Impurity bound states in a compensated quantum well

Superlattices and Microstructures, Vol. 30, No. 3, 2001 doi:10.1006/spmi.2001.1003 Available online at http://www.idealibrary.com on

Impurity bound states in a compensated quantum well Y. P. VARSHNI† Department of Physics, University of Ottawa, Ottawa, Canada K1N 6N5 (Received 7 July 2001)

A detailed study of the bound-state properties of an impurity in a compensated semiconductor quantum well is presented using the screened potential of a minority impurity ion in a compensated semiconductor due to Schechter (1981). Accurate eigenenergies for the first 15 states are obtained for this potential as a function of the screening parameter λ by numerical integration of the two-dimensional (2D) Schrödinger equation. The energies are found to decrease with increasing values of the screening parameter λ in all cases. The variation of splitting between adjacent levels for the same value of n with the screening parameter is also studied. c 2001 Academic Press

Key words: compensated quantum wells, impurity bound-state energies.

1. Introduction Doped semiconductors have been of considerable interest both from the point of view of their fundamental physics and their technological applications. In recent years it has become possible to make compensated Alx Ga1−x As–GaAs quantum well heterostructures [1] and their properties have been investigated. de Andrada e Silva and da Cunha Lima [2] have analyzed a lightly doped and compensated quantum well within a semiclassical impurity-band model. These authors have used a Monte Carlo simulation to study the single-particle density of states in an n-type Ga1−x Alx /As/GaAs infinite well. They have also investigated the Fermi level, the charge distribution, and the distribution of the electric fields at neutral donors. Emmel, de Andrada e Silva and da Cunha Lima [3] have presented a calculation of the density of states of electrons bound to donor impurities in a lightly doped and compensated quantum well. They have used a quasiclassical treatment suitable for low compensation in which they apply the dipole model. Colchesqui et al. [4] have investigated the macroscopic effect of the compensation on the binding energy of an electron bound to a donor in an n-type quantum well using a semiclassical model which is valid in the low-concentration regime. Emmel and da Cunha Lima [5] have calculated the infrared absorption coefficients for intra-donor transitions inside a lightly doped and compensated quantum well of Ga1−x Alx /As/GaAs assuming a random distribution of shallow impurities. These authors [6] have also examined the effect of compensation on the absorption coefficient for shallow donors in a quantum well by Monte Carlo simulation. In the present paper we have investigated the bound states of a minority impurity in a compensated semiconducting quantum well using the screened potential derived by Schechter [7]. We have calculated accurate eigenenergies for the first 15 states for this potential as a function of the screening parameter λ by numerical † Author to whom correspondence should be addressed. E-mail: [email protected]

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c 2001 Academic Press

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Superlattices and Microstructures, Vol. 30, No. 3, 2001 Table 1: Energy (in effective Rydbergs) of the 1s state as a function of λ

λ

1s

1.0 2.0 3.0 4.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0 75.0 80.0 85.0 90.0

−3.2807 −2.6948 −2.2153 −1.8224 −1.5008 −0.5826 −0.2382 −0.1026 −0.04624 −0.02171 −0.01055 −0.00529 −0.00272 −0.00144 −0.00078 −0.00043 −0.00024 −0.00014 −0.00008 −0.00004 −0.00003 −0.00002

integration of the two-dimensional (2D) Schrödinger equation. We have also studied the variation of splitting between adjacent levels for the same value of n with the screening parameter. The organization of the paper is as follows. In Section 2, a brief outline of the theory is given. The results are given in Section 3 and discussed.

2. Theory Schechter [7] has derived the screened potential of a minority impurity ion in a compensated semiconductor below freeze-out following the theory of Falicov and Cuevas [8]. If N D represents donors and N A acceptors per unit volume, then the compensaton K is given by K = N A /N D .

