- Email: [email protected]
Quantum rings in tilted magnetic fields
ARTICLE IN PRESS
Physica E 33 (2006) 370–375 www.elsevier.com/locate/physe
Quantum rings in tilted magnetic fields J. Planellesa,, J.I. Climentea,b, F. Rajadella a
Departament de Cie`ncies Experimentals, Universitat Jaume I, Box 224, E-12080 Castello´, Spain b CNR-INFM S3, Via Campi 213/A, 41100 Modena, Italy Received 13 April 2006; accepted 13 April 2006 Available online 6 June 2006
Abstract The electronic states of semiconductor quantum rings (QRs) under tilted magnetic fields are studied in the framework of the effective mass and envelope function approximations. For an axial field, the orbital Zeeman contribution prevails leading to the well-known Aharanov–Bohm spectrum, but it slowly decreases as the magnetic field direction declines. For an in-plane field, only the diamagnetic shift survives and it leads to the formation of double quantum well solutions, this result being relevant for experimental techniques which use in-plane magnetic fields to determine the spin of QR ground states. We also investigate the magnetic response of partially overlapped QRs, which are characteristic of high-density samples of self-assembled rings, and find that the spectrum is quite sensitive to ring coupling. r 2006 Elsevier B.V. All rights reserved. PACS: 72.21.La; 73.22.f; 73.22.Dj; 75.75.þa Keywords: Quantum rings; Electronic states; Magnetization; Tilted magnetic field
1. Introduction Quantum rings (QRs) have received a great deal of attention from researchers in the condensed matter field mainly due to their magnetic properties [1–3]. When these non-simply connected structures are pierced by a magnetic field, the Aharonov–Bohm (AB) effect [4] leads to the appearance of persistent currents and periodic oscillations of the carriers energy levels. Several works in the literature have investigated these phenomena (see e.g. Refs. [5,6] and references therein), which have been confirmed experimentally in metallic and semiconductor QRs both in the mesoscopic [7,8] and nanoscopic [9] regimes. Noteworthy, all these studies have been carried out for QRs submitted to magnetic fields applied perpendicular to the ring plane (axial magnetic fields). To the best of our knowledge, no theoretical work has dealt with the response of QRs to tilted or in-plane fields yet, so that the angle of inclination of the magnetic field (y) remains as an unexplored degree of Corresponding author.
E-mail address: [email protected] (J. Planelles). 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.04.004
freedom in these systems. Knowledge on the physics of QRs submitted to tilted magnetic fields is further motivated by recent experiments which use in-plane fields to determine the electron ground state spin in mesoscopic semiconductor QRs [10]. Such experiments are based on a technique originally conceived for quantum dots (QDs) [11], where the response of conduction band energy levels to in-plane fields is well known [12–14]. Theoretical understanding of the response in QRs is of clear interest so as to confirm to which extent this technique can be safely applied in these structures. In this paper, we study the effect of tilted magnetic fields, from the axial to the in-plane limit, on the single-electron energy levels of semiconductor QRs. Since, the vertical confinement in these structures is much stronger than the lateral one, we employ a two-dimensional Hamiltonian to describe the low-lying energy levels. Note that similar models have proved successful in order to explain the fundamental physics of QRs in the presence of axial magnetic fields [2,9,15]. We show that for an axial field the orbital Zeeman contribution prevails leading to the usual AB spectrum, but it slowly decreases as the angle between
ARTICLE IN PRESS J. Planelles et al. / Physica E 33 (2006) 370–375
the magnetic field and the growth direction increases. For an in-plane magnetic field only the diamagnetic shift survives and it leads to the formation of double quantum well solutions. This effect is found even in nanoscopic QR devices and moderate values of the field, and it may have important implications in the applicability of in-plane magnetic fields to determine the spin of QR ground states. Finally, we investigate the magnetic response of partially overlapped QRs, which are usually found in high-density samples of self-assembled QRs [3,16]. As we shall see, while the bottom part of the spectrum preserves the single QR behavior, the higher energy region shows severe differences.
371
z
y
B
x
2. Theoretical considerations We study the conduction band energy levels of semiconductor QRs within the effective mass and envelope function approximations. We label ðx; yÞ the in-plane coordinates of the QR and write the Hamiltonian in atomic units as H¼
1 ðp þ AÞ2 þ V ðx; yÞ, 2m
(1)
where m stands for the electron effective mass, A is the vector potential and V ðx; yÞ represents a finite scalar potential which confines the electron within an annular finite region of the space. A tilted magnetic field, declined an angle y from the axial axis z, may be formally decomposed as a superposition of an in-plane magnetic field applied along the x-axis, Bin ¼ ðB sin y; 0; 0Þ (which may be derived from Ain ¼ ð0; 0; yÞB sin y, see Appendix), and an axial magnetic field, Baxial ¼ ð0; 0; B cos yÞ. Within the Coulomb gauge [17], the latter term may be derived from Aaxial ¼ ðy; x; 0Þ12B cos y. Therefore, the Hamiltonian in the presence of a tilted magnetic field finally reads: H¼
1 B2 cos2 y 2 ðp^ 2x þ p^ 2y Þ þ ðx þ y2 Þ 2m 8m B2 sin2 y 2 B cos y q q y þ y i x þ V ðx; yÞ. ð2Þ 2m 2m qy qx
It is worth noting that in the limit of in-plane magnetic field only, the Hamiltonian is of the form: H¼
1 B2 2 2 2 ^ ^ ð p þ p Þ þ y þ V ðx; yÞ, y 2m x 2m
(3)
which describes a harmonic oscillator motion along the y coordinate. This characteristic behavior results from the projection of the free electron orbits induced by the inplane magnetic field on the infinitesimally thin ðx; yÞ plane, as represented in Fig. 1. The eigenvalue equation of the abovementioned Hamiltonian has been solved numerically using a finite-difference method on a two-dimensional grid (x; y) extended far beyond the QR limits. This discretization yields an eigenvalue problem of a huge asymmetric complex sparse
Fig. 1. Schematic representation of a free electron classical orbit originated by a magnetic field applied along the x-axis and its projection on the ðx; yÞ plane, where the electrons are restricted to move. The motion restricted to the ðx; yÞ plane corresponds to a harmonic oscillator on the yaxis.
