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The effect of hydrostatic pressure on binding energy of impurity states in spherical quantum dots
ARTICLE IN PRESS
Physica E 28 (2005) 225–229 www.elsevier.com/locate/physe
The effect of hydrostatic pressure on binding energy of impurity states in spherical quantum dots A. John Peter Arulmigu Kalasalingam College of Engineering, Krishnankoil-626 190, India Received 23 February 2005; accepted 9 March 2005
Abstract The binding energy of shallow hydrogenic impurities in spherical quantum dots in the influence of pressure is calculated using a variational approach within the effective mass approximation. The binding energy is computed for GaAs/Ga1xAlxAs structures as a function of the dot size. The results show that the impurity binding energy (i) increases with the reduction in dot sizes for a given pressure (ii) increases with the pressure is increased for a given dot (iii) the energy increases to a maximum value at 50 A˚ and then decreases as the size of the dot increases beyond 50 A˚ when the realistic model is considered and (iv) the ionization energies are higher as the pressure becomes stronger. All the calculations have been carried out with finite and infinite models and the results are compared with available data in the literature. r 2005 Elsevier B.V. All rights reserved. PACS: 72.20.My; 71.30.+h; 73.40.Lq; 73.20.Dx; 73.20.Hb Keywords: Impurity state; Donor energy; Quantum dot
1. Introduction In the last two decades there has been a tremendous research activity on low dimensional semiconductor systems (LDSS). Eventhough the physics of one, two dimensional systems have drawn the attention of theoreticians for a long time, only since the late 1970s these systems could be realized, Tel.: +91 04 56 328 9042; fax: +91 04 56 328 9322.
E-mail address: [email protected].
with the development of semiconductor growth techniques like the molecular beam epitaxy (MBE) and metal oxide chemical vapour deposition (MOCVD). The quantum size effects in low dimensions have been extensively carried out both experimentally [1] and theoretically [2]. Accurate resolution of the sizes of quantum dots (QDs) is important for most of their device applications, since QDs emission energies, number of excited states, energy sublevels, charging energies, Coulomb interactions, and other optoelectronic properties
1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.03.018
ARTICLE IN PRESS 226
A. John Peter / Physica E 28 (2005) 225–229
are determined by their confinement dimensions. The measurements leading to electronic and optical properties of such microstructures of low temperature photoluminous are made possible under high static pressure [3]. The same study has been carried out under the atmospheric pressure in the same heterostructures [4]. The effect of hydrostatic pressure on the binding energy of donor impurities in quantum well (QW) structures has been calculated by Elabsy [5]. The impurity plays a fundamental role in some physical properties such as optical and transport phenomena at low temperature. The binding energy and the density of states of shallow impurities in cubic [6] and in spherical quantum dot [7] have been calculated as a function of dot size. The study of impurity states in semiconductor states is imperative as the addition of impurities can change the properties of any quantum device dramatically. These effect have been studied by Zhigang et al. [8] and Szwacka [9]. This kind of study of binding energy of the impurity makes it possible to fabricate the low threshold laser diodes [10]. It is found that the binding energy increases with increasing external hydrostatic pressure for a given QW thickness and temperature. Mercy et al. [11] have found that the carrier concentration could be decreased with increasing pressure when the samples are cooled to low temperatures in hydrostatic pressure transport studies of modulated doped QWs. QDs are an ideal system to observe how applied pressure changes in electronic levels can be influenced by changes in electron–phonon, lattice covalency, and donor levels. The dependence of the photoluminescence on hydrostatic pressure up to 12 GPa and laser excitation intensity ranging from 300 to 3000 W cm2 has been studied at 10 K [12]. Several studies have probed structural phase transitions in QDs at high pressure. At low pressure (o1 GPa), the optical properties of solvated nanomaterials are modulated by pressure induced electronic level tuning, particularly for surface and trap states [13]. In the present work, calculations of binding energies of the donor impurities in GaAs QD, placed at the centre are performed using the effective mass approximation within a variational
scheme. The effect of hydrostatic pressure on the binding energies of shallow donors in the GaAs QD with Ga1xAlxAs barriers is investigated. A systematic study of variation of pressure as the function of dot size has been attempted for both the finite and infinite confinement of a spherical QD. The results are then compared with the existing data available. The method followed is presented in Section 2 while the results and discussion are provided in Section 3.
