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Magnetic field effect on the binding energy of a hydrogenic impurity in cylindrical quantum dot
Physica B 293 (2000) 137}143
Magnetic "eld e!ect on the binding energy of a hydrogenic impurity in cylindrical quantum dot R. Charrour, M. Bouhassoune, M. Fliyou*, A. Nougaoui Laboratoire de Dynamique et d'Optique des Mate& riaux, Faculte& des Sciences, Universite& Mohamed 1CP, Oujda, Morocco Received 16 December 1999; received in revised form 28 February 2000; accepted 15 March 2000
Abstract A systematic study of the ground state binding energy of a hydrogenic impurity in cylindrical quantum dot subjected to an external strong magnetic "eld is presented. Calculations are performed within the e!ective-mass approximation using the variational procedure and considering an in"nite con"ning potential on all surfaces of the system. The binding energy is calculated as a function of the dot size (radius and height), applied magnetic "eld and donor impurity position. We found that the impurity binding energy depends strongly on the impurity position and magnetic "eld strength of the dot. Our results are in good agreement with those obtained from low-dimensional systems. The presented results can be useful to control the spatial distribution of the donor center in the microstructure. 2000 Elsevier Science B.V. All rights reserved. PACS: 73.20.Dx; 73.20.Fz Keywords: Semiconductors; Quantum dots; Impurity level; Binding energy; Magnetic "eld
1. Introduction Owing to the advances in nanofabrication technology, it has been possible to manufacture highquality semiconductor nanostructure with good optical properties. The interest on these systems resides essentially in their non-linear optical properties and the possibility to realize high-performance optoelectronic devices [1]. Owing to the break in translation symmetry, the optoelectronic properties of the nanostructure differ from those of the bulk semiconductors. The quantum con"nement and low dimensionality lead * Corresponding author. Permanent address: ENS De`partement de physique, B.P. 5206 Benssouda, Fe`s, Morocco. E-mail address: #[email protected] (M. Fliyou).
to an enhancement of the density of states in such structure, so the electrons are partially or fully quantized into a discrete spectrum of energy level. The study of the hydrogenic impurities is one of the main problems in semiconductor low-dimensional systems because the presence of the impurity states in this nanostructure in#uences greatly both the electronic mobility and their optical properties. Since the quantum size becomes salient when the dimensions of the system are reduced, the electron move only in a smaller space and spends most of its time close to the impurity ion. Consequently, the e!ective strength of the coulomb interaction increases, and the binding energy of the electron should be larger in lower dimensions. The binding energy calculations of hydrogenic impurities in quantum wells [2}4], quantum well wires [5}8]
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 4 9 5 - 6
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R. Charrour et al. / Physica B 293 (2000) 137}143
and quantum dots [9}16] have been performed on several occasions. This problem has been studied by variational techniques [17,18], tight-binding self-consistent linear screening calculation [19,20], and the strong con"nement approach [21,22] because it has no exact solution. It is found that the strong electronic con"nement in quantum dot systems re#ects itself in the ground state energy and in the impurity binding energy, which are higher than those found in the quantum wires and quantum wells. Magnetic "eld has become an interesting probe for studying the physical properties of low-dimensional structures, both from the theoretical and technological point of view. There has been a considerable amount of work, during the past decade, devoted to the study of the behavior of energy levels of shallow impurities in bulk semiconductors and their heterostructures such as QW [23}25], QWW [26}30] and QD [31}34] in the presence of a magnetic "eld. The problem of the bound electron to an impurity, located on the axis of the wire, in a cylindrical quantum wire with in"nite and "nite potential barriers and in the presence of a uniform magnetic "eld; as a function of the width of the quantum wire was treated by Branis et al. [35]. They have used a variational method in which the trial wave function contains a hydrogenic part and the appropriate con#uent hypergeometric function. This con#uent hypergeometric function is the radial solution of an electron in an in"nite potential cylindrical wire, in the presence of any magnetic "eld, applied parallel to the wire axis with appropriate boundary conditions on the wire surface. Fereyra et al. [36] have reported a study of the dependence of the impurity related energy corrections on the impurity position and the magnetic "eld in parabolic quantum dot by using the strong con"nement approach. They have shown that the impurity related binding energy depends strongly on the impurity position in the dot and the magnetic "eld strength. Xiao et al. [32] have reported the calculation of the ground state binding energy of hydrogenic o! center donor in spherical quantum dot in the presence of magnetic "eld by using a variational method. In their study, the in#uence of magnetic "eld is introduced as an exponential term in the trial wave function.