matrix that has been solved in turn by employing the iterative Arnoldi factorization [18]. 3. Tilting the magnetic field We investigate GaAs QRs embedded in an Al0:3 Ga0:7 As matrix. The AlGaAs material acts as a barrier for the conduction band electrons confined in the ring structure. We then use an effective mass m ¼ 0:067 and a bandoffset of 0.25 eV [19]. The inner radius of the ring is rin ¼ 12 nm and the outer one is rout ¼ 16 nm. Synthesis of QRs with similar dimensions has been reported by various authors [3]. A range of 0–20 T for the external magnetic field is considered, as these are currently attainable fields in many experimental setups. Fig. 2 (upper panels) illustrates the electron energy levels in a QR under axial (y ¼ 0 ), tilted (y ¼ 45 ) and in-plane (y ¼ 90 ) magnetic fields. The equivalent results for a QD with radius r ¼ 16 nm are shown in Fig. 2 (lower panels) for comparison [20]. In a first instance, we focus on the axial magnetic field. In this configuration the axial symmetry of the confining potential is not destroyed by the field, and the magnetic term in Eq. (2) is composed by two competing terms, namely the orbital Zeeman and the diamagnetic term, which depend linearly and quadratically on the field, respectively. The corresponding results are the usual AB spectrum for the QR [1,2], and a Fock–Darwinlike spectrum for the QD [2]. The differences between the magnetic response of the two structures arise mainly from the magnetic flux trapping in the QR [4–6]. A completely different picture is found when an in-plane magnetic field is applied. In this case, the axial symmetry of the structures is broken by the external field, there is no flux penetration through the ring hole (i.e., no AB effect) and only the diamagnetic term survives. Remarkably, even in the
ARTICLE IN PRESS J. Planelles et al. / Physica E 33 (2006) 370–375
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θ = 0°
θ = 45°
θ = 90°
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0
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10 B (T)
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0
5
10 B (T)
Fig. 2. Low-lying electron energy levels in a QR (upper panels) and a QD (lower panels) vs. magnetic fields tilted by an angle y with respect to the axial direction of the structure.
absence of AB features, significant differences between QR and QD spectra are observed. In particular, we note that with increasing magnetic fields consecutive QR energy levels are arranged in pairs. This fundamental difference is due to the inner hole of the QR which, together with the magnetic field-induced wave function compression along the y direction, leads to the formation of quasi-degenerate double quantum well solutions along the x axis. Obviously, the higher the kinetic energy of the states the stronger the magnetic field required to reach the one-dimensional behavior. This effect has an outcome in the eigenfunctions, as illustrated in Fig. 3, where the lowest-lying wave functions of QR and QD states at B ¼ 20 T are compared. It can be seen that, while the number of nodes on the x-axis increases unit by unit for the QD, typical double-well pairs of even/odd states are formed in the QR case. Finally, when the magnetic field is declined 45 , an intermediate behavior of the system is observed in Fig. 2. In this case, the spectrum roughly resembles that of the y ¼ 0 configuration, but the AB periods are now larger and the formation of double-quantum-well solutions starts becoming visible for the lowest-lying energy levels at high fields. A more detailed insight into the transition from the axial to the in-plane configurations is shown in Fig. 4, where the
lowest-lying QR energy levels vs. magnetic field angle y at B ¼ 7 T are represented. When y ¼ 0 , the inter-level energy spacing is the same as in the absence of magnetic field, because for axial magnetic fields B ¼ 7 T corresponds to one AB oscillation (see Fig. 2). As the magnetic field declines the inter-level spacing decreases, so that first a situation where pairs of consecutive energy levels are degenerate is reached (y ¼ 60) and then the energy levels are reversed. This evolution is to some extent equivalent to moving from B ¼ 7 to 0 T in the QR energy spectrum of Fig. 2 when y ¼ 0 . Thus, the degeneracy point corresponds to the case where half flux quantum is piercing the ring, and therefore the energy levels are equivalent to those of y ¼ 0 at B ¼ 3:5 T. Note that halving of the flux quantum takes place when the linear term of Eq. (2) is exactly halved (cos y ¼ 12), but deviations from this value may be expected when the QR is thicker because of the magnetic field penetration in the ring region [6,21]. On the other hand, the y ¼ 90 limit in Fig. 4 differs from the B ¼ 0 T limit of axial magnetic field inasmuch as the inplane diamagnetic term is relevant. It is worth stressing that the formation of double quantum well solutions is predicted here for a small, nanoscopic QR submitted to moderate in-plane magnetic
ARTICLE IN PRESS J. Planelles et al. / Physica E 33 (2006) 370–375
373
Energy (meV)
220
180
140
100 0
30
θ (degrees)
60
90
Fig. 4. Low-lying electron energy levels in a QR submitted to an external magnetic field of B ¼ 7 T vs. the declination angle with respect to the axial direction.