2. Theory In the effective mass approximation, the Hamiltonian for a hydrogenic donor impurity in a GaAs/Ga1xAlxAs QD under the influence of hydrostatic pressure is given by 8 r2 > > > < 2m ðPÞ þ V ðr; PÞ; rpR; (1) H¼ > r2 > > þ V ðr; PÞ; rXR; : 2m ðPÞ where r is the electron–donor distance, m ðPÞ is the effective mass of an electron in GaAs as a function of hydrostatic pressure P, V ðr; PÞ is the confining potential, R is the dot radius. The ground state energy of the system E1 is obtained by solving the Schro¨dinger equation, H 1 F1 ¼ E 1 F1 .
(2)
The confining potential due to the discontinuity of band edges at the GaAs/Ga1xAlxAs interface at the dot radius is as follows: ( 0; rpR; V ðrÞ ¼ (3) V ðrÞ; rXR:
2.1. Effect of pressure The application of hydrostatic pressure modifies the lattice constants, dot size, barrier height, effective masses and dielectric constants. These values are obtained in the following way.
ARTICLE IN PRESS A. John Peter / Physica E 28 (2005) 225–229
The variation of well width with pressure is given by RðPÞ ¼ R0 ð1 1:5082 103 PÞ where P is in GPa, R0 is the zero pressure width of the QD, taking in to account using ðda=dPÞ ¼ 2:6694 104 ao where ao is the lattice constant of GaAs [14]. The variation of dielectric constant with pressure is given as ðPÞ ¼ 13:13 0:088P, where P is in GPa [8]. The effective mass in the well and barrier region change as
The transcendental equation aRðPÞ þ br tan aRðPÞ ¼ 0 when the impurity is present at the centre of the dot, the Hamiltonian of the system is given by, 8 r2 e2 > > > < 2m ðPÞ ðPÞr þ V ðr; PÞ; rpR; (6) H¼ > r2 e2 > > þ V ðr; PÞ; rXR: : 2m ðPÞ ðPÞr The ground state energy of the system E1 is obtained by solving the Schro¨dinger equation, H 2 F2 ¼ E 2 F2 .
m ðPÞ ¼ m ð0Þ expð0:078PÞ; where P is in GPa [14]. The total band gap difference between GaAs and Ga1xAlxAs as a function of x is given by DE g ðx; PÞ ¼ DE g ðxÞ þ PDðxÞ where DE g ðxÞ ¼ 1:555x þ 0:37x2 in eV is the variation of the energy gap difference [15] and DðxÞ is the pressure coefficient of the band gap given by DðxÞ ¼ ½ð1:3 103 Þx eV=Kbar,
Since the inclusion of impurity potential leads to a nonseparable differential equation which cannot be solved analytically it is necessary to use a variational approach to calculate the eigen function and eigen value of the Hamiltonian for the ground state. The trial wave function of the hydrogenic impurity is given by 8 N 2 sinðarÞ expðlrÞ > > ; rpR; < r c¼ N 2 sinðaRÞ expðbRÞ exp½ðb þ lÞr
> > ; rXR; : r (7)
where 1 kbar is 0.1 GPa and the potential barrier, as a function of Al concentration x, is given by V ðrÞ ¼ 0:6DE g ðx; PÞ
227
(4)
using these variations the donor ionization energies are obtained, for different pressures, for both finite and infinite barriers using variational method followed below. 2.2. Finite barrier problem Eq. (2) can be solved exactly using wave function, given by 8 N 1 sinðarÞ > > ; rpR; < r (5) c¼ N 1 sinðaRÞ expðbRÞ expðbrÞ > > ; rXR; : r where a ¼ ½2m ðPÞE 1 1=2 and b ¼ ½2m ðPÞðV ðrÞ E 1 1=2 .
where l is the variational parameter and N 2 is the normalization constant. The binding energy of the hydrogenic impurity is defined as the ground energy of the system without the impurity present minus the impurity state energy. The ionization energy, E ion ¼ E 1 hHimin .
(8)
The evaluation of hHi in each case is straight forward. As the expressions obtained for hHi are too lengthy; we refrain from giving them here.
