Oscillator strength of a two-dimensional D2 ion in magnetic fields

A.S. Santosa,*, L. Ioriattib, J.J. De Grootec

aDepartamento de Fısica, Universidade Federal de Sao Carlos, Rod. Washington Luiz, Km 235, 13565-905 Sao Carlos, SP, BrazilbInstituto de Fısica de Sao Carlos, Universidade de Sao Paulo, Caixa Postal 369, 13560-970 Sao Carlos, SP, Brazil

cLaboratorio Interdisciplinar de Computacao Cientifica, Faculdades COC, Rua Abraao Issa Halack 980, 14096-175 Ribeirao Preto, SP, Brazil

Received 18 September 2003; received in revised form 8 October 2003; accepted 18 October 2003 by C.E.T. Goncalves da Silva

Abstract

In the present paper we determine the oscillator strength of two-dimensional (2D) D2 ions under the influence of a static

magnetic field. The results are important for the analysis of the optical transitions observed in semiconductor quantum wells.

We have applied the ab initio procedure Hyperspherical Adiabatic Approach, based on the use of hyperspherical coordinates.

This approach uses an adiabatic separation of the total wave function that allows accurate energies determination from

molecular-like potential curves. The convergence is obtained in a systematic way by the inclusion of non-adiabatic couplings

without the need of adjustable parameters.

q 2003 Elsevier Ltd. All rights reserved.

PACS: 73.20.Hb; 78.66. 2 w

Keywords: A. Quantum wells; C. Impurities in semiconductors; D. Electronic states (localized)

1. Introduction

Quasi-two-dimensional (2D) semiconductor structures,

as quantum wells, can be currently grown in laboratory by

alternating layers of different semiconductor materials.

Electrons bound to impurities can be confined in these

structures, increasing significantly the binding energy of the

system. In appropriately doped quantum wells, a quasi-2D

D2 ion is formed when an extra electron becomes bound to a

neutral shallow donor. The D2 ion in semiconductors is

analog to the H2 ion in atomic physics. This is one of the

simplest systems where the electronic correlation is

important, since the nuclear charge is screened by the

inner electron. Its experimental identification in magneto-

optical spectra of selectively doped GaAs/GaAlAs quantum

wells, by Huant et al. [1], originated a series of experimental

[2,3] and theoretical works [5–12].

In the literature, there are some theoretical calculations

for the bound states of a D2 ion in quasi-2D structures.

Using a variational approach, Phelps and Bajaj [4] have

calculated the binding energy of the 2D D2 ion. They have

found a value nearly 10 times greater than the binding

energy of the equivalent system in three-dimensional (3D).

Pang and Louie [5] have calculated the bound states of D2

ion in an static magnetic field in quantum wells using a

Monte Carlo method. Their results fairly agree with the

measured data of Huant et al. [1]. For a strictly 2D D2 ion,

Larsen and McCann [6] have demonstrated that in the limit

of infinite magnetic field the bound states can be exactly

found. They have found that there are only four bound

states: the spin singlet ground state and the three spin triplet

states. By means of a variational approach, Larsen and

McCann [7] have calculated the transition energy of the

singlet ground state to the lowest singlet excited state for a

2D D2 ion in a magnetic field of arbitrary strength.

Magnetopolaron corrections were calculated by Dzyubenko

and Sivachenko [8] at high fields, by R. Chen et al. [9] and

Shi et al. [10], which are consistent with the experimental

data of Huant et al. [1]. For a D2 ion in a quantum well, the

energies and oscillator strengths for transitions from the

0038-1098/$ - see front matter q 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ssc.2003.10.011

Solid State Communications 129 (2004) 325–330

www.elsevier.com/locate/ssc

* Corresponding author. Address: Instituto de Fısica de Sao

Carlos, Universidade de Sao Paulo, Caixa, Postal 369, 13560-970

Sao Carlos, SP, Brazil. Tel.: þ55-162-739-877; fax: þ55-162-739-

777.