contain at least one hundred electrons). We expect this phenomenon to become relevant for the low-lying orbitals, though. 4. Effect of ring coupling
Fig. 3. Contours of the lowest-electron states wave functions in a QD (left column) and QR (right column) under strong in-plane magnetic fields.
fields (B 10 T for the lowest-energy levels in Fig. 2). Obviously, this effect should also be present in more voluminous QRs but at weaker fields. This imposes a natural limit to the applicability of in-plane fields in order to determine the ground state spin in QRs using the experimental technique described in Ref. [11]. Such technique infers the spin quantum numbers of the occupied orbitals at B ¼ 0 from the energy differences between consecutive levels submitted to finite in-plane magnetic fields. For large QRs under in-plane fields, the low-lying levels will rapidly arrange in degenerate pairs of orbitals (as in Fig. 2) and then the ground state spin will no longer be representative of the energy structure at B ¼ 0. This is likely to occur e.g. in the QR investigated in Ref. [10], which is about an order of magnitude larger than the one we consider in our simulations. Still, the results of such experiment do not reflect the formation of doublequantum-well solutions, which may be due to the fact that the spin–orbitals probed are very excited (the QR is said to
High-density samples of self-assembled QRs often exhibit clusters of partially overlapping rings [3,16]. In this section we study the effect of the overlap on the magnetic response of the QRs. To this end, we model a cluster composed by five QRs with C 4v symmetry (see inset in Fig. 5), and consider axial, tilted (y ¼ 45 ) and in-plane magnetic fields. The QRs in the cluster have the same dimensions as in the previous section (ri ¼ 12, re ¼ 16 nm), and the distance from the origin of the central QR to that of the external QRs is set to d ¼ 28 nm, so that the ring coupling is not severe. At B ¼ 0, the confining potential has C 4v symmetry. The symmetry is lowered to C 4 in the presence of an axial magnetic field and it is further lowered when the field declines. This is reflected in the corresponding energy spectra of Fig. 5. The most symmetric (C 4 ) spectrum, corresponds to axial magnetic field. This spectrum shows crossings similar to those occurring in isolated QRs, but also anti-crossings coming from the cluster symmetry. Thus, we can see that the spectrum is splitted into non-crossing sets of four states. Within each set, the states cross repeatedly one another as B increases. Every set contains one instance of each of the four C 4 symmetries, namely A, B, E þ , and E . Additionally, the two lowest sets show a regular pattern as well as an important splitting of energies produced by the magnetic field. This behavior is due to the fact that the involved
ARTICLE IN PRESS J. Planelles et al. / Physica E 33 (2006) 370–375
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θ = 0°
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110
100
90
80 0
5
10 B (T)
15
20 0
5
10 B (T)
15
20 0
5
10 B (T)
15
20
Fig. 5. Low-lying electron energy-levels in a cluster of five coupled QRs vs. magnetic fields tilted by an angle y with respect to the axial direction of the structure. The geometry of the cluster is shown in the inset, with dark (light) regions representing AlGaAs (GaAs).