3. Results and discussion The characteristic parameters of GaAs and Ga1xAlxAs, wherein x is taken as 0.3, used in the calculations are presented in Table 1. The variation of subband energies for different hydrostatic pressures and dot sizes are given in Table 2.
ARTICLE IN PRESS A. John Peter / Physica E 28 (2005) 225–229
228
Table 1 Characteristic parameters of GaAs and Ga0.7Al0.3As Pressure (GPa)
Barrier height (meV)
m (a.u)
0 5 10 15 20
227.88 216.18 204.48 192.78 181.08
13.13 12.69 12.25 11.81 11.37
0.067 0.099 0.146 0.216 0.319
Table 2 Subband energy (meV) Dot size (A˚)
10 20 30 40 50
Pressure (GPa) 0
5
10
15
20
209.44 168.67 130.56 103.36 81.6
193.12 146.88 108.8 84.32 65.28
174.08 127.84 89.76 65.28 51.68
157.76 106.08 73.44 51.68 38.08
138.72 89.76 57.12 40.8 29.92
The subband energy decreases when the pressure increases for all the well width. This is due the variation of mass and the dielectric constant with the change in pressure. Also the subband energy increases when the width decreases for all the pressure. Fig. 1 presents the ionization energy, for a shallow donor impurities, as a function of impurity concentration with and without the hydrostatic pressure for different dot sizes. The behaviour of the variation of ionization energy with the size of the dot is a well-known behaviour. The decrease in ionization energy with the increase of well width is a common feature [16,17]. For a lower dot size 100 A˚, the ionization energy is always higher than that of a well of R ¼ 1000 A˚. A small dot yields larger ionization energy for a given pressure which decides the barrier height. As hydrostatic pressure becomes stronger the ionization effect becomes dominant which reduces tunnelling. The ionization energy is higher for a given dot size when the hydrostatic pressure is applied. This is due to the additional confinement due to the pressure. The ionization energy reaches a bulk value (5.3 meV) when L - 600 A˚.
Fig. 1. Variation of ionization energy with dot size.
Fig. 2. Variation of ionization energy with well width for a finite dot for two different pressures.
Fig. 2 shows the ionization energy as a function of dot size with and without applying the hydrostatic pressure a finite barrier. For a
ARTICLE IN PRESS A. John Peter / Physica E 28 (2005) 225–229
229
given pressure which decides the barrier height. As the pressure becomes stronger the ionization effect becomes dominant which reduces tunnelling. To conclude it is demonstrated that the binding energy is computed for a GaAs/Ga1xAlxAs QD in the influence of pressure within the effective mass approximation. This is an attempt to investigate the donor binding energy theoretically on such systems when pressure is applied. And it is hoped that the present work would stimulate further experimental activities in semiconductor in QD nanostructures.
Fig. 3. Variation of ionization energy with the pressure for two different dot size.
particular dot size the ionization energy of donor electrons are higher when the pressure becomes stronger. In contrast to infinite well model, the ionization energy approaches the bulk value in both the limits of L ! 0 and L ! 1. Similar behaviour is exhibited in other low dimensional systems such as QD [18,19]. The ionization energy as a function of dot size for two different pressure is shown in Fig. 2 in a more realistic model. It is seen that in all the cases, the energy increases to a maximum value at 50 A˚ and then decreases as the size of the dot increases beyond 50 A˚. The ionization energies are higher as the hydrostatic pressure becomes stronger. For larger radius it shows 3D behaviour. For extremely small dot sizes the particle is less localized due to tunnelling effects; however, the pressure increases the localization. In this region the tunnelling effect dominates. In the other extreme, where the dot size is larger, the pressure induced localization wins over tunnelling. This net effect results in the variation shown in Fig. 2. In Fig. 3 we have presented the variation of ionization energy with pressure for the different dot sizes. The variation of ionization energy with the size of the dot is a well-known behaviour. A small dot yields larger ionization energy for a