(1)

At low temperatures and for compensations not too small, the number of carriers is much smaller than the number of acceptors. Some of the donors would be ionized and the released electrons would attach themselves to acceptors, which are thus fully ionized negatively. Thus, below freeze-out, there are practically no electrons. It is energetically favorable for the ionized donors to be clustered around the minority impurity ions, thus screening their field. Schechter [7] obtained the following expression for the screened potential: Z ∞ 2 ea (2 + u 2 )1/2 φ(R) = sin(Ru), (2) du π KR 0 (1 + u 2 )

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Table 2: Energy (in effective Rydbergs) of 2s and 2p states as a function of λ

λ

2s

5p

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.5 3.0 3.5 4.0

−0.3704 −0.3067 −0.2521 −0.2055 −0.1660 −0.1329 −0.1054 −0.08299 −0.06488 −0.05048 −0.03917 −0.03039 −0.02362 −0.01843 −0.01445 −0.01139 −0.00904 −0.00722 −0.00580 −0.00469 −0.00176 −0.00074 −0.00032 −0.00012

−0.3696 −0.3039 −0.2462 −0.1960 −0.1527 −0.1157 −0.08476 −0.05947 −0.03947 −0.02435 −0.01360 −0.00661 −0.00260 −0.00073 −0.00007

where R = ar and a 3 = 8π(N D − N A ). The limiting values of Rφ(R) are as follows [7]: ea lim Rφ(R) = (3) R→0 K √ 2 ea 2 lim Rφ(R) = . (4) R→∞ π K R Close to the impurity the screening is of Debye type, but far from the impurity, as eqn (4) shows, the screened potential decreases much more slowly, as 1/r 2 . Equation (2) is not very convenient for practical applications. Varshni [9] was able to represent it by an equivalent empirical expression which is numerically very close to eqn (2) and also satisfies eqns (3) and (4). Thus the potential energy between the impurity and a hole (or a positron) could then be expressed as √  2  2 e2 a π −0.416R 2(1 − e−0.019R ) V (R) = − e + . (5) π KR 2 R

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Superlattices and Microstructures, Vol. 30, No. 3, 2001 Table 3: Energy (in effective Rydbergs) of 3s, 3p and 3d states as a function of λ

λ

3s

3p

3d

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20

−0.1239 −0.09464 −0.07103 −0.05230 −0.03778 −0.02684 −0.01887 −0.01326 −0.00940 −0.00677 −0.00369 −0.00212 −0.00125 −0.00075 −0.00044 −0.00024 −0.00010

−0.1238 −0.09397 −0.06973 −0.05031 −0.03515 −0.02372 −0.01547 −0.00984 −0.00617 −0.00386 −0.00152 −0.00058 −0.00016

−0.1232 −0.09195 −0.06572 −0.04410 −0.02684 −0.01383 −0.00509 −0.00072

Table 4: Energy (in effective Rydbergs) of 4s, 4p, 4d and 4f states as a function of λ

λ

4s

4p

4d

4f

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.24 0.28 0.32 0.36 0.40 0.44 0.48 0.52

−0.06689 −0.05434 −0.04370 −0.03474 −0.02729 −0.02117 −0.01625 −0.01236 −0.00936 −0.00710 −0.00420 −0.00259 −0.00164 −0.00105 −0.00065 −0.00038 −0.00018 −0.00004

−0.06686 −0.05422 −0.04347 −0.03438 −0.02678 −0.02054 −0.01549 −0.01152 −0.00848 −0.00621 −0.00337 −0.00189 −0.00107 −0.00058 −0.00026 −0.00003

−0.06676 −0.05388 −0.04278 −0.03328 −0.02525 −0.01858 −0.01318 −0.00895 −0.00579 −0.00356 −0.00119 −0.00034 −0.00005

−0.06661 −0.05332 −0.04161 −0.03139 −0.02258 −0.01514 −0.00907 −0.00439 −0.00125

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Table 5: Energy (in effective Rydbergs) of 5s, 5p, 5d, 5f and 5g states as a function of λ

λ

5s

5p

5d

5f

5g

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.24 0.28 0.32 0.36 0.40 0.44

