# Energy levels of a parabolically confined quantum dot

in the presence of spin-orbit interaction

###### Abstract

We present a theoretical study of the energy levels in a parabolically confined quantum dot in the presence of the Rashba spin-orbit interaction (SOI). The features of some low-lying states in various strengths of the SOI are examined at finite magnetic fields. The presence of a magnetic field enhances the possibility of the spin polarization and the SOI leads to different energy dependence on magnetic fields applied. Furthermore, in high magnetic fields, the spectra of low-lying states show basic features of Fock-Darwin levels as well as Landau levels.

###### pacs:

72.15.Lh,71.70.Ej,73.63.Hs## I introduction

The state-of-the-art material engineering and nano-fabrication techniques have made it possible to realize advanced semiconductor devices at atomic scales, such as quantum dots in which the electron motion along all directions is quantized and conducting electrons are confined within the nanometer distances. In such a system, few electrons are confined within a point-like structure so that it can behave as an artificial atom and, consequently, be used as electronic and optical devices such as memory chip Thornton , quantum computer Loss ; Benjamin ; Lent ; Tougaw , quantum cryptography Molotkov , quantum-dot laser Saito , etc. In recent years, the electronic, transport, optical and optoelectronic properties of spin-degenerate quantum dots have been intensively investigated.

On the other hand, the progress made in realizing spin polarized electronic systems on the basis of diluted magnetic semiconductors and narrow-gap semiconductor nanostructures has opened up a field of spin-electronics (or spintronics). As has been pointed out in a good review edited by Wolf and Awaschalom Wolf , due to unique nature of the SOI in electronic materials, quantum transport of electrons in a spin polarized system differs fundamentally from that in a spin-degenerate device. Thus, the spin-dependent effects can offer new mechanisms and schemes for information storage and for increasing the speed of data processing. As a result, the electronic devices, such as spin-transistor FET , spin-waveguide wang , spin-filter Tko , etc., have been proposed. Moreover, optical methods for injection, modulation and detection of spin polarized electrons will eventually become the target for the development of spin polarized nano-optoelectronic devices. At present, one of the major challenges for the application of the spintronic systems as working devices is to optimize the spin life times of the carriers in the devices. It has been realized that in contrast to the diluted magnetic semiconductors in which the SOI is induced by the presence of an external magnetic field, spin splitting of carriers can be achieved in narrow-gap semiconductor nanostructures even in the absence of the magnetic field. Experimental data have indicated that in narrow-gap semiconductor based quantum wells, such as in InAlAs/InGaAs heterostructures, the higher-than-usual zero-magnetic-field spin splitting (or spontaneous spin splitting) can be realized by the inversion asymmetry of the microscopic confining potential due to the presence of the heterojunction scha . This kind of inversion asymmetry corresponds to an inhomogeneous surface electric field and, hence, this kind of spin-splitting is electrically equivalent to the Rashba spin-splitting or Rashba effect Rashba . From the fact that InGaAs-based quantum dots are normally fabricated using InAlAs/InGaAs heterojunctions, one would expect that the Rashba spin splitting can be observed in these quantum dot systems and spin polarized quantum dots can therefore be achieved. Practically, the spin-split quantum dots can be made from, e.g., an InAlAs/InGaAs heterostructure with a negative bias applied to the side gate Tarucha .

In contrast to a spin-split quantum well structure in which the Rashba effect can be easily identified by, e.g., the magnetotransport experiments via measuring the Shubnikov-de Hass (SdH) oscillations scha ; Grundler , the spintronic effects in a quantum dot cannot be easily observed using conventional transport measurements. Normally, optical measurements, such as optical absorption and transmission bnm , cyclotron-resonance Krahne , etc., can be used to determine the energy spectrum of a quantum dot. Although recently there are some theoretical work published regarding the Rashba effect in quantum dots in the presence of magnetic fields prl ; Bulgakov ; Reed , the effect of the SOI on energy spectrum of a quantum dot has not yet been fully analyzed. It should be noted that in Ref. prl, , the SOI was taken as a perturbation. In order to understand how SOI affects the energy levels of a parabolically confined quantum dot, we feel that more theoretical work is needed and it is the prime motivation of the present work. In this paper, we present a tractable approach to calculate energy spectrum of an InGaAs-based quantum dots with the inclusion of SOI induced by the Rashba effect. We would like to examine the important and interesting consequences such as the exchange of the energy states due to SOI and the enhancement of the spin polarization by the presence of the magnetic fields.

In Section II the theoretical approach is developed in calculating the electronic subband structure of a parabolically confined quantum dot in the presence of SOI and a magnetic field. The numerical results are presented and discussed in Section III and the conclusions obtained from this work are summarized in Section IV.

