- Email: [email protected]

Physica B 348 (2004) 66–72

Density of electronic states in impurity-doped quantum well wires J. Sierra-Ortegaa,b,*, I.D. Mikhailovc, F.J. Betancurc a

Department of Physics and Astronomy, and CMSS Program, Ohio University, Athens, OH 45701, USA b Departamento de F!ısica, Universidad del Magdalena, A.A. 731 Santa Marta, Colombia c Departamento de F!ısica, Universidad Industrial de Santander, A.A. 678 Bucaramanga, Colombia Received 22 July 2003; received in revised form 18 September 2003; accepted 18 November 2003

Abstract We analyze the electronic states in the cylindrical quantum well-wire (QWW) with randomly distributed neutral, D0 and negatively charged D donors. In order to calculate the ground state energies of the off-center donors D0 and D as a function of the distance from the axis of QWW we use the recently developed fractal dimension method in which the problems are reduced to a similar one for a hydrogen-like atom and a negative-hydrogen-like ion respectively, in an isotropic effective space with variable fractional dimension. The numerical trigonometric sweep method and the threeparameter Hylleraas trial function are used to solve these problems. We present a novel curves for the density of impurity states in a cylindrical QWWs with square-well, parabolic and soft-edge barrier potentials. Additionally we analyze the effect of the repulsive core on the density of the impurity states. r 2003 Published by Elsevier B.V. PACS: 73.20.Dx; 73.20.Hb Keywords: Semiconductor nanostructures; Quantum wires; Fractal dimension; Shallow donor; Binding energy and density of the states

1. Introduction The spectra of donors conﬁned in quantum-well wires (QWWs) have drawn much attention from theorists in the last two decades [1–10]. Initially, the effects of the cross-sectional form [1] and the donor position [2] on the neutral donors (D0 ) binding energies in QWWs with inﬁnite barrier height have been studied by using the variational *Corresponding author. Departmento de F!ısica, Universidad del Magdalena, A.A. 731 Santa Marta, Colombia. Tel.: +57-76323095; fax: +57-7-6323477. 0921-4526/$ - see front matter r 2003 Published by Elsevier B.V. doi:10.1016/j.physb.2003.11.072

method. Later on, these calculations have been extended to the model with ﬁnite barrier [3–9] for which the effects of the donor position [3], dielectric mismatch [7] and the magnetic ﬁeld [8,9] on the donor binding energy, the density of the impurity states [5,6] and the probability of infrared transitions [5–7] have been analyzed. An alternative fractional-dimensional method to analyze excitonic spectra of semiconductor quantum wells (QWs), quantum well-wires (QWWs) and superlattices (SLs), in which a real anisotropic environment is considered as it were isotropic with reduced non-integer dimension, has been proposed

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by Lefebvre with co-investigators [10]. As it was shown by Oliveira and collaborators [11] this approach also provides an elegant and convenient way to calculate the donor binding energies in QWs, QWWs, SLs and quantum dots (QDs). Recently, to reveal the nature of the fractional dimension we propose a variational procedure that reduces the wave equation for donors placed in anisotropic media to one for hydrogenic atom in an effective isotropic space with the Jacobian nonpower dependence on the electron–ion separation [12]. We ﬁnd that this procedure which we call fractal-dimensional method can be considered as reﬁnement of the approach developed in the papers [10,11] and it provides a simple and efﬁcient algorithm for calculating the binding energies of on- and off-center neutral and negatively charged donors in QWs, QWWs and QDs with a high accuracy [12]. One of the advantages of our method consists in that it gives a uniﬁed formulation for the donor spectra suitable for heterostructures with arbitrary geometry and different conﬁning potential shape. In this work we apply the fractal-dimension method to study the off-center D0 and D ground state binding energies and the density of the impurity states for QWWs with different conﬁning potential shape. We present the calculation results for a cylindrical GaAs/Ga0.7Al0.3As QWW with square-well, parabolic, soft-edge-barrier, and repulsive core potentials.