E-mail address: assantos@df.ufscar.br (A.S. Santos).

singlet state M ¼ 0 to M ¼ 21; 1 as a function of the

magnetic field was obtained numerically by Riva et al. [13].

The hyperspherical (HS) approach has been recently

applied to the calculation of the ground and an excited state

of the 2D D2 ion in a magnetic field [11,12]. The

introduction of the HS approach to the study of 2D atomic

systems has been motivated by the precise results obtained

for correlated 3D systems [14–27]. The choice of such

coordinates allows the adiabatic separation of the Schro-

dinger partial differential equation using the unique HS

radial variable as an adiabatic parameter for the HS angular

equation. Such procedure, called Hyperspherical Adiabatic

Approach (HAA), has been used to describe atomic spectra

with potential curves and non-adiabatic couplings, in a

similar way as the Born–Oppenheimer approach. Differ-

ently of the variational procedure, the HAA is an ab initio

approach and all physical intuition is furnished by the

potential curves structure. In a previous paper [12], the HS

was used to calculate bound state energies for the 2D D2 ion

in a magnetic field. The system of HS equations was solved

by the use of a modified HS angular variable that allows the

analytical solution of the HS angular equation by the

Frobenius method, increasing the precision of the results.

The accuracy of the angular wave functions and the

introduction of the non-adiabatic couplings in the radial

equations allow the precise calculation of system properties,

as the oscillator strengths, which is the main objective of this

work. For the 2D D2 ion it was not possible to find any

results in the literature for the oscillator strengths for

transitions in a magnetic field.

The following sections present a brief discussion of the

HS approach and the results obtained for the oscillator

strengths of the 2D D2 ion in the presence of a magnetic

field.

2. The two-dimensional hyperspherical adiabatic

approach

For the 2D D2 ion the Schrodinger equation in the

presence of a static magnetic field, is given by"72

1 þ 722 þ

2

r1

þ2

r2

22

l~r1 2 ~r2l2

g2

4ðr2

1 þ r22Þ

2gðLz1þ Lz2

Þ þ 2E

#Cð~r1; ~r2Þ ¼ 0 ð1Þ

where

g ¼"vc

2 Ry

is a function of the magnetic field on the cyclotron

frequency vc ¼ eB=mc: The Lzi; i ¼ 1; 2 is the electron

azimuthal angular momentum operator in units of ":

Energies are measured in Ry; the hydrogenic impurity

Rydberg constant in three dimensions and distances in units

of the 3D Bohr radius a0: Atomic units (a.u.) are defined by

set m ¼ e ¼ " ¼ a0 ¼ 1: Only spin-singlet states are being

considered in this work.

The HS coordinates are defined as a function of the radial

polar coordinates, given by

R2 ¼ r21 þ r2

2

tan a ¼r1

r2

; ð2Þ

which is equivalent to the transformation of Cartesian

coordinates to polar ones. With the spherical angles f1;f2;

which are not changed, the system is now described by three

angular coordinates V ¼ {a;f1;f2}; and only one radial R:

The HS radius R and the hyperangle a take into account the

correlation effects of the system.

The Schrodinger equation in HS coordinates is

›2

›R2þ

1

4R2þ

1

R2UðR;VÞ2

g2R2

4þ 2e

" #cðR;VÞ

¼ 0 ð3Þ

with the re-normalization

cðR;VÞ ¼ R3=2ðsin a cos aÞ1=2CðR;VÞ; ð4Þ

which results in the normalizationðlcðR;VÞl2dRdadf1df2 ¼ 1 ð5Þ

The total energy E is redefined as e ¼ E 2 gM=2; where M

is the total azimuthal quantum number. In HS coordinates,

the 2D Schrodinger equation dependency in magnetic field g

is only on the radial equation.