## Ii approaches

The device system under investigation is a typical quantum dot formed on top of a narrow-gap semiconductor heterojunction (such as an InGaAs/InAlAs-based heterostructure) grown along the -direction, and the lateral confinement is formed in the -plane. A perpendicular magnetic field is applied along the growth-direction of the quantum dot. We consider the situation where the SOI is mainly induced by the presence of the InGaAs/InAlAs heterojunction due to the Rashba effect. In such a case, the lowest order of the SOI can be obtained from, e.g., a band-structure calculation scha . It should be noted that when the SOI is induced by the Rashba effect due to the presence of the heterojunction and when the magnetic field is applied along the -direction, the electronic subband structure along the growth-direction (or the -axis) depends very little on the SOI. Therefore, under the effective-mass approximation, the electronic structure along the -plane and in the -direction in a quantum dot can be treated separately. Including the contribution from SOI, the single-electron Hamiltonian describing the electronic system in the -plane can be written as

(1) |

where the Zeeman term is neglected for simplicity. Here, is the electron effective mass in the -plane, with is the momentum operator, is the vector potential induced by the magnetic field which is taken in the symmetric gauge here for convenience, and is the confining potential of the quantum dot along the -plane with . Furthermore, is the Rashba parameter which measures the strength of the SOI. Due to the Pauli spin matrices , this Hamiltonian is a matrix. After rewriting this Hamiltonian in the cylindrical polar coordinate, we find that the solution of the corresponding Schrödinger equation is in the form of , where is a good quantum number and is an azimuthal angle to the -axis. Thus, the eigenfunction and eigenvalue of the system can be determined by solving

(2) |

with

and

Here, is the cyclotron frequency, is the radius of the ground cyclotron orbit, and is the effective trapping potential. It should be noted that the solution of Eq. (2) does not depend on , because the eigenfunction and eigenvalue can be obtained, in principle, by solving

(3) |

where . Eq. (3) is a 4th-order differential equation. To the best of our knowledge, there is no simple and analytical solution to this equation. In this paper, we therefore attempt a tractable approach to solve the problem.

For the case of a parabolically confined quantum dot, we have the confining potential where is the characteristic frequency of the confinement. For convenience, we define and . After setting and assuming , Eq. (2) readily becomes

(4) |

Here, , , , and . In this paper, we are interested in finding the energy spectrum of a quantum dot in the presence of the SOI and of a magnetic field. We assume that the electron wavefunctions are in the form

and

with being the associated Laguerre polynomial. Thus, introducing into Eq. (4) and carrying out the normalization, we obtain two coupled equations which determine the energy and the coefficients and

(5) |

and

(6) |

where is the energy of a parabolically confined quantum dot in the absence of SOI,

when for and for and, otherwise,

with . These results suggest that the SOI in a quantum dot can result in band-mixing and shifting. Using sequence relations given by Eqs. (5) and (6), the electron energy as well as the coefficients and can be determined.

## Iii numerical results and discussion

In the present study, we examine the dependence of the energy levels and the occupation of electrons to different states on the strength of the SOI and of the magnetic field in a few-electron quantum dot system. We take the Rashba parameter to be eVm, in conjunction with recent experimental data realized in InAlAs/InGaAs heterostructures Saito ; Grundler . The size effects of the dot are also considered. For weakly and strongly confined dots, we take the typical values of the confining potential as meV and meV, respectively. The condition of electron number conservation is applied to determine the Fermi energy of the system. For demonstration of how electrons occupy to the different energy levels in the presence of SOI, we assume there are ten electrons in the dot. Moreover, in the present study the effect of the SOI on lifetimes of an electron in different states is not taken into consideration Governale . The inclusion of the many-body effects induced by Coulomb interaction in a quantum dot is a higher order effect to affect the energy spectrum, although the Coulomb interaction can result in an exchange enhancement which may affect the results qualitatively Pfan . Below, we discuss the effect of the SOI on energy levels of the low-lying states at zero and non-zero magnetic fields. For convenience of the discussion, we label the energy states with quantum numbers , where d or (u or ) referring to the down (up) spin branch of the energy states.

