D − centre in a quantum well in the presence of a strong magnetic field

Available online at http://www.idealibrary.com ondoi:10.1006/spmi.2000.0905Superlattices and Microstructures, Vol. 29, No. 1, 2001

D centre in a quantum well in the presence of a strong magnetic field

B. S. MONOZONPhysics Department, State Marine Technical University, 3 Lotsmanskaya Str., 190008 St. Petersburg,

Russia

(Received 6 June 2000)

An analytical approach to the problem of a negatively charged donor in a quantum wellin the presence of a strong magnetic field directed perpendicular to the heteroplanes isdeveloped. The double-adiabatic approximation is used. Firstly, the transverse motion ofthe electron caused by the strong magnetic field is much faster than that in the direction ofthe field. In the second stage the state of the outer electron is governed by the short-rangeadiabatic potential formed by the impurity centre and tightly bounded inner electron. Thedependencies of the binding energy upon the magnitude of the magnetic field, the widthof the well and the position of the impurity within the well are derived in an explicit form.The analytical results coincide with those obtained numerically. In principle, the presentedapproach is suitable for the description of a charged donor and an exciton in the quantumwell in the presence of electric and strong magnetic fields.

c 2001 Academic PressKey words: charged donor, quantum well, magnetic field, binding energy.

1. IntroductionDuring the last decade, the subject of a negatively charged donor (D) in quasi-two-dimensional sys-

tems has been studied extensively both experimentally and theoretically. Much of this work has concentratedon GaAs/GaAlAs structures; particularly on isolated single quantum wells (QWs) subjected to an externalmagnetic field directed perpendicular to the heteroplanes. The reason for this is that the magnetic field signif-icantly increases the stability of the charged donor. Since the D centres were identified in magneto-opticalspectra by Huant et al. [1], a considerable amount of literature concerning the theoretical work has beendeveloped. The prime object is to calculate the binding energy of the D donor as a function of the mag-nitude of the magnetic field, the width of the QW and the position of the impurity centre within the QW.Using the Monte Carlo method, Pang and Louie [2] calculated the binding energy of the ground state of theD positioned at the centre of the narrow QW. Mueller et al. [3] and Dunn et al. [4] studied the problemby the variational approach based on the Chandrasekhar-type [5] and Gaussian-like trial functions, respec-tively. Dzubenko and Sivachenko [6] formed the wavefunction of the D from the wavefunctions of the freeelectron corresponding to several Landau levels and the ground size-quantization level. A simple variationalprocedure in which the Gaussian-like trial wavefunction is modulated by the wavefunction of the groundstate in the QW was put forward in [7]. Michailov et al. [8], in their variational scheme, reduced the problemof the D centre to the problem of a neutral donor with an additional effective charge. In an effort to focusE-mail: [email protected]

07496036/01/010017 + 08 $35.00/0 c 2001 Academic Press

18 Superlattices and Microstructures, Vol. 29, No. 1, 2001

on the transitions between singlet and triplet states of the Dand cyclotronresonance transitions, Riva et al.[9] studied numerically the states of D shifted from the mid-point of the QW. These states were observedexperimentally in [10]. Numerical results obtained in the above-mentioned papers are close to both eachother and experimental data.

However, it is pointed out [7] that the numerical approach to the D problem requires a lot of computa-tional effort. In parallel with this the physics of the problem is kept in the dark. In this connection, along withnumerical calculations, analytical study methods are of great interest because they enable the basic physicsof the problem to be kept in clear view.

In this paper, an analytical approach to the problem of a negatively charged donor in a QW in the pres-ence of a strong magnetic field is developed. The effect of the magnetic field directed perpendicular to theheteroplanes is taken to be much greater than that of the Coulomb field of the impurity. Note that extremelystrong magnetic fields, in excess of 40 T, become applicable in the experiment [11]. The dependencies ofthe binding energy of the D centre upon the magnitude of the magnetic field, the width of the QW and theposition of the impurity within the well are found in an explicit form. The presented results are in line withthose obtained by numerical methods.

2. General theoryThe z-axis is chosen to lie along the direction of the uniform magnetic field B which is applied perpen-

dicular to the heteroplanes. The QW is treated as a square well of width d bounded by infinite barriers at theplanes z = d/2. The parameters relevant to the calculation are the impurity Bohr radius (a0), the magneticlength (aB) and the distance of the impurity centre (b) from the mid-point of the QW that is taken to be thepoint z = 0. They are defined, as usual, by

a0 = 4pi0h2

e2aB =

h

eBwhere is the dielectric constant and is the electron effective mass. We take the energy bands to beparabolic, nondegenerate and separated by a wide energy gap.

