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The effect of hydrostatic pressure on metal–insulator transition in quantum well semiconductor systems
Solid State Communications 130 (2004) 155–158 www.elsevier.com/locate/ssc
The effect of hydrostatic pressure on metal –insulator transition in quantum well semiconductor systems A. John Petera,*, K. Navaneethakrishnanb a
Department of Physics, Arulmigu Kalasalingam College of Engineering, Krishnankoil, 626 190 Viruthunagar, India b School of Physics, Madurai Kamaraj University, Madurai 625-021, India Received 2 January 2004; received in revised form 4 February 2004; accepted 5 February 2004 by A.K. Sood
Abstract Ionization energies of a shallow donor in a quantum well of GaAs/Ga12xAlxAs superlattice system in the influence of pressure using a variational procedure within the effective mass approximation are obtained. The vanishing of ionization energy triggering a Mott transition is observed within the one-electron approximation. The effects of Anderson localization, exchange and correlation in the Hubbard model are included in this model. It is found that the ionization energy (i) increases as well width increases for a given pressure (ii) increases when well width increases and (iii) the critical concentration at which the metal– insulator transition occurs is increased when pressure is applied. All the calculations have been carried out with finite and infinite barriers and the results are compared with available data in the literature. q 2004 Elsevier Ltd. All rights reserved. PACS: 71.55; 71.38. þ h; 71.30. þ h; 72.15.Rn; 73.40.Qv; 73.61. 2 r; 72.20.My; 71.30. þ h; 73.40.Lq Keywords: D. Metal–insulator transition; D. Quantum well system; D. Impurity state; D. Donor energy
1. Introduction The ideal system for studying two dimensional carriers in semiconductors is GaAs/Ga12xAlxAs quantum well system (QW). The measurements leading to electronic and optical properties of such microstructures of low temperature photoluminous are made possible under high static pressure [1]. The same study has been carried out under the atmospheric pressure in the same heterostructures [2]. The effect of hydrostatic pressure on the binding energy of donor impurities in QW structures has been calculated by Elabsy [3]. It is found that the binding energy increases with increasing external hydrostatic pressure for a given QW thickness and temperature. Mercy et al. [4] have found that the carrier concentration could be decreased with increasing pressure when the samples are cooled to low temperatures * Corresponding author. Tel.: þ 91-4563289042; fax: þ 914563289322. E-mail address: [email protected] (A.J. Peter). 0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2004.02.007
in Hydrostatic pressure transport studies of modulated doped QWs. The metal – insulator transition (MIT) is usually achieved at low temperatures by increasing the doping concentrations. It leads to an increase of overlap of wave functions and as a result, a delocalization of electrons occurs at a critical concentration, where MIT is achieved. This MIT was confirmed by many experiments for various materials with the critical concentration varying eight orders of magnitude (1014 – 1022 cm23) [5]. However, we still have some unsolved and controversial problems even though there exists some experimental and theoretical researches on MIT for a long time. The metal– non-metal transition in GaAs– Ga12xAlxAs heterostructures has been investigated using electrical conductivity and Hall measurements in the presence of a magnetic field and hydrostatic pressure [6]. Perry et al. [7] have investigated many body effects of 2D electron gas to determine the pressure dependence of the effective mass and to monitor the MIT by combining magnetic field with high pressure. In the present work, a systematic study of variation of
156
A.J. Peter, K. Navaneethakrishnan / Solid State Communications 130 (2004) 155–158
pressure as the function of well width has been attempted for both the infinite and finite confinement potential model. Calculations of vanishing of ionization energies of the donor impurities are performed using the effective mass approximation within a variational scheme in GaAs/Ga12xAlxAs structures. The binding energy of the donor is obtained for a particular well width within the variational framework and thereby vanishing of donor energy is obtained for a critical concentration. The relationship between the present model and the Mott criterion in terms of Hubbard model are also brought out; 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.