When a perpendicular d.c. magnetic field is applied to a quantum dot, the nature of spin-dependent energy spectrum can become even richer in terms of physics due to the coupling of the magnetic field to the confining potential of the quantum dot and to the potential induced by SOI. It should be noted that in the present study, in order to examine the net contribution to the energy spectrum of a dot from the SOI, we have neglected the effects induced by the Zeeman splitting. In Figs. 1 and 2 we plot the low-lying energy levels as a function of the Rashba parameter for different magnetic fields B = 1 T and 3 T, respectively, at a fixed quantum dot confinement 1 meV. In the absence of the Zeeman splitting, the SOI induced by the Rashba effect at finite magnetic field can lift the magnetic degeneracy of electrons. Namely, the states with the number can have different energies. We find that in this situation, the occupancy of electrons to different states has some unique features. On one hand, the presence of the magnetic field can increase the energy gaps among the states with the quantum number . Thus, similar to a strongly confined dot, the magnetic field may increase the critical value above which the system is fully spin-polarized. This effect can be seen at a relatively low-B-field (see Fig. 1) where eVm is found. On the other hand, when the magnetic field predominates the electronic energy spectra, the energy states with the negative magnetic quantum number and the spin-down branches are more preferable for an electron to stay. As a result, the presence of a B-field can increase the spin-polarization of a quantum dot. This effect is more pronounced in the presence of a higher magnetic field (see Fig. 2 where eVm) so that the states with positive quantum number and spin-up orientation have higher energies relative to the states. Furthermore, after comparing the values obtained in Figs. 1 and 2, we note that the presence of the magnetic field can result in a lower value of and, consequently, the stronger spin-polarization can be achieved in a dot in the presence of the SOI and of a magnetic field.

In Figs. 3, we show the spectra of low-lying states as a function of the magnetic field at a fixed Rashba parameter eVm for a weakly confined dot meV. We find that in such a case, because is relatively small, the electronic subband energy in the high-B field regime is mainly determined by the strength of the magnetic field through . The energy spectra at low-B fields differ slightly from this dependence (see the insert in Fig. 3). Thus, similar to a spin-degenerate quantum dot, in the high-B field regime the energy of a state increases with B field and levels with positive magnetic quantum number are always higher than those with negative . The results shown in the insert in Fig. 3 indicate that the cross-over of the energy states with different numbers occurs when T. When T, these states can be assembled to the energy groups of E 14, 42, 70, 96, 124 meV, etc., corresponding to the states with different numbers (see captions in Fig. 3), and these states are affected weakly by the SOI. In the presence of very high magnetic fields ( T in Fig. 3), the effects of the SOI and the confinement of the dot can be largely suppressed and the energy levels are therefore determined mainly by . Although the effect of SOI induced by Dresselhaus splitting is not included in the present study, these features enable us to compare qualitatively our results with those obtained by Voskoboynikov Vosko01 . Our analyses give an explicit expression of energy levels and our results clearly show the effects of SOI on energy spectra for different sizes of quantum dots.

For case of a strongly confined dot with meV, the B-field dependence of the energy levels are shown in Fig. 4 at a fixed eVm. From this figure, we see clearly that spin-up and -down levels with the same number are in pairs, where the energy difference between the up and down states is determined mainly by the confinement potential and the strength of the SOI. It can be seen further from Fig. 4 that in the presence of SOI, a magnetic field still plays a role in lowering the energies of those states with negative numbers. These levels lowered by the magnetic field are known as the Fock-Darwin Fock ; Darwin levels and their basic features are not affected significantly by the SOI.

The results shown in Fig. 3 and 4 suggest that at relatively high-B fields so that , the effect of SOI on energy spectrum of a quantum dot can be neglected. For case of a weakly confined dot, free electron (Landau-)Landau type levels can be used to describe the energy spectra of the system. When the cyclotron energy is larger than the confining potential of a dot, there is a hybridization of the Landau levels from the spatial confinement.

## Iv Concluding Remarks

In this paper, we have examined how the Rashba SOI affects the energy levels of a parabolically confined quantum dot. A non-perturbative approach to deal with SOI in a quantum dot has been developed. The main theoretical results obtained from this study are summarized as follows.

In the presence of a perpendicular magnetic field, we enter a regime with different competing energies, where magnetic potential, confining potential of the dot and potential induced by the SOI are coupled. As a result, the presence of the magnetic field can result in much richer features of the spintronic properties in a quantum dot. The energy spectra and the value of in a dot depend strongly on the strength of the magnetic field. In different B-field regimes, the energy levels of spin-modified states have different dependence of the magnetic field.

We have found that a coefficient with plays a role in switching the SOI. When a strong effect of the SOI on energy levels can be observed, whereas the effects of SOI can be neglected when . Thus, (1) the strong spin polarization can be achieved in a weakly confined dot at zero and non-zero magnetic fields; (2) the characteristics of the Fock-Darwin scheme and the Landau type levels are revealed when the magnetic field is strong enough; and (3) the effective SOI in the system decreases with increasing magnetic field and confining potential of the dot.