states mainly localize in the central ring of the cluster. On the contrary, the three upper sets are basically located in the external rings and hence exhibit less ordered features. Similar behavior is found for tilted and in-plane magnetic fields, where the low-lying states again retrieve the singleQR results (i.e., they localize in the central ring) while the excited states are more chaotic (they localize in the external rings). Aside from the symmetry issues, which are characteristic of the particular geometry of the cluster under study, it follows from Fig. 5 that moderate coupling between adjacent QRs significantly influences the magnetic response. The lowest part of the spectrum may preserve the single-QR behavior, but the high-energy part changes dramatically. 5. Conclusions We have studied the single-electron energy levels in a QR under magnetic fields with arbitrary direction. It has been shown that for an axial magnetic field the orbital Zeeman contribution prevails, leading to the well-known AB phenomena, but it slowly decreases as the angle between the magnetic field and the growth direction increases. For an in-plane field only the diamagnetic shift survives and it leads to the formation of one-dimensional double quantum well solutions at high fields. This result, which constitutes a fundamental difference between the response of QRs and QDs to in-plane magnetic fields, is relevant with regard to the applicability of experimental techniques which use inplane magnetic fields to determine the spin of QR ground states. Finally, we have shown that while the bottom of the energy spectra of high-density samples of self-assembled QRs resembles that of isolated QRs, the higher energy region shows severe differences. Acknowledgments Financial support from MEC-DGI Project CTQ200402315/BQU and UJI-Bancaixa Project P1-B2002-01 is
gratefully acknowledged. This work has been supported in part by the EU under the TMR network ‘‘Exciting’’ (J.I.C.). Appendix The Hamiltonian of an electron under an in-plane magnetic field, strongly confined in the vertical direction and loosely confined in-plane by a potential V ¼ V k ðx; yÞ þ V ? ðzÞ, may be written employing a Coulomb gauge defined by A ¼ ð0; 0; yÞB as 1 ðp þ AÞ2 þ V k ðx; yÞ þ V ? ðzÞ 2m 1 ¼ ðp2 þ p2k þ A2 þ 2ApÞ þ V k ðx; yÞ þ V ? ðzÞ 2m z 1 ¼ ðp2 þ 2Bypz Þ þ V ? ðzÞ 2m z 1 þ ðp2 þ B2 y2 Þ þ V k ðx; yÞ. 2m k
H¼
ð4Þ
Now we introduce the flatness of the system: due to the strong confining potential V ? ðzÞ, the wavefunction is always zero except at a very narrow slab around z ¼ 0. Then, an effective decoupling of the vertical directions occurs: Cðx; y; zÞ ¼ ZðzÞFðx; yÞ. Additionally, there are very many in-plane states with energy between the ground and first excited state in the vertical direction so that all the states of interest are approximately described by wavefunctions of the form Cðx; y; zÞ ¼ Z0 ðzÞFn ðx; yÞ, where Z0 ðzÞ is the ground state eigenfunction of H z ¼ p2z =2m þ V ? ðzÞ. Left-multiplying the eigenvalue equation of Hamiltonian (4) by Z 0 ðzÞ and integrating over z yields, hZ 0 ðzÞ jHjCðx; y; zÞi 2 p ¼ Fn ðx; yÞ Z0 ðzÞ z þ V ? ðzÞ Z0 ðzÞ 2m By þ hZ 0 ðzÞ jpz jZ 0 ðzÞi m
ARTICLE IN PRESS J. Planelles et al. / Physica E 33 (2006) 370–375
1 þ hZ 0 ðzÞ jZ 0 ðzÞi ðp2 þ B2 y2 Þ 2m k þ V k ðx; yÞ Fn ðx; yÞ
¼ EhZ 0 ðzÞ jZ 0 ðzÞiFn ðx; yÞ.
ð5Þ
Since, hZ 0 ðzÞ j½ðp2z =2m Þ þ V ? ðzÞjZ 0 ðzÞi ¼ E 0z , hZ 0 ðzÞ jZ 0 ðzÞi ¼ 1 and, for symmetry reasons, hZ 0 ðzÞ jpz jZ0 ðzÞi ¼ 0, we end up with 1 2 2 2 ðp þ B y Þ þ V k ðx; yÞ Fn ðx; yÞ 2m k ¼ ðE E 0z ÞFn ðx; yÞ,
ð6Þ
which is the equation employed in this work. Note that using a bonafide gauge A ¼ ðz; 0; 0ÞB we would not be able to make this effective separation of variables, which is crucial to study the isolated in-plane system. References [1] S. Viefers, P. Koskinen, P.S. Deo, M. Manninen, Physica E 21 (2004) 1 and references therein. [2] T. Chakraborty, Quantum Dots, Elsevier Science B.V., Amsterdam, 1999. [3] B.C. Lee, O.P. Voskoboynikov, C.P. Lee, Physica E 24 (2004) 87 and references therein. [4] Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485. [5] S. Olariu, II. Popescu, Rev. Mod. Phys. 57 (1985) 339. [6] J. Planelles, J.I. Climente, J.L. Movilla, in: Symmetry, Spectroscopy and SCHUR, Proceedings of the Prof. Brian G. Wybourne Commemorative Meeting, N. Copernicus University Press, Torun, 2006. See also cond-mat/0506691.