In the present paper, we propose to study the magnetic "eld and the impurity position dependence of the binding energy of hydrogenic impurity in cylindrical quantum dot. We employ the variational scheme. In the absence of the impurity, the wave function is the product of two exact eigenfunction of the cylindrical quantum dot Hamiltonian along the z-axis and in the (x}y) plane without any restriction on the magnetic "eld strength. The organisation of the paper is as follows. In Section 2, we present the theoretical calculation of the binding energy. The discussion of the results and the conclusion are given in Section 3.
2. Formalism In the framework of an e!ective-mass approximation, the Hamiltonian for a hydrogenic donor in cylindrical quantum dot, of radius R and height (H"2d), with in"nite potential barrier at all the surfaces under an applied uniform magnetic "eld along the z direction is
1 e e H" P! A ! #<(r), 2mH c e "r!r "
(1)
where "r!r ""((o!o )#(z!z ), r (r) is the impurity ion (electron) position measured from the center of the quantum dot, e is the dielectric permittivity, mH is the electron e!ective mass, A(r) is the potential vector of the magnetic "eld and <(r) is the con"nement potential considered zero inside the dot and in"nite outside it. The potential vector is written as A(r)"(B;r), with B"Bz. In cylindrical coordinates, the components of the vector potential are A "A "0 and M X A "(Bo). P The Hamiltonian of the system can be written in cylindrical coordinates and in the reduced units, such as the e!ective Bohr radius aH and Rydberg RH to measure distances and energies, respectively, as 2 H"H(o, u)#H(z)#<(r)! , ((o!o )#(z!z ) (2)
R. Charrour et al. / Physica B 293 (2000) 137}143
139
where
This wave function is given by
H(o, u)t (o, u)
(r)"Nt (o, u)t (z) ;exp(!a((o!o )#(z!z )), (7) where N is the normalization constant. The ground state energy corresponds to l"1, m"0 and n"1. The donor binding energy E of the ground state is obtained by subtracting the minimized energy E from the lowest subband energy E .
E "1 (r, a"0)"H(o, u)#H(z) #<(r)" (r, a"0)2
"!
R 1R 1 R # # t (o, u) Ro o Ro o Ru
co # t (o, u)#c¸ t (o, u) X 4 "E t (o, u),
(2a)
R H(z)t (z)"! t (z)"E t (z). Rz
(2b)
In the in"nite potential well approximation, Eq. (2a) [35] and Eq. (2b) can be solved analytically, thus their eigenfunctions are
co t (o, u)"exp(im u) exp ! 4
co K 2
co ; F !a , "m"#1; , K J 2 t (z)"cos(k z), X
(3) (4)
where ¸ is the z-component of the angular moX mentum operator (in units of g), c"gu /2RH is the dimensionless measure of the magnetic "eld, u is the cyclotron frequency, m"0, $1, 2, 2, and l"1, 2, 3, 2 F (!a , "m"#1; co/2) is the K J general form of the con#uent hypergeometric function. The values of the k and a are determined X K J by the boundary conditions that the wave function vanishes at z"$d and o"R, respectively
cR !a , "m"#1; "0 K J 2
(5)
np k " , n"1, 2, 3, 2. X 2d
(6)
F
"
For the more details on the eigenfunction t (o, u) see Ref. [35]. The inclusion of the impurity potential leads to a nonseparable di!erential equation. Using the variational method, it is possible to associate a trial wave function, which is an approximated eigenfunction of the Hamiltonian described in Eq. (2).
