phys. stat. sol. (a) 190, No. 3, 781 785 (2002) Magneto-Excitons in Semiconductor Quantum Rings I. Galbraith 1 ), F. J. Braid, and R. J. Warburton Department of Physics, Heriot-Watt University, Edinburgh, UK (Received September 4, 2001; accepted September 10, 2001) Subject classification: 71.35.Ji; 73.21. b; 78.67. n; S7.12 In this paper we present calculations of the magneto-excitonic absorption spectrum for two different quantum ring geometries. Calculations are performed using a direct diagonalization of the single particle wavefunctions. Two different behaviours including the novel appearance of a ground state exciton with a finite angular momentum are described and calculations of the conditional correlation function are used to explore the nature of the exciton peaks. Magneto-excitons in semiconductor quantum rings present a fascinating example of how quantum-confined geometry can influence the optical and electrical properties of semiconductors. By interrupting the growth appropriately, rings of diameter approximately 500 A can be formed in the GaAs/InAs material system [1]. Typical exciton binding energies in these rings are 30 mev, which is much larger than the single particle energy spacings. Hence the Coulomb interaction mixes the discrete valence and conduction levels such as to reduce the average electron hole separation. The application of a magnetic field perpendicular to the rings shifts the single particle energy levels dramatically, in particular when exactly one flux quantum passes through the ring, the lowest lying electron energy level is the angular momentum, l e ¼ 1, state. An interesting question then arises as to the nature of the exciton in this case since the magnetic field is driving the electron and hole in opposite senses while the influence of the Coulomb attraction, V eh, is to keep the pair as close as possible. This problem has been tackled in various levels of approximation. Chaplik [2] and Römer et al. [3] considered analytical approximations to the case of infinitely narrow confining rings and found that Aharonov-Bohm-like oscillations should appear in the ground state exciton oscillator strength. Using more realistic, finite confinement potential and a numerical approach Song et al. [4] and Hu et al. [5] showed that such oscillations do not appear when finite thickness ring potentials are used. We present here calculations of the optical and angular momentum characteristics of two different ring geometries, showing distinctly different properties. We need to solve the Schr odinger equation for the electron and hole motion, including the Coulomb interaction and the magnetic field applied perpendicular to the ring ðh e 0 þhh 0 þv ehþ Yðr e ; r h Þ¼EYðr e ; r h Þ ; where H e 0 ¼ 1 2m* e ðp e eaþ 2 þv e ðr e Þ ; 1 ) Corresponding author; Tel.: (+44) 131 451 3066; Fax: (+44) 131 451 3136; e-mail: I.Galbraith@hw.ac.uk # WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2002 0031-8965/02/19004-0781 $ 17.50þ.50/0
782 I. Galbraith et al.: Magneto-Excitons in Semiconductor Quantum Rings is the Hamiltonian of a single electron in the ring potential, V e ðr e Þ, and similarly, H h 0, for the hole. Expanding the solution as Yðr e ; r h Þ¼ P n; m A nm w e n ðr eþ w h m ðr hþ ; where w e ðr e Þ and w h ðr h Þ are the wavefunctions of the single electron and hole in the quantum ring and exploiting the orthogonality of the single particle wavefunctions we get ðe e p þ Eh q EÞA pq þ P nm A nm ÐÐ w e p ðr e Þ w h q ðr hþv eh ðr e r h Þ w e n ðr eþ w h m ðr hþ dr e dr h ¼ 0 ; where E e p ; Eh q are the single particle energies. We account for the finite width of the confinement potential in the growth direction using a form factor which multiplies the Coulomb potential in momentum space, reducing the problem to a quasi-two dimensional one. Numerically solving the eigenvalue problem of Eq. (1) [6] gives the transition energies E a and oscillator strengths F a, via F a ¼ P 2 ð A a ij M ij ; M ij ij ¼ d l e rw e i 0 lh j i ðrþ wh j ðrþ dr ; where li e and lj h are the orbital angular momentum quantum numbers of the single particle states. Assuming a Gaussian line shape we can construct the absorption spectrum from the energy positions and oscillator strengths. The novelty in the system is clearly dictated by the confining potentials for the electrons and holes in the ring. As precise parameters, such as the ring size, composition, bandstructure and strain for the rings grown to date are rather poorly known we will explore the influence of different confinement regimes on the optical spectrum. We choose as a first example the ring potential, V e ðr e Þ¼ð1=2Þ m e ðw e=r e Þ 2 ðre 2 R2 e Þ2, and similarly V h ðr h Þ. These consist of an offset potential well having a minimum at a radius R eðhþ ¼ 200 A and having a soft potential turnover in the center. We use hw e ¼ 14 mev and m e ¼ 0:067m 0 for the electron and some low lying electron single particle wave functions are shown as the inset to Fig. 1. The main part of Fig. 1 depicts the energetic positions of these single particle electronic states as a function of magnetic field applied perpendicular to the ring. As can be seen the ring geometry has a profound effect in producing an l e ¼ 1 ground state