decreases𝜖𝜖𝜖𝜖 confinements: c and increasing m and0 decreasing . This effective 3.044 and = 5.412𝑎𝑎𝑎𝑎 R ω had been𝑅𝑅𝑅𝑅∗ plotted . The binding ∗energy, (BE), shows a ωgreat c 0 𝑟𝑟𝑟𝑟 For = 3.044 and d=0 a∗*, theω bindingω energy∗ increases𝑅𝑅𝑅𝑅 asω 0the magnetic𝑅𝑅𝑅𝑅 field energy (BE) . 𝜖𝜖𝜖𝜖 𝑅𝑅𝑅𝑅 ω0 ∗ enhancementTo see as the the effect pressures of pressureincreases and for temperature fixed values parameters of confinement on the frequencybinding energy of the 0 Int. J. Nano Dimens., 10 (4): 375-390, Autumn 2019 387 strengthω increases.𝑅𝑅𝑅𝑅 This behavior persists for all d-values. However, the binding energy becausedonor of increasing impurity, m * weand have decreasing inserted the . pressureThis effective and temperaturemass and dielectric dependent In Fmaterialig.7a ,7b , the behavior of the donor binding energy had been shown as a function of decreases as the impurity (d) increases. We can see a significant decrease𝑟𝑟𝑟𝑟 in the binding pressure dependence lead to lower the electron kinetic𝜖𝜖𝜖𝜖 energy while the coulomb energy the magnetic field strength , for different values of parameters : = * * = 3.044 c 0 ١٠ energy as d increases froincreases.m d=0.1 a to d=0.5 a . For example, at and ∗ ω ω 0 = 2 the binding energy reduces significantly from 4.62499 ωR* to 2.55177𝑅𝑅𝑅𝑅 R*. ١١ ∗ Fig.13a and 13b, the binding energy as a function of magnetic field strength for d=0.5 c Thisω result𝑅𝑅𝑅𝑅 is due to the great reduction in the coulomb impurity energy (Eq. 4), as we ∗ and various confinement, had been shown. We have plotted in Fig.14 the dependence 𝑎𝑎𝑎𝑎of mentioned earlier. The same qualitative behavior can also be observed in Table 4 with the donor binding energy against the temperature for d=0.5 and various confinement different quantitative behavior for = 5.412 . ∗ frequencies = 3.044 and = 5.412 R and the rest𝑎𝑎𝑎𝑎 parameter are fixed. The ∗ 0 ∗ ∗ ωω0 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ω0 To see the effects of pressurebehavior and is temperaturein qualitative parametersagreement with on thethe bindingresults expl energyained ofin Fig.the 6. donor impurity, we have insertedIn F igthe.15 apressure and 15b, andwe showtemperature similar qualitativedependent behaviormaterials as given in figures

3.10 a and b (d=0 a*). Fig.15a and 15b show the effect of pressure and impurity position

comparison between ١٠ d=0.5 ) on the (BE) as a function of magnetic field strength. The) ∗ two d-values𝑎𝑎𝑎𝑎 (d=0 and 0.5 a*) show that, as we increase, d, the binding energy increases

١٣

Fig. 1. The ground state energy of the quantum heterostructure for fixed value of magnetic field * * strength (ωc )= 2 R and parabolic confinment strength ω0 = 5.412 R against the number of basis for a) donor impurity at the origin (d = 0 a* ) and b) donor impurity at (d = 0.5 a* ). Fig. 2. The ground state energy of the quantum heterostructure for absence and presence of

the impurity against the magnetic field strength ωc the dashed line with impurity and solid line * for no impurity cases for ω0 = 5.412 R . * Fig. 3. The ground-state for fixed value of ωc = 2 R against the distance for two ** ω00 , ( ω= 3.044 R dashed line and ω0 = 5.412 R ) the solid line).