p #c(2a #1), 2d
(8)
E "min [1 (r)"H" (r)2]
? p "min #c(2a #1)!a 2d ? a p P ! 2RcP !P !P # P !2 . 2 P P (9)
The integrals P are de"ned by G Rc x P " exp ! 2 \ p Rc ; F !a , 1; x cos t 2 2
;exp[!2a(R(x!x )#d(t!t )]x dx dt, (9a)
P "
Rcx pt x(x!x ) exp ! cos 2 2
\ exp[!2a(R(x!x )#d(t!t )] ; (R(x!x )#d(t!t ) 1 Rcx ; F !a , 1; 2 2
#a F 1!a , 2 ;
; F !a , 1;
Rcx 2
Rcx 2
dx dt,
(9b)
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R. Charrour et al. / Physica B 293 (2000) 137}143
P "
\
Rc x exp ! 2
Rc p ; F !a , 1; x cos t 2 2 ;
exp[!2a(R(x!x )#d(t!t )] dx dt, (R(x!x )d(t!t ) (9c)
P "
\
Rc exp ! x 2
p Rc ; F !a , 1; x cos t 2 2
exp[!2a(R(x!x )#d(t!t )] ; (R(x!x )#d(t!t ) ;(x!x ) dx dt, (9d) Rc P " exp ! x 2 \ Rc ; F !a , 1; x sin(pt) 2
exp[!2a(R(x!x )#d(t!t )] ; (R(x!x )#d(t!t ) ;(t!t ) dx dt. (9e) 3. Numerical results and discussion We have calculated the values of the binding energy E of a donor, where the impurity is dis placed away from the center to the edge both along the axial or radial directions in the presence of a uniform magnetic "eld applied along the z-axis as a function of a structure size and the magnetic "eld strength. We have displayed in Fig. 1, the ground state binding energy versus the dot radius for a "xed height H and di!erent strength of the magnetic "eld c when the impurity is placed at the center of the quantum dot (x "o /R; t "z /d). This "gure re #ects the competition between the magnetic "eld e!ect and the spatial con"nement e!ect. For a very small radius R, the strong geometric con"nement
Fig. 1. Binding energy of an on-center donor impurity E as a function of the dot radius for a "xed height H"1aH and several values of the magnetic "eld.
leads the electronic wave function to be more compressed in the quantum dot. The binding energy is signi"cant and relatively insensitive to the magnetic "eld since the electron spatial localisation prevails over the magnetic "eld con"nement. To the vicinity of 1aH, the e!ect of the magnetic "eld begins to be apparent and the curves corresponding to di!erent strength of magnetic "eld tend to deviate from each other, as the dot radius rises, reaching asymptotically to the quantum well case values. For a large values of the radius, R'1aH, the magnetic "eld governs the behavior of the binding energy because it overcomes the spatial localisation. On the other hand, increasing the strength of the applied "eld decreases the cyclotron radius for the electron relative to the dot radius and con"nes the electron to be close to the impurity center. Consequently, the binding energy is enhanced in proportion with an increase of the magnetic "eld strength in the region of the large dot radius. The same results are found by Xiao et al. [32] in spherical quantum dot and Branis et al. [35] in the cylindrical quantum well wire. Here, we have just limited our work to the in"nite height potential barrier. However, in the case of the "nite height
R. Charrour et al. / Physica B 293 (2000) 137}143
potential barrier, the binding energy increases as the radius is reduced, reaches a maximum and decreases as RP0. In addition, the e!ect of the external "eld becomes remarkable only in the large radius R region, as an in"nite barrier case. We should signal that with our model, we can describe any low-dimensional structure going from 2D (two-dimensional) to 0D (zero-dimensional) by taking into account the ratio between the height H and the radius R of the structure, that is (H/R);1 for a QW (2D), (H/R) 1 for a QWW (1D) and (H/R) 1 for a QD (0D). The exact nature of dependence of the binding energy on the impurity position along the radial
141
direction, axial magnetic "eld strength and quantum system size (radius and height) inside the cylindrical structure is presented in Fig. 2. From Figs. 2(a)}(c), several remarks can be noticed. First of all, it is apparent that the binding energy is greatest for any given c, H and radius R when the impurity is in the center of the structure (x "0; t "0). The further we remove the impurity from the center, the lower the binding energy becomes. This behavior is due to the increased spread in the radial probability density when the impurity is displaced from the axis of the system. Results showing a similar nature, in the absence of the magnetic "eld, have been reported by Bose et al. [15,16] in the case
Fig. 2. The binding energies' dependence is with respect to the radial impurity position o /R (measured from the center of the structure) with z "0, for three values of magnetic "eld strength (c"0 (solid curve); c"2 (broken curve); c"3 (doted curve)). (a) The structure is a quantum dot with R"1.5aH and H"1.5aH. (b) The structure is a quantum well wire R"1.5aH and H"8aH. (c) The structure is a quantum well with R"8aH and H"1aH.
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R. Charrour et al. / Physica B 293 (2000) 137}143
of spherical QD and Brown et al. [37] in cylindrical QWW. Furthermore, the introduction of magnetic "eld enhances the binding energy especially in the large radius system and in the case of on center impurity position.