In the effective mass approximation, the Hamiltonian for a hydrogenic donor impurity in a GaAs/Ga12xAlxAs quantum well under the influence of hydrostatic pressure is given by "2 e2 72 2 þ V0 ðz; PÞ p 2m ðPÞ 1ðPÞr
ð1Þ
where mp ðPÞ and 1ðPÞ are the effective mass and the dielectric function of GaAs as functions of hydrostatic pressure P and V0 ðz; PÞ represents the confinement potential. The Thomas– Fermi screening function is used for 1ðPÞ which is given by 121 ðPÞ ¼
DEg ðx; PÞ ¼ DEg ðxÞ þ PDðxÞ where DEg ðxÞ ¼ 1:555x þ 0:37x2 in eV is the variation of the energy gap difference [9] and DðxÞ is the pressure coefficient of the band gap given by DðxÞ ¼ ½2ð1:3 £ 1023 ÞxeV=kbar where 1 kbar is 0.1 GPa and the potential barrier, as a function of Al concentration x; is given by VðxÞ ¼ 0:6 DEg ðx; PÞ
2. Theory
H¼2
where P is in GPa [8]. The total band gap difference between GaAs/Ga12xAlxAs as a function of x is given by
exp½2lðPÞr 10 ðPÞ
where 10 ðPÞ is the static dielectric constant and lðPÞ is the screening parameter given by lðPÞ ¼ 12pmp ðPÞð3 p2 Þ22=3 NðPÞ1=3 =10 ðPÞ; where NðPÞ is the impurity concentration. 2.1. Effect of pressure The application of hydrostatic pressure modifies the lattice constants, well width barrier height, effective masses and dielectric constants. These values are obtained in the following way. The variation of well width with pressure is given by LðPÞ ¼ L0 ð1 2 1:5082 £ 1023 PÞ where P is in GPa, L0 is the zero pressure width of the QW, taking into account using ðda=dPÞ ¼ 22:6694 £ 1024 a0 ; where a0 is the lattice constant of GaAs [8]. The variation of dielectric constant with pressure is given as 1ðPÞ ¼ 13:13 2 0:088P where P is in GPa [8]. The effective mass in the well and barrier region change as mp ðPÞ ¼ mp ð0Þexpð0:078PÞ
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 We have considered a quantum well of GaAs sandwiched between two barriers of Ga12xAlxAs. The trial wave function for the ground state is chosen as
C ¼ N wðzÞexp½2dðr2 þ z2 Þ1=2
ð2Þ
where N is the normalization constant and d is the variational parameter, with 8 N cos az 2L=2 , lzl # L=2 > > < wðzÞ ¼ N cosðaL=2ÞexpðbL=2Þexpð2bzÞ lzl $ L=2 > > : N cosðaL=2ÞexpðbL=2ÞexpðbzÞ lzl $ 2L=2
with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mp ðPÞEðPÞ a¼ "2 and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mp ðPÞ½V0 ðzÞ 2 EðPÞ b¼ "2 EðPÞ may be obtained as the first root of the transcendental equation
b ¼ a tan½aLðPÞ=2 which comes by the applications of boundary conditions in the z direction. Since an exact solution of the Schro¨dinger equation with the Hamiltonian in Eq. (1) is not possible a variational approach has been adopted. As the calculations are straight forward, we refrain from giving the detailed expression here. The ionization energy is given by Eion ¼ EðPÞ 2 kHlmin ; where H is from Eq. (1).
A.J. Peter, K. Navaneethakrishnan / Solid State Communications 130 (2004) 155–158
2.3. Infinite barrier
Table 1 Characteristic parameters of GaAs and Ga0.7Al0.3As
With a trial wave function for the ground state of the impurity we have used pz C ¼ N1 exp½2d0 ðr2 þ z2 Þ1=2 cos L where N is the normalization constant and d0 is the variational parameter. The effect of a random set of potential wells was first pointed out by Anderson [10]. Assuming a distribution of impurities given by Pðri Þ ¼ A expð2d1 ri Þ; where d1 represents random parameter, which corresponds to the probability of finding an impurity at ri if another impurity is already available at the origin, an impurity potential Vd ðrÞ ¼ ð2e2 =10 rÞexpð2lrÞ þ ðe2 =10 rÞexpð2grÞ þ ðe2 d1 =10 2 e2 d31 =210 g2 þ e2 d21 r=210 2 e2 d31 r=210 gÞexpð2grÞ
157
ð5Þ
where g ¼ d1 þ l is obtained as in Ref. [11] the same procedure for calculating the ionization energy, with the inclusion impurity potential using Eq. (5), is given by Eion ¼ Esub 2 kHlmin ð6Þ where Esub ¼ "2 =2mp ðPÞðp=LðPÞÞ2 : The calculations are carried out with exchange and correlation among electrons in the inclusion of Hubbard model from Ref. [12] through the variation of effective mass with concentration. These calculations are based on the one electron Hamiltonian in which the many body effects enter through the dielectric screening. Since the effects of correlation among the electrons is significant in the impurity band formation which ultimately decides the semiconductor metal transition, the correlation effects are introduced in the Hubbard model taking the variation of effective mass as a function of the inter impurity separation [12]. The value of kHlmin is obtained variationally after evaluating the integrals numerically. In the actual calculations, the donor ionization energies are obtained when P ¼ 0; for different well width. For a particular concentration and for a particular pressure and well width the kHlmin value was obtained by varying d0 : this was repeated for different pressures, well widths and concentration, keeping the random parameter d1 constant.