Fig. 4. The ground-state binding energy against ωc , where * * * * ( ω0 = 3.044 R for dashed line, and ω0 = 5.412 R for solid line ) (a) d = 0 a , (b) d = 0.1 a , and (c) d=0.5a* . * Fig. 5. The binding energy for d=0 a at constant pressure (P=10 kbar) as a function of ωc for ** three temperatures (5K, 100K, and 200K) for (a ) ω00= 3.044 R and b) ω= 5.412 R . S. Abuzaid* et al. * Fig. 6. The binding energy change for d=0 a at constant pressure (P=10 kbar) and ωc = 2 R * with respect to the temperature for( ω0 = 5.412 R * solid line and for ω0 = 3.044 R for the dashed line ). . * Fig. 7. The variation of ground-state binding energy for d=0a against the ωc at fixed temperature (20K) and for three different values of pressure (0, 10, and 20 kbar) a) ** ω00= 3.044 Rb ) ω= 5.412R . Fig. 8. The variation of ground-state binding energy for d = 0 a* against the pressure at fixed * ** temperature (20K) and ωc = 2 R for ( ω00= 3.044 R and ω= 5.412R ). Fig. 9. The ground-state binding energy for d=0.1a* at constant pressure (P=10 kbar) as a

function of ωc and for three temperatures (5K, 100K, and 200K): ** a) ω00= 3.044 R and b) ω= 5.412 R . Fig. 10. The ground-state binding energy for d = 0.1 a* at constant pressure (P=10 kba and * * ωc = 2 R , and temperature for (ω0 = 3.044 R dashed line * and ω0 = 5.412 R for the solid line). * Fig. 11. The variation of binding energy for d=0.1a against the ωc at fixed temperature (20K) ** for three pressure values (0, 10, and 20 kbar) a) for ω00= 5.412 Rb ) and for ω= 3.044 R . Fig. 12. The variation of ground-state binding energy for d=0.1a* as a function of the pressure * ** at fixed temperature (20K) and ωc = 2 R for ω00= 3.044 R and ω= 5.412 R . * Fig. 13. The binding energy for d=0.5a at constant pressure (P=10 kbar) as a function of ωc for three temperatures (5K, 100K, and 200K). a) ω= 3.044 R ** and b) ω= 5.412 R . 00 * = * Fig. 14. The binding energy for d=0.5 a at constantFigure pressure 14 (P=10 kbar) and ωc 2 R against * * temperature for( ω0 = 3.044 R for the dashed line and for ω0 = 5.412 R solid line).

Fig. 15.a

١٤

Fig. 15.b

* Fig. 15. The binding energy for d=0.5a againstFigure the 15 ωc at fixed temperature (20 K) for three ** pressure values (0, 10, and 20 kbar): a) for ( ω00= 3.044 Rb ) and ω= 5.412 R ). Fig. 16. The variation of ground-state binding energy for d = 0.5 a* against the pressure at fixed ** temperature (20K) and ω2c = for ω00= 3.044 R and ω= 5.412 R . * mass and dielectricTable pressure 1. The dependence dependence of the physical lead parameters function on the pressure of and magnetic temperature. field strength for d=0.5a to lower the electronTable 2 . The kinetic ground state energy (m=0) energy while (in R*) the computedand by various the exact confinement, diagonalization method had been shown. We 1 against expansion method for various range of magnetic field strength (1γ = 6.75B (Tesla)). the dependence of the ١٥ coulomb energy increases.N have plotted in Fig. 14 Fig. 13a and 13b,Table 3 the. The donor binding impurity energy ground state as aenergy (E(donorR*), and bindingdonor impurity energy binding againstenergy the temperature * (BE(R )) against the magnetic field strength ωc and various impurity position (d) for * ω0 = 3.044 R . 388 Table 4. The donor impurity energy (E(R*), and donor impurity bindingInt. J. Nanoenergy Dimens., (BE(R*)) against 10 (4): 375-390, Autumn 2019 * the magnetic field strength ωc and various impurity position (d) for ω0 = 5.412 R . energy of the donor Hamiltonian enhances as the confinement strength ( ) increases