It is observed, also, that the ground state binding energy E increases with the reduction of the vol ume of the structure, being larger when the two dimensions (R and H) are diminished simultaneously, namely, E is more important in the QD (2(a)) than in QWW (2(b)) and QW (2(c)). This agrees with the results of Moreno et al. [27]. Fig. 2(a), in which we have reproduced the situation of a QD taking H"R"1.5aH, shows that the binding energy for the impurity located in the edge of the quantum dot is totally insensitive to the increase of the magnetic "eld strength. Similar theoretical results for a spherical [32] and parabolic quantum dot [36] have been reported. The same physical phenomena can be observed in the QWW system (2(b)) and QW system (2(c)). Fig. 3 illustrates the binding energy as a function of the axial impurity position z for o "0 with di!erent magnetic "eld strength in the QD system (3(a)) and QW system (3(b)). It can be seen that E is degenerate for symmetrical positions with respect to the center of transversal section of the cylindrical system. Also, this degeneracy is not broken by an application of the magnetic "eld in the z-direction contrary to the electric "eld. For the QD system (3(a)), the E increases in proportion with the ex ternal "eld and it is insensitive to the axial applied "eld for the impurity near the top or bottom end faces (z "$d). This is agreeing with the result of the Ref. [36]. In Fig. 3(b) we have reproduced the situation of a QW, that is a structure with small H and large R. As for the case of the QD described in Fig. 3(a), we remark that the ground state binding energy is higher for an impurity on center and enhances with the increase of the magnetic "eld. This validates the result plotted by Ribeiro et al. [33] in Fig. 4.
4. Summary
Fig. 3. The binding energies dependence is with respect to the radial impurity position z /d (measured from the center of the structure) with o "0, for three values of magnetic "eld strength (c"0 (solid curve); c"2 (broken curve); c"5 (doted curve)). (a) The structure is a quantum dot with R"1.5aH and H"1.5aH. (b) The structure is a quantum well with R"8aH and H"1aH.
In the present work we have studied the in#uence of the geometrical con"nement and the e!ect of an applied magnetic "eld on the binding energy of the o!-center donor impurity in cylindrical quantum system. The calculations were performed within the e!ective-mass approximation and using a variational method. The electronic con"nement was
R. Charrour et al. / Physica B 293 (2000) 137}143
modeled by an in"nite potential well on all surfaces. We found that the ground state binding energy depends strongly on the spatial con"nement, applied magnetic "eld strength and an impurity position. This behavior is a consequence of the modi"cation of the wave function nature under the magnetic and geometric con"nement. In addition, the change of the impurity position modi"es the spread of the electron wave function. Since the impurity binding energy is insensitive to the application of a magnetic "eld when the donor is located near the surfaces, this result can be useful in the technological application. Indeed, we can control the spatial distribution of the donor center in the microstructure.
Acknowledgements This work has been supported by the Programsin-Aide for Scienti"c Research (PARS). Physique 16 Oujda and Physique 03 Fe`s and under AI No. 97/046/SM.