3. Results and discussion The Characteristic parameters of GaAs and Ga12xAlxAs, 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 well widths are given in Table 2. 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.
Pressure (GPa)
Barrier height (meV)
1
mp (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
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 well widths. ˚ , the ionization energy is For a lower well width 100 A ˚ for any always higher than that of a well of L ¼ 1000 A donor concentration. It is found that the metallic transition occurs at 6.8 £ 1013 and 5.5 £ 1016 cm22 for well widths ˚ , respectively, when the applied L ¼ 1000 and 100 A pressure is 20 GPa. These values are one to two orders higher than that of those values obtained in the absence of pressure. We have some experimental data available. For 2D GaAs/GaAlAs structures one has Nc < 3 £ 1010 cm22 for holes [13]. In CdTe/Cd MnTe, Cain et al. [13] have observed Mott transition at Nc < 2:2 £ 1012 cm22 for a ˚ . This shows that the application of quantum well of 90 A hydrostatic pressure is leading to a more confinement of the impurity electrons. So that MIT occurs always at higher concentrations when the pressure is applied. The ionization energy obtained as a function of well width is given in Fig. 2. The decrease in ionization energy with the increase of well width is a common feature [2,12]. The ionization energy is higher for a given well width 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 ! 1000 A ˚ when the The same bulk value is obtained for L ! 1400 A applied pressure is 20 GPa. Also the MIT can occur, where the ionization energy becomes zero, by minimizing the ˚ without applying Hamiltonian up to the well width of 60 A pressure [14].When the pressure is applied the Hamiltonian ˚ below which it was not can be minimized only up to 100 A to minimize our expression for kHlmin : This is in support of the scaling theory of MIT [15]. Table 2 Subband energy (meV) ˚) Well width (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
158
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Fig. 3. Variation of ionization energy with well width for a finite well for two different pressures. Fig. 1. Variation of donor ionization energy with donor concentration in the infinite well model.
Fig. 3 shows the ionization energy as a function of well width with and without applying the hydrostatic pressure a finite barrier. For a particular well width the ionization energy of donor electrons are higher when the pressure becomes stronger. Also the critical concentration at which the MIT occurs is increased to two to three orders more. 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 quantum dot [16,17]. In summary, the results show that phase transition is not obtainable in strictly 2D, a result which in agreement with the scaling theory for phase transition [15]. It also follows that strictly 2D behavior is not possible in quantum well systems. This is due to the fact that all quantum well nanostructures have finite potential barriers in which, when the well dimension approaches zero, tunneling effect becomes important making the system 3D-like. Hence the experimental observation of metallic transition by Kravchenko et al. [18], in a strictly 2D electron gas system, should be explained using some other mechanisms like traps, etc. as suggested by Altshuler et al. [19]. It is hoped that the present work will stimulate further experimental activity in semiconductor– metal transition in quantum well
Fig. 2. Variation of ionization energy with well width.
