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16 Surface Science 229 ( 1990) 16-20 North-Holland SPIN-REVERSED GROUND STATE AND ENERGY GAP IN THE FRACTIONAL QUANTUM HALL EFFECT Tapash CHAKRABORTY Max-Planck-Institut ftir Festkijrperforschung, Heisenbergstrasse I, D- 7000 Stuttgart SO, Fed. Rep. of Germany Received 11 July 1989; accepted for publication 14 September 1989 The role of reversed spins in the fractional quantum Hall effect is briefly reviewed. The ground-state spin configurations for various tilling fractions and the spin-reversed quasi-particles are discussed. Some new results for the spin-reversed excitations are presented and compared with recent experimental results. 1. Introduction In this paper, I shall review the current status of one of the interesting aspects of the fractional quantum Hall effect (FQHE) - the role of reversed spins in the incompressible many-electron ground state and in the quasi-particle (QP )-quasi-hole (QH ) energy gap [ 11. Some new results obtained recently in collaboration with Pietilginen will also be presented [ 21. Although the theoretical work on this important problem began soon after the original experimental discovery [ 31 of FQHE and its explanation by Laughlin [4], the experimental support of the theoretical predictions on reversed spins [ 5-71 came only recently [ 8,9]. In the standard theory (for a review, see refs. [ 1,5] ) developed to explain the FQHE at u= 1/m filling of the lowest Landau level (m is an odd integer) [ 4 1, electrons were considered to be in a fully spin-polarized state (here V= 27&p is the filling factor of the Landau level, p is the area1 density of twodimensional electrons, and lo = (fic/&) ‘I2 is the magnetic length). Considering the high magnetic field involved at these filling fractions, such a choice was quite reasonable. For the higher-order filling fractions observed in the experiments, this approach requires modifications. For example, at + filling of the lowest Landau level, Halperin [ 51 proposed a spinunpolarized ground-state wave function. The small g-factor in GaAs prompted Halperin to predict that, when the magnetic field is not too large, such a state 0039-6028/90/$03.50 (North-Holland) 0 Elsevier Science Publishers B.V. might be energetically preferred. Subsequent calculations revealed that in the absence of Zeeman energy, this state has indeed the lowest energy [ 1,6]. It was also found that, even for a fully spin-polarized ground state, one can obtain spin-reversed excitations [ 71, which might manifest itself in the linear behaviour of the activation energy observed in the experiment [ lo] (the Zeeman energy is linear in magnetic field B, while the Coulomb energy goes as B”‘). 2. Ground state The results to be described below are based on the calculations done for finite electron systems in a periodic rectangular geometry [ 111, suitably modified to deal with spin [ 6 1. The total energy of the twodimensional interacting electron system consists of a Coulomb term (which depends on the spin quantum number S, and the component of magnetic field normal to the electron plane), and also the Zeeman term. The ground state energy is calculated for different S by numerically diagonalizing the Hamiltonian. The spin state having the lowest energy will be the preferred ground state. In table 1, I present the ground state energy (in units of e2/&, E being the background dielectric constant ) for a few filling fractions. Similar results for other filling fractions are available in our earlier work [ 6 1. The Zeeman energy is not included in these results. T. Chakraborty/Spin-reversed Table 1 Potential energy per particle for four-electron systems (for f and $) and six-electron systems (for 4 and 3) for various values of spin polarization; the Zeeman energy is not included v Potential s=o f -0.4135 Ground energy S=l s=2 s=3 state f ! -0.5331 -0.4464 -0.4120 -0.5291 -0.4410 -0.4152 -0.5257 -0.4403 -0.5232 - Polarized Unpolarized Unpolarized 3 -0.5074 -0.5096 -0.5044 -0.50104 Partially polarized 1-l ground state and energy gap in FQHE It is interesting to note that even allowing spin degrees of freedom, the ground state at Y= 4 is