On the basis that the energy levels in spin-degenerate quantum dot systems have been well studied using optical and optoelectronic measurements, we believe these experimental techniques can also be used to examine the energy spectra of a quantum dot with SOI. We therefore hope that the theoretical results obtained in this study can be verified experimentally.

Although the present work deals with single-particle properties of a quantum dot in the presence of SOI, some many-body effects induced by Coulomb interaction can be investigated on the basis of these results. For example, using the energy spectrum and wavefunction obtained from this study, we can calculate the collective excitation modes and fast-electron optical spectrum using, e.g., a random-phase approximation Xu . However, for the case of a spin-split quantum dot, these further studies require numerical calculations considerably and we therefore do not attempt them in the present work.

###### Acknowledgements.

We thank T.F. Jiang and P.G. Luan for valuable discussions. W.X. is a Research Fellow of the Australian Research Council and C.S.T. is a Staff Scientist of the National Center for Theoretical Sciences (NCTS) in Taiwan. This work was also supported by the National Science Council of Taiwan under Grant No. 91-2119-M-007-004 (NCTS).## References

- (1) T.J. Thornton, Rep. Prog. Phys. 58, 311 (1995).
- (2) D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 (1998).
- (3) S. Benjamin and N.F. Johnson, Appl. Phys. Lett. 70, 2321 (1997).
- (4) C.S. Lent and P.D. Tougaw and W.Porod, Appl. Phys. Lett. 62, 714(1993).
- (5) P.D. Tougaw and C.S. Lent, J. Appl. Phys. 75, 1818 (1994).
- (6) S.N. Molotkov and S.S. Nazin, JEPT Lett. 63, 687 (1996).
- (7) H. Saito, K. Nishi, I. Ogura, S. Sugou, and Y. Sugomito, Appl. Phys. Lett. 69, 3140 (1996).
- (8) S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von Molnár, M.L. Roukes, A.Y. Chtchelkanova, and D.M. Treger, Science 294, 1488 (2001).
- (9) B. Datta, S. Das, Appl. Phys. Lett. 56, 665 (1990)
- (10) X. F. Wang, P. Vasilopoulos, and F.M. Peeters, Phys. Rev. B 65, 165217 (2002).
- (11) T. Koga, J. Nitta, H. Takayanagi, and S. Datta, Phys. Rev. Lett. 88, 126601 (2002).
- (12) Th. Schäpers, G. Engels, J. Lange, Th. Klocke, M. Hollfelder, and H. Lüth, J. Appl. Phys. 83, 4324 (1998).
- (13) Y.A. Bychkov and E.I. Rashba, J. Phys. C 17, 6039 (1984).
- (14) S. Tarucha, D.G. Austing, T. Honda, R.J. van der Hage, and L.P. Kouwenhoven, Phys. Rev. Lett. 77, 3613 (1996).
- (15) D. Grundler, Phys. Rev. Lett. 84, 6074 (2000).
- (16) B.N. Murdin, A.R. Hollingworth, M. Kamal-Saadi, R.T. Kotitschke, C.M. Ciesla, C.R. Pidgeon, P.C. Findlay, H.P.M. Pellemans, C.J.G.M. Langerak, A. C. Rowe, R. A. Stradling, and E. Gornik, Phys. Rev. B 59, R7817 (1999).
- (17) R. Krahne, V. Gudmundsson, C. Heyn, and D. Heitmann, Phys. Rev. B 63, 195303 (2001).
- (18) B.I. Halperin, A. Stern, Y. Oreg, J.N.H.J. Cremers, J.A. Folk, and C.M. Marcus, Phys. Rev. Lett. 86, 2106 (2001).
- (19) E.N. Bulgakov and A. F. Sadreev, JETP Lett. 73, 505 (2001).
- (20) M.A. Reed, J.N. Randall, R.J. Aggarwal, R.J. Matyi, T.M. Moore, and A. E. Wetsel, Phys. Rev. Lett. 60, 535 (1988).
- (21) M. Governale, Phys. Rev. Lett. 89, 206802 (2002).
- (22) D. Pfannkuche, V. Gudmundsson, and P. A. Maksym, Phys. Rev. B 47, 2244 (1993).
- (23) O. Voskoboynikov, C.P. Lee, and O. Tretyak, Phys. Rev. B 63, 165306 (2001).
- (24) V. Fock, Z. Physik 47, 446 (1928).
- (25) C.G. Darwin, Proc. Cambridge Philos. Soc. 27, 86 (1930).
- (26) L. Landau, Z. Physik 64, 629 (1930).
- (27) For case of a spin-split 2DEG, see, e.g., W. Xu, Appl. Phys. Lett. 82, 724 (2003).