The operator UðR;VÞ is called HS angular operator, and

is defined as

UðR;VÞ ¼›2

›a22

L2z12 1=4

sin2 a2

L2z22 1=4

cos2 aþ

2ZR

sin a

þ2ZR

cos a2

2Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 sin 2a cos f12

p ; ð6Þ

where cos f12 ¼ r1r2: Within the HAA [14], an eigenvalue

problem for the angular operator is solved considering the

hyperradial variable R as a parameter. Since this operator is

Hermitian, for each value of R it has a complete set of

eigenfunctions FmðR;VÞ: This functions are called channel

functions, and are given by

UðR;VÞFmðR;VÞ ¼ UmðRÞFmðR;VÞ; ð7Þ

whose eigenvalues are the potential curves UmðRÞ: The

angular channel functions are used as a basis for the total

wave function as

cðR;VÞ ¼Xm

FmðRÞFmðR;VÞ: ð8Þ

This basis choice is based on the simple linear dependence

A.S. Santos et al. / Solid State Communications 129 (2004) 325–330326

of the angular operator with respect to R and also on the

good adiabatic characteristic of this variable compared with

the angular ones. The expansion quality is verified in the fast

convergence attained.

The radial components FmðRÞ and the bound states

energies e are determined from the set of coupled radial

equations

d2

dR2þ

UmðRÞ þ 1=4

R22

g2R2

4þ 2e

!FmðRÞ

þXy

2Pmy ðRÞd

dRþ Qmy ðRÞ

� Fy ðRÞ ¼ 0 ð9Þ

where

Pmy ðRÞ ¼ FmðR;aÞD ��� ›

›RFy ðR;aÞi ð10Þ

Qmy ðRÞ ¼ FmðR;aÞD ��� ›2

›R2Fy ðR;aÞi ð11Þ

are the non-adiabatic couplings. This procedure changes the

solution of a partial differential equation to the solution of an

infinite set of radial coupled ordinary equations. As a

consequence it allows the calculation of the resonant states

as bound states of excited potential curves by removing the

non-adiabatic couplings between open and closed channels.

At this point the method provides a hierarchy of

approximations. The first one is the extreme uncoupled

adiabatic approach (EUAA), obtained by neglecting all the

non-adiabatic couplings and by solving the radial equation

for the lowest potential curve. It furnishes a lower bound for

the bound state energy. The inclusion of diagonal couplings

furnishes an upper bound for the energy and is called

uncoupled adiabatic approximation (UAA). With the

inclusion of non-diagonal couplings, the coupled adiabatic

approach (CAA), the accuracy of the calculation is

improved until the desired precision. This calculation has

been shown to be precise for the D2 ion and other few body

systems [12,16,23] due to the good choice of the adiabatic

variable R: The calculation of the bound states are also

precise, even for a small number of coupled equations [12].

3. Angular equation

The solution of the eigenvalue problem for the angular

operator is obtained analytically for R ¼ 0 and R !1: For

small values of R the solution can be expanded on the HS

harmonics, the analytical R ¼ 0 eigenstates [15]. For large

values of R the hydrogen-like behavior predominates,

changing the channel function into very located functions

at the a contour points 0 and p=2 [14,15]. To avoid different

basis expansion for different values of R; a direct numerical

solution may be performed, which allows precise results.

The channel functions can be expanded in the coupled

angular eigenfunctions of the azimuthal angular momentum

operators of the particles 1 and 2,

FmðR;a;VÞ ¼1

2p

Xm1m2

epaðsin aÞlm1lþ1=2ðcos aÞlm2 lþ1=2

£ GmMm1m2

ðR;aÞexpðim1f1Þexpðim2f2Þ: ð12Þ

The exponential term with

p ¼ 2ZR

nm

was included to take into account the correct R !1

hydrogen-like behavior of the angular channels [18]. The

index nm is defined as nm ¼ N þ 1=2; where N ¼ 1; 2;… is

the asymptotic hydrogenic principal quantum number.

