Physica B 292 (2000) 153}159 Magnetic "eld e!ects on the binding energy of hydrogen impurities in quantum dots with parabolic con"nements V Lien Nguyen *, M Trinh Nguyen , T Dat Nguyen Theoretical Department, Institute of Physics, P.O Box 429, Bo Ho, Hanoi 10 000, Viet Nam Physics Faculty, Hanoi National University, 90 Nguyen-Trai Str., Thanh-Xuan, Hanoi, Viet Nam Received February 2000 Abstract Using a very simple trial function with only one variational parameter, the e!ects of parabolic con"ning potentials and magnetic "elds on the binding energy of hydrogen impurities in quantum dots are investigated in detail For a comparison, the perturbation calculations are also performed in the limit cases of weak and strong con"nements The obtained results are suggested to be used for shallow donor impurities in GaAs-type quantums dots 2000 Elsevier Science B.V All rights reserved Keywords: Binding energy; Hydrogen impurity; Quantum dots; Magnetic "eld Introduction The aim of this work is to study the con"nement and the magnetic "eld e!ects on the binding energy of hydrogen impurities in quantum dots (QDs) with parabolic con"ning potentials The physics of impurity states developed since early days of the semiconductor science [1,2], has recently received renewed attention in relation to low-dimensional semiconductor structures such as quantum wells, quantum wires, and quantum dots While for quantum wells the binding energy of hydrogen impurities was investigated in great detail [3}5], the problem is much less studied for quasizero-dimensional systems of QDs Until now, almost all studies on binding energy of hydrogen * Corresponding author Tel.: #84-4-843-5917; fax: #84-48349050 E-mail address: nvlien@bohr.ac.vn (V Lien Nguyen) impurities in QDs have exclusively been limited to the e!ect of con"ning potentials: the square (in"nite or "nite) potentials [6}11], or parabolic potentials [12}15] The most important feature of all these models [6}15] is their spherical symmetry that allows one to reduce the problem to solving a simpler equation of the radial variable, which could even be solved by the NUMEROV [15] A breakage of the spherical symmetry may be caused by di!erent factors such as the dot shapes, asymmetric con"ning potentials, or external "elds Recently, we have calculated the binding energy of hydrogen impurities in two types of QDs, spherical QDs with parabolic con"nements and disk-like QDs with parabolic lateral con"nements, in an external electric "eld [16] The electric "eld destroys a symmetry of the problem (the spherical symmetry of spherical QDs, or the cylindrical symmetry of disklike QDs), and the new behaviors of the binding energy, depending on the relative strength of two, 0921-4526/00/$ - see front matter 2000 Elsevier Science B.V All rights reserved PII: S - ( 0 ) 0 - 154 V Lien Nguyen et al / Physica B 292 (2000) 153}159 con"ning and electric "eld, potentials, have been recognized It should be noted here that the results of a numerical self-consistent solution of the Poisson and Schrodinger equations in the Hartree approximation performed by Kumar et al [17] strongly support the parabolic form of con"ning potentials for QDs fabricated from GaAs/AlGaAs heterostructures The e!ect of an external magnetic "eld on the energy spectrum and on the related optical transitions is certainly the most important tool for the study of the electronic states This is why the magnetic "eld e!ects were extensively studied for the impurities in bulk semiconductors [2] For semiconductor QDs we not "nd any work dealing with the problem, though the magnetic "eld e!ects on the conduction electron energy levels have been studied at length [18,19] The e!ective mass Hamiltonian of a hydrogen atom in a QD, with a con"ning potential < in an ! external magnetic "eld B has the standard form e  e H" p! A ! #< , ! c r 2mH (1) where mH is the e!ective mass, e is the elementary charge, p is the momentum, is the dielectric constant of the QD material, and A is the vector potential of the magnetic "eld B (B"rot A) The atom is here assumed to be located at the center of QD, which is also chosen as the origin of coordinates system The spin term is not included in the Hamiltonian of Eq (1) since it simply produces a constant shift of energies In the model, the polarization and image charge e!ects are also assumed to be neglected that should describe, for example, the QDs fabricated from GaAs/AlGaAs heterostructures [18] The binding energy is generally de"ned as E "E !E , (2)  ! where E and E are ground-state energies of the !  