Perturbation calculation of donor states in a spherical quantum dot

PII: Solid-State Electronics Vol. 42, No. 9, pp. 1661±1663, 1998 # Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0038-1101/98/$ - see front matter S0038-1101(98)00126-9 PERTURBATION CALCULATION OF DONOR STATES IN A SPHERICAL QUANTUM DOT C. BOSE1 and C. K. SARKAR2 1 Department of Electronics and Telecommunication Engineering, Jadavpur University, Calcutta 700 032, India 2 Department of Physics, B.E. College (D.U.), Shibpur, Howrah 711 103, India (Received 19 August 1997; in revised form 2 February 1998) AbstractÐAn attempt is made to derive the donor states associated with the ground and a few higher levels in spherical quantum dot (QD) using the perturbation method. The binding energies are computed for GaAs QD as functions of the dot size and the impurity position. The results show that the impurity binding energies decrease with the increase in dot dimension. The binding energies are also found to show a strong dependence on the location of the impurity within the dot. For s-levels, the donor binding energy is maximum for an on-centre impurity, while for p and d-levels, the binding energy maxima occur for a donor located o€ the dot centre. # 1998 Published by Elsevier Science Ltd. All rights reserved 1. INTRODUCTION spherical QD can be expressed as The presence of impurities in semiconductors introduces bound states in the forbidden gap which strongly a€ect both the optical and transport properties of semiconductors. Therefore, the understanding of impurity states in semiconductors is of utmost importance. Although the impurity states within bulk semiconductors had been exhaustively studied, the same within the quantum well (QW) structures was initiated only in the last decade[1±4]. Recently similar studies have been extended to quasi-one-dimensional and quasi-zero-dimensional structures like quantum well wires (QWWs)[5,6] and quantum dots (QDs), respectively[7,8]. The e€ect of the geometry of QDs[9] and that of the cross-sectional forms of QWWs[10] on the impurity binding energy have also been investigated. However, the investigators have so far only concentrated their interests on the impurity ground state, using mostly variational techniques in the e€ective mass approximation. In the present analysis, we employ a perturbation method to calculate the donor states associated with few higher-lying levels, including the lowest one in a spherical QD having a square potential well of in®nite barrier. The impurity binding energies are theoretically estimated for QD of wide-gap semiconductors and computed for GaAs, a typical representative. 2. THEORY In the e€ective mass approximation, the Hamiltonian of a single hydrogenic impurity in a Hˆ p2 e2 ‡ V r† ÿ 2m* 4pEjr ÿ ri j 1† where e and m* are, respectively, the electronic charge and e€ective mass, p is the momentum, E is the dielectric constant of the dot material and ri gives the location of the impurity with respect to the centre of the dot. V(r) describes the potential pro®le, i.e. V(r) is zero for r < R and in®nite for r>R, R being the radius of the dot. The last term in Equation (1) is the Coulomb interaction term due to the hydrogenic impurity and acts as a perturbation over the original Hamiltonian, consisting of the remaining terms of the same equation. The eigenfunction of the Hamiltonian in the absence of the impurity is given by cnlm ˆ fnl r†Ylm y, f† 2† where fnl(r) gives the radial part and Ylm(y, f) gives the angle dependent part. n, l and m are the principal, angular momentum and magnetic quantum numbers, respectively. fnl(r) for a spherical QD with square well potential is given by r 1 2 Jl‡1=2 knl r† fnl r† ˆ 3† R r rJl‡3=2 knl R† Jg, being the Bessel function of order g. The binding energy of the hydrogenic impurity is de®ned as the di€erence between the energy states without and with the impurity present, for a particular level. Thus, the impurity binding energies are given by the correction term, obtained from the per- 1661 1662 C. Bose and C. K. Sarakr turbation method, as  1† DE nlm ˆ cnlm * ÿ e2 c 4pEjr ÿ ri j nlm  4† The term 1/v r ÿ ri v can be expanded in spherical harmonics as mˆ‡l 1 X 1 4p rl< X ˆ Ylm * y 0 , f 0 †Ylm y, f† l‡1 jr ÿ ri j lˆ0 2l ‡ 1 r> mˆÿl 5† where r< (r>) is the smaller (larger) of r and ri. From the above expression we get 1† DE nlm ˆÿ e2 4pE R f 0 2 nl r†Ilm r2 dr 6† where Fig. 1. The binding energy of the four lowest donor states as a function of the donor position within an in®nite barrier GaAs quantum dot of radius 5 nm 1 ‡l X 4p rl< X Ilm ˆ Ylm * y 0 , f 0 † 2l ‡ 1 rl‡1 > mˆÿl lˆ0 r 2l ‡ 1† 2l ‡ 1† 2l ‡ 1†  ÿ1† 4p    l l l l l l  0 0 0 ÿm m m  ÿm† 7† where  l ÿm l m l m  and  l l 0 0 l 0  in Equation (7) are the 3ÿj symbols. The properties of 3ÿj symbols and spherical harmonics ®nally yield Ilm in the present problem as 2l X 4p rl< Ilm ˆ Yl0 y 0 , f 0 † 2l ‡ 1 rl‡1 > lˆ0 r 2l ‡ 1† 2l ‡ 1† 2l ‡ 1†  ÿ1† 4p    l l l l l l  0 0 0 ÿm 0 m  even for spherical dots of radius larger than twice the Bohr radius[12]. The material parameters used here for GaAs yield the Bohr radius as 104 AÊ. Accordingly, the perturbation calculation is suciently accurate even for dot radius 0200 AÊ. In the present analysis we have, therefore, restricted our calculation to QD of radius 200 AÊ. In Figs 1±3, the donor binding energies for several excited levels along with the ground level are plotted as a function of impurity position and dot size. The binding energies in these ®gures are expressed in terms of Rydberg's constant Ry, which is 5.24 meV for donors (electrons) in GaAs. In Fig. 1 we present the binding energies associated with the four lowest states of the QD as a function of the donor position. In the case of s-states (i.e. n00 states), the binding energy is maxi- ÿm† 8† where Yl0 y 0 , f 0 † is the average value of Yl0(y', f'). Equation (6) together with Equations (3) and (8) give the donor states bound to various excited levels in a spherical QD with square well potential. 