for 3 T < B < 7.2 T and l e ¼ 2 ground state for B > 7.2 T. This corresponds well with the number of flux quanta passing through the ring [2]. Similar results are found for the hole states, for which we pffiffiffiffiffiffiffiffiffiffiffiffiffiffi choose hw h ¼ hw e m e =m h, with m h ¼ 0:2m 0 to reflect the reduced in-plane heavy-hole mass due to the confinement in the growth direction. As the strength of the confinement decreases the minima of the non-zero angular momentum curves become less pronounced so this novel nature requires the electron to be confined away from the ring center. The absorption strength for this ring as a function of magnetic field is depicted in Fig. 2. The energy zero corresponds to the bottom of the confining potential wells and the ground state exciton is bound by 18 mev in this structure. The lowest exciton energy shows a parabolic field dependence (/ 19B 2 mev) but there is no reflection of the ð1þ
phys. stat. sol. (a) 190, No. 3 (2002) 783 Fig. 1. Electron energy levels in a quantum ring as a function of magnetic field. The insert shows the lowest lying wavefunctions which are offset vertically by their confinement energy underlying non-zero angular momentum electron ground state for any magnetic field. The total angular momentum operator, L, commutes with the Hamiltonian because of the cylindrical symmetry of the problem and is therefore quantized. However the electron and hole components of that total are not quantized and in fact, calculated from the eigenvectors of Eq. (1) increase monotonically with increasing field, remaining equal and opposite for the optically active states. So with finite width rings there is sufficient Coulomb inter-mixing to eliminate the strong field dependence of the single particle states. Our calculations also compute the states with finite total angular momentum and we find that the L ¼ 0 state always has the lowest energy. Fig. 2. Quantum ring absorption spectrum as a function of magnetic field
784 I. Galbraith et al.: Magneto-Excitons in Semiconductor Quantum Rings We can further explore the nature of each absorption resonance by computing the conditional correlation function, P c ðr Þ¼ j Yðr ; r þ Þj Ð 2 j Yðr; rþ Þj 2 dr ; i.e. given a hole at r þ what is the probability of observing an electron at r.infig.3 this is shown for the B ¼ 0 exciton ground state with r þ directly below the peak. As can be seen the electron is clearly localized around the hole, this localization being achieved by mixing equal measures of states with l ¼1; 2; 3 :::. This picture does not alter much as the magnetic field is increased. The higher energy states in Fig. 2 exhibit a complex interplay between the magnetic field and Coulomb interaction which lead to a variety on energy shifts and anti-crossings. Looking at the correlation function we can identify those peaks in the transmission for which the electrons and holes are tightly correlated and those whose motion is essentially determined by the single particle properties. A quite different behaviour emerges when the confinement for electrons and holes is different. In particular when the inner potential for the hole is weak and the hole is essentially confined in a dot-like potential, whilst the electron is confined in a finite radius ring. To model this case we take exactly the same parameters as above, except that we use a simple parabolic confining potential for the holes. The exciton energies and their total angular momenta for B ¼ 7 T are plotted in Fig. 4. As can be seen, these states form a discrete parabolic band with the lowest energy states for each L being set apart from the rest. This picture is exactly analogous to the normal exciton dispersion curves for the center of mass motion. Interestingly here the lowest state is for L ¼ 1 i.e. an optically dark state with a finite center of mass angular momentum. This is shown most clearly in the insert. The optical spectra for this case show again nothing dramatic as a function of the magnetic field as the absorption only probes the optically active states and is insensitive to the existence of a dark ground state. In this paper we have presented calculations for two different quantum ring geometries demonstrating two different behaviours including the novel appearance of a Fig. 3. Conditional correlation function for the ground state exciton in the ring at B ¼ 0. The limit of the plot region is at a radius of 50 nm
phys. stat. sol. (a) 190, No. 3 (2002) 785 Fig. 4. Excitonic state energies versus total angular momentum for B ¼ 7:0 T when the hole is confined in a dot-like potential ground state exciton with a finite angular momentum. Progress in growth techniques and nano-optics will allow more detailed experimental studies of such effects in single quantum rings. References [1] A. Lorke, R. J. Luyken, A. O. Govorov, J. P. Kotthaus, J. M. Garcia, and P. M. Petroff, Phys. Rev. Lett. 84, 2223 (2000). [2] A. Chaplik, Sov. Phys. JETP Lett. 62, 900 (1995). [3] R. A. Römer and M. E. Raikh, Phys. Rev. B 62, 7045 (2001). [4] J. Song and S. E. Ulloa, Phys. Rev. B 63, 5302 (2001). [5] H. Hu, J.-L. Zhu, D.-J. Li, and J.-J. Xiong, Phys. Rev. B 63, 5307 (2001). [6] T. Chakraborty and P. Pietiläinen, Phys. Rev. B 50, 8460 (1994).