Composite Fermion Theory of Excitations in the Fractional Quantum Hall Effect

Solid State Communications 135 (2005) 602–609 www.elsevier.com/locate/ssc Composite fermion theory of excitations in the fractional quantum Hall effect J.K. Jaina,*, K. Parkb, M.R. Petersona, V.W. Scarolab a104 Davey Laboratory, Physics Department, The Pennsylvania State University, University Park, PA 16802, USA bPhysics Department, Condensed Matter Theory Center, University of Maryland, College Park, MD 20742, USA Received 28 February 2005; accepted 1 May 2005 by the guest editors Available online 13 May 2005 Abstract Transport experiments are sensitive to charged ‘quasiparticle’ excitations of the fractional quantum Hall effect. Inelastic Raman scattering experiments have probed an amazing variety of other excitations: excitons, rotons, bi-rotons, trions, flavor altering excitons, spin waves, spin-flip excitons, and spin-flip rotons. This paper reviews the status of our theoretical understanding of these excitations. q 2005 Elsevier Ltd. All rights reserved. PACS: 71.10.Pm; 73.43.Kf Keywords: A. Composite fermion; D. Fractional quantum Hall effects 1. Introduction have revealed a strikingly rich structure with many kinds of excitations. Transport experiments told us quite early on that there is There has also been substantial theoretical progress on a gap to charged excitations in the fractional quantum Hall this issue, and an excellent description of the neutral effect (FQHE) [1]. The longitudinal resistance was seen to excitations has been achieved in terms of a particle hole pair behave like rxxZexp(KD/2kBT) in a range of temperature, of composite fermions (CFs), which dovetails nicely with and the quantity D is interpreted as the minimum energy our understanding of the incompressible ground states as an required to create a charged excitation. Subsequently, the integral number of filled quasi-Landau levels of composite excitations were probed by optical means [2–6] as well as by fermions. This paper will review the status of our qualitative ballistic phonon absorption [7–9]. These and subsequent and quantitative understanding of the excitations of the optical experiments have proved to be a treasure-trove of FQHE states. The reader is referred to the original articles information on charge-neutral as well as high energy for technical details. excitations not accessible in transport experiments and The character of neutral excitations is of great interest in its own right, and their explanation is a critical test for theory. There is another motivation for their consideration. * Corresponding author. Tel.: C1 814 863 1162; fax: C1 814 865 Much effort has been invested into a calculation of the gaps 3604. to charged excitations. However, a pesky factor of two E-mail address: [email protected] (J.K. Jain). discrepancy between theory and experiment has persisted 0038-1098/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2005.04.033 J.K. Jain et al. / Solid State Communications 135 (2005) 602–609 603 gs over the years, believed to be caused by disorder for which a the electron ground state at filling factor n, Fn , and its ex quantitatively reliable theoretical treatment is not available exciton, Fn : at the moment. In contrast, disorder is not likely to affect the Y gs Z K 2p gs energy of a localized neutral excitation as significantly, Fn=ð2nC1Þ PLLL ðzj zkÞ Fn (1) because of its much weaker dipolar coupling to disorder. j!k The coupling is further diminished because the disorder in Y ex Z K 2p ex modulation-doped samples is typically smooth on the scale Fn=ð2nC1Þ PLLL ðzj zkÞ Fn (2) of the size (on the order of a magnetic length) of many j!k spatially localized neutral excitations. We will find that the where z Zx Kiy is the position of the jth particle, and P theoretical predictions, without including disorder effects, j j j LLL denotes projection of the wave function into the lowest agree reasonably well with experiment. Landau level. The explicit form of these wave functions has been given in the literature [14]. The composite fermion interpretation ofQ Fn follows since, multiplication by the K 2p 2. The composite fermion exciton Jastrow factor j!kðzj zkÞ is tantamount to attaching 2p vortices to each electron, thus, converting it into a composite A composite fermion [10–13] is the bound state of an fermion. These wave functions have been found to be quite electron and an even number of quantized vortices (some- accurate in tests against exact diagonalization results times modeled as the bound state of an electron and an even available for small systems [10,14]. We will not discuss number of magnetic flux quanta, with a flux quantum the field theoretical approaches to study the excitations; the reader is referred to the articles by Lopez and Fradkin, [15] defined as f0Zhc/e). Composite fermions are to FQHE what Cooper pairs are to superconductivity. The interacting He, Simon and Halperin, [16] and Murthy and Shankar [17]. electrons at Landau level filling factor nZn/(2pnG1), n and The HamiltonianÂÃP for the many electron system is given 1 by HZP s Vðr Þ P V r p being integers, transform into weakly interacting compo- LLL 2 j k jk LLL, where ( ) is the effec- site fermions at an effective filling n*Zn. The ground state tive two-dimensional electron–electron interaction. For a Z 2 corresponds to n filled CF Landau levels (LLs), shown strictly two-dimensional system V(rjk) e /3rjk, where 3 is schematically in Fig. 1(a). The CF theory not only explains the dielectric constant of the background material. A the origin of a gap at these fractional fillings, which are quantitative correction comes from the finite transverse precisely the observed fractions, but also gives a natural extent of the electron wave function, which alters the form insight into the excitations. Fig. 1(b) shows a typical neutral of the effective two-dimensional interaction at short excitation, which is a particle-hole pair of composite distances. The effective interaction can be calculated once fermions, called the CF exciton. It is analogous to the the transverse wave function is known, which in turn is familiar exciton at integral fillings. Jain’s wave functions for determined by self-consistently solving the Schro¨dinger and theCFgroundstateandtheCFexcitonarereadily Poisson equations, taking into account the interaction effects constructed by analogy to the known wave functions of through the local density approximation including the exchange correlation potential [18]. The energy of the exciton, h exj j exi h gsj j gsi ex Z Fn H Fn K Fn H Fn D ex ex gs gs (3) hFn jFn i hFn jFn i is computed by Monte Carlo methods in the spherical geometry [19]. All results below are thermodynamic extrapolations, unless mentioned otherwise.p Theffiffiffiffiffiffiffiffiffiffiffiffi energies 2 Z Z = are quoted in units of e /3l0, where l0 c eB is the magnetic length. Scarola, Park and Jain [20] determined the dispersion of the CF exciton for 1/3, 2/5, and 3/7, corresponding to one, two, and three filled CF-LLs, respectively. The dispersion, as shown in Fig. 2, is rather complicated in general, reflecting the complex density profiles of the CF-quasiparticle and the CF-quasihole. In particular, it has several minima, which are called ‘rotons’ (This name has its origin 4 Z Fig. 1. (a) The incompressible ground state at nZ2/5, which in analogy to superfluid He. The minimum at n 1/3 was represents two filled CF-Landau levels. (b) An exciton of composite earlier found by Girvin, MacDonald and Platzman [21] in a fermions. Composite fermions are depicted as electrons carrying single mode approximation). It might seem that rotons vortices. would not be accessible in light scattering due to their large Download English Version: https://daneshyari.com/en/article/9801934 Download Persian Version: https://daneshyari.com/article/9801934 Daneshyari.com.

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