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[7] V. Chandrasekhar, R.A. Webb, M.J. Brady, M.B. Ketchen, W.J. Gallagher, A. Kleinsasser, Phys. Rev. Lett. 67 (1991) 3578. [8] A. Fuhrer, S. Lu¨scher, T. Ihn, T. Heinzel, K. Ensslin, W. Wegscheider, M. Bichler, Nature 413 (2001) 822. [9] M. Bayer, M. Korkusinski, P. Hawrylak, T. Gutbrod, M. Michel, A. Forchel, Phys. Rev. Lett. 90 (2003) 186801. [10] T. Ihn, A. Fuhrer, K. Ensslin, W. Wegscheider, M. Bichler, Physica E 26 (2005) 225. [11] S. Lindemann, T. Ihn, T. Heinzel, W. Zwerger, K. Ensslin, K. Maranowski, A.C. Gossard, Phys. Rev. B 66 (2002) 195314. [12] B. Meuer, D. Heitmann, K. Ploog, Surface Sci. 305 (1994) 610. [13] T. Pyragiene, A. Matulis, Physica E 8 (2000) 230. [14] D. Bellucci, M. Rontani, G. Goldoni, E. Molinari, Phys. Rev. B, submitted. [15] D.V. Bulaev, V.A. Geyler, V.A. Margulis, Phys. Rev. B 69 (2004) 195313; A. Emperador, M. Pi, M. Barranco, A. Lorke, Phys. Rev. B 62 (2000) 4573; A. Lorke, R.J. Luyken, A.O. Govorov, J.P. Kotthaus, J.M. Garcia, P.M. Petroff, Phys. Rev. Lett. 84 (2000) 2223; H. Hu, J.L. Zhu, J.J. Xiong, Phys. Rev. B 62 (2000) 16777; M. Manninen, M. Koskinen, S.M. Reimann, B. Mottelson, Eur. Phys. J. D 16 (2001) 381. [16] B.C. Lee, C.P. Lee, Nanotechnology 15 (2004) 848. [17] The Coulomb gauge does not determines Aaxial completely, we could select A0axial ¼ ð0; x; 0ÞB cos y instead. [18] W.E. Arnoldi, Quart. J. Appl. Math. 9 (1951) 17; Y. Saad, Numerical Methods for Large Scale Eigenvalue Problems, Halsted Press, New York, 1992; R.B. Morgan, Math. Comput. 65 (1996) 1213. [19] M. Yamagiwa, N. Sumita, F. Minami, N. Koguchi, J. Lumin. 108 (2004) 379. [20] In this work we discuss only the difference between QRs and QDs magnetic response. Dissimilarities at zero magnetic field have been analyzed in S.S. Li, J.B. Xia, J. Appl. Phys. 89 (2001) 3434. [21] O. Voskoboynikov, Y. Li, H.M. Lu, C.F. Shih, C.P. Lee, Phys. Rev. B 66 (2002) 155306.
Physica E 33 (2006) 370–375 www.elsevier.com/locate/physe
Quantum rings in tilted magnetic fields J. Planellesa,, J.I. Climentea,b, F. Rajadella a
Departament de Cie`ncies Experimentals, Universitat Jaume I, Box 224, E-12080 Castello´, Spain b CNR-INFM S3, Via Campi 213/A, 41100 Modena, Italy Received 13 April 2006; accepted 13 April 2006 Available online 6 June 2006
Abstract The electronic states of semiconductor quantum rings (QRs) under tilted magnetic fields are studied in the framework of the effective mass and envelope function approximations. For an axial field, the orbital Zeeman contribution prevails leading to the well-known Aharanov–Bohm spectrum, but it slowly decreases as the magnetic field direction declines. For an in-plane field, only the diamagnetic shift survives and it leads to the formation of double quantum well solutions, this result being relevant for experimental techniques which use in-plane magnetic fields to determine the spin of QR ground states. We also investigate the magnetic response of partially overlapped QRs, which are characteristic of high-density samples of self-assembled rings, and find that the spectrum is quite sensitive to ring coupling. r 2006 Elsevier B.V. All rights reserved. PACS: 72.21.La; 73.22.f; 73.22.Dj; 75.75.þa Keywords: Quantum rings; Electronic states; Magnetization; Tilted magnetic field
1. Introduction Quantum rings (QRs) have received a great deal of attention from researchers in the condensed matter field mainly due to their magnetic properties [1–3]. When these non-simply connected structures are pierced by a magnetic field, the Aharonov–Bohm (AB) effect [4] leads to the appearance of persistent currents and periodic oscillations of the carriers energy levels. Several works in the literature have investigated these phenomena (see e.g. Refs. [5,6] and references therein), which have been confirmed experimentally in metallic and semiconductor QRs both in the mesoscopic [7,8] and nanoscopic [9] regimes. Noteworthy, all these studies have been carried out for QRs submitted to magnetic fields applied perpendicular to the ring plane (axial magnetic fields). To the best of our knowledge, no theoretical work has dealt with the response of QRs to tilted or in-plane fields yet, so that the angle of inclination of the magnetic field (y) remains as an unexplored degree of Corresponding author.
E-mail address: [email protected] (J. Planelles). 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.04.004
freedom in these systems. Knowledge on the physics of QRs submitted to tilted magnetic fields is further motivated by recent experiments which use in-plane fields to determine the electron ground state spin in mesoscopic semiconductor QRs [10]. Such experiments are based on a technique originally conceived for quantum dots (QDs) [11], where the response of conduction band energy levels to in-plane fields is well known [12–14]. Theoretical understanding of the response in QRs is of clear interest so as to confirm to which extent this technique can be safely applied in these structures. In this paper, we study the effect of tilted magnetic fields, from the axial to the in-plane limit, on the single-electron energy levels of semiconductor QRs. Since, the vertical confinement in these structures is much stronger than the lateral one, we employ a two-dimensional Hamiltonian to describe the low-lying energy levels. Note that similar models have proved successful in order to explain the fundamental physics of QRs in the presence of axial magnetic fields [2,9,15]. We show that for an axial field the orbital Zeeman contribution prevails leading to the usual AB spectrum, but it slowly decreases as the angle between
ARTICLE IN PRESS J. Planelles et al. / Physica E 33 (2006) 370–375
the magnetic field and the growth direction increases. For an in-plane magnetic field only the diamagnetic shift survives and it leads to the formation of double quantum well solutions. This effect is found even in nanoscopic QR devices and moderate values of the field, and it may have important implications in the applicability of in-plane magnetic fields to determine the spin of QR ground states. Finally, we investigate the magnetic response of partially overlapped QRs, which are usually found in high-density samples of self-assembled QRs [3,16]. As we shall see, while the bottom part of the spectrum preserves the single QR behavior, the higher energy region shows severe differences.
