DSpace at VNU: Magnetic field effects on the binding energy of hydrogen impurities in quantum dots with parabolic confinements

Magnetic "eld e!ects on the binding energy of hydrogen impurities in quantum dots with parabolic con"nements Theoretical Department, Institute of Physics, P.O.. Box 429, Bo Ho, Hanoi 10

* Corresponding author Tel.: 843-5917; fax:

#84-4-8349050.

E-mail address: nvlien@bohr.ac.vn (V Lien Nguyen).

Magnetic "eld e!ects on the binding energy of hydrogen impurities in quantum dots with parabolic con"nements

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 8 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 compari-son, 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

1 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

quasi-zero-dimensional systems of QDs Until now,

al-most all studies on binding energy of hydrogen

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 break-age 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 elec-tric "eld [16] The elecelec-tric "eld destroys a sym-metry of the problem (the spherical symsym-metry of spherical QDs, or the cylindrical symmetry of disk-like 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 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 5 0 3 - 2

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

Pois-son and Schrodinger equations in the Hartree

ap-proximation 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 do not "nd any work

deal-ing 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

H" 1

2mH
p! e

c A

!e

er#<!, (1)

where mH is the e!ective mass, e is the elementary

charge, p is the momentum,e is the dielectric

con-stant of the QD material, and A is the vector

poten-tial 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

Hamil-tonian 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

neglect-ed 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

calcu-lating 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

2 Variational calculations

2.1 Theory

Choosing the symmetric gauge in the cylindrical

coordinates: A" B[0,0,o], and de"ning the para-bolic con"ning potential as <!"b(o#z), the

Hamiltonian of Eq (1) for the ground state (zero magnetic quantum number), denoted as H, could

be written in the dimensionless form

2

(o#z#

1

4co#b(o#z),

(3) whereb and c are positive parameters, measuring magnitudes of the con"ning and the magnetic po-tentials, respectively In the e!ective atom units used throughout this work the energy is measured and the length is in units of the e!ective Bohr radius nothing but

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/b and

¸+"1/c will also be used in equivalence to b and

c 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 separat-ing the variables:

(5) (6) Each of Eqs (4) and (5) describes a harmonic oscil-lator (two- and one-dimensional, respectively), and

therefore, the ground-state energy E of the total

Hamiltonian H is already known

E"b#2(b#c. (7)

In the presence of the Coulomb term, the

Hamil-tonian 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

sugges-ted as the following:

W"C exp[!(Co#bz)/2]exp[!a((o#z)],

(8)

where C is the normalization constant, a is only the

variational parameter, andC (as the measure of the

total transverse potential) is de"ned by

C"b#c. (9)

Without magnetic "eld (c"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(!br)

was approximately replaced by a simple polynomial

with the introduction of one more variational

para-meter 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

1W"H"W2

1W"W2 "2C#b!a

#(a!2)I!2aCI#2aI

where

I"



r dr exp(! br!2ar),

IL"[p/(C!b)]



r L\ dr exp(!Cr!2ar)

;Er (C!b], n"1, 2, 3

with Er"(x) being the imaginary error function

[20,21]

For given values of the parameters b and c, 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 b and c, 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] m H"0.067 m, e"12.9, a " 10.19 nm, and therefore R"5.478 meV For

these values of material parameters, the value of the strengthc"1 is corresponding to a magnetic"eld

of +6.68 ¹, and to the length ¸+"1 a

While the magnetic "eld dependence of the

en-ergy E is well de"ned by Eq (7), Fig 1 shows how the ground-state energy E! depends on the

mag-netic "eld c for various values of the length

¸!: ¸!"1, 1.5, 2, 3, and 5 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!(c) 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!(c) in

Fig 1 follows the general behavior: in the limit of

small "elds E!(c)Jc, while in the opposite limit

of high "elds E!(c) becomes linear to c 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 c)0.5 for the weak "eld regime and the

asymp-totic behavior E!(c)+(2/3)c for the "eld depend-ence of E! in the high "eld regime.