2. Theory Within the framework of the effective-mass theory, the dimensionless Hamiltonian of a D ion S-states in a QWW can be written as 2 H# ¼ H# 0 ðr1 ; Z ¼ 1Þ þ H# 0 ðr2 ; Z ¼ 1Þ þ ; r12 2Z 2 ; i ¼ 1; 2; ð1Þ H# 0 ðri ; ZÞ ¼ ri þ V ðri Þ j r i fj where, the parameter Z is equal to zero for the free electron and one for the donor, f and ri ¼ ðqi ; zi Þ; ði ¼ 1; 2Þ designate the 3D-vectors of the ion and the electrons 1 and 2 positions, respectively. The vectors f and ri are measured from the

67

center of the heterostructure, 2=r12 is the interaction between the two electrons, r12 ¼ jr1 r2 j and V ðrÞ is some conﬁning potential. The effective Bohr radius a0 ¼ e_2 =m e2 ; the effective Rydberg Ry ¼ e2 =2a0 e have been taken as units of length and energy, respectively. The ground state wave functions f0 ðqÞ for a free electron, CD0 ðrÞ for a neutral donor and CD ðr1 ; r2 Þ for a negatively charged donor, which correspond to the lowest energies E0 ; EðD0 Þ and EðD Þ in QWW, can be found as solution of the respective eigenvalue problems: H# 0 ðr; Z ¼ 0Þf0 ðqÞ ¼ E0 f0 ðqÞ;

ð2aÞ

H# 0 ðr; Z ¼ 1ÞCD0 ðrÞ ¼ EðD0 ÞCD0 ðrÞ;

ð2bÞ

# D ðr1 ; r2 Þ ¼ EðD ÞCD ðr1 ; r2 Þ: HC

ð2cÞ

Once all energies, E0 ; EðD0 Þ and EðD Þ are calculated, the binding energies Eb ðD0 Þ and Eb ðD Þ for these donors may be obtained in a straightforward way through: Eb ðD0 Þ ¼ E0 EðD0 Þ; Eb ðD Þ ¼ E0 þ EðD0 Þ EðD Þ:

ð3Þ

As the wave equation (2a) for the free electron ground state energy E0 and its wave function f0 ðqÞ is separable it can be solved numerically by using the trigonometric sweep method [13]. To solve Eq. (2b) we assume that in the system of reference with the origin placed in the donor position, the ground state off-center D0 trial function can be presented as CD0 ðrÞ ¼ f0 ðq þ fr ÞFD0 ðrÞ;

ð4Þ

where, the correlation function FD0 ðrÞ depends only on the electron–ion separation. Considering the function FD0 ðrÞ as a variational we carry out the following Euler–Lagrange equation by using the variational principle [12]: 1 d dF 0 ðrÞ 2 JðrÞ D FD0 ðrÞ JðrÞ dr dr r ð5Þ ¼ ½E E0 FD0 ðrÞ;

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J. Sierra-Ortega et al. / Physica B 348 (2004) 66–72

where the Jacobian JðrÞ is given by

to the parameters a; mina;b;ZAð0;1Þ EðD ; a; b; ZÞ:

JðrÞ ¼ 4pr2 PðrÞ; PðrÞ ¼ Z p qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z 1 2p r2 sin2 y þ z2r þ 2rzr sin y cos j sin y dy: dj f02 4p 0 0

