An analytical model of fractional overshooting
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We predict resistance anomalies to be observed at high mobility two dimensional electron systems (2DESs) in the fractional quantized Hall regime, where the narrow (L <10 ?m) Hall bar is defined by top gates. An analytic calculation scheme is used to describe the formation of integral and fractional incompressible strips. We incorporate the screening properties of the 2DES, together with the effects of perpendicular magnetic field, to calculate the effective widths of the current carrying channels. The many-body effects are included to our calculation scheme through the energy gap obtained from the well accepted formulation of the composite fermions. We show that, the fractional incompressible strips at the edges, assuming different filling factors, become evanescent and co-exist at certain magnetic field intervals yielding an overshoot at the Hall resistance. Similar to that of the integral quantized Hall effect. We also provide a mechanism to explain the absence of 1/3 state at the Fabry-Perot interference experiments. Yet, an un-investigated sample design is proposed to observe and enhance the fragile effects like interference and overshooting based on our analytical model.Keywords:
Composite fermion
STRIPS
Overshoot (microwave communication)
Most of the fractions observed to date belong to the sequences $\ensuremath{\nu}=n/(2pn\ifmmode\pm\else\textpm\fi{}1)$ and $\ensuremath{\nu}=1\ensuremath{-}n/(2pn\ifmmode\pm\else\textpm\fi{}1)$, $n$ and $p$ integers, understood as the familiar integral quantum Hall effect of composite fermions. These sequences fail to accommodate, however, many fractions such as $\ensuremath{\nu}=4/11$ and $5/13$, discovered recently in ultrahigh mobility samples at very low temperatures. We show that these ``next generation'' fractional quantum Hall states are accurately described as the fractional quantum Hall effect of composite fermions.
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In the standard hierarchical scheme the daughter state at each step results from the fractional quantum Hall effect of the quasiparticles of the parent state. In this paper a new possible approach for understanding the fractional quantum Hall effect is presented. It is proposed that the fractional quantum Hall effect of electrons can be physically understood as a manifestation of the integer quantum Hall effect of composite fermionic objects consisting of electrons bound to an even number of flux quanta.
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We show a generic formation of the primary magnetorotons in the collective modes of the observed "unconventional" fractional quantum Hall effect states of the composite fermions at the filling factors 4/11, 4/13, 5/13, 5/17, and 3/8 at very low wave vectors with anomalously low energies which do not have any analog to the conventional fractional quantum Hall states. Rather slow decay of the oscillations of the pair-correlation functions in these states is responsible for the low-energy magnetorotons. This is a manifestation of the distinct topology predicted previously for these fractional quantum Hall effect states. Experimental consequences of our theory are also discussed.
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Composite fermion
Landau quantization
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We investigate the edge structure of a system of composite fermions confined in a wire. Using the composite fermion picture of the fractional quantum Hall effect, we have studied the electronic edge structure of different fractional quantum Hall liquids. For strong confinement and in the case of the 2/3 quantum Hall state we find that a droplet of charge with filling factor 1 appears at the edge of the system. In the case of the 1/3 quantum Hall state the droplet of charge corresponds to filling factor 2/5. For very smooth boundaries, we obtain the existence of plateaus and droplets of charge at fractional filling factors. We present an expression for the fractional Hall resistance as a function of the composite fermion edge states, which is a generalization of the Landauer-Buttiker formula for the integer quantum Hall effect.
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Bulk two-dimensional electron systems in a strong perpendicular
magnetic field exhibit the fascinating phenomenon of fractional
quantum Hall effect. Composite fermion theory was developed in the
process of understanding the fractional quantum Hall effect and was
proven to work successfully for the FQHE and even beyond. In this
dissertation, we explore the effect of the strong correlation
between electrons in several cases. All of them belong to the
category of 2DES in strong perpendicular magnetic field and they are
listed below:
(i)A fractional quantum Hall island surrounded by a bulk fractional
quantum Hall state with a different filling factor. Specifically, we
study the resonant tunneling composite fermions through their
quasibound states around the island. A rich set of possible
transitions are found and the possible relevance to an interesting
experiment is discussed. Also, we discuss the subtlety of separating
the effect of fractional braiding statistics from other factors.