In the effective mass approximation the equation describing the D donor formed by the impurity centreand two electrons at positions ri (i , zi )(i = 1, 2) has the form{

i=1,2

[1

2

(i hi + e2 [Bri ]

)2e2

4pi0|ri bez |]+ e

2

4pi0|r2 r1|

}9(r1, r2) = E9(r1, r2) (1)

where ez is the unit vector.By solving this equation subject to the boundary conditions

9

(1,

d2; 2,

d2

)= 0 (2)

the energy E and wavefunction 9 can be found in principle. In the strong magnetic field limit for whichaB/a0 1 (3)

the solution to (1) describing the singlet ground state of the D is taken in the form

9(r1, r2) = 120(1)0(2)[ f0(z1)(z2)+ f0(z2)(z1)]. (4)

In the above expression the function

0() = 12piaB

exp(

2

4a2B

)(5)

Superlattices and Microstructures, Vol. 29, No. 1, 2001 19

describes the transverse motion of an electron of energy E0 = heB2 (ground Landau level) in the xy plane.The function f0(z) is the ground state wavefunction of the inner electron bound tightly by the impuritycentre (D0 state). The function f0(z) satisfies the equation

h2

2d2 f0(z)

dz2+ V (z) f0(z) = 30 f0(z) (6)

with the boundary conditionsf0(d/2) = 0 (7)

and with

V (z) = e2

4pi0

|0()|22 + (z b)2 d. (8)

It is convenient to introduce the notation 30 = R/20 where R = e2/8pi0a0 is the impurity Rydbergconstant and 0( a0so that the ground state has quasi-Coulomb character and energy 30 < 0 [12]. In addition, we assume thatthe impurity centre is separated from the edge of the QW by a distance greater than the effective radius of theground state which in turn implies that d/2 |b| a00. It was shown in [12] that in this case the effect ofthe boundaries of the QW on the electron states is exponentially small. Furthermore, we use the solution toeqn (6) for the bulk semiconductor (d = ) obtained originally in [13]. The expression for the wavefunctionf0(z) is given by

f0(z) = 1a00

exp(|z b|

a00

). (9)

The quantum number 0(


f0 z

d 2z2

0b

0a0

z1d 2

U zW

z

V z

0

Fig. 1. A sketch of the potentials V (z) (8) and U (z) (13) and energies of the inner (30) and outer (W ) electrons. The function f0(z)is the wavefunction of the ground state of the inner electron.

The wavefunction (z) satisfies the boundary conditions

(d/2) = 0. (15)The binding energy of the D in the QW E is defined, as usual, by the difference between the sum of the

energies of the neutral donor (D0) and the electron in the QW (E0 + 30) + (E0 + h2pi2/2d2) and theenergy of the D centre E . Using (14) we have

E = h2pi2

2d2W. (16)

A sketch of the effective potentials V (z) (8) and U (z) (13), the resulting wavefunction f0(z) (9) andenergies 30 in (6) and W in (12) is shown in Fig. 1.

3. Results and discussionUnder the condition 0 < 1 the effective radius of the ground state a00 is less than the Bohr radius a0.

Using this condition the analytical calculations are continued. At the zeroth approximation we take in (13)f 20 (z) = (z b). It follows from (13) that

U (z) =

e2

4pi0aB at z = b e2a2B4pi0|zb|3 for |z b| aB

(17)

and the potential U (z) has the form of the finite well of effective width aB . As the magnetic length aB ismuch less than the Bohr radius we replace the potential U (z) (13) by the -type function potential. Leavingaside the effect of the QW boundaries we set

U (z) = h2q0

(z b) (18)where the parameter q0 is defined by the ground energy level W0 in the bulk semiconductor, namely

q0 = (2W0/h2)1/2.


Further, we consider the wide QW for which the energies W are negative (W < 0). The solution toeqn (12) satisfying the boundary conditions (15) is given by

(z) = +d/2d/2

GW (z, z)U (z)(z)dz (19)

where the Green function of the free electron in the QW GW (z, z) has the form

GW (z, z) = 2h2q sinh(qd) sinh[

q(

d2 z

)]sinh

[q(

d2+ z

)]for z > z (20)

with q = (2W/h2)1/2. The expression for the Green function for the region z < z can be obtained fromeqn (20) by replacing z by z and vice versa. The wavefunction (z) and Green function GW (z, z) bothsatisfy the boundary conditions (15).