fully spin-polarized (the inclusion of the Zeeman energy would make this state even more favourable relative to the other spin states). This is in line with the fully antisymmetric wave function proposed by Laughlin [ 41 for Y= f . For the other tilling fractions, v=p/q with q odd and p> 1, spin-reversed states are found to be energetically favourable over the fully spin-polarized states in the absence of Zeeman energy [ 6,121. A pictorial representation of the low magnetic field spin assignments obtained theoretically for various tilling fractions can be given as netic field is tilted from the direction normal to the electron plane [ 8,9]. Dramatic changes occurring at different filling factors with tilt angle are ascribed [ 81 to various spin polarizations consistent with the above predictions. In the other experiment [ 9 1, a sharp change in the dependence of the activation energy on tilt angle was observed for v=g (electronhole symmetric to v=$). This is described as a transition from a spin-unpolarized ground state at small angles to a polarized state at larger angles. The linear behaviour of the activation energy at the two ground states were identified with the Zeeman energy, and they appear because of the presence of spin-reversed quasi-particles and quasi-holes [ 15 1, to be discussed below. In fig. 1, I present the results for the ground-state energy (per particle) for (a) v=$ and (b) v=$, -0.4445 -’ -0.4455 - \ I I Ground v=2/5 ‘\ \,4-electrons ‘\ EG -0.4465 - -0.4475 - (a) ‘\ ‘\ State I I ~‘~~---_!_; I ‘\\ ,l I 30 I I I I I 40 50 6.0 70 80 B ITI -0.528 (here t?, # and 1 corresponds to fully spin-polarized, spin-unpolarized and partially spin-polarized states respectively). The spin assignments for 2 > v> 1 can be obtained from the electron-hole symmetry which, in the absence of Landau level mixing is between v and 2- v [ 131. The absence of electron-hole symmetry between f and 3 was first noticed in the tilted-field measurements of FQHE by Haug et al. [ 141. The different spin polarization at different filling factors has been explained in terms of competition between Coulomb repulsion, exchange and interaction of unlike spins [ 12 1. The spin assignments for various filling fractions discussed above have received strong support from two recent experiments on FQHE where the mag- -0.537 6-electrons v=2/3 t - (b) L-L 5.0 I 1 15.0 10.0 I 20.0 1 25.0 B (T) Fig. 1. Ground-state energy per article B (Tesla) for (a) v=: and (b) polarizations. (in units of e2/clo ) versus v=f for different spin 18 T. ChakrabortyLYpin-reversed ground state and energy gap in FQHE where the Zeeman energy (g= 0.4) contributions are also included. In both cases, we observe a cross-over point in the magnetic field below which the spin unpolarized state (SC 0) is energetically favoured. The fully spin-polarized state (S= NJ2, N, is the electron number) is favoured for magnetic fields beyond the cross-over point. In the case of f and f, the ground-state energy is remarkably insensitive to the system size [ 111. Therefore, a similar situation in the present case is expected. From these results, it is clear that, when we study the QP-QH gap, we should consider the two different ground states for two regions of the magnetic field. For the spin-unpolarized ground state, the spin-reversed QP-QH excitation gap has the lowest energy. For the polarized ground state, the gap for spin-reversed QP (or QH) and the other partner spin-polarized, is close to zero. The energy gap for spin-reversed QP and QH is even negative. At first sight, this is quite disturbing. However, we should remember that, in the absence of Zeeman energy, the fully spin-polarized ground state is not the preferred ground state. The Zeeman term will add a positive contribution to the energy gap thereby increasing its value. The spin-reversed QP-QH gaps for two different ground states (spin-unpolarized and fully spin-polarized) at v=$ are depicted in fig. 2a. They are the 3. Energy gap It was first pointed out by Laughlin [4] that the elementary charged excitations in the quantized Hall states are quasi-particles with fractional charge, more precisely, the charge &e/q for a filling fraction of ZJ=p/q. The energy required to create a pair of quasiparticles of opposite charge well separated from each other is then related to the discontinuity of the slope of the energy by [ 16 ] 0.0100 ;&b-_&G@ ,.’ 