(ii) Correlated states of a quantum dot, at high magnetic fields,
assuming four electrons with two components. Such a dot can be
realized by reducing the two lateral dimensions of a 2DES
tremendously. Both the liquid states and crystallites (the latter
occurring at large angular momenta) of four electrons in terms of
composite fermions are considered. Residual interaction between
composite fermions is shown to leads to complex spin correlations.
(iii) Bilayer quantum Hall effect at total filling $
u_T=5$.
This can accommodate an excitonic superfluid state at small layer
separations just like at $
u_T=1$. At large layer separations, however, $
u_T=5$ state evolves into
uncorrelated $
u=5/2$ fractional quantum Hall states in both
layers, in contrast to uncorrelated composite Fermi sea in $
u_T=1$ case. We focus on finding the critical layer separation at
which the correlation between electrons on different layers are
destroyed. Effects due to the finite width of the layers are also
considered.
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We present a mean-field theory of composite fermion edge states and their transport properties in the fractional and integer quantum Hall regimes. Slowly varying edge potentials are assumed. It is shown that the effective electrochemical potentials of composite fermions at the edges of a Hall bar differ, in general, from those of electrons, and an expression for the difference is obtained. Composite fermion edge states of three different types are identified. Two of these types have no analog in previous theories of the integer or fractional quantum Hall effect. The third type includes the usual integer edge states. The direction of propagation of the edge states is consistent with experimental observations. The present theory yields the experimentally observed quantized Hall conductances at the bulk Landau-level filling fractions \ensuremath{\nu}=p/(mp\ifmmode\pm\else\textpm\fi{}1), where m=0, 2, and 4, and p=1,2,3, . . . . It also explains the results of experiments that involve conduction across smooth potential barriers and through adiabatic constrictions, and of experiments that involve selective population and detection of edge channels in the fractional quantum Hall regime. The relationship between the present work and Hartree theories of composite fermion edge structure is discussed.
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Numerical studies by W\'ojs, Yi and Quinn showed that an unconventional fractional quantum Hall effect is possible at filling factors $\nu=$ 1/3 and 1/5, provided the interparticle interaction has an unusual form for which the energy of two fermions in the relative angular momentum three channel dominates. The interaction between composite fermions in the second $\Lambda$ level (composite fermion analog of the electronic Landau level) satisfies this property, and recent studies have supported unconventional fractional quantum Hall effect of composite fermions at $\nu^*=$4/3 and 5/3, which manifests as fractional quantum Hall effect of electrons at $\nu=$ 4/11, 4/13, 5/13 and 5/17. I investigate in this article the nature of the fractional quantum Hall states at $\nu=$ 4/5, 5/7, 6/17 and 6/7, which correspond to composite fermions at $\nu^*=$ 4/3, 5/3 and 6/5, and find that all these fractional quantum Hall states are conventional. The underlying reason is that the interaction between composite fermions depends substantially on both the number and the direction of the vortices attached to the electrons. I also study in detail the various spin polarized states at 6/17 and 6/7 and predict the critical Zeeman energies for the spin phase transitions between them. I calculate the excitation gaps at 4/5, 5/7, 6/7 and 6/17 and compare them against recent experiments.
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Zeeman energy
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I demonstrate that the wavefunction for a nu = n+ tilde{nu} quantum Hall state with Landau levels 0,1,...,n-1 filled and a filling fraction tilde{nu} quantum Hall state with 0 < tilde{nu} \leq 1 in the nth Landau level can be obtained hierarchically from the nu = n state by introducing quasielectrons which are then projected into the (conjugate of the) tilde{nu} state. In particular, the tilde{nu}=1 case produces the filled Landau level wavefunctions hierarchically, thus establishing the hierarchical nature of the integer quantum Hall states. It follows that the composite fermion description of fractional quantum Hall states fits within the hierarchy theory of the fractional quantum Hall effect. I also demonstrate this directly by generating the composite fermion ground-state wavefunctions via application of the hierarchy construction to fractional quantum Hall states, starting from the nu=1/m Laughlin states.
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