Introducing this expansion into the HS angular equation

results in the following system of coupled equations,(›2

›a2þ 2½ðlm1lþ 1=2Þcot a

2 ðlm2lþ 1=2Þtan aþ p�›

›aþ

2R

sin aþ

2R

cos aþ p2

þ 2p½ðlm1lþ 1=2Þcot a2 ðlm2lþ 1=2Þtan a�

2 ðlm1lþ lm2lþ 1Þ2 2 UmðRÞ

)G

mMm1

ðR;aÞ

22R

cos a

Xj;m0

1

tanj aðsin aÞlm01l2lm1lðcos aÞlM2m0

1l2lm2l

£ CmðM;m1;m01; jÞG

m

Mm01

ðR;aÞ ¼ 0; ð13Þ

where the coupling constant Cm is given by the expression

CmðM;m1;m01; jÞ ¼ 22j

Xj

n¼n0

£ ð2Þk22n ðn þ jÞ!

n 2 Dm1

2

� !

n þ Dm1

2

� !

j 2 n

2

� !

j þ n

2

� !

where n0 ¼ j 2 2� j

2

�; k ¼

j 2 n

2and Dm1 ¼ m1 2 m0

1:

An efficient procedure to solve this system of coupled

differential equations was developed in Ref. [18] and it is

utilized in this work to determine numerically exact solutions.

4. Oscillator strengths

In this section we develop the expression for the

oscillator strength in the dipole approximation with light

polarization ex: The oscillator in the length form is

fL ¼ 2vlDLl2; ð14Þ

where v is the photon energy and DL is the electric dipole

A.S. Santos et al. / Solid State Communications 129 (2004) 325–330 327

matrix element, defined as

DL ¼ kCE0

ðR;VÞlex·ð~r1 þ ~r2ÞlCEðR;VÞl: ð15Þ

Substituting the adiabatic expansion of CE one obtains,

DL ¼Xmn

kFE0

m ðRÞFmðR;VÞlRðsin a cos f1 þ cos a

£ cos f2ÞlÞlFEn ðRÞFnðR;V ¼

Xmn

kFE0

m ðRÞlRImnðRÞlFEn ðRÞl

ð16Þ

where the matrix element Imn is given by

ImnðRÞ ¼ kFmðR;VÞlðsin a cos f1 þ cos a

£ cos f2ÞlFnðR;VÞl ¼1

2

Xm1m2

£

(ðsin a½G

mMm1m2

ðGnM0 ðm1þ1Þm2

þ GnM0 ðm121Þm2

Þ�da

þð

cos a�GmMm1m2

ðGnM0m1ðm2þ1Þ þ Gn

M0m1ðm221ÞÞ�da

)

ð17Þ

The calculation of the channel functions allows obtaining

these matrix elements. At this point is very important the

quality of the wave functions for the precision of the

obtained results.

5. Results and discussion

In this section we initially present the results obtained for

the D2 spin-singlet bound states in order to analyze the

method efficiency. Firstly the HS angular equation is solved,

providing the potential and coupling terms necessary for the

radial equation. The HS angular equation is independent of

the system energy, allowing a unique calculation to obtain

the set of HS potential curves UmðRÞ and non-adiabatic

couplings Pmy ðRÞ and Qmy ðRÞ: However, this set of coupled

equations is solved for each value of the adiabatic parameter

R; whose effects on the channel function bring some

difficulties which have to be treated properly in order to

provide an efficient numerical calculation. To avoid

different basis expansion for different values of R; a direct

numerical solution may be performed, which allows precise

results, but with a significant increase of the processing time

as more coupled channels are taken into account. The

procedure adopted in this work [18] consists in an analytical

solution leading to fast and precise numerical calculations

for the angular solutions. The use of the z ¼ tan a=2 variable

changes the angular equation with trigonometric parameters

into an equation with rational coefficients, allowing the use

of Frobenius method with a fast convergent power series for

all values of R [18]. The accuracy of these results can be

analyzed by obtaining the bound state energy of the D2 ion.