ð6Þ Here zr is the distance up to axis from the donor position and the function PðrÞ gives the density probability of ﬁnding electron over a sphere of radius r centered at donor location. Eq. (5) . coincides with the Schrodinger equation for a hydrogen atom in an effective space with the locally variable dimension [12]. Once the function f0 ðqÞ is found, the Jacobian may then be calculated in a straightforward way through (6). We calculate the values of the Jacobian in 1000 nodes of a grid and store them in computer memory to evaluate afterwards the Jacobian in any point by using interpolating cubic spline. Finally to deﬁne the donor energy we solve the wave Eq. (5) by means the trigonometric sweep method [13]. A similar procedure can be also used for calculating the ground state binding energies of negatively charged donors in different heterostructures, by choosing a variational function of the form: CD ðr1 ; r2 Þ ¼ f0 ðq1 Þf0 ðq2 ÞFD ðjr1 fj; jr2 fj; r12 Þ; ð7Þ where, the envelope function FD= takes into account the effect of the correlation between the electrons within the heterostructure produced by the Coulomb interaction. To estimate the D ground state energy we use a simple threeparameter Hylleraas trial function [14]: FD ðr1 ; r2 ; r12 Þ ¼ eas ð1 þ bt2 þ ZuÞ;

ð8Þ

where, a; b and Z are variational parameters and s ¼ r1 þ r2 ; t ¼ r1 r2 ; u ¼ r12 are the coordinates of Hylleraas. By using the well-known procedure of calculation of the multiple integrals in coordinates of Hylleraas [14] we obtain an explicit expression for the variational energy [12] as a function of the parameters a; b and Z: The variational D ground state energy EðD Þ can be found by minimizing this energy with respect

b

and

Z:EðD Þ ¼

3. Model of the conﬁnement To describe the conﬁnement potential in a GaAs/Ga1xAlxAs QWW we assume that the variation of the Al concentration, x along the radius r is determined by a smoothly varying function, xðrÞ and the conﬁnement potential is given by Cen and Bajaj [15] V ðrÞ ¼ 0:6 1:36xðrÞ þ 0:22x2 ðrÞ ðeV Þ: If we additionally assume that at the axis of QWW there is a cylindrical island of the radius Ri with locally increased Al concentration Al substituting Ga embedded in a cylindrical GaAs matrix of the radius Re then the conﬁnement potential for such heterostructure can be described by means of the following relation: V ðrÞ ¼ Vi Wðr; Ri ; W Þ þ Ve Wðr; Re ; W Þ;

ð9Þ

where Ri and Vi are the radius and the potential of the repulsive core; Re and Ve are the exterior radius and the barrier height of the QWW and the non-abrupt version of a step function deﬁned as

8 > 0; zoz0 W ; > <

2 2 2 Wðz; z0 ; W Þ ¼ ðz z0 Þ =W 1 ; z0 W pzoz0 ; > > : 1; zXz0 :

ð10Þ The values of Vi and Ve depend on the Al concentrations in the repulsive core and in the barrier and the parameter W can be associated with thickness of the transition region in the junctions. One can analyze different potential shapes by varying in Eq. (9) the barrier heights Vi and Ve ; radii Re and Ri and the thickness of the transition regions W :

4. Results and discussion In Fig. 1(a) we display the transverse conﬁnement potential V ðrÞ without repulsive core ðVi ¼ 0Þ given by expression (9) for a cylindrical QWWs with the exterior radius Re ¼ 1a0 and the barrier

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40

square-well * W=0.01a0

Re=1a0* Ve=40Ry*, Vi=0

parabolic bottom * W=1a0 Vi=0; Ve=40Ry

Eb / Ry*

V(r)/Ry*

30

20

*

*

Re=1a0

D

0

0 0.2

0.4

(a)

0.6 r/a0*

V(r)/Ry*

1.0

1.0

0.5

parabolic bottom W=0.4a0*

1.5

2.0

ξp / a0*

2

Ve=40Ry*, Vi=20Ry* Re=1a0*, Ri=0.2a0*

0

D

soft-edge barrier W=0.1 a0*

20

0.0

(a) square-well W=0.01a0*

40 Vi=20Ry*; Ve=40Ry* Ri=0.2a0*; Re=1a0*

0.8

Eb / Ry*

0.0

W=0.01a0* W=0.4a0* 1

10

D

0

0 0.0

(b)

W=0.01a0* W=0.3a0* W=1a0*

0

D

2

soft-edge-barrier * W=0.3a0

10

30

69

0.2

0.4

0.6 r/a *

0.8

1.0

0

0

(b)

1

2

3

ξp / a0*

Fig. 1. Electron conﬁnement potential in GaAs/Ga0.7 Al0.3As QWW with different thickness of the transition region W without (a) and with (b) repulsive core obtained in according to Eq. (9).