0 from 3.044 R* to 5.412R*. This energy behavior agrees with our expectation.ω

Furthermore, we have investigated the impurity ground -state eigenenergy (E) and the

binding energy (BE) against the magnetic field strength , for specific values of

c 0 and versus values of (d). Fig.4a ,4b,4c,4d show the dependenceω s of the ground stateω

binding energy (BE) on the magnetic field strength , for different values of impurity

c distances : (a) d=0 , (b) d=0.1 , and (c) d=0.5 ω and two confinement frequencies ∗ ∗ ∗ ( =3.044 R* and 𝑎𝑎𝑎𝑎 =5.412R*)𝑎𝑎𝑎𝑎. The binding energy𝑎𝑎𝑎𝑎 (BE), against the magnetic field S. Abuzaid et al. energy of the donor Hamiltonian enhances0 as the confinement0 strength ( ) increases strength𝜔𝜔𝜔𝜔 , is greatly reduced𝜔𝜔𝜔𝜔 as the distances (d) increases as shown by Figs. 2, 3 and 4. 0 from 3.044 R* to 5.412R*. This energy behavior agrees with our expectation.ω In addition, the figures demonstrate that the binding energy increases as magnetic field

Furthermore, we have investigated the and impurity the parabolic ground confinement -state eigenenergy strengths (E) and increase. the binding energy (BE) against the ωmagneticc field strength , for specific values𝜔𝜔𝜔𝜔0 of In Table 3, we have listed the donor impurity energy and binding energy as a c 0 and versus values of (d). Fig.4a ,4b,4c,4d show the dependenceω s of the ground stateω function of magnetic field strength for = 3.044 and for various values of d. ∗ binding energy (BE) on the magnetic field strength , for different valuesc of0 impurity For = 3.044 and d=0 a*, theω bindingω energy increases𝑅𝑅𝑅𝑅 as the magnetic field c ω∗ distances : (a) d=0 , (b) d=0.1 , and0 (c) d=0.5 and two confinement frequencies ∗ strength∗ ω increases.𝑅𝑅𝑅𝑅 This∗ behavior persists for all d-values. However, the binding energy ( =3.044 R* and 𝑎𝑎𝑎𝑎 =5.412R*)𝑎𝑎𝑎𝑎. The binding energy𝑎𝑎𝑎𝑎 (BE), against the magnetic field decreases as the impurity (d) increases. We can see a significant decrease in the binding 0 0 strength𝜔𝜔𝜔𝜔 , is greatly reduced𝜔𝜔𝜔𝜔 as the distances (d) increases as shown by Figs. 2, 3 and 4. * * energy as d increases from d=0.1 a to d=0.5 a . For example,* at = 3.044 and Fig. 15. The binding energy for d=0.5a against the ωc at fixed temperature (20 K) for three In addition, the figures demonstrate that the binding energy increasespressure as magnetic values (0, field 10, and 20 kbar): a) for ( ω= 3.044 Rb**∗ ) and ω= 5.412 R ). 0* 00* = 2 the binding energy reduces significantly from 4.62499 ωR to 2.55177𝑅𝑅𝑅𝑅 * R . Fig. 16. The variation of ground-state bindingFigure energy 16 for d = 0.5 a against the pressure at fixed and the parabolic confinement strengths∗ increase. temperature (20K) and ω2= for ω = 3.044 R ** and ω= 5.412 R . ωc 𝑅𝑅𝑅𝑅 c 00 This result is due to the great reductionTable 1. Thein thedependence coulomb of theimp physicalurity energyparameters (Eq. on 4),the pressureas we and temperature. c 0 ω 𝜔𝜔𝜔𝜔 Table 2. The ground state (m=0) energy (in R*) computed by the exact diagonalization method

In Table 3, we have listedmentioned the donor earlier. impurity The energy same qualitativeand binding1 behavior energy canas aalso be observed in Table 4 with * against expansion method for various range of magneticincrease field the strength temperature (1γ = 6.75B (Tesla)). from T=5K to T= 200 K for d=0.5 a and variousN confinement frequencies * * ,respectively* calculated at ωc =2.0 R , ω = 5.412 R function of magnetic field strengthdifferent for quantitative = 3.044 behavior and forforTable various 3=. The5. 412donor values impurity . of d. ground state energy (E(R ), and donor impurity binding energy 0 and * and the rest (BE(R )) against the magnetic field strength ωandc and various pressure impurity P=10 position Kbar.Also (d) for the donor impurity parameter are∗ fixed. The behavior∗ is in qualitative * c 0 0 = * For = 3.044 and d=0 a , theω bindingω energy increases𝑅𝑅𝑅𝑅 asω the magnetic𝑅𝑅𝑅𝑅 field ω0 3.044 Rbinding. energy is increasing function of pressure