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[10] F.J. Ribeiro, A. Latge`, Phys. Rev. B 50 (1994) 4913. [11] U. Woggon, S.V. Gaponenko, Phys. Stat. Sol. B 189 (1995) 285. [12] D.S. Chuu, C.M. Hsiao, W.N. Mei, Phys. Rev. B. 46 (1992) 3898. [13] J.L. Zhu, J. Wu, R.T. Fu, H. Chen, Y. Kawazoe, Phys. Rev. B 55 (1997) 1673. [14] Q. Fanyao, A.L.A. Fonseca, O.A.C. Nunes, J. Appl. Phys. 82 (1997) 1236. [15] C. Bose, C.K. Sarkar, Physica B 253 (1998) 238. [16] C. Bose, J. Appl. Phys. 83 (1998) 3089. [17] J.L. Zhu, J.H. Zhao, J.J. Xiong, Phys. Rev. B 50 (1994) 1832. [18] R. Tsu, D. Babic, Appl. Phys. Lett. 64 (1994) 1806. [19] M. Lannoo, C. Delerue, G. Allan, Phys. Rev. Lett. 74 (1995) 3415. [20] G. Allan, C. Delerue, M. Lannoo, E. Martin, Phys. Rev. B 52 (1995) 11982. [21] J.M. Ferreyra, C.R. Proetto, Phys. Rev. B 52 (1995) R2309. [22] C.R. Proetto, Phys. Rev. Lett. 76 (1996) 2824. [23] F.A.P. Osorio, M.Z. Maialle, O. Hipolito, Phys. Rev. B 57 (1998) 1644. [24] A. Sali, M. Fliyou, L. Roubi, Loumrhari, J. Phys: Condens. Matter 11 (1999) 2427. [25] E.P. Sinyavskii, S.M. Sokovnich, F.I. Pasechnik, Phys. Stat. Sol. B 209 (1998) 55. [26] Z.G. Xiao, J.Q. Zhu, F.L. He, Phys. Stat. Sol. B 191 (1995) 401. [27] J.J. Vivas-Moreno, N. Porras-Montenegro, Phys. Stat. Sol. B 210 (1998) 723. [28] P. Villamil, N. Porras-Montenegro, J. Phys: Condens. Matter 10 (1998) 10599. [29] P. Villamil, N. Porras-Montenegro, J.C. Granada, Phys. Rev. B 59 (1999) 1605. [30] S.T. Perez-Merchancano, G.E. Marques, Phys. Stat. Sol. B 212 (1999) 375. [31] M. El Said, Phys. Stat. Sol. B 193 (1996) 105. [32] Z.G. Xiao, J.Q. Zhu, F.L. He, J. Appl. Phys. 79 (1996) 9181. [33] F.J. Ribeiro, A. Latg, M. Pacheco, Z. Barticevic, J. Appl. Phys. 82 (1997) 270. [34] C.Y. Chen, P.W. Jin, W.S. Li, D.L. Lin, Phys. Rev. B 56 (1997) 14913. [35] S.V. Branis, G. Li, K.K. Bajaj, Phys. Rev. B 47 (1993) 1316. [36] J.M. Ferryra, P. Bosshard, C.R. Proetto, Phys. Rev. B 55 (1997) 13682. [37] J.W. Brown, H.N. Spector, J. Appl. Phys. 59 (1985) 1179.
Magnetic "eld e!ect on the binding energy of a hydrogenic impurity in cylindrical quantum dot R. Charrour, M. Bouhassoune, M. Fliyou*, A. Nougaoui Laboratoire de Dynamique et d'Optique des Mate& riaux, Faculte& des Sciences, Universite& Mohamed 1CP, Oujda, Morocco Received 16 December 1999; received in revised form 28 February 2000; accepted 15 March 2000
Abstract A systematic study of the ground state binding energy of a hydrogenic impurity in cylindrical quantum dot subjected to an external strong magnetic "eld is presented. Calculations are performed within the e!ective-mass approximation using the variational procedure and considering an in"nite con"ning potential on all surfaces of the system. The binding energy is calculated as a function of the dot size (radius and height), applied magnetic "eld and donor impurity position. We found that the impurity binding energy depends strongly on the impurity position and magnetic "eld strength of the dot. Our results are in good agreement with those obtained from low-dimensional systems. The presented results can be useful to control the spatial distribution of the donor center in the microstructure. 2000 Elsevier Science B.V. All rights reserved. PACS: 73.20.Dx; 73.20.Fz Keywords: Semiconductors; Quantum dots; Impurity level; Binding energy; Magnetic "eld
1. Introduction Owing to the advances in nanofabrication technology, it has been possible to manufacture highquality semiconductor nanostructure with good optical properties. The interest on these systems resides essentially in their non-linear optical properties and the possibility to realize high-performance optoelectronic devices [1]. Owing to the break in translation symmetry, the optoelectronic properties of the nanostructure differ from those of the bulk semiconductors. The quantum con"nement and low dimensionality lead * Corresponding author. Permanent address: ENS De`partement de physique, B.P. 5206 Benssouda, Fe`s, Morocco. E-mail address: #[email protected] (M. Fliyou).