The result 24.48150 a.u. [12] obtained with two coupled

radial channels is comparable within 0.04% to the

variational result of Phelps and Bajaj 24.4801 a.u. [4].

With three coupled channels the energy is slightly lower, i.e.

24.48167 a.u., which is coherent with the comparison of the

3D H2 HS calculations with the same variational basis size.

For the D2 system only one bound state has been found,

which means that in order to observe experimental

transitions from the ground state to an excited state M ¼ 1

an external field is necessary. By introducing a static

magnetic field the energies of the system can be obtained

from the HS Schrodinger equation with the inclusion of a

simple hyperradial quadratic term [12], since the HS angular

equation is independent of the magnetic field. The resulting

potential curves are shown in Fig. 1.

The results shown in Tables 1 and 2 are in agreement

with the variational results found in the literature [7].

The oscillator strength of the transition from the lower

states of M ¼ 0 to M ¼ 1 can now be determined. Initially

the angular channels are used in the calculation of the matrix

elements ImnðRÞ; which as the potential curves, are

independent of the energy and the magnetic field. In Fig. 2

the components between the lowest radial channels are

shown for different values of R: These components are input

functions to the dipole matrix elements (Eq. (16)).

To conclude the calculation it is also necessary to obtain

the normalized radial components, which has to be done for

each value of the magnetic field parameter. Figs. 3 and 4

show the behavior of the normalized radial components of

two coupled radial channels as dimensionless parameter g

varies. For both M ¼ 0 and M ¼ 1 states the magnetic field

tends to localize the wave functions more and more with the

increasing of the field values. This means that the electrons

are closer and closer to each other with the increasing of the

magnetic field. For M ¼ 0 the peak of the radial function is

Fig. 1. The lowest M ¼ 0 hyperspherical potential curve for the D2

ion, in which is added the effect of the hyperradial parabolic term

due to the magnetic field.

A.S. Santos et al. / Solid State Communications 129 (2004) 325–330328

more localized at R ¼ 0:75 a:u: approximately while for

M ¼ 1 the peak changes in the range R ¼ 0:8–2:5 a:u:

because this state is weakly bound, resulting in a spread

wave function.

The results for the oscillator strength are shown in

Table 3 and Fig. 5. For small values of g the overlap

between the M ¼ 0 and M ¼ 1 HS radial components are

small because the M ¼ 1 wave function is localized further

that the M ¼ 0: As the magnetic field is increased the M ¼ 1

and M ¼ 0 radial wave functions become located at smaller

values of R; increasing the overlap between the wave

functions and the oscillator strength increases up to a

maximum value for approximately g ¼ 1:4: After this

maximum, the oscillator strength starts to decrease. The

corresponding magnetic field relates to the most intense line

for this transition. For small values of the field the Coulomb

energy is greater than the magnetic energy ðgp 1Þ and the

Coulomb potential dominates. The attractive electron–

donor potential increases faster then the repulsive electron–

electron potential and the oscillator strength increases with

the increasing of the binding energy of the system. For g .1 the Coulomb and magnetic energies are of same

magnitude. The magnetic field parabolic potential starts to

dominate for g . 1 and the oscillator strength decreases due

to the diminished binding energy. For values of the field

sufficiently large, the Coulomb interaction may be con-

sidered a perturbation when compared to the magnetic

interaction. To compare to an analytical asymptotic

behavior larger values of g would be necessary. For very

large values of g the oscillator strength should approach a

constant value, as the effects of the perturbation due to the

Coulomb interaction vanishes.