To see the agreementeffects of pressure with the andTable results temperature 4. The donorexplained impurity parameters in energy Fig. on (E6.( Rthe*), and binding donor impurityenergy bindingof the energy (BE(R*)) against ∗ for fixed values of temperature and magnetic 0 In Fig. 15a and 15b, the we magnetic show field similar strength qualitative ω and various impurity position (d) for ω= 5.412 R * . strengthω increases.𝑅𝑅𝑅𝑅 This behavior persists for all d-values. However, the binding energy c field. For strong magnetic0 field strength, the donor donor impurity,behaviors we have as given inserted in Figs.the pressure3.10 a andand btemperature (d=0 a*). dependent material binding energies enhances significantly for any decreases as the impurity (d) increases. We Fig.can see 15a a significant and 15b decrease show thein the effect binding of pressure a* hydrostatic pressure and temperature values as and impurity position (d=0.5 ) on the (BE) * * = 3.044 we expected. ١٠ energy as d increases from d=0.1 a to d=0.5as a . functionFor example, of at magnetic fieldand strength. The comparison between two d-values∗ (d=0 and = 2 ω0* 𝑅𝑅𝑅𝑅 * CONFLICT OF INTEREST the binding energy reduces significantly0.5 a*) show from that, 4.62499 as we R increase,to 2.55177 d, R the. binding ∗ The authors declare that they have no c energy increases due to the great reduction in the Thisω result𝑅𝑅𝑅𝑅 is due to the great reduction in the coulomb impurity energy (Eq. 4), as we competing interests. coulomb attraction energy (Eq. 4). In Fig.16, we mentioned earlier. The same qualitative behaviordisplay can the also results be observed of the in donorTable 4 binding with energy * REFERENCES for d=0.5 a . The behavior shown agrees with the [1] Bastard G., (1981), Hydrogenic impurity states in a quantum different quantitative behavior for = 5.412 . * (BE) behavior given in Fig.8 (for d=0 a with same well: A simple model. Phys. Rev. B. 24: 4714-4718. ∗ Applications of silver ١٦ ,(reason). [2] Poonam V., Sanjiv K. M., (2019 0 ω 𝑅𝑅𝑅𝑅 To see the effects of pressure and temperature parameters on the binding energy of the nanoparticles in diverse scctors. Int. J. Nano. Dimens.10: CONCLUSION 18-36. donor impurity, we have inserted the pressureIn conclusion,and temperature the exact dependent diagonalization material method [3] Porras-Montenegro N., Perez-Merchancano S., latge A., (1993), Binding energies and density of impurity states in had been used to solve the donor impurity spherical GaAs‐(Ga, Al)As quantum dots. J. Appl. Phys. 74: Hamiltonian in a heterostructure subjected to 7624-7628. ground-state [4] Ciftja O., (2013), Understanding electronic systems in an applied magnetic field. The١٠ energy of GaAs/AlGaAs heterostructure had been semiconductor quantum dots, Phys. Scrip. 88: 058302- computed. Furthermore, the impurity effect on 058306. [5] Liang S., Xie W. F., (2011), The hydrostatic pressure and the ground-state energy had been shown. In temperature effects on a hydrogenic impurity in a spherical addition, the influence of the hydrostatic pressure, quantum dot. Eur. Phys. J. B. 81: 79-84. temperature and magnetic field on the binding [6] Holtz O., Qing Xiang Zh., (2004), Impurities confined in energy of the donor impurity can be summarized quantum structures, 1st ed, (Springer), 5-10. as follows: the donor impurity binding energy [7] Dingle R., Stormer H. L., Gossard A. C.,Wiegmann W., (1978), is a decreasing function of temperature for Electron mobilities in modulation‐doped semiconductor heterojunction superlattices.Appl. Phys. Lett. 33: 665-671. fixed values of pressure and magnetic field. For [8] Zhu J.-L., (1989), Exact solutions for hydrogenic donor states example the donor impurity binding energy in a spherically rectangular quantum well. Phys. Rev. B. 39: decrease from BE=8.3 R * to BE= 7.8 R * as we 8780-8786.

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