to an enhancement of the density of states in such structure, so the electrons are partially or fully quantized into a discrete spectrum of energy level. The study of the hydrogenic impurities is one of the main problems in semiconductor low-dimensional systems because the presence of the impurity states in this nanostructure in#uences greatly both the electronic mobility and their optical properties. Since the quantum size becomes salient when the dimensions of the system are reduced, the electron move only in a smaller space and spends most of its time close to the impurity ion. Consequently, the e!ective strength of the coulomb interaction increases, and the binding energy of the electron should be larger in lower dimensions. The binding energy calculations of hydrogenic impurities in quantum wells [2}4], quantum well wires [5}8]
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 4 9 5 - 6
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R. Charrour et al. / Physica B 293 (2000) 137}143
and quantum dots [9}16] have been performed on several occasions. This problem has been studied by variational techniques [17,18], tight-binding self-consistent linear screening calculation [19,20], and the strong con"nement approach [21,22] because it has no exact solution. It is found that the strong electronic con"nement in quantum dot systems re#ects itself in the ground state energy and in the impurity binding energy, which are higher than those found in the quantum wires and quantum wells. Magnetic "eld has become an interesting probe for studying the physical properties of low-dimensional structures, both from the theoretical and technological point of view. There has been a considerable amount of work, during the past decade, devoted to the study of the behavior of energy levels of shallow impurities in bulk semiconductors and their heterostructures such as QW [23}25], QWW [26}30] and QD [31}34] in the presence of a magnetic "eld. The problem of the bound electron to an impurity, located on the axis of the wire, in a cylindrical quantum wire with in"nite and "nite potential barriers and in the presence of a uniform magnetic "eld; as a function of the width of the quantum wire was treated by Branis et al. [35]. They have used a variational method in which the trial wave function contains a hydrogenic part and the appropriate con#uent hypergeometric function. This con#uent hypergeometric function is the radial solution of an electron in an in"nite potential cylindrical wire, in the presence of any magnetic "eld, applied parallel to the wire axis with appropriate boundary conditions on the wire surface. Fereyra et al. [36] have reported a study of the dependence of the impurity related energy corrections on the impurity position and the magnetic "eld in parabolic quantum dot by using the strong con"nement approach. They have shown that the impurity related binding energy depends strongly on the impurity position in the dot and the magnetic "eld strength. Xiao et al. [32] have reported the calculation of the ground state binding energy of hydrogenic o! center donor in spherical quantum dot in the presence of magnetic "eld by using a variational method. In their study, the in#uence of magnetic "eld is introduced as an exponential term in the trial wave function.
In the present paper, we propose to study the magnetic "eld and the impurity position dependence of the binding energy of hydrogenic impurity in cylindrical quantum dot. We employ the variational scheme. In the absence of the impurity, the wave function is the product of two exact eigenfunction of the cylindrical quantum dot Hamiltonian along the z-axis and in the (x}y) plane without any restriction on the magnetic "eld strength. The organisation of the paper is as follows. In Section 2, we present the theoretical calculation of the binding energy. The discussion of the results and the conclusion are given in Section 3.
2. Formalism In the framework of an e!ective-mass approximation, the Hamiltonian for a hydrogenic donor in cylindrical quantum dot, of radius R and height (H"2d), with in"nite potential barrier at all the surfaces under an applied uniform magnetic "eld along the z direction is
1 e e H" P! A ! #<(r), 2mH c e "r!r "
(1)
where "r!r ""((o!o )#(z!z ), r (r) is the impurity ion (electron) position measured from the center of the quantum dot, e is the dielectric permittivity, mH is the electron e!ective mass, A(r) is the potential vector of the magnetic "eld and <(r) is the con"nement potential considered zero inside the dot and in"nite outside it. The potential vector is written as A(r)"(B;r), with B"Bz. In cylindrical coordinates, the components of the vector potential are A "A "0 and M X A "(Bo). P The Hamiltonian of the system can be written in cylindrical coordinates and in the reduced units, such as the e!ective Bohr radius aH and Rydberg RH to measure distances and energies, respectively, as 2 H"H(o, u)#H(z)#<(r)! , ((o!o )#(z!z ) (2)
R. Charrour et al. / Physica B 293 (2000) 137}143
139
where
This wave function is given by
H(o, u)t (o, u)
(r)"Nt (o, u)t (z) ;exp(!a((o!o )#(z!z )), (7) where N is the normalization constant. The ground state energy corresponds to l"1, m"0 and n"1. The donor binding energy E of the ground state is obtained by subtracting the minimized energy E from the lowest subband energy E .
E "1 (r, a"0)"H(o, u)#H(z) #<(r)" (r, a"0)2
"!