371
z
y
B
x
2. Theoretical considerations We study the conduction band energy levels of semiconductor QRs within the effective mass and envelope function approximations. We label ðx; yÞ the in-plane coordinates of the QR and write the Hamiltonian in atomic units as H¼
1 ðp þ AÞ2 þ V ðx; yÞ, 2m
(1)
where m stands for the electron effective mass, A is the vector potential and V ðx; yÞ represents a finite scalar potential which confines the electron within an annular finite region of the space. A tilted magnetic field, declined an angle y from the axial axis z, may be formally decomposed as a superposition of an in-plane magnetic field applied along the x-axis, Bin ¼ ðB sin y; 0; 0Þ (which may be derived from Ain ¼ ð0; 0; yÞB sin y, see Appendix), and an axial magnetic field, Baxial ¼ ð0; 0; B cos yÞ. Within the Coulomb gauge [17], the latter term may be derived from Aaxial ¼ ðy; x; 0Þ12B cos y. Therefore, the Hamiltonian in the presence of a tilted magnetic field finally reads: H¼
1 B2 cos2 y 2 ðp^ 2x þ p^ 2y Þ þ ðx þ y2 Þ 2m 8m B2 sin2 y 2 B cos y q q y þ y i x þ V ðx; yÞ. ð2Þ 2m 2m qy qx
It is worth noting that in the limit of in-plane magnetic field only, the Hamiltonian is of the form: H¼
1 B2 2 2 2 ^ ^ ð p þ p Þ þ y þ V ðx; yÞ, y 2m x 2m
(3)
which describes a harmonic oscillator motion along the y coordinate. This characteristic behavior results from the projection of the free electron orbits induced by the inplane magnetic field on the infinitesimally thin ðx; yÞ plane, as represented in Fig. 1. The eigenvalue equation of the abovementioned Hamiltonian has been solved numerically using a finite-difference method on a two-dimensional grid (x; y) extended far beyond the QR limits. This discretization yields an eigenvalue problem of a huge asymmetric complex sparse
Fig. 1. Schematic representation of a free electron classical orbit originated by a magnetic field applied along the x-axis and its projection on the ðx; yÞ plane, where the electrons are restricted to move. The motion restricted to the ðx; yÞ plane corresponds to a harmonic oscillator on the yaxis.
matrix that has been solved in turn by employing the iterative Arnoldi factorization [18]. 3. Tilting the magnetic field We investigate GaAs QRs embedded in an Al0:3 Ga0:7 As matrix. The AlGaAs material acts as a barrier for the conduction band electrons confined in the ring structure. We then use an effective mass m ¼ 0:067 and a bandoffset of 0.25 eV [19]. The inner radius of the ring is rin ¼ 12 nm and the outer one is rout ¼ 16 nm. Synthesis of QRs with similar dimensions has been reported by various authors [3]. A range of 0–20 T for the external magnetic field is considered, as these are currently attainable fields in many experimental setups. Fig. 2 (upper panels) illustrates the electron energy levels in a QR under axial (y ¼ 0 ), tilted (y ¼ 45 ) and in-plane (y ¼ 90 ) magnetic fields. The equivalent results for a QD with radius r ¼ 16 nm are shown in Fig. 2 (lower panels) for comparison [20]. In a first instance, we focus on the axial magnetic field. In this configuration the axial symmetry of the confining potential is not destroyed by the field, and the magnetic term in Eq. (2) is composed by two competing terms, namely the orbital Zeeman and the diamagnetic term, which depend linearly and quadratically on the field, respectively. The corresponding results are the usual AB spectrum for the QR [1,2], and a Fock–Darwinlike spectrum for the QD [2]. The differences between the magnetic response of the two structures arise mainly from the magnetic flux trapping in the QR [4–6]. A completely different picture is found when an in-plane magnetic field is applied. In this case, the axial symmetry of the structures is broken by the external field, there is no flux penetration through the ring hole (i.e., no AB effect) and only the diamagnetic term survives. Remarkably, even in the
ARTICLE IN PRESS J. Planelles et al. / Physica E 33 (2006) 370–375
372
θ = 0°
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Fig. 2. Low-lying electron energy levels in a QR (upper panels) and a QD (lower panels) vs. magnetic fields tilted by an angle y with respect to the axial direction of the structure.