Substituting expressions (18) and (20) into eqn (19) and then setting z = b in (19) we obtain the transcen-dental equation

2q0q sinh(qd)

sinh[

q(

d2 b

)]sinh

[q(

d2+ b

)]= 1. (21)

By solving eqn (21) the parameter q and the energy W can be found as function of the width of the QWand the position of the impurity centre b.

In order for the dependence of the energy W0 and parameter q0 on the magnetic field to be studied qualita-tively we solve the eqn (12) with the potential U (z) (13) for the bulk semiconductor (d = ). It is convenientto introduce the following notations:

= R

W0; u = 2(z b)

a0; g0 = 2

a0; g12 = 2|1 2|

a0.

Equation (12) then becomes

(u)+ [0|(u2 + g20)1/2|0 1, 2|(u2 + g212)1/2|1, 2](u)14(u) = 0 (22)

where 0||0 and 1, 2||1, 2 are averages with respect to the functions 0() and 0(1)0(2), re-spectively. Setting in (22) u g0, g12 2aB/a0 and then neglecting the term proportional to |u|3, weobtain

(u) = A exp( 12 u) (23)where A is a constant.

In the region u 1, an iteration method is performed by double integration of eqn (22) using the trialfunction (0)(u) satisfying the boundary conditions

(0)(0) = B; d(0)(0)du

= 0where B is a constant.

A comparison of the coefficients is then made between the result of the integration and the expansion offunction (23) for u 1. When terms of the same order are equated, a set of linear algebraic equations arefound. The system of these equations is solved by the determinantal method to give the following expressionfor the quantum number :

= 1 + C

[1 2

2

(pi

2

)](24)


where

= 1pi2

2pi0

d1 2pi

0d2

0

x1dx1

0x2dx2 exp[(x21 + x22)] ln[x21 + x22 2x1x2 cos(1 2)]

(25)and where

= 1pi2

2pi0

d1 2pi

0d2

0

x1dx1

0x2dx2 exp[(x21 + x22)][x21 + x22 2x1x2 cos(1 2)]1/2.

(26)Integration in (25) and (26) can be performed in an explicit form to give the following values for the

parameters = ln 2 C and = 21/2pi1/2 which in turn leads to the expression for the parameter q0:

q0 = (ln 2)[

a0

(1+

2pi(

2 1)

)]1.

For the binding energy of the D in the bulk semiconductor |W0| scaled to the energy R we have|W0|

R= (ln 2)2

(1+

2pi(

2 1)

)2. (27)

It follows from eqn (27) that as magnetic field increases in magnitude, the binding energy of D increasesboth in the bulk semiconductor and in the QW. This dependence coincides qualitatively with those obtainednumerically by a variational approach [2, 79]. A quantitative comparison requires the numerical study ofeqn (12) in which the potential U (z) is calculated using the wavefunction f0(z) (9).

As the consideration of the bulk semiconductors is beyond the scope of this paper, then following theprincipal idea of the -function type potential (18) we further assume that that the parameter q0 is determinedfrom the experiment or numerical calculations. Below we focus on the dependence of the binding energy Eon the displacement of the impurity centre b and the width of the QW d.

For the small distances b for which qb 1 the shift of the binding energy1E can be found in an explicitform. Substituting the solution to eqn (21) for qb 1 in (16) we obtain

1E(b) = h2q(0)2

2[q(0)b]2 (28)

where

= 4q0q(0) sinh(q(0)d)(1 q0d + q(0)d coth(q(0)d)) (29)

and where q(0) is the solution to eqn (21) for b = 0.It is clear from eqn (28) that the shift of the impurity from the mid-point of the QW leads to a decrease

in binding energy. An impurity located at the centre of the QW (b = 0) produces the largest binding energy.This result is confirmed by numerical calculations performed in [9]. The wider the QW, the less the shift1E(b) (28). Under the condition q0d 1 the factor in (28) becomes exponentially small, namely 8 exp(q0d). The shift of the binding energy 1E(b) (28) for the parameters = 5, d/a0 = 4 as a functionof the displacement of the impurity b is given in Fig. 2.

In order for presented analytical results and those obtained numerically [7] to be compared, suitable valuesfor the parameters of the GaAs/Ga1x Alx As (x = 0.25) QW are needed for the case of a strong magneticfield ( = 5) for a well of depth V0 = 34.9R. Thus we take = 0.067m0, = 12.5, a0 = 98.7 andR = 5.83 meV. Figure 3 shows the dependence of the binding energy E upon the width of the well d for


0

0.1

0.2

0.3

0.4

0.1 0.2 0.3 0.4 0.5 0.6

E R

b d2

Fig. 2. The shift of the binding energy 1E scaled to the Rydberg constant R as a function of the displacement of the impurity b for theparameters = 5 and d/a0 = 4.