R 1R 1 R # # t (o, u) Ro o Ro o Ru
co # t (o, u)#c¸ t (o, u) X 4 "E t (o, u),
(2a)
R H(z)t (z)"! t (z)"E t (z). Rz
(2b)
In the in"nite potential well approximation, Eq. (2a) [35] and Eq. (2b) can be solved analytically, thus their eigenfunctions are
co t (o, u)"exp(im u) exp ! 4
co K 2
co ; F !a , "m"#1; , K J 2 t (z)"cos(k z), X
(3) (4)
where ¸ is the z-component of the angular moX mentum operator (in units of g), c"gu /2RH is the dimensionless measure of the magnetic "eld, u is the cyclotron frequency, m"0, $1, 2, 2, and l"1, 2, 3, 2 F (!a , "m"#1; co/2) is the K J general form of the con#uent hypergeometric function. The values of the k and a are determined X K J by the boundary conditions that the wave function vanishes at z"$d and o"R, respectively
cR !a , "m"#1; "0 K J 2
(5)
np k " , n"1, 2, 3, 2. X 2d
(6)
F
"
For the more details on the eigenfunction t (o, u) see Ref. [35]. The inclusion of the impurity potential leads to a nonseparable di!erential equation. Using the variational method, it is possible to associate a trial wave function, which is an approximated eigenfunction of the Hamiltonian described in Eq. (2).
p #c(2a #1), 2d
(8)
E "min [1 (r)"H" (r)2]
? p "min #c(2a #1)!a 2d ? a p P ! 2RcP !P !P # P !2 . 2 P P (9)
The integrals P are de"ned by G Rc x P " exp ! 2 \ p Rc ; F !a , 1; x cos t 2 2
;exp[!2a(R(x!x )#d(t!t )]x dx dt, (9a)
P "
Rcx pt x(x!x ) exp ! cos 2 2
\ exp[!2a(R(x!x )#d(t!t )] ; (R(x!x )#d(t!t ) 1 Rcx ; F !a , 1; 2 2
#a F 1!a , 2 ;
; F !a , 1;
Rcx 2
Rcx 2
dx dt,
(9b)
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P "
\
Rc x exp ! 2
Rc p ; F !a , 1; x cos t 2 2 ;
exp[!2a(R(x!x )#d(t!t )] dx dt, (R(x!x )d(t!t ) (9c)
P "
\
Rc exp ! x 2
p Rc ; F !a , 1; x cos t 2 2
exp[!2a(R(x!x )#d(t!t )] ; (R(x!x )#d(t!t ) ;(x!x ) dx dt, (9d) Rc P " exp ! x 2 \ Rc ; F !a , 1; x sin(pt) 2
exp[!2a(R(x!x )#d(t!t )] ; (R(x!x )#d(t!t ) ;(t!t ) dx dt. (9e) 3. Numerical results and discussion We have calculated the values of the binding energy E of a donor, where the impurity is dis placed away from the center to the edge both along the axial or radial directions in the presence of a uniform magnetic "eld applied along the z-axis as a function of a structure size and the magnetic "eld strength. We have displayed in Fig. 1, the ground state binding energy versus the dot radius for a "xed height H and di!erent strength of the magnetic "eld c when the impurity is placed at the center of the quantum dot (x "o /R; t "z /d). This "gure re #ects the competition between the magnetic "eld e!ect and the spatial con"nement e!ect. For a very small radius R, the strong geometric con"nement
Fig. 1. Binding energy of an on-center donor impurity E as a function of the dot radius for a "xed height H"1aH and several values of the magnetic "eld.
leads the electronic wave function to be more compressed in the quantum dot. The binding energy is signi"cant and relatively insensitive to the magnetic "eld since the electron spatial localisation prevails over the magnetic "eld con"nement. To the vicinity of 1aH, the e!ect of the magnetic "eld begins to be apparent and the curves corresponding to di!erent strength of magnetic "eld tend to deviate from each other, as the dot radius rises, reaching asymptotically to the quantum well case values. For a large values of the radius, R'1aH, the magnetic "eld governs the behavior of the binding energy because it overcomes the spatial localisation. On the other hand, increasing the strength of the applied "eld decreases the cyclotron radius for the electron relative to the dot radius and con"nes the electron to be close to the impurity center. Consequently, the binding energy is enhanced in proportion with an increase of the magnetic "eld strength in the region of the large dot radius. The same results are found by Xiao et al. [32] in spherical quantum dot and Branis et al. [35] in the cylindrical quantum well wire. Here, we have just limited our work to the in"nite height potential barrier. However, in the case of the "nite height
R. Charrour et al. / Physica B 293 (2000) 137}143
potential barrier, the binding energy increases as the radius is reduced, reaches a maximum and decreases as RP0. In addition, the e!ect of the external "eld becomes remarkable only in the large radius R region, as an in"nite barrier case. We should signal that with our model, we can describe any low-dimensional structure going from 2D (two-dimensional) to 0D (zero-dimensional) by taking into account the ratio between the height H and the radius R of the structure, that is (H/R);1 for a QW (2D), (H/R) 1 for a QWW (1D) and (H/R) 1 for a QD (0D). The exact nature of dependence of the binding energy on the impurity position along the radial
141
direction, axial magnetic "eld strength and quantum system size (radius and height) inside the cylindrical structure is presented in Fig. 2. From Figs. 2(a)}(c), several remarks can be noticed. First of all, it is apparent that the binding energy is greatest for any given c, H and radius R when the impurity is in the center of the structure (x "0; t "0). The further we remove the impurity from the center, the lower the binding energy becomes. This behavior is due to the increased spread in the radial probability density when the impurity is displaced from the axis of the system. Results showing a similar nature, in the absence of the magnetic "eld, have been reported by Bose et al. [15,16] in the case
Fig. 2. The binding energies' dependence is with respect to the radial impurity position o /R (measured from the center of the structure) with z "0, for three values of magnetic "eld strength (c"0 (solid curve); c"2 (broken curve); c"3 (doted curve)). (a) The structure is a quantum dot with R"1.5aH and H"1.5aH. (b) The structure is a quantum well wire R"1.5aH and H"8aH. (c) The structure is a quantum well with R"8aH and H"1aH.