absence of AB features, significant differences between QR and QD spectra are observed. In particular, we note that with increasing magnetic fields consecutive QR energy levels are arranged in pairs. This fundamental difference is due to the inner hole of the QR which, together with the magnetic field-induced wave function compression along the y direction, leads to the formation of quasi-degenerate double quantum well solutions along the x axis. Obviously, the higher the kinetic energy of the states the stronger the magnetic field required to reach the one-dimensional behavior. This effect has an outcome in the eigenfunctions, as illustrated in Fig. 3, where the lowest-lying wave functions of QR and QD states at B ¼ 20 T are compared. It can be seen that, while the number of nodes on the x-axis increases unit by unit for the QD, typical double-well pairs of even/odd states are formed in the QR case. Finally, when the magnetic field is declined 45 , an intermediate behavior of the system is observed in Fig. 2. In this case, the spectrum roughly resembles that of the y ¼ 0 configuration, but the AB periods are now larger and the formation of double-quantum-well solutions starts becoming visible for the lowest-lying energy levels at high fields. A more detailed insight into the transition from the axial to the in-plane configurations is shown in Fig. 4, where the
lowest-lying QR energy levels vs. magnetic field angle y at B ¼ 7 T are represented. When y ¼ 0 , the inter-level energy spacing is the same as in the absence of magnetic field, because for axial magnetic fields B ¼ 7 T corresponds to one AB oscillation (see Fig. 2). As the magnetic field declines the inter-level spacing decreases, so that first a situation where pairs of consecutive energy levels are degenerate is reached (y ¼ 60) and then the energy levels are reversed. This evolution is to some extent equivalent to moving from B ¼ 7 to 0 T in the QR energy spectrum of Fig. 2 when y ¼ 0 . Thus, the degeneracy point corresponds to the case where half flux quantum is piercing the ring, and therefore the energy levels are equivalent to those of y ¼ 0 at B ¼ 3:5 T. Note that halving of the flux quantum takes place when the linear term of Eq. (2) is exactly halved (cos y ¼ 12), but deviations from this value may be expected when the QR is thicker because of the magnetic field penetration in the ring region [6,21]. On the other hand, the y ¼ 90 limit in Fig. 4 differs from the B ¼ 0 T limit of axial magnetic field inasmuch as the inplane diamagnetic term is relevant. It is worth stressing that the formation of double quantum well solutions is predicted here for a small, nanoscopic QR submitted to moderate in-plane magnetic
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Fig. 4. Low-lying electron energy levels in a QR submitted to an external magnetic field of B ¼ 7 T vs. the declination angle with respect to the axial direction.
contain at least one hundred electrons). We expect this phenomenon to become relevant for the low-lying orbitals, though. 4. Effect of ring coupling
Fig. 3. Contours of the lowest-electron states wave functions in a QD (left column) and QR (right column) under strong in-plane magnetic fields.
fields (B 10 T for the lowest-energy levels in Fig. 2). Obviously, this effect should also be present in more voluminous QRs but at weaker fields. This imposes a natural limit to the applicability of in-plane fields in order to determine the ground state spin in QRs using the experimental technique described in Ref. [11]. Such technique infers the spin quantum numbers of the occupied orbitals at B ¼ 0 from the energy differences between consecutive levels submitted to finite in-plane magnetic fields. For large QRs under in-plane fields, the low-lying levels will rapidly arrange in degenerate pairs of orbitals (as in Fig. 2) and then the ground state spin will no longer be representative of the energy structure at B ¼ 0. This is likely to occur e.g. in the QR investigated in Ref. [10], which is about an order of magnitude larger than the one we consider in our simulations. Still, the results of such experiment do not reflect the formation of doublequantum-well solutions, which may be due to the fact that the spin–orbitals probed are very excited (the QR is said to
High-density samples of self-assembled QRs often exhibit clusters of partially overlapping rings [3,16]. In this section we study the effect of the overlap on the magnetic response of the QRs. To this end, we model a cluster composed by five QRs with C 4v symmetry (see inset in Fig. 5), and consider axial, tilted (y ¼ 45 ) and in-plane magnetic fields. The QRs in the cluster have the same dimensions as in the previous section (ri ¼ 12, re ¼ 16 nm), and the distance from the origin of the central QR to that of the external QRs is set to d ¼ 28 nm, so that the ring coupling is not severe. At B ¼ 0, the confining potential has C 4v symmetry. The symmetry is lowered to C 4 in the presence of an axial magnetic field and it is further lowered when the field declines. This is reflected in the corresponding energy spectra of Fig. 5. The most symmetric (C 4 ) spectrum, corresponds to axial magnetic field. This spectrum shows crossings similar to those occurring in isolated QRs, but also anti-crossings coming from the cluster symmetry. Thus, we can see that the spectrum is splitted into non-crossing sets of four states. Within each set, the states cross repeatedly one another as B increases. Every set contains one instance of each of the four C 4 symmetries, namely A, B, E þ , and E . Additionally, the two lowest sets show a regular pattern as well as an important splitting of energies produced by the magnetic field. This behavior is due to the fact that the involved
ARTICLE IN PRESS J. Planelles et al. / Physica E 33 (2006) 370–375
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Fig. 5. Low-lying electron energy-levels in a cluster of five coupled QRs vs. magnetic fields tilted by an angle y with respect to the axial direction of the structure. The geometry of the cluster is shown in the inset, with dark (light) regions representing AlGaAs (GaAs).