3. RESULTS AND DISCUSSION To compute the derived impurity binding energies, we use the material parameters for GaAs[11]. For the perturbation calculation to be valid, the size quantization energy should be greater than the Coulomb interaction between the electron and the impurity. This implies that the radius of the QD must be suciently smaller than the Bohr radius (RB). However, the estimation shows that the perturbation calculation yields error of only about l%, Fig. 2. The splitting of 1p and 1d donor states as a function of the donor position within an in®nite barrier GaAs quantum dot of radius 5 nm Donor states in a spherical quantum dot Fig. 3. The binding energy of the two lowest donor states as a function of the radius of an in®nite barrier GaAs quantum dot for the impurity position ri=0 (solid curves) and 2 nm (broken curves). The curves (a) and (b) correspond, respectively, to the 100 and 110 states mum for an on-centre impurity and decreases sharply as the impurity moves away from the dot centre. It can be seen from Fig. 1 that for n = 2 the binding energy assumes higher values in the vicinity of the dot centre, but falls o€ more rapidly compared to that for n = 1. The binding energies for 1p-state (i.e. 110 state) and 1d-state (i.e. 120 state) are seen to be relatively less sensitive to donor position than that for s-states. For both 1p- and 1dstate, the binding energy initially increases, reaches a maximum and subsequently decreases as the impurity position is allowed to change from the centre to the edge of the dot. It can be further noted that the maximum for the 1d-state is less prominent and is located farther away from the centre of the dot as compared to that of the 1p-state. The observed variations can be attributed to the nature of the wavefunctions corresponding to the s-, pand d-states. In the QD, the p-states are triply degenerate whereas the d-states are 5-fold degenerate. The Coulomb interaction, however, lifts this degeneracy due to which the donor state corresponding to 1pstate splits into two levels and that corresponding to 1d-state splits into three levels. These splittings are shown in Fig. 2. The observed variations indicate that for states with m = l (i.e. for 111 and 122 states in Fig. 2), the binding energies assume their maximum values for a centre-doped dot. On the other hand, the states corresponding to l$ m (i.e. 110, 120 and 121 states) are found to possess maximum binding energies for impurity located somewhere between the centre and edge of the dot, the location and the prominence of the maximum depending on the particular value of l and m. 1663 Figure 3 illustrates the binding energies of the two lowest states in a spherical GaAs QD as a function of the dot radius for di€erent impurity positions. In a QD with the impurity located centrally, the binding energies for both the s- and p-states decrease as the dot radius increases. For an o€centre impurity, the variation in binding energy with radius becomes ¯attened for both s- and pstates with the plot for p-state exhibiting a maximum slightly displaced from R = ri. For R = ri, the impurity is at the edge of the dot for which the binding energy is minimum. However, as R increases the impurity e€ectively shifts towards the dot centre, indicating an increase in the binding energy. An increase in dot radius will also mean a spread of the wavefunction which will bring down the binding energy. These two opposing e€ects dictate the variation in the binding energy for the o€centre impurity, whereas, only the last term is relevant for the, on-centre impurity. The above considerations explain the variations observed in Fig. 3. 4. CONCLUSION In conclusion, we have calculated the binding energies for the ground-state and a few adjacent excited states for a shallow hydrogenic donor in spherical in®nite barrier GaAs QDs with squarewell potential, following a perturbation method. The computed result shows that the impurity binding energy is quite sensitive to the impurity position. The binding energies of the p- and d-levels are shown to split as the location of the donor is varied within the QD. The strong dependence of the di€erent donor states on the dot size is also established. AcknowledgementsÐThe work is ®nancially supported by the UGC and the CSIR, India. The authors are grateful to Dr M. K. Bose for valuable suggestions. REFERENCES 1. Bastard, G., Phys. Rev. B, 1981, 24, 4714. 2. Bastard, G., Surf. Sci., 1982, 113, 165. 3. Masselink, W. T., Chang, Y. C. and Morkoc, H., Phys. Rev. B, 1983, 28, 7373. 4. Oliveira, L. E. and Falicov, I. M., Phys. Rev. B, 1986, 34, 8676. 5. Brown, J. and Spector, N., J. Appl. Phys., 1986, 59, 1179. 6. Weber, G., Schulz, P. A. and Oliveira, L. E., Phys. Rev. B, 1988, 38, 2179. 7. Porras-Montenegro, N. and Perez-Merchancano, S. T., Phys. Rev. B, 1992, 46, 9780. 8. Porras-Montenegro, N., Perez-Merchancano, S. T. and Latge, A., J. Appl. Phys., 1993, 74, 7624. 9. Ribeiro, F. J. and Latge, A., Phys. Rev. B, 1994, 50, 4913. 10. Bryant, G. W., Phys. Rev. B, 1985, 31, 7812. 11. Adachi, S., J. Appl. Phys., 1985, 58, R±1. 12. Ekimov, A. I., Kudryavtsev, I. A., Ivanov, M. G. and Efros, A. L., Sov. Phys. Solid State, 1989, 31(8), 1385.