References [1] P. Ramvall, S. Tanaka, S. Nomura, P. Riblet, Y. Aoyagi, Appl. Phys. Lett. 73 (1998) 1104. [2] A. Corella Madueno, R.A. Rosas, J.L. Marine, R. Riera, Phys. Low-Dimens. Semicond. Struct. 5/6 (1999) 75. [3] B. Welber, M. Cardona, C.K. Kim, S. Rodriquez, Phys. Rev. B 12 (1975) 5729. [4] R.L. Greene, K.K. Bajaj, Sol. Stat. Commun. 45 (1983) 825. [5] A.M. Elabsy, Phys. Scr. 48 (1993) 376. [6] F.J. Ribeiro, A. Latge´, Phys. Rev. B 50 (1994) 4913. [7] J.L. Zhu, J.J. Xiong, B.L. Gu, Phys. Rev. B 41 (1990) 6001. [8] Zhigang Xiao, Jiqian Zhu, Fenglai He, Phys. Stat. Sol. (b) 191 (1995) 401. [9] T. Szwacka, J. Phys.: Condens. Matter 8 (1996) 10521. [10] K.M. Groom, A.I. Tartakovskii, D.J. Mowbray, M.S. Skolnick, P.W. Smowton, M. Hopkinson, G. Hill, Appl. Phys. Lett. 8 (2002) 1. [11] J.M. Mercy, C. Bousquet, J.L. Robert, A. Raymond, G. Gregoris, J. Beerens, J.C. Portal, P.M. Frijlink, P. Delescluse, J. Chevrier, N.T. Linh, Surf. Sci. 142 (1984) 298. [12] Z.X. Liu, O.G. Schmidt, U.D. Venkateswaran, K. Eberl, K. Syassen, Semicond. Sci. Technol. 15 (2000) 155. [13] R.W. Meulenberg, G.F. Strouse, Phys. Rev. B 66 (2002) 035317. [14] A. Beneditctal, B. Sukumar, K. Nananeethakrishnan, Phys. Stat. Sol. B 178 (1993) 167. [15] S. Adachi, J. Appl. Phys. 58 (1985) R1. [16] R.L. Greene, K.K. Bajaj, Sol. Stat. Commun. 45 (1983) 825. [17] A.J. Peter, K. Navaneethakrishnan, Physica E 15 (2002) 153. [18] V. Ranjan, V.A. Singh, J. Appl. Phys. 89 (2001) 6415. [19] Sr.G. Jayam, K. Navaneethakrishnan, Solid State Commun. 126 (2003) 681.
Physica E 28 (2005) 225–229 www.elsevier.com/locate/physe
The effect of hydrostatic pressure on binding energy of impurity states in spherical quantum dots A. John Peter Arulmigu Kalasalingam College of Engineering, Krishnankoil-626 190, India Received 23 February 2005; accepted 9 March 2005
Abstract The binding energy of shallow hydrogenic impurities in spherical quantum dots in the influence of pressure is calculated using a variational approach within the effective mass approximation. The binding energy is computed for GaAs/Ga1xAlxAs structures as a function of the dot size. The results show that the impurity binding energy (i) increases with the reduction in dot sizes for a given pressure (ii) increases with the pressure is increased for a given dot (iii) the energy increases to a maximum value at 50 A˚ and then decreases as the size of the dot increases beyond 50 A˚ when the realistic model is considered and (iv) the ionization energies are higher as the pressure becomes stronger. All the calculations have been carried out with finite and infinite models and the results are compared with available data in the literature. r 2005 Elsevier B.V. All rights reserved. PACS: 72.20.My; 71.30.+h; 73.40.Lq; 73.20.Dx; 73.20.Hb Keywords: Impurity state; Donor energy; Quantum dot
1. Introduction In the last two decades there has been a tremendous research activity on low dimensional semiconductor systems (LDSS). Eventhough the physics of one, two dimensional systems have drawn the attention of theoreticians for a long time, only since the late 1970s these systems could be realized, Tel.: +91 04 56 328 9042; fax: +91 04 56 328 9322.