0.0025 .x 0 (1) E~=f(El”+-iFl”_). Defining the chemical potential _ 9.p. 1 + q.h 1 v=2/5 4 - electrons ,’ ,’ ,’ l’ ‘I. ,’ --I_ -.._ 2.. (al I I 30 4.0 I 50 I / 60 I 70 80 I 90 B (T) as p=~o(~)+~a~o/a~, 6-electrons (2) where Eo( v) is the energy per particle at V, the QPQH gap is P& =-&+tl/lq)(E~ -J%) l(vk -v) . (3) Here v?=NJ(N,T 1) and E,=E,(v,), and N, is the number of flux quanta related to the filling fraction [ 11. The spin-reserved QP and QH excitation energy gaps are obtained by evaluating Ei in a system where the spin difference from the ground state is unity (spin-l excitations). The ground-state energy (per particle) at v is taken to be spin-polarized or spin-reversed, depending on the magnetic field region of fig. 1. Let us first consider the case of v=$, and also the situation where the Zeeman energy is not included. Ground State s=3 0 5.0 15.0 10.0 20.0 B(T) Fig. 2. Lowest energy quasi-particle-quasi-hole gap (in units of e’/tl,) ) as a function of magnetic field B (in Tesla) for (a) u=: and (b) v= 5. Above and below the cross-over point, two different ground states form fig. I, are considered. The gap is nonexistent in the shaded region, as discussed in the text. T. ChakrabortyBpin-reversed ground state and energy gap in FQHE lowest energy excitations in the presence of Zeeman energy. The results are qualitatively similar to the observed behaviour at v=: [ 91. In comparing the present results with those of ref. [ 91, it should be remembered that here I assume the magnetic field to be perpendicular to the electron plane, while in the experiment, the magnetic field was tilted from the direction normal to the plane. The tilted field is however expected to change primarily the Zeeman energy, but the Coulomb energy is likely to remain unchanged. Our theoretical result therefore is in agreement with the model proposed in ref. [ 91 that the observed sharp change in the slope of the activation energy corresponds to spin-reversed excitations from the different ground states. From the results discussed above , we identify the excitations as a spin-reversed QP-QH pair [ 15 1. The results for the QP-QH gap for v= 3 is depicted in fig. 2b. Just as in fig. 2a, we plot only the lowest energy excitations as a function of magnetic field, with the Zeeman energy included. In this case, the situation is clearly different from that of v=$ Below the cross-over point, the preferred ground state is spin-unpolarized. The lowest energy excitations in this state involve a spin-polarized QP-spin-reversed QH pair. The energy gap decreases rapidly and vanishes before the cross-over point is reached. From this point onward, the discontinuity in the chemical potential is in fact negative, indicating that FQHE is unstable in this region of magnetic field. Beyond the cross-over point, the spin-unpolarized state is no longer the ground state and the energy gap is to be calculated from the fully spin-polarized ground state. In this case, spin-reversed QP-spin-polarized QH pair excitations have the lowest energy. The energy gap steadily increases with the magnetic field. Therefore, between the two ground states exhibiting FQHE, there is a gapless domain, where the FQHE state is not stable. Such a gapless domain is not present at v=$. A somewhat similar situation has indeed been observed for v=! (electron-hole symmetric to 3 ) in the experiments of ref. [ 8 1. The activation energy for this filling fraction is found to decrease rapidly with increasing magnetic fields. The gap then vanishes for a small region of magnetic field. The gap reappears with increasing magnetic fields. In our calculation, the effect is purely a result of the Zeeman 19 energy, which stabilizes different ground states in two regions of magnetic field. The absence of such a gapless domain for 3, both in the experimental observation and in our calculation is noteworthy. I tend to take the novel feature of the energy gap for 3 in our present calculation as