Fig. 2. D0 and D ground state binding energy as a function of the distance from the donor position to axis of a GaAs/ Ga0.7Al0.3As cylindrical QWW without (a) and with (b) repulsive core and with different transition region thickness W :

height Ve ¼ 40Ry for three different values of W corresponding to the square-well ðW ¼ 0:01a0 Þ; soft-edge ðW ¼ 0:3a0 Þ and parabolic ðW ¼ 1:0a0 Þ barrier proﬁles. Similar curves for a GaAs/ Ga0.7Al0.3As QWW with the repulsive core of the radius Ri ¼ 0:2a0 and interior barrier height Vi ¼ 20Ry ; with W ¼ 0:01a0 ; 0:1a0 and 0:4a0 are displayed in Fig. (1b). In Fig. 2 we show the calculation results of the off-center D0 and D ground state binding energy as a function of the distance from the donor position to axis for a GaAs/Ga0.7Al0.3As cylindrical QWW without (Fig. 2a) and with (Fig. 2b) repulsive core whose conﬁnement potentials are presented in Fig. 1a and b, respectively. It is seen that the energies for different shapes of potential differ signiﬁcantly only as the donor is close to the

axis and the difference between energies diminishes quickly as the donor approaches to the barrier. In all cases the binding energies decrease when the donor center is displaced from the axis of QWW. The reason for this is that the electron tends even in the case off-center donor position, to be localized at the axis of QWW due to the lateral strong conﬁnement, and the electron–donor separation increases with the increase of zr : Therefore as the donor position falls outside the limits of QWW, the binding energy is equal approximately to energy of the electron–donor electrostatic interaction. By using a simple electrostatic model, the uncertainty relation and the virial theorem one 0 can estimate, pﬃﬃﬃ for example the binding energy of D as Eb E 2=zr ðRy Þ: This relation is in excellent agreement for all donor positions in the barrier

ARTICLE IN PRESS J. Sierra-Ortega et al. / Physica B 348 (2004) 66–72

r

To calculate function (9) we use the inverse cubic interpolation for the curves E ¼ Eb ðzr Þ presented in Fig. 2 and the calculation results for the densities of the impurity states in relative units are shown in Figs. 3 and 4. In Fig. (3a) we present the densities of the impurity states for neutral donor in GaAs/ Ga0.7Al0.3As cylindrical QWWs with different

1.0

g(E) (1/ Ry*)

with the curves presented in Fig. (2a). Similar results are displayed in Fig. (2b), for the QWW model with repulsive core of radius 0:2a0 and the repulsive potential (20Ry) corresponding approximately to the 15% of the Al concentration at the axis of QWW. It is seen that for this model the behavior of the curves for the donor positions close to the axis of QWW is similar to the curves in Fig. (2a) with only essential difference, the donor binding energy in this region very weakly depends on the donor position, in the region zr o0:5a0 for D0 it is almost constant and for D a weakly pronounced maximum is observed. As we will show below this behavior of the binding energy curve provides an additional singularity in the density of the impurity states. Besides, one can see that for large distances zr the donor binding energy in QWWs with repulsive core diminishes slower than corresponding energies in Fig. (2a). It is due to the fact that the repulsive core replaces the electron location in QWW from the axis to the barrier reducing a distance between electron in well and a donor in the barrier. Assuming that the circular cross-section of the QWW is not too small we can consider the distribution of the impurities as homogeneous and ﬁnd the number of donors dN located within the ring of the radii zr and zr þ dzr as dN ¼ 1 2npzr dzr ¼ 2npzr dEðzr Þ=dzr dE; where n is the density of the impurity per unit area of the cross section and E ¼ Eb ðzr Þ is the donor binding energy which depends on the donor position. The density of the impurity states per unit binding energy is given by dzr ðEb Þ gðEb Þ ¼ 2pnzr ðEb Þ dEb dEb ðzr Þ1 : ¼ 2pnzr ðEb Þ ð11Þ dz