E-mail address: [email protected].
with the development of semiconductor growth techniques like the molecular beam epitaxy (MBE) and metal oxide chemical vapour deposition (MOCVD). The quantum size effects in low dimensions have been extensively carried out both experimentally [1] and theoretically [2]. Accurate resolution of the sizes of quantum dots (QDs) is important for most of their device applications, since QDs emission energies, number of excited states, energy sublevels, charging energies, Coulomb interactions, and other optoelectronic properties
1386-9477/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.03.018
ARTICLE IN PRESS 226
A. John Peter / Physica E 28 (2005) 225–229
are determined by their confinement dimensions. The measurements leading to electronic and optical properties of such microstructures of low temperature photoluminous are made possible under high static pressure [3]. The same study has been carried out under the atmospheric pressure in the same heterostructures [4]. The effect of hydrostatic pressure on the binding energy of donor impurities in quantum well (QW) structures has been calculated by Elabsy [5]. The impurity plays a fundamental role in some physical properties such as optical and transport phenomena at low temperature. The binding energy and the density of states of shallow impurities in cubic [6] and in spherical quantum dot [7] have been calculated as a function of dot size. The study of impurity states in semiconductor states is imperative as the addition of impurities can change the properties of any quantum device dramatically. These effect have been studied by Zhigang et al. [8] and Szwacka [9]. This kind of study of binding energy of the impurity makes it possible to fabricate the low threshold laser diodes [10]. It is found that the binding energy increases with increasing external hydrostatic pressure for a given QW thickness and temperature. Mercy et al. [11] have found that the carrier concentration could be decreased with increasing pressure when the samples are cooled to low temperatures in hydrostatic pressure transport studies of modulated doped QWs. QDs are an ideal system to observe how applied pressure changes in electronic levels can be influenced by changes in electron–phonon, lattice covalency, and donor levels. The dependence of the photoluminescence on hydrostatic pressure up to 12 GPa and laser excitation intensity ranging from 300 to 3000 W cm2 has been studied at 10 K [12]. Several studies have probed structural phase transitions in QDs at high pressure. At low pressure (o1 GPa), the optical properties of solvated nanomaterials are modulated by pressure induced electronic level tuning, particularly for surface and trap states [13]. In the present work, calculations of binding energies of the donor impurities in GaAs QD, placed at the centre are performed using the effective mass approximation within a variational
scheme. The effect of hydrostatic pressure on the binding energies of shallow donors in the GaAs QD with Ga1xAlxAs barriers is investigated. A systematic study of variation of pressure as the function of dot size has been attempted for both the finite and infinite confinement of a spherical QD. The results are then compared with the existing data available. The method followed is presented in Section 2 while the results and discussion are provided in Section 3.
2. Theory In the effective mass approximation, the Hamiltonian for a hydrogenic donor impurity in a GaAs/Ga1xAlxAs QD under the influence of hydrostatic pressure is given by 8 r2 > > > < 2m ðPÞ þ V ðr; PÞ; rpR; (1) H¼ > r2 > > þ V ðr; PÞ; rXR; : 2m ðPÞ where r is the electron–donor distance, m ðPÞ is the effective mass of an electron in GaAs as a function of hydrostatic pressure P, V ðr; PÞ is the confining potential, R is the dot radius. The ground state energy of the system E1 is obtained by solving the Schro¨dinger equation, H 1 F1 ¼ E 1 F1 .
(2)
The confining potential due to the discontinuity of band edges at the GaAs/Ga1xAlxAs interface at the dot radius is as follows: ( 0; rpR; V ðrÞ ¼ (3) V ðrÞ; rXR:
2.1. Effect of pressure The application of hydrostatic pressure modifies the lattice constants, dot size, barrier height, effective masses and dielectric constants. These values are obtained in the following way.
ARTICLE IN PRESS A. John Peter / Physica E 28 (2005) 225–229
The variation of well width with pressure is given by RðPÞ ¼ R0 ð1 1:5082 103 PÞ where P is in GPa, R0 is the zero pressure width of the QD, taking in to account using ðda=dPÞ ¼ 2:6694 104 ao where ao is the lattice constant of GaAs [14]. The variation of dielectric constant with pressure is given as ðPÞ ¼ 13:13 0:088P, where P is in GPa [8]. The effective mass in the well and barrier region change as
The transcendental equation aRðPÞ þ br tan aRðPÞ ¼ 0 when the impurity is present at the centre of the dot, the Hamiltonian of the system is given by, 8 r2 e2 > > > < 2m ðPÞ ðPÞr þ V ðr; PÞ; rpR; (6) H¼ > r2 e2 > > þ V ðr; PÞ; rXR: : 2m ðPÞ ðPÞr The ground state energy of the system E1 is obtained by solving the Schro¨dinger equation, H 2 F2 ¼ E 2 F2 .