−0.03536 −0.02466 −0.01666 −0.01088 −0.00693 −0.00438 −0.00283 −0.00189 −0.00129 −0.00088 −0.00042 −0.00022 −0.00012 −0.00006 −0.00004 −0.00002

−0.03533 −0.02456 −0.01647 −0.01062 −0.00661 −0.00406 −0.00252 −0.00161 −0.00105 −0.00068 −0.00029 −0.00013 −0.00006 −0.00003

−0.03525 −0.02427 −0.01592 −0.00983 −0.00566 −0.00308 −0.00164 −0.00087 −0.00045 −0.00022 −0.00004

−0.03510 −0.02377 −0.01498 −0.00847 −0.00404 −0.00148 −0.00040

−0.03489 −0.02306 −0.01362 −0.00644 −0.00162

0

Relative Splitting

-0.2

-0.4

-0.6

-0.8

-1

0

0.2

0.4

0.6

0.8 λ

1

1.2

1.4

1.6

Fig. 1. Relative splitting as a function of the screening parameter λ for n = 2 states. The curve represents [E(2 p) − E(2s)]/E(2s).

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Superlattices and Microstructures, Vol. 30, No. 3, 2001 0

Relative Splitting

-0.2

-0.4

-0.6

-0.8

-1

0

0.2

0.4

λ

0.6

0.8

Fig. 2. Relative splitting as a function of the screening parameter λ for n = 3 states. The curves from right to left are in this order: [E(3 p) − E(3s)]/E(3s), [E(3d) − E(3 p)]/E(3 p).

This potential is called the compensated semiconductor impurity (CSI) potential. In terms of r , eqn (5) becomes √  2 2  e2 −0.416ar 2 2 (1 − e−0.019a r ) V (r ) = − e + . (6) Kr π ar Hereafter, in this paper, we shall use atomic units such that the unit of length is a0 = K h¯ 2 /m ∗ e2 , and the unit of energy is effective Rydberg = m ∗ e4 /2K 2 h¯ 2 , where m ∗ is the effective mass. Also we shall use λ = aa0 ; we shall call λ the screening parameter. In these units the CSI potential can be written as √  2 2  1 2 2 (1 − e−0.019λ r ) V (r ) = − e−0.416λr + . (7) r π λr For an impurity in 2D, we have axial symmetry and the angular and center-of-mass coordinate dependence of the wavefunction can be separated off so that the radial part of the 2D Schrödinger equation in polar coordinates (r, θ) can be written as   d 2 R(r ) 1 d R(r ) 2µ h¯ 2 m 2 + + E − V (r ) − R(r ) = 0, (8) r dr 2µ r 2 dr 2 h¯ 2 where µ is the reduced mass of the hole and m is the quantum number specifying the angular momentum of the hole. On substituting p R(r ) = F(r )/ (r ) (9)

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0

Relative Splitting

-0.2

-0.4

-0.6

-0.8

-1

0

0.1

0.2

λ

0.3

0.4

0.5

Fig. 3. Relative splitting as a function of the screening parameter λ for n = 4 states. The curves from right to left are in this order: [E(4 p) − E(4s)]/E(4s), [E(4d) − E(4 p)]/E(4 p), and [E(4 f ) − E(4d)]/E(4d).

one gets  2  m − 1/4 2µ d 2 F(r ) − F(r ) + 2 {E − V (r )} F(r ) = 0. (10) dr 2 r2 h¯ For 2D systems, the quantum number m is analogous to the quantum number ` in the three-dimensional (3D) case, and the levels are oftened labelled in an analogous fashion. We shall follow the same practice here.