Table 1

Ground state energy convergency within the hyperspherical

approximation, compared with the variational result of Ref. [7] in

Rydberg ðRyÞ

g EEUAA EUAA ECAA ðmmax ¼ 2Þ Evar

0.0 24.575361 24.474651 24.481501 24.478

0.5 24.440382 24.339634 24.347200 24.346

1.0 24.132499 24.035238 24.042433 24.042

2.0 23.258710 23.170252 23.175104 23.172

4.0 20.999903 20.922835 20.924967 20.917

Table 2

The M ¼ 1 lower state energy convergency within the hyper-

spherical approximation, compared with the variational result of

Ref. [7] in Ry

g EEUAA EUAA ECAA ðmmax ¼ 2Þ Evar

0.5 23.492268 23.420811 23.426676 23.430

1.0 22.875823 22.769353 22.784363 22.789

2.0 21.506703 21.364191 21.395346 21.399

4.0 1.493931 1.659955 1.61159 1.615

Fig. 2. Matrix element ImnðRÞ obtained from the hyperspherical

angular solutions, for different values of the radial parameter.

Fig. 3. Hyperspherical radial density of probability as a function of

g for the D2 ion M ¼ 0 state.

Fig. 4. Hyperspherical radial density of probability as a function of

g for the D2 ion M ¼ 1 state.

A.S. Santos et al. / Solid State Communications 129 (2004) 325–330 329

6. Conclusion

In this work, the oscillator strengths for the electronic

transitions of the 2D D2 ion as a function of a static

magnetic field are obtained. The HAA is shown to be an

efficient method to perform the calculations considering its

precision and convergence control. With a unique potential

curve a good estimate for the eigenstates can be obtained.

Our calculation has been extended to two coupled radial

channels to obtain slightly better results. For such weakly

bound system, excited states become bound in the presence

of a magnetic field. The HS method is appropriated to the

calculation of the transitions among such states, since the

magnetic potential adds only a quadratic correction to

the Hamiltonian in terms of the HS radial variable. The

quality of the eigenstate solutions is determined based on the

accuracy of the energy calculation, in very good agreement

with precise variational calculations. The consistency of the

results is also based on the HS approach efficiency shown in

previous 3D energy and oscillator strengths calculations

[25].

The 2D approximation may be employed to describe the

experimental situation found in practice in quantum wells of

20–400 A wide. Thus, the limiting case studied here, of the

D2 at the center of a strictly 2D quantum well, is useful as

an insight for the case of finite size barriers associated with

narrow wells.

Acknowledgements

The authors would like to acknowledge Fundacao de

Amparo a Pesquisa do Estado de Sao Paulo (FAPESP,

Brazil) for the financial support given to this work.

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Table 3

Two-dimensional D2 ion oscillator strength for the M ¼ 0 to M ¼ 1

transition in the presence of a magnetic field

g DL DE (a.u.) Osc

0.2 0.4973673 0.6737379 0.3333308

0.4 0.5347634 0.8429598 0.4821256

0.6 0.5226032 0.9941982 0.5430592

0.8 0.5022049 1.1316361 0.5708194

1.0 0.4815533 1.2580697 0.5834766

1.2 0.4624720 1.3754795 0.5883760

1.4 0.4452718 1.4853367 0.5889864

1.6 0.4298502 1.5887718 0.5871188

1.8 0.4160025 1.6866743 0.5837852

2.0 0.4035169 1.7797578 0.5795814

2.2 0.3922031 1.8686032 0.5748694

2.4 0.3818981 1.9536902 0.5698764

2.6 0.3724649 2.0354188 0.5647478

2.8 0.3637888 2.1141269 0.5595768

3.0 0.3557741 2.1901016 0.5544252

3.2 0.3483403 2.2635897 0.5493324

3.4 0.3414198 2.3348045 0.5443246

3.6 0.3349550 2.4039320 0.5394176

3.8 0.3288971 2.4711348 0.5346216

4.0 0.3232040 2.5365567 0.5299416

Fig. 5. Oscillator strength of the M ¼ 0 to M ¼ 1 transition for the

2D D2 ion in the presence of a magnetic field calculated with

different values of g:

A.S. Santos et al. / Solid State Communications 129 (2004) 325–330330