m ðPÞ ¼ m ð0Þ expð0:078PÞ; where P is in GPa [14]. The total band gap difference between GaAs and Ga1xAlxAs as a function of x is given by DE g ðx; PÞ ¼ DE g ðxÞ þ PDðxÞ where DE g ðxÞ ¼ 1:555x þ 0:37x2 in eV is the variation of the energy gap difference [15] and DðxÞ is the pressure coefficient of the band gap given by DðxÞ ¼ ½ð1:3 103 Þx eV=Kbar,
Since the inclusion of impurity potential leads to a nonseparable differential equation which cannot be solved analytically it is necessary to use a variational approach to calculate the eigen function and eigen value of the Hamiltonian for the ground state. The trial wave function of the hydrogenic impurity is given by 8 N 2 sinðarÞ expðlrÞ > > ; rpR; < r c¼ N 2 sinðaRÞ expðbRÞ exp½ðb þ lÞr
> > ; rXR; : r (7)
where 1 kbar is 0.1 GPa and the potential barrier, as a function of Al concentration x, is given by V ðrÞ ¼ 0:6DE g ðx; PÞ
227
(4)
using these variations the donor ionization energies are obtained, for different pressures, for both finite and infinite barriers using variational method followed below. 2.2. Finite barrier problem Eq. (2) can be solved exactly using wave function, given by 8 N 1 sinðarÞ > > ; rpR; < r (5) c¼ N 1 sinðaRÞ expðbRÞ expðbrÞ > > ; rXR; : r where a ¼ ½2m ðPÞE 1 1=2 and b ¼ ½2m ðPÞðV ðrÞ E 1 1=2 .
where l is the variational parameter and N 2 is the normalization constant. The binding energy of the hydrogenic impurity is defined as the ground energy of the system without the impurity present minus the impurity state energy. The ionization energy, E ion ¼ E 1 hHimin .
(8)
The evaluation of hHi in each case is straight forward. As the expressions obtained for hHi are too lengthy; we refrain from giving them here.
3. Results and discussion The characteristic parameters of GaAs and Ga1xAlxAs, wherein x is taken as 0.3, used in the calculations are presented in Table 1. The variation of subband energies for different hydrostatic pressures and dot sizes are given in Table 2.
ARTICLE IN PRESS A. John Peter / Physica E 28 (2005) 225–229
228
Table 1 Characteristic parameters of GaAs and Ga0.7Al0.3As Pressure (GPa)
Barrier height (meV)
m (a.u)
0 5 10 15 20
227.88 216.18 204.48 192.78 181.08
13.13 12.69 12.25 11.81 11.37
0.067 0.099 0.146 0.216 0.319
Table 2 Subband energy (meV) Dot size (A˚)
10 20 30 40 50
Pressure (GPa) 0
5
10
15
20
209.44 168.67 130.56 103.36 81.6
193.12 146.88 108.8 84.32 65.28
174.08 127.84 89.76 65.28 51.68
157.76 106.08 73.44 51.68 38.08
138.72 89.76 57.12 40.8 29.92
The subband energy decreases when the pressure increases for all the well width. This is due the variation of mass and the dielectric constant with the change in pressure. Also the subband energy increases when the width decreases for all the pressure. Fig. 1 presents the ionization energy, for a shallow donor impurities, as a function of impurity concentration with and without the hydrostatic pressure for different dot sizes. The behaviour of the variation of ionization energy with the size of the dot is a well-known behaviour. The decrease in ionization energy with the increase of well width is a common feature [16,17]. For a lower dot size 100 A˚, the ionization energy is always higher than that of a well of R ¼ 1000 A˚. A small dot yields larger ionization energy for a given pressure which decides the barrier height. As hydrostatic pressure becomes stronger the ionization effect becomes dominant which reduces tunnelling. The ionization energy is higher for a given dot size when the hydrostatic pressure is applied. This is due to the additional confinement due to the pressure. The ionization energy reaches a bulk value (5.3 meV) when L - 600 A˚.
Fig. 1. Variation of ionization energy with dot size.
Fig. 2. Variation of ionization energy with well width for a finite dot for two different pressures.