a physical effect, and not an artifact of our finite-size calculation. Another interesting result for 3 is that the spin-reserved QP seems to persist for a rather large magnetic field. Having made the comparison of our theoretical results with the experimental observations in refs. [ 8,9 ] I wish to make a few cautionary remarks. In our earlier work, we noticed that, while the ground-state energy is insensitive to the system size, the QP-QH gap is in fact, system-size dependent [ 1,7]. Therefore, the absolute value of the different QP-QH gap discussed above, might change with increasing system size. The finite-thickness correction [ 1 ] is also known to influence the size of the gap. Larger system sizes are difficult to handle numerically, especially in the presence of reversed spins. The physical characteristics of the energy gap in the presence of spin reversal observed here are however, expected to remain unchanged when these corrections are made. We have made similar calculations for other filling fractions (v=? and :) [ 21. The gapless domain was also found to appear in these cases. For these two fractions, we observed several other interesting features in the energy gap. Besides the usual transition from one ground state to the other as seen here, we also observed in the QP-QH gap, transitions from one spin state to the other in the same ground state 121. While the current experimental results for v=$ (electron-hole symmetric to 2) seem to indicate the presence of such transitions [ 8 1, more experimental work is needed to clarify the situation. Verification of such transitions would provide a means to distinguish between the spin reversal of the quasi-particles and quasi-holes from activation energy measurements. Finally, just in the case of v=$, the energy gap for : was found to vanish at a finite magnetic: field. Experimental attempts to observe its possible re-appearance would be very interesting. In closing, the tilted-field experiments of ref. [ 8,9 ] have brought in a wealth of information about the various spin-reversed ground states and excitations. 20 T. ChakrabortyLYpin-reversed ground state and energy gap in FQHE They have provided convincing evidence for the existence of spin-reversed quasi-particles introduced theoretically in ref. [ 7 1. Finite-size systems are found to be very effective in explaining several interesting features observed in these experiments. Acknowledgements Most of the work described here has been done in collaboration with Pekka Pietilainen (Oulu, Finland). The work on v= 3 was inspired by a conversation with Bob Clark (Oxford), during the QHE meeting in Schleching, Germany. References [ I] Tapash Chakraborty and P. Pietillinen, The Fractional Quantum Hall Effect (Springer, Heidelberg, 1988). [ 2 ] Tapash Chakraborty and P. Pietillinen, to be published. [ 3 ] D.C. Tsui, H.L. Stormer and A.C. Gossard, Phys. Rev. Lett. 48 (1982) 1559. [4] R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. Helv. Phys. Acta 56 (1983) 75. [6] Tapash Chakraborty and F.C. Zhang, Phys. Rev. B 29 (1984) 7032; B 30 (1984) 7320. [ 71 Tapash Chakraborty, P. Pietillinen and F.C. Zhang, Phys. Rev. Lett. 57 ( 1986) 130; Tapash Chakraborty and P. Pietillinen, Phys. Ser. T 14 (1986) 58; see also: Tapash Chakraborty, Phys. Rev. B 34 ( 1986) 2926; R. Morfand B.I. Halperin, Phys. Rev. B 33 ( 1986) 222 1. [ 81 R.G. Clark, S.R. Haynes, A.M. Suckling, J.R. Mallet& J.J. Harris and C.T. Foxon, Phys. Rev. Lett. 62 ( 1989) 1536. [9] J.P. Eisenstein, H.L. Stiirmer, L. Pfeiffer and K.W. West, Phys. Rev. Lett. 62 (1989) 1540. [lo] G.S. Boebinger, A.M. Chang, H.L. Stonner and D.C. Tsui, Phys. Rev. Lett. 55 (1985) 1606. [ 111D. Yoshioka, B.I. Halperin and P.A. Lee, Phys. Rev. Lett. 50 (1983) 1219. [ 121 P.A. Maksym, J. Phys. Condensed Matter 1 ( 1989) L6299. [ 131 F.C. Zhang and Tapash Chakraborty, Phys. Rev. B 34 (1986) 7076. [ 141 R.J. Haug, K. von Klitzing, R.J. Nicholas, J.C. Maan and G. Weimann, Phys. Rev. B 36 (1987) 4528; see also J.E. Fumeaux, D.A. Syphers and A.G. Swanson, Phys. Rev. L&t. 63 (1989) 1098. [ 151 Tapash Chakraborty, to be published. [ 161 B.I. Halperin, Surf. Sci. 170 (1986) 115. [ 51B.I. Halperin,