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R. Charrour et al. / Physica B 293 (2000) 137}143
of spherical QD and Brown et al. [37] in cylindrical QWW. Furthermore, the introduction of magnetic "eld enhances the binding energy especially in the large radius system and in the case of on center impurity position.
It is observed, also, that the ground state binding energy E increases with the reduction of the vol ume of the structure, being larger when the two dimensions (R and H) are diminished simultaneously, namely, E is more important in the QD (2(a)) than in QWW (2(b)) and QW (2(c)). This agrees with the results of Moreno et al. [27]. Fig. 2(a), in which we have reproduced the situation of a QD taking H"R"1.5aH, shows that the binding energy for the impurity located in the edge of the quantum dot is totally insensitive to the increase of the magnetic "eld strength. Similar theoretical results for a spherical [32] and parabolic quantum dot [36] have been reported. The same physical phenomena can be observed in the QWW system (2(b)) and QW system (2(c)). Fig. 3 illustrates the binding energy as a function of the axial impurity position z for o "0 with di!erent magnetic "eld strength in the QD system (3(a)) and QW system (3(b)). It can be seen that E is degenerate for symmetrical positions with respect to the center of transversal section of the cylindrical system. Also, this degeneracy is not broken by an application of the magnetic "eld in the z-direction contrary to the electric "eld. For the QD system (3(a)), the E increases in proportion with the ex ternal "eld and it is insensitive to the axial applied "eld for the impurity near the top or bottom end faces (z "$d). This is agreeing with the result of the Ref. [36]. In Fig. 3(b) we have reproduced the situation of a QW, that is a structure with small H and large R. As for the case of the QD described in Fig. 3(a), we remark that the ground state binding energy is higher for an impurity on center and enhances with the increase of the magnetic "eld. This validates the result plotted by Ribeiro et al. [33] in Fig. 4.
4. Summary
Fig. 3. The binding energies dependence is with respect to the radial impurity position z /d (measured from the center of the structure) with o "0, for three values of magnetic "eld strength (c"0 (solid curve); c"2 (broken curve); c"5 (doted curve)). (a) The structure is a quantum dot with R"1.5aH and H"1.5aH. (b) The structure is a quantum well with R"8aH and H"1aH.
In the present work we have studied the in#uence of the geometrical con"nement and the e!ect of an applied magnetic "eld on the binding energy of the o!-center donor impurity in cylindrical quantum system. The calculations were performed within the e!ective-mass approximation and using a variational method. The electronic con"nement was
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modeled by an in"nite potential well on all surfaces. We found that the ground state binding energy depends strongly on the spatial con"nement, applied magnetic "eld strength and an impurity position. This behavior is a consequence of the modi"cation of the wave function nature under the magnetic and geometric con"nement. In addition, the change of the impurity position modi"es the spread of the electron wave function. Since the impurity binding energy is insensitive to the application of a magnetic "eld when the donor is located near the surfaces, this result can be useful in the technological application. Indeed, we can control the spatial distribution of the donor center in the microstructure.
Acknowledgements This work has been supported by the Programsin-Aide for Scienti"c Research (PARS). Physique 16 Oujda and Physique 03 Fe`s and under AI No. 97/046/SM.
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