nanostructures. The calculation with a better screening function such as Hartree in conjunction with a more realistic model is underway. To conclude, it is demonstrated that the MI transition is not possible in a hydrostatic pressure supporting the scaling theory of localization [15]. References [1] B. Welber, M. Cardona, C.K. Kim, S. Rodriquez, Phys. Rev. B 12 (1975) 5729. [2] R.L. Greene, K.K. Bajaj, Solid State Commun. 45 (1983) 825. [3] A.M. Elabsy, Phys. Scr. 48 (1993) 376. [4] 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. [5] H. Fritzche, L.F. Friedman, T.P. Tunstall (Eds.), The Metal– Non-Metal Transition in Disordered System, University of St Andrews, Scotland, 1978, p. 130. [6] L. Robert, A. Raymond, L. Konczewicz, C. Bousquet, W. Zawadzki, F. Alexandre, I.M. Masson, J.P. Andre, P.M. Frijlink, Phys. Rev. B 33 (1986) 5935–5938. [7] H. Perry, W. Zhou, J.M. Worlock, Phys. Rev. B 42 (1990) 9657. [8] A. Beneditctal, B. Sukumar, K. Nananeethakrishnan, Phys. Status Solidi B 178 (1993) 167. [9] S. Adachi, J. Appl. Phys. 58 (1985) R1. [10] P.W. Anderson, Phys. Rev. 109 (1958) 1492. [11] A.J. Peter, K. Navaneethakrishnan, Phys. Status Solidi B 220 (2000) 897. [12] A.J. Peter, K. Navaneethakrishnan, Phys. E E15 (2002) 153. [13] N. Cain, M.O. Neil, J.E. Nicholls, T. Stirner, W.E. Hagston, D.E. Ashenford, J. Lumin. 75 (1997) 269. [14] A.J. Peter, K. Navaneethakrishnan, Solid State Commun. 120 (2001) 393. [15] E.A. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673. [16] V. Ranjan, V.A. Singh, J. Appl. Phys. 89 (2001) 6415. [17] S.R. Geradin Jayam, K. Navaneethakrishnan, Solid State Commun. 126 (2003) 681. [18] S.V. Kravchenko, W. Mason, J.E. Furneaux, V.M. Pudalov, Phys. Rev. Lett. 75 (1995) 910. [19] B.L. Altshuler, D.L. Maslov, V.M. Pudalov, Phys. Status Solidi B 218 (2000) 193.
The effect of hydrostatic pressure on metal –insulator transition in quantum well semiconductor systems A. John Petera,*, K. Navaneethakrishnanb a
Department of Physics, Arulmigu Kalasalingam College of Engineering, Krishnankoil, 626 190 Viruthunagar, India b School of Physics, Madurai Kamaraj University, Madurai 625-021, India Received 2 January 2004; received in revised form 4 February 2004; accepted 5 February 2004 by A.K. Sood
Abstract Ionization energies of a shallow donor in a quantum well of GaAs/Ga12xAlxAs superlattice system in the influence of pressure using a variational procedure within the effective mass approximation are obtained. The vanishing of ionization energy triggering a Mott transition is observed within the one-electron approximation. The effects of Anderson localization, exchange and correlation in the Hubbard model are included in this model. It is found that the ionization energy (i) increases as well width increases for a given pressure (ii) increases when well width increases and (iii) the critical concentration at which the metal– insulator transition occurs is increased when pressure is applied. All the calculations have been carried out with finite and infinite barriers and the results are compared with available data in the literature. q 2004 Elsevier Ltd. All rights reserved. PACS: 71.55; 71.38. þ h; 71.30. þ h; 72.15.Rn; 73.40.Qv; 73.61. 2 r; 72.20.My; 71.30. þ h; 73.40.Lq Keywords: D. Metal–insulator transition; D. Quantum well system; D. Impurity state; D. Donor energy
1. Introduction The ideal system for studying two dimensional carriers in semiconductors is GaAs/Ga12xAlxAs quantum well system (QW). The measurements leading to electronic and optical properties of such microstructures of low temperature photoluminous are made possible under high static pressure [1]. The same study has been carried out under the atmospheric pressure in the same heterostructures [2]. The effect of hydrostatic pressure on the binding energy of donor impurities in QW structures has been calculated by Elabsy [3]. It is found that the binding energy increases with increasing external hydrostatic pressure for a given QW thickness and temperature. Mercy et al. [4] have found that the carrier concentration could be decreased with increasing pressure when the samples are cooled to low temperatures * Corresponding author. Tel.: þ 91-4563289042; fax: þ 914563289322. E-mail address: [email protected] (A.J. Peter). 0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2004.02.007
in Hydrostatic pressure transport studies of modulated doped QWs. The metal – insulator transition (MIT) is usually achieved at low temperatures by increasing the doping concentrations. It leads to an increase of overlap of wave functions and as a result, a delocalization of electrons occurs at a critical concentration, where MIT is achieved. This MIT was confirmed by many experiments for various materials with the critical concentration varying eight orders of magnitude (1014 – 1022 cm23) [5]. However, we still have some unsolved and controversial problems even though there exists some experimental and theoretical researches on MIT for a long time. The metal– non-metal transition in GaAs– Ga12xAlxAs heterostructures has been investigated using electrical conductivity and Hall measurements in the presence of a magnetic field and hydrostatic pressure [6]. Perry et al. [7] have investigated many body effects of 2D electron gas to determine the pressure dependence of the effective mass and to monitor the MIT by combining magnetic field with high pressure. In the present work, a systematic study of variation of
156
A.J. Peter, K. Navaneethakrishnan / Solid State Communications 130 (2004) 155–158
pressure as the function of well width has been attempted for both the infinite and finite confinement potential model. Calculations of vanishing of ionization energies of the donor impurities are performed using the effective mass approximation within a variational scheme in GaAs/Ga12xAlxAs structures. The binding energy of the donor is obtained for a particular well width within the variational framework and thereby vanishing of donor energy is obtained for a critical concentration. The relationship between the present model and the Mott criterion in terms of Hubbard model are also brought out; 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.