1.1

1.0

0.9

0.8

2.5 3.0 3.5 4.0 4.5 5.0

Sz.B.B [7]

E R

d a0

Fig. 3. The dependence of the binding energy E of the charged donor positioned at the mid-point of the QW (b = 0) upon the width ofthe well d for the parameter = 5. The variational result obtained in [7] is indicated.

the case of the impurity positioned at the centre of the well (b = 0). This dependence is given by eqn (16)corrected for the case of the finite depth V0, namely

E = h2pi2

2d2

(1 4h

d

2V0

)W.

The binding energy decreases with the increase in the width d of the well. This is in accordance withthe results of the numerical approaches [14, 6, 7]. Nevertheless, the wide quantum well strongly affectsthe binding energy. For the parameter = 5 the binding energy in the well of width d = 5a0 (E =0.856R) substantially exceeds that in the bulk semiconductor (E = 0.591R [7]). Good agreement betweenour analytical calculations and numerical data presented in [7] at d/a0 = 4 is found. Other results obtainedin [7] correspond to the intermediate and narrow QWs (d/a0 2) for which the energies W are positive. A


discrepancy in the binding energy is of the same order as those obtained by the various numerical methods [2,3, 6, 7]. It is therefore concluded that the method developed can be readily extended to the problem of thecharged donor in the presence of electric and strong magnetic fields. In this case the Green function (20)should be replaced by the Green function of the electron in the presence of an electric field in the QW. Inaddition, this approach can be used to study the singlettriplet transitions described theoretically in [9] andfor the charged exciton in the QW.

4. ConclusionIn summary, we have developed an analytical approach to the problem of the charged donor (D) posi-

tioned anywhere in the reasonable wide QW in the presence of a strong magnetic field. The double-adiabaticapproximation is used. The dependencies of the binding energy of the ground singlet state upon the width ofthe QW and the position of the impurity within the well are derived explicitly. It is shown that if the mag-netic field increases in magnitude, the binding energy also increases. With increasing distance between theimpurity centre and the mid-point of the QW the binding energy decreases. The narrowing of the QW leadsto an increase in the binding energy. All these results coincide completely with those obtained numericallyby variational methods. The developed approach can be extended comfortably to the problem of a chargeddonor and an exciton in the QW in the presence of electric and strong magnetic fields.AcknowledgementsWe are grateful to Professor P. A. Braun for useful discussion and Prof. I. V. Komarovand Mr G. B. Turchinivich for help in our calculations.

References[1] S. Huant, S. P. Najda, and B. Etienne, Phys. Rev. Lett. 65, 1486 (1990).[2] T. Pang and S. G. Louie, Phys. Rev. Lett. 65, 1635 (1990).[3] E. R. Mueller, D. M. Larsen, and J. Waldman, Phys. Rev. Lett. 68, 2204 (1992).[4] J. L. Dunn, E. P. Pearl, and C. A. Bates, J. Phys.: Condens. Matter 5, 7815 (1993).[5] S. Chandrasekhar, J. Astrophys. 100, 176 (1944).[6] A. B. Dzyubenko and A. Yu. Sivachenko, Sov. Phys. JETP Lett. 57, 487 (1993).[7] T. Szwacka, J. Blinowski, and J. Betancur, J. Phys.: Condens. Matter 7, 4489 (1995).[8] I. D. Mikhailov, F. J. Betancur, J. H. Marin, and L. E. Oliveira, Phys. Status Solidi (b) 210, 605.[9] C. Riva, V. A. Schweigert, and F. M. Peeters, Phys. Rev. B57, 15392 (1998).

[10] Z. X. Jiang, B. D. McCombe, Jia-Lin Zhu, and W. Schaff, Phys. Rev. B56, R1692 (1997).[11] V. F. Aguekian, B. S. Monozon, C. A. Bates, J. L. Dunn, T. Komatsu, N. Miura, and K. Uchida, Phys.

Rev. B56, 1479 (1997).[12] B. S. Monozon and A. G. Zhilich, Sov. Phys. JETP 73, 1066 (1991).[13] H. Hasegawa and R. E. Howard, J. Phys. Chem. Solids 21, 173 (1961).

IntroductionGeneral theoryFig. 1

Results and discussionFig. 2Fig. 3

ConclusionReferences

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D − centre in a quantum well in the presence of a strong magnetic field