In Fig 2 the binding energy E is plotted as

a function of the "eld parameter c for the same

Fig 1 The variational ground-state energy E! as a function of

the magnetic strength c for QDs with various con "nement

lengths ¸! : ¸!"1, 1.5, 2, 3, and 5 (from top) All the curves of

¸!' 5 (not shown) are practically coincided with that of

¸!" 5.

Fig 2 The variational binding energy E as a function of the

magnetic strengthc The data are resulted from E of Eq (7) and

E! in Fig 1 for the same values of the length ¸!: 1, 1.5, 2, 3, and

5 (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 b,¸\! , 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

the limit of weak con"nements the "eld dependence

of the binding energy should be linear, E (c)Jc, at

weak "elds Such a linear region could really be recognized in the curve with largest ¸! (¸!"10 meansb" in Eq (7)) in Fig 2 For other curves

of smaller ¸! two e!ects of con"ning potential and

of magnetic "eld are mixed with the totally e!ective strengthC of Eq (9) and a linear region of E (c) at

weak "elds is no more seen

We would mention that in the limit of zero "eld

our results of E (b), describing the e!ect of

con"n-ing potential alone on the bindcon"n-ing 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 forb"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 coincid-ence of two results forb"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 bind-ing energy

3 Perturbation calculations in the limit cases

As was shown in the previous section, for con"n-ing 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 "! 2

r

and the perturbation part H"co#br (11) with both e!ective con"nement and magnetic "eld strengthsb and c assumed to be small, b;1 and c;1

Fig 3 Two binding energies E (variational, solid lines) and

E of Eq (12) (perturbation, dashed lines) are compared in the

weak con"nement regime, ¸!"5, 7, and 10 (from top), and in

weak "eld region, c)0.15.

The ground-state solution of the Hamiltonian

H  is well known with the eigenstate and the

eigenvalue being

!1, respectively Using these unperturbation

solu-tions 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#3b#c/6

and further, from Eqs (2), (7) the binding energy

is evaluated in the same approximation as the

following:

E,E!E! "1#2C#b!3b!c/6,

(12) whereC is de"ned as in Eq (9) Thus, in the

frame-work 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 3 the perturbation binding energy Eof

Eq (12) is plotted (dashed lines) as a function of the

e!ective magnetic "eld strengthc for c)0.15, and

for three values of the e!ective con"ning strength

b : , , 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 ofc and b are also presented for

a comparison It is clear that even for the case of

b" 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 c"0.001, 0.05, 0.1, and 0.15, respectively The smaller the b (weaker con"nement), the closer to each other two corresponding curves become However, it should be noted that from the "eld of c+0.1, where the length 2¸+ becomes smaller than

5, for all three cases of ¸!"5, 7, and 10 in Fig 3 the magnetic "eld potential becomes stronger than the con"ning one in the Hamiltonian of

Eq (3), and therefore, the relative di!erences be-tween two results, peturbation and variational, will

be determined by the magnetic "eld rather than by the con"ning potential that results in similar be-haviors of all the curves atc*0.1 It is here useful

to recall that for the GaAs-QDs the valuec"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 ofc)0.15 Moreover, an agreement between the results, ob-tained 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 poten-tials on the binding energy even in the limit of strong con"nements To see how two approxima-tions, perturbation and variational, are in agree-ment in this limit of c<1 we performed

perturbation calculations of the energy E!, consid-ering the Coulomb potential (!2/r) as a

perturba-tion The unperturbation Hamiltonian is then nothing but H of Eqs (4)}(6), and therefore, the corresponding ground-state solution is already known with the eigenvalue given in Eq (7) and the eigenfunction of the form

t"(Cb/p)exp[!(Co#bz)/2]. (13)

In this case since the unperturbation energy is

exactly coincided with the energy E the

bind-ing energy seems entirely to be de"ned by the perturbation correction (with opposite signs) This correction could be evaluated by the standard pro-cedure of calculating perturbation energies, using

Fig 4 The zero-"eld binding energies E (variational, solid

line), and E of Eq (14) (perturbation, dashed line) are in

comparison plotted versus the length ¸! for the strong

con"ne-ment regime of ¸! ranging from 0.1 to 0.8.

the unperturbation solution of Eqs (7) and (13)

Thus, to the "rst-order approximation we obtain

the binding energy in the strong con"nement limit

E"4Cb



drexp(!Cr)

(C!b Er"[r (C!b].