In the effective mass approximation, the Hamiltonian for a hydrogenic donor impurity in a GaAs/Ga12xAlxAs quantum well under the influence of hydrostatic pressure is given by "2 e2 72 2 þ V0 ðz; PÞ p 2m ðPÞ 1ðPÞr
ð1Þ
where mp ðPÞ and 1ðPÞ are the effective mass and the dielectric function of GaAs as functions of hydrostatic pressure P and V0 ðz; PÞ represents the confinement potential. The Thomas– Fermi screening function is used for 1ðPÞ which is given by 121 ðPÞ ¼
DEg ðx; PÞ ¼ DEg ðxÞ þ PDðxÞ where DEg ðxÞ ¼ 1:555x þ 0:37x2 in eV is the variation of the energy gap difference [9] and DðxÞ is the pressure coefficient of the band gap given by DðxÞ ¼ ½2ð1:3 £ 1023 ÞxeV=kbar where 1 kbar is 0.1 GPa and the potential barrier, as a function of Al concentration x; is given by VðxÞ ¼ 0:6 DEg ðx; PÞ
2. Theory
H¼2
where P is in GPa [8]. The total band gap difference between GaAs/Ga12xAlxAs as a function of x is given by
exp½2lðPÞr 10 ðPÞ
where 10 ðPÞ is the static dielectric constant and lðPÞ is the screening parameter given by lðPÞ ¼ 12pmp ðPÞð3 p2 Þ22=3 NðPÞ1=3 =10 ðPÞ; where NðPÞ is the impurity concentration. 2.1. Effect of pressure The application of hydrostatic pressure modifies the lattice constants, well width barrier height, effective masses and dielectric constants. These values are obtained in the following way. The variation of well width with pressure is given by LðPÞ ¼ L0 ð1 2 1:5082 £ 1023 PÞ where P is in GPa, L0 is the zero pressure width of the QW, taking into account using ðda=dPÞ ¼ 22:6694 £ 1024 a0 ; where a0 is the lattice constant of GaAs [8]. The variation of dielectric constant with pressure is given as 1ðPÞ ¼ 13:13 2 0:088P where P is in GPa [8]. The effective mass in the well and barrier region change as mp ðPÞ ¼ mp ð0Þexpð0:078PÞ
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 We have considered a quantum well of GaAs sandwiched between two barriers of Ga12xAlxAs. The trial wave function for the ground state is chosen as
C ¼ N wðzÞexp½2dðr2 þ z2 Þ1=2
ð2Þ
where N is the normalization constant and d is the variational parameter, with 8 N cos az 2L=2 , lzl # L=2 > > < wðzÞ ¼ N cosðaL=2ÞexpðbL=2Þexpð2bzÞ lzl $ L=2 > > : N cosðaL=2ÞexpðbL=2ÞexpðbzÞ lzl $ 2L=2
with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mp ðPÞEðPÞ a¼ "2 and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mp ðPÞ½V0 ðzÞ 2 EðPÞ b¼ "2 EðPÞ may be obtained as the first root of the transcendental equation
b ¼ a tan½aLðPÞ=2 which comes by the applications of boundary conditions in the z direction. Since an exact solution of the Schro¨dinger equation with the Hamiltonian in Eq. (1) is not possible a variational approach has been adopted. As the calculations are straight forward, we refrain from giving the detailed expression here. The ionization energy is given by Eion ¼ EðPÞ 2 kHlmin ; where H is from Eq. (1).