3. Results and discussion The Schrödinger equation for potential (7) is, of course, not analytically solvable. It was recognized that if a variational method were to be used for the present problem, many of the integrals will have to be calculated numerically because of the form of V (r ). Further the variational method gives only an upper bound to the energy, and the values obtained for excited states can be liable to large uncertainties. In view of these factors, it was thought best to carry out the necessary calculations by direct numerical integration of the Schrödinger equation. Numerical integration of the Schrödinger equation, if properly implemented and fully tested, can give eigenenergies of high accuracy. The exact eigenenergies for the potential (7) were determined by numerical integration of the Schrödinger equation (10) using Numerov’s method and a logarithmic mesh [10–13]. Calculations were carried out for a large number of values of λ for all m levels for n = 1 to 5. Some representative results are shown for n = 1 to 5 in Tables 1 to 5. It will be noticed from Tables 1 to 5 that the energies decrease with increasing values of the screening parameter λ in all cases. It will also be seen that the ordering of the levels, in general, is quite similar to that

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Superlattices and Microstructures, Vol. 30, No. 3, 2001 0

Relative Splitting

-0.2

-0.4

-0.6

-0.8

-1

0

0.1

0.2 λ

0.3

0.4

Fig. 4. Relative splitting as a function of the screening parameter λ for n = 5 states. The curves from right to left are in this order: [E(5 p) − E(5s)]/E(5s), [E(5d) − E(5 p)]/E(5 p), [E(5 f ) − E(5d)]/E(5d), and [E(5g) − E(5 f )]/E(5 f ).

of such other screened potentials like the screened Coulomb potential [14, 15] and the Hulthén potential [16] in 3D. The behavior of the energy splitting for different m values as λ increases is of interest. We define a quantity, relative splitting, between adjacent levels for the same value of n as follows: Relative splitting = [E(n, m + 1) − E(n, m)]/E(n, m).

(11)

In Figs 1 to 4 we show the relative splittings as a function of λ for n = 2 to 5. It will be noticed from these figures that at low values of λ the levels for different m values are practically degenerate. Also that the relative splitting increases rapidly with increase in λ, and this rate of increase depends on the m value, being greater for larger m. Acknowledgements—This work was supported in part by a research grant from the Natural Sciences and Engineering Research Council of Canada to the author.

References [1] R. W. Kaliski, N. Holonyak, Jr, K. C. Hsieh, D. W. Nam, R. D. Burnham, J. E. Epler, R. L. Thornton, and T. L. Paoli, Appl. Phys. Lett. 49, 1390 (1986). [2] E. A. de Andrada e Silva and I. C. da Cunha Lima, Phys. Rev. B39, 10101 (1989); Mod. Phys. Lett. B3, 815 (1989). [3] P. D. Emmel, E. A. de Andrada e Silva, and I. C. da Cunha Lima, Phys. Rev. B40, 3394 (1989).

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[4] B. C. F. Colchesqui, P. D. Emmel, E. A. de Andrada e Silva, and I. C. da Cunha Lima, Phys. Rev. B40, 12513 (1989). [5] P. D. Emmel and I. C. da Cunha Lima, Mater. Sci. Forum 65-66, 129 (1990). [6] P. D. Emmel and I. C. da Cunha Lima, Solid State Commun. 79, 431 (1991). [7] D. Schechter, Phys. Rev. B24, 3610 (1981). [8] L. M. Falicov and M. Cuevas, Phys. Rev. 164, 1025 (1967). [9] Y. P. Varshni, Phys. Rev. B48, 10870 (1993). [10] C. Froese, Can. J. Phys. 41, 1895 (1963). [11] F. Y. Hajj, H. Kobeisse, and N. R. Nassif, J. Comput. Phys. 16, 150 (1974). [12] C. Froese Fischer, The Hartree-Fock Method for Atoms (John Wiley, New York, 1977). [13] S. E. Koonin and D. C. Meredith, Computational Physics (FORTRAN version) (Addison-Wesley, New York, 1990). [14] F. J. Rogers, H. C. Graboske, Jr, and D. J. Harwood, Phys. Rev. A1, 1577 (1970). [15] D. Singh and Y. P. Varshni, Phys. Rev. A29, 2895 (1984). [16] Y. P. Varshni, Phys. Rev. A41, 4682 (1990).