(14)

In Fig 4, the binding energy Eof Eq (14) for

the zero-"eld case is presented (dashed lines) as

a function of the con"ning length ¸! in comparison

with the corresponding variational binding energy

E , calculated in the previous section The relative

di!erences between two energies seem to be as large

as +1.4%, 5%, 7%, and 11% for ¸!"0.1, 0.3, 0.5,

and 0.8, respectively The discrepancy between two

approximations certainly increases with increasing

¸! Note that it is not interesting to study the

magnetic "eld e!ect in this limit since for such

strong con"nements as those in Fig 4 (¸!)0.8)

the e!ect of magnetic "elds realized usually in

ex-periments is relatively small

Originally, the concept of strong and weak

con-"nement regimes was introduced by Efros and

Ef-ros [23] in relation to the exiton problems Ekimov

et al [24] suggested that in calculating the exiton

binding energy in the strong con"nement regime,

the perturbation method could be used with an

error of +1% for square-well QDs of sizes as large

as 2a This estimation, certainly, is not related to

the present problem of impurities in parabolic

con-"ning potentials Our results in Fig 4 suggest that

a perturbation approach could perhaps be used for calculating the impurity binding energy in the re-gime of such strong con"nements as of ¸!(0.2 Thus, while Fig 3 could really be seen as a sup-port for using the perturbation method as a good approximation in the weak con"nement regime,

in the opposite regime of strong con"nements of Fig 4 we, however, have no such belief, except the limit region of very small con"ning lengths

4 Conclusion

We calculated the e!ects of both the parabolic con"ning potential and the magnetic "eld on the binding energy of hydrogen impurities in parabolic con"nement QDs, using mainly a variational method A very simple trial function with only one variational parameter was suggested that gives for the binding energy a con"ning potential depend-ence, which agrees very well with those of previous publications in the limit of zero "eld In the pres-ence of a magnetic "eld both the ground-state

en-ergy E! and the binding enen-ergy E are increased,

but the e!ect depends on the con"nement strength: the weaker the con"nement, the stronger the mag-netic "eld e!ect While the con"nement e!ect on

E! could be considered negligibly small, when the

con"ning length ¸!*5, the binding energy E still

decreases with increasing ¸! and the magnetic "eld

dependence behavior of E at large ¸!"10 is

similar to that for the bulk semiconductors The perturbation calculations are also performed in the limit regimes of weak and strong con"nements

A good coincidence of two results, variational and perturbation, gives a con"dence in the chosen trial function, and therefore in the variational results presented

In reality, the etched GaAs/AlGaAs QDs often have the shape of cubes However, since the size of cubes is always much greater than the length ¸!

of parabolic con"ning potentials [18], the e!ect of

cubical boundaries on the binding energy E is

relatively small, and the present model of QDs with

parabolic con"nements should then be applied.

Concerning the impurity position, we believe that

for a given magnetic "eld the binding energy

de-creases as the impurity moves away from the dot

center in the same way as is shown in Ref [13] for

the case of zero "eld Thus, it is hoped that our

results might provide useful insights on

experi-mental investigations of shallow donor impurities

in GaAs-type QDs and stimulate further

theoret-ical interest in the problem

Acknowledgements

One of the authors (NVL) thanks Professor Peter

Thomas for the kind hospitality at Physics

Depart-ment, Philipps-University Marburg, where this

work was "nally completed This work was partly

supported by the collaboration fund from the Solid

State Group of Lund University, Sweden

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