A.J. Peter, K. Navaneethakrishnan / Solid State Communications 130 (2004) 155–158
2.3. Infinite barrier
Table 1 Characteristic parameters of GaAs and Ga0.7Al0.3As
With a trial wave function for the ground state of the impurity we have used pz C ¼ N1 exp½2d0 ðr2 þ z2 Þ1=2 cos L where N is the normalization constant and d0 is the variational parameter. The effect of a random set of potential wells was first pointed out by Anderson [10]. Assuming a distribution of impurities given by Pðri Þ ¼ A expð2d1 ri Þ; where d1 represents random parameter, which corresponds to the probability of finding an impurity at ri if another impurity is already available at the origin, an impurity potential Vd ðrÞ ¼ ð2e2 =10 rÞexpð2lrÞ þ ðe2 =10 rÞexpð2grÞ þ ðe2 d1 =10 2 e2 d31 =210 g2 þ e2 d21 r=210 2 e2 d31 r=210 gÞexpð2grÞ
157
ð5Þ
where g ¼ d1 þ l is obtained as in Ref. [11] the same procedure for calculating the ionization energy, with the inclusion impurity potential using Eq. (5), is given by Eion ¼ Esub 2 kHlmin ð6Þ where Esub ¼ "2 =2mp ðPÞðp=LðPÞÞ2 : The calculations are carried out with exchange and correlation among electrons in the inclusion of Hubbard model from Ref. [12] through the variation of effective mass with concentration. These calculations are based on the one electron Hamiltonian in which the many body effects enter through the dielectric screening. Since the effects of correlation among the electrons is significant in the impurity band formation which ultimately decides the semiconductor metal transition, the correlation effects are introduced in the Hubbard model taking the variation of effective mass as a function of the inter impurity separation [12]. The value of kHlmin is obtained variationally after evaluating the integrals numerically. In the actual calculations, the donor ionization energies are obtained when P ¼ 0; for different well width. For a particular concentration and for a particular pressure and well width the kHlmin value was obtained by varying d0 : this was repeated for different pressures, well widths and concentration, keeping the random parameter d1 constant.
3. Results and discussion The Characteristic parameters of GaAs and Ga12xAlxAs, 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 well widths are given in Table 2. 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.
Pressure (GPa)
Barrier height (meV)
1
mp (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
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 well widths. ˚ , the ionization energy is For a lower well width 100 A ˚ for any always higher than that of a well of L ¼ 1000 A donor concentration. It is found that the metallic transition occurs at 6.8 £ 1013 and 5.5 £ 1016 cm22 for well widths ˚ , respectively, when the applied L ¼ 1000 and 100 A pressure is 20 GPa. These values are one to two orders higher than that of those values obtained in the absence of pressure. We have some experimental data available. For 2D GaAs/GaAlAs structures one has Nc < 3 £ 1010 cm22 for holes [13]. In CdTe/Cd MnTe, Cain et al. [13] have observed Mott transition at Nc < 2:2 £ 1012 cm22 for a ˚ . This shows that the application of quantum well of 90 A hydrostatic pressure is leading to a more confinement of the impurity electrons. So that MIT occurs always at higher concentrations when the pressure is applied. The ionization energy obtained as a function of well width is given in Fig. 2. The decrease in ionization energy with the increase of well width is a common feature [2,12]. The ionization energy is higher for a given well width 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 ! 1000 A ˚ when the The same bulk value is obtained for L ! 1400 A applied pressure is 20 GPa. Also the MIT can occur, where the ionization energy becomes zero, by minimizing the ˚ without applying Hamiltonian up to the well width of 60 A pressure [14].When the pressure is applied the Hamiltonian ˚ below which it was not can be minimized only up to 100 A to minimize our expression for kHlmin : This is in support of the scaling theory of MIT [15]. Table 2 Subband energy (meV) ˚) Well width (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
158
A.J. Peter, K. Navaneethakrishnan / Solid State Communications 130 (2004) 155–158
Fig. 3. Variation of ionization energy with well width for a finite well for two different pressures. Fig. 1. Variation of donor ionization energy with donor concentration in the infinite well model.
Fig. 3 shows the ionization energy as a function of well width with and without applying the hydrostatic pressure a finite barrier. For a particular well width the ionization energy of donor electrons are higher when the pressure becomes stronger. Also the critical concentration at which the MIT occurs is increased to two to three orders more. 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 quantum dot [16,17]. In summary, the results show that phase transition is not obtainable in strictly 2D, a result which in agreement with the scaling theory for phase transition [15]. It also follows that strictly 2D behavior is not possible in quantum well systems. This is due to the fact that all quantum well nanostructures have finite potential barriers in which, when the well dimension approaches zero, tunneling effect becomes important making the system 3D-like. Hence the experimental observation of metallic transition by Kravchenko et al. [18], in a strictly 2D electron gas system, should be explained using some other mechanisms like traps, etc. as suggested by Altshuler et al. [19]. It is hoped that the present work will stimulate further experimental activity in semiconductor– metal transition in quantum well
Fig. 2. Variation of ionization energy with well width.
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