Re=1a0* Ve=40Ry*; Vi=0 W=0.01a0*

0.5

W=0.3a0*

0

D

W=1a0* 0.0 2

3

(a)

4

Eb / Ry* Re=1a0*; Ri=0.2a0* Ve=40Ry*; Vi=20Ry*

3

g(E) (1/ Ry*)

70

W=0.01a0*

2

W=0.4a0* 0

D

1

0 1.0

(b)

1.5

2.0

2.5

Eb / Ry*

Fig. 3. Density of the D0 states in a GaAs/Ga0.7Al0.3As cylindrical QWW with different shapes of the potential without (a) and with (b) repulsive core

shapes of conﬁnement potential. The right-side of our curve for a square-well potential, is in a good agreement with the results obtained previously by Porras-Montenegro et al. [5]. The leftside of our curves are different from those obtained previously by other authors because of we include in our calculations all positions of the impurities in the barrier which provide the left-side tail that tends to inﬁnity as Eb -0: One can observe in Fig. 3(a) a successive displacement of the right-side threshold of the density of the impurity states to larger energies with increasing of the transition region thickness which transforms the shape of the potential from square-well to parabolic bottom. The curves of the density of the impurity states for the model with repulsive core are presented in Fig. 3b. It is seen that the repulsive core provides essential modiﬁcation of the curves of the density of the impurity

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Ve=40Ry*; Vi=0 R0=1a0*

g(E) (1/ Ry*)

8 6

D W=0.3a0*

4

W=0.01a0*

2

W=1a0* 0

(a)

0.2

0.4

0.6

0.8

Eb / Ry*

g(E) (1/ Ry*)

12

Re=1a0*; Ri=0.2a0* Ve=40Ry*; Vi=20Ry* W=0.4a0*

71

very pronounced peak due to existence an additional maximum in the above mentioned dependence when the donor is located between the repulsive core and the barrier (see Fig. 2b). Similar effects have been found in Ref. [16] under a strong electric ﬁeld. In conclusion, we have presented a simple method to calculate the ground state binding energies for D0 and D off-center donors in QWWs, which provides a uniﬁed formulation of the problem for any conﬁnement potential shape. By using this procedure we calculate the binding energies and density of the impurity states for D0 and D in QWWs with square-well, soft-edgebarrier, and parabolic bottom QWWs with and without repulsive core.

8

D

Acknowledgements 4

W=0.01a0* 0 0.2

(b)

0.4

0.6

Eb / Ry* Fig. 4. Similar to Fig. 3, except for D :

states related to the appearance of the singularities and the ﬁne structure close to the right-side threshold. These singularities are due to the unusual behavior of the binding energy dependence on the donor position as it can be seen from Fig. 2b, where the essential part of the corresponding curves is nearly horizontal and the value of the derivative dEðzr Þ=dzr is close to zero. In Fig. 4 we present the densities of the impurity states for D in GaAs/Ga0.7Al0.3As cylindrical QWWs. One can observe more pronounced ﬁne structure in the curves of the density of the impurity states for D in comparison with the similar curves for D0 (except the parabolic bottom potential case). All curves of the D impurity states have a singularity in the region of the righthand threshold due to the maximum in the D binding energy dependence on the donor position distance from the axis (see curves in Fig. 2). In the curves for QWW with repulsive core appears a

We would like to thank S. Ulloa for stimulating discussions about our method. One of us, J. S-O., expresses his sincere thanks to the Department of Physic and Astronomy of Ohio University and to all of the members in the S. Ulloa’s group for their kind hospitality during his stay in Athens. This work was partially ﬁnanced by the Universidad Industrial de Santander (UIS), through the General Researches (DIF) code No. (5111).

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