states mainly localize in the central ring of the cluster. On the contrary, the three upper sets are basically located in the external rings and hence exhibit less ordered features. Similar behavior is found for tilted and in-plane magnetic fields, where the low-lying states again retrieve the singleQR results (i.e., they localize in the central ring) while the excited states are more chaotic (they localize in the external rings). Aside from the symmetry issues, which are characteristic of the particular geometry of the cluster under study, it follows from Fig. 5 that moderate coupling between adjacent QRs significantly influences the magnetic response. The lowest part of the spectrum may preserve the single-QR behavior, but the high-energy part changes dramatically. 5. Conclusions We have studied the single-electron energy levels in a QR under magnetic fields with arbitrary direction. It has been shown that for an axial magnetic field the orbital Zeeman contribution prevails, leading to the well-known AB phenomena, but it slowly decreases as the angle between the magnetic field and the growth direction increases. For an in-plane field only the diamagnetic shift survives and it leads to the formation of one-dimensional double quantum well solutions at high fields. This result, which constitutes a fundamental difference between the response of QRs and QDs to in-plane magnetic fields, is relevant with regard to the applicability of experimental techniques which use inplane magnetic fields to determine the spin of QR ground states. Finally, we have shown that while the bottom of the energy spectra of high-density samples of self-assembled QRs resembles that of isolated QRs, the higher energy region shows severe differences. Acknowledgments Financial support from MEC-DGI Project CTQ200402315/BQU and UJI-Bancaixa Project P1-B2002-01 is
gratefully acknowledged. This work has been supported in part by the EU under the TMR network ‘‘Exciting’’ (J.I.C.). Appendix The Hamiltonian of an electron under an in-plane magnetic field, strongly confined in the vertical direction and loosely confined in-plane by a potential V ¼ V k ðx; yÞ þ V ? ðzÞ, may be written employing a Coulomb gauge defined by A ¼ ð0; 0; yÞB as 1 ðp þ AÞ2 þ V k ðx; yÞ þ V ? ðzÞ 2m 1 ¼ ðp2 þ p2k þ A2 þ 2ApÞ þ V k ðx; yÞ þ V ? ðzÞ 2m z 1 ¼ ðp2 þ 2Bypz Þ þ V ? ðzÞ 2m z 1 þ ðp2 þ B2 y2 Þ þ V k ðx; yÞ. 2m k
H¼
ð4Þ
Now we introduce the flatness of the system: due to the strong confining potential V ? ðzÞ, the wavefunction is always zero except at a very narrow slab around z ¼ 0. Then, an effective decoupling of the vertical directions occurs: Cðx; y; zÞ ¼ ZðzÞFðx; yÞ. Additionally, there are very many in-plane states with energy between the ground and first excited state in the vertical direction so that all the states of interest are approximately described by wavefunctions of the form Cðx; y; zÞ ¼ Z0 ðzÞFn ðx; yÞ, where Z0 ðzÞ is the ground state eigenfunction of H z ¼ p2z =2m þ V ? ðzÞ. Left-multiplying the eigenvalue equation of Hamiltonian (4) by Z 0 ðzÞ and integrating over z yields, hZ 0 ðzÞ jHjCðx; y; zÞi 2 p ¼ Fn ðx; yÞ Z0 ðzÞ z þ V ? ðzÞ Z0 ðzÞ 2m By þ hZ 0 ðzÞ jpz jZ 0 ðzÞi m
ARTICLE IN PRESS J. Planelles et al. / Physica E 33 (2006) 370–375
1 þ hZ 0 ðzÞ jZ 0 ðzÞi ðp2 þ B2 y2 Þ 2m k þ V k ðx; yÞ Fn ðx; yÞ
¼ EhZ 0 ðzÞ jZ 0 ðzÞiFn ðx; yÞ.
ð5Þ
Since, hZ 0 ðzÞ j½ðp2z =2m Þ þ V ? ðzÞjZ 0 ðzÞi ¼ E 0z , hZ 0 ðzÞ jZ 0 ðzÞi ¼ 1 and, for symmetry reasons, hZ 0 ðzÞ jpz jZ0 ðzÞi ¼ 0, we end up with 1 2 2 2 ðp þ B y Þ þ V k ðx; yÞ Fn ðx; yÞ 2m k ¼ ðE E 0z ÞFn ðx; yÞ,
ð6Þ
which is the equation employed in this work. Note that using a bonafide gauge A ¼ ðz; 0; 0ÞB we would not be able to make this effective separation of variables, which is crucial to study the isolated in-plane system. References [1] S. Viefers, P. Koskinen, P.S. Deo, M. Manninen, Physica E 21 (2004) 1 and references therein. [2] T. Chakraborty, Quantum Dots, Elsevier Science B.V., Amsterdam, 1999. [3] B.C. Lee, O.P. Voskoboynikov, C.P. Lee, Physica E 24 (2004) 87 and references therein. [4] Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485. [5] S. Olariu, II. Popescu, Rev. Mod. Phys. 57 (1985) 339. [6] J. Planelles, J.I. Climente, J.L. Movilla, in: Symmetry, Spectroscopy and SCHUR, Proceedings of the Prof. Brian G. Wybourne Commemorative Meeting, N. Copernicus University Press, Torun, 2006. See also cond-mat/0506691.
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