Electron localization in low-density quantum rings F. Pederiva, A. Emperador, and E. Lipparini Dipartimento di Fisica, and INFM, Universita` di Trento, I-38050 Povo, Trento, Italy ~Received 5 June 2002; published 23 October 2002! We present a systematic study of ground and excited states of the six-electron nanoscopic ring by fixed-node diffusion Monte Carlo calculations for a wide range of ring diameters and strengths of the harmonic confine- ment. We compare the density and correlation energies to the predictions of local spin density approximation theory, Hartree-Fock theory, and a model in which electrons are localized along the vertices of an hexagon, and analyze the electron-electron pair-correlation functions. We find evidence for a Wigner crystallization transition as the density is lowered. Conversely, evidence for a spin polarization transition is not found. DOI: 10.1103/PhysRevB.66.165314 PACS number~s!: 73.61.2r, 85.35.Be I. INTRODUCTION Recently, the availability of nanoscopic semiconductor ring structures 1–4 has stimulated a strong interest in the prop- erties of quantum rings. One of the most interesting argu- ments in the field concerns the possibility that in the low- density limit a quantum ring represents a physical realization of a Wigner molecule. A consequence of the crossover from weak localization to strong localization in these quasi-one- dimensional semiconductor structures might be the occur- rence of a metal-insulator transition similar to the one clearly observed in two-dimensional ~2D! heterostructures, 5–7 with possible interesting applications in microelectronics. Another interesting research area is the study of magnetic properties of quantum rings, in particular the appearance of Aharonov-Bohm oscillations of the energy and of the persis- tent current in the ring. Such oscillations have been observed 8 in mesoscopic rings in a GaAlAs/GaAs hetero- structure and studied within many theoretical approaches. 9–21 The formation of Wigner molecules in two-electron quan- tum rings was first argued in Refs. 22 and 23. More recently, Koskinen et al. 24 performed configuration-interaction calcu- lations of the many-body spectra of quantum rings that con- tain up to six electrons, using the Hamiltonian 3,15 H 5 ( i 51 N S 2 \ 2 2 m * „ i 2 1 1 2 m * v 0 2 ~ r i 2R 0 ! 2 D 1 e 2 e ( i , j N 1 u r i 2r j u , ~1! where N is the number of electrons in the ring, m * is the mass of the electron, and e is the dielectric constant of the semiconductor. In Eq. ~1!, the radius of the ring, R 0 , and the strength of the harmonic confinement, v 0 , can be related to the one-dimensional density parameter r s 1 D which describes the electron density r 51/(2 r s 1 D ) along the ring ( R 0 5Nr s 1 D / p ), and to a dimensionless parameter C F 5v 0 32m * r s 1 D2 / p 2 \ , which is the ratio between the confine- ment potential width and the Fermi energy for 1D free fer- mions and which measures the degree of one dimensionality. Throughout this paper we use effective atomic units. For the GaAs rings we consider here, e 512.4 and m * 50.067m e , and the effective Bohr radius a 0 * and effective Hartree H * are .97.93 Å and .11.86 meV, respectively. Fixing the number of electrons at 6 and going from a ( r s 1 D 52 a 0 * , C F 54) ring to a narrower, lower-density ( r s 1 D 56 a 0 * , C F 525) ring, Koskinen et al. found that the many-body spectra show a more and more defined rotational-vibrational band struc- ture. The structure has been compared with the one com- puted for a model of a rotating planar hexagonal molecule of localized vibrating electrons, finding very strong agreement. The hypothesis of the formation of Wigner molecules was then reinforced by finite-temperature path integral Monte Carlo calculations 25 in rings with up to six electrons with ( r s 1 D .2.6a 0 * , C F .24) and by an exact calculation 26 in the two-electron ring. It should be noticed that, at values of r s 1 D which are ac- cessible to exact diagonalization techniques ( r s 1 D < 6) and for a ring confining six electrons, clear signals of the forma- tion of a Wigner molecule have been observed only for high values of C F , which is when the system is quasi one dimen- sional. Furthermore, for all calculations reported the ground state remains unpolarized. The region of the phase diagram in the plane ( r s 1 D , C F ) which corresponds to a two- dimensional system with relatively small C F and high r s 1 D is still unexplored. This region is quite interesting since recent diffusion Monte Carlo calculations 27 in the bulk for the two- dimensional electron gas predict a polarization transition at r s 2 D .26 and polarized liquid crystallization at r s 2 D P @ .35,41# . The density parameters r s 1 D and r s 2 D can be re- lated to each other by looking at the confining potential. In fact, a more realistic model of a quantum ring is given in terms of the self-consistent potential generated by a thick ring of uniformly distributed positive charge defined by an inner radius R in and by an outer radius R out , and neutraliz- ing the electrons charge, as illustrated in the left panel of Fig. 1. It turns out that a parabolic confinement with a proper value of v 0 fits very well the jellium ring potential in the region between R in and R out . An example is shown in the right panel of Fig. 1. Of course, one can use the inverse procedure and find the values of R in and R out corresponding to a given v 0 . In this way one can relate the one-dimensional Wigner-Seitz radius to the effective two-dimensional density, parametrized by r s 2 D 51/A pr , where r 5N / p ( R 0 2 2R I 2 ). As an example, in the case C F 54, which will be considered in PHYSICAL REVIEW B 66, 165314 ~2002! 0163-1829/2002/66~16!/165314~6!/$20.00 ©2002 The American Physical Society 66 165314-1