Hamiltonian of Eq (1) with and without the Coulomb term, respectively In this work for calculating the binding energy E of hydrogen impurities in QDs with parabolic con"ning potentials in an external magnetic "eld we mainly use a variational method, but in the limit cases of weak and strong con"nements some perturbation results are also included for a comparison Variational calculations 2.1 Theory Choosing the symmetric gauge in the cylindrical coordinates: A"B[0, 0, ], and de"ning the para bolic con"ning potential as < " ( #z), the ! Hamiltonian of Eq (1) for the ground state (zero magnetic quantum number), denoted as H , could  be written in the dimensionless form H "! ! #  # ( #z),  ( #z (3) where and are positive parameters, measuring magnitudes of the con"ning and the magnetic potentials, respectively In the e!ective atom units used throughout this work the energy is measured in units of the e!ective Rydberg R "mHe/2    and the length is in units of the e!ective Bohr radius a "  /mHe The magnetic "eld strength is then nothing but ,( eB/mHc)/2R Thus, and are  two parameters, characterizing the problem in the study, and the binding energy will be calculated as the functions of them Below, conveniently, two lengths ¸ and ¸ de"ned as ¸ "1/ and ! + ! ¸ "1/ will also be used in equivalence to and + as the measures of corresponding potentials (measuring the spatial scales of potentials, the length ¸ is sometimes explained as the e!ective ! radius of QD [13,14,18], while ¸ is often called + the magnetic length) Without the Coulomb potential the Hamiltonian of Eq (3) could easily be solved exactly by separating the variables: H "H #H , (4)  M X H "! #( # ) , (5) M M  H "! # z (6) X X Each of Eqs (4) and (5) describes a harmonic oscillator (two- and one-dimensional, respectively), and therefore, the ground-state energy E of the total  V Lien Nguyen et al / Physica B 292 (2000) 153}159 Hamiltonian H is already known  E " #2( #  (7)   In the presence of the Coulomb term, the Hamiltonian of Eq (3) could not be solved analytically The variational method is widely accepted as a good approximation for calculating the ground state energy In order to choose an adequate trial function one should remark on the main features of the Hamiltonian H : at very small distances (close  to the impurity) the Coulomb potential should be dominant, while at large distances the harmonic oscillator potentials play a more important role Reasonably, the trial function could then be suggested as the following: "C exp[!( # z)/2]exp[!a(( #z)], (8) where C is the normalization constant, a is only the variational parameter, and (as the measure of the total transverse potential) is de"ned by " #  (9)  Without magnetic "eld ( "0) the trial function of Eq (8) is exactly coincided with that used by Xiao et al [12] and Bose [13] In Varshni's trial function [15] the exponentially con"ning factor exp(! r) was approximately replaced by a simple polynomial with the introduction of one more variational parameter Note, however, that in the model of Ref [15] besides the parabolic con"ning potential there exists also an in"nite square potential well Substituting the trial function of Eq (8) into the Hamiltonian of Eq (3) we obtain "H "  "2 # !a " where  (a!2)I !2a I #2aI   # , (10) I   r dr exp(! r!2ar),   I "[ /( ! )] rL\ dr exp(! r!2ar) L  ;Er"[r( ! ], n"1, 2, I"  155 with Er"(x) being the imaginary error function [20,21] For given values of the parameters and , minimizing the energy of Eq (10) with respect to the variational parameter a, we will obtain the energy E , and further, from the energy E of Eq !  (7) the binding energy E will be determined Such calculations have been performed for large ranges of values of and , and the obtained results are shown in the next sub-section 2.2 Numerical results It should be mentioned again that all the energies as well as the lengths that appear in the results shown below are measured in the atomic units For de"nition, taking GaAs as a typical QD material, one has [22] mH"0.067 m , "12.9, a "  10.19 nm, and therefore R "5.478 meV For  these values of material parameters, the value of the strength "1 is corresponding to a magnetic "eld of +6.68 ¹, and to the length ¸ "1 a + While the magnetic "eld dependence of the energy E is well de"ned by Eq (7), Fig shows how  the ground-state energy E depends on the mag! netic "eld for various values of the length ¸ : ¸ "1, 1.5, 2, 3, and The most impressive ! ! feature found in the "gure is that with increasing ¸ the energy E at the beginning falls steeply, and ! ! then ceases to fall further at ¸ +5 (all the curves of ! E ( ) for ¸ '5 are indistinguishably close to that ! ! for ¸ "5 shown in Fig 1) This unambiguously ! means that for spherical QDs with a parabolic con"ning potential the con"nement e!ect on the ground-state energy becomes negligibly small, when the dot `e!ective sizea ¸ is as large as 5a or more ! For any ¸ in the study the curve of E ( ) in ! ! Fig follows the general behavior: in the limit of small "elds E ( )J , while in the opposite limit ! of high "elds E ( ) becomes linear to The widths ! of these limit regions depend on ¸ For the case of ! large ¸ "5 (it could be seen as the limit of weak ! con"nement) a rough estimation gives the region )0.5 for the weak "eld regime and the asymptotic behavior E ( )+(2/3) for the "eld depend! ence of E in the high "eld regime ! In Fig the binding energy E is plotted as a function of the "eld parameter for the same 156 V Lien Nguyen et al / Physica B 292 (2000) 153}159 Fig The variational ground-state energy E as a function of ! the magnetic strength for QDs with various con"nement lengths ¸ : ¸ "1, 1.5, 2, 3, and (from top) All the curves of ! ! ¸ '5 (not shown) are practically coincided with that of ! ¸ "5 ! the limit of weak con"nements the "eld dependence of the binding energy should be linear, E ( )J , at weak "elds Such a linear region could really be recognized in the curve with largest ¸ (¸ "10 ! ! means "  in Eq (7)) in Fig For other curves  of smaller ¸ two e!ects of con"ning potential and ! of magnetic "eld are mixed with the totally e!ective strength of Eq (9) and a linear region of E ( ) at weak "elds is no more seen We would mention that in the limit of zero "eld our results of E ( ), describing the e!ect of con"ning potential alone on the binding energy, are in very good agreement with those of Refs [12}15] For example, our calculations give for E the values of 1.48946, 1.68020, and 1.84963 for "0.2, 0.3, and 0.4, respectively, while the corresponding values obtained in Ref [12] for the case of the largest radius of hard boundary (R"7) are 1.49063, 1.68022, and 1.84963 [12,15] The coincidence of two results for "0.4 certainly implies that at such strong con"nements the distance of 7a could be considered in"nite, and therefore, the hard boundary located there does not yet a!ect the binding energy Perturbation calculations in the limit cases Fig The variational binding energy E as a function of the magnetic strength The data are resulted from E of Eq (7) and  E in Fig for the same values of the length ¸ : 1, 1.5, 2, 3, and ! ! (from top) The lowest curve of ¸ "10 is added to show the ! e!ect of con"ning potential on E for QDs of large ¸ ! values of the con"ning potential length ¸ as in ! Fig 1, except the lowest curve of ¸ "10 Though ! the energy E ceases to depend on the length ¸ at ! ! ¸ *5, the energy E that resulted from Eq (2) ! with the term E depending on ,¸\, certainly  ! continues to decrease with increasing ¸ as shown ! by this curve of ¸ "10 ! Note that, as is well known for the bulk materials [2], and as can be seen from Eqs (2), (7), and (10), in As was shown in the previous section, for con"ning potentials with the length ¸ '3 the e!ect of ! con"nements on the ground-state energy E seems ! to be very weak, and therefore one can suggest to use a perturbation approach for calculating E in ! this limit Taking such an opportunity we write the Hamiltonian of Eq (3) in the form H "H #H  ,    where H is the unperturbation Hamiltonian of  a hydrogen atom, H "! !  r and the perturbation part H  "  # r (11)   with both e!ective con"nement and magnetic "eld strengths and assumed to be small, ;1 and ;1 V Lien Nguyen et al / Physica B 292 (2000) 153}159 The ground-state solution of the Hamiltonian H is well known with the eigenstate and the  eigenvalue being "(1/( )exp(!r) and E " !1, respectively Using these unperturbation solutions and the perturbation Hamiltonian H  of Eq  (11) the standard and simple calculations lead to the "rst-order approximation for the ground-state energy E  ,E #E  "!1#3 # /6 ! and further, from Eqs (2), (7) the binding energy is evaluated in the same approximation as the following: E  ,E !E  "1#2 # !3 ! /6,  ! (12) where is de"ned as in Eq (9) Thus, in the framework of the "rst-order perturbation approximation we obtain a very simple expression for the binding energy in the limit of weak con"nements In Fig the perturbation binding energy E  of Eq (12) is plotted (dashed lines) as a function of the e!ective magnetic "eld strength for )0.15, and for three values of the e!ective con"ning strength :  ,  , and  (correspondingly, ¸ "5, 7, and    ! 10) In this "gure the binding energies E , obtained by the variational method in the previous section for the same values of and are also presented for a comparison It is clear that even for the case of "  two curves are very close to each other: an  estimation gives the relative di!erences between them as (0.1%, 0.2%, 0.5%, and 1% for the "elds "0.001, 0.05, 0.1, and 0.15, respectively The smaller the (weaker con"nement), the closer to each other two corresponding curves become However, it should be noted that from the "eld of +0.1, where the length 2¸ becomes smaller than + 5, for all three cases of ¸ "5, 7, and 10 in ! Fig the magnetic "eld potential becomes stronger than the con"ning one in the Hamiltonian of Eq (3), and therefore, the relative di!erences between two results, peturbation and variational, will be determined by the magnetic "eld rather than by the con"ning potential that results in similar behaviors of all the curves at *0.1 It is here useful to recall that for the GaAs-QDs the value "0.15 corresponds to a "eld of +1¹ Thus, our calculations suggest that for QDs with con"ning potentials of ¸ *5 the perturbation ! method could be used for investigating the e!ect of a magnetic "eld on the binding energy of Hydrogen impurities at least in the range of "elds of )0.15 Moreover, an agreement between the results, obtained by the two methods could also be seen as a bene"t for the chosen trial function of Eq (8) Lastly, we would mention that Bose and Sarkar [14] have recently used the perturbation method to investigate the e!ect of parabolic con"ning potentials on the binding energy even in the limit of strong con"nements To see how two approximations, perturbation and variational, are in agreement in this limit of