Fig. 2 shows the ionization energy as a function of dot size with and without applying the hydrostatic pressure a finite barrier. For a
ARTICLE IN PRESS A. John Peter / Physica E 28 (2005) 225–229
229
given pressure which decides the barrier height. As the pressure becomes stronger the ionization effect becomes dominant which reduces tunnelling. To conclude it is demonstrated that the binding energy is computed for a GaAs/Ga1xAlxAs QD in the influence of pressure within the effective mass approximation. This is an attempt to investigate the donor binding energy theoretically on such systems when pressure is applied. And it is hoped that the present work would stimulate further experimental activities in semiconductor in QD nanostructures.
Fig. 3. Variation of ionization energy with the pressure for two different dot size.
particular dot size the ionization energy of donor electrons are higher when the pressure becomes stronger. In contrast to infinite well model, the ionization energy approaches the bulk value in both the limits of L ! 0 and L ! 1. Similar behaviour is exhibited in other low dimensional systems such as QD [18,19]. The ionization energy as a function of dot size for two different pressure is shown in Fig. 2 in a more realistic model. It is seen that in all the cases, the energy increases to a maximum value at 50 A˚ and then decreases as the size of the dot increases beyond 50 A˚. The ionization energies are higher as the hydrostatic pressure becomes stronger. For larger radius it shows 3D behaviour. For extremely small dot sizes the particle is less localized due to tunnelling effects; however, the pressure increases the localization. In this region the tunnelling effect dominates. In the other extreme, where the dot size is larger, the pressure induced localization wins over tunnelling. This net effect results in the variation shown in Fig. 2. In Fig. 3 we have presented the variation of ionization energy with pressure for the different dot sizes. The variation of ionization energy with the size of the dot is a well-known behaviour. A small dot yields larger ionization energy for a
References [1] P. Ramvall, S. Tanaka, S. Nomura, P. Riblet, Y. Aoyagi, Appl. Phys. Lett. 73 (1998) 1104. [2] A. Corella Madueno, R.A. Rosas, J.L. Marine, R. Riera, Phys. Low-Dimens. Semicond. Struct. 5/6 (1999) 75. [3] B. Welber, M. Cardona, C.K. Kim, S. Rodriquez, Phys. Rev. B 12 (1975) 5729. [4] R.L. Greene, K.K. Bajaj, Sol. Stat. Commun. 45 (1983) 825. [5] A.M. Elabsy, Phys. Scr. 48 (1993) 376. [6] F.J. Ribeiro, A. Latge´, Phys. Rev. B 50 (1994) 4913. [7] J.L. Zhu, J.J. Xiong, B.L. Gu, Phys. Rev. B 41 (1990) 6001. [8] Zhigang Xiao, Jiqian Zhu, Fenglai He, Phys. Stat. Sol. (b) 191 (1995) 401. [9] T. Szwacka, J. Phys.: Condens. Matter 8 (1996) 10521. [10] K.M. Groom, A.I. Tartakovskii, D.J. Mowbray, M.S. Skolnick, P.W. Smowton, M. Hopkinson, G. Hill, Appl. Phys. Lett. 8 (2002) 1. [11] J.M. Mercy, C. Bousquet, J.L. Robert, A. Raymond, G. Gregoris, J. Beerens, J.C. Portal, P.M. Frijlink, P. Delescluse, J. Chevrier, N.T. Linh, Surf. Sci. 142 (1984) 298. [12] Z.X. Liu, O.G. Schmidt, U.D. Venkateswaran, K. Eberl, K. Syassen, Semicond. Sci. Technol. 15 (2000) 155. [13] R.W. Meulenberg, G.F. Strouse, Phys. Rev. B 66 (2002) 035317. [14] A. Beneditctal, B. Sukumar, K. Nananeethakrishnan, Phys. Stat. Sol. B 178 (1993) 167. [15] S. Adachi, J. Appl. Phys. 58 (1985) R1. [16] R.L. Greene, K.K. Bajaj, Sol. Stat. Commun. 45 (1983) 825. [17] A.J. Peter, K. Navaneethakrishnan, Physica E 15 (2002) 153. [18] V. Ranjan, V.A. Singh, J. Appl. Phys. 89 (2001) 6415. [19] Sr.G. Jayam, K. Navaneethakrishnan